Lecture notes

Lecture Notes

– Intermediate Microeconomics

Xu [email protected]

Department of Economics, Texas A&M University

November 12, 2010

Contents

1 Introduction 5

1.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Demand-supply analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Consumer Behavior 11

2.1 Preference and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Marginal Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Indifference Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Marginal Rate of Substitution . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Producer Behavior in the Competitive Market 27

3.1 Producer Behavior with single input . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Producer Behavior with two inputs . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Production Technology with two inputs . . . . . . . . . . . . . . . . . 41

3.2.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Long Run Equilibrium in Competitive Market . . . . . . . . . . . . . . . . . . 47

3

4 CONTENTS

4 Monopoly 53

4.1 Question in front of a monopolist . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Profit Maximization for a monopolist . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Profit Maximizing Condition . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Duopoly 61

5.1 Cournot Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Stackelberg Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Bertrand Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 1

Introduction

1.1 Prologue

This short article serves as an introduction to the course ECON 323, Intermediate Microe-conomics. In this article, I will present the major topics to be discussed in this course andwill review some basics of the demand-supply analysis.

The goal for economics is to understand how economy works. The approach we take is verysimilar to the one used by natural scientists. In order to understand the aggregate ”behavior”of the economy, we first examine the working of its components,i.e., various types of market,such as commodity markets, labor market, capital market, etc.1 After we obtain a goodunderstanding of how each market works, we investigate how one market is linked to theothers. By examining the inter-dependence of different markets, we are able to discuss theworking of the whole economy.

In this course, we only concentrate on the working of markets, mainly the commodity market.The analysis of how one market is linked to the others is left to the intermediate Macro course.

Again, the approach we take to understand the working of a market is by examining itscomponents,i.e., people who participate in the market. We group them by the functionthey play. The most relevant classification for us is ”buyer” and ”seller”, or ”consumer” and”producer”. Thus the first topic is the theory which explains how an individual consumer acts(Part II) and how an individual producer acts(Part III). On the basis of this, we investigate theinteractions between consumers and producers and try to understand how price is determined.The mechanism under which consumers interact with producers depends on the structure ofmarket, which also influences the determination of price.

Let me give you an example to illustrate this point.

Suppose there are two isolated islands . In island A, there is a lake rich in fishes. In island B,there is a field of corn. Let us call the people who live in island A fishermen, and in island Bfarmers. Even though two islands are separated, in between there lies a small island whichhas no inhabitant. Since fishermen in island A also want to eat some corn and farmers in

1Sometime this methodology is called Reductionism

5

6 CHAPTER 1. INTRODUCTION

island B want to eat some fish, they come to the small island between them to trade. Thisis the ”market”.

• If the corn and the fish traded in the market are homogeneous, i.e., without any typesof difference, such as quality(A very unrealistic assumption.), and if there is a largenumber of fishermen and farmers in the market so that no one alone has the powerto influence the market price, we think this market is competitive. The device we useto analyze the working of the market here is Demand-Supply analysis. We will reviewthe basics later. What you have not learned is how the demand and supply curves arederived, which will be discussed in the first topic.

• What if the corn and fish traded in the market are not homogeneous? For example,some farmers in island B are able to produce the corn of better quality than the rest.In this case how the market functions?

• What if fishermen in island A act collusively, i.e., forming a monopoly for fish or amonopsony for corn? In this case fishermen as a whole act as one person, having thepower to influence the market price. In part IV and V we will deal with this type ofsituation.

1.2 Demand-supply analysis

In this section, we will start with an example as an illustration of what demand curve andsupply curve are, and then I will elaborate a little bit on the concept of equilibrium andcomparative static analysis. In the end, we will discuss the concept of elasticity.

1.2.1 Equilibrium

From our two-island story, we have seen that the fishermen in island A and the farmers inisland B are willing to trade with each other. The fishermen in island A are the buyers forcorn and the suppliers for fish. Now, let us consider the market for corn. The followingschedule tells us what the total amount of corn demanded by the fishermen in island A isunder a given price.

Price(Unit: fish per corn) Quantity demanded for corn1 102 83 64 45 26 0

This schedule tells us that when farmers in island B ask one fish for each corn the fishermenas a whole in island A want to have 10 corns; if the farmers ask 3 fishes for each corn, the

1.2. DEMAND-SUPPLY ANALYSIS 7

fishermen will only demand 6 corns. We can notice that as the price decreases, the quantitydemanded for corn increases.

We can represent this schedule by a linear function, i.e.,

Qdc = 12− 2Pc

, where Qdc is the quantity demanded for corn and Pcis the price of corn in terms of fish.

We can also draw this schedule into a two-dimensional curve. In the horizontal axis, we useQ to denote the quantity demanded. In the vertical axis, we use P to denote the price. Eachpair in the table above corresponds to a point in such a two-dimensional plane. We draw aline which transverses all the points, which gives us the demand curve for corn. It should bedownward sloping, which says the higher the price the lower the quantity demanded.

Now let’s consider the farmers in island B, the suppliers of corn. The following schedule tellsus what the total amount of corn supplied by the farmers in island B is under a given price.

Price(Unit: fish per corn) Quantity supplied for corn1 22 43 64 85 106 12

This schedule tells us that when the fishermen in island A offer one fish for each corn thefarmers as a whole in island B are willing to provide 2 corns; if the fishermen offer two fishesfor each corn, the farmers will increase their supply to 4 corns. We can notice as the pricedecreases, in contrast with the demand curve, the quantity supplied decreases as well. We candraw a line to represent such schedule in a two-dimensional plane as we do for the demandschedule. And the functional representation is,

Qsc = 2Pc

, where Qdc is the quantity supplied for corn.

Now, look at two schedules. We notice that only at price 3 fishes per corn, they can reach anagreement in the sense that the quantity demanded is equal to the quantity supplied. Whatif the price is one fish per corn? When the price is 1 fish per corn, the fishermen in islandA as a whole want to have 10 corns while only 2 corns are provided by the farmers. Thisimplies under this price the need of some fishermen in island A who want to buy some cornis not satisfied. What should they do? They can go to negotiate with the farmers. Fromthe demand curve, we see that some of them are willing to offer a higher price for each corn.By offering a favorable term(a higher price), the unsatisfied fishermen will find some farmersin island B who want to provide more. This process will continue until the price reaches 3fishes per corn. What if the price is 5 fishes per corn? In this case, the fishermen demandless than the farmers are willing to provide. Thus some farmers can not sell out their corns.By offering a lower price, they will have more buyers. This process will continue until the


price reaches 3 fishes per corn. Only when the price is exactly 3 fishes per corn, everyone issatisfied.

Formally, we call the price as the equilibrium price provided it makes the quantity demandedequal to the quantity supplied. (Warning: I did not say the demand curve is equivalent withthe supply curve.) In the graph, the intersection of the demand curve and the supply curvegives us the equilibrium price and quantity.

Equilibrium in general is the situation where no individual has any incentive to change theirdecisions. In this case, equilibrium is the situation when quantity demanded is equal to thequantity supplied. Most of economists believe that the equilibrium state is a resting pointsuch that the economy will finally achieve. Whenever the economy is not at the equilibriumstate, there always exists a tendency for the economy to move toward the equilibrium. Thushistorically economists also call the equilibrium state as the stationary state.

Take the market for corn we analyzed above for example. Only when the price is 3 fishes percorn everyone is satisfied and no one is willing to deviate from that state. When the price isnot 3 fishes per corn, either the buyers will offer a higher price or the suppliers will requesta lower price. In other words, the tendency is present toward the equilibrium price 3 fishesper corn.

We have mentioned that as long as the equilibrium state is reached, the economy will staythere forever. Does that mean the economy will not change at all after that? How can we usesuch a static method to analyze a changing world? And nobody will believe the existence ofsuch a non-changing world. The world is always changing. So how?

1.2.2 Comparative Statics

Before answering this question, it is necessary to introduce a pair of concepts exogenous andendogenous variables. In this world, we have some observations, and they make us curiousabout why they are so. Therefore, theorists are trying to set up models to explain them. Ina model, there are some variables we are trying to know how their equilibrium values aredetermined. We call them endogenous variables. And there are some other variables whichfor the present purpose we are not trying to explain but instead whose value we take asgiven. In our two-island example, even though we did not specify what exogenous variablesare, they actually determine the demand and supply schedules. For instances, the weatherdetermines how many corn the farmers can collect each day which affects how many cornthey want to trade regardless of the price. Given the demand schedule and supply schedule,we can determine the equilibrium price and quantity. So you see the price and quantity inour case are the endogenous variables.

Come back to the question: How can we use such a static method to analyze the phenomenain a changing world?

We first classify all the changes as changes in exogenous variables. As the exogenous variablesvary, the demand and supply curve might shift (NOT the change in quantities !), which inthe end determines the new equilibrium price. This type of process is called the ComparativeStatic analysis.

Let’s see some examples.

1.2. DEMAND-SUPPLY ANALYSIS 9

1.2.3 Elasticity

In this section, we briefly review a useful concept, elasticity. Here we focus on the priceelasticity. Price elasticity of demand (supply) measures the sensitivity of quantity demanded(supplied) in response to the changes in price. For example, in the case of the market for cornfrom our two-island story, you might want to know if the price falls 1% by what percentagethe quantity demanded will increase and the quantity supplied will fall. If the quantitydemanded (supplied) increases (falls) to a large degree in the percentage sense, we call thedemand (supply) is fairly sensitive to the price and the demand curve is elastic. Otherwisewe call it inelastic.

Formally, the following is the definition,

ε =∆Q/Q∆P/P

Remarks:

• In most of cases, the price elasticity of demand is negative due to the ”Law of demand”,which says as the price increases, the quantity demanded falls. But it is NOT alwaystrue. We will see some counter examples in Part II.

• More precise definition for elasticity calls for the use of calculus. See Textbook Page46 footnote 1.

• The price elasticity is evaluated at some point. (See Example below for details.)

• When the absolute value of the elasticity is large, it implies the demand curve or supplycurve at the point where the elasticity is evaluated is elastic.

|ε| = 1 0 ≤ |ε| < 1 |ε| > 1Unitary elasticity Inelastic Elastic

Example:

Price(Unit: fish per corn) Quantity demanded for corn Elasticity

2 8 −2/81/2 = −0.5

3 6 −2/61/3 = −1

4 4 −2/41/4 = −2

5 2 −2/21/5 = −5

Price(Unit: fish per corn) Quantity supplied for corn Elasticity

2 4 2/41/2 = 1

3 6 2/61/3 = 1

4 8 2/81/4 = 1

5 10 2/101/5 = 1

Chapter 2

Consumer Behavior

In this article, we will focus on the theory which explains how consumers make their choices.Let’s first take a look at the big picture of the theory.

In front of the consumers, there are a bunch of choices for them to pick. Take a collegestudent for example. He can have a lunch at Subway, Mcdonald’s, PizzaHut, or Jin’s Cafe.He can buy a Honda, toyota, or Fold.... But not all the choices are affordable for him.Theset of the choices that consumers can pick from is restricted by the resources they own,for example, money. We think they have a preference ordering over all the choices. Thepreference ordering says choice A is better than choice B,choice C is worse than choice D,etc. According to such preference ordering, they will pick a ”best” affordable choice.

In a word, somehow we believe that individuals behave as if they were maximizing thesatisfaction resulted from their actions. Another way of saying is that they are calculatinggains and pains when they are making choices, and choose the best one to maximize the gainand minimize the pain.

A set of questions might arise: what do I mean by gains and pains? materialistic or psy-chological? Does the theory suggests that people only care about money? How to explaingenerous donations and charity activities?

The theory does not say anything about what kinds of gains and pains that people arecalculating at all. It can be materialistic or psychological. Neither does the theory suggestthat different people have the same preference. What kind of preference that people haveis not the question that economists are trying to answer. Economists only assume thereexists such a preference ordering but the specific content is left to be open so that it canaccommodate the variety of tastes among people.

ExampleSuppose one fisherman in island A on July 3rd has five fishes, and the market price of fish is2 corns per fish. Then we know the affordable choices for him are as follows.

11

12 CHAPTER 2. CONSUMER BEHAVIOR

Fish Corn0 101 82 63 44 25 0

Notice that choices like 6 fishes, 4 fishes and 3 corns, or 11 corns are not affordable. Why?suppose the man wants to have 4 fishes and 3 corns. He has five corns. Eat 4 of 5, only 1fish is left. Since the price of fish is 2 corns per fish, the maximal amount of fish he can haveis 2. Thus 4 fishes and 3 corns are not affordable for him.

Now suppose he has the following preference ordering over the choices.

Desirability Fish CornSuper Best 4 2Second Best 3 4

good 5 0OK 2 6

Just so so 1 8Worst 0 10

According to the table above, the optimal choice for the man in island A is to demand 4fishes and 2 corns when the price of fish is 2 apples per fish.

You may notice that the optimal choice made by the man is essentially dependent on threethings.

1. subjective valuation over the choices : this provides a criterion for individuals to decidewhich choice is best for them.

2. the market price: this determines affordable choices that individuals can pick.

3. the initial endowment(or income) (5 fishes in this case.)

To address this point, we can think of the following changes. Suppose now, the market priceof fish is 3 corns per fish. The set of all affordable choices for the man in island A is changedto the one in below.

Fish Corn0 151 122 93 64 35 0

2.1. PREFERENCE AND UTILITY 13

Since now the man can choose some combinations of fish and corn which are not affordableunder the previous price level,i.e., 2 corns per fish, the optimal choice he makes will bedifferent from the one he made before. Now consider a different change. Suppose the priceof fish is still 2 corns per fish, but the man in island A is endowed with 6 fishes. The new setof all affordable choices is as follows.

Fish Apple0 121 102 83 64 45 26 0

Again, the total resources also affect the set of all affordable choices.

In below, we study the preference ordering in more details. And then we move on to discussthe budget constraint which shapes the set of choices affordable for individuals. In the end,we explain how to derive optimal choices.

2.1 Preference and Utility

Let X =choices, denotes the set of all the possible choices. In this course, we only considertwo-commodity case. Thus, the elements in X are pairs. In our two-island example, there aretwo commodities, fish and corn. In this case, the set of all possible choices contains elementslike, (5 fishes, 1 corn), (3 fishes, 2 corns), etc. Formally, X = {(a, b) : a, b ∈ R+}, where R+

denotes nonnegative real numbers. The first coordinate indicates the amount of fish and thesecond the amount of corn. For example, 5 fishes and 1 corn can represented by (5,1). Thus,we can associate each element in Xwith a point in a two-dimensional plane.

In the rest of the course, we assume divisibility of the quantity of goods. In other words, anyreal number can denotes certain quantity of one good, even though we know 1/3 of a car isnot a car any more, and can be not sold in the real world. However, we have this assumptionfor the convenience of our analysis. Therefore, X = {(a, b) : a, b ∈ R+}, where R+ denotesnonnegative real numbers. Graphically, X is simply the first quadrant.

Preference is defined over X which specifies a relation between two pairs. For example,(4,2) is better than (3,4), which says 4 fishes and 2 corns combination is better than thecombination of 3 fishes and 4 corns. Or (3,4) and (5,0) are the same, which means 3 fishesand 4 corns are the same with 5 fishes. Note, different people might have different preferenceorderings. For example. One guy might prefer the combination (3,4) to (4,2) while the otherguy might choose the opposite. Thus in essence, the preference order reflects people’s tastesand subjective valuation of commodities.

There are several axioms on preference.We require all preferences should satisfy following fiveaxioms.


1. completeness All pairs in X are comparable.

2. transitivity If pair A is better than pair B, and pair B is better than pair C, thenpair A is better than pair C. For example, for someone if the 4 fishes and 2 applescombination is better the combination of 3 fishes and 4 apples, and 3 fishes and 4apples combination is better than 5 fishes, then the 4 fishes and 2 apples combinationis better than 5 fishes.

3. continuity A technical condition which preference is representable via a real function.(Not required to know. If interested, come to me.)

4. monotonicity The more the better. For example, (4,2) is better than (1,1).

5. convexity Diversification is desirable. Technically, it says for example, (1,2) is betterthan both (0,4) and (2,0) when (0,4) and (2,0) are the same. In other words, if you areindifferent to eat 4 fishes and 2 apples, the mixture of them makes you happier. Noticethat 1/2 of (0,4) and 1/2 of (2,0) is (1,2).

With axiom 1-3, we can prove there exists a function u : X → R such that function upreserves the ordering. We call such function, utility function, which assigns a real numberto each pair, and call such number as the utility from consuming such pair. For example:

pairs(fish,apple) utility(5,0) 6(3,4) 10(4,2) 12(4,5) 15

... ...

Thus, we see u(5,0)=6, u(3,4)=10.

What do I mean by ” utility function preserves the ordering?” If pair (4,2) is better than(3,4), then u(4,2)=12 > u(3,4)=10. Formally, if according to the preference ordering pair Ais better than pair B , then u(A) > u(B). Thus the utility actually represents the satisfactionfrom consuming one certain combination of goods. If you still remember, I mentioned thetheory somehow suggests that individuals behave as if they were maximizing the satisfactionthey gain from consuming. Here we can see that maximizing the satisfaction is equivalentwith maximizing the utility. In other words, picking the ” best ” choice is equivalent withchoosing the pair which yields the highest utility.

Now the question is Does it matter if we change the number but keep the relative relationintact? For example,

pairs(fish,apple) utility(5,0) 90(3,4) 100(4,2) 134(4,5) 156

... ...


In this case, you see the utility from consuming 3 fishes and 4 apples is 100, different fromthe previous case. So yes, we have a different utility function, but we did not change therelative relations. For example, (4,2) is still better than (3,4). So even though, two utilityfunctions might give us two different numbers for one pair, as long as they reflect the samepreference ordering, they will induce the same behavior. We will come to this again when wediscuss how to derive the optimal choice.

2.1.1 Marginal Utility

Now we need to introduce the concept of marginal utility. Here I just give you the definition,and we will use this concept later. Marginal utility means the additional amount of utilityyou can gain from consuming one more unit of one good. For example,

pairs(fish,corn) utility Marginal Utility (per fish)(2,4) 4 NA(3,4) 10 6(4,4) 14 4(5,4) 16 2

... ...

In this example, when the amount of fish is increased from 2 to 3, the utility is increasedfrom 4 to 10. 6 per fish is the marginal utility evaluated at pair(2,4). (Warning: when youcalculate the marginal utility for one good, for example fish, you need to keep the amount ofother goods constant, say corn in our case. When you say marginal utility, you should alwaysspecify which commodity you are talking about and it is evaluated at which point.)

Now let’s present the mathematical definition of marginal utility.

2.1.2 Indifference Curve

Now we are ready to introduce the useful tool for our analysis, i.e., indifference curve. Indif-ference curve collects all the pairs which give the same utility.

Example 1: Linear utility function Suppose the utility function is

u(x1, x2) = x1 + x2

, where x1 denotes the quantity consumed of commodity 1 and x2 denotes the quantityconsumed of commodity 2. What this function does is for each pair of x1 and x2 it gives anumber(which is the utility from consuming such pair) by summing up these two numbers.Let’s see some examples to illustrate this.

pairs (commodity 1, commodity 2) utility(5,0) 5(3,4) 7(4,5) 9

... ...


Figure 2.1: Example 1: Linear Indiffernce Curve

Now let’s look for the indifference curve the points on which give utility 10 according thisutility function.


... ...

We can draw this as a straight line in a two-dimensional plane. See Figure 1.

Example 2 Suppose the utility function is

u(x1, x2) = x1 ∗ x2

, where x1 denotes the quantity consumed of commodity 1 and x2 denotes the quantityconsumed of commodity 2.Let’s see the pairs which give utility 10.

pairs (commodity 1, commodity 2) utility(10,1) 10(5,2) 10

(4,2.5) 10... ...

Example 3: Leontif Utility Function Suppose the utility function is

u(x1, x2) = min{x1, x2}

, where x1 denotes the quantity consumed of commodity 1 and x2 denotes the quantityconsumed of commodity 2. What this function does is for each pair of x1 andx2it gives theminimum of the two. For example, u(3, 4) = 3, and u(4, 5) = 4.


Figure 2.2: Example 2

Let’s see the pairs which give utility 10.


... ...(10,11) 10(10,12) 10(10,13) 10

... ...

Remarks on indifference curve.

1. Two indifference curves which have different utility level never intersect.

2. The indifference curve which has higher utility level will always lies above the one whichhas lower utility level. We have this because we assume ”the more the better”.

2.1.3 Marginal Rate of Substitution

Now we are ready to introduce the concept, Marginal Rate of Substitution, (MRS).

If one unit of a good is given up, in order to keep the utility the same MRS means theamount of the other good that needs to increase to compensate the loss. This implies we can


Figure 2.3: Example 3: Leontif Indifference Curve

calculate MRS from the indifference curve since along an indifference curve the utility is thesame.

Shape of Indifference CurveThe next question is the shape of indifference curve. In the rest of the course, most of time,we assume the strict convexity of the preference, therefore, the indifference curve should havethe shape similar to the one we see in Example 2. We have already mentioned the economicmeaning of the convexity of the preference. Now let’s see the consequence of this assumptionon the MRS.

Marginal Rate of Substitution and Marginal UtilityNow it is a good place to give you the formula to calculate MRS. My purpose is not to teachyou the math, but to link the concept of marginal utility with MRS.

MRS1→2 =MU1

MU2

, where MRS1→2 means the marginal rate of substitution of commodity 1 with respect tocommodity 2, and MU1 and MU2 means the marginal utility of commodity 1 and commodity2 respectively. WHY?

suppose now you consume one additional unit of commodity 1, how much utility you gain?That is MU1 by the definition of marginal utility. In order to keep the utility the same, youhave to reduce some amount of commodity 2. How many? First you have to reduce MU1

this much of utility resulted from consuming one more unit of commodity 1. We know if youreduce one unit of commodity 2, we will lose MU2 this much of utility, and therefore, to lose1 unit of utility, we have to reduce 1

MU2this amount of commodity 2. Thus to reduce MU1

this much of utility, you have to reduce MU1× 1MU2

this amount of commodity 2. And wait,the amount of commodity 2 that needs to be reduced if one more unit of commodity 1 isincreased in order to keep the utility the same, emmmmm...., what is that? Oh that is MRS.:-).

2.2. BUDGET CONSTRAINT 19

Figure 2.4: The Budget Set

2.2 Budget Constraint

In this section, we learn how to derive the set of all affordable choices for an individual, andthe set is called the budget set. There are two things exogenous for them, the prices andtotal resources.

Let’s first take a look at the general setup for deriving the budget set. Suppose the person weare considering has income I in terms of dollars. And there are commodities in front of him.We use X1 and X2 to denote the amount of commodity 1 and commodity 2 respectively. Weuse P1 and P2 to denote the price of commodity 1 and commodity 2 in terms of dollars. It isnot hard to realize that total use can not exceed the total resources. That says money spenton commodity 1 and commodity 2 can not exceed the total income, I. Thus, we should have,

P1 ×X1 + P2 ×X2 ≤ I

This inequality restricts the choices of X1 and X2. Let’s represent this in a two-dimensionalplane. Recall, we have assumed that ”the more the better”, which implies the individualswill always choose the combination on the boundary of the budget set, i.e., the budget line.(see Figure 4)

It is the good point to introduce the concept of relative price. The relative price of com-modity 1 in terms of commodity 2 is the maximal amount of commodity 2 you can have if


you give up one unit of commodity 1. In our general setup, the relative price of commodity1 in terms of commodity 2 is P1

P2 . Why?

If you give up one unit of commodity 1, then we save P1dollar, and you can use this many ofdollars to buy commodity. We know with 1 dollar, we can buy 1

P2this amount of commodity

2. And then with P1dollar, we can buy P1 × 1P2

this amount of commodity 2. By definitionthis is relative price of commodity 1 in terms of commodity 2. Graphically what is P1

P2 ? Thatis the absolute value of the slope of the budget line!!

Now I want to talk about the changes in the budget set. As I mentioned before, in generalthere are two things which affect the budget set, prices and income. We are considering fourtypes of changes.

2.2.1 Applications

1. Intertemporal substitution

Let’s imagine the following situation. When you are just born, you are thinking you mightearn some money when you are young, say I, and you would earn nothing when you are old.And you notice there is a program which helps you save some money. And this programpromises to pay you an interest when you are old for each dollar you invest into it when youare young.We use R to denote the gross interest rate, which means, if you invested 1 dollarwhen you were young, it will return you R dollar including the principal plus the interests.Suppose there is only one good to consume, and we use Cy to denote the quantity of the goodthat you consume when you were young and Co to denote the quantity of the good that youconsume when you are old. And the price of this good is 1 $ per unit and remains the samethroughout your life time. So the situation looks like as follows,

young oldIncome($) I 0consumption(quantity) Cy Co

price of consumption goods($ per unit) 1 1

The question is to write down the life-time budget constraint you are facing. When you areyoung, there are two options for you, spend some money on the consumption and save somemoney. Let’s use S to denote saving expressed in terms of dollars. Therefore, the moneyspent on the consumption plus the saving should not exceed the total income I, i.e.,

1× Cy + S ≤ I

, where recall the price of the good is 1 $ per unit. When you are old, the total resourcesavailable to you are saving times the gross interest rate, i.e., R× S. Therefore, we see

1× Co ≤ R× S

, which says the money spent on consumption when we are old can not exceed the savingplus interest earnings. Now do some algebra, we combine two inequalities, and we get,

Cy +Co

R≤ I

2.2. BUDGET CONSTRAINT 21

Basically, there are two things in the individual’s mind who are making such intertemporaldecision, how much to consume today and how much to consume tomorrow. Definitely theirdecision depends on how impatient they are for delaying the consumption. If I am tellingyou now I’ll give you a brand new car tomorrow, you probably will be quite excited. Whatif I am telling you yes I’ll give you a new car, but 20 years from now? I bet you will be lessexcited. On the other hand, their decision also depends on how much they will ”lose” if theychoose to consume earlier, since if they postpone their consumption and save them, they willalways earn some interests later. And this is captured in the gross interest rate. When theinterest rate is high, they will lose more if they choose to consume earlier. So in this sense,the price of consuming today in terms of consuming tomorrow is the gross interest rate. Weoften call this the relative price of consuming today to consuming tomorrow. According tothe definition of relative price, why so? Suppose you give up one unit of consumption today,how much you save? 1 × 1 = 1 dollar. And you can invest this amount of money into theprogram which returns you 1 × R dollars when you were old. And how many consumptiongoods you can buy by using this amount of money? That is R

1 . Thus the relative price ofconsuming today to consuming tomorrow should be R, which is the gross interest rate.

2. Consumption and Leisure

Suppose now you are in the following situation. We all know each day there are 24 hours.Let T = 24. And suppose w denotes the wage rate, $ per hour. We assume there is only onegood to consume and the price of it is 1 $ per unit. And we use C to denote the quantityof the good that you consume and L to denote the hours you choose to have a rest or havea fun, but not work. So the question is to write down the budget constraint. Now we caneasily see the money you earn, that is, (T −L)×w, the hours you work times the wage rate.And the money spent on consumption should not exceed the money you earn. Therefore, wehave,

C ≤ (T − L)× w

Do some algebra, we see that

C + L× w ≤ T × w

Again. What is the price of leisure in terms of dollar? Yes the wage rate. If you choose tosleep at home for a hour, you are actually giving up the opportunity to work for a hour whichearns you 1× w dollar.

Now let’s have some complication.

1. suppose if the working hours exceed 1/3 of T, that is 8 hours, you will be given aone-time bonus say B.

2. suppose if you decide to work outside the regular working hours, that is 8, you are givenextra dollars, say τ , for every hour you work beyond the regular hours.

3. suppose the worker union forces the congress to pass law which forbids the citizens towork longer than 8 hours each day.

2.3. UTILITY MAXIMIZATION 23

Figure 2.6: Utility Maximization: Graphical Presentation

2.3 Utility Maximization

In the section of preference and utility, I have mentioned that the theory suggests thatindividuals behave as if they were maximizing the satisfaction resulted from consuming,and the utility function captures the satisfaction individuals gain from consuming certaincommodity pairs, like 3 fishes and 4 corns, or 4 fishes and 2 corns. So given the set of allpossible choices, the individual will choose a pair which maximize the utility function. Inother words, the individual will choose a pair which has the highest utility among all thosepairs within the budget set. Our goal is to know under certain prices and income what theindividual’s optimal choice will be.

We first take a look at the graphical presentation and then I will give some applications. Inthe end, I give the mathematical presentation.(which is definitely not required but good toknow.)

We have already know the absolute value of the slope of indifference curve evaluated at somepoint is the MRS evaluated at this point. From the graph, we observe that if the point isthe optimal choice, the MRS evaluated at this point, i.e., the absolute value of the slope ofthe indifference curve, is equal to absolute value of the slope of the budget line, which isthe relative price. This equality has quite a lot of economics to learn. Recall the MRS ofcommodity 1 in terms of commodity 2 is equal to MU1

MU2. And the relative price is P1

P2. Then


we see,MU1

MU2=P1

P2

Do some algebra, we see that,MU1

P1=MU2

P2

What is this? If you spent 1 $ on commodity 1, we can have how many commodity 1, that is,1P1

, if you have one additional commodity 1, how much additional utility you can gain, thatis, 1

P1×MU1. In a word, the term MU1

P1means if you spend 1 $ on commodity 1, the amount

of utility you can have. Suppose, MU1P16= MU2

P2, let ’s assume that MU1

P1> MU2

P2. what will you

do? Move 1 $ from consuming commodity 2, and use it to consume commodity 1, becausethat additional dollar will gain your more utility. Utill when you will stop move from one tothe other. utill they are equal.

ExampleNow, let’s see an example. Consider a consumer’ s choice of fish and corn. Suppose his utilityfunction is

u(f, c) = f × c

According to his utility function, the marginal utility of fish MUf can be calculated by usingthe following formula,

MUf = c

AndMUc = f

And thus, MRSf→c the marginal rate of substitution of fish with respect to corn is,

MRSf→c =c

f

Notice this utility does not have the property of decreasing marginal utility butthe property of diminishing marginal rate of substitution.

Suppose the consumer has 8 dollars, and the price of fish is 2 $ per fish and the price of cornis 1$ per corn. The following choices are on the budget line.

pairs(fish,corn) utility MUf MUcMUf

Pf|MUc

PcMRSf→c

(4,0) 0 0 4 0 4 0(3.5, 1) 3.5 1 3.5 0.5 3.5 1

3.5(3, 2) 6 2 3 1 3 2/3

(2.5, 3) 7.5 3 2.5 1.5 2.5 6/5(2, 4) 8 4 2 2 2 2

(1.5, 5) 7.5 5 1.5 2.5 1.5 10/3(1, 6) 6 6 1 3 1 6(0.5,7) 3.5 7 0.5 3.5 0.5 14(0,8) 0 8 0 4 0 ∞

Now let’s have some applications. Back to our example2 (Consumption and Leisure) in theBudget Constrain section.

2.3. UTILITY MAXIMIZATION 25

Figure 2.7: Utility Maximization: Applications

Chapter 3

Producer Behavior in theCompetitive Market

In this lecture, we will see a simple model which attempts to understand producer behaviorin the competitive market.

Let’s first take a look at the big picture of the model. We define a producer as an entity ofa production technology which can transform certain amounts of inputs into certain amountof output. Take a construction company for example. The company needs some constructionworkers first; second it also needs some materials like concrete, woods, iron, etc; third, someconstruction devices like drills, are also needed. All these are inputs for the constructioncompany. And the output will be buildings. We will use a function to describe the quan-titative relation between inputs and outputs, which simply says a certain amount of inputscan produce a certain amount output. We call this production technology. In the model weare considering, the production technology is exogenous for producers. In other words, theproducers will take the technology as given. And the activities like R&D are excluded fromthe model. We are interested in the question–how producers choose the quantity of outputgiven a production technology and the market prices of its outputs and inputs. We thinkthe producers choose the quantity of output so as to maximize its profit. In other words, wethink the profit maximization is the objective that firms are trying to achieve. Is that true?...

ExampleLet’s consider an Island where people want to produce baskets. Suppose in order to producebaskets, they only need labor as input, and he is given the following production technology.

Labor (hours) Basket (quantities)1 22 53 74 85 8.5... ...

27

28 CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET

The table above simply says: if the man works one hour he can make 1 basket; if he workstwo hours, 5 baskets will made. It is often convenient to write down the inverse relation ofproduction which says in order to produce one unit of output the minimal amount of inputis needed. Suppose the inverse relation is as follows,

Basket (quantities) Labor1 50 min2 one hour3 one hour and 15 min4 one hour and 35 min5 two hours... ...

The table above says: if the man wants to make 1 basket he needs to work 50 minutes; if theman wants to make 2 baskets he needs to work one hour.

Now suppose the market price of labor is 1$ per hour and the market price of basket is 0.4 $per basket. The question for us is how much to produce? Now first let’s calculate the cost ofproduction. Since the only input is labor, and we know in order to produce a certain amountof baskets, how many hours are needed, and then we can calculate the market value of thelabor used to produce such amount of baskets.

Basket (quantities) Labor Total Cost (dollars)1 50 min 5/62 one hour 13 one hour and 15 min 1.254 one hour and 35 min 1.595 two hours 2... ... ...

The table above simply says that if 1 basket is produced, one hour of labor is used and it hasthe worth of 5/6 dollars; if two baskets are produced, two hours of labor, worth 1$ are used.At the same time we can calculate if 1 basked is sold, what the revenue will be. Since themarket price of basket is 0.4$ per unit, we know,

Basket (quantities) Revenue (dollars)1 0.42 0.83 1.24 1.65 2... ...

Everyone knows that the profit is just the difference between revenue and cost. Then we seethat,

3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT 29

Basket (quantities) Revenue ($) Cost($) Profit($)1 0.4 5/6 -0.432 0.8 1 -0.23 1.2 1.25 -0.054 1.6 1.59 0.015 2 2 0

From the information we know so far, the man in island C will choose to produce 4 baskets,which gives the highest profit.

This example illustrates how we will proceed in later sections. First we study the productiontechnology from which we can derive the cost function for the producer. And then we moveon to characterize the profit-maximizing choice.

Now one more problem is left before we move on. In the title of this lecture, you see a term ”Competitive Market”. What do I mean by that? How that influences our analysis?. Actuallyit is a big assumption. We will discuss this issue later in profit maximization section.

3.1 Producer Behavior with single input

In this section, we only consider the case where there is only one input for the production. Wecan describe a production technology basically in two ways: write the production functiony = f(x) and draw the curve in the graph. In this section, we only give the graphicalpresentation.

First let’s consider the concept of marginal product (or sometimes we call marginal return).The marginal product of the input is just the additional amount of output which can beproduced if the one more unit of the input is added. Now suppose I give you a productiontechnology presented by a curve. How can you find out the marginal product? Take thefollowing curve for example. (Figure 1)

We can see that the slope of the production function is just the marginal product. If themarginal product is decreasing as the quantity of input is increasing, we call this productiontechnology has the diminishing marginal return property. Graphically, that means, the slopeof the production function will be decreasing. If the marginal product is increasing as thequantity of input is increasing, we call this production technology has the increasing marginalreturn property. The economic meaning of diminishing marginal return is that as the quantityof input used for production is increasing, its productivity is actually decreasing, since itgenerates less output if one more unit of input is used.

Now let me give you all the possible production technology we will consider in this section.See Figure 2.

Here, I want to introduce the concept of inverse function of production function, say, x =f−1(y), where x is input and y is output. The meaning of this function is that given a certainlevel of output,y, the minimal amount of input x is needed to produce such amount of output.


Figure 3.1: Marginal Product : An example

Figure 3.2: Production Function


Figure 3.3: Inverse Function

Figure 3.4: Inverse Function


The next question is: given a production function, how can I draw the inverse function of itin the graph. See Figure 4.

3.1.1 Cost

There are several types of cost. There are three relevant for us. The important thing aboutthem is that they are all functions of the quantity of output. In other words, they will varyas the quantity of output changes.

Total CostTotal cost: as its name suggests, it just the total cost associated with production. In ourcase, it only comes from the cost of buying input. Without loss of generality, we can assumethe price of input is 1 $ per unit. Therefore, the total cost of production is just 1 times thequantity of input used to produce certain amount of output.Back to our basket-producing example.

Basket (quantities) Labor Total Cost (dollars)1 50 min 5/62 one hour 13 one hour and 15 min 1.254 one hour and 35 min 1.595 two hours 2

From the example above, we can see that the inverse function of production is used to cal-culate the total cost.

Average CostAverage cost means in average what is the cost for producing one unit of output, i.e., thetotal cost/the number of output.Example:

Basket (quantities) Labor Total Cost ($) Average Cost($ per unit)1 50 min 5/6 5/62 one hour 1 0.53 one hour and 15 min 1.25 0.424 one hour and 35 min 1.59 0.39755 two hours 2 0.4

Marginal CostMarginal cost means the additional cost if one more unit is produced.Example:


Figure 3.5: Cost: Example1

Basket (quantities) Labor Total Cost (dollars) Marginal Cost(Dollars per unit)1 50 min 5/6 NA2 one hour 1 1/63 one hour and 15 min 1.25 0.254 one hour and 35 min 1.59 0.345 two hours 2 0.41

Now the question is: graphically given a production function how to figure out the costfunction. And then how to calculate the average cost and marginal cost. We consider twospecial cases. See Figure 5 and Figure 6. From Figure 6, we see there is a well-known resultbetween average cost and marginal cost. In the presence of fixed cost, the average cost reachesits minimum as it is equal to the marginal cost.


Figure 3.6: Cost: Example2


Figure 3.7: Graphical presentation of Cost

3.1.2 Profit Maximization

I have mentioned in this model we assume that the firms are trying to maximize their profit byusing the quantity of output. Now in this section, we want to have a concrete characterizationas we did in the theory of demand. I said in most of cases, the optimal choice by a typicalconsumer who is trying to maximize his/her utility should satisfy the condition (sometime wecall it the marginal condition), i.e., the marginal rate of substitution is equal to the relativeprice. Here we also have such concrete characterization.

Before going to derive such profit-maximizing condition, let’s represent the profit in the graph.

Everyone knows that the profit is just the difference between revenue and cost. Now let’s seewhat the cost should be associated with a certain level of output. see Figure 7.

Now suppose the market of price of the output is p $ per unit. let’s see what the revenueshould be in the graph when the output is q. see Figure 8.

Now in order to find out the profit, we can combine Figure 7 with Figure 8. See Figure 9.

Now we are ready to derive the condition for profit maximization. let’s first state the condi-tion,

marginal cost = the price of output

, NOTE: we have to pay very attention to the fact that we have assumed the price of inputis one. If the price of input is not one, the condition should be

marginal cost = the relative price of output in terms of input

. I need to give several remarks on this condition:

1. this condition presupposes the existence of maximization. It is possible that the maxi-mum does not exist at all.


Figure 3.8: Graphical presentation of revenue

Figure 3.9: Graphical presentation of profit


2. we know that the marginal cost varies as the quantity of output changes. Therefore wecould find the quantity which satisfies this condition. That quantity is the one whichmaximizes the profit.

3. There is one more assumption for the statement to be true, i.e., we are considering theproducers in the Competitive Market. What do you mean by Competitive Market? If amarket is perfectly competitive, we mean the producers in such market are price-takers.In other words, they ”think” their action can not influence the price. This point is verysubtle. You may say well definitely their action can influence the price; for example, ifone producer cuts its production, the price will go up as long as the demand remains thesame. You are right. They actually can influence the price but I assume they feel thatthey can not. Is that a bad assumption? Not really, when the number of producers inthis market goes to infinity, any single producer has very tiny influence on the marketprice. And therefore they feel they can not influence the price. For example, whenyou go to HEB, when you are buying 1 gallon of milk, are you thinking that if youbuy one more gallon, the price of milk will mark up significantly? I think you won’tbecause your purchasing takes up only a tiny part of the demand. This idea applies tothe producer side as well. If you are a producer of basket in island C, and there aremillions of and tons of competitors out there, will you think you can influence the priceof basket? I don’t think so because your selling only forms a tiny part of the supplyof basket. You will see an amazing justification of this assumption later. I can brieflymention it here. Later we will discuss monopoly. Definitely the monopolist is not aprice-taker, since it is the only producer and thus apparently it can influence the price.Furthermore, when we model the behavior of monopolist, we think they actually setthe price, usually higher than the price in competitive market. And then we will seethe so-called duopoly, where there are two producers. And we can see the market pricein duopoly which is also higher than the price in competitive market but lower thanthe price set by monopolist. Question: as we are adding more producers and as theproducers’ number goes to infinity, will the price converges to the price in competitivemarket? The answer is yes. We will see the details later.

Now let’s give the justification of profit-maximizing condition. As before, we prove by con-tradiction. Suppose it it not true. And suppose

marginal cost < the price of output

In such situation, what the producer will do? If the producer decides to produce one moreunit, the cost associated with this one more unit (NOT the total cost) is just the marginalcost. By selling such one more unit, the additional revenue is just the price of output. Andif marginal cost < the price of output, the producer thinks it is profitable to produce onemore unit, and it gains more profit, and thus if the profit has been achieved the maximum,the marginal cost should NOT be lower than the price of output otherwise there is a roomto gain more profit.And now suppose

marginal cost > the price of output

In such situation, if the producer reduce production of one unit, it saves the cost of producingthat one unit, i.e., the marginal cost, and at the same time it loses the revenue, i.e., the priceof output. Since marginal cost > the price of output, it implies that actually producing


Figure 3.10: Graphical presentation of profit maximization

this unit and selling it does not gain any profit but instead it causes loss. Thus if the profithas been achieved the maximum, the marginal cost should NOT be higher than the price ofoutput otherwise there is a room to save the loss by reducing production of one unit.

Now let’s show graphically the profit-maximizing condition stated above is true. see Figure10.

Now let’s see one exception see Figure 11.

Now let’s have another graphical presentation of the profit maximization. See Figure 12.

3.2 Producer Behavior with two inputs

In this section, we consider the case where the production technology requires two inputs.The same as before, we first specify the production technology, derive the cost function, andthen by profit-maximization hypothesis we find the optimal quantity of output. As long aswe know the cost function, we back to the analysis we have done in the previous sectionwhere there is one single input. There will be no essential difference between this section andthe previous one in the manner of deriving optimal quantity of output. The differences takeplace only in deriving the cost function. Therefore, we only talk about how to derive costfunction when there are two inputs. And then we should be able to know how to find theoptimal quantity of output with two inputs.

Let’s back to our story: the producer of basket. Now suppose producing baskets needs laborand bamboo as inputs.

Suppose the producer has the following technology,

3.2. PRODUCER BEHAVIOR WITH TWO INPUTS 39

Figure 3.11: profit maximization : exception

Figure 3.12: profit maximization : Another Graphical Presentation


labor(hours) bamboo(quantities) Basket(quantities)30 mins 1 1

1 hr 1 11 hr 2 22 hr 3 33 hr 2 21 hr 4 2... ... ...

I want to use the example to illustrate the idea of deriving cost function when there are twoinputs. Recall, the cost function simply says in order to produce certain amount of outputwhat the total cost should be to produce that amount. Suppose now the producer wants toproduce 6 baskets. The first step is to find out all the combinations of labor and bamboosuch that producing 6 units of basket is possible. According to the technology the producerhas, suppose we have the following combinations,

labor(hours) bamboo(quantities) Basket(quantities)3 hr 6 64 hr 6 65 hr 6 66 hr 6 63 hr 7 63 hr 8 6... ... ...

Now suppose if the price of labor is 1$ per hour as before, and the price of bamboo is 0.5$per unit. Then we can know the total cost of all combinations which are able to produce 6units of basket. In contrast with the case where there is one single input, here we have abunch of choices to produce certain amount of output. The question is : what is the rulefor the producer to pick one combination of inputs given the quantity of output? Here weassume that the producer will choose a combination which is the cheapest. In other words,the producer is minimizing the cost while he/her is choosing the combination of inputs toproduce certain amount of output.

labor(hours) bamboo(quantities) Cost ($)3 hr 6 3+3=64 hr 6 4+3=75 hr 6 5+3=86 hr 6 6+3=93 hr 7 3+3.5=6.53 hr 8 3+4=7... ... ...

From the table above, we see that the producer will work 3 hours and use 6 units of bambooto produce 6 baskets, and this costs 6 $ which is the minimal among all other productionchoices.


Now suppose the producer wants to produce 8 units of basket. In this case, we can also findout all the combinations which makes this possible, compare them by their costs, and choosethe cheapest one. Finally we know producing 8 units of baskets will cost the producer 8$.

labor(hours) bamboo(quantities) basket(quantities) Cost ($)4 hr 8 8 4+4=84 hr 9 8 4+4.5=8.55 hr 8 8 5+4=96 hr 8 8 6+4=104 hr 10 8 4+5=94 hr 11 8 4+5.5=9.5... ... ... ...

Thus we see we could associate any level of output with a minimal cost,and then we derivethe cost function. Suppose we have the following cost function for the man in island C.

basket(quantities) Cost ($)1 12 23 34 46 68 8... ...

Actually, I did NOT make up this table randomly. What we observe is a linear cost functionwhich is resulted from the special technology function we are using implicitly behind the seriesof table I gave above. In general, if the production function is Leontief, the cost functionderived from it is linear. We will get into that later in more details.

3.2.1 Production Technology with two inputs

Suppose we consider the production technology with two inputs, capital and labor. Suppose,we use K to denote capital stock and L to denote labor. Everyone knows that the moneypaid for the service provided by capital is called interest and the money paid for the laborservice is called wage. We use r to denoted the interest, $ per unit and w to denote the wagerate, $ per unit.

For example, suppose, capital is a machine, if it is used for a hour, the producer needs to pay400$, therefore the interest is 400$ per hour of usage.

Suppose we write the production function as follows,

q = f(K,L)

, where q is the quantity of output. Then there are several major concepts I want to introduce.


Marginal ProductWe have already seen the definition of marginal product in the case of single input. We canapply the same definition here with some caution. The marginal product of capital is justthe additional amount of output produced if one more unit of capital is added. The marginalproduct of labor is just the additional amount of output produced if one more unit of laboris added. Take the construction company for example,

Capital labor output Marginal Product of CapitalMachine(Hour of Usage per day) Worker(persons per day) buildings per month buildings per hour of usage per day

2 hrs 10 0.5 NA3 hrs 10 0.9 0.44 hrs 10 1.2 0.35 hrs 10 1.4 0.26 hrs 10 1.5 0.1

... ... ... ...

Notice, while calculating the marginal product of capital,the amount of labor remains thesame. And we also notice that as the usage of the machine per day is increasing, the marginalproduct of capital is decreasing. Let’s take a look at an example of the marginal product oflabor.

labor Capital output Marginal Product of LaborWorker(persons per day) Machine(Hour of Usage per day) buildings per month buildings per person per day

11 3 hrs 1 NA12 3 hrs 1.5 0.513 3 hrs 1.8 0.314 3 hrs 2 0.215 3 hrs 2.1 0.1... ... ... ...

IsoquantIsoquant is a curve which collects all the combinations of inputs which produce the sameamount of output.Let’s look at the example of the construction company. If the construction company needs toproduce one building in a month, the following combinations of capital and labor are possiblechoices,

labor(persons per day) machine(hours of usage per day)11 3 hrs10 3.5 hrs9 4.1 hrs8 4.9 hrs7 6 hrs... ...

Marginal Rate of Technical SubstitutionThe marginal rate of technical substitution (MRTS) of capital with respect to labor means if


Figure 3.13: MTRS

the one unit of capital is reduced in order to produce the same amount of output, the amountof labor needs to be added. The marginal rate of technical substitution of labor with respectto capital means if the one unit of labor is reduced in order to produce the same amount ofoutput, the amount of capital needs to be added.

Let’s look at the example of the construction company.

labor(persons per day) machine(hours of usage per day) MRTS of labor11 3 hrs NA10 3.5 hrs 0.59 4.1 hrs 0.68 4.9 hrs 0.87 6 hrs 1.1... ...

What we observe is that as the quantity of labor used for production is decreasing it needsmore extra capital, additional amount of capital to make 1 building within a month for thecompany. In other words, the quantity of labor is increasing, the marginal rate of technicalsubstitution of labor is decreasing. This is called diminishing marginal rate of technical sub-stitution.Now the question is to find the MRTS from the isoquant.The shape of isoquant, and its consequence on MRTS. see Figure 21.

The following is a formula you may find analogous to the one we have seen about marginalrate of substitution in the theory of demand,

MRTSK→L =MPK

MPL

where MTRSK→L is the marginal rate of technical substitution, MPK and MPL are marginalproduct of capital and labor respectively.


Figure 3.14: The shape of Isoquant

Why this is true? The question is left to you to think about.

Returns to scaleSuppose the quantities of capital and labor are doubled. The question is: will the quantityof output be doubled? If the output more than doubles, there are increasing returns to scale.If the output less than doubles, there are decreasing returns to scale. If the output exactlydoubles, there are constant returns to scale. We can see that Returns to scale is the rateat which output increases as inputs are increased proportionately. Take the constructioncompany for example. If there are increasing returns to scale, we should have followingsituations,

labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 2.5 150%33 9 200% 3.8 280%... ... ... ... ...

Constant Returns to scale,

labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 2 100%33 9 200% 3 200%... ... ... ... ...

Decreasing Returns to scale,


labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 1.8 80%33 9 200% 2.5 150%... ... ... ... ...

3.2.2 Cost Minimization

Our goal here is to derive the cost function. We have already known that given a certainlevel of output, there are a bunch of possible combinations of inputs to produce that amountof output and those choices are on the same isoquant. By cost-minimization hypothesis, wethink the producer will choose a combination of inputs which has the lowest total cost. Whatwe are heading for is to find a condition which can characterize the cost-minimizing choiceof inputs given a certain level of output. And we will show how to find the cost-minimizingchoice graphically.

In order to do that, we need to introduce the concept of Isocost. Isocost is a straight linewhich collects all the combinations of inputs, capital and labor which have the same total cost.For example, the producer of basket. Suppose, the price of labor is 1$ per hour as before, andthe price of bamboo is 0.5$ per unit. And we are now looking for the combinations of laborand bamboo such that the total cost is 6$. It is not hard to check the following combinationscost 6$ in total,

labor(hrs) bamboo(quantities) total cost($)0 12 61 10 62 8 63 6 64 4 65 2 66 0 6

Definitely there are many more others. From the table above, we can see the isocost withthe total cost 6$ should be a straight line and we draw it in the graph.What if the cost is 7? The the combinations should be as follows,

labor(hrs) bamboo(quantities) total cost($)0 14 71 12 72 10 73 8 74 6 75 4 76 2 77 0 7


Figure 3.15: Isocost

Figure 3.16: Cost Minimization

What we observe is that the isocost with the cost 7$ is parallel to the isocost with the cost6$. This really is because the prices of inputs remains the same. Now the question is: whatdoes the slope of isocost mean in economic terms?

Now we are ready to show how to find the cost-minimizing choice of inputs graphically, giventhe prices of inputs and the quantity of output to be produced. Step1Given the prices of inputs, we can draw a series of isocost with different costs.Step2Given the quantity of output, we can draw the isoquant, the combinations of inputs on whichare able to produce that amount of output.Step3Looking for an isocost which is the tangent line of the isoquant.see Figure 23.

3.3. LONG RUN EQUILIBRIUM IN COMPETITIVE MARKET 47

From the graph, we observe a condition, which is also analogous to the one we find in utilitymaximization.

MTRS = Relative Price

Why this is true? Similar argument.

3.3 Long Run Equilibrium in Competitive Market

This section, we finally touch upon some equilibrium concept, but it is a partial equilibriumconcept, in the sense that we only consider one market with all other markets being exogenous.Take the construction company for example, we consider in this case, how the price of building(output) is determined while the prices of inputs, like wage rate or the rent for the machinesare exogenous for our analysis and we are NOT attempting to understand how the prices ofthem are determined.

The question we are trying to answer is : suppose there is a competitive market we areconsidering, and suppose there are a bunch of producers in this market. Given the demandcurve, we are looking for the equilibrium price and quantity in this market.

Assumptions of a perfectly competitive market

1. Price Taking: all producers and consumers are price takers.

2. Free entry and Exit

3. Product Homogeneity : this implies all producers are facing the same cost function.

4. The factor industry will not be influenced by the output industry.

Conditions for the long-run equilibrium

1. all producers are maximizing profit

2. no producer has any incentive to either enter or exit the industry

With the assumption we made above, and from the conditions for the long-run equilibrium,we see that in the equilibrium, there will be zero profit for each producer. Why? If the profitis not zero, more producers will be attracted into the industry. And thus by the condition 2,only when the profit is zero, we have the equilibrium conditions satisfied.

We are ready to use the equilibrium conditions to find the equilibrium price in a competitivemarket.So how can we find the equilibrium quantity and equilibrium price?

1. First we derive the supply curve of a typical producer


Figure 3.17: Cost Function for example 1

Figure 3.18: Marginal Cost and Average Cost for example 1

2. then we see what the supply curve of the whole industry should be.

3. the intersection of the demand curve and supply curve gives us the equilibrium quantityand equilibrium price.

Example 1

Suppose we are considering the market of basket in island C. And each producer has thefollowing type of cost function which is linear. see Figure 24. Now we are looking for theprice level such that for each producer the profit is zero. Since all producers have the samecost function, we consider a representative producer. This cost just tells us that for a typicalproducer producing 3 baskets costs 6$, 4 baskets 8$, 5 baskets 10$, so on and so forth.What do you think the marginal cost and average cost should be in this case? It should bea horizontal line with the vertical coordinate being 2. see Figure 25.Now the question is which supply curve for each producer? Now suppose the price of output

is 3, what is happening? In order to gain more profit, the producer will produce as much as


Figure 3.19: Individual Supply curve for example 1

Figure 3.20: Supply Curve and equilibrium for example 1

possible, that is, infinity. And this is true for any price higher than 2. What if the price ofoutput is 1? The produce will not produce anything, since it is losing money. From above,we can the supply curve of an individual producer should be as follows, see Figure 26.

From here we can derive the supply curve of basket for the market. When the price ofbasket is above 2, the quantity supplied will be infinity. When the price of basket is below2, the quantity supplied will zero. When the price is exactly equal to 2, the supply willbe anything.(It does NOT mean it will be infinity. It only means producer will be equallysatisfied with any level of output simply because all of them bring zero profit.) So the supplycurve should be as follows. See Figure 27. We see it should be a horizontal line. Thencombining with the demand curve, we find the equilibrium quantity and equilibrium price.What we can see is that the price is the one which makes a typical producer gain zero profitand the quantity is somehow determined by the demand curve. To see this, suppose for somereason, the demand curve shifts up. The equilibrium quantity will increase.

Example 2Now let’s look at a different example. Suppose in this basket industry, all producer have thesome cost function indicated by Figure 28. We have already known the marginal cost and


Figure 3.21: Cost Function for example 2

Figure 3.22: Marginal Cost and Average Cost for example 2

average cost for this case. see Figure 29.

Now let’s see what the supply curve for any individual producer is. Suppose the price we findin Figure 29 is p. Then we see when the price of output is larger than and equal to p, thequantity produced should make the marginal cost equal to the price. Therefore, the marginalcost above the price p part is the supply curve. When the price is lower than p, the producerwill produce nothing. see Figure 30

Now what do you think the supply curve of the whole basket industry should be? When theprice is larger than p, each producer wants to produce something, and more importantly theyare gaining non-zero profit, which attract more and more producers. Since we have assumedfree entry, as long as the profit is not zero, there will be infinitely many producers enteringthe market which makes the total supply of baskets amount to infinity even though eachproducer produces something finite. When the price is lower than p, no producer will stayin the market. So we conclude that only when the price is p, the total supply will anything.(First, it does NOT mean each producer is indifferent with all level of output. Instead each


Figure 3.23: Individual Supply curve for example 2

Figure 3.24: Supply curve for example 2

one of them will produce q. It does suggest that producers are indifferent to enter or exit themarket.) Therefore, the supply curve of the whole industry should be as follows, see Figure31.

From this two examples, we could see in the competitive market, the supply curve is a hori-zontal line. And the market price is equal to the marginal cost, and the equilibrium quantityis determined by the demand side. And all producers gain zero profit in the equilibrium.

You might ask me why the supply curve is a horizontal line, not a upwards sloping curve? Ifyou relax any assumption we made, we will get a upwards sloping curve.

Chapter 4

Monopoly

In this lecture, we are trying to understand how monopolists make their production decision.

4.1 Question in front of a monopolist

Suppose, there is a market where there is only a single supplier. We are interested in twoquestions:

1. Will this producer behave in a different manner comparing with a typical producer ina competitive market?

2. Will the equilibrium price and quantity in such a market be different from the ones ina competitive market?

Recall the behavioral assumption we made for a typical producer in a competitive market isthat it takes the price of output and inputs as given. We have to give up this assumptionwhen we are analyzing the behavior of a monopolist. Since the monopolist by definition isthe single supplier, and thus it definitely feels it has the power to influence the price. So wethink a monopolist will consider the impact of its action (decision of how to produce) on themarket price.

Now the next question is : do you think the monopolist will choose the quantity of outputarbitrarily? No. The monopolist will try to maximize the profit as any producer does. Thedifference between the producer in a competitive market and a monopolist is the monopolistwill use its power to influence the market price of output and thus the revenue.

Let’s take a look at an example. There is a single producer of basket, who definitely is lonely.And suppose the demand curve of basket in front of this lonely monopolist is described bythe equation,

p = 9− q

where p denotes the price and q denotes the quantity. we can also describe this demandschedule in a table as follows.

53

54 CHAPTER 4. MONOPOLY

Figure 4.1: Demand Curve

Price of Basket Demand for Basket($ per basket) (Quantities)

1 82 73 64 56 37 28 19 0... ...

We can also draw the line on the graph. see Figure 1.

Now we can express the relation between price and quantity in the other way (see the tablebelow), which is sometime called the inverse demand function. This function simply tells youif the quantity demanded is this much, what the price should be so that the consumers willdemand that much. In the table below, we see that when the quantity is 8, the price shouldbe 1$ per basket so that the consumers will demand 8 baskets.

Quantity Demanded for Basket Price of Basket(Quantities) ($ per basket)

8 17 26 35 43 62 71 80 9... ...

4.1. QUESTION IN FRONT OF A MONOPOLIST 55

Now why this inverse demand function is relevant for the lonely monopolist in island C? Thispiece of information is important for the monopolist to know what the revenue will be if thehe produces certain amount of baskets. Suppose, he wants to produce 6 baskets, and then heknows that the price will be 3$ per basket. Because, if the price is lower than 3, the quantitydemanded will be more than 6; and if the price is higher than 3, the quantity demanded willbe less than 6, according to the table we have above. And thus, the revenue for producing 6baskets will be 6 times 3, 18 $. Now let’s work for all levels of output.

Quantity produced Price of Basket Revenue(Quantities) ($ per basket) ($)

8 1 87 2 146 3 185 4 204 5 203 6 182 7 141 8 80 9 0... ... ...

Now the monopolist is trying to calculate the profit for producing all levels of output andthen he can decide which level of output will generate the highest profit. Suppose the costfunction is linear for the monopolist, and its functional form is

c(q) = q

where q is the quantity. we can describe the cost function in the table below.

Quantity produced Total Cost(Quantities) ($)

8 87 76 65 54 43 32 21 10 0... ...

Now, we combine the revenue and cost.


Quantity produced Price of Basket Revenue Total Cost Profit(Quantities) ($ per basket) ($) ($) ($)

8 1 8 8 07 2 14 7 76 3 18 6 125 4 20 5 154 5 20 4 163 6 18 3 152 7 14 2 121 8 8 1 70 9 0 0 0

Then we see the monopolist will produce 4 baskets which generates the highest profit.

4.2 Profit Maximization for a monopolist

4.2.1 Revenue

In this lecture, we only consider the following type of demand curve,

p(q) = a− b× q

where q is the quantity and p is the price,and a, b > 0 constant. And then we see the revenuefunction should be

R = p× q = p(q)× q = (a− b× q)× q = a× q − b× q2

We need to introduce the concept marginal revenue.Marginal revenue is just the additional revenue the producer will have if one more unit ofoutput is produced. Precisely it should be first order derivative of the revenue function, wesee then,

MR = a− 2× b× q

Now let’s draw the demand curve and marginal revenue in the same graph. see Figure 2.

4.2.2 Profit Maximizing Condition

The same as before, we are trying to find the condition which characterize the profit-maximizing choice. Let me first state the condition.

Marginal Revenue = Marginal Cost

Then let me explain why this is true for profit-maximizing quantity of output. Suppose,it is not true and Marginal Revenue > Marginal Cost. What will be happening? Themonopolist will find it is profitable to produce one more unit. What if Marginal Revenue <

4.2. PROFIT MAXIMIZATION FOR A MONOPOLIST 57

Figure 4.2: Demand Curve and Marginal Revenue

Marginal Cost? The monopolist will find it is profitable to reduce one more unit. Do youthink this condition can be applied to the producer in a competitive market? The answer isyes.

4.2.3 Examples

Example 1 Suppose the monopolist has the following type of cost function.See Figure 3. Andsuppose the monopolist is facing a demand curve in Figure 3. And then we see what themarginal cost and average cost should be. Now we are looking for the quantity which makesthe marginal revenue equal to the marginal cost by looking at the intersection of two curves.After finding out the quantity produced, we are looking for the market price by looking atthe demand curve, and finally we can calculate the maximal profit earned by the monopolist.l cost by looking at the intersection of two curves. After finding out the quantity produced,we are looking for the market price by looking at the demand curve, and finally we cancalculate the maximal profit earned by the monopolist.

I use this example to illustrate the steps to find the optimal quantity of output produced bya monopolist when he is facing a demand curve. Generally, we should,

1. Draw the demand curve, marginal revenue, marginal cost, and average cost in the samegraph.

2. Find the intersection of marginal cost and marginal revenue. Then we see the quantityproduced by a monopolist.

3. Plug in the quantity produced by a monopolist into the inverse demand function,(Orgraphically find the price which corresponds to the quantity in the demand curve.)

4. Finally we can calculate the profit earned by the monopolist.



Example 2 Suppose the monopolist has the following type of cost function. See Figure 4.And suppose the monopolist is facing a demand curve in Figure 4. And then we see whatthe marginal cost and average cost should be.

From this example, we can have some idea about how the monopolistic market differs fromthe competitive market. The equilibrium price in the monopolistic market will be higher,and the quantity supplied will be lower.

Example 3 Suppose the monopolist has a linear cost function,

c(q) = 2q

Then we see that the marginal cost is 2. And suppose the demand curve the monopolist isfacing is

p(q) = 10− q

Thus the revenue function should be,

R = (10− q)× q = 10q − q2

The the marginal revenue function should be,

MR = 10− 2q

4.2. PROFIT MAXIMIZATION FOR A MONOPOLIST 59




By setting the marginal revenue equal to the marginal cost, we find that

10− 2q = 2

q = 4

Thus we see the quantity which maximizes the profit earned by the monopolist is 4. whatabout the market price? Plug in the quantity into the demand function, we see that themarket price is 6. And how much does this monopolist earn? The profit should be thedifference between revenue and cost, that is, 6× 4− 2× 4 = 16.

Now we are ready to compare the monopolistic market and competitive market. What wehave already known in the competitive market the price should be equal to marginal cost,that is, 2, and the quantity should be determined by the demand curve, that is, 8. And allproducers are gaining zero profit. See table below. See Figure 5.

– Monopoly Competitive MarketEquilibrium price 6 2

Equilibrium quantity 4 8Profit 16 0

Chapter 5

Duopoly

In this lecture, we will see three types of duopoly model. The situation we are considering is amarket where there are only two producers. We are interested in the following question, howproducers in such market behave differently from a typical producer in a competitive marketand a monopolist, and how the equilibrium price and quantities differ from the competitiveand monopolistic market.

5.1 Cournot Duopoly

This model is named after French economist, Antoine Augustin Cournot (1801-1877) .The story for the cournot duopoly is the following. Suppose there are only two producers,say Brian and Justin, in a market, say the market of basket. The goods they are producingare homogeneous, that is, indistinguishable from the consumers’ point of view. And thereforethey will the share the market. It means, the number of baskets produced by Brian plus theone produced by Justin is the total supply of basket. The decision each producer has to makeis just simply to choose a quantity of output to produce. And their goal is just to maximizethe profit. But while they are making this decision, they have to consider how many basketstheir competitor will produce since the total supply of baskets will influence the market price.In constrast with a monopolist, each producer here only has some partial influence on themarket price because of the presence of a competitor. For example, if Brian tries to raise theprice by reducing the quantity, but at the same time Justin is increasing the quantity, whichfinally offsets the effort to raise the price made by Brian. Even though each producer cannot completely influence the price, in constrast with a typical producer in the competitivemarket, they still have some influence.

Now let’s take a look at a numerical example.Suppose Brian and Justin have the same cost function,

c(q) = 2× q

which is a linear function. We see the marginal cost for both them is 2$ per unit. Now

61

62 CHAPTER 5. DUOPOLY

suppose the demand curve they are facing is,

p(q) = 11− q

In order to illustrate how each producer makes the decision, let’s take a look at Brian’sdecision. (Since the structure of the problem is kind of symmetric, we can work out Justin’sproblem in the similar way.) In Brian’s mind, he is thinking ”if Justin produce this much,how much should I produce? Suppose Justin is producing 5 baskets. Then I have followingtable. From the table below, I should produce 2 baskets if Justin is producing 5 baskets. SinceI have the maximal profit in this case.”

Baskets produced by Brian Baskets produced by Justin Total Supply Market price Revenue Cost Profit0 5 5 6 0 0 01 5 6 5 5 2 32 5 7 4 8 4 43 5 8 3 9 6 34 5 9 2 8 8 05 5 10 1 5 10 -5

Then Brian can find out the best response to all possible numbers of baskets Justin is pro-ducing. Then he has the following table,

Baskets produced by Justin Best Response by Brian0 4.51 42 3.53 34 2.55 26 1.57 18 0.59 0

For Justin, we can have exactly the same table by symmetry for the problem. In other words,when Justin is making the decision, he is also doing that sort of thinking as we did above,that is, try to find a best response to Brian’s decision in terms of the quantity to produce.

Baskets produced by Brian Best Response by Justin0 4.51 42 3.53 34 2.55 26 1.57 18 0.59 0

5.1. COURNOT DUOPOLY 63

Now the question is how much they will actually produce. Suppose Brian will produce 5baskets, then Justin want to produce 2 baskets. If if Brian knows that Justin is producing2 baskets, he actually produces 3.5 baskets, but if Justin knows that Brian now wants toproduce 3.5 baskets, he won’t produce 2 any more....

Where can we end such argument which appears to be endless? Now suppose Brian wantsto produce 3 baskets, in response then Justin wants to produce 3 baskets as well. WhenBrian knows Justin is producing 3 baskets, he is satisfied with the decision he made, that is,producing 3 baskets.

From the example above, we know two things, first, how each producer makes decision, theyare actually looking for the best response to their own expectation of their competitor’s deci-sion; second, how we come up with a solution concept for this type of problem, that is, eachproducer is satisfied with the present solution and has no incentive to move.

Now we give the formal presentation of this model. Suppose we have two producers, Brianand Justin. They have the same cost function,

c(q) = c× q

where c is a constant which is equal to the marginal cost. And they are facing the samedemand function,

p(q) = a− q

where a, b > 0 constant. We use q1 to denote the quantity of baskets produced by Brian, andq2 to denote the quantity of baskets produced by Justin. The problem for Brian is to chooseq1 given q2 to maximize the profit.

First, we know q1 + q2 is the total quantity supplied, then we know the market price shouldbe p = a− (q1 + q2), therefore, the total revenue for Brian given q2 is R = (a− q2)× q1 − q21,then the marginal revenue for Brian is MR = (a− q2)− 2× q1. Let the marginal cost equalto marginal revenue. We have (a− q2)− 2× q1 = c. Do some algebra, we see that,

q1 =a− c

2− q2

2

We call this function, the best response function by Brian, which is the function of the numberof baskets produced by Justin. We can also derive the best response function by Justin inthe exactly same way, we can have,

q2 =a− c

2− q1

2

See Figure 1.

Now the question is what the equilibrium concept is. The equilibrium concept we will use iscalled Nash equilibrium by John Nash. But for me the idea has been known among economistsfor a quite long time.


Figure 5.1: Best Response Function

Figure 5.2: Equilibrium: Example

”. . . , the general idea of equilibrium, refers to a certain type of relationshipbetween the plans of different members of a society. It refers to, that is, the casewhere these plans are fully adjusted to one another, so that it is possible for allof them to be carried out because the plans of any one member are based on theexpectation of such actions on the part of the other members as are contained inthe plans which those others are making at the same time.”-By Friedrich A. Hayek ”The Pure Theory of Capital” Midway Reprint 1975, p18.

Let’s transform the quote into our language. Brian and Justin are making decision on thebasis of their expectation of how many baskets the other will produce. In other words,their action depends on the other’s action. The equilibrium refers to the situation where alltheir plan can be carried out when their expectation becomes true. For example, let’s makea = 11 and c = 2, then we back to the numerical example I gave at the very beginning.Suppose Justin anticipates that Brian will produce 2 baskets. In figure 2, or by the formulaabove,we see that he will produce 3.5 baskets. We see this plan made by Justin is based onhis expectation of how many baskets Brian will produce. If Brian anticipates that Justin willproduce 3.5, he will produce 2.75 which is not what Justin anticipates. In other words, theirplans are NOT compatible.

So how can we find the equilibrium? The intersection of best response function. Now let’s

5.1. COURNOT DUOPOLY 65

Figure 5.3: Equilibrium: Example

verify this. In the same example as above, if Justin believes that Brian will produce 3 baskets,then Justin will produce 3 baskets. If Brian knows Justin is producing 3 baskets, he will dowhat Justin anticipates. Then their plans are fully adjusted. See Figure 3.

So we can find the equilibrium by looking at the intersection of two best response functions.Mathematically, we can have two equations and two variables, we can solve them.

q1 =a− c

2− q2

2q2 =

a− c2− q1

2

we have,

q1 = q2 = (a− c)/3

let’s see a numerical example. Demand function: p(q) = 10−q, cost function: c(q) = 2q. Theresult we have is each of two producers will produce 8/3 , and market price is 10−8/3−8/3 =14/3. Let’s compare this with the monopolistic market and competitive market.

– Monopoly Competitive Market Cournot DuopolyEquilibrium price 6 2 ≈ 4.6

Equilibrium quantity 4 8 ≈ 5.3Profit for each producer 16 0 ≈ 6.76

The conclusion is the price in cournot duopoly will be higher than the competitive marketand lower than the monopoly, and the quantity will be lower than the competitive marketand higher than the monopoly.

Now suppose the number of producer becomes three. What is the equilibrium price andquantity? What if the number of producer becomes 100? ...


5.2 Stackelberg Duopoly

The same as Cournot duopoly, Stackelberg duopoly also considers a market with two pro-ducers. The difference is in Stackelberg duopoly one producer is the leader in the market andthe other is the follower. In Cournot duopoly, two producers choose the quantity of outputsimultaneously. But here, the leader will choose the quantity first, and then it is the fol-lower’s turn to choose. The situation becomes different because when the follower is makingthe decision, the leader’s decision has become a given condition for him/her in Stackelbergduopoly while in Cournot duopoly two producers also need to consider what their competitorwill do but they only have anticipations of their competitor’s action.

Let’s take a look at an example. Suppose there are two producers, Brian and Justin, in themarket of basket. Suppose Brian is the leader and he will choose the quantity of basket first.Suppose, the demand curve they are facing is p(q) = 10 − q. And the cost function theyhave is c(q) = 2q. The real thing between them is : On one hand, since Brian is the leaderand therefore he can take a large portion of demand, and this will force Justin to restrict theproduction otherwise the price will be too low to earn some profit; on the other hand, Justincan also give Brian some trouble by increasing the production such that both of them willnot earn some profit. For example, Suppose Brian chooses to produce 7 baskets, notice thisis a large portion of demand, since when the total quantity supplied is 8 baskets the marketprice will be equal to the marginal cost. Now how Justin deals with this? If Justin produces1 baskets, the profit he can earn is zero. He can earn some profit only when he produces lessthan 1 baskets. Let’s calculate what the best decision for him is in this case.

Baskets produced by Justin Baskets produced by Brian Total Supply Market price Revenue Cost Profit0 7 7 3 0 0 0

0.25 7 7.25 2.75 0.6875 0.5 0.18750.5 7 7.5 2.5 1.25 1 0.250.75 7 7.75 2.25 1.6875 1.5 0.1875

1 7 8 2 2 2 0

Then we see that Justin will produce 0.5 baskets and he will earn 0.25 $ if Brian chooses toproduce 7 basket. On the other hand, Brian will earn 3.5 $. Justin might think he can domuch better. He goes to threaten Brian, saying ”If you produce more than 4 baskets, I willpunish you by producing more baskets to make you lose money.” For example, in the case wejust talked about, if Brian produced 7 baskets, what Justin can do to make Brian crazy is toproduce more than 1 basket, say 2 baskets. In a result, the market price will be 1 and Brianactually loses 7 $ while Justin just loses 2$. One thing here we have to pay attention to isBrian has already made the decision while Justin is making his choice. For Brian , after hemade the choice, it it irreversible. So if Brian is trying to take a large portion of the demand,he is also taking the risk of being punished by Justin.

Now the question is will Brian be threatened ? Put it in the other way, will Brian take muchcredit of what Justin is saying? in other words, will Brian believe Justin will what he is saying?

5.3. BERTRAND DUOPOLY 67

The Answer is NO. What Justin said is called Empty Threat. The threat is empty becauseJustin will not do what he said. Since we have known if Brian produces 7 baskets, the bestchoice for Justin is to produce 0.5 baskets, not 2, or something else. Of course if Justin isemotionally unpredictable person, we can not expect him to do something reasonable. Butthe point is the reasonable choice for Justin in that case is to produce 0.5 baskets Not 2.

5.3 Bertrand Duopoly

In this section, we consider Bertrand duopoly model named after Joseph Louis Bertrand, aFrench mathematician. In this model, we consider a market of two producers as before. Here,the choice variable for the producer is NOT the quantity of output anymore. Instead, theproducer is choosing the price at which level its product is sold. The demand for its productis dependent on its own price and its competitor’s price. The goal for the producer is still tomaximize profit.

Let’ s take a look at an example. Suppose there are two producers, Brian and Justin, in themarket of basket. The demand curve they are commonly facing is p(q) = 10 − q. Supposethey have the same cost function, c(q) = 2q. Brian and Justin will set their prices simultane-ously. The producer with the lower price will take over the whole demand. If the prices arethe same, two producers share the demand. For example, if for each basket Brian charges2$ but Justin charges only 1$, consumers will go to buy baskets produced by Justin. Andthe quantity will be 9, according to the demand curve. If both of them charges 2$ for eachbasket, the total quantity demanded will be 8, and each one of them will get a half of 8, thatis, 4. So we see that the price set by one producer is dependent on his anticipation of theprice that his competitor charges. Now let’s me illustrate how Brian and Justin are makingthe decision on the basis of their anticipation of what the other will do. Suppose Brian thinksJustin will charge 4$ per basket, any price lower than 4 will be a good choice for him, say3.5. But if Justin knows that Brian will charge 3.5$ for each basket, Justin will cut its price,say 3$ per basket. But if Brian knows Justin will cut the price to 3, he will lower the pricefurther. So this process will continue. Until....?? Until the price is zero? No until the profitis zero. When ? when the price reaches 2$ per basket. Why? Suppose Brian knows thatJustin will charge 2$ per basket. Then he has incentive to undercut its competitor’s price ?NO ! If Brian lowers the price to 1$ per basket, he takes over the whole demand (9 baskets),the revenue he gains is 9$, what about the cost? 18$ !!!, Brian is losing money. It is true forany price lower than 2. So if Brian anticipates Justin will charge 2 dollars for each basket,he does not have incentive to undercut the price, will he charge more than 2? To answer thisquestion. Suppose Brian charges 2 dollars also. The profit he gains is zero. If he charges morethan 2 dollars, no consumer will come to him and thus he gains nothing. So he is actuallyindifferent to these two choices.

Now the question is what the equilibrium should be? The Nash equilibrium here should be:both Brian and Justin charge 2 dollars for each basket. Why? If Brian knows Justin willcharge 2$, he is indifferent between setting price at 2 and higher. If Brian also charges 2$,Justin will not have incentive to change his decision, still setting price at 2. So this is a


”resting point”.

Now let me give the formal presentation of the model.

Suppose the cost function for both two producers is

c(q) = c× q

I use q1 to denote the demand for Brian’s product, and q2 the demand for Justin’s product.I use p1 to denote the price set by Brian, and p2 the price set by Justin. So the demand forBrian’s product is dependent on the price set by Brian and Justin, we have

q1 = a− p1 + 1/2× p2

This demand function for Brian’s product says that the higher the price set by Brian, thelower the demand for his product; the higher the price set by Justin, the higher the demandfor Brian’s product. Symmetrically, we have the demand curve for Justin’s product

q2 = a− p1 + 1/2× p1

There is one thing we need to pay attention to is that for each producer the price set by itscompetitor is exogenous; in other words, each producer takes the price set by its competitoras given.

Step 1 The first step is to derive the best response function for each producer. Let’s firstlook at Brian’s problem. Given the price set by Justin, p2, and facing the demand curve forhis own productq1 = a − p1 + 1/2p2, Brian is trying to maximize the profit by setting thepricep1.

Now what is the revenue?

R = p1 × q1 = p1 × (a− p1 + 1/2p2)

What is the cost?C = 2× q1 = c× (a− p1 + 1/2p2)

What is the profit?

Profit = R− C = p1 × (a− p1 + 1/2p2)− c× (a− p1 + 1/2p2)

Then we haveProfit = (p1 − c)(a− p1 + 1/2p2)

Let’s take the derivative of profit function with respect to p1, we have

(a− p1 + 1/2p2)− (p1 − c)

setting it to zero, we can obtain the price which maximizes the profit for Brian when p2 isgiven,

p1 =a+ c

2+p2

4

5.3. BERTRAND DUOPOLY 69

Figure 5.4: Best Response Function in Bertrand Duopoly

This is the best response function by Brian. We can have the best response function byJustin by doing the similar thing,

p2 =a+ c

2+p1

4

Let’s draw the best response function in the graph. As before, the equilibrium should be theintersection of the best response functions. We can know then, in the equilibrium the priceset by two producers should be the same, that is, 2(a+c)

3 . m should be the intersection of thebest response functions. We can know then, in the equilibrium the price set by two producersshould be the same, that is, 2(a+c)

3 .

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