Economics Office Word Document

8/3/2019 Economics Office Word Document

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Practical importance of elasticity of demand

Elasticity of Demand :- " Elasticity of demand is the rate at which the quantitydemanded changes with a change in price."

In other words we can say that elasticity of demand is the relationship between theproportionate change in price and the proportionate change in quantity demanded.

FORMULA : ED = Proportionate change in quantity demanded/Proportionatechange in price.

PRACTICAL IMPORTANCE OF ELASTICITY OF DEMAND

This concept has a great practical importance in the sphere of government financeand in the commerce and trade due to the following reasons :

1. IMPORTANCE FOR FINANCE MINISTER :- Before imposing the taxes financeminister has to keep in view the elasticity of demand of various goods. If thedemand is inelastic, he can increases the tax and thus can collect large revenue.

2. IMPORTANCE FOR THE MONOPOLIST :- If the monopolist finds that thedemand for his product is inelastic, he will fix the price at a higher level, otherwisehe will lower the price.

3. FIXATION OF WAGES :- If a demand of labour is inelastic, it is easy to rise theirwages otherwise not.

4. INTERNATIONAL TRADE :- If the demand of commodity is inelastic then heavyduties can be imposed on its import heavy duties can be imposed on its import andexport.

5. IMPORTANCE FOR THE PRODUCER :- Producer will study elasticity of demand
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before fixing the price of his commodities. Secondly, If the demand for a commodityis inelastic the producer will spend a large amount on advertisement for increasingthe sale.

6. RATE OF FOREIGN EXCHANGE :- The rate of foreign exchange is alsoconsidered on the elasticity of exports and imports of the country.

7. TERMS OF TRADE :- The terms of trade between two countries are based onthe elasticity of demand of the traded goods.

8. IMPORTANCE FOR THE BUSINESSMAN :- When the demand of good iselastic , businessman increase his sale by lowing the price. If the demand is elasticthen he fixes high prices.

9. JOINT PRODUCT COST PROBLEM :- Sometimes it is very difficult to know theseparate cost of each factor of joint products. Here elasticity of demand becomesvery helpful in determining the cost of each factor of production.

10. IMPORTANCE FOR COMMUNICATION INDUSTRY :- The concept of elasticityis practically used in fixing the rates and fares of transfer of goods.

11. LAW OF INCREASING RETURN AND DEMAND :- When small industry isworking under the law of increasing return, its demand should be elastic. So it willlower the price and increase the sale.

In economics, elasticity is the measurement of how changing one economic variable affects others. Forexample:

"If I lower the price of my product, how much more will I sell?"

"If I raise the price, how much less will I sell?"

"If we learn that a resource is becoming scarce, will people scramble to acquire it?"In more technical terms, it is the ratio of thepercentage change in one variable to the percentage changein another variable. It is a tool for measuring the responsiveness of a function to changes in parameters ina unit less way. Frequently used elasticity include price elasticity of demand,price elasticity of supply,income elasticity of demand,elasticity of substitution between factors of production and elasticity ofintertemporal substitution.

Elasticity is one of the most important concepts in neoclassical economic theory. It is useful inunderstanding the incidence of indirect taxation, marginal concepts as they relate to the theory of the firm,and distribution of wealth and different types of goods as they relate to the theory of consumer choice.Elasticity is also crucially important in any discussion ofwelfare distribution, in particularconsumersurplus, producer surplus, orgovernment surplus.

In empirical work an elasticity is the estimated coefficient in a linear regression equation where both thedependent variable and the independent variable are in natural logs. Elasticity is a popular tool amongempiricists because it is independent of units and thus simplifies data analysis.

Generally, an elastic variable is one which responds a lot to small changes in other parameters. Similarly,an inelastic variable describes one which does not change much in response to changes in otherparameters. A major study of the price elasticity of supply and theprice elasticity of demand for USproducts was undertaken by Hendrik S. Houthakker and Lester D. Taylor.
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Mathematical definitionElasticity of a function

The definition of elasticity is based on the mathematical notion ofpoint elasticity.In general, the "x-elasticity ofy", also called the "elasticity ofy with respect tox", is:

The approximation becomes exact in the limit as the changes become infinitesimal in size.

The absolute value operator is for simplicity generally the sign of the elasticity is understood as beingalways positive or always negative. However, sometimes the elasticity is defined without the absolutevalue operator, when the sign may be either positive or negative or may change signs. A context wherethis use of a signed elasticity is necessary for clarity is the cross-price elasticity of demand theresponsiveness of the demand for one product to changes in the price of another product; since the

products may be eithersubstitutes orcomplements, this elasticity could be positive or negative.

Specific elasticity

Elasticity of demand

Price elasticity of demand

Price elasticity of demand measures the percentage change in quantity

demanded caused by a percent change in price. As such, it measures the

extent of movement along the demand curve. This elasticity is almost always

negative and is usually expressed in terms of absolute value (i.e. as positivenumbers) since the negative can be assumed. In these terms, then, if the

elasticity is greater than 1 demand is said to be elastic; between zero and

one demand is inelastic and if it equals one, demand is unit-elastic. A

perfectly elastic demand curve is horizontal (with an elasticity of infinity)

whereas a perfectly inelastic demand curve is vertical (with an elasticity of

0).

Income elasticity of demand

Income elasticity of demand measures the percentage change in demand

caused by a percent change in income. A change in income causes thedemand curve to shift reflecting the change in demand. IED is a

measurement of how far the curve shifts horizontally along the X-axis.

Income elasticity can be used to classify goods as normal or inferior. With a

normal good demand varies in the same direction as income. With an inferior

good demand and income move in opposite directions.

Cross price elasticity of demand
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Cross price elasticity of demand measures the percentage change in demand

for a particular good caused by a percent change in the price of another

good. Goods can be complements, substitutes or unrelated. A change in the

price of a related good causes the demand curve to shift reflecting a change

in demand for the original good. Cross price elasticity is a measurement of

how far, and in which direction, the curve shifts horizontally along the x-axis.A positive cross-price elasticity means that the goods are substitute goods.

Cross elasticity of demand between firms

Cross elasticity of demand for firms, sometimes referred to as conjectural

variation, is a measure of the interdependence between firms. It captures the

extent to which one firm reacts to changes in strategic variables (price,

quantity, location, advertising, etc.) made by other firms.

Elasticity of intertemporal substitution

Combined Effects

It is possible to consider the combined effects of two or more determinant of

demand. The steps are as follows: PED = (Q/P) x P/Q. Convert this to the

predictive equation: Q/Q = PED(P/P) if you wish to find the combined effect

of changes in two or more determinants of demand you simply add the

separate effects: Q/Q = PED(P/P) + YED(Y/Y)[12]

Remember you are still only considering the effect in demand of a change in

two of the variables. All other variables must be held constant. Note also that

graphically this problem would involve a shift of the curve and a movement

along the shifted curve.

Elasticity of supply

Price elasticity of supply

The price elasticity of supply measures how the amount of a good firms wish

to supply changes in response to a change in price. In a manner analogous to

the price elasticity of demand, it captures the extent of movement along the

supply curve. If the price elasticity of supply is zero the supply of a good

supplied is "inelastic" and the quantity supplied is fixed.

Elasticity of scale: Returns to scale

Elasticity of scale or output elasticities measure the percentage change in

output induced by a percent change in inputs. A production function or

process is said to exhibit constant returns to scale if a percentage change in

inputs results in an equal percentage in outputs (an elasticity equal to 1). It

exhibits increasing returns to scale if a percentage change in inputs results in
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greater percentage change in output (an elasticity greater than 1). The

definition ofdecreasing returns to scale is analogous.

Applications

The concept of elasticity has an extraordinarily wide range of applications in economics. In particular, an

understanding of elasticity is fundamental in understanding the response ofsupply and demand in amarket.

Some common uses of elasticity include:

Effect of changing price on firm revenue. See Markup rule.

Analysis of incidence of the tax burden and other government policies. SeeTax incidence.

Income elasticity of demand can be used as an indicator of industry health,future consumption patterns and as a guide to firms investment decisions.See Income elasticity of demand.

Effect of international trade and terms of trade effects. See MarshallLerner

condition andSingerPrebisch thesis. Analysis ofconsumption and saving behavior. See Permanent income

hypothesis.

Analysis ofadvertising on consumer demand for particular goods. SeeAdvertising elasticity of demand

Variants

In some cases the discrete (non-infinitesimal) arc elasticity is used instead. In other cases, such asmodified duration in bond trading, a percentage change in output is divided by a unit (not percentage)

change in input, yielding a semi-elasticity instead.

Arc elasticity

Arc elasticity is the elasticity of one variable with respect to another between two given points.

Formula

They arc elasticity ofx is defined as:

where the percentage change is calculated relative to the midpoint
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The midpoint arc elasticity formula was advocated by R. G. D. Allen due to the following properties: (1)it is symmetric with respect to the two prices and two quantities, (2) it is independent of the units ofmeasurement, and (3) it yields a value of unity if the total revenues at two points are equal.

Arc elasticity is used when there is not a general function for the relationship of two variables. Therefore,point elasticity may be seen as an estimator of elasticity; this is because point elasticity may beascertained whenever a function is defined.

For comparison, they point elasticity ofx is given by:

Application in economics

The P arc elasticity of Q is calculated as

The percentage is calculated differently from the normal manner of percent change. This percentchange uses the average (or midpoint) of the points, in lieu of the original point as the base.

Example

Suppose that you know of two points on a demand curve (Q1,P1) and (Q2,P2). (Nothing else might beknown about the demand curve.) Then you obtain the arc elasticity (a measure of the price elasticity ofdemand and an estimate of the elasticity of a differentiable curve at a single point) using the formula

Suppose we measure the demand for hot dogs at a football game. Let's say that after halftime we lowerthe price, and quantity demanded changes from 80 units to 120 units. The percent change, measuredagainst the average, would be (120-80)/((120+80)/2))=40%.

Normally, a percent change is measured against the initial value. In this case, this gives (12-8)/8= 50%.The percent change for the opposite trend, 120 units to 80 units, would be -33.3%. The midpoint formulahas the benefit that a movement from A to B is the same as a movement from B to A in absolute value.(In this case, it would be -40%.)

Suppose that the change in the price of hot dogs was from $3 to $1. The percent change in price measuredagainst the midpoint would be -100%, so the price elasticity of demand is (40%/-100%) or -40%. It iscommon to use the absolute value of price elasticity, since for a normal (decreasing) demand curve theyare always negative. Thus the demand of the football fans for hot dogs has 40% elasticity, and is thereforeinelastic.

Elasticity of a function

n mathematics, elasticity of a positive differentiable functionfof a positive variable (positive input,positive output) at pointx is defined as[1]
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or, in terms of percentage change

Intuitively, it is thepercentage change in output for apercentage change in input percentages onlymaking sense if the quantities are all positive.[2]Formally, it is the ratio of the incremental change of thelogarithm of a function with respect to an incremental change of the logarithm of the argument. Thisdefinition of elasticity is also called point elasticity, and is the limit ofarc elasticity between two points.

Elasticity is widely used in economics; see elasticity (economics) for details.

Rules

Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Letf, gbedifferentiable. Then

The chain rule is similar to the derivative

The derivative can be expressed in terms of elasticity as

Let a and b be constants.

Then

,
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, and also

.

ForHomogeneous functions

Estimating Point Elasticities

PED can also be expressed as (dQ/dP)/Q/P or the ratio of the marginal function to the average functionfor a demand curve Q = f( P). This relationship provides an easy way of determining whether a point on ademand curve is elastic or inelastic. The slope of a line tangent to the curve at the point is the marginalfunction. The slope of a secant drawn from the origin through the point is the average function. If theslope of the tangent is greater than the slope of the secant (M > A) then the function is elastic at thepoint.]If the slope of the secant is greater than the slope of the tangent then the curve is inelastic at thepoint. If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing

angles formed by the lines and the horizontal axis. If the marginal angle is numerically greater than theaverage angle then the function is elastic at the point. If the marginal angle is less than the average anglethen the function is inelastic at that point. If you follow the convention adopted by economist and plot theindependent variable on the vertical axis and the dependent variable on the horizontal axis then themarginal function will be dP/dQ and the average function will be P/Q meaning that you are deriving thereciprocal of elasticity. Therefore opposite rules would apply. The tangency line slope would be dP/dQand the slope of the secant would be the numerical value P/Q. This method is not limited to demandfunctions it can be used with any functions. For example a linear supply curve drawn through the originhas unitary elasticity (if you use the method the marginal function is identical to the slope). If a linearsupply function intersects the y axis then the marginal function will be less than the average and thefunction is inelastic at any point and becomes increasingly inelastic as one moves up the curve. With asupply curve that intersects the x axis then the slope of the curve will exceed the slope of the secant at all

point meaning that the M > A the slope is elastic and will become increasingly elastic as one moves upthe slope. Again this assumes that the dependent variable is drawn on the Y axis.

Semi-elasticity

A semi-elasticity (or semielasticity) gives the percentage change inf(x) in terms of a change (notpercentage-wise) ofx. Algebraically, the semi-elasticity S of a functionfat pointx is

An example of semi-elasticity is modified duration in bond trading.

The terms Semi-elasticity is also sometimes used for the change iff(x) in terms of a percentage change inx[9]which would be
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Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if theargument is multiplied by a factor, then the result is multiplied by some power of this factor. Moreprecisely, if : V Wis a functionbetween two vector spaces over a fieldF, and kis an integer, then is said to be homogeneous of degree kif

for all nonzero Fand vV. When the vector spaces involved are over the real numbers, a slightlymore general form of homogeneity is often used, requiring only that (1) hold for all > 0.

Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used inthe definition ofsheaves on projective space in algebraic geometry. More generally, ifSVis anysubset that is invariant under scalar multiplication by elements of the field (a "cone"), then anhomogeneous function from Sto Wcan still be defined by (1).

Linear functions

Any linearfunction : V Wis homogeneous of degree 1, since by the definition of linearity

for all Fand vV. Similarly, any multilinearfunction : V1 V2 ... Vn Wis homogeneous ofdegree n, since by the definition of multilinearity

for all Fand v1V1, v2V2, ..., vnVn. It follows that the n-th differential of a function :X Y

between two Banach spacesXand Yis homogeneous of degree n.

Homogeneous polynomials

Monomials in n variables define homogeneous functions :Fn F. For example,

is homogeneous of degree 10 since
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.

The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. Forexample,

is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneousfunctions.

Polarization

A multilinear functiong: V V ... VFfrom the n-th Cartesian product ofVwith itself to thegroundfieldFgives rise to an homogeneous function : VFby evaluating on the diagonal:

The resulting function is a polynomial on the vector space V.

Conversely, ifFhas characteristic zero, then given an homogeneous polynomial of degree n on V, thepolarization of is a multilinear functiong: V V ... VFon the n-th Cartesian product ofV. Thepolarization is defined by

These two constructions, one of an homogeneous polynomial from a multilinear form and the other of amultilinear form from an homogeneous polynomial, are mutually inverse to one another. In finitedimensions, they establish an isomorphism ofgraded vector spaces from the symmetric algebra ofV to

the algebra of homogeneous polynomials on V.

Rational functions

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions offof the affine cone cut out by the zero locus of the denominator. Thus, iffis homogeneous of degree m andgis homogeneous of degree n, thenf/gis homogeneous of degree m n away from the zeros ofg.

Non-Examples

Logarithms

The natural logarithmf(x) = lnx scales additively and so is not homogeneous.

This can be proved by noting thatf(5x) = ln 5x = ln 5 +f(x),f(10x) = ln 10 +f(x), andf(15x) = ln 15 +f(x).

Therefore such that .

Affine functions

Affine functions (the functionf(x) =x + 5 is an example) do not scale multiplicatively.

Some polynomials
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The functionf(x) =x2 + 2x + 1 does not scale multiplicatively.

Positive homogeneity

In the special case of vector spaces over the real numbers, the notation of positive homogeneity oftenplays a more important role than homogeneity in the above sense. A function : V\ {0} Ris positive

homogeneous of degree kif

for all > 0. Here kcan be any complex number. A (nonzero) continuous function homogeneous ofdegree kon Rn \ {0} extends continuously to Rn if and only if Re{k} > 0.

Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Supposethat the function : Rn \ {0} Ris continuously differentiable. Then is positive homogeneous ofdegree kif and only if

This result follows at once by differentiating both sides of the equation (y) = k(y) with respect to and applying the chain rule. The converse holds by integrating.

As a consequence, suppose that : Rn Ris differentiable and homogeneous of degree k. Then its first-

order partial derivatives are homogeneous of degree k 1. The result follows from Euler's

theorem by commuting the operator with the partial derivative.

Homogeneous distributions

A compactly supported continuous function on Rn is homogeneous of degree kif and only if

for all compactly supported test functions and nonzero real t. Equivalently, making a change of variabley = tx, is homogeneous of degree kif and only if

for all tand all test functions . The last display makes it possible to define homogeneity ofdistributions.A distribution Sis homogeneous of degree kif

for all nonzero real tand all test functions . Here the angle brackets denote the pairing betweendistributions and test functions, and t : R

n Rn is the mapping of scalar multiplication by the realnumbert.

Application to differential equations

The substitution v =y/x converts the ordinary differential equation
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whereIandJare homogeneous functions of the same degree, into the separable differential equation

Production functionIn microeconomics and macroeconomics, a production function is a function that

specifies the output of a firm, an industry, or an entire economy for all combinations

ofinputs. This function is an assumed technological relationship, based on the

current state ofengineering knowledge; it does not represent the result of economic

choices, but rather is an externally given entity that influences economic decision-

making. Almost all economic theories presuppose a production function, either onthe firm level or the aggregate level. In this sense, the production function is one of

the key concepts ofmainstreamneoclassical theories. Some non-mainstream

economists, however, reject the very concept of an aggregate production function
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Concept of production functionsIn micro-economics, a production function is a function that specifies the output of a firm for allcombinations of inputs. A meta-production function (sometimes metaproduction function) compares thepractice of the existing entities converting inputs into output to determine the most efficient practiceproduction function of the existing entities, whether the most efficient feasible practice production or themost efficient actual practice production.[3]clarification neededIn either case, the maximum output of atechnologically-determined production process is a mathematical function of one or more inputs. Putanother way, given the set of all technically feasible combinations of output and inputs, only thecombinations encompassing a maximum output for a specified set of inputs would constitute theproduction function. Alternatively, a production function can be defined as the specification of theminimum input requirements needed to produce designated quantities of output, given availabletechnology. It is usually presumed that unique production functions can be constructed for every

production technology.By assuming that the maximum output technologically possible from a given set of inputs is achieved,economists using a production function in analysis are abstracting from the engineering and managerialproblems inherently associated with a particular production process. The engineering and managerialproblems oftechnical efficiency are assumed to be solved, so that analysis can focus on the problems ofallocate efficiency. The firm is assumed to be making allocate choices concerning how much of eachinput factor to use and how much output to produce, given the cost (purchase price) of each factor, theselling price of the output, and the technological determinants represented by the production function. Adecision frame in which one or more inputs are held constant may be used; for example, (physical) capital
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may be assumed to be fixed (constant) in the short run, and labor and possibly other inputs such as rawmaterials variable, while in the long run, the quantities of both capital and the other factors that may bechosen by the firm are variable. In the long run, the firm may even have a choice of technologies,represented by various possible production functions.

The relationship of output to inputs is non-monetary; that is, a production function relates physical inputsto physical outputs, and prices and costs are not reflected in the function. But the production function is

not a full model of the production process: it deliberately abstracts from inherent aspects of physicalproduction processes that some would argue are essential, including error, entropy or waste. Moreover,production functions do not ordinarily model thebusiness processes, either, ignoring the role ofmanagement. (For a primer on the fundamental elements of microeconomic production theory, seeproduction theory basics).

The primary purpose of the production function is to address allocative efficiency in the use of factorinputs in production and the resulting distribution of income to those factors. Under certain assumptions,the production function can be used to derive a marginal product for each factor, which implies an idealdivision of the income generated from output into an income due to each input factor of production.

Specifying the production function

A production function can be expressed in a functional form as the right side ofQ = f(X1,X2,X3,...,Xn)

where:

Q = quantity of output

X1,X2,X3,...,Xn = quantities of factor inputs (such as capital, labour, land or raw

materials).

IfQ is not a matrix (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not encompassjoint production, which is a production process that has multiple co-products. On the other hand, iffmaps

from Rn to Rk then it is a joint production function expressing the determination ofkdifferent types ofoutput based on the joint usage of the specified quantities of the n inputs.

One formulation, unlikely to be relevant in practice, is as a linear function:

Q = a + bX1 + cX2 + dX3 + ...

where a,b,c, and dare parameters that are determined empirically.

Another is as a Cobb-Douglas production function:

The Leontief production function applies to situations in which inputs must be used in fixed proportions;starting from those proportions, if usage of one input is increased without another being increased, outputwill not change. This production function is given by

Other forms include the constant elasticity of substitution production function (CES), which is a

generalized form of the Cobb-Douglas function, and the quadratic production function. The best form of
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the equation to use and the values of the parameters (a,b,c,...) vary from company to companyand industry to industry. In a short run production function at least one of theX's (inputs) is fixed. In thelong run all factor inputs are variable at the discretion of management.

Production function as a graph

Quadratic Production Function

Any of these equations can be plotted on a graph. A typical (quadratic) production function is shown inthe following diagram under the assumption of a single variable input (or fixed ratios of inputs so the canbe treated as a single variable). All points above the production function are unobtainable with currenttechnology, all points below are technically feasible, and all points on the function show the maximumquantity of output obtainable at the specified level of usage of the input. From the origin, through pointsA, B, and C, the production function is rising, indicating that as additional units of inputs are used, thequantity of output also increases. Beyond point C, the employment of additional units of inputs producesno additional output (in fact, total output starts to decline); the variable input is being used too intensively.With too much variable input use relative to the available fixed inputs, the company is experiencing

negative marginal returns to variable inputs, and diminishing total returns. In the diagram this isillustrated by the negative marginal physical product curve (MPP) beyond point Z, and the decliningproduction function beyond point C.

From the origin to point A, the firm is experiencing increasing returns to variable inputs: As additionalinputs are employed, output increases at an increasing rate. Both marginal physical product (MPP, thederivative of the production function) and average physical product (APP, the ratio of output to thevariable input) are rising. The inflection point A defines the point beyond which there are diminishingmarginal returns, as can be seen from the declining MPP curve beyond point X. From point A to point C,
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the firm is experiencing positive but decreasing marginal returns to the variable input. As additional unitsof the input are employed, output increases but at a decreasing rate. Point B is the point beyond whichthere are diminishing average returns, as shown by the declining slope of the average physical productcurve (APP) beyond point Y. Point B is just tangent to the steepest ray from the origin hence the averagephysical product is at a maximum. Beyond point B, mathematical necessity requires that the marginalcurve must be below the average curve (Seeproduction theory basics for further explanation.).

Stages of production

To simplify the interpretation of a production function, it is common to divide its range into 3 stages. InStage 1 (from the origin to point B) the variable input is being used with increasing output per unit, thelatter reaching a maximum at point B (since the average physical product is at its maximum at that point).Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm willalways operate beyond this stage.

In Stage 2, output increases at a decreasing rate, and the average and marginal physical product aredeclining. However, the average product of fixed inputs (not shown) is still rising, because output is risingwhile fixed input usage is constant. In this stage, the employment of additional variable inputs increasesthe output per unit of fixed input but decreases the output per unit of the variable input. The optimuminput/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 1. In Stage 3, too much variableinput is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense thattheir presence on the margin obstructs the production process rather than enhancing it. The output per unitof both the fixed and the variable input declines throughout this stage. At the boundary between stage 2and stage 3, the highest possible output is being obtained from the fixed input.

Shifting a production function

By definition, in the long run the firm can change its scale of operations by adjusting the level of inputsthat are fixed in the short run, thereby shifting the production function upward as plotted against thevariable input. If fixed inputs are lumpy, adjustments to the scale of operations may be more significantthan what is required to merely balance production capacity with demand. For example, you may only

need to increase production by a million units per year to keep up with demand, but the productionequipment upgrades that are available may involve increasing productive capacity by 2 million units peryear.
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Shifting a Production Function

If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduceits scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, theproduction function will shift down. The beginning of stage 2 shifts from B1 to B2. The (unchanged)profit-maximizing output level will now be in stage 2.

Homogeneous and homothetic production functions

There are two special classes of production functions that are often analyzed. The production function Q=f(X1,X2) is said to be homogeneous of degree n, if given any positive constant k,f(kX1,kX2) = k

nf(X1,X2).Ifn > 1, the function exhibits increasing returns to scale, and it exhibits decreasing returns to scale ifn 1, decreasing ifb + c+ ... < 1, and constant ifb + c + ... = 1.

If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". Alinearly homogeneous production function with inputs capital and labour has the properties that themarginal and average physical products of both capital and labour can be expressed as functions of thecapital-labour ratio alone. Moreover, in this case if each input is paid at a rate equal to its marginalproduct, the firm's revenues will be exactly exhausted and there will be no excess economic profit.

Homothetic functions are functions whose marginal technical rate of substitution (the slope of theisoquant, a curve drawn through the set of points in say labour-capital space at which the same quantity ofoutput is produced for varying combinations of the inputs) is homogeneous of degree zero. Due to this,along rays coming from the origin, the slopes of the isoquants will be the same. Homothetic functions areof the formF(h(X1,X2)) whereF(y) is a monotonically increasing function (the derivative ofF(y) ispositive (dF/ dy > 0)), and the function h(X1,X2) is a homogeneous function of any degree.

Aggregate production functions

In macroeconomics, aggregate production functions for whole nations are sometimes constructed. Intheory they are the summation of all the production functions of individual producers; however there aremethodological problems associated with aggregate production functions, and economists have debatedextensively whether the concept is valid.

Criticisms of production functions

There are two major criticisms of the standard form of the production function. On the history ofproduction functions, see Mishra (2007).

On the concept of capital

During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production

functions. (See the Capital controversy.) Although the criticism was directed primarily at aggregateproduction functions, microeconomic production functions were also put under scrutiny. The debatebegan in 1953 when Joan Robinson criticized the way the factor input capital was measured and how thenotion of factor proportions had distracted economists.

According to the argument, it is impossible to conceive of capital in such a way that its quantity isindependent of the rates ofinterest and wages. The problem is that this independence is a precondition ofconstructing an isoquant. Further, the slope of the isoquant helps determine relative factor prices, but thecurve cannot be constructed (and its slope measured) unless the prices are known beforehand.
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On the empirical relevance

As a result of the criticism on their weak theoretical grounds, it has been claimed that empirical resultsfirmly support the use of neoclassical well behavedaggregate production functions. Nevertheless, AnwarShaikhhas demonstrated that they also has no empirical relevance, as long as alleged good fit outcomesfrom an accounting identity, not from any underlying laws of production/distribution.

Natural resourcesOften natural resources are omitted from production functions. When Solow and Stiglitz sought to makethe production function more realistic by adding in natural resources, they did it in a manner thateconomist Georgescu-Roegen criticized as a "conjuring trick" that failed to address the laws ofthermodynamics, since their variant allows capital and labour to be infinitely substituted for naturalresources. NeitherSolow norStiglitz addressed his criticism, despite an invitation to do so in theSeptember 1997 issue of the journal Ecological Economics. For more recent retrospectives, see Cohenand Harcourt [2003] and Ayres-Warr (2009).
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