Lecture Notes
– Intermediate Microeconomics
Xu Hu
huxu85@tamu.edu
Department of Economics, Texas A&M University
November 12, 2010
2
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
2.1
2.2
2.3
11
Preference and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.1
Marginal Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1.2
Indifference Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1.3
Marginal Rate of Substitution . . . . . . . . . . . . . . . . . . . . . . .
17
Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.1
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3 Producer Behavior in the Competitive Market
3.1
3.2
3.3
27
Producer Behavior with single input . . . . . . . . . . . . . . . . . . . . . . .
29
3.1.1
Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.1.2
Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Producer Behavior with two inputs . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2.1
Production Technology with two inputs . . . . . . . . . . . . . . . . .
41
3.2.2
Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
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 Microeconomics. In this article, I will present the major topics to be discussed in this course and
will review some basics of the demand-supply analysis.
The goal for economics is to understand how economy works. The approach we take is very
similar 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 good
understanding of how each market works, we investigate how one market is linked to the
others. By examining the inter-dependence of different markets, we are able to discuss the
working 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 its
components,i.e., people who participate in the market. We group them by the function
they 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 the
interactions between consumers and producers and try to understand how price is determined.
The mechanism under which consumers interact with producers depends on the structure of
market, 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 B
farmers. Even though two islands are separated, in between there lies a small island which
has no inhabitant. Since fishermen in island A also want to eat some corn and farmers in
1
Sometime 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. This
is the ”market”.
• If the corn and the fish traded in the market are homogeneous, i.e., without any types
of difference, such as quality(A very unrealistic assumption.), and if there is a large
number of fishermen and farmers in the market so that no one alone has the power
to influence the market price, we think this market is competitive. The device we use
to analyze the working of the market here is Demand-Supply analysis. We will review
the basics later. What you have not learned is how the demand and supply curves are
derived, 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 a
monopsony for corn? In this case fishermen as a whole act as one person, having the
power to influence the market price. In part IV and V we will deal with this type of
situation.
1.2
Demand-supply analysis
In this section, we will start with an example as an illustration of what demand curve and
supply curve are, and then I will elaborate a little bit on the concept of equilibrium and
comparative 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 in
island B are willing to trade with each other. The fishermen in island A are the buyers for
corn and the suppliers for fish. Now, let us consider the market for corn. The following
schedule tells us what the total amount of corn demanded by the fishermen in island A is
under a given price.
Price(Unit: fish per corn)
1
2
3
4
5
6
Quantity demanded for corn
10
8
6
4
2
0
This schedule tells us that when farmers in island B ask one fish for each corn the fishermen
as 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 quantity
demanded 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 Pc is the price of corn in terms of fish.
We can also draw this schedule into a two-dimensional curve. In the horizontal axis, we use
Q to denote the quantity demanded. In the vertical axis, we use P to denote the price. Each
pair in the table above corresponds to a point in such a two-dimensional plane. We draw a
line which transverses all the points, which gives us the demand curve for corn. It should be
downward 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 tells
us what the total amount of corn supplied by the farmers in island B is under a given price.
Price(Unit: fish per corn)
1
2
3
4
5
6
Quantity supplied for corn
2
4
6
8
10
12
This schedule tells us that when the fishermen in island A offer one fish for each corn the
farmers as a whole in island B are willing to provide 2 corns; if the fishermen offer two fishes
for each corn, the farmers will increase their supply to 4 corns. We can notice as the price
decreases, in contrast with the demand curve, the quantity supplied decreases as well. We can
draw a line to represent such schedule in a two-dimensional plane as we do for the demand
schedule. 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 an
agreement in the sense that the quantity demanded is equal to the quantity supplied. What
if the price is one fish per corn? When the price is 1 fish per corn, the fishermen in island
A as a whole want to have 10 corns while only 2 corns are provided by the farmers. This
implies under this price the need of some fishermen in island A who want to buy some corn
is not satisfied. What should they do? They can go to negotiate with the farmers. From
the 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 farmers
in island B who want to provide more. This process will continue until the price reaches 3
fishes per corn. What if the price is 5 fishes per corn? In this case, the fishermen demand
less 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
8
CHAPTER 1. INTRODUCTION
price reaches 3 fishes per corn. Only when the price is exactly 3 fishes per corn, everyone is
satisfied.
Formally, we call the price as the equilibrium price provided it makes the quantity demanded
equal to the quantity supplied. (Warning: I did not say the demand curve is equivalent with
the supply curve.) In the graph, the intersection of the demand curve and the supply curve
gives us the equilibrium price and quantity.
Equilibrium in general is the situation where no individual has any incentive to change their
decisions. In this case, equilibrium is the situation when quantity demanded is equal to the
quantity supplied. Most of economists believe that the equilibrium state is a resting point
such that the economy will finally achieve. Whenever the economy is not at the equilibrium
state, there always exists a tendency for the economy to move toward the equilibrium. Thus
historically 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 per
corn everyone is satisfied and no one is willing to deviate from that state. When the price is
not 3 fishes per corn, either the buyers will offer a higher price or the suppliers will request
a lower price. In other words, the tendency is present toward the equilibrium price 3 fishes
per corn.
We have mentioned that as long as the equilibrium state is reached, the economy will stay
there forever. Does that mean the economy will not change at all after that? How can we use
such a static method to analyze a changing world? And nobody will believe the existence of
such 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 and
endogenous variables. In this world, we have some observations, and they make us curious
about why they are so. Therefore, theorists are trying to set up models to explain them. In
a model, there are some variables we are trying to know how their equilibrium values are
determined. We call them endogenous variables. And there are some other variables which
for the present purpose we are not trying to explain but instead whose value we take as
given. In our two-island example, even though we did not specify what exogenous variables
are, they actually determine the demand and supply schedules. For instances, the weather
determines how many corn the farmers can collect each day which affects how many corn
they 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 in
our case are the endogenous variables.
Come back to the question: How can we use such a static method to analyze the phenomena
in a changing world?
We first classify all the changes as changes in exogenous variables. As the exogenous variables
vary, the demand and supply curve might shift (NOT the change in quantities !), which in
the end determines the new equilibrium price. This type of process is called the Comparative
Static analysis.
Let’s see some examples.
1.2. DEMAND-SUPPLY ANALYSIS
1.2.3
9
Elasticity
In this section, we briefly review a useful concept, elasticity. Here we focus on the price
elasticity. 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 corn
from our two-island story, you might want to know if the price falls 1% by what percentage
the quantity demanded will increase and the quantity supplied will fall. If the quantity
demanded (supplied) increases (falls) to a large degree in the percentage sense, we call the
demand (supply) is fairly sensitive to the price and the demand curve is elastic. Otherwise
we call it inelastic.
Formally, the following is the definition,
=
∆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 always
true. We will see some counter examples in Part II.
• More precise definition for elasticity calls for the use of calculus. See Textbook Page
46 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 supply
curve at the point where the elasticity is evaluated is elastic.
|| = 1
Unitary elasticity
0 ≤ || < 1
Inelastic
|| > 1
Elastic
Example:
Price(Unit: fish per corn)
Quantity demanded for corn
Elasticity
2
8
3
6
4
4
5
2
−2/8
1/2 = −0.5
−2/6
1/3 = −1
−2/4
1/4 = −2
−2/2
1/5 = −5
Price(Unit: fish per corn)
Quantity supplied for corn
Elasticity
2
4
3
6
4
8
5
10
2/4
1/2 = 1
2/6
1/3 = 1
2/8
1/4 = 1
2/10
1/5 = 1
10
CHAPTER 1. INTRODUCTION
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 college
student 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.The
set 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. The
preference 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 the
satisfaction resulted from their actions. Another way of saying is that they are calculating
gains and pains when they are making choices, and choose the best one to maximize the gain
and minimize the pain.
A set of questions might arise: what do I mean by gains and pains? materialistic or psychological? Does the theory suggests that people only care about money? How to explain
generous donations and charity activities?
The theory does not say anything about what kinds of gains and pains that people are
calculating at all. It can be materialistic or psychological. Neither does the theory suggest
that different people have the same preference. What kind of preference that people have
is not the question that economists are trying to answer. Economists only assume there
exists such a preference ordering but the specific content is left to be open so that it can
accommodate the variety of tastes among people.
Example
Suppose one fisherman in island A on July 3rd has five fishes, and the market price of fish is
2 corns per fish. Then we know the affordable choices for him are as follows.
11
12
CHAPTER 2. CONSUMER BEHAVIOR
Fish
0
1
2
3
4
5
Corn
10
8
6
4
2
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 1
fish is left. Since the price of fish is 2 corns per fish, the maximal amount of fish he can have
is 2. Thus 4 fishes and 3 corns are not affordable for him.
Now suppose he has the following preference ordering over the choices.
Desirability
Super Best
Second Best
good
OK
Just so so
Worst
Fish
4
3
5
2
1
0
Corn
2
4
0
6
8
10
According to the table above, the optimal choice for the man in island A is to demand 4
fishes 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 three
things.
1. subjective valuation over the choices : this provides a criterion for individuals to decide
which 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 price
of fish is 3 corns per fish. The set of all affordable choices for the man in island A is changed
to the one in below.
Fish
0
1
2
3
4
5
Corn
15
12
9
6
3
0
2.1. PREFERENCE AND UTILITY
13
Since now the man can choose some combinations of fish and corn which are not affordable
under the previous price level,i.e., 2 corns per fish, the optimal choice he makes will be
different from the one he made before. Now consider a different change. Suppose the price
of fish is still 2 corns per fish, but the man in island A is endowed with 6 fishes. The new set
of all affordable choices is as follows.
Fish
0
1
2
3
4
5
6
Apple
12
10
8
6
4
2
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 discuss
the 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 consider
two-commodity case. Thus, the elements in X are pairs. In our two-island example, there are
two commodities, fish and corn. In this case, the set of all possible choices contains elements
like, (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 the
second 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, any
real number can denotes certain quantity of one good, even though we know 1/3 of a car is
not a car any more, and can be not sold in the real world. However, we have this assumption
for the convenience of our analysis. Therefore, X = {(a, b) : a, b ∈ R+ }, where R+ denotes
nonnegative 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 the
combination of 3 fishes and 4 corns. Or (3,4) and (5,0) are the same, which means 3 fishes
and 4 corns are the same with 5 fishes. Note, different people might have different preference
orderings. For example. One guy might prefer the combination (3,4) to (4,2) while the other
guy might choose the opposite. Thus in essence, the preference order reflects people’s tastes
and subjective valuation of commodities.
There are several axioms on preference.We require all preferences should satisfy following five
axioms.
14
CHAPTER 2. CONSUMER BEHAVIOR
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, then
pair A is better than pair C. For example, for someone if the 4 fishes and 2 apples
combination is better the combination of 3 fishes and 4 apples, and 3 fishes and 4
apples combination is better than 5 fishes, then the 4 fishes and 2 apples combination
is 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 better
than both (0,4) and (2,0) when (0,4) and (2,0) are the same. In other words, if you are
indifferent to eat 4 fishes and 2 apples, the mixture of them makes you happier. Notice
that 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 u
preserves the ordering. We call such function, utility function, which assigns a real number
to each pair, and call such number as the utility from consuming such pair. For example:
pairs(fish,apple)
(5,0)
(3,4)
(4,2)
(4,5)
...
utility
6
10
12
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 A
is better than pair B , then u(A) > u(B). Thus the utility actually represents the satisfaction
from consuming one certain combination of goods. If you still remember, I mentioned the
theory somehow suggests that individuals behave as if they were maximizing the satisfaction
they gain from consuming. Here we can see that maximizing the satisfaction is equivalent
with maximizing the utility. In other words, picking the ” best ” choice is equivalent with
choosing the pair which yields the highest utility.
Now the question is Does it matter if we change the number but keep the relative relation
intact? For example,
pairs(fish,apple)
(5,0)
(3,4)
(4,2)
(4,5)
...
utility
90
100
134
156
...
2.1. PREFERENCE AND UTILITY
15
In this case, you see the utility from consuming 3 fishes and 4 apples is 100, different from
the previous case. So yes, we have a different utility function, but we did not change the
relative relations. For example, (4,2) is still better than (3,4). So even though, two utility
functions might give us two different numbers for one pair, as long as they reflect the same
preference ordering, they will induce the same behavior. We will come to this again when we
discuss 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 utility
you can gain from consuming one more unit of one good. For example,
pairs(fish,corn)
(2,4)
(3,4)
(4,4)
(5,4)
...
utility
4
10
14
16
...
Marginal Utility (per fish)
NA
6
4
2
In this example, when the amount of fish is increased from 2 to 3, the utility is increased
from 4 to 10. 6 per fish is the marginal utility evaluated at pair(2,4). (Warning: when you
calculate the marginal utility for one good, for example fish, you need to keep the amount of
other goods constant, say corn in our case. When you say marginal utility, you should always
specify 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. Indifference 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 quantity
consumed of commodity 2. What this function does is for each pair of x1 and x2 it gives a
number(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)
(5,0)
(3,4)
(4,5)
...
utility
5
7
9
...
16
CHAPTER 2. CONSUMER BEHAVIOR
Figure 2.1: Example 1: Linear Indiffernce Curve
Now let’s look for the indifference curve the points on which give utility 10 according this
utility function.
pairs (commodity 1, commodity 2)
(10,0)
(5,5)
(4,6)
...
utility
10
10
10
...
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 quantity
consumed of commodity 2.
Let’s see the pairs which give utility 10.
pairs (commodity 1, commodity 2)
(10,1)
(5,2)
(4,2.5)
...
utility
10
10
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 quantity
consumed of commodity 2. What this function does is for each pair of x1 andx2 it gives the
minimum of the two. For example, u(3, 4) = 3, and u(4, 5) = 4.
2.1. PREFERENCE AND UTILITY
17
Figure 2.2: Example 2
Let’s see the pairs which give utility 10.
pairs (commodity 1, commodity 2)
(11,10)
(12,10)
(13,10)
...
(10,11)
(10,12)
(10,13)
...
utility
10
10
10
...
10
10
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 which
has 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 the
amount of the other good that needs to increase to compensate the loss. This implies we can
18
CHAPTER 2. CONSUMER BEHAVIOR
Figure 2.3: Example 3: Leontif Indifference Curve
calculate MRS from the indifference curve since along an indifference curve the utility is the
same.
Shape of Indifference Curve
The 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 have
the shape similar to the one we see in Example 2. We have already mentioned the economic
meaning of the convexity of the preference. Now let’s see the consequence of this assumption
on the MRS.
Marginal Rate of Substitution and Marginal Utility
Now it is a good place to give you the formula to calculate MRS. My purpose is not to teach
you the math, but to link the concept of marginal utility with MRS.
M RS1→2 =
M U1
MU2
, where M RS1→2 means the marginal rate of substitution of commodity 1 with respect to
commodity 2, and M U1 and M U2 means the marginal utility of commodity 1 and commodity
2 respectively. WHY?
suppose now you consume one additional unit of commodity 1, how much utility you gain?
That is M U1 by the definition of marginal utility. In order to keep the utility the same, you
have to reduce some amount of commodity 2. How many? First you have to reduce M U1
this much of utility resulted from consuming one more unit of commodity 1. We know if you
reduce one unit of commodity 2, we will lose M U2 this much of utility, and therefore, to lose
1 unit of utility, we have to reduce M1U2 this amount of commodity 2. Thus to reduce M U1
this much of utility, you have to reduce M U1 × M1U2 this amount of commodity 2. And wait,
the amount of commodity 2 that needs to be reduced if one more unit of commodity 1 is
increased 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, and
the set is called the budget set. There are two things exogenous for them, the prices and
total resources.
Let’s first take a look at the general setup for deriving the budget set. Suppose the person we
are 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. We
use P1 and P2 to denote the price of commodity 1 and commodity 2 in terms of dollars. It is
not hard to realize that total use can not exceed the total resources. That says money spent
on 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-dimensional
plane. Recall, we have assumed that ”the more the better”, which implies the individuals
will 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 commodity 1 in terms of commodity 2 is the maximal amount of commodity 2 you can have if
20
CHAPTER 2. CONSUMER BEHAVIOR
you give up one unit of commodity 1. In our general setup, the relative price of commodity
1 in terms of commodity 2 is PP12 . Why?
If you give up one unit of commodity 1, then we save P1 dollar, and you can use this many of
dollars to buy commodity. We know with 1 dollar, we can buy P12 this amount of commodity
2. And then with P1 dollar, we can buy P1 × P12 this amount of commodity 2. By definition
this is relative price of commodity 1 in terms of commodity 2. Graphically what is PP12 ? That
is 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 general
there are two things which affect the budget set, prices and income. We are considering four
types of changes.
2.2.1
Applications
1. Intertemporal substitution
Let’s imagine the following situation. When you are just born, you are thinking you might
earn 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 program
promises to pay you an interest when you are old for each dollar you invest into it when you
are young.We use R to denote the gross interest rate, which means, if you invested 1 dollar
when 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 good
that you consume when you were young and Co to denote the quantity of the good that you
consume when you are old. And the price of this good is 1 $ per unit and remains the same
throughout your life time. So the situation looks like as follows,
Income($)
consumption(quantity)
price of consumption goods($ per unit)
young
I
Cy
1
old
0
Co
1
The question is to write down the life-time budget constraint you are facing. When you are
young, there are two options for you, spend some money on the consumption and save some
money. Let’s use S to denote saving expressed in terms of dollars. Therefore, the money
spent 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 resources
available 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 saving
plus interest earnings. Now do some algebra, we combine two inequalities, and we get,
Cy +
Co
≤I
R
2.2. BUDGET CONSTRAINT
21
Basically, there are two things in the individual’s mind who are making such intertemporal
decision, how much to consume today and how much to consume tomorrow. Definitely their
decision depends on how impatient they are for delaying the consumption. If I am telling
you now I’ll give you a brand new car tomorrow, you probably will be quite excited. What
if I am telling you yes I’ll give you a new car, but 20 years from now? I bet you will be less
excited. On the other hand, their decision also depends on how much they will ”lose” if they
choose to consume earlier, since if they postpone their consumption and save them, they will
always earn some interests later. And this is captured in the gross interest rate. When the
interest 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. We
often call this the relative price of consuming today to consuming tomorrow. According to
the 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 the
program which returns you 1 × R dollars when you were old. And how many consumption
goods you can buy by using this amount of money? That is R1 . Thus the relative price of
consuming 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 one
good to consume and the price of it is 1 $ per unit. And we use C to denote the quantity
of the good that you consume and L to denote the hours you choose to have a rest or have
a fun, but not work. So the question is to write down the budget constraint. Now we can
easily 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, we
have,
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 to
sleep at home for a hour, you are actually giving up the opportunity to work for a hour which
earns 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 a
one-time bonus say B.
2. suppose if you decide to work outside the regular working hours, that is 8, you are given
extra 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 to
work longer than 8 hours each day.
22
CHAPTER 2. CONSUMER BEHAVIOR
Figure 2.5: Example 2
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 that
individuals behave as if they were maximizing the satisfaction resulted from consuming,
and the utility function captures the satisfaction individuals gain from consuming certain
commodity pairs, like 3 fishes and 4 corns, or 4 fishes and 2 corns. So given the set of all
possible choices, the individual will choose a pair which maximize the utility function. In
other words, the individual will choose a pair which has the highest utility among all those
pairs within the budget set. Our goal is to know under certain prices and income what the
individual’s optimal choice will be.
We first take a look at the graphical presentation and then I will give some applications. In
the end, I give the mathematical presentation.(which is definitely not required but good to
know.)
We have already know the absolute value of the slope of indifference curve evaluated at some
point is the MRS evaluated at this point. From the graph, we observe that if the point is
the optimal choice, the MRS evaluated at this point, i.e., the absolute value of the slope of
the indifference curve, is equal to absolute value of the slope of the budget line, which is
the relative price. This equality has quite a lot of economics to learn. Recall the MRS of
U1
P1
commodity 1 in terms of commodity 2 is equal to M
M U2 . And the relative price is P2 . Then
24
CHAPTER 2. CONSUMER BEHAVIOR
we see,
P1
M U1
=
M U2
P2
Do some algebra, we see that,
M U2
M U1
=
P1
P2
What is this? If you spent 1 $ on commodity 1, we can have how many commodity 1, that is,
1
P1 , if you have one additional commodity 1, how much additional utility you can gain, that
is, P11 × M U1 . In a word, the term MPU1 1 means if you spend 1 $ on commodity 1, the amount
of utility you can have. Suppose, MPU1 1 6= MPU2 2 , let ’s assume that MPU1 1 > MPU2 2 . what will you
do? Move 1 $ from consuming commodity 2, and use it to consume commodity 1, because
that additional dollar will gain your more utility. Utill when you will stop move from one to
the other. utill they are equal.
Example
Now, let’s see an example. Consider a consumer’ s choice of fish and corn. Suppose his utility
function is
u(f, c) = f × c
According to his utility function, the marginal utility of fish M Uf can be calculated by using
the following formula,
M Uf = c
And
M Uc = f
And thus, M RSf →c the marginal rate of substitution of fish with respect to corn is,
M RSf →c =
c
f
Notice this utility does not have the property of decreasing marginal utility but
the 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 corn
is 1$ per corn. The following choices are on the budget line.
pairs(fish,corn)
(4,0)
(3.5, 1)
(3, 2)
(2.5, 3)
(2, 4)
(1.5, 5)
(1, 6)
(0.5,7)
(0,8)
utility
0
3.5
6
7.5
8
7.5
6
3.5
0
M Uf
0
1
2
3
4
5
6
7
8
M Uc
4
3.5
3
2.5
2
1.5
1
0.5
0
M Uf
Pf
0
0.5
1
1.5
2
2.5
3
3.5
4
| MPUc c
4
3.5
3
2.5
2
1.5
1
0.5
0
M RSf →c
0
1
3.5
2/3
6/5
2
10/3
6
14
∞
Now let’s have some applications. Back to our example2 (Consumption and Leisure) in the
Budget Constrain section.
2.3. UTILITY MAXIMIZATION
Figure 2.7: Utility Maximization: Applications
25
26
CHAPTER 2. CONSUMER BEHAVIOR
Chapter 3
Producer Behavior in the
Competitive Market
In this lecture, we will see a simple model which attempts to understand producer behavior
in the competitive market.
Let’s first take a look at the big picture of the model. We define a producer as an entity of
a production technology which can transform certain amounts of inputs into certain amount
of output. Take a construction company for example. The company needs some construction
workers first; second it also needs some materials like concrete, woods, iron, etc; third, some
construction devices like drills, are also needed. All these are inputs for the construction
company. And the output will be buildings. We will use a function to describe the quantitative relation between inputs and outputs, which simply says a certain amount of inputs
can produce a certain amount output. We call this production technology. In the model we
are considering, the production technology is exogenous for producers. In other words, the
producers will take the technology as given. And the activities like R&D are excluded from
the model. We are interested in the question–how producers choose the quantity of output
given a production technology and the market prices of its outputs and inputs. We think
the producers choose the quantity of output so as to maximize its profit. In other words, we
think the profit maximization is the objective that firms are trying to achieve. Is that true
?...
Example
Let’s consider an Island where people want to produce baskets. Suppose in order to produce
baskets, they only need labor as input, and he is given the following production technology.
Labor (hours)
1
2
3
4
5
...
Basket (quantities)
2
5
7
8
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 works
two hours, 5 baskets will made. It is often convenient to write down the inverse relation of
production which says in order to produce one unit of output the minimal amount of input
is needed. Suppose the inverse relation is as follows,
Basket (quantities)
1
2
3
4
5
...
Labor
50 min
one hour
one hour and 15 min
one hour and 35 min
two hours
...
The table above says: if the man wants to make 1 basket he needs to work 50 minutes; if the
man 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 of
production. Since the only input is labor, and we know in order to produce a certain amount
of baskets, how many hours are needed, and then we can calculate the market value of the
labor used to produce such amount of baskets.
Basket (quantities)
1
2
3
4
5
...
Labor
50 min
one hour
one hour and 15 min
one hour and 35 min
two hours
...
Total Cost (dollars)
5/6
1
1.25
1.59
2
...
The table above simply says that if 1 basket is produced, one hour of labor is used and it has
the 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 the
market price of basket is 0.4$ per unit, we know,
Basket (quantities)
1
2
3
4
5
...
Revenue (dollars)
0.4
0.8
1.2
1.6
2
...
Everyone knows that the profit is just the difference between revenue and cost. Then we see
that,
3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT
Basket (quantities)
1
2
3
4
5
Revenue ($)
0.4
0.8
1.2
1.6
2
Cost($)
5/6
1
1.25
1.59
2
29
Profit($)
-0.43
-0.2
-0.05
0.01
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 production
technology from which we can derive the cost function for the producer. And then we move
on 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?. Actually
it 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. We
can describe a production technology basically in two ways: write the production function
y = f (x) and draw the curve in the graph. In this section, we only give the graphical
presentation.
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 be
produced if the one more unit of the input is added. Now suppose I give you a production
technology presented by a curve. How can you find out the marginal product? Take the
following curve for example. (Figure 1)
We can see that the slope of the production function is just the marginal product. If the
marginal product is decreasing as the quantity of input is increasing, we call this production
technology has the diminishing marginal return property. Graphically, that means, the slope
of the production function will be decreasing. If the marginal product is increasing as the
quantity of input is increasing, we call this production technology has the increasing marginal
return property. The economic meaning of diminishing marginal return is that as the quantity
of input used for production is increasing, its productivity is actually decreasing, since it
generates 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 certain
level of output,y, the minimal amount of input x is needed to produce such amount of output.
30
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Figure 3.1: Marginal Product : An example
Figure 3.2: Production Function
3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT
Figure 3.3: Inverse Function
Figure 3.4: Inverse Function
31
32
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
The next question is: given a production function, how can I draw the inverse function of it
in the graph. See Figure 4.
3.1.1
Cost
There are several types of cost. There are three relevant for us. The important thing about
them is that they are all functions of the quantity of output. In other words, they will vary
as the quantity of output changes.
Total Cost
Total cost: as its name suggests, it just the total cost associated with production. In our
case, it only comes from the cost of buying input. Without loss of generality, we can assume
the price of input is 1 $ per unit. Therefore, the total cost of production is just 1 times the
quantity of input used to produce certain amount of output.
Back to our basket-producing example.
Basket (quantities)
1
2
3
4
5
Labor
50 min
one hour
one hour and 15 min
one hour and 35 min
two hours
Total Cost (dollars)
5/6
1
1.25
1.59
2
From the example above, we can see that the inverse function of production is used to calculate the total cost.
Average Cost
Average cost means in average what is the cost for producing one unit of output, i.e., the
total cost/the number of output.
Example:
Basket (quantities)
1
2
3
4
5
Labor
50 min
one hour
one hour and 15 min
one hour and 35 min
two hours
Total Cost ($)
5/6
1
1.25
1.59
2
Average Cost($ per unit)
5/6
0.5
0.42
0.3975
0.4
Marginal Cost
Marginal cost means the additional cost if one more unit is produced.
Example:
3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT
33
Figure 3.5: Cost: Example1
Basket (quantities)
1
2
3
4
5
Labor
50 min
one hour
one hour and 15 min
one hour and 35 min
two hours
Total Cost (dollars)
5/6
1
1.25
1.59
2
Marginal Cost(Dollars per unit)
NA
1/6
0.25
0.34
0.41
Now the question is: graphically given a production function how to figure out the cost
function. And then how to calculate the average cost and marginal cost. We consider two
special cases. See Figure 5 and Figure 6. From Figure 6, we see there is a well-known result
between average cost and marginal cost. In the presence of fixed cost, the average cost reaches
its minimum as it is equal to the marginal cost.
34
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Figure 3.6: Cost: Example2
3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT
35
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 by
using the quantity of output. Now in this section, we want to have a concrete characterization
as we did in the theory of demand. I said in most of cases, the optimal choice by a typical
consumer who is trying to maximize his/her utility should satisfy the condition (sometime we
call it the marginal condition), i.e., the marginal rate of substitution is equal to the relative
price. 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 see
what 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 revenue
should 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 condition,
marginal cost = the price of output
, NOTE: we have to pay very attention to the fact that we have assumed the price of input
is 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 maximum does not exist at all.
36
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Figure 3.8: Graphical presentation of revenue
Figure 3.9: Graphical presentation of profit
3.1. PRODUCER BEHAVIOR WITH SINGLE INPUT
37
2. we know that the marginal cost varies as the quantity of output changes. Therefore we
could find the quantity which satisfies this condition. That quantity is the one which
maximizes the profit.
3. There is one more assumption for the statement to be true, i.e., we are considering the
producers in the Competitive Market. What do you mean by Competitive Market? If a
market 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 very
subtle. You may say well definitely their action can influence the price; for example, if
one producer cuts its production, the price will go up as long as the demand remains the
same. You are right. They actually can influence the price but I assume they feel that
they can not. Is that a bad assumption? Not really, when the number of producers in
this market goes to infinity, any single producer has very tiny influence on the market
price. And therefore they feel they can not influence the price. For example, when
you go to HEB, when you are buying 1 gallon of milk, are you thinking that if you
buy one more gallon, the price of milk will mark up significantly? I think you won’t
because your purchasing takes up only a tiny part of the demand. This idea applies to
the producer side as well. If you are a producer of basket in island C, and there are
millions of and tons of competitors out there, will you think you can influence the price
of basket? I don’t think so because your selling only forms a tiny part of the supply
of basket. You will see an amazing justification of this assumption later. I can briefly
mention it here. Later we will discuss monopoly. Definitely the monopolist is not a
price-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 set
the price, usually higher than the price in competitive market. And then we will see
the so-called duopoly, where there are two producers. And we can see the market price
in duopoly which is also higher than the price in competitive market but lower than
the price set by monopolist. Question: as we are adding more producers and as the
producers’ number goes to infinity, will the price converges to the price in competitive
market? 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 contradiction. 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 more
unit, the cost associated with this one more unit (NOT the total cost) is just the marginal
cost. By selling such one more unit, the additional revenue is just the price of output. And
if marginal cost < the price of output, the producer thinks it is profitable to produce one
more 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 room
to 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 producing
that one unit, i.e., the marginal cost, and at the same time it loses the revenue, i.e., the price
of output. Since marginal cost > the price of output, it implies that actually producing
38
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
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 profit
has been achieved the maximum, the marginal cost should NOT be higher than the price of
output 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 Figure
10.
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, and
then by profit-maximization hypothesis we find the optimal quantity of output. As long as
we know the cost function, we back to the analysis we have done in the previous section
where there is one single input. There will be no essential difference between this section and
the previous one in the manner of deriving optimal quantity of output. The differences take
place only in deriving the cost function. Therefore, we only talk about how to derive cost
function when there are two inputs. And then we should be able to know how to find the
optimal quantity of output with two inputs.
Let’s back to our story: the producer of basket. Now suppose producing baskets needs labor
and bamboo as inputs.
Suppose the producer has the following technology,
3.2. PRODUCER BEHAVIOR WITH TWO INPUTS
Figure 3.11: profit maximization : exception
Figure 3.12: profit maximization : Another Graphical Presentation
39
40
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
labor(hours)
30 mins
1 hr
1 hr
2 hr
3 hr
1 hr
...
bamboo(quantities)
1
1
2
3
2
4
...
Basket(quantities)
1
1
2
3
2
2
...
I want to use the example to illustrate the idea of deriving cost function when there are two
inputs. Recall, the cost function simply says in order to produce certain amount of output
what the total cost should be to produce that amount. Suppose now the producer wants to
produce 6 baskets. The first step is to find out all the combinations of labor and bamboo
such that producing 6 units of basket is possible. According to the technology the producer
has, suppose we have the following combinations,
labor(hours)
3 hr
4 hr
5 hr
6 hr
3 hr
3 hr
...
bamboo(quantities)
6
6
6
6
7
8
...
Basket(quantities)
6
6
6
6
6
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 6
units of basket. In contrast with the case where there is one single input, here we have a
bunch of choices to produce certain amount of output. The question is : what is the rule
for the producer to pick one combination of inputs given the quantity of output? Here we
assume 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 to
produce certain amount of output.
labor(hours)
3 hr
4 hr
5 hr
6 hr
3 hr
3 hr
...
bamboo(quantities)
6
6
6
6
7
8
...
Cost ($)
3+3=6
4+3=7
5+3=8
6+3=9
3+3.5=6.5
3+4=7
...
From the table above, we see that the producer will work 3 hours and use 6 units of bamboo
to produce 6 baskets, and this costs 6 $ which is the minimal among all other production
choices.
3.2. PRODUCER BEHAVIOR WITH TWO INPUTS
41
Now suppose the producer wants to produce 8 units of basket. In this case, we can also find
out all the combinations which makes this possible, compare them by their costs, and choose
the cheapest one. Finally we know producing 8 units of baskets will cost the producer 8$.
labor(hours)
4 hr
4 hr
5 hr
6 hr
4 hr
4 hr
...
bamboo(quantities)
8
9
8
8
10
11
...
basket(quantities)
8
8
8
8
8
8
...
Cost ($)
4+4=8
4+4.5=8.5
5+4=9
6+4=10
4+5=9
4+5.5=9.5
...
Thus we see we could associate any level of output with a minimal cost,and then we derive
the cost function. Suppose we have the following cost function for the man in island C.
basket(quantities)
1
2
3
4
6
8
...
Cost ($)
1
2
3
4
6
8
...
Actually, I did NOT make up this table randomly. What we observe is a linear cost function
which is resulted from the special technology function we are using implicitly behind the series
of table I gave above. In general, if the production function is Leontief, the cost function
derived 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 money
paid for the service provided by capital is called interest and the money paid for the labor
service is called wage. We use r to denoted the interest, $ per unit and w to denote the wage
rate, $ per unit.
For example, suppose, capital is a machine, if it is used for a hour, the producer needs to pay
400$, therefore the interest is 400$ 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.
42
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Marginal Product
We have already seen the definition of marginal product in the case of single input. We can
apply the same definition here with some caution. The marginal product of capital is just
the additional amount of output produced if one more unit of capital is added. The marginal
product of labor is just the additional amount of output produced if one more unit of labor
is added. Take the construction company for example,
Capital
Machine(Hour of Usage per day)
2 hrs
3 hrs
4 hrs
5 hrs
6 hrs
...
labor
Worker(persons per day)
10
10
10
10
10
...
output
buildings per month
0.5
0.9
1.2
1.4
1.5
...
Marginal Product of Capital
buildings per hour of usage per day
NA
0.4
0.3
0.2
0.1
...
Notice, while calculating the marginal product of capital,the amount of labor remains the
same. And we also notice that as the usage of the machine per day is increasing, the marginal
product of capital is decreasing. Let’s take a look at an example of the marginal product of
labor.
labor
Worker(persons per day)
11
12
13
14
15
...
Capital
Machine(Hour of Usage per day)
3 hrs
3 hrs
3 hrs
3 hrs
3 hrs
...
output
buildings per month
1
1.5
1.8
2
2.1
...
Marginal Product of Labor
buildings per person per day
NA
0.5
0.3
0.2
0.1
...
Isoquant
Isoquant is a curve which collects all the combinations of inputs which produce the same
amount of output.
Let’s look at the example of the construction company. If the construction company needs to
produce one building in a month, the following combinations of capital and labor are possible
choices,
labor(persons per day)
11
10
9
8
7
...
machine(hours of usage per day)
3 hrs
3.5 hrs
4.1 hrs
4.9 hrs
6 hrs
...
Marginal Rate of Technical Substitution
The marginal rate of technical substitution (MRTS) of capital with respect to labor means if
3.2. PRODUCER BEHAVIOR WITH TWO INPUTS
43
Figure 3.13: MTRS
the one unit of capital is reduced in order to produce the same amount of output, the amount
of labor needs to be added. The marginal rate of technical substitution of labor with respect
to capital means if the one unit of labor is reduced in order to produce the same amount of
output, the amount of capital needs to be added.
Let’s look at the example of the construction company.
labor(persons per day)
11
10
9
8
7
...
machine(hours of usage per day)
3 hrs
3.5 hrs
4.1 hrs
4.9 hrs
6 hrs
...
MRTS of labor
NA
0.5
0.6
0.8
1.1
What we observe is that as the quantity of labor used for production is decreasing it needs
more extra capital, additional amount of capital to make 1 building within a month for the
company. In other words, the quantity of labor is increasing, the marginal rate of technical
substitution of labor is decreasing. This is called diminishing marginal rate of technical substitution.
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 marginal
rate of substitution in the theory of demand,
M RT SK→L =
M PK
M PL
where M T RSK→L is the marginal rate of technical substitution, M PK and M PL are marginal
product of capital and labor respectively.
44
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Figure 3.14: The shape of Isoquant
Why this is true? The question is left to you to think about.
Returns to scale
Suppose the quantities of capital and labor are doubled. The question is: will the quantity
of 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 exactly
doubles, there are constant returns to scale. We can see that Returns to scale is the rate
at which output increases as inputs are increased proportionately. Take the construction
company for example. If there are increasing returns to scale, we should have following
situations,
labor
11
22
33
...
machine
3
6
9
...
percentage increase of inputs
NA
100%
200%
...
buildings
1
2.5
3.8
...
percentage increase of output
NA
150%
280%
...
buildings
1
2
3
...
percentage increase of output
NA
100%
200%
...
Constant Returns to scale,
labor
11
22
33
...
machine
3
6
9
...
percentage increase of inputs
NA
100%
200%
...
Decreasing Returns to scale,
3.2. PRODUCER BEHAVIOR WITH TWO INPUTS
labor
11
22
33
...
3.2.2
machine
3
6
9
...
percentage increase of inputs
NA
100%
200%
...
45
buildings
1
1.8
2.5
...
percentage increase of output
NA
80%
150%
...
Cost Minimization
Our goal here is to derive the cost function. We have already known that given a certain
level of output, there are a bunch of possible combinations of inputs to produce that amount
of output and those choices are on the same isoquant. By cost-minimization hypothesis, we
think the producer will choose a combination of inputs which has the lowest total cost. What
we are heading for is to find a condition which can characterize the cost-minimizing choice
of inputs given a certain level of output. And we will show how to find the cost-minimizing
choice graphically.
In order to do that, we need to introduce the concept of Isocost. Isocost is a straight line
which 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, and
the price of bamboo is 0.5$ per unit. And we are now looking for the combinations of labor
and bamboo such that the total cost is 6$. It is not hard to check the following combinations
cost 6$ in total,
labor(hrs)
0
1
2
3
4
5
6
bamboo(quantities)
12
10
8
6
4
2
0
total cost($)
6
6
6
6
6
6
6
Definitely there are many more others. From the table above, we can see the isocost with
the total cost 6$ 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)
0
1
2
3
4
5
6
7
bamboo(quantities)
14
12
10
8
6
4
2
0
total cost($)
7
7
7
7
7
7
7
7
46
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
Figure 3.15: Isocost
Figure 3.16: Cost Minimization
What we observe is that the isocost with the cost 7$ is parallel to the isocost with the cost
6$. This really is because the prices of inputs remains the same. Now the question is: what
does the slope of isocost mean in economic terms?
Now we are ready to show how to find the cost-minimizing choice of inputs graphically, given
the prices of inputs and the quantity of output to be produced. Step1
Given the prices of inputs, we can draw a series of isocost with different costs.
Step2
Given the quantity of output, we can draw the isoquant, the combinations of inputs on which
are able to produce that amount of output.
Step3
Looking 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 utility
maximization.
M T RS = Relative P rice
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 equilibrium
concept, 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 machines
are exogenous for our analysis and we are NOT attempting to understand how the prices of
them are determined.
The question we are trying to answer is : suppose there is a competitive market we are
considering, and suppose there are a bunch of producers in this market. Given the demand
curve, 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 profit
is 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 competitive
market.
So how can we find the equilibrium quantity and equilibrium price?
1. First we derive the supply curve of a typical producer
48
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
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 quantity
and equilibrium price.
Example 1
Suppose we are considering the market of basket in island C. And each producer has the
following type of cost function which is linear. see Figure 24. Now we are looking for the
price level such that for each producer the profit is zero. Since all producers have the same
cost function, we consider a representative producer. This cost just tells us that for a typical
producer 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 be
a 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
3.3. LONG RUN EQUILIBRIUM IN COMPETITIVE MARKET
49
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 of
output 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 of
basket is above 2, the quantity supplied will be infinity. When the price of basket is below
2, the quantity supplied will zero. When the price is exactly equal to 2, the supply will
be anything.(It does NOT mean it will be infinity. It only means producer will be equally
satisfied with any level of output simply because all of them bring zero profit.) So the supply
curve should be as follows. See Figure 27. We see it should be a horizontal line. Then
combining 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 profit
and the quantity is somehow determined by the demand curve. To see this, suppose for some
reason, the demand curve shifts up. The equilibrium quantity will increase.
Example 2
Now let’s look at a different example. Suppose in this basket industry, all producer have the
some cost function indicated by Figure 28. We have already known the marginal cost and
50
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
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 find
in Figure 29 is p. Then we see when the price of output is larger than and equal to p, the
quantity produced should make the marginal cost equal to the price. Therefore, the marginal
cost above the price p part is the supply curve. When the price is lower than p, the producer
will produce nothing. see Figure 30
Now what do you think the supply curve of the whole basket industry should be? When the
price is larger than p, each producer wants to produce something, and more importantly they
are gaining non-zero profit, which attract more and more producers. Since we have assumed
free entry, as long as the profit is not zero, there will be infinitely many producers entering
the market which makes the total supply of baskets amount to infinity even though each
producer produces something finite. When the price is lower than p, no producer will stay
in 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
3.3. LONG RUN EQUILIBRIUM IN COMPETITIVE MARKET
51
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 the
market.) Therefore, the supply curve of the whole industry should be as follows, see Figure
31.
From this two examples, we could see in the competitive market, the supply curve is a horizontal line. And the market price is equal to the marginal cost, and the equilibrium quantity
is 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? If
you relax any assumption we made, we will get a upwards sloping curve.
52
CHAPTER 3. PRODUCER BEHAVIOR IN THE COMPETITIVE MARKET
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 two
questions:
1. Will this producer behave in a different manner comparing with a typical producer in
a competitive market?
2. Will the equilibrium price and quantity in such a market be different from the ones in
a competitive market?
Recall the behavioral assumption we made for a typical producer in a competitive market is
that it takes the price of output and inputs as given. We have to give up this assumption
when we are analyzing the behavior of a monopolist. Since the monopolist by definition is
the single supplier, and thus it definitely feels it has the power to influence the price. So we
think a monopolist will consider the impact of its action (decision of how to produce) on the
market price.
Now the next question is : do you think the monopolist will choose the quantity of output
arbitrarily? No. The monopolist will try to maximize the profit as any producer does. The
difference between the producer in a competitive market and a monopolist is the monopolist
will 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 by
the equation,
p=9−q
where p denotes the price and q denotes the quantity. we can also describe this demand
schedule in a table as follows.
53
54
CHAPTER 4. MONOPOLY
Figure 4.1: Demand Curve
Price of Basket
($ per basket)
1
2
3
4
6
7
8
9
...
Demand for Basket
(Quantities)
8
7
6
5
3
2
1
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 table
below), which is sometime called the inverse demand function. This function simply tells you
if the quantity demanded is this much, what the price should be so that the consumers will
demand that much. In the table below, we see that when the quantity is 8, the price should
be 1$ per basket so that the consumers will demand 8 baskets.
Quantity Demanded for Basket
(Quantities)
8
7
6
5
3
2
1
0
...
Price of Basket
($ per basket)
1
2
3
4
6
7
8
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? This
piece of information is important for the monopolist to know what the revenue will be if the
he produces certain amount of baskets. Suppose, he wants to produce 6 baskets, and then he
knows that the price will be 3$ per basket. Because, if the price is lower than 3, the quantity
demanded will be more than 6; and if the price is higher than 3, the quantity demanded will
be less than 6, according to the table we have above. And thus, the revenue for producing 6
baskets will be 6 times 3, 18 $. Now let’s work for all levels of output.
Quantity produced
(Quantities)
8
7
6
5
4
3
2
1
0
...
Price of Basket
($ per basket)
1
2
3
4
5
6
7
8
9
...
Revenue
($)
8
14
18
20
20
18
14
8
0
...
Now the monopolist is trying to calculate the profit for producing all levels of output and
then he can decide which level of output will generate the highest profit. Suppose the cost
function is linear for the monopolist, and its functional form is
c(q) = q
where q is the quantity. we can describe the cost function in the table below.
Quantity produced
(Quantities)
8
7
6
5
4
3
2
1
0
...
Now, we combine the revenue and cost.
Total Cost
($)
8
7
6
5
4
3
2
1
0
...
56
CHAPTER 4. MONOPOLY
Quantity produced
(Quantities)
8
7
6
5
4
3
2
1
0
Price of Basket
($ per basket)
1
2
3
4
5
6
7
8
9
Revenue
($)
8
14
18
20
20
18
14
8
0
Total Cost
($)
8
7
6
5
4
3
2
1
0
Profit
($)
0
7
12
15
16
15
12
7
0
Then we see the monopolist will produce 4 baskets which generates the highest profit.
4.2
4.2.1
Profit Maximization for a monopolist
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 revenue
function should be
R = p × q = p(q) × q = (a − b × q) × q = a × q − b × q 2
We need to introduce the concept marginal revenue.
Marginal revenue is just the additional revenue the producer will have if one more unit of
output is produced. Precisely it should be first order derivative of the revenue function, we
see 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 profitmaximizing choice. Let me first state the condition.
M arginal Revenue = M arginal Cost
Then let me explain why this is true for profit-maximizing quantity of output. Suppose,
it is not true and M arginal Revenue > M arginal Cost. What will be happening? The
monopolist will find it is profitable to produce one more unit. What if M arginal Revenue <
4.2. PROFIT MAXIMIZATION FOR A MONOPOLIST
57
Figure 4.2: Demand Curve and Marginal Revenue
M arginal Cost? The monopolist will find it is profitable to reduce one more unit. Do you
think this condition can be applied to the producer in a competitive market? The answer is
yes.
4.2.3
Examples
Example 1 Suppose the monopolist has the following type of cost function.See Figure 3. And
suppose the monopolist is facing a demand curve in Figure 3. And then we see what the
marginal cost and average cost should be. Now we are looking for the quantity which makes
the 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 at
the 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 can
calculate the maximal profit earned by the monopolist.
I use this example to illustrate the steps to find the optimal quantity of output produced by
a 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 same
graph.
2. Find the intersection of marginal cost and marginal revenue. Then we see the quantity
produced by a monopolist.
3. Plug in the quantity produced by a monopolist into the inverse demand function,(Or
graphically find the price which corresponds to the quantity in the demand curve.)
4. Finally we can calculate the profit earned by the monopolist.
58
CHAPTER 4. MONOPOLY
Figure 4.3: Example 1
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 what
the marginal cost and average cost should be.
From this example, we can have some idea about how the monopolistic market differs from
the 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 is
facing is
p(q) = 10 − q
Thus the revenue function should be,
R = (10 − q) × q = 10q − q 2
The the marginal revenue function should be,
M R = 10 − 2q
4.2. PROFIT MAXIMIZATION FOR A MONOPOLIST
Figure 4.4: Example 2
59
60
CHAPTER 4. MONOPOLY
Figure 4.5: Example 3
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. what
about the market price? Plug in the quantity into the demand function, we see that the
market price is 6. And how much does this monopolist earn? The profit should be the
difference between revenue and cost, that is, 6 × 4 − 2 × 4 = 16.
Now we are ready to compare the monopolistic market and competitive market. What we
have 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 all
producers are gaining zero profit. See table below. See Figure 5.
–
Equilibrium price
Equilibrium quantity
Profit
Monopoly
6
4
16
Competitive Market
2
8
0
Chapter 5
Duopoly
In this lecture, we will see three types of duopoly model. The situation we are considering is a
market where there are only two producers. We are interested in the following question, how
producers in such market behave differently from a typical producer in a competitive market
and a monopolist, and how the equilibrium price and quantities differ from the competitive
and 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 producing
are homogeneous, that is, indistinguishable from the consumers’ point of view. And therefore
they will the share the market. It means, the number of baskets produced by Brian plus the
one produced by Justin is the total supply of basket. The decision each producer has to make
is just simply to choose a quantity of output to produce. And their goal is just to maximize
the profit. But while they are making this decision, they have to consider how many baskets
their 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 the
market price because of the presence of a competitor. For example, if Brian tries to raise the
price by reducing the quantity, but at the same time Justin is increasing the quantity, which
finally offsets the effort to raise the price made by Brian. Even though each producer can
not completely influence the price, in constrast with a typical producer in the competitive
market, 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’s
decision. (Since the structure of the problem is kind of symmetric, we can work out Justin’s
problem 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 following
table. From the table below, I should produce 2 baskets if Justin is producing 5 baskets. Since
I have the maximal profit in this case.”
Baskets produced by Brian
0
1
2
3
4
5
Baskets produced by Justin
5
5
5
5
5
5
Total Supply
5
6
7
8
9
10
Market price
6
5
4
3
2
1
Revenue
0
5
8
9
8
5
Cost
0
2
4
6
8
10
Profit
0
3
4
3
0
-5
Then Brian can find out the best response to all possible numbers of baskets Justin is producing. Then he has the following table,
Baskets produced by Justin
0
1
2
3
4
5
6
7
8
9
Best Response by Brian
4.5
4
3.5
3
2.5
2
1.5
1
0.5
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
0
1
2
3
4
5
6
7
8
9
Best Response by Justin
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
5.1. COURNOT DUOPOLY
63
Now the question is how much they will actually produce. Suppose Brian will produce 5
baskets, then Justin want to produce 2 baskets. If if Brian knows that Justin is producing
2 baskets, he actually produces 3.5 baskets, but if Justin knows that Brian now wants to
produce 3.5 baskets, he won’t produce 2 any more....
Where can we end such argument which appears to be endless? Now suppose Brian wants
to produce 3 baskets, in response then Justin wants to produce 3 baskets as well. When
Brian 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, they
are actually looking for the best response to their own expectation of their competitor’s decision; second, how we come up with a solution concept for this type of problem, that is, each
producer 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, Brian
and 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 same
demand function,
p(q) = a − q
where a, b > 0 constant. We use q1 to denote the quantity of baskets produced by Brian, and
q2 to denote the quantity of baskets produced by Justin. The problem for Brian is to choose
q1 given q2 to maximize the profit.
First, we know q1 + q2 is the total quantity supplied, then we know the market price should
be p = a − (q1 + q2 ), therefore, the total revenue for Brian given q2 is R = (a − q2 ) × q1 − q12 ,
then the marginal revenue for Brian is M R = (a − q2 ) − 2 × q1 . Let the marginal cost equal
to marginal revenue. We have (a − q2 ) − 2 × q1 = c. Do some algebra, we see that,
q1 =
a − c q2
−
2
2
We call this function, the best response function by Brian, which is the function of the number
of baskets produced by Justin. We can also derive the best response function by Justin in
the exactly same way, we can have,
q2 =
a − c q1
−
2
2
See Figure 1.
Now the question is what the equilibrium concept is. The equilibrium concept we will use is
called Nash equilibrium by John Nash. But for me the idea has been known among economists
for a quite long time.
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CHAPTER 5. DUOPOLY
Figure 5.1: Best Response Function
Figure 5.2: Equilibrium: Example
”. . . , the general idea of equilibrium, refers to a certain type of relationship
between the plans of different members of a society. It refers to, that is, the case
where these plans are fully adjusted to one another, so that it is possible for all
of them to be carried out because the plans of any one member are based on the
expectation of such actions on the part of the other members as are contained in
the 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 the
basis 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 all
their plan can be carried out when their expectation becomes true. For example, let’s make
a = 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 formula
above,we see that he will produce 3.5 baskets. We see this plan made by Justin is based on
his expectation of how many baskets Brian will produce. If Brian anticipates that Justin will
produce 3.5, he will produce 2.75 which is not what Justin anticipates. In other words, their
plans 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 do
what 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 q2
a − c q1
− q2 =
−
2
2
2
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. The
result 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.
–
Equilibrium price
Equilibrium quantity
Profit for each producer
Monopoly
6
4
16
Competitive Market
2
8
0
Cournot Duopoly
≈ 4.6
≈ 5.3
≈ 6.76
The conclusion is the price in cournot duopoly will be higher than the competitive market
and lower than the monopoly, and the quantity will be lower than the competitive market
and higher than the monopoly.
Now suppose the number of producer becomes three. What is the equilibrium price and
quantity? What if the number of producer becomes 100? ...
66
5.2
CHAPTER 5. DUOPOLY
Stackelberg Duopoly
The same as Cournot duopoly, Stackelberg duopoly also considers a market with two producers. The difference is in Stackelberg duopoly one producer is the leader in the market and
the other is the follower. In Cournot duopoly, two producers choose the quantity of output
simultaneously. But here, the leader will choose the quantity first, and then it is the follower’s turn to choose. The situation becomes different because when the follower is making
the decision, the leader’s decision has become a given condition for him/her in Stackelberg
duopoly while in Cournot duopoly two producers also need to consider what their competitor
will 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 the
market 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 they
have is c(q) = 2q. The real thing between them is : On one hand, since Brian is the leader
and therefore he can take a large portion of demand, and this will force Justin to restrict the
production otherwise the price will be too low to earn some profit; on the other hand, Justin
can also give Brian some trouble by increasing the production such that both of them will
not earn some profit. For example, Suppose Brian chooses to produce 7 baskets, notice this
is a large portion of demand, since when the total quantity supplied is 8 baskets the market
price will be equal to the marginal cost. Now how Justin deals with this? If Justin produces
1 baskets, the profit he can earn is zero. He can earn some profit only when he produces less
than 1 baskets. Let’s calculate what the best decision for him is in this case.
Baskets produced by Justin
0
0.25
0.5
0.75
1
Baskets produced by Brian
7
7
7
7
7
Total Supply
7
7.25
7.5
7.75
8
Market price
3
2.75
2.5
2.25
2
Revenue
0
0.6875
1.25
1.6875
2
Cost
0
0.5
1
1.5
2
Profit
0
0.1875
0.25
0.1875
0
Then we see that Justin will produce 0.5 baskets and he will earn 0.25 $ if Brian chooses to
produce 7 basket. On the other hand, Brian will earn 3.5 $. Justin might think he can do
much better. He goes to threaten Brian, saying ”If you produce more than 4 baskets, I will
punish you by producing more baskets to make you lose money.” For example, in the case we
just talked about, if Brian produced 7 baskets, what Justin can do to make Brian crazy is to
produce more than 1 basket, say 2 baskets. In a result, the market price will be 1 and Brian
actually loses 7 $ while Justin just loses 2$. One thing here we have to pay attention to is
Brian has already made the decision while Justin is making his choice. For Brian , after he
made the choice, it it irreversible. So if Brian is trying to take a large portion of the demand,
he is also taking the risk of being punished by Justin.
Now the question is will Brian be threatened ? Put it in the other way, will Brian take much
credit of what Justin is saying? in other words, will Brian believe Justin will what he is saying?
5.3. BERTRAND DUOPOLY
67
The Answer is NO. What Justin said is called Empty Threat. The threat is empty because
Justin will not do what he said. Since we have known if Brian produces 7 baskets, the best
choice for Justin is to produce 0.5 baskets, not 2, or something else. Of course if Justin is
emotionally unpredictable person, we can not expect him to do something reasonable. But
the point is the reasonable choice for Justin in that case is to produce 0.5 baskets Not 2.
5.3
Bertrand Duopoly
In this section, we consider Bertrand duopoly model named after Joseph Louis Bertrand, a
French mathematician. In this model, we consider a market of two producers as before. Here,
the choice variable for the producer is NOT the quantity of output anymore. Instead, the
producer is choosing the price at which level its product is sold. The demand for its product
is dependent on its own price and its competitor’s price. The goal for the producer is still to
maximize profit.
Let’ s take a look at an example. Suppose there are two producers, Brian and Justin, in the
market of basket. The demand curve they are commonly facing is p(q) = 10 − q. Suppose
they have the same cost function, c(q) = 2q. Brian and Justin will set their prices simultaneously. The producer with the lower price will take over the whole demand. If the prices are
the same, two producers share the demand. For example, if for each basket Brian charges
2$ but Justin charges only 1$, consumers will go to buy baskets produced by Justin. And
the quantity will be 9, according to the demand curve. If both of them charges 2$ for each
basket, the total quantity demanded will be 8, and each one of them will get a half of 8, that
is, 4. So we see that the price set by one producer is dependent on his anticipation of the
price that his competitor charges. Now let’s me illustrate how Brian and Justin are making
the decision on the basis of their anticipation of what the other will do. Suppose Brian thinks
Justin will charge 4$ per basket, any price lower than 4 will be a good choice for him, say
3.5. 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 price
further. So this process will continue. Until....?? Until the price is zero? No until the profit
is zero. When ? when the price reaches 2$ per basket. Why? Suppose Brian knows that
Justin 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 for
any 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 this
question. Suppose Brian charges 2 dollars also. The profit he gains is zero. If he charges more
than 2 dollars, no consumer will come to him and thus he gains nothing. So he is actually
indifferent 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 will
charge 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
68
CHAPTER 5. DUOPOLY
”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 for
Brian’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, the
lower the demand for his product; the higher the price set by Justin, the higher the demand
for 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 its
competitor is exogenous; in other words, each producer takes the price set by its competitor
as given.
Step 1 The first step is to derive the best response function for each producer. Let’s first
look at Brian’s problem. Given the price set by Justin, p2 , and facing the demand curve for
his own productq1 = a − p1 + 1/2p2 , Brian is trying to maximize the profit by setting the
pricep1 .
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?
P rof it = R − C = p1 × (a − p1 + 1/2p2 ) − c × (a − p1 + 1/2p2 )
Then we have
P rof it = (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 is
given,
a + c p2
p1 =
+
2
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 by
Justin by doing the similar thing,
p2 =
a + c p1
+
2
4
Let’s draw the best response function in the graph. As before, the equilibrium should be the
intersection of the best response functions. We can know then, in the equilibrium the price
set by two producers should be the same, that is, 2(a+c)
3 . m should be the intersection of the
best response functions. We can know then, in the equilibrium the price set by two producers
should be the same, that is, 2(a+c)
3 .