mortorella

Chapter TwoIntroduction to the Theory of Consumer Behavior

2.1 UTILITYConsumer theory relies only on the assumption that consumers can provide relative rankings of market baskets. It is often useful to assign numerical values to individual baskets. Using this numerical approach, it is possible to describe consumer preferences by assigning scores to the levels of satisfaction associated with each indifference curve. Generally, the word utility means “benefit or well-being.” In the language of economics, the concept of utility refers to the numerical score representing the satisfaction that a consumer gets from a market basket. In other words, utility is a device used to simplify the ranking of market baskets. Ex: If buying three copies of a textbook gives, you more pleasure than buying one shirt, then it is said that the books give you more utility than the shirt.

2.1.i UTILITY FUNCTIONSA utility function is a formula that assigns a level of utility to each market basket. For example, that a consumer’s utility function for food (F) and clothing (C) is u(F,C,)=F+2C. In this case, a market basket consisting of 8 units of food and 3 units of clothing generates a utility of 8 + (2) (3) = 14. Consumer is therefore indifferent between this market basket and a market basket containing 6 units of food and 4 units of clothing (6+(2)(4) = 14).Not all kinds of preferences can be represented by a utility function. For example, suppose someone had intransitive preferences so that . Then a utility function for these preferences would have to consist of numbers u(A), and u(B) and u(C) such that u(A)>u(B)>u(C)>u(A). However, this is impossible. So such kind of perverse cases not considered for construction of utility function. The figure 1 and illustration gives the details of constructing a utility function. A utility function is a way to label the indifference curves such that higher indifference curves get larger numbers. To do this, draw the diagonal line illustrated and label each indifference curve with its distance from the origin measured among the curve.How do you know this as a utility function? It is not difficult to see that preferences are monotonic then the line through the origin must intersect every indifference curve exactly once. Thus, every bundle is getting larger labels – and that is all it takes to be a utility function. This is one way to find labeling of indifference curves, at least as long as preferences are monotonic. This at least shows that ordinal utility function is quite general: nearly any kind of “reasonable” preferences can be represented by a utility function.

1

Measures distance from origin

Indifference curves

Figure 1: Constructing a utility function from indifference curves. Draw a diagonal line and label each indifference curve with how fat it is from the origin measured along the line.

X2

X1

2.1.ii ORDINAL UTILITYA utility function that generates ranking in order of most to least preferred of market baskets is called an ordinal utility function. In fact, numerical values are arbitrary, interpersonal comparisons of utility are impossible. The only property of a utility assignment that is important is how it orders the bundles of goods. The magnitude of the utility function is only important insofar as it ranks the different consumption bundles; the size of the utility difference between any two consumption bundles does not matter. Because of this emphasis on ordering bundles of goods, this kind of utility is referred to as ordinal utility.Consider the following table 1. where several different ways of assigning utilities to three bundles of goods is illustrated, all of which order the bundles in the same way. In this example, the consumer prefers A to B and B to C. All of the way indicated are valid utility functions that describe the same preferences because they all have the property that A is assigned a higher number than B, which in turn in assigned a higher number than C.Table : 1 Different ways to assign utilities.

Bundles

U1 U2 U3

A 3 17 -1B 2 10 -2C 1 0.002 -3

Since only the ranking of the bundles matters, there can be no unique way to assign utilities to bundles of goods.

2.1. iii CARDINAL UTILITYA utility function that describes by how much one market basket is preferred to another is called cardinal utility function. Unlike ordinal utility functions, a cardinal utility function attaches to market baskets numerical values that cannot arbitrarily be doubled or tripled without altering the differences between values of various market baskets.There are some theories of utility that attach a significance to the magnitude of utility. These are known as cardinal utility theories. In a theory of cardinal utility, the size of the utility difference between two bundles of goods is supposed to have some sort of significance.In fact, there is no hard and fast rule to measure utility. It is almost not possible to say, whether a person gets twice as much satisfaction from one market value as from another. Not it is not known whether one person gets twice as much satisfaction as another from consuming the same basket. In fact as far as theory is concerned this constraint is not important. Because the objective is to understand consumer behavior all that matters is knowing how consumers rank different baskets.

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2.1. iv Marginal UtilityConsider a consumer who is consuming some bundle of goods, (x1,x2). How does this consumer’s utility change as we give a little more of good 1? This rate of change is called the marginal utility with respect to good 1. We write it as MU1 and think of it as being ratio,

That measures the rate of change in utility (∆U) associated with a small change in the amount of good 1 (∆x1). Good 2 is held fixed.

This definition implies that to calculate the change in utility associated with a small change in consumption of good1, just multiply the change in consumption by the marginal utility of the good.

∆U = MU1∆X1.

The marginal utility with respect to good 2 is defined in a similar manner:

When computing the marginal utility with respect to good 2 keep the amount of good 1 constant. We can calculate the change in utility associated with a change in the consumption of good2 by the formula ∆U = MU2∆X2.Marginal utility depends on the particular utility function that we use to reflect the preference ordering and its magnitude has no particular significance. Because of conceptual problems, economists have abandoned the old-fashioned view of utility as being a measure of happiness. Instead the theory of consumer behavior has been reformulated entirely in terms of consumer preferences, and utility is seen only a s a way to describe preferences.

2.1.ii. Indifference Curves

Consumer preferences are graphically shown with the use of indifference curves.Definition: An indifference curve represents all combinations of market baskets that provide a consumer with the same level of satisfaction. Given the three assumptions about preferences, it is known that a consumer can always indicate either a preference for one market basket over another or indifference between the two. By using this information, it is possible to rank all potential consumer choices. In order to appreciate this principle in graphic form, assume that there are only two goods available for consumption: food F and clothing C. In this case, all market baskets describe combinations of food and clothing that a person might wish to consume. The following table provides baskets containing various amounts of food and clothing.Table1. Alternative Market Baskets for Food and Clothing.

MARKET BASKET

UNITS OF FOOD UNITS OF CLOTHING

3

A 20 30

B 10 50

D 40 20 E 30 40 G 10 20

H 10 40

The following figure shows the same baskets listed in the table above.

Figure 1 Describing Individual Preferences

The horizontal axis measures the number of units of food purchased each week; the vertical axis measures the number of units of clothing. Market basket A, with 20 units of food and 30 units of clothing, is preferred to basket G because A contains more food and more clothing (third assumption non-satiation). Similarly, market basket E, which contains even more food and even more clothing preferred to A. In fact, it is easy to compare all market baskets in the two shaded areas (such as E and G) to A because they all contain either more or less of both food and clothing. Note, however, that B contains more clothing but less food than A. Likewise, D contains more food but less clothing than A. Therefore, comparisons of market baskets A with baskets B, D, and H are not possible without more information about the consumer’s ranking.

The following figure 2 shows an indifference curve, labeled U1, that passes through points A, B, and D. This curve indicates that the consumer is indifferent among these three market baskets.

It shows that in moving from market basket A to market basket B, the consumer feels neither better nor worse off in giving up 10 units of food to 20 units of clothing. Likewise, the consumer is indifferent between points A and D. He or she will give up 10 units of clothing to obtain 20 units of food. On the other hand, consumer prefers A to H, which lies below u1.

Indifference curve in the figure 2 slopes downwards from left to right. To understand why this must be the case, suppose instead it sloped upward from A to E. This would violate the assumption that more of any commodity is preferred less. Because market basket E has more of food and clothing than market basket A, it must be preferred to A and therefore cannot be on the same indifference curve as A. In fact, any market

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10 20 30 40

Clothing 50

40

30

20

10(Units/week)

*B *H *E *A *G *D

Food(Units perWeek)

basket lying above to the right of indifference curve u1 in figure 2 is preferred to any market basket on u1.

Figure 2 AN INDIFFERENCE CURVE

2.1.iii. CHARACTERISTICS OF INDIFFERENCE CURVESIndifference curves have certain characteristics that reflect assumptions about consumer behavior. In fact, one of the major uses of indifference curves is to examine the kinds of consumer behavior implied by different preferences, prices, and incomes. For simplicity, assume there are only two goods, Food and Clothing.1. An indifference curve passes through each point in the commodity space.2. Indifference curves cannot intersect.3. Indifference curves are negatively sloped.4. Indifference curves are convex in shape.5. The higher or further to the right is an indifference curve, the higher the bundles on that curve are in the consumer’s preference ordering, that is, baskets on higher indifference curves re preferred to bundles on lower indifference curves.

2.1.iv. THE MARGINAL RATE OF SUBSTITUTION (MRS)To quantify the amount of one good that a consumer will give up to obtain more of another, we use a measure called the marginal rate of substitution (MRS). The MRS of food F for clothing C is the amount of clothing that a person is willing to give up to obtain one additional unit of food. Suppose, for instance, the MRS is 3. This means that the consumer will give up 3 units of clothing to obtain 1 additional unit of food. If the MRS is 1/2, the consumer is willing to give up only ½ unit of clothing. Thus, the

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10 20 30 40 Food

*B *H *E *A *G *D u1

Food(Units per week)

Clothing(Units/week) 50

40

30

20

MRS measures the value that the individual places on 1 extra unit of one good in terms of another.

Figure 3. The Marginal Rate of Substitution

In figure 3 note that clothing appears on the vertical axis and food on the horizontal axis. When we describe the MRS, we must be clear about which good we are giving up and which we are getting more of. We define MRS in consistent terms as the amount of the good on the vertical axis that the consumer is willing to give up to obtain 1 extra unit of the good on the horizontal axis. In figure 3 the MRS refers to the amount of clothing that the consumer is willing to give up to obtain an additional unit of food. If we denote the change in clothing buy ∆C and the change in food by ∆F, the MRS can be written as -∆C/∆F. We add the negative to make the marginal rate of substitution a positive number (∆C is always negative; the consumer give up clothing to obtain additional food.)

Thus, the MRS at any point is equal in magnitude to the slope of the indifference curve. In Figure 3 for example, the MRS between points A and B is 6: The consumer is willing to give up 6 units of clothing to obtain 1 additional unit of food. Between points B and D, however, the MRS is 4: With these quantities of food and clothing, the consumer is willing to give up only 4 units of clothing to obtain 1 additional unit of food.

CONVEXITY Also observe in figure 3 that the MRS falls as we move down the indifference curve. This is not a coincidence. This decline in the MRS reflects an important characteristic of consumer preferences. To understand this, we will add an additional assumption regarding consumer preferences to the three that we discussed earlier in the chapter:

2.1.v Diminishing marginal rate of substitution: (Assumption 4 of Consumer Preferences) Indifference curves are convex or bowed inward. The term convex means that the slope of the indifference curve increase (i.e., becomes less negative) as we move down along the curve. In other words, an indifference curve is convex if the MRS diminishes along the curve. The indifference curve in Fig 3 is convex. As we have seen, starting with market basket A in Figure and moving to basket B, the MRS of Food F for Clothing C is -∆C/∆F = -(-6)/1 = 6. However, when we start at basket B and move from B to D, the MRS fall to 4. If we start at basket D and move to E, the MRS is 2. Starting at E and moving to G, we get an MRS of 1. As food consumption increases, the slope of the indifference curve falls in magnitude. Thus the MRS also falls. (With

6

-6

1 -4

1 -2

Cloth 16Units/wee

10

06

04 02

A

B

D

EG

0 1 2 3 4 5 Food Units/week

-1

non-convex preferences, the MRS increases as the amount of the good measured on the horizontal axis increases along any indifference curve. This unlikely possibility might arise if one or both goods are addictive. For example, the willingness to substitute an addictive drug for other goods might increase as the use of the addictive drug increased).

Is it reasonable to expect indifference curves to be convex? Yes, as more and more of one good is consumed, we can expect that a consumer will prefer to give up fewer and fewer units of a second good to get additional units of the first one. As we moved down the indifference curve in figure 3 and consumption of food increases, the additional satisfaction that a consumer gets from still more food will diminish. Thus, he will give unless and less clothing to obtain additional food.

Another way of describing this principle is to say that consumers generally prefer balanced market baskets to market baskets that contain all of one good and none of another. Note from Figure 3 that a relatively balanced market basket containing 3 units of food and 6 units of clothing (basket D) generates as much satisfaction as another market basket containing 1 unit of food and 16 units of clothing (Basket A). It follows that a balanced market basket containing (for example) 6 units of food and 8 units of clothing will generate a higher level of satisfaction.

2.1.vi. PERFECT SUBSTITUTES AND COMPLIMENTSThe shape of an indifference curve describes the willingness of a consumer to substitute one good for another. An indifference curve with a different shape implies a different willingness to substitute.

The goods are substitutes when an increase in the price of one leads to an increase in the quantity demanded of the other. In general, perfect substitutes when the marginal rate of substitution of one for the other is a constant. The indifference curves describing the trade-off between the consumption of the goods are straight lines. The slope of the indifference curves need not be-1 in the case of perfect substitutes. For more illustration, see the fig 3.a.

This fig 3.a shows preferences of a consumer for apple juice and orange juice. These two goods are perfect substitutes for a consumer because he is entirely indifferent between having a glass of one or the other. In this case, the MRS of apple juice for orange juice is 1: consumer is always willing to trade 1 glass of one for 1 glass of the other.Goods are compliments when an increase in the price of one leads to a decrease in the quantity demanded of the other. Two goods are perfect compliments when the indifference curves for both are shaped as right angles. The following figure 3.b illustrates a consumer’ preferences for left shoe and right shoe. For a consumer, the two goods are perfect complements because a left shoe will not increase her satisfaction unless he/she can obtain the matching right shoe. In this case, the MRS of left shoes for right shoes is zero whenever there are more right shoes than left shoes; consumer will not give up any left shoes to get additional right shoes. Correspondingly, the MRS is infinite whenever there are more left shoes than right because consumer will give up all but one of his/her excess left shoes than obtain an additional right shoe. Fig. 3.a Perfect substitutes

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3.b. Perfect Compliments.

Bads so far, all the examples mentioned above have involved only commodities that have been considered as “goods” – i.e., cases in which more of a commodity is preferred to less. But in the world there are also bad commodities which are preferred by certain groups of consumers. Bads are those commodities less of them is preferred to more. Air Pollution and asbestos in housing insulation are some of the examples of this kind of commodities. To analyze the consumer preferences for these commodities, they have to be redefined under study so that the consumer tastes are represented as the preference for less of the bad. This reversal turns the bads into goods. For instance, instead of preference for air pollution, it can be discussed as the preference for clean air, which can measure the degree of reduction in air pollution. Similarly, instead of referring to asbestos as a bad, it can be referred to corresponding good, the removal of asbestos.

With this simple adaptation, all four of the basic assumptions of consumer theory continue to hold.

2.2 BUDGET CONSTRAINTSThe budget constraints are that consumers face because of their limited incomes. People are compelled to determine their behavior in light of limited financial resources. For the theory of consumer behavior, this means that each consumer has a maximum amount that can be spent per period of time. The consumer’s problem is to spend this amount in the way they yield maximum satisfaction.

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1 2 3 4 Orange Juice (glasses)

Apple Juice (glasses) 4 3 2 1

1 2 3 4 Orange Juice (glasses)

Apple Juice (glasses) 4 3 2 1

The Consumption Basket of a consumer (x1,x2) is simply a list of two numbers that tells us how much the consumer is choosing to consume of good 1, x1, and how much the consumer is choosing to consume of good 2, x2. Sometimes it is convenient to denote the consumer’s bundle by a single symbol like X, where X is simply an abbreviation for the list of two numbers (x1,x2).

We suppose that we can observe the prices of the two goods, (p1,p2) and the amount of money the consumer has to spend m, then the budget constraint of the consumer can be written asp1x1 + p2x2 < m -------------------------------------------------(1)

Here p1x1 is the amount of money the consumer is spending on good 1, and p2x2 is the amount of money the consumer is spending on good 2. The budget constraint of the consumer requires that the amount of money spent on the two goods be no more than the total amount the consumer has to spend. The consumer’s affordable consumption bundles are those that do not cost any more than m.

i. Two Goods Are Often Enough.The two-goods assumption is more general. Because it is often interpreted as one of the goods are representing everything else the consumer might want to consume. For example, if we are interested in studying a consumer’s demand for milk, let x1 measure his or her consumption of milk in quarts per month. Then let x2 stand for everything else the consumer might want to consume.

When you adopt this interpretation, it is convenient to think of good 2 as being the money that the consumer can use to spend on other goods. Under this interpretation, the price of good 2 will automatically be 1, since the price of one birr is one birr. Thus, the budget constraint will take the form

p1x1+x2 < m -------------------------------------------------------(2)

This expression simply says that the amount of money spent on good 1, p1x1, plus the amount of money spent on all other goods, x2, must be no more than the total amount of money the consumer has to spend, m.Good 2 represents a composite good that stands for everything else that the consumer might want to consume other than good 1. Such a composite good is invariably measured in birr to be spent on goods other than good 1. As far as the algebraic form of the budget constraint is concerned, equation (2) is just a special case of the formula given in the equation (1)., with p2 = 1, so everything that we have to say about the budget constraint in general will hold under composite good interpretation.

ii. Properties of the Budget Set.The budget line is the set of bundles that cost exactly m:p1x1 + p2x2 = m----------------------------------------------(3)These are the bundles of goods that just exhaust the consumer’s income. The budget set is depicted in Fig 1. The heavy line is the budget line – the bundles that cost exactly m – and the bundles below this line are those that cost strictly less than m.

Figure 1 Budget Line.

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X2

Vertical intercept = m/p2

The budget set consists of all bundles that are affordable at the given prices and income.

Rearrange the budget line in equation (3) to get the formula

x2 = m/p2 – p1/p2 * x1 ----------------------------------------------------(4)

This is the formula for a straight line with a vertical intercept of m/p2 and a slope of –p1/p2. The formula tells how many units of good 2 the consumer needs to consume in order to just satisfy the budget constraint if he or she consuming x1 units of good 1.

The slope of the budget line measures the rate at which the market is willing to substitute good 1 for good2. For example that the consumer is going to increase her consumption of good 1 by ∆x1.

How much will consumption of good 2 have to change in order to satisfy her budget constraint?

p1x1 + p2x2 = m and p1(x1+∆x1) + p2(x2+∆x2) = m.

Subtracting the first equation from the second gives

p1∆x1 +p2∆x2 = 0.

This says that the total value of the change in consumption must be zero. Solving for ∆x2/∆x1, the rate at which good 2 can be substituted for good 1 while still satisfying the budget constraint, gives

This is just slope of the budget line. The negative sign is there since ∆x1 and ∆x2 must always have opposite signs. Because consumption of more of good1 leads to consumption of less of good 2 and vice versa, as long as the consumer continue to satisfy the budget constraint.

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Budget line; slope = -p1/p2

Horizontal Intercept = m/p1 X1

Fig . 1

iii. How the Budget Line ChangesWhen prices and incomes change, the set of goods that a consumer can afford changes as well. To find out, how these changes affect budget set? First consider changes in income. It easy to see from equation (4) that all increase in income will increase the vertical intercept and not affect the slope of the line. Thus an increase in income will result in a parallel shift outward of the budget line as in the following figure 2. Similarly, decrease in income will cause a parallel shift inward.

To identify changes in prices? First, consider increasing price 1 while holding price 2 and income fixed. According to equation (4) increase in p1 will not change the vertical intercept, but it will make the budget line steeper since p1/p2 will become larger.

What happens to the budget line when we change the prices of good 1 and good 2 at the same time? For example, if the prices of both goods 1 and 2 gets doubled, both the horizontal and vertical intercepts shift inward by a factor of one-half, and therefore the budget line shifts inwards by one-half as well. Multiplying both prices by two is just like dividing income by 2.

To show this algebraically. Suppose our original budget line is

p1x1 + p2x2 = m

Now suppose that both prices become t times as large. Multiplying both prices by t yields

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X2

m’/p1

m/p2

m/p1 m’/p1 X1

Fig. 2

m/p`1 m/p1 X1Fig. 3

Increasing Price. If good 1 becomes more expensive, the budget line becomes steeper.

X2 m/p2

Slope =-p1/p2

Budget Lines

Slope = -p1/p2

tp1x1 +t p2x2 = m

However, this equation is same as

p1x1 + p2x2 = m/t.

Thus multiplying both prices by a constant amount t is just like dividing income by the same constant t. It follows that if we multiply both prices by t and we multiply income by t. then budget line will not change at all.

What happens if both prices go up and incomes go down?If m decreases, p1, and p2 both increase, then the intercepts m/p1 and m/p2 must both decrease. This means that the budget line will shift inward. How about the slope of budget line? If price 2 increases more than price 1, so that – p1/p2 decreases ( in absolute value) then the budget line will be flatter; if price 2 increases less than 1, the budget line will be steeper.

The budget line is defined by two prices and one income, but one of these variables is redundant. We could peg one of the prices, or the income, to some fixed value, and adjust the other variables to describe exactly the same budget set. Thus the budget linep1x1 + p2x2 = m

exactly the same budget line as

p1/p2. x1+x2 =m/p2

orp1/m. x1 +p2/m. x2 = 1.

Since the first budget line results from dividing everything by p2, and the second budget line results from dividing everything by m. In the first case, we have pegged p2

= 1, and in the second case, we have pegged m = 1. Pegging the price of one of the goods or income to 1 and adjusting the other prices to 1, as we did above we often refer to that as the numeraire price. The numeraire price is the price relative to which we are measuring the other price and income. It will occasionally be convenient to think of one of the goods and being a numeraire good, since there will then be one less price to worry about.

2.3 CONSUMER PREFERENCESThe theory of consumer behavior begins with three basic assumptions about people’s preferences for one market basket versus another. These assumptions hold for most people in most situations.

2.3.i. Assumptions:1. Completeness: Preferences are assumed to be complete. In other words, consumers can compare and rank all possible baskets. Thus, for any two market baskets A and B, a consumer will prefer A to B, will prefer B to A, or will be indifferent between the two. By indifferent we mean that a person will be equally satisfied with either basket. Note that these preferences ignore costs. A consumer might prefer steak to hamburger but by hamburger because it is cheaper.

2. Transitivity: Preferences are transitive. Transitivity means that if a consumer prefers basket A to basket B and basket B to basket C, then the consumer also prefers

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A to C. For example, if a Ford is preferred to a Toyota and a Toyota to a Chevrolet, then Ford is also preferred to a Chevrolet. Transitivity is normally regarded as necessary for consumer consistency.

3. More is better than less (Nonsatiation): Goods are assumed to be desirable – i.e., to be good. Consequently, consumers always prefer more of any good to less. In addition, consumers are never satisfied or satiated; more is always better, even if just a little better. This assumption is made for academic reasons; namely, it simplifies the graphical analysis. Of course, some goods such as air pollution may be undesirable, and consumers will always prefer less.

These three assumptions form the basis of consumer theory. They do not explain consumer preferences, but they do impose a degree of rationality and reasonableness on them.

2.4 CONSUMER CHOICE

Given preferences and budget constraints, it is possible to determine how individual consumers choose how much of each good to buy. The basic assumption is that consumers make this choice in a rational way – that they choose goods to maximize the satisfaction they can achieve, given the limited budget available to them.

The maximizing market basket must satisfy two conditions:

1. It must be located on the budget line.2. It must give the consumer the most preferred combination of goods and services.

These two conditions reduce the problem of maximizing consumer satisfaction to one of picking point on the budget line.

We can graphically illustrate the solution to the consumer’s choice problem.

Indifference curve

Fig 1 Maximizing consumer satisfaction.

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Budget line

A

D

B

U3

U2

U1

Food

Clothing

The above figure 1 shows how the problem is solved. Here, three indifference curves describe a consumer’s preferences for food and clothing. Remember that of the three curves, the outer most curve u3, yields the greatest amount of satisfaction, curve u2the next greatest amount, and curve u1 the least.

The point B on indifference curve u1 is not the most preferred choice, because a reallocation of income in which more is spent on food and less on clothing can increase the consumer’s satisfaction. In particular, by moving to point A, the consumer spends the same amount of money and achieves the increased level of satisfaction associated with indifference curve u2 like the basket associated with D on indifference curve u3, achieve a higher level of satisfaction but cannot be purchased with the available income. Therefore, A maximizes the consumer’s satisfaction.

*If you know the consumer choices that a consumer made, how to determine his preferences?

REVEALED PREFERENCES THEORY The basic idea is simple. If a consumer chooses one market basket over another, and if the chosen market basket is more expensive than the alternative, then the consumer must prefer the chosen market basket.

Consider figure 2 where a consumer’s demanded bundle (x1, x2) and another arbitrary bundle (y1, y2) i.e., beneath the consumer’s budget line are depicted. The bundle (y1, y2) is certainly an affordable purchase at the given budget-the consumer could have bought it, if preferred, and would even have had money left over. Since (x1, x2) is the optimal bundle, it must be better than anything else that the consumer could afford. Hence, in particular it must be better than (y1, y2).

The same argument holds for any bundle on or underneath the budget line other than the demanded bundle. Since it could have been bought at the given budget but was not, then what was bought must be better. Here is where we use the assumption that there is a unique demanded bundle for each budget. If preferences are not strictly convex, so that indifference curves have flat spots, it may be that some bundles that are on the budget line might be just as good as the demanded bundle.

In figure 2 all of the bundles underneath the budget line are revealed worse than the demanded bundle (x1, x2). This is because they could have been chose, but were rejected in favor of (x1, x2).

Let (x1, x2) be the bundle purchased at prices (p1, p2) when the consumer has income m.

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Figure 2. Revealed Preferences: Two Budget Lines X1

The bundle (x1,x2), that the consumer chooses is revealed preferred to the bundle (y1,y2), a bundle that he could have chosen.

* *

(X1,X2)

L1

*(Y1,Y2)

X2

Since (x1, x2) is actually bought at the given budget, it must satisfy the budget constraint with equality

.

Putting these two equation together, the fact that (y1, y2) is affordable at the budget (p1, p2, m) means that

.

If the above inequality is satisfied and (y1, y2) is actually a different bundle from (x1, x2), then (x1, x2) is directly revealed to (y1, y2).

Important point is that the left hand side of this inequality is the expenditure on the bundle that is actually chosen at prices (p1, p2). Thus, revealed preference is a relation that holds between the bundle that is actually demanded at some budget and the bundles that could have been demanded at that budget. When we say that X is revealed preferred to Y, it means that X is chosen when Y could have been chose; that is, that

Optimum of the ConsumerA consumer is said to be at optimum when he/she maximizes utility subject to income constraint.

Consider U = f(x), price of Px.The consumer can either buy an additional unit of X or keep the income, which was to be spent on that additional unit of X unspent. Both give some satisfaction to the consumer. To determine the optimum of the consumer will need to compare the MUx

and Px.

MUx which the consumer will have to pay.

Px What the consumer has to pay.

MUx > Px the consumer can raise his/her satisfaction by purchasing more units of good of X.

MUx <Px The consumer can raise his level of satisfaction by reducing consumption of x.

MUx = Px Optimum of the consumer. When the utility function of the consumer contains more than one good. U = f (x1,x2,….xn). The optimum of the consumer receives the following conditions to be fulfilled.

The condition.

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The satisfaction the consumer derives by spending one unit of money x1.

Spending one unit of money should result in the same satisfaction regardless of the good it is spent on. If spending one unit of money results in higher satisfaction when it is spent on x1 compared to other goods. The consumer can raise his satisfaction by spending more on x1 and less on other unit the above condition is fulfilled.

Appendix: Mathematical Derivation of Optimum of Consumer Though the measurement of marginal utility is unnecessary the optimality condition. We arrive at through the ordinal approach is the same that of the cardinal approach. Mathematically the optimum of the consumer can be derived by maximizing the utility subject to the budget constraint.

max.u(x1 . x2)

subject to P1X1+P2X2 < m to combine the two use what is called a Langragean function.

L = U (X1/X2) – λ (P1X1+P2X2 – m)

The Lagrangian theorem states that the optimum of the consumer should fulfill the following three first order conditions.

Condition 1.

Condition 2.

Condition 3.

From (1) MUX1 = λ P1

From (2) MUX2 = λP2

for the Cobb-Douglas utility function

u (X1.X2) = X1c X2

d

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P2cX2=P1dX1

From (condition 3)

where

The proposition of income spent on x1 at the consumer optimum.

a in the proportion of m spent on X1

1-a in the proportion of m spent on X2

U = f (X1,X2,……………..,Xn)

17

Proposition of m spent on X1 and X2 respectively.

The % of income which is spent on the X1.

The % of income which is spent on X2.

Example:

Suppose a consumer spends his/ her entire income on food (x1) and clothing (x2) and 25% of the total income is spend on food. If price of x1 = 2, prices of X2=3, m = 200, estimate function & determine the optimum quantity of x1 & x2.

u (x1. x2) = X1 ¼ , X2 ¾

x1= 0.25*200/2 = 25, x2 = 0.75*200/3=50

2.5 OFFER CURVES OF INCOME AND PRICE CONSUMPTION

1. IntroductionThe consumer’s demand function gives the optimal amounts of each of the goods as a function of the prices and income faced by the consumer. The demand function is written as x1=x1 (p1,p2,m)

x2=x2 (p1,p2,m).

The left hand side of each equation stands for the quantity demanded. The right-hand side of each equation is the function that relates the prices and income to that quantity.

Studying how a choice responds to changes in the economic environment is known as comparative statics. “Comparative” means that we want to compare situations: before and after the change in the economic environment. Statics means that we are not concerned with any adjustment process may be involved in moving from one choice to another, others be examined in equilibrium choice. In the case of consumer, there are two things in the model that affect optimal choice: price and income. Two comparative statics questions in consumer theory therefore involve investigating how demand changes when prices and income changes.

2. Normal and Inferior GoodsNormal goods are those goods when their demand for each good would increase when income increases. If good 1 is a normal good, then the demand for it increases when income increases, and decreases when income decreases. For a normal good, the quantity demanded always changes in the same way as income changes. Graphical representation is shown in the following figure 1

18

Optimal choices

Budget lines

Indifference curves

X1

X2

Fig. 1 Normal Goods: The demand for both goods increases when income increases.If something is called normal, there must be possibility of being abnormal. where an increase of income result in a reduction in the consumption of one of the goods. Such goods are called as Inferior Goods. There are many goods for which demand decreases as income increases: example, gruel, bologna, shacks, or nearly any kind of low quality good.

Whether a good is inferior or not depends on the income level that we examining. It might very well be that very poor people consume more bologna as their income increases. But after a point, the consumption of bologna would probably decline as income continued to increase. Since in real life the consumption of goods can increase when income increases. Fig 2 shows the graphical presentation of inferior good.

Fig. 2 An Inferior Good: Good 1 is an inferior good, which means that the demand for it decreases when income increases.

3. Income Offer Curves and Engel CurvesIncome offer curve is constructed by shifting the budget line outward. This curve illustrates the bundles of goods that are demanded at the different levels of income, as depicted in figure 3.a. For each level of income, m, there will be some optimal choice for each of the goods. First focus on good 1 and consider the optimal choice of each set of prices and income, x1(p1.p2.m). This is simply the demand function for good1. If prices of goods 1 and 2 are held fixed and look at how demand changes because of change income, for that purpose Engel Curve is generated. The Engel curve is a graph of the demand for

19

Budget linesOptimal choices

X1

X2

one of the goods as a function of income, with all prices being held constant . See figure 3.b

3.a Income Offer Curve x1 3.b Engel Curve x1

The income offer (for the consumption) curve is obtained by connecting successive optimum points arising from increasing income. The curve is sloped positively, when X1 and X2 are normal goods. But if one of the goods is inferior then the curve is negatively sloped.

4. The Price Offer Curve and the Demand Curve Suppose the price of good 1 change while p2 and income are fixed. Geometrically this involves pivoting the budget line. To construct price offer curve optimal points be connected together as illustrated in figure 4.a.This curve represents the bundles that would be demanded at different prices for good 1.

In other words, hold the price of good2 and money income fixed, and for each different value of p1 plot the optimal level of consumption of good1. The result is the demand curve depicted in figure 4.b. The demand curve is a plot of the demand function, x1(p1, p2, m), holding p2 and m fixed at some predetermined values.

Ordinarily, when the price of a good increases, the demand for the good will decrease. Thus the price and quantity of a good will move in opposite directions, which means that the demand curve will typically have a negative slope. In terms of rates of change,

, which simply says that demand curves usually have a negative slope.

20

How demand changes as income changes. The income offer curve (or income expansion path) shown in panel A depicts the optimal choice at different levels of income and constant prices. When you plot the optimal choice of good 1 against income, m, we get the Engel curve, depicted in panel B.

X2

Indifference curves

Income offer curveEngel Curve

m

Indifference curves

Price Offer Curve

Demand Curve

4.a Price Offer Curve 4.b Demand Curve X1

The price offer curve and demand curve. Panel A contains a price offer curve, which depicts the optimal choices as the price of good1, changes. Panel B contains the associated demand curve, which depicts a plot of the optimal choice of good 1 as a function of its price.

X1

X2 P1

The price offer curve is obtained as a locus of successive optimum points resulting from successive changes in price of one of the goods keeping income and prices of the price offer curve shows you what happens in the optimal of consumer consumption with respect to the decrease in price of X1.

2.6 INCOME AND SUBSTITUTION EFFECTS (Slutsky Equation)Slutsky equation is the equation showing how the effect on demand for a good of a change in price can be decomposed into the substitution effect, which shows the effect of a change in relative prices at an unchanged level of real income, and the income effect, which shows the effect of change in real income holding prices constant.

When the price of a good changes, there are two sorts of effects: the rate at which you can exchange one good for another changes, and the total purchasing power of your income is altered. For example, if good1 becomes cheaper, it means that you have to give up less of good 2 to purchase good1. The change in the price of good 1has changes the rate at which the market allows you to ‘substitute’ good 2 for good 1. The trade-off between the two goods that the market presents the consumer has changed.

At the same time, if good1 becomes cheaper it means that your money income will buy more of good 1. “The purchasing power of your money has gone up although the number of dollars you have is the same, the amount that they will buy has increased.The Substitution Effect: The part of the effect of a price change on demand due to the change in relative prices, assuming that the consumers compensated sufficiently to remain at the same level of utility.The change in demand due to the change in the rate of exchange between the two goods is called substitution effect.

X2

21

*

X1 x1 Figure 1. Pivot and Shift. When the price of good 1 changes and income stays fixed, the budget line pivots around the vertical axis. We will view this adjustment as occurring in two stages: first pivot the budget line around the original choice, and then shift this line outward to the new demanded bundle.

X2 Original Budget Line

Final Choice

Final Budget line

Indifference Curves

Original Choice

What are the economic meanings of the pivoted and shifted budget lines? This “pivot-shift” operation gives us a convenient way to decompose the change in demand into two pieces. The first step- the pivot- is a movement where the slope of the budget line changes while its purchasing power stays constant, while the second step is a movement where the slope stays constant and the purchasing power changes. This decomposition is only a hypothetical construction – the consumer simply observes a change in price and chooses a new bundle of goods in response. But in analyzing how the consumer’s choice changes, it is useful to think of budget line change in two stages – first pivot and then the shift.

First, consider the pivoted line. Here is a budget line with the same slope and thus the same relative prices as the final budget line. However, the money income associated with this budget line is different. Since the vertical intercept is different. Since the original consumption bundle (x1,x2) lies on the pivoted budget line, that consumption bundle is just affordable. The purchasing power of the consumer has remained constant in the sense that the original bundle of goods is just affordable at the new pivoted line.

Calculate how much we have to adjust money income in order to keep the old bundle just affordable. Letm1 be the amount of money income that will just make the original consumption bundle affordable; this will be the amount of money income associated with the pivoted budget line. Since (x1, x2) is affordable at both (p1, p2, m|), we have

Subtracting the second equation from the first gives

.

This equation says that the change in money income necessary to make the old bundle affordable at the new prices is just the original amount of consumption of good 1 times the change in prices.

Letting represents the change in price 1, and represents the

change in income necessary to make the old bundle just affordable, we have

Now we have a formula for the pivoted budget line: it is just the budget line at the new price with income change by ∆m. Note that if the price of good 1 goes down, a consumer’s purchasing power goes up, so we will have to decrease the consumer’s income in order to keep purchasing power fixed. Similarly, when a price goes up, purchasing power constant must be positive.

The substitution effect is the change in the demand for good1 when the price of good 1 changes to p|

1 and, at the same time, money income changes to m|:

22

In order to determine the substitution effect, use the consumer’s demand function to calculate the optimal choices at (p1

1,m1) and (p1,m). The change in demand for good 1 may be large or small, depending on the shape of the consumer’s indifference curves.

The substitution effect is sometimes called the change in compensated demand. The idea is that the consumer is being compensated for a price rise by having enough income given back to him to purchase his old bundle. Of course, if the price goes down he is compensated by having money taken away from him.

The Income EffectThe change in demand due to having more purchasing power is called income effect. This is the part of the response in the demand for a good to a change in its price which is due to the rise in the real income of consumer resulting from a price decrease.A parallel shift of the budget line is the movement that occurs when income changes while relative prices remain constant. This second stage of price adjustment is called income effect. Simply change the consumer’s income from m| to m, keeping the prices constant at (p1

1, p2). In figure 2 this change moves us from the point Y to Z. It is natural to call this last movement the income effect because changing the income while keeping the prices fixed at the new prices.More precisely, the income effect, , is the change in the demand for good 1 when we change income from m1, to m, holding the price of good 1 fixed at p`1:

.

When the price of a good decreases, we need to decrease income in order to keep purchasing power constant. If the good is a normal good, then this decrease in income will lead to a decrease in income will lead to a decrease in demand. If the good is an inferior good, then the decrease in income will lead to an increase in demand.

The Total Change in DemandThe total change in demand Δx1, is the change in demand due to the change in price, holding income constant;

23

0 Substitution Income X1

effect effect Figure 2: Substitution effect and Income effect. The pivot gives the substitution effect and the shift gives the income effect.

X2

m/p2

m1/P2

x y

z

Indifference Curves

.

This can be broken up into substitution effect and the income effect. In terms of symbols defined above,

This equation is called Slutsky’s Identity (Named after Russian economist Eugen Slutskey, 1880-1948). In words, this equation says that the total change in demand equals the substitution effect plus the income effect. It is called identity because; it is true for all values of p1, p|

1, m and m|. The first and fourth terms on the right hand side cancel out, so the right hand side is identically equal to the left hand side.

The content comes from the interpretation of two terms on the right hand side: substitution effect and the income effect. While substitution effect must always be negative-opposite the change in the price-the income effect can go either way. Thus the total effect may be positive or negative. However, if we have normal good, then the substitution effect and the income effect work in the same direction. An increase in price means that the demand will go down due to the substitution effect. If price goes up, it is like decrease in income, which, for a normal good, means a decrease in demand. The Law of Demand: If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases. This follows directly from the Slutsky equation: if the demand increases when income increases, we have normal good. In addition, if we have a normal good then the substitution effect and the income effect reinforce each other, and an increase in price definitely reduce demand.Some Examples with Perfect compliments and substitutes. Slutsky’s decomposition is illustrated in the fig 3. When we pivot the budget line around the chosen point, the optimal choice at the new budget line is the same as at the old one – this means that the substitution effect is zero. The change in demand is due entirely to the income effect.The case of perfect substitutes illustrated in figure 4. When we tilt the budget line, the demand bundle jumps from the vertical axis to the horizontal axis. This is

24

because of the entire change in the demand due to the substitution effect.

To conclude, Slutsky equation says that the total change in demand is the sum of the substitution effect and the income effect.

A Numerical Example: Calculating the Substitution Effect

Suppose that the consumer has a demand function for milk of the form .

Originally his income is Br.120 per week and the price of milk is Br.3 per quart. Thus

his demand for milk will be quarts per week. Now suppose that the price

of milk falls to Br.2 per quart. Then his demand at this new price will be

quarts of milk per week. The total change in demand is +2 quarts a week.

In order to calculate the substitution effect, we must first calculate how much income would have to change in order to make the original consumption of milk just affordable when the price of milk is Br.2 a quart. We apply the formula

25

Pivot Shift

Original budget line

Final budget line

Income effect = Total effectX1

Indifference curvesX2

Figure 3Perfect Compliments. Slutsky decomposition with perfect compliments

Indifference curves

Final budget line

Final Choice

Original Choice

Original budget line

X1

X2

Substitution effect = total effect

Figure 4Perfect substitutes. Slutsky decomposition with perfect substitutes.

Thus the level of income necessary to keep purchasing power constant is what is the consumer’s demand for milk at the new price, Br.2

per quart, and this level of income? Just plug numbers into the demand function to find.

Thus the substitution effect is .

Calculating the Income Effect.

Thus the income effect for this problem is .

Since milk is a normal good for this consumer, the demand for milk increases when income increases.

2.7 DERIVATION OF CONSUMER DEMANDThe derivation of demand is based on the axiom of diminishing marginal utility. The marginal utility of commodity x may be depicted by a line with a negative slope (fig. 2). Geometrically the marginal utility of x is the slope of the total utility function U=f(qx). The total utility increases, but at a decreasing rate, up to quantity x and then starts declining (fig. 1).

Accordingly the marginal utility of x declines continuously, and becomes negative beyond quantity x. If the marginal utility is measured in monetary units, the demand curve for x is identical to the positive segment of the marginal utility curve. At x t the marginal utility is MU1 (fig. 3). This is equal to P1, by definition. Hence, at P1 the consumer demand x1 quantity (fig. 4).

26

0 x qx Fig. 1

TU

Ux

qx

MUx

MUx

0 x

Fig. 2

MUx

MU1

MU2

MU3

0 x1 x2 x3 x qx

Fig. 40 x1 x2 x3

Fig. 3 MUx

Px

P1

P2

P3

Similarly, at x2 the marginal utility is MU2, which is equal to P2. Hence, at P2 the consumer will buy x2, and so on. The negative section of MU curve does not form part of the demand curve, since negative quantities do not make sense in economics.

Graphical Presentation of the Equilibrium of the Consumer:

Given the indifference map of the consumer and his budget line, the equilibrium is defined by the point of tangency of the budget line with the highest possible indifference curve (point e in fig. 5).

At the point of tangency the slopes of the budget line (Px/Py) and of the indifference curve (MRSx,y=MUx/MUy) are equal:

Thus the first-order condition is denoted graphically by the point of tangency of the two relevant curves. The second-order condition is implied by the convex shape of the indifference curves. The consumer maximizes his utility by buying x* and y* of the two commodities.

Mathematical derivation of the equilibrium:

Given the market prices and his income, the consumer aims at the maximization of his utility. Assume that there are n commodities available to the consumer, with given market prices P1,P2,…….Pn. The consumer has a money income (Y) which he spends on the available commodities.

Formally, the problem may be stated as follows:

Maximise U = f(q1,q2,….,qn)

Subject to

27

0 x * B XFig. 5

e

Y A

Y*

We use the ‘Langrangian multipliers’ method for the solution of this constrained maximum.The steps involved in this method may be outlined as follows:

(a) Rewrite the constraint in the form

(q1P1+q2P2+…+qnPn - Y) = 0

(b) Multiply the constraint by a constant λ, which is the Lagrangian multiplier.

λ(q1P1+q1P2+…..qnPn – Y) = 0.

(c) Subtract the above constraint from the utility function and obtain the ‘composite function’.

= U – λ(q1P1+q2P2+…..qnPn – Y)

It can be shown that maximization of the ‘composite function implies maximization of the utility function.The first condition for the maximization of a function is that its partial derivatives be equal to zero. Differentiating Φ with respect to q1,…qn and λ and equating to zero we find

. . . .

From these equation we obtain

. . .

Substituting and solving for λ we find

Alternatively, we may divide the preceding equation corresponding to commodity x, by the equation which refers to commodity y, and obtain

28

We observe that the equilibrium conditions are identical in the cardinalist approach and in the indifference curves approach. In both theories we have

Thus, although in the indifference – curves approach cardinality of utility is not required. The MRS requires knowledge of the ratio of the marginal utilities, given that the first order condition for any two commodities may be written as

Hence, the concept of marginal utility is implicit in the definition of the slope of the indifference curves, although its measurement is not required by this approach. What is preceded is a diminishing marginal rate of substitution, which of course does not require diminishing marginal utilities of the commodities involved in the utility function.

2.8 THE CONSUMER’S SURPLUSThe consumer’s surplus is a concept introduced by Marshall, who maintained that it can be measured in monetary units, and is equal to the difference between the amount of money that a consumer actually pays to buy a certain quantity of commodity x, and the amount that he would be willing to pay for this quantity rather than do with it.

Graphically the consumer’s surplus may be found by his demand curve for commodity x and the current market price. Assume that the consumer’s demand for x is a straight line (AB in Figure 1) and the market price is P. At this price the consumer buys q units of x and pays an amount (q).(P) for it. However, he would be willing to pay p1 for q1, P2 for q2, P3 for q3 and so on. The fact that the price in the market is lower than the price he would be willing to pay for the initial units of x implies that his actual expenditure is less than he would be willing to spend to acquire the quantity q. This difference is the consumer’s surplus, and is the area of the triangle PAC in figure 1.

Figure 1

29

Px

A P1

P2

P3

P

0 q1 q2 q3 q B

C

The Marshallian consumer’s surplus can also be measured by using indifference curves analysis. In figure 2 the good measured on the horizontal axis is x, while on the vertical axis we measure the consumer’s money income. The budget line of the consumer is MM1 and its slope is equal to the price of the commodity x (since the price of one unit of monetary unit is 1). Given Px, the consumer is in equilibrium at E: he buys OQ quantity of x and pays AM` of his income for it, being left with OA amount of money to spend on all other commodities.

Next, find the amount of money which consumer would be willing to pay for OQ quantity of x rather than do without it.

This is attained by drawing an indifference curve passing through M. Under Marshallian assumption that the MU of money income is constant, this indifference curve will be vertically parallel to the indifference curve I1; the indifference curves will have the same slope at any given quantity of x. for example, at Q the slope of I1 is the same as the I0

Slope I1 for Q units of X = MRS x,M =

(Given that MUM = 1)

Similarly

Slope I0 for Q units of X = MRS x,M =

Given that the quantity of x is the same at E and B, the two slopes are equal.

The indifference curve I0 shows that the consumer would be willing to pay A1M for the quantity OQ, since point B shows indifference of the consumer between having OQ of x and OA` of income to spend on other goods, or having none of x and spending all his income M on other goods. In other words A`M is the amount of money that the consumer would be willing to pay for OQ rather than do without it.

The difference A`M - AM = AA` = EB is the difference between what the consumer actually pays and what he would be willing to pay for OQ of x. This difference is the Marshallian Consumer Surplus.Other Interpretation of Consumer’s Surplus

30

M Income

M1

A

A|

I1

E

B

O Q M| X Fig. 2

There is also another way to think about consumer’s surplus. Suppose that the price of the discrete goods is p. Then the value that the consumer places on the first unit of consumption of that good is r1, but he only has to pay p for it. This gives him a surplus of r1-p on the first unit of consumption. He values the second unit of consumption of at r2, but again he only has to pay p for it. This gives him a surplus of r2-p on that unit. If we add this up- over all n units the consumer chooses, we get his total consumer’s surplus:

Since the sum of the reservation prices just gives us the utility of consumption of good 1. We can also write this as

From Consumer’s surplus to Consumers surplus:So fare we are considering the case of a single consumer. If several consumers are involved we can ad up each consumer’s surplus across the consumers to create an aggregate measure of the consumers’ surplus. Observe carefully the distinction between the two concepts: consumer’s surplus refers to the surplus of single consumer: consumers’ surplus refers to the sum of the surpluses across a number of consumers.

Consumers’ surplus serves as a convenient measure of the aggregate gains from trade, just as consumer’s surplus serves as a measure of the individual gains from trade.

Interpreting the changes in Consumer’s Surplus

In Figure 3 the change in consumer’s surplus associated with a change in price. The change in consumer’s surplus is the difference between two roughly triangular regions and will there has trapezoidal shape. The trapezoid is further composed of two regions, the rectangle indicated buy R and the roughly triangular region indicated by T.

The rectangle measures the loss in surplus due to the fact that the consumer is now paying more for all the units he continues to consume. After the price increases the consumer continues to consumer x|| units of the good and each unit of the good is now

31

0 X11 X1 XFig. 3 Change in consumer’s surplus. The change in consumer’s surplus will be the difference between two roughly triangular areas, and thus will have a roughly trapezoidal shape.

R

P

P11

P1

Demand Curve

T

Change in consumer surplus

more expensive by p|| - p1. This means he has to spend (p|| - p|) x|| more money than he did before just to consume x|| units of the good.

This is not entire welfare loss. Due to the increase in the price of the x good, the consumer has decided to consumer less of it than he was before. The triangle T measures the value of the lost consumption of the x good. The total loss to the consumer is the sum of these two effects: R measures the loss from having to pay more for the units he continues to consume, and T measures the loss from the reduced consumption.

2.9 THE MARKET DEMAND1. Derivation of the Market DemandThe market demand for a given commodity is the horizontal summation of the demand of the individual consumers. In other words, the quantity demanded in the market at each price is the sum of the individual demands of all consumers at that price.

Table1 Individual and Market DemandPrice demand Quantity

demanded by consumer A

Quantity demanded by consumer B

Quantity demanded by consumer C

Quantity demanded by consumer D

Market

2 107 40 4 45 186 72 24 5 30 1310 50 14 10 15 1114 29 8 5 10 618 6 4 2 0 020 3 3 0 0 0

Economic theory does not define any particular form of the demand curve. Market demand is sometimes shown in textbooks as a straight line (linear demand curve) and sometimes as a curve convex to the origin. The linear demand curve (Fig 1) may be written in the formQ = b0 – b1P

32

0 10 20 30 40 50 60 70 80 90 100 110 Q Fig. 1

P

20

16

14

10

6

2

*

* ** * * * * * * * * * * ** * * *

and implies a constant slope, but a changing elasticity at various prices. The most common form of a non-linear demand curve is the so called constant – elasticity – demand curve, which implies constant elasticity at all prices; its mathematical form is

Q = b0.pb1

Where b1 is the constant price elasticity.

2. DETERMINANTS OF DEMANDTraditionally the most important determinants of the market demand are considered to be the price of the commodity in question, the prices of other commodities, consumer’s income and tastes. The result of a change in the price of the commodity is shown by a movement from one point to another on the same demand curve. Thus these factors are called shift factors, and the demand curve is drawn under the ceteris paribus assumption, that the shift factors are constant. The distinction between movements along the curve and shifts of the curve is convenient for the graphical presentation of the demand function. Conceptually, however, demand should be thought of as being determined by various factors (is multivariate) and the change in any one of these factors changes the quantity demanded.

Apart from the above determinants, demand is affected by numerous other factors, such as the distribution of income, total population and its composition, wealth, credit availability, stocks and habits. The last two factors allow for the influence of past behavior on the present, thus rendering demand analysis dynamic.

2.10 ELASTICITY OF DEMANDThere are as many, elasticities of demand as its determinants. The most important of these are (i) the price elasticity, (ii) the income elasticity, (iii) the cross-elasticity of demand.(i) The Price Elasticity of Demand:The price elasticity is a measure of the responsiveness of demand to changes in commodity’s own price. If the changes in price are very small, we use a measure of the responsiveness of demand the point elasticity of demand. If the changes in price are not small, we use the arc elasticity of demand as the relevant measure.

33

D

0 x1 x2 Q Fig:2 Movement along the demand curve as the price

Of x changes.

P

P1 P2

0 x1 x2 x3 QFig 3: Shifts of the demand curve as, for example Income increases.

or

If the demand curve is linear

Its slope is . Substituting in the elasticity formula, we obtain

This implies that the elasticity changes at the various points of the linear-demand curve. Graphically the point elasticity of a linear demand curve is shown by the ratio of the segments of the line to the right and to the left of the particular point. In figure

1 the elasticity of the linear demand curve at point F is the ratio

From Fig.1 we see that

ΔP = P1P2 = EF ΔQ = Q1Q2 = EF|

P = OP1 Q = OQ1

If we consider very small changes in P and Q, then

Thus substituting in the formula for the point elasticity, we obtain

From the figure we can also see that the triangles FEF| and FQ1D| are similar (because each corresponding angle is equal). Hence

Thus

Furthermore the Triangles DP1F and FQ1D| are similar, so that34

0 Q1 Q2 D| O

Figure 1

P D

P1

P2

F

F1

E

Re Arranging we obtain

Thus the price elasticity at point F is

Given the graphical measurement of point elasticity it is obvious that at the mid-point of a linear demand curve ep = 1 (Point M in Fig.2)

At any point to the right of M the point elasticity is less than unity (ep < 1); finally at any point to the left of M, ep > 1. At point D the , while as point D` the ep = 0. The price elasticity is always negative because of the inverse relationship between Q and P implied by the ‘law of demand’. However, traditionally the negative sign is omitted when writing the formula of the elasticity.The range of values of elasticity are If ep = 0 the demand is perfectly inelasticif ep=1 the demand has unitary elasticityIf , the demand is perfectly elastic.If 0<e<1, we say that the demand inelasticif 1<e< , the demand is elastic.Figures 3, 4, and 5.

35

0 ep =0 0 ep =1 0 Fig 3 Fig 4 Fig 5

M

ep=1

ep<1

ep>1

ep=0

ep→∞

0 QFig. 2 D|

P D

D

D

DP

The basic determinants of the elasticity of demand of a commodity with respect to its own price are:(1) The availability of substitutes; the demand for a commodity is more elastic if there are close substitutes for it.(2) The nature of the need that the commodity satisfies. In general, luxury goods are price elastic, while necessities are price inelastic.(3) The time period. Demand is more elastic in the long run.(4) The number of uses to which a commodity can be put. The more the possible uses of a commodity the greater its price elasticity will be.(5) The proportion of income spent on the particular commodity.The above formula for the price elasticity is applicable only for extremely small changes in the price. If the price changes appreciably, we use the following formula, which measures the arc elasticity of demand: ep = ΔQ P1+P2/2 = ΔQ (P1+P2) ΔP Q1+Q2/2 ΔP (Q1+Q2)The arc elasticity is a measure of the average elasticity, i.e., the elasticity at the mid point of the chord that connects the two points (A and B) on the demand curve defined by the initial and the new price levels (fig 6). It should be clear that the measure of the arc elasticity is an approximation of the true elasticity of the section AB or if the demand curve, which is used when we know only the two points A and B from the demand curve is, the poorer the liner approximation attained by the arc elasticity formula.

.

(ii) The Income Elasticity of Demand.The income elasticity is defined as the proportionate change in the quantity demanded resulting from a proportionate change in Income. Symbolically we may write

ey = dQ/Q = dQ Y dY/Y dY QThe income elasticity is positive for normal goods. A commodity is considered to be a luxury if its income elasticity is greater than unity. A commodity is a necessity if its income elasticity is small (less than unity, usually).The main determinants of income elasticity are:1. The nature of the need that the commodity covers: the percentage of income spent on food declines as income increases (this is known as ‘Engel’s Law’ and has some times been used as a measure of welfare and of the development stage of an economy.)

36

Aarc elasticityA

B

D

O Q1 Q2 Q Figure 6

P

P1

P2

D

2. The initial level of income of a country. For example, a TV set is a luxury in an underdeveloped, poor country while it is a ‘necessity’ in a country with high per capita income.3. The Time period, because consumption patterns adjust with a time-lag to changes in income.

(iii) The Cross-Elasticity of DemandThe cross-elasticity of demand is defined as the proportionate change in the quantity demanded of x resulting from a proportionate change in the price of y. Symbolically we have

The sign of the cross elasticity is negative if x and y are complementary goods, and positive if x and y are substitutes. The higher the value of the cross-elasticity the stronger will be the substitutability or complemetarity of x and y.

The main determinant of cross-elasticity is the nature of the commodities relative to their use. If two commodities can satisfy equally well the same need, the cross-elasticity is high, and vice-versa.

Chapter ThreeTheory of production

3.1 The Production Functioni. The Production Function for a single product:The production function is a purely technical relation which connects factor-inputs and outputs. It describes the laws of proportion, which is the transformation of factor inputs into products (outputs) at any particular time period. The production function represents the technology of a firm of an industry, of the economy as a whole. The production function includes all the technically efficient methods of production.

The production function in traditional economic theory assumes the form. The factor ν, returns to scale, refer to the long run analysis of the laws

of production. The efficiency parameter γ refers to the entrepreneurial-organizational aspects of production.

Graphically, the production function is usually presented as a curve on two dimensional graphs. Changes of the relevant variables are shown either by movements along the curve that depicts the production function, or by shifts of this curve. The most commonly used diagrams of a single product are shown in figures 1a and 1b.

37

X=f(L) k3, ν3, γ3

X=f(L) k1, ν1, γ1

X X=f(K) L3, ν3, γ3

X=f(K) L1, ν1, γ1

X

Fig.1a. Fig.1b.

Each curve shows the relation between X and L for given K, ν, and γ. As labor increases, we move along the curve depicting the production function. If capital increase, the production function X= f(L) moves upwards.The marginal product of a factor is defined as the change in output resulting from a very small change of this factor, keeping all other factors constant.

Mathematically the marginal product of each factor is the partial derivative of

production function with respect to this factor. and

Graphically the marginal product of labor is shown by the slope of the production function. X=f1(L) K, ν, γ and the marginal product of capital is shown by the slope of the production function X = f2 (K) L, ν, γ. The theory of production concentrates on levels of employment of the factors over which their marginal products are positive but decrease: in fig.2 the range of employment of L examined by the theory of production is AB over that range MPL>0

but .

Similarly in fig.3 the range of employment of capital examined by the theory of

production is CD over that range MPK>0 but .

X

A method of production is a combination of factor inputs required for the production of one unit of output. Usually a commodity may be produced by various methods of

38

L0

K0

0 Fig. 2 A| B| L

X|

production. For example, a unit of commodity x may be produced by the following process:

Process P1 Process P2 Process P3

Labor units 2 3 1

Capital Units 3 2 4

Activities may be presented graphically by the length of lines from the origin to the point determined by the labor and capital inputs. The three processes above are shown in Fig. 3.1.

A BLabor units 2 3

Capital Units 3 3

A method of production A is technically efficient relative to any other method B, if A uses less of at least one factor and no more from the other factors compared with B. Method B is technically inefficient as compared with A. The basic theory of production concentrates only on efficient methods. Inefficient methods will not be used by rational entrepreneurs.

A BLabor units 2 1Capital Units 3 4

Process A uses less of some factor(s) and more of some other(s) as compared with any other process B, then A and B cannot be directly compared on the criterion of technical efficiency. For example, the activities are not directly comparable. Both process are considered as technically efficient and are included in the production function (the technology). When one of them will be chosen at any particular time depends on the prices of factors. The theory of production describes the laws of production. The choice of any particular technique is economic one, based on prices, and not a technical one. Note that! a technically efficient method is not necessarily economically efficient. There is difference between technical and economic efficiency.

An isoquant includes all the technically efficient methods (or all the combinations of factors of production) for producing a given level of output. The production isoquant may assume various shapes depending on the degree of substitutability of factors. They are broadly classified as linear isoquant, Input-output isoquant, Kinked Isoquant and Convex Isoquant.

39

0 1 2 3 4 L Fig 1

K 4 3 2

Linear isoquant: assumes perfect substitutability of factors of production: a given commodity may be produced by using only capital, or only labor, or by an infinite combination of K and L. See figure. 2

Input-output isoquant: assumes strict complimentarity (that is zero substitution) of the factors of production. There is only one method of production for any one commodity. The isoquant takes the shape of right angle (fig. 3). This type of isoquant is also called ‘Leontief isoquant’ after Leontief, who invented the input output analysis.

Kinked isoquant: assumes limited substitutability of K and L. There are only a few processes for producing any one commodity. Substitutability of the factors is possible only at the kinks. This form is also called ‘activity analysis-isoquant’ or “linear programming isoquant”, because it is basically used in linear programming. Smooth, convex Isoquant: assumes continuous substitutability of K and L only over a certain range, beyond which factors cannot substitute each other. The isoquant appears as a smooth curve, convex to the origin. See fig 5. The production function describes not only a single isoquant, but the whole array of isoquants, each of which shows a different level of output. It shows how output varies as the factor inputs change.

Cobb-Douglas Production Function: is the most popular in applied research, because it is easiest to handle mathematically. The Cobb-Douglas function is of the form

40

L0

X

K

Fig.2 Linear Isoquant

L0

K

X

Fig.3 Input-Output Isoquant

X

P1 P2

P3

P4

0L

K

Fig 4. Linear Programming Isoquant

A

B

L0

Fig. 5 Convex Isoquant

K

To indicate magnitude of production function we allow parameters to take arbitrary values. The parameter b measures, the scale of production: how much out put we would get if we used one unit of each input. The parameter b1 and b2 measure how the amount of output responds to the changes in the input.

1. The Marginal Product of factors

a. The

=

=

Where APL = the average product of labor

b. Similarly

2. Marginal rate of substitutionMRSL,K = X/ L = b1 (X/L) = b1 * K X/K b2 (X/K)b2 L

3.2 TECHNOLOGYInputs to production are called factors of production. Factors of production are often classified into broad categories such as land, labor, capital, and raw materials. Capital goods are those inputs to production that are themselves produced goods. Capital goods are machines of one sort or another: tractors, buildings, computers, or whatever.

Sometimes capital is used to describe the money used to start up or maintain a business. Financial capital is the term that used to describe this concept of capital goods, or physical capital, for produced factors of production.

Inputs and outputs as being measured in flow units: a certain amount of labor per week and a certain number of machine hours per week will produce a certain amount of output a week.

Technology changes as knowledge of new and more efficient methods of production becomes available. In addition, new inventions may result in the increase of the efficiency of all methods of production. At the same time some techniques may become inefficient and drop out from the production function. These changes in technology constitute technological progress.

Graphically the effect of innovation in processes is shown with an upward shift of the production function (fig. 1), or down ward movement of production isoquant (fig. 2). This shift shows that the same output may be produced by less factor inputs, or more output may be obtained with the same inputs.

410

X

X1

X

X1=f(L) K

X=f(L)

X0

X0

0 L

Fig. 1 Fig. 2

Technical progress may also change the shape of isoquant. Hicks has distinguished three types of technical progress, depending on effect on the rate of substitution of the factors of production.

Capital-deepening technical progressTechnical progress is capital-deepening or capital using, along a line on which K/L ratio is constant, the MRSL,K increases. This implies that technical progress increases the marginal product of capital by more than the marginal product of the labor. The ratio of marginal products (which is the MRSL,K) decreases in absolute value. But taking into account that the slope of the isoquant is negative, this sort of technical progress increases the MRSL,K . The slope of the shifting isoquant becomes less steep along any given radius. The capital-deepening technical progress is shown in fig.3.

Fig3. Capital-deepening technical progressLabor-deepening technical progressTechnical progress is labour deepening, if along a radius through the origin with constant K/L ratio), the MRSL,K decreases. This implies that the technical progress increases the MPL faster than the MPK. Accordingly the MRSL,K, being the ratio of the marginal products / , increases in absolute value. The downwards-shifting isoquant becomes steeper along any given radius through the origin. This is show in the figure 4

42

L

A

A1

A11

LO

K

B

B1

B11

K

Fig. 4 Labour-deepening technical progress Neutral – technical progressTechnical progress is neutral if it increases the marginal product of both factors by the same percentage, so that the MRSL,K (along any radius) remains constant. The isoquant shifts downwards parallel to itself. This is shown in the fig. 5.

Fig. 5 Neutral Technical progressProperties of TechnologyFirst, technologies are monotonic. If you increase the amount of at least one of the inputs, it should be possible to produce at least as much out put as you were producing originally. This is sometimes referred to as the property of free disposal: if the firm can dispose costlessly any of inputs, having extra inputs around can not disturb it.Second, the technology is Convex. This means that if you have two ways to produce y units of output, (x1, x2) and (z1, z2) when their weighted average will produce at least y units of output. One argument for convex technologies goes as follows. Suppose that you have a way to produce 1 unit of output using L1 units of factor (Labour) 1 and K2 units of factor (capital) 2 and that you have another way to produce 1 unit of output using L2 units of factor 1 and K2 units of factor2. These two ways are called as production techniques.

Furthermore, let us suppose that you are free to scale the output up to arbitrary amounts so that (100a1, 100b2) will produce 100 units of output. But now note that if you have (Technique A) 25L1+75L2, units of factor 1 (Labor) and (Technique B) 25K1+75K2 units of factor 2 (Capital) you can still produce 100 units of output: just produce 25 units of the output using the “a” technique and 75 units of the output using “b” technique.This is depicted in the fig.6 by choosing the level at which you operate each of the two activities, you can produce a given amount of output in a variety of different ways. In particular, every input combination along the line connecting (100b1. 100b2) will be a feasible way to produce 100 units of output.

43

X2

100a2

100b2

LO

C

C1

C11

L

K

0

(25a1+75b1, 25a2+75b2)

In this kind of technology, where you can scale the production process up and down easily and where separate production processes do not interface with each other, convexity is a very natural measure.

3.3 LAWS OF PRODUCTION

IntroductionThe laws of production describe the technically possible ways of increasing the level of production. Output may increase in various ways.Output can be increased by changing all factors of production. Clearly this is possible only in the long run. Thus the laws of returns to scale refer to the long run analysis of production.

In the short run output may be increased by using more of the variable factor(s), while capital (and possibly other factors as well) are kept constant. The marginal product of the variable factor(s) will decline eventually as more and more quantities of this factor are combined with the other constant factors. The expansion of output with one factor (at least) constant is described by the law of diminishing returns of the variable factor, which is often referred to as the law of variable proportions.

Product LinesTo analyze the expansion of output we need a third dimension, since along the two dimensional diagram we can depict only the isoquant along which the level of output is constant. Instead of introducing a third dimension it is easier to show the change of output by shifts of the isoquant and use the concept of product lines to describe the expansion of output.Product Line shows the (physical) movement from one isoquant to another as we change both factors or a single factor. A product curve is drawn independent of the prices of factors of production. It does not imply any actual choice of expansion, which is based on the prices of factors and is shown by the expansion path.The product line describes the technically possible alternative paths of expanding output. What path will actually be chosen by the line will depend on the prices of factors.The product curve passes through the origin if all factors are variable. If only one factor is variable (the other being kept constant) the product line is straight line parallel to the axis of the variable factor (Fig. 1). The K/L ratio diminishes along the product line.

44

100 a1 100 b2 X1Fig. 6 Convexity: If you can operate production activities independently, then weighted averages of production plans will also be feasible. Thus the isoquants will have a convex shape.

Among all possible product lines of particular interest are the so called isoclines. An isocline is the locus of points of different isoquants at which the marginal rate of substitution (MRS) of factors is constant. If the production function is homogeneous the isoclines are straight lines through the origin. Along any of one isocline the K/L ratio is constant (as the MRS of the factors). Of course the K/L ratio (and the MRS is different technologies (Fig.2).

3.3.1 LAWS OF RETURNS TO SCALE: Long-run Analysis of Production.The long run expansion of output may be achieved by varying all factors. In the long run all factors are variable. The laws of returns to scale refer to the effects of scale relationships.

In the long run output may be increased by changing all factors by the same proportion, or by different proportions. Traditional theory production concentrates on the first case that is the study of output as all inputs change by the same proportion. The term returns to scale refers to the changes in output as all factors change by the same proportion.Suppose we start from an initial level of inputs and output.

and we increase all the factors by the same proportion k, we will clearly obtain a new level of output X*, higher than the original level X0,

If X* increases by the same proportion k as the inputs, we say that there are constant returns to scale.If X* increases less than proportionally with the increase in the factors we have decreasing returns to scale.If X* increases more than proportionally with the increase in the factors, we have increasing returns to scale.

Returns to scale and Homogeneity of the Production Function

Suppose we increase both factors of the functionX0 = f(L,K)

by the same proportion k, and we observe the resulting new level of output X.X* = f(kL, kK)

If k can be factored out (that is may be taken out of the brackets as a common factor), then the new level of output X* can be expressed as a function of k and the initial level of output

45

Fig.1 Product line for K given L

N

L

K

Product Lines

X* = kvf(L,K) or X* = kv.X0

And the production function is called homogeneous. If k cannot be factored out, the production is non-homogeneous. Thus a homogeneous function is a function such that if each of the inputs is multiplied by k, then k can be completely factored out of the function. The power v of k is called the degree of homogeneity of the function and is a measure of the returns to scale.

If v = 1 we have constant returns to scale. This production function some times called linear homogeneous.

If v < 1 we have decreasing returns to scale.

If v > 1 we have increasing returns to scale.

Graphical Representation of the Returns to scale for a homogeneous Production FunctionThe returns to scale may be shown graphically by the distance (on an isocline) between successive multiple level of output isoquants, that is isoquants that show levels of output which are multiples of some base level of output, e.g., X, 2X, 3X,.. etc.

Constant Returns to Scale: Along any isocline the distance between successive multiple isoquants is constant. Doubling the factor inputs achieve double the level of the initial output; tripling inputs achieves triple output, and so on (fig. 2)

Decreasing Returns to Scale: The distance between consecutive multiple isoquants increases. By doubling the inputs, output increases by less than twice its original level in fig. 3 the point a defined by 2K and 2L, lies on an isoquant below the one showing 2x. Increasing returns to scale. The distance between consecutive multiple-isoquants decreases. By doubling the inputs, output is more than doubled. In fig. 4 doubling K and L leads to point b which lies on an isoquant above the one denoting 2x.

46

3X

2X

x

0 L 2L 3L L Fig.2 Constant Returns to Scale: Oa = Ob = Oc

K

3K

2K

K

Returns to scale are usually assumed to be the same everywhere on the production surface, which is the same along all the expansion product line. All processes are assumed to show the same returns over all ranges of output: constant returns everywhere, decreasing returns everywhere, or increasing returns everywhere. However, the technological conditions of production may be such that returns to scale may vary over different ranges of output. Over some range we may have constant returns to scale may vary over different ranges output.

With a non-homogeneous production function returns to scale may be increasing, constant, or decreasing, but their measurement and graphical presentation is not as straight forward as in the case of the homogeneous production function. The isoclines will be curves over the production surface and along each one of them the K/L ratio varies. Homogeneity is assumed to simplify the statistical work. Homogeneity however is a special assumption. In some cases a very restrictive one. When the technology shows increasing or decreasing returns to scale it may or may not imply a homogeneous production function.

3.3.2 THE LAW OF VARIABLE PROPORTIONS: Short-run AnalysisIf one of the factors of the production (usually capital K) is kept constant. The marginal product of the variable factor (Labour L) will diminish after a certain range of production. The traditional theory of production concentrates on the ranges of output over which the marginal products of the factors are positive but diminishing. The ranges of increasing returns (to a factor) and the range of negative productivity are not equilibrium ranges of output. If production function is homogeneous with constant or decreasing returns to scale everywhere on the production surface, the productivity of the variable factor will necessarily be diminishing. If, however, the production function exhibits increasing returns to scale, the diminishing returns arising from the decreasing marginal product of the variable factor (labor) may be offset, if the returns to scale are considerable.

47

3X

2X

x

0 L 2L L Fig.3 Decreasing Returns to Scale: Oa < Ob < Oc

K

2K

K

3X

2X

x

0 L 2L L Fig.4 Increasing Returns to Scale: Oa > Ob>< Oc

K

2K

K

This however, is rare. In general the productivity of single-variable factor is diminishing.Let us examine the law of variable proportions or the law of diminishing productivity (returns) in some detail.

b

c

If the production function is homogeneous with constant returns to scale everywhere, the returns to a single-variable factor will be diminishing. This is implied by the negative slope and the convexity of the isoquants. With constant returns to scale everywhere on the production surface, doubling both factors (2K, 2L) leads to a doubling of output. In fig.5 point b on the isocline OA lays on the isoquant 2x. However, if we keep K constant (at the level ) and we double only the amount of L, we reach point c, which clearly lies on a lower isoquant than 2X. If we wanted to double output with the initial capital , we would require L* units of labour. Clearly L* > 2L. Hence doubling L, with K constant, less than doubles output. The variable factor L exhibits diminishing productivity (diminishing returns). If the production function is homogeneous with decreasing returns to scale, the returns to a single – variable factor will be, a fortiori, diminishing. Since returns to scale are decreasing, doubling both factors will less than double output. In fig.6 we see that with 2L and 2K output reaches the level d which is on a lower isoquant than 2x. If we double only labor while keeping capital constant, output reaches the level c, which lies on a still lower isoquant.

b d

a c

b

a c a b c2X

48

0 L 2L L* L Fig. 5

Product Line

K

2K

A

2x

Product Line

0 L 2L L Fig.6

2X

K

2K

X

0 L 2L L Fig. 7

K

0 L 2L L Fig. 8

K

X

If the production function shows increasing returns to scale, the returns to the single variable factor L will in general be diminishing (Fig.7), unless the positive returns to scale are so strong as to offset the diminishing marginal productivity of the single variable factor. Fig. 8 shows the rare case strong returns to scale which offset the diminishing productivity of L.

3.4 CHOICE OF OPTIMAL COMBINATION OF FACTORS OF PRODUCTIONIn this section we shall show the use of the production function in the choice of the optimal combination of factors by the firm. In part A we will examine two cases in which the firm is faced with a single decision, namely maximizing output for a given cost and minimizing cost subject to a given output. Both these decisions comprise cases of constrained profit maximization in a single period.In above cases it is assumed that the firm can choose the optimal combination of factors, that it can employ any amount of any factor in order to maximize its profits. This assumption is valid if the firm is new, or if the firm is in the long-run. However, an existing firm may be pressurized, due to pressure of demand, to expand its output in the short-run, when at least one factor, usually capital, is constant. Assumptions:1. The goal of the firm is profit maximization – that is, the maximization of the

difference CR where П = profits, R = revenue, and C = cost.2. The price of output is given, Px.

3. The prices of factors are given:

w is the given wage rate,

r is the given price of

capital services (rental price of machinery).

A. Single Decision of the FirmThe problem facing the firm is that of constrained profit maximization, which may take one of the following forms:

a. Maximize profit П, subject to a cost constraint. In this case total cost and prices

are given (C,

w ,

r , Px), and the problem may be stated as follows.

max CR CXPx Clearly maximization of П is achieved in this case if cost C is minimized, given that C and Px are given constants by the assumption.

b. Maximize profit П, for a given level of output. For example, a contractor wants to build a bridge (X is given) with the maximum profit. In this we have

max CR CXP x

Clearly maximization of П is achieved in this case is cost C is minimized, given that X and Px are given constants by assumption.

For a graphical presentation of the equilibrium of the firm (its profit maximizing position) we will use the isoquant map (fig. 1) and the isocost-line(s) fig.2.

The slope of an isoquant is: -KX

LX

MP

MPMRS

L

K

K

LKL

/

/,

The isocost line is defined by the cost equation ))(())(( LwKrC , where w = wage rate, and r = price of capital services

49

The isocost line is the locus of all combinations of factors the firm can purchase with a given monetary cost outlay.The slope of the isocost line is equal to the ratio of the price of the factors of production. Slope of isocost line = w/r

Case 1: Maximization of output subject to a cost constraint (Financial constraint)Assume (a) A given production function X = f(L,K,v,γ)and (b) given factor prices w, r, for labor and capital respectively.The firm is in equilibrium when it maximizes its output given its total cost outlay and the prices of the factors, w and r.In fig.3 the maximum level of output the firm can produce, given the cost constraint, is X2 defined by the tangency of the isocost line, and the highest isoquant. The optimal combination factors of production is K2 and L2, for prices w and r. Higher levels of output (to the right of e) are desirable but not attainable due to the cost constraint.

Other points on AB0, below it lie on a lower isoquant than X2. Hence X2 is the maximum output possible under the above assumptions (of given cost outlay, given production function, and given factor prices). At the point of tangency (e) the slope of the isocost line (w/r) is equal to the slope of the isoquant (MPL/MPK). This constitutes the first condition for equilibrium. The second condition is that the isoquants be convex to the origin. In summary: the conditions for equilibrium of the firm are:

(a) Slope of isoquant = Slope of isocost, KLK

L MRSKX

LX

MP

MP

r

w,/

/

50

Fig. 2

A

B LL

K K

Fig. 1

r

C

0 L2 B LFig. 3

K

A

K2

X3

X2

X

e

Fig. 2

A

B LL

K K

Fig. 1

r

C

(b) The isoquants must be convex to the origin; if the isoquant is concave the point of tangency of the isocost curves does not define an equilibrium position Fig.3Output X2 (depicted by the concave isoquant) can be produced with lower cost at e2 which lies on a lower isocost curve than e. (with concave isoquant we have a cornel solution).

Formal derivation of the equilibrium conditions: The equilibrium conditions may be obtained by applying calculus and solving a constraint maximum problem which may be stated as follows. The rational entrepreneur seeks the maximization of his output, given his total-cost outlay and the prices of factors, formally:Maximize X = Subject to C = wL + rK (cost constraint)This is a problem of constrained maximum and the above conditions for the equilibrium of firm may be obtained from its solution.We can solve this problem by using Lagrangian multipliers. The solution involves the following steps: Rewrite the constraint in the form Multiply the constraint by a constant λ which is the Lagrangian multiplier:The Lagrangian multipliers are undefined constraints which are used for solving constraint maxima or minima. Their value is determined simultaneously with the values of the other unknown (L and K in our example). There will be as many Lagrangian multipliers as there are constraints in the problem. From the composite function )( rKwLCX It can be shown that maximization of the function implies maximization of the output.

The first condition for the maximization of a function is that its partial derivatives be equal to zero. The partial derivatives of the above function with respect to L, K, and λ are:

(1)

0)(

rK

X

k

(2)

(3)

Solving the first two equations for λ we obtain

or w

MP

w

LX L

/

or

51

0 e2 L Fig. 4

K

e1

Isoquant X2

e

The two equations must be equal then

or

This firm is in equilibrium when it equates the ratio of the marginal productivities of factors to the ratio of their prices.

It can be shown that the second-order conditions for equilibrium of the firm require that the marginal product curves of the two factors have a negative slope.The slope of the marginal product curve of labor is the second derivative of the production function:

Slope of MPL curve = , similarly, for capital Slope of MLK curve =

The second order conditions are

& and

These conditions are sufficient for establishing the convexity of the isoquants.

Case 2: Minimization of cost for a given level of outputThe conditions for equilibrium of the firm are formally the same as in case 1. That is there must be tangency of the (given) isoquant and the lowest possible isocost curve, and the isoquant must be convex. However, the problem is conceptually different in the case of cost minimization. The entrepreneur wants to produce a given output for example, a bridge, a building, or X tons of a commodity with the minimum cost outlay.

In this case we have a single isoquant (fig.5) which denotes the desired level of output, but we have a set of isocost curves (fig. 6). Curves closer to the origin show a lower total-cost outlay. The isocost lines are parallel because they are drawn on the assumption of constant prices of factors: since w and r do not change, all the isocost curves have the same slope w/r.

The firm minimizes its costs by employing the combination of K and L determined by

the point of tangency of the _

X isoquant with the lowest isocost line (fig.7). Points

below e are desirable because they show lower cost but are not attainable for output_

X . Points above e show higher costs. Hence point e is the least – cost point, the

point denoting the least-cost combination of the factors K and L for producing X.

52

0 L 0 L 0 L L Fig. 5 Fig.6 Fig.7

K

_

X

K

K

K

Clearly the conditions for equilibrium (least cost) are the same as in case 1, that is equality of the slopes of the isoquant and the isocost curves, and convexity of the isoquant.

Formally:Minimize:Subject to:Steps we have to follow are: Rewrite the constraint in the form: Multiply the constraint by the Lagrangian multiplier λ. From the composite function or Take the partial derivatives of with respect to L, K and λ and equal to zero.

,

from the first two equations we obtain: ,

Dividing through these expressions we find:

This condition is the same as in case 1 above. The second sufficient conditions, concerning the convexity of the isoquants, is fulfilled by the assumption of negative

slopes of the marginal product of factors as in case 1, that is , and

CHOICE OF OPTIMAL EXPANSION PATH

We distinguish two cases: expansion of output with all factors variable (the long run), and expansion of output with some factor(s) constant (short run).Optimal expansion path in the long runIn the long run all factors of production are variable. There is no limitation (technical or financial) to the expansion of output. The firm’s objective is the choice of the optimal way of expanding its output, so as to maximize profits. With given factor prices (w, r) and given production function, the optimal expansion path is determined by the points of tangency of successive isocost lines and successive isoquants.

If he production function is homogeneous the expansion path will be straight line through the origin whose slope (which determines the optimal K/L ratio) depends on the ratio of the factor prices.

53

In fig. 8 the optimal expansion will be OA defined by the locus of points of tangency of the isoquants with successive parallel isocost lines with a slope of w/r. If prices increases the isocost lines become flatter (for example, with slope of w’/r’), and the optimal expansion path will be the straight line OB. Of course, if the ratio of prices of factors was initially w/r and subsequently changes to w’/r’, the expansion path changes: initially the firm moves along OA, but after the change in the factor prices it moves along OB.

If the production function is non-homogeneous the optimal expansion path will not be a straight line, even if the ratio of prices of factors remains constant. This is shown w/r ratio with MRSL,K, which the same on a curved isocline.Optimal Expansion path in the short run: In the short run, capital is constant and the firm is coerced to expand along a straight line parallel to the axis on which we measure the variable factor L. With prices of factors constant the firm does not maximize its profits in the short run, due to the constraint of the given capital. This situation is shown in fi.10. The optimal expansion path would be OA were it possible to increase K. Given the capital equipment, the firm can expand only along KK in the short run.

CHAPTER FOURTHEORY OF COST

IntroductionCost functions are derived functions. They are derived from the production function, which describes the available efficient methods of production at any one time.

54

0 L 0 L Fig. 8 Fig. 9

K K A

B

w/r, w’/r’

_

K_

K

A

O L

K

Fig. 10

Economic theory distinguishes between short-run and long-run costs. Short-run costs are the costs over a period during which some factors of production (usually capital equipment and management) are fixed. The long-run costs are those costs over a period long enough to permit the change of all factors of production. In the long-run all factors become variable.

Both in the short-run and in the long-run, total cost is a multivariable function, that is, total cost is determined by many factors. Symbolically long-run cost function can be written as C = f (X,T,Pf)and the short-run cost function as C = f (X,T,Pf,K)Where C = Total Cost, X = Output, T = Technology, P f = Prices of factors, K = Fixed factor(s).

Graphically, costs are shown on two-dimensional diagrams. Such curves imply that cost is a function of output, C=f(X), ceteris paribus. The clause ceteris paribus implies that all other factors which determine costs are constant. If these factors do change, their effect on costs is shown graphically by a shift of the cost curve. This is the reason why determinants of cost. Other than output, are called shift factors. Mathematically there is no difference between the various determinants of costs. The distinction between movements along the cost curve (when output changes) and shifts of the curve (when the other determinants change) is convenient only pedagogically, because it allows the use of two-dimensional diagrams. But it can be misleading when studying the determinants of costs. It is important to remember that if the cost curve shifts, this does not imply that the cost function is indeterminate.

The factor technology is itself a multidimensional factor, determined by the physical quantities of factor inputs, the quality of the factor inputs, the efficiency of the entrepreneur, both in organizing the physical side of the production (technical efficiency of the entrepreneur), and in making the correct economic choice of techniques (economic efficiency of the entrepreneur). Thus, any change in these determinants (e.g., the introduction of a better method of organization of production, the application of an educational program to the existing labor) will shift the production function, and hence will result in a shift of the cost curve. Similarly the improvement of raw materials or the improvement in the use of the same raw materials will lead to a shift downwards of the cost function.

The short-run costs are the costs at which the firm operates in any one period. The long-run costs are planning costs or ex ante (based on prior assumptions or expectations) costs, in that they present the optimal possibilities for expansion of the output and thus help the entrepreneur is in a long-run situation, in the sense that he can choose any one of a wide range of alternative investments, defined by the state of technology. After the investment decision is taken and funds are tied up in fixed-capital equipment, the entrepreneur operates under short-run conditions; he is on a short cost curve.

A distinction is necessary between internal (to the firm) economies of scale and external economies. The internal economies are build into the shape of the long-run cost curve, because they accrue to the firm from its own action as it expands the level of its output. The external economies arise outside the firm, from improvement (or depreciation) of the environment in which the firm operates. Such economies external to the firm may be realized from actions of other firms in the same or in

55

another industry. The important characteristic of such economies is that they are independent of the actions of the firm, they are external to it. Their effect is a change in the prices of the factors employed by the firm (or in a reduction in the amount of inputs per unit of output), and thus cause a shift of the cost curves, both the short-run and the long-run.

In summary: while the internal economies of scale relate only to the long-run and are built into the shape of the long-run cost curve, the external economies affect the position of the cost curves; both the short-run and the long-run cost curves will shift if external economies affect the prices of the factors and/or the production function.

Any point on a cost curve shows the minimum cost at which a certain level of output may be produced. This is the optimality implied by the points of accost curve. Usually the above optimality is associated with the long-run cost curve. However, a similar concept may be applied to the short-run, given the plant of the firm in any one period.

Short-run costs The Traditional Theory

Traditional Theory distinguishes between the short run and the long run. The short run is the period during which some factor(s) is fixed; usually capital equipment and entrepreneurship are considered as fixed in the short run. The long run is the period over which all factors become variable. In the traditional theory of the firm total costs are split into two groups: total fixed costs and total variable costs:

TC = TFC + TVCThe fixed costs include:

(a)Salaries of administrative staff(b)Depreciation (wear and tear) of machinery(c) Expenses for building depreciation and repairs(d)Expenses for land maintenance and depreciation

Another element that may be treated in the same way as fixed costs is the normal profit, which is a lump sum, including a percentage return on fixed capital and allowance for risk.The variable costs include:

(a)The raw materials(b)The cost of direct labor(c) The running expenses of fixed capital, such as fuel ordinary repairs and routine

maintenance.

Total fixed cost is graphically denoted by a straight line parallel to the output axis (fig.1). The total variable cost is the traditional theory of the firm has broadly an inverse – S shape (fig. 2) which reflects the law of variable proportions. According to this law, at the initial stages of production with a given plant, as more of the variable factor(s) is employed, its productivity increases and the average variable factor(s) employed, its productivity increases and the average variable cost falls.

56O X O X O X Fig. 1 Fig. 2 Fig. 3

C

TFC

TVCTC

TVC

TFC

This continues until optimal combination of the fixed and variable factors is reached. Beyond this point as increased quantities of the variable factor(s) are combined with the fixed factor(s) the productivity of the variable factor(s) declines (and the AVC rises). By adding the TFC and TVC we obtain total cost curves. The average fixed cost is found by dividing TFC by the level of output: AFC = TFC XGraphically the AFC is a rectangular hyperbola, showing at all its points the same magnitude, that is, the level of TFC. The average variable cost is similarly obtained by dividing the TVC with the corresponding level of output: AVC = TVC

X

d

c a da b b

c

Graphically the AVC at each level of output is derived from the slope of a line drawn from the origin to the point on the TVC curve corresponding to the particular level of output. For example fig.5 the AVC at X1 is the slope of the ray 0a, the AVC at X2 is the slope of a ray 0b and so on. The slope of the ray through the origin declines continuously until the ray becomes tangent to the TVC curve at c. SAVC falls initially as the productivity of the variable factor(s) increases, reaches a minimum when the plant is operated optimally (with the optimal combination of fixed and variable factors), and rises beyond that point fig.6. The ATC is obtained by dividing the TC by the corresponding level of output:

Graphically the ATC curve is derived in the same way as the SVAC. The ATC at any level of output is the slope of the straight line from the origin to the point on the TC curve corresponding to that particular level of output (fig.7). The shape of the ATC reaches a minimum at the level of optimal operation of the plant (XM) and subsequently rises again (fig.8). The U shape of both the AVC reflects the law of variable proportions or law of eventually decreasing returns to the variable factor(s) of production. The marginal cost is defined as the change in TC which results from a unit change in output. Mathematically the marginal cost is the first derivative of the TC

function. Denoting total cost by C and output by X we have .

57

0 x1 x2 x3 x4 X Fig. 5

C

O x1 x2 x3 x4 X Fig. 6

CTVC

SAVC

TC

O x1 x2 xM xL X Fig. 8

SATC

Graphically the MC is the slope of the TC curve (which of course is the same at point as the slope of the TVC). The slope of a curve at any one of its points is the slope of the tangent at that point. With an inverse S shape of the TC (and TVC) the MC curve will be U-shaped. In fig.9, we observe that the slope of the tangent to the total-cost curve declines gradually, until it becomes parallel to the X-axis (with its slope being equal to zero at this point), and then starts rising. Accordingly we picture the MC curve in fig.10 as U shaped.

In summary: the traditional theory of costs postulates that in the short run the cost curves (AVC, ATC and MC) are U shaped, reflecting the law of variable proportions. In the short run with a fixed plant there is a phase of increasing productivity (falling unit costs) and a phase of decreasing productivity (increasing unit costs) of the variable factor(s). Between these two phases of plant operation there is a single point at which unit costs are at a minimum. When this point on the SATC is reached the plant is utilized optimally, that is with the optimal combination (proportions) of fixed and variable factors.The relationship between ATC and AVCThe AVC is a part of the ATC, given ATC=AFC+AVC. Both AVC and ATC are U-shaped, reflecting the law of variable proportions. However, the minimum point of the ATC occurs to the right of the minimum point of the AVC (fig.11). This is due to the fact that ATC includes AFC, and the latter falls continuously with increase in output. After the AVC has reached its lowest point and starts rising, its rise is over a certain range offset by the fall in the AFC, so that the ATC continues to fall despite the increase in AVC. However, the rise in AVC eventually becomes greater than the fall in the AFC so that the ATC starts increasing. The AVC approaches the ATC asymptotically as X increases.

58

O x1 x2 xM xL X Fig.7

TC SMC

0 XA X Fig.9

C

O XA X Fig.10

C

SMC SATC

SAVC

AFC

O X1 X2 X Fig.11

a

In fig.11 the minimum AVC is reached at X1 at while the ATC is at its minimum at X2. Between X1 and X2 the fall in AFC more than offsets the rise in AVC so that the ATC continues to fall. Beyond X2 the increase in AVC is not offset by the fall in AFC, so that ATC rises.The relationship between MC and ATC: The MC cuts the ATC and the AVC at their lowest points. We will establish this relation only for the ATC and MC, but the relation between MC, but the relation between MC and AVC can be established on the same lines of reasoning. The MC is the change in the TC for producing an extra unit of output. Assume that we start from a level of n units of output. If increase the output by one unit the MC is the change in total cost resulting from the production of the (n+1)th unit.

The AC at each level of output is found by dividing TC by X. Thus the AC at the level

of Xn is

and the AC at the level Xn+1 is

Clearly,

Thus: 1. If the MC of the (n+1)th unit is less than Can (the AC of the previous n units) the ACn+1 will be smaller than the ACn. 2. If the MC of the (n+1)th unit is higher than ACn (the AC of the previous n units) the ACn+1 will be higher than the ACn.

So long as the MC lies below the AC curves, it pulls the latter downwards when the MC rises above the AC, it pulls the latter upwards. In fig.11 to the left of a the MC lies below the AC curve, and hence the latter falls downwards. To the right of a the MC curve lie above the AC curve, so that AC rises. It follows that at point a, where the intersection of the MC and AC occurs, the Ac has reached its minimum level.

The Modern Theory of Short-run CostsThe U-shaped cost curves of the traditional theory have been questioned by various writers both on theoretical a priori and on empirical grounds. As early as 1939 George Stigler suggested that the short-run average variable cost has a flat stretch over a range of output which reflects the fact that firms build plans with some flexibility in their productive capacity. The reasons for this reserve capacity have been discussed in detail by various economists. As in the traditional theory, short-run costs are distinguished into average variable costs (AVC) and average fixed costs (AFC).

The Average Fixed CostThis is the cost of indirect factors that is the cost of the physical and personal organization of the firm. The fixed costs include the costs for:

59

(a) the salaries and other expenses of administrative staff(b) the expenses for maintenance of buildings,(c) the wear and tear of machinery (standard depreciation allowances),(d) the expenses for maintenance of buildings,(e) The expenses for the maintenance of land on which the plant is installed and

operates..The planning of the plant (or the firm) consists in deciding the size of these fixed, indirect factors, which determine the size of the plant, because they set limits to its production. Direct factors such as labor and raw materials are assumed not to set limits on size; the firm can acquire them easily with a figure for the level of output which entrepreneur anticipates selling, and he will choose the size of plant which will allow him to produce this level of output more efficiently and with the maximum flexibility. The plant will wants to have some reserve capacity for various reasons.

The businessman will want to be able to meet sensational and cyclical fluctuations in his demand. Such fluctuations cannot always be met efficiently by a stock-inventory policy. Reserve capacity will allow the entrepreneur to work with more shifts and with lower costs than a stock – piling policy.

Reserve capacity will give the businessman greater flexibility for repairs for broken down machinery without disrupting the smooth flow of the production process.

The entrepreneur will want to have more freedom to increase his output if demand increases. All businessmen hope for growth. In view of anticipated increases in demand the entrepreneur builds some reserve capacity, because he would not like to let all new demand go to his rivals, as this may endanger his future old on the market. It also gives him some flexibility for minor alternations of his product, in view of changing tastes of customers.

Technology usually makes it necessary to build into the plant some reserve capacity. Some basic types of machinery may not be technically fully employed when combined with other small types of machines in certain numbers, more of which may not be required given the specific size of the chosen plant. Also such basic machinery may be difficult to install due to time – lags in the acquisition. The entrepreneurs will thus buy from the beginning such a ‘basic’ machine which allows the highest flexibility, in view of future growth in demand, even though this is a more expensive alternative now. Furthermore some machinery may be so specialized as to be available only to order, which takes time. In this case such machinery will be bought in excess of the minimum required at present numbers, as a reserve.

60

O XA XB XFig.15

A B

a

b

Some reserve capacity will always be allowed in the land and buildings, since expansion of operations may be seriously limited if new land or new buildings have to be acquired.

Finally, there will be some reserve capacity on the organizational and administrative level. The administrative staff will be hired at such numbers as to allow some increase in the operations of the firm.In summary: the businessman will not necessarily choose the plant which will give him today the lowest cost, but rather that equipment which will allow him the greatest possible flexibility, for minor alternations of his product or his technique.

Under these conditions the AFC curve will be as in fig.15. The firm has some largest-capacity units of machinery which set an absolute limit to the short-run expansion of output (boundary b in fig. 15). The firm has also small-unit machinery, which sets a limit to expansion (boundary A in fig. 15). this is not an absolute boundary, because the firm can increase its output in the short run(until the absolute limit B is encountered), either by paying overtime to direct labor for working longer hours (in this case AFC shown by the dotted line in fig.15), or by buying some additional small-unit types of machinery (in this case the AFC curve shifts upwards, and starts falling again as shown by the line ab in fig.15).

The Average variable costAs in the traditional theory, the average variable cost of modern microeconomics includes the cost of:

(a)Direct labor which varies with output,(b)Raw materials,(c) Running expenses of machinery.

The SAVC in modern theory has a saucer-type shape that is broadly shaped but has a flat stretch over a range of output (fig.16). The flat stretch corresponds to the built-in the plant reserve capacity. Over this stretch the SAVC is equal to the MC, both being constant per unit of output. To the left of the flat stretch the MC lies below the SAVC, while to the right of the flat stretch the MC rises above the SAVC. The falling part of the SAVC shows the reduction in costs due to the better utilization of the fixed factor and the consequent increase in skills and productivity of the variable factor (labor) with better skills the wastes in raw materials are also being reduced and a better utilization of the whole plant is reached.The increasing part of the SAVC reflects reduction in labor productivity due to the longer hours of work, the increasing cost of labor due to overtime payment (which is higher than the current wage), the wastes in materials and the more frequent break down of machinery as the firm operates with overtime or with more shifts.

61

O X Fig. 16

MC

SAVC

MC

C

The innovation of modern microeconomics in this field is the theoretical establishment of a short-run SAVC curve with a flat stretch over a certain range of output (fig.18). It should be clear that this reserve capacity is planned in order to give the maximum flexibility in the operation of the firm. It is completely different from the excess capacity which arises with the U-shaped costs of the traditional theory for the firm. The traditional theory assumes that each plant is designed without any flexibility: it is designed to produce optimally only a single level of output (XM in fig. 17). If the firm produces an output X smaller than XM there is excess (unplanned) capacity, equal to the difference XM – X. This excess capacity is obviously undesirable because it leads to higher unit costs.

In the modern theory of costs the range of output X1, X2 in fig. 18 reflects the planned reserve capacity which does not lead to increases in costs. The firm anticipates using its plant sometimes closer to X1 and at others closer to X2. On the average the entrepreneur expects to operate his plant within the X1X2 range. Usually firms consider that the ‘normal’ level of utilization of their plant is somewhere between two-thirds and three-quarters of their capacity, that is, at a point closer to X2 than X1. The level of utilization of the plant which firms consider as ‘normal’ is called ‘the load factor’ of the plant.The Average Total CostThe average total cost is obtained by adding the average fixed (inclusive of normal profit) and the average variable costs at each level of output. The ATC is shown in fig.19. The ATC curves falls continuously up to the level of output (X2) at which the reserve capacity is exhausted. Beyond that level ATC will start rising. The MC will intersect the average total-cost curve at its minimum point (which occurs to the right of the level of output XA at which the flat stretch of the AVC ends).Mathematically the cost-output relation may be written in the form C = b0 + b1XTC = TFC + TVC The TC is a straight line with a positive slope over the range of reserve capacity (fig. 20).

62

O X XM X Fig. 17 O X1 X2 X Fig. 18

Excess capacity

SAVC

C

Reserve Capacity

SAVC C

C

MC SATC

SAVC

AFCMC

The AFC is a rectangular hyperbola: AFC = bo

X

The AVC is a straight line parallel to the output axis:

The ATC is falling over the range of reserve capacity:

The MC is a straight line which coincides with the AVC: MC=

Thus the range of reserve capacity we have MC=AVC=b1, while ATC falls continuously over this range (fig. 21). Note that the above total cost function does not extend to the increasing part of costs, that is it does not apply to ranges of output beyond the reserve capacity of the firm.

Long-run costsLong-run Costs of the Traditional Theory: The Envelope CurveIn the long – run all factors are assumed to become variable. It is known that the long-run cost curve is a planning curve, in the sense that it is a guide to the entrepreneur in his decision to plan the future expansion of his output. The long run average cost curve is derived from short-run cost curves. Each point on the LAC corresponds to a point on a short-run cost curve, which is tangent to the LAC at that point. Let us examine in detail how the LAC is derived from the SAC curves.

Assume, as a first approximation, that the available technology to the first at a particular point of time includes three methods of production, each with a different plant size: a small plant, medium plant and large plant. The small plant operates with costs denoted by the curve SAC1, the medium size plant operates with the costs on SAC2 and the large size plant gives rise to the costs shown on SAC3 (fig.1).

63

O Fig. 19 XA X

C

range of reserve capacity Fig. 21

TFC

TVC

TC

O Fig. 20 X O

X X

SACSAVC=MC

AFC

C

C1 C|

1

C| 2

C2 C3

SATC1

SATC2

SATC3

If the firm plans to produce output X1 it will choose the small plant. If it plants to produce X2 it will choose medium size plant. If it wishes to produce X3 it will choose the large size plant. If the firm starts with the small plant and its demand gradually increases, it will produce at lower costs (up to level x1). Beyond that point costs start increasing. If its demand reaches the level X”1 the firm can either continue to produce with the small plant or it can install the medium size plant. The decision at this point depends on not on costs but on the firm’s expectations about its future demand. If the firm expects that the demand will expand further thatnX”1 it will install the medium plant because with this plant outputs larger than X”1 are produced with a lower cost. Similar considerations hold for the decision of the firm when it reaches the level X”2. If it expects its demand to stay constant at this level, the firm will not install the large plant, given that it involves a larger investment which is profitable only if demand expands beyondX”2. For example, the level of output X3 is produced at a cost C3 with the large plant, while it costs C’3 if produced with medium size plant (C’2 >C3).Now if we relax the assumption of the existence of only three plants and assume that the available technology includes many plant sizes, each suitable for a certain level of output, the points of intersection of consecutive plants are more numerous. In the limit, if we assume that there are a very large number of plants, we obtain a continuous curve, which is the planning LAC curve of the firm. Each point of this curve shows the minimum (optimal) cost for producing the corresponding level of output. The LAC curve is the locus of points denoting the least cost of producing the corresponding output. It is a planning curve because on the basis of this curve the firm decides what plant to set up in order to produce optimally the expected level of output. The firm chooses the short-run plant which allows it to produce the anticipated output at the least possible cost. In the traditional theory of the firm the LAC curve is U-shaped and it is often called the ‘envelope curve’ because it envelopes the SRC curves Fig. 2 Let us examine the U shape of the LAC. This shape reflects the laws of returns to scale. According to these laws the unit costs of production decreases as plant size increases, due to the economies of scale which the larger plant size make possible. The traditional theory of the firm assumes that economies of scale exist only up to a certain size of plant, which is known as the optimum plant size, because with this plant size all possible economies of scale are fully exploited. If the plant increases further than this optimum size there are diseconomies of scale, arising from managerial inefficiencies. It is argued that management becomes highly complex, managers are overworked and the decision making process becomes less efficient. The turning-up of the LAC curve is due to managerial diseconomies of scale, since the technical diseconomies can be avoided by duplicating the optimum technical plant size.

64

0 X1 X|1 X||

1 X2 X|2 X3 X Fig. 1

CLAC

A serious assumption of the traditional U-shaped cost curves is that each plant size is designed to produce optimally a single level of output (e.g. 1000 units of X). Any departure from that X no matter how small (e.g., an increase by 1 unit of X) leads to increased costs. The plant is completely inflexible. There is no reserve capacity, not even to meet seasonal variations in demand. As a consequence of this assumption the LAC curve ‘envelopes’ the SRAC. Each point of the LAC is a point of tangency with the corresponding SRAC curve. The point of tangency occurs to the falling part of the SRAC curves for points lying to the left of the minimum point of the LAC: since the slope of LAC is negative up to M the slope of the SRAC curves must also be negative, since at the point of their tangency the two curves have the same slope. The point of tangency for outputs larger than XM occurs to the rising part of the SRAC curves: since the LAC rises, the SAC must rise at the point of their tangency with the LAC. Only at the minimum point M of the LAC is the corresponding SAC also at a minimum. Thus at the falling pat of the LAC the plants are not worked to full capacity to the rising part of the LAC the plants are overworked; only at the minimum point A is the (short-run) plant optimally employed.

We stress once more the optimality implied by the LAC planning curve: each point represents the least unit-cost for producing the corresponding level of output. Any point above the LAC is inefficient in that it shows a higher cost for producing the corresponding level of output. Any point below the LAC is economically desirable because it implies a lower unit-cost, but it is not attainable in the current state of technology and with the prevailing market prices of factors of production.

The long-run marginal cost is derived from the SRMC curves, but does not envelope them. The LRMC is formed from the points of intersection of the SRMC curve with vertical lines (to the x-axis) drawn from the points of tangency of the corresponding SAC curves and the LRA cost curve (fig. 3). The LMC must be equal to the SMC for the output at which the corresponding SAC is tangent to LAC. For levels of X to the left of tangency a the SAC>LAC . At the point of tangency SAC=LAC. As we move from a position of inequality of SRAC and LRAC to a position of equality. Hence the change in total cost (i.e. the MC) must be smaller for the short-run curve than for the long-run curve. Thus, LMC> SMC to the left of a. For an increase in output beyond X1 (e.g.X||

1), the SAC>SMC. That is we move from the position a of equality of the two costs to the position b where SAC is greater than LAC. Hence the addition to total cost (= MC) must be larger for the short run curve than for the long run curve. Thus, LMC<SMC to the right of a.Since to the left of a, LMC>SMC, and to the right of a, LMC<SMC, it follows that a, LMC = SMC. If we draw a vertical line from a to the X – axis the point at which it intersects the SMC (point A for SAC1) is a point of the LMC.

If we repeat this procedure for all points of tangency of SRAC and LAC curves to the left of the minimum point of the LAC, we obtain points of the section of the LMC which

65

XO Fig. 2 XM

M

lies below the LAC. At the minimum point M the LMC intersects the LAC. To the right of M the LMC lies above the LAC curve. At point M we have SACM = SMCM = LAC = LMCThere are various mathematical forms which give rise to U-shaped unit cost curves. The simplest total cost function which would incorporate the law of variable proportions is the cubic polynomial

C = b0 + b1X – b1X2 + b3X3

TC = TFC + TVC

The AVC is: AVC =

The MC is: MC =

The ATC is:

The TC curve is roughly S-shaped, while the ATC, the AVC and the MC are all U-shaped; the MC curve intersects the other two curves at their minimum points.

Long run costs in modern micro economic theory: The L – Shaped scale Curve.These are distinguished into production costs and managerial costs. All costs are variable in the long run and they give rise to a long-run cost curve which is roughly L-shaped. The production costs fall continuously with increases in output. At very large scales of output managerial costs may rise. But the fall in production costs more than offsets the increase in the managerial costs, so that the total LAC falls with increases in scale.

Production costsProduction costs fall steeply to begin with and then gradually as the scale of production increases. The L-shape of the production cost curve is explained by the technical economies of large-scale production. Initially these economies are substantial, but after a certain level of output is reached all or most of these

66

LAC

LMC

0 X|1 X1 X||

1 X2 XM XFig. 3

SMC1

SMC2

SMCM

M

a

C

economies are attained and the firm is said to have reached the minimum optimal scale, given the technology of the industry. If new techniques are invented for larger scales of output, they must be cheaper to operate. But even with the existing known techniques some economies can always be achieved at larger outputs:

(a)Economies from further decentralization and improvement in skills;(b)Lower repairs costs may be attained if the firm reaches a certain size;(c) The firm, especially if it is multiproduct, may well undertake itself the production

of some of the materials or equipment which it needs instead of buying them from other firms.

Managerial Costs: In the modern management science for each plant size there is a corresponding organizational administrative set-up appropriate for the smooth operating of that plant. There are various levels of management, each with its appropriate kind of management technique. Each management technique is applicable to a range of output. There are small-scale as well as large-scale organizational techniques. The costs of different techniques of management first fall up to a certain plant size. At very large scales of output managerial costs may rise, but very slowly.

In summary: Production costs fall smoothly at very large scales, while managerial costs may rise only slowly at very large scales. Modern theorists seems to accept that the fall in technical costs more than offsets the probable rise of managerial costs, so that the LRAC curve falls smoothly or remains constant at very large scales of output.

We may draw the LAC implied by the modern theory of costs as follows. For each short-run period we obtain the SRAC which includes production costs, administration costs. Other fixed costs and an allowance for normal profit. Assume that we have a technology with four plant sizes, with costs falling as size increases. We said that in business practice it is customary to consider that a plant is used normally when it operates at a level between two-thirds and three-quarters of capacity.

Following this procedure, and assuming that the typical load factor of each plant is two-thirds of its full capacity (limit capacity), we may draw the LAC curve by joining the points on the SATC curves corresponding to the two-thirds of the full capacity of each plant sizes the LAC curve will be continuous (fig.4).

The characteristic of this LAC curve is that (a) it does not turn up at very large scales of output; (b) it is not the envelope of the SATC curves, but rather intersects them (at the level of output defined by the typical load factor of each plant). If, LAC falls

67

0 Fig. 4 X Output

C Cost

LAC

SATC4

SATC3

SATC2

SATC1

2/32/3

2/3

2/3

continuously (though smoothly at very large scales of output), the LMC will lie below the LAC at all scales (fig. 5). If there is a minimum optimal scale of plant (x in fig. 6) at which all possible scale economies are reaped, beyond that scale the LAC remains constant. In this case the LMC lies below the LAC until the minimum optimal scale is reached, and coincides with the LAC beyond the U-shaped costs of traditional theory.

CHAPTER FIVE

PERFECT COMPETITION5.1 Short-run Equilibrium of the Firm and the IndustryPerfect competition is a market structure by a complete absence of rivalry among the individual firms. Thus, perfect competition in economic theory has a meaning diametrically opposite to the everyday use of this term. In practice businessmen use the word competition as synonymous to rivalry. In theory perfect competition implies no rivalry.The model of perfect competition is based on following assumption.I. Assumptions:1. Large numbers of sellers and buyers: The industry or market includes a large

number of firms (and buyers), so that each individual firm, however large, implies only a small part of the total quantity offered in the market. The buyers also numerous; that no monopsonistic power can affect the working of the market. Under these conditions each firm alone cannot affect the price in the market by changing its output.

2. Product homogeneity: The industry is defined as a group of firms producing a homogeneous product. The technical characteristics of the product as well as the services associated with its sale and delivery are identical. There is no way in which a buyer could differentiate among the products of different firms.

3. Free entry and exit of firms: There is no better to entry or exit from the industry. Entry or exit may take time, but firms have freedom of movement in and out of the industry. This assumption is supplementary to the assumption of large numbers.

4. Profit Maximization: The goal of all firms is profit maximization. No other goals are pursued.

5. No government regulation: No government intervention is there in the market (tariffs, subsidies, rationing of production or demand and so on are ruled out).The above assumptions are sufficient for the firm to be a price-taker and have an infinitely elastic demand curve. The market structure in which the above assumptions are fulfilled is called pure competition. It is different from perfect competition which requires the fulfillment of the following additional assumptions.

6. Perfect mobility of factors of production: factors of production are free to move from one firm to another throughout the economy. Workers can move between jobs, which imply that skills can be learned easily. Raw materials and

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LACLMC

0 Fig. 5 X

C

0 Fig. 6 x X Minimum optimal scale

C

LAC = LMC

other resources are not monopolized and labour is not unionized. There is perfect competition in markets of factors of production.

7. Perfect knowledge: it is assumed that all sellers and buyers have complete knowledge of the prevailing and future period conditions of the market. Information is free and costless. Uncertainty about the future development in the market is ruled out. Under the above assumptions we will examine the equilibrium of the firm and the industry in the short run and in the long run.

II. Short-run equilibriumIn order to determine the equilibrium of the industry we need to derive the market supply. This requires the determination of the supply of the individual firms, since the market supply is the sum of the supply of all the firms in the industry.A. Short-run equilibrium of the Firm: The firm is in equilibrium when it maximizes

its profits (П), defined as the difference between total cost and total revenue: П = TR – TC

Given that the normal rate of profit is included in the cost items of the firm, П is the profit above the normal rate of return on capital and the remuneration for the risk bearing function of the entrepreneur. The firm is in equilibrium when it produces the output that maximizes the difference between total receipts and total costs. The equilibrium of the firm may be shown graphically in two ways. Either by using the TR and TC curves, or the MR and MC curves. In fig.1 we show the total revenue and total cost curves of a firm in a perfectly competitive market. The total revenue curve is a straight line through the origin; showing that the price is constant at all levels of output. The firm is a price-taker and can sell any amount of output at the going market price, with its TR increasing proportionately with its sales. The slope of the TR curve is the marginal revenue. It is constant and equal to the prevailing market price, since all units are sold at the same price. Thus in pure competition MR = AR = P. The shape of the total-cost curve reflects the U shape of the average-cost curve, that is, the law of variable proportions. The firm maximizes its profit at the output XC, where the distance between the TR and TC curves is the greatest. At lower and higher levels of output total profit is not maximized: at levels smaller than XA and larger than XB the firm has losses.

The total-revenue-total-cost approach is awkward to use when firms are combined together in the study of the industry. The alternative approach, which is based on marginal cost and marginal revenue, uses price as an explicit variable, and shows clearly the behavioral rule that leads to profit maximization. In fig.2 we show the average and marginal cost curves of the firm together with its demand curve.

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0 XA XC XB X Fig. 1

П Max

TCTR

C R

0 XC X X Fig. 2

SATC

SMC

B

The demand curve is also the average revenue curve and the marginal revenue curve of the firm in a perfectly competitive market. The marginal cost cuts the SATC at its minimum point. Both curves are U-shaped, reflecting the law of variable proportions which is operative in the short run during which the plant is constant. The firm is in equilibrium (maximizes its profit) at the level of output defined by the intersection of the MC and the MR curves (point e in fig. 2). To the left of e profit has not reached its maximum level because each unit of output to the left of Xe brings to the firm revenue which is greater than its marginal cost. To the right of Xe each additional unit of output costs more than the revenue earned by its sale, so that a loss is made and total profit is reduced. In summary:a. If MR>MC, total profit hasn’t been maximized and it pays the firm to expand its

output. b. If MC>MR, the level of total profit is being reduced and it pays the firm to cut its

output.c. If MC = MR, short-run profits are maximized.Thus the first condition for the equilibrium of the firm is that marginal cost be equal to marginal revenue. However, this condition is not sufficient, since it may be fulfilled and yet the firm may not be in equilibrium. In fig. 3 we observe that the condition MC = MR is satisfied at point e|, yet clearly the firm is not in equilibrium, hence profit is maximized at Xe> Xe

| The second condition for equilibrium requires that the MC must cut the MR curve from below, i.e., the slope of the MC must be steeper than the slope of the MR curve. In the fig. 3 the slope of MC is positive at e, while the slope of the MR curve is zero at all levels of output. Thus at e both conditions for equilibrium are satisfied

i. MC = MR andii. (slope of MC) > (slope of MR).

It should be noted that the MC is always positive, because the firm must spend some money in order to produce an additional unit of output. Thus at equilibrium the MR is also positive.

SMC SATC e

0 X|e Xe

Fig. 3

70

P=MR

P A

e

e|

P C

C P

The fact that a firm is in short run equilibrium does not necessarily mean that it makes excess profits. Whether the firm makes excess profits or losses depends on the level of the ATC at the short-run equilibrium. If the ATC is below the price at equilibrium (fig. 4) the firm earns excess profits (equal to the area PABe). If, however, the ATC is above the price (fig. 5) the firm makes a loss (equal to the area FPeC). In the latter case the firm will continue to produce only if it covers its variable costs. Otherwise it will close down, since by discontinuing its operations the firm is better off: it minimizes its losses. The point at which the firm covers its variable costs is called the closing down point. In fig. 6 the closing-down point of the firm is denoted by point w. If price falls below Pw the firm does not cover its variable costs and is better off it closes down.

Mathematical Derivation of the equilibrium of the FirmThe firm aims at the maximization of its profit

, Where, = Profit, R = Total Revenue, C = Total Cost Clearly R = f1(X) and C = f2(X), given the price P.(a) The first order condition for the maximization of a function is that its first derivative be equal to zero. Differentiating the total-profit function and equating to zero we obtain

or

The term is the slope of the total revenue curve, that is, the marginal revenue. The term is the slope of the total cost curve, or the marginal cost. Thus the first-order condition for profit maximization is MR = MC.

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SATC

FMR P

SMC

O Xe X Fig. 4

P C

P

A

e

B

0 Xe X Fig. 5

SMC

MR

SATCPC

C

e

0 Xw X Fig. 6

AFC

SAVC

SATC

SMC

Pw

P C

Given that MC>0, MR must also be positive at equilibrium. Since MR = P the first order condition may be written as MC = P.(b) The second order condition for a maximum requires that the second derivative of function be negative (implying that after its highest point the curve turns downwards).

The second order derivative of the total-profit function is

This must be negative if the function has been maximized, that is ,

which yields the condition, , but is the slope of the MR curve and

is the slope of the MC must cut the MR curve from below. In pure competition the slope of the MR curve is zero, hence the second-order condition is

simplified as follows, , which reads: the MC curve must have a positive slope, or

the MC must be rising.

B. Short-run Equilibrium of the Industry: Given the market demand and the market supply, the industry is in equilibrium at that price which clears the market that is at the price at which the quantity demanded is equal to the quantity supplied. In fig. 7 the industry is in equilibrium at price P, at which the quantity demanded and supplied is Q. However, this will be short run equilibrium, if at the prevailing price firms are making excess profits fig. 8 or losses Fig. 9. In the long run, firms that make losses and cannot readjust their plant will close down. Those that make excess profits will expand their capacity, while excess profits will also attract new firms into the industry. Entry, exit and readjustment of the remaining firms in the industry will lead to a long-run equilibrium in which firms will just be earning normal profits and there will be no entry or exit from the industry.

5.3 Long-run equilibrium of the firm and the industry

C. Equilibrium of the Firm in the Long Run: In the long run firms are in equilibrium when they have adjusted their plant so as to produce at the minimum point of their long-run AC curve, which is tangent (at this point) to the demand curve defined by the market price.

In the long run the firms will be earning just normal profits, which are included in the LAC. If they are making excess profits new firms will be attracted in the industry; this will lead to a fall in price (a downward shift in the individual demand curves) and an upward shift of the cost curves due to the increase of the prices of factors as the industry expands. These changes will continue until the LAC is tangent to the demand curve defined by the market price. If the firms make losses in the long run they will

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D

D S’

S

0 Q X 0 Xe X 0 Xe X Fig. 7 Fig. 8 Fig. 9

P

P’

P C

P’

SMCSATC

e

B

leave the industry, price will rise and costs may fall as industry contracts until the remaining firms cover the total cost firms inclusive of the normal rate of profit. In fig. 1 we show how firms adjust to their long-run equilibrium position. If the price is P, the firm is making excess profits working with the plant whose cost is denoted by SAC1. It will therefore have an incentive to build new capacity and it will move along its LAC. At the same time new firms will be entering the industry attracted by the excess profits. As the quantity supplied in the market increases (by the increased production of expanding old firms and by the newly established ones) supply curve in the market will shift to the right and price will fall until it reaches the level of P1 (in fig.1) at which the firm and the industry are in the long-run equilibrium. The LAC in fig.2 is the final-cost curve including any increase in the prices of factors that may have taken place as the industry expanded.

The condition for the long-run equilibrium of the firm is that the marginal cost be equal to the price and to the long-run average cost LMC = LAC = PThe firm adjusts its plant size so as to produce that level of output at which the LAC is the minimum possible, given the technology and the prices of factors of production. At equilibrium the short-run marginal cost equal to the long-run marginal cost and the short-run average cost is equal to the long-run average cost. Thus given the above equilibrium condition, we have: SAC = SMC = LAC = LMC = P = MR

This implies that at the minimum point of the LAC the corresponding (short-run) plant is worked at its optimal capacity, so that the minima of the LAC and SAC coincide. On the other hand, the LMC cuts the LAC at its minimum point and the SMC cuts the SAC at its minimum point. Thus, at the minimum point of the LAC the above equality between short-run and long-run costs is satisfied.

D. Equilibrium of the Industry in the Long-run: The industry is in long-run equilibrium when a price is reached at which all firms are in equilibrium (proceeding at the minimum point of their LAC curve and making just normal profits). Under these conditions there is no further entry or exit of firms in the industry, given the technology and factor prices. The long-run equilibrium of the industry is shown in

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0 Q Q1 X 0 X Fig.1 Fig.2

P C

P

P1

P C

P

P1

DS

S1

S1

LMC

LACSAC1

SMC1

SMC

SAC

fig. 3. At the market price, P, the firms produce at their minimum cost, earning just normal profits. The firm is in equilibrium because at the level of output X.

LMC = SMC = P = MR. This equality ensures that the firm maximizes its profits. At the price P the industry is in equilibrium because profits are normal and all costs are covered so that there is no incentive for entry or exit. That the firms earn just normal profit (neither excess profits nor losses) is shown by the equality; LAC= SAC= PThis is observed at the minimum point of the LAC curve. With all firms in the industry being in equilibrium and with no entry or exit, the industry supply remains stable, and, given the market demand (DD|) in fig.3, the price P is a long run equilibrium price.Since the price in the market is unique, this however, does not mean that all firms in the industry have the same minimum long-run average cost. This, however, does not mean that all firms are of the same size or have the same efficiency, despite the fact that their LAC is the same in equilibrium. The more efficient firms employ more productive factors of production and/or more able managers. These more efficient factors must be remunerated for their higher productivity; otherwise they will be bid off by the new entrants in the industry. In other words, as the price rises in the market the more efficient firms earn a rent which they must pay to their superior resources. Thus rents of more efficient factors become costs for the individual firm, and hence the LAC of the more efficient firms’ shifts upwards as the market price rises, even if the factor prices for the industry as a whole remain constant as the industry expands. In this situation the LAC of the old, more efficient, firms must be redrawn so as to be tangent at the higher market price. The LMC of the old firms is not affected by the rents accruing to its more productive factors. (It will be shifted only if the prices of factors for the industry in general increase.)

Fig. 3Thus, the more efficient firms will be in equilibrium, producing that output at which the redrawn LAC is at its minimum (at which point the LAC is cut by the initial LMC given that factor prices remain constant). Under these conditions, with the superior, more productive resources properly costed at their opportunity cost, all firms have the same unit cost in their long-run equilibrium. This is shown in fig 4. At the initial price P0 the second firm as not in the industry as it could not cover its cots at that price. However, at the new price, P1, firm B enters the industry, making just normal profits. The established firm A earns rents which are imputed costs, so that its LAC shifts upwards and it reaches a new long-run equilibrium producing a higher level of output (X|

A).

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D|

DS|

S

P = MR

0 X 0 X

LACSAC

LMC

SMC

P

XQ

P C

P C

S|

S D|

D

LMCA LAC|A

LACA

LMCA

LACB

0 X 0 XA A|A X 0 XB X

Market Equilibrium Equilibrium of a more efficient firm Equilibrium of a new entrant Fig. 4

P1

P0

P1

P0

P1

P

Chapter-SixMonopoly Market Structure

Introduction:In a perfectly competitive market each firm's production is such a small proportion of industry out put that an individual firm has to influence on the market price. As we have seen the competitive firm is a price taker. In this unit we will see the opposite extreme to a purely competitive firm of pure monopoly a single seller in an industry. The sources of monopoly, price and out put determination in the short and long run periods, price discrimination by a monopoly and comparison of the price and out put with perfect competition are some of the points to be discussed in this unit.DefinitionA monopoly is a market structure where there is only one firm that produces and sells a particular commodity or service and there are no close substitutes available. Since the monopoly is the seller in the market, the industry is a single firm industry and it has no direct competitors. However it does not necessarily mean that it is a guarantee to get an abnormal profit. Monopoly power only guarantees that the monopolist can make the best of whatever demand and cost conditions exist with out fear of the entrant of new competing firms. Source and Types of MonopolyThe rise and existence of monopoly is related to the factors, which prevents the entry of new firms. The different barriers to entry that are the causes of monopoly are described below.i. Legal Restrictions: A monopoly, which are created for the interest of the public.

For example the public utility sectors such as water supply, postal, telegraph and telephone services, radio and TV services, generation and distribution of electricity such monopolies are known as public monopolies

ii. Control over key raw materials: some firms may get monopoly power if they posses certain scarce & key raw materials that are essential for the production of certain goods or if the supply of a commodity is localized in a single place. This type monopoly is known as raw material monopoly. For example India possesses manganese mines, the extraction of diamonds is controlled by South Africa

iii. Efficiency: A primary and technical reason for growth of monopolies is economies of scale. The most efficient plant (probably large size firm), which can produce at minimum cost, could eliminate the competitors by cutting down its price for a short period and can acquire monopoly power. Monopolies created through efficiency are known as natural monopolies.

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iv. Patent rights: - The government has granted firms a patent right, for producing a commodity of specified quantity and character so that firms will have exclusive rights to produce the specified commodity. Such monopolies are called patent monopolies.

Demand, Marginal Revenue and cost curves under MonopolyIn the analysis of consumer behavior you have seen that the demand curve is generally down ward sloping showing inverse relation ship between price and quantity demanded. In perfectly competitive market the industry faces down ward sloping demand curve; firms face a horizontal demand curve because of the existence of large number of producers and homogeneity of the product, the firm can not exert power on the total industry supply.

The monopoly industry on the other hand is a single firm industry. A monopoly firm there fore faces a down ward sloping demand curve. It implies given the demand curve, a monopoly firm has the option to choose between prices to be charged or out put to be sold. But he cannot simultaneously control both the price and the level of out put. He can either decide the level of out put, and leave the price of the out put to be determined by consumer demand or he can fix the price and leave the level of out put to be decided by the demand for the product at that price. One of the fundamental differences between a monopolist and a competitor is there fore the demand (AR) and marginal revenue curves they face. In the case of perfectly competitive market MR = AR=P=D. But in the case of down ward sloping demand curve of monopoly marginal revenue curve falls twice as much as the fall of average revenue curves i.e. the slope of MR is twice as steep as the average revenue curve. The following figure illustrates this relationship

AR and RM curves for Monopoly

DM is the demand curve and DT is the marginal revenue curve, which bisects the quantity demanded OM. Thus the distance OT = TMThis can be shown mathematically as follows assuming linear demand function

1. The demand functionP = a - bx where a = constant, x=quantity demanded

2. The total revenue isR = Px

76

D

D=AR=P

QuantityMR

0T M

= (a - bx)x = ( substituting P = a -bx) = ax -bx2

3. The Average revenue

AR =

Thus the demand curve is also the AR curve of the monopolist with slope = -b4. The marginal revenue (the first derivative of R)

=a-2bxThat is the MR is a straight line with the same intercept (a) as the demand curve, but twice as steep ( i.e slope = -2b)

Note: - the general relation between P and MR is found as follows R = Px

(Product rule of differentiation, you have learned in your quantitative for economists I course)

(But is negative due to the inverse relation between

demand & price)

P = MR+x

Thus the marginal revenue is smaller than price at all levels of out put.

The nature and shape of cost curves confronting a monopolist are similar to those faced by a perfectly competitive firm. Because cost depends on the production function and input prices, irrespective of whether a firm is a monopoly or perfectly competitiveShort Run Equilibrium of MonopolyAs we know, in the short run period the scale of plant is fixed and the firm is unable to change it. If the monopolist desires to produce more, he can make an intensive use of the variable input. Since the monopolist has to incur fixed costs in the short run, the minimum price acceptable by him must be equal to his average variable cost. For this reason the equilibrium condition is the same as we have explained under perfect competition i.e. equilibrium occurs when marginal revenue equals marginal cost. But for the monopolist, MR does not equal price. The short run price and out put determination under monopoly, and also the firm’s equilibrium are illustrated in the following figure.

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The monopolist firm is in equilibrium at point E where SMC interests the MR curve

from below. The profit maximizing (equilibrium) out put is Qe and price is 0P1. At OQe level of out put, the average cost is OP2 (or QeB). Thus the monopolist's per unit abnormal profit is equal to AB, which is the difference between the price OP1, and the corresponding average cost of production (OP2). The shaded area; P1ABP2 represents the total monopoly profit.Total revenue = AR x out put sold = OP1 x OQe

= Area OP1AQeTotal Cost = AC x out put produced

= OP2 x OQe= Area OP2 BQe

Profits = TR-TC= Area OP1 AQe - Area OP2 BQe= Area P1AB P2

Does a monopolistic firm always earn an abnormal profit? Have you said no? Good. As we explain in the introduction part a monopolistic position does not guarantee above normal return always weather the monopolist gets profit or not, depends up on the conditions of demand and costs. It is possible that his entire short run average cost curve lies above the demand curve or AR curve so that he has to incur a loss. The following figure illustrates such a situation.

P1 D ATC

P2 B AVCP3 N

Qe

78

SMC

SAC

MR

A

B

P1

P2

0Qe

AR=DE

B

000

AR=DMR

As the figure prevails marginal cost equals marginal revenue at point E where Qe level of out put is produced and OP2 is the equilibrium price. But average total cost is OP1

(or QeD). ThusTotal revenue = OP2 x OQe Total cost = OP1 x OQe

= OP2BQe = Area OP1 DQeTotal revenue exceeds total cost; hence, the firm makes a loss of P2 P1DB. The monopolist would produce rather than shut down in the short run, since price exceeds the AVC (OP3). If demand decreases so that the monopolist cannot cover all variable cost at any price, the firm would shut down and lose only fixed cost.

Long Run Equilibrium of MonopolyContrary to perfect competition pure economic profit is not eliminated in the long run under monopoly... As we already discuss a monopoly exists if there is only one firm in the market. This statement implies that entry in to the market is closed. If a monopolist should earn a pure profit in the short run, no other producer can enter the market in the hope of sharing whatever profit potential exists in the long run. Thus a monopolist will continue to earn abnormal profit even in the long run. The magnitude of the long run profits will depend up on the cost condition under which he has to operate production and the demand curve he has to face in the long run.

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If it’s AR> SMC1, it earns a short run profit at out put Oq1 as shown in the following figure (area P1 P2AB). The firm would therefore not only continue in the business but would also expand its business to the size that yields maximum profit in the long run (a plant with SMC2 and SAC2)

Long run Equilibrium of monopoly

As depicted in the figure, the point of intersection between LMC and MR curves determine the equilibrium out put at Oq2, price Op3. The total long run profit has been shown by the area P4P3cd(Total revenue (OP3Cq2)-Total cost (OP4Dq2))

Under perfect competition we have seen that in the long run every firm has be of optimum size where its long run average cost (LAC) is minimum. Does a monopolistic firm always produce at the optimum size in the long run? In the cost of monopoly, the equilibrium level of out put (where LMC=MR) may or may not have the lowest long run average cost. There are three possibilities to exist depending on the cost structure of the firm and demand for the product.

i. Under utilization of capacity:- A case in which the market size and cost conditions lead to maximize profit at lower than optimum size (i.e LMC &MR intersects out puts less than where LAC is at its minimum

ii. Over utilization of capacity :- A case where in the market is so large that it forces the monopolist to maximize profit by over utilizing the capacity ( at out puts greater than where LAC is at its minimum)

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iii. Optimum size of the plant: - This is the case in which the market size and cost conditions allow. The monopolist to maximize long run profit at exactly equal to the optimum plant.

Price DiscriminationA producer, mostly likely a monopolist need not always charge a single price to his customers since he is the only producer in the market, he has a control over the supply of the product. He can charge different prices to different consumers or in different markets. Thus when the same product is sold at different price to different consumers, it is called price discrimination. The two most important points to note about the definition of price discrimination are: first, exactly the same products must have different prices. A trip from Bahir Dar to Gondar is not the same as a trip from Bahir Dar to Dessie because transportation costs are different and this difference raises the price of the trip. Second, in order for price discrimination to exist, production costs must be equal. If costs are different, a profit maximizing firm who sets MR=MC will usually charge different price for a product. This price difference is also due tot cost difference not discrimination.

Consumers are discriminated in respect of price on the basis of their incomes, geographical location, Age, sex, quantity they purchase, frequency of visits to the shop etc. For instance, price discrimination on the basis of age in airways, railways, Cinema shows, Musical concerts, charging lower price for teenagers. Doctors has charged different price for rich and poor persons. But to implement price discrimination effectively the following conditions has to be full filled.

1. There must be separate markets, so that no reselling can take place from a low price market to a high price market. It is most commonly effective for goods that can not be easily traded (exchange of services for example, a poor receiving medical care at relatively low price can not resell his or her operation to another patient, but a lower price buyer of some raw material could resell it to some one in the higher price market.

2. Differences in the elasticity of demand. It is the difference in price elasticity that provides opportunity for price discrimination. If price elasticity of demand in different markets is the same, price discrimination would not be important (gain full).

Degrees of price discrimination The main objective of price discrimination is to maximize profit more than that the firm could obtain by charging the same price defined by the equation of his MC and MR. The degree of price discrimination, there fore, refers to the extent to which a seller can divide the market and can take advantage of it in extracting the consumers surplus. Accordingly there are three degrees of price discrimination practiced by monopolists

a. First degree price discrimination

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The discriminatory pricing that attempts to take away the entire consumer surplus is called first-degree discrimination. Under first price discrimination, the firm treats each individual's demand separately and each consumer is assumed as a separate market.

b. Second -Degree price Discrimination.

First-degree price discrimination is expense to implement. Because dealing each individual demand curve needs vast information and it is costly. Second degree price discrimination is some what simpler because it requires the firm to consider groups of consumers. This is discrimination on the basis of quantity purchased and intends to take only the major part of consumer surplus rather than the entire.

c. Third degree Price discriminationSelling the same product with different price in different markets having demand curves with different elasticity is called third degree price discrimination. Profit in each market would be maximum only when his MR=MC in each market. The monopolist there for divides his out put between the markets so that in all markets his MR=MC. In this case suppose that the monopolist has to sell his product in only two markets A and B. As a result, the monopolist must allocate out put between the two markets in such proportion that the necessary condition for profit maximization (i.e. MR = MC) is satisfied. The equilibrium condition is satisfied i.e. MC=MRa =MRb (common MC equal individual MR)Social cost of Monopoly Power

So far we have seen how out put and price is determined in the case of perfect competition and monopoly. And we have examined the quantity supplied under perfect competition & monopoly; the price charged by a perfectly competitive and monopolistic firm. Economists are always criticized monopoly firms in that it is less efficient than competitive firms and it causes social welfare loss and distortions in resource allocation. The most common reason for criticism of monopoly is that price is higher and out put is lower than in a perfectly competitive market. These fore, we compare the long run price and out put under monopoly and perfect competition using the following graphical analysis assuming a constant cost industry ( so that LAC =LMC)

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0 q1 q2

Price and output under monopoly & perfect competition As it is prevailed in the above figure given the cost and revenue conditions, the perfectly competitive industry will produce Oq2 at which is LAC = LMC= AR. Its Price will be OP1.On the other hand, the monopoly firm produces an out put where LMC = MR. Monopoly firm produces Oq1 and charges price OP2. Thus, if both monopoly and competitive industries are faced with identical cost conditions, the out put under competitive condition is higher than under monopoly (0q2 >0q1), and price in the competitive industry is lower than in monopoly (OP1<OP2). Perfect competition is therefore more desirable from social welfare angle. The loss of social welfare is measured in terms of loss of consumer surplus. The total consumer surplus equals the difference between the total utility which a society gains from and the total price which he pays for a given quantity of goods as you have seen in module one.

Consumer surplus under competitive market- The total utility from oq2 = area OALq2

- Total price paid by the society = area OP1Lq2

- Consumer surplus = area OALq2 = area OP1Lq2

= area ALP1

Consumer surplus under monopoly- Total utility from Oq2 = area OAJq1

- Total price paid by them society = OP2Jq1- Consumer surplus = area OAJq1-areaOP2q1

= area AJP2 Thus loss of consumer surplus under monopoly equals

Area ALP1 -Area AJP2 = P2JLP1

Of this total loss of consumer surplus (P2JLP1), P2JkP1 is extracted by the monopolist as profit, the remaining JKL goes to none and it is termed as dead weight loss to the society

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LAC=LMC=MR=AR=DD perfect