1 Optimization and Allocation
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 5, 2007
Lecture 1
Overview: Themes, Types of Markets, Economic
Measurement, Economic Analysis
Microeconomics is a branch of economics that studies how individuals and
firms make decisions to allocate limited resources, typically in markets where
goods or services are being bought and sold.
Outline
1. Chap 1: Optimization and Allocation
2. Chap 1: Definition and Various Type of Markets
3. Chap 1: Economic Measurement
4. Chap 1: Economic Analysis
1
Optimization and Allocation
Consumer theory. Maximize preference (with limited income or time)
Producer theory. Maximize profit (with limited capital)
2
Definition and Various Type of Markets
Market. A place where buyers and sellers come together to exchange some
product or good.
Product and Factor Markets
Market
Product Market
Factor Market
Buyers
individuals
firms
Sellers
firms
individuals
Table 1: Product and Factor Markets.
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3 Economic Measurement
2
In a factor market, buyers are firms who need to hire workers and borrow
money for capital expenditure, and sellers are individuals who provide labor
and save money in banks.
Types of Markets Based on Influence on Price
Market Type
Competitive
Monopolistic
Oligopoly
Monopoly
Monopsony
Oligopsony
Products
homogeneous
heterogeneous
Sellers
many
many
a few
one
many
many
Buyers
many
many
many
many
one
a few
Table 2: Types of Markets Based on Influence on Price.
Table 2 shows different markets based on product differentiation and influ­
ence on price. Influence on price increases in moving from Competitive markets
to Monopoly.
3
Economic Measurement
Flow and Stock Variables
Stock variables. Not measured with respect to time. e.g. price, wealth, in­
ventories.
Flow variables. Measured per some unit of time. e.g. production, consump­
tion, income.
Two additional flow variables:
Expenditure.
EXPENDITURE = PRICE × CONSUMPTION.
Revenue.
REVENUE = PRICE × PRODUCTION.
Prices
Nominal price. The absolute or current dollar price of a good or service when
it is sold.
Real price. The price relative to an aggregate measure of prices or constant
dollar price. It also measures prices relative to others. Price after adjust­
ment for inflation.
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4 Economic Analysis
3
CPI(Consumer Price Index). Total spending on a market basket of
goods.
Formula of inflation rate:
(Gross) Inflation rate =
CPI (current year)
.
CPI (base year)
Formula of real price:
Real price =
Nominal price (current year)
,
Inflation rate (base year to current year)
or
Real price =
Nominal price (current year)
.
CPI(current)/CPI(base)
Example. For instance, the average tuition of college:
Year
1970
1990
2002
Nominal Price
2,530
12,018
18,273
CPI
38.8
130.7
181.0
Real Price (base year 1970)
2,530
3,569
3,917
Table 3: Average Tuition of College 1970 to 2002.
Notice that from 1970 to 2002 nominal price increases by 7 times but real
price increases by 1.5 times.
4
Economic Analysis
Positive analysis. Study the relationship of cause and effect (Questions that
deal with explanation and prediction).
Normative analysis. Analysis examining questions of what ought to be (Of­
ten supplemented by value judgments).
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1 Demand and Supply Curves
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 7, 2007
Lecture 2
The Basics of Supply and Demand
MARKET
⎧
⎫
⎪
⎪ BUYERS =⇒ DEMAND ⎪
⎪
⎨
⎬
⎪
⎪
⎩
SELLERS =⇒ SUPPLY
EQUILIBRIUM
⎪
⎪
⎭
Outline
1. Chap 2: Demand and Supply Curves
2. Chap 2: Equilibrium in the Market
3. Chap 2: Government Interventions
1
Demand and Supply Curves
Quantity Demanded and Quantity Supplied
QD (Quantity demanded). Depends on price.
QD = D(P ).
(1.1)
QS (Quantity supplied). Depends on price.
QS = D(P ).
Notes:
(1.2)
1. Market demand/supply is the sum of individual demands/supplies.
2. Assume individuals are price takers who cannot affect price.
Demand and Supply Curves
From Equations (1.1) and (1.2), draw demand curves and supply curves as
follows:
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Figure 1: Supply curve. Price higher, Figure 2: Demand curve.
Price
quantity supplied more.
higher, quantity demanded less.
Figure 3: Shift in supply curve.
Figure 4: Shift in demand curve.
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2 Equilibrium in the Market
3
Supply curve
See Figure 1 and Figure 3:
1. Change in price causes change in quantity supplied, on the graph, there
is movement along the curve accordingly.
2. Change in something other than price causes change in supply, on the
graph, the supply curve shifts.
Example. Production cost falls → supply curve S shifts to S’ (See Fig­
ure 3).
Demand curve
See Figure 2 and Figure 4:
1. Change in price causes change in quantity demanded, on the graph, there
is movement along the curve accordingly.
2. Change in something other than price causes change in demand, on the
graph, the demand curve shifts.
Example. People’s income increases → demand curve D shifts to D’ (Fig­
ure 4).
Substitutes and Complements
Substitutes. Increase in the price leads to an increase in the demand of the
other.
Example (Italian and French bread). Price of Italian bread increases, de­
mand of French bread increases.
Complements. Increase in the price leads to a decrease in the demand of the
other.
Example (Pasta and pasta sauce). Price of pasta increases, demand of
pasta sauce decreases.
2
Equilibrium in the Market
Equilibrium state:
• No shortage
• No surplus
• Equilibrium price clears the market.
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Figure 5: Demand and Supply curves. Equilibrium state.
Refer to Figure 5. (P0 , Q0 ) is the equilibrium state, which is the intersection
point of the demand and supply curves.
Price
Supply
=⇒ Change in equilibrium
Change in
Demand
Quantity
Surplus and Shortage
Surplus. Price P1 is higher than P0 and will fall down.
Shortage. Price P2 is lower than P0 and will raise up.
Comparative Static Analysis and Comparative Dynamics
Comparative static analysis. Compares the new and old equilibrium and
not the actual path through time of the change.
Comparative dynamic analysis. Traces out the path over time.
This course will cover primarily Comparative Static analysis.
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3 Government Interventions
5
Figure 6: Decrease in raw material prices.
Examples
Example (Decrease in raw material prices). Raw material prices �→ Supply
�→ Price � and Quantity � (Figure 6).
Example (Increase in income). Income �→ Demand�→ Price � and Quantity
� (Figure 7).
Dual shifts in supply and demand
When supply and demand change simultaneously, the impact on the equilibrium
price and quantity is determined by the size and direction of the changes and
the slope of two curves.
3
Government Interventions
How can government help sellers? Discuss two methods.
Problem Description
Assume that
QD = 10 − P,
QS = −2 + P.
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Figure 7: Increse in income.
The original equilibrium point is
P0 = 6,
QD0 = QS0 = 4,
and the revenue before government intervention is:
REV EN U E = P0 × QD0 = 6 × 4 = 24.
The government’s goal: increase sellers’ revenue.
Price Floor
The first method: set a price floor. Assume the lowest price is set to be 8, thus:
QD = 2,
QS = 6.
The revenue after using method 1 is:
REVENUE = P × QD = 8 × 2 = 16 < 24.
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3 Government Interventions
7
Subsidy
The second method: provide subsidy.
Customers get a 2 unit price refund per unit quantity bought, thus the
quantity demanded changes:
QD = 10 − (P − 2) = 12 − P.
The new intersection point is
P = 7,
QD = QS = 5.
The revenue after using method 2 is:
REVENUE = P × QD = 7 × 5 = 35 > 24.
For this example, providing subsidies achieves the government’s goal to increase
seller’s revenue, but setting price floor does not and even makes the revenue
less.
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1 Price Elasticity of Demand
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 10, 2007
Lecture 3
Elasticities of Demand
Elasticity. Elasticity measures how one variable responds to a change in an­
other variable, namely the percentage change in one variable resulting a
one percentage change in another variable. (The percentage change is
independent of units.)
Outline
1. Chap 2: Price Elasticity of Demand
2. Chap 2: Income Elasticity of Demand
3. Chap 2: Cross Price Elasticity of Demand
4. Chap 2: Comparison of Elasticity Over Short Run and Long Run
1
Price Elasticity of Demand
Price elasticity of demand. Price elasticity of demand measures the per­
centage change in quantity demanded resulting from one percentage change
in price.
D
EE
%△QP
=
=
%△P
△Q
Q
△P
P
.
Example Calculation
Figure 1 shows a demand curve:
Q(P ) = 8 − 2P.
When the price changes from 2 to 1, the price elasticity of demand is:
EPD |p=2→1 =
ΔQ
Q
ΔP
P
=
2
4
−1
2
= −1.
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1 Price Elasticity of Demand
2
Figure 1: Price Elasticity of Demand.
If the direction of change is opposite, from 1 to 2, then the price elasticity of
demand is:
ΔQ
−2
1
Q
D
EP |P =1→2 = ΔP = 61 = − .
3
1
P
The two quantities are different. To solve this conflict, consider small changes
in P and Q, and define:
dQ
P dQ
Q
EPD = dP =
.
Q dP
P
Thus, at the point P = 2, the price elasticity of demand is:
EPD |P =2 =
P dQ
2
= × (−2) = −1.
Q dP
4
Properties of Price Elasticity of Demand
1. Price elasticity of demand is usually a negative number.
2. |EP | > 1 indicates that the good is price elastic, perhaps because the good
has many substitutes; |EP | < 1 indicates that the good is price inelastic,
perhaps because the good has few substitutes.
3. Given a linear demand curve, EP is not a constant along the curve. For
example, for curve in Figure 1, EP = −∞ at top portion, but zero at
bottom portion.
4. Discuss two extreme situations: |EP | = 0, quantity independent of price
Figure 2 and |EP | = ∞, quantity very sensitive to price. See Figure 3.
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2 Income Elasticity of Demand
3
Figure 2: Extreme demand elas- Figure 3: Extreme demand elas­
ticity. |EP | = 0, quantity inde- ticity. |EP | = −∞, quantity very
pendent of price.
sensitive to price.
5. The constant elasticity demand function is
Q = aP b ,
since
EP =
dQ P
P
aP b
= abP b−1 = b
= b.
dP Q
Q
Q
Refer to Figure 4.
6. How do total consumer expenditure change when the price of a good
changes?
dExp
d(P QD (P ))
dQ
=
=Q+P
= Q(1 + EP ) = Q(1 − |EP |).
dP
dP
dP
• If |EP | > 1, total expenditure decreases when price increases;
• If |EP | < 1, total expenditure increases when price increases.
Example (Cell phone). People need to do business in the morning, so EP is
low, so cell phone companies increase the rate while customers will expend
more; but EP is high in the evening since people do not have to talk, so
cell phone companies lower the rate to encourage customer expenditure.
2
Income Elasticity of Demand
Income elasticity of demand. Income elasticity of demand measures the per­
centage change in quantity demanded resulting from one percentage change
in income. Similarly,
dQ
I dQ
Q
EI = dI =
.
Q
dI
I
The income elasticity of demand is usually positive.
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3 Cross Price Elasticity of Demand
4
Figure 4: Constant Demand Elasticity.
3
Cross Price Elasticity of Demand
Cross price elasticity of demand. Cross price elasticity of demand measures
the percentage change in quantity demanded of a good (x) resulting from
one percentage change in price of another good (y).
EQxP y =
dQx
Qx
dPy
Py
=
Py dQx
.
Qx dPy
• If y is a substitute of x, the cross price elasticity of demand is positive.
• If y is a complement of x, the cross price elasticity of demand is negative.
4
Comparison Between Elasticity Over Short Run
and Long Run
Is demand more elastic in the long run or short run?
Consumption goods. For consumption goods, the demand is more elastic in
the long run. Because people need goods for daily life and buy them
constantly, the short run demand is inelastic. Faced with high prices in
the long run, they may change habits or find more substitutes.
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4 Comparison Between Elasticity Over Short Run and Long Run
5
Durable goods. For durable goods, the demand is more elastic in the short
run. Consider cars. If price of of cars increase, in the short run people
might use their current cars longer. In the long run, though, people have
to replace their cars.
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1 Price Elasticity of Supply
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 12, 2007
Lecture 4
Price Elasticity of Supply; Consumer
Preferences
Outline
1. Chap 2: Elasticity - Price Elasticity of Supply
2. Chap 3: Consumer Behavior - Consumer Preferences
1
Price Elasticity of Supply
Price elasticity of supply. The percentage change in quantity supplied re­
sulting from one percentage change in price.
EPS =
dQS
QS
dP
P
=
P dQS
.
QS dP
In the short run, if price increases, firms will want to produce more but cannot
hire workers and buy machines immediately, thus the supply is less elastic. In
contrast, supply is more elastic in the long run.
Example (Example in Elasticities of Demand). Assume the quantity demanded
is
QD = 14 − 3P + I + 2PS − PC .
• P - Price
• I - Income
• PS - Price of substitute
• PC - Price of complement
Calculate EPD , EI , EQPS and EQPC when P = 1, I = 10, PS = 2 and PC = 1.
Solution:
Given the values of variables, the quantity demanded is:
QD = 14 − 3 × 1 + 10 + 2 × 2 − 1 = 24.
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2 Consumer Preferences
2
The elasticities are
EPD =
P dQD
1
1
=
× (−3) = − ,
QD dP
24
8
EI =
I dQ
10
5
=
×1=
,
Q dI
24
12
EQPS =
EQPC =
2
PS dQ
2
1
=
×2= ,
Q dPS
24
6
PC dQ
1
1
=
× (−1) = − .
Q dPC
24
24
Consumer Preferences
Consumer behavior
�
Consumer preferences
Budget constraints
�
=⇒
• What amount and types of goods will be purchased.
• Origin of demand, how to decide demand.
Topics
1. Preference
2. Indifference Curve, Marginal Rate of Substitution (MRS)
3. Utility Functions
Preference
Notation
• A ≻ B: A is preferred to B.
• A∼ B: A is indifferent to B.
Basic assumptions for preferences
• Completeness - can rank any basket of goods.
(always possible to decide preference or indifference)
• Transtivity - A≻B and B≻C implies A≻ C.
This assumption seems obvious, but can have contradiction (see example
below).
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2 Consumer Preferences
Good A
Good B
Good C
Property I
3
2
1
3
Property II
1
3
2
Property III
2
1
3
Table 1: Example of contradiction of transitivity.
Example (A contradiction of transtivity). Chart below lists 3 goods and
3 properties, assume that people will prefer one to another if 2 properties
are better. Table 1. Actually A ≻ B, B ≻ C and C ≻ A - this loop
contradicts the assumption.
• Non-satiation - more is better. (Monotonicity) Assume we discuss goods,
since in general, more is not always better.
• Convexity - given two indifferent bundles, always prefer the average to
each of them. In Figure 1, the average point C is more preferred to A or
B.
Figure 1: Convexity of indifference curve.
Indifference Curve, Marginal Rate of Substitution (MRS)
Properties of indifference curves
• Downward sloping: if not, non-satiation violated. Refer to Figure 1.
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2 Consumer Preferences
4
Figure 2: Compare the Shapes of Indifference Curve.
• Cannot cross: if not, non-satiation and transitivity cannot be satisfied
simultaneously. In Figure 1, assume there is another indifference curve
through A and D.
A ∼ B, A ∼ D =⇒ B ∼ D.
However,
B≻D
in this figure. Contradiction exists.
• Shape: describes how willing one is to substitute one good for another.
See Figure 2.
Marginal rate of substitution (MRS)
Marginal rate of substitution (MRS). How many units of Y one is willing
to give up in order to get one more unit of X.
−Δy
−dy
=
Δx
dx
People prefer a balanced basket of goods.
• MRS decreasing.
• Preferred set is convex.
• The left one in Figure 2 makes more sense in the real world.
Perfect substitution. MRS is constant.
Perfect complements. Indifference curves are shaped as right angles.
Example (Perfect complements). Buying shoes. People need both the left one
and the right one.
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2 Consumer Preferences
5
Figure 3: Perfect Substitution and Perfect Complements.
Figure 4: Indifference Curve with Utility Function u(x, y) = xy.
Utility Functions
Utility function. Assigns a level of utility to each basket of consumption.
Example (A sample utility function).
u(x, y) = xy.
For example, (5,5) is indifferent to (25,1) and (1,25).
Ordinal utility function. Ranks the preferences, but does not indicate how
much one is preferred to another.
Cardinal utility function. Describes the extent to which one of the bundles
is preferred to another. Only the ordinal utility function is required in
this course.
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1 Utility Function, Deriving MRS
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 14, 2007
Lecture 5
Deriving MRS from Utility Function, Budget
Constraints, and Interior Solution of
Optimization
Outline
1. Chap 3: Utility Function, Deriving MRS
2. Chap 3: Budget Constraint
3. Chap 3: Optimization: Interior Solution
1
Utility Function, Deriving MRS
Examples of utility:
Example (Perfect substitutes).
U (x, y) = ax + by.
Example (Perfect complements).
U (x, y) = min{ax, by}.
Example (Cobb-Douglas Function).
U (x, y) = Axb y c .
Example (One good is bad).
U (x, y) = −ax + by.
An important thing is to derive MRS.
M RS = −
dy
= |Slope of Indifference Curve|.
dx
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1 Utility Function, Deriving MRS
2
10
9
8
7
U(x,y)=ax+by=Const
y
6
5
4
3
2
1
0
0
2
4
6
8
10
x
Figure 1: Utility Function of Perfect Substitutes
10
9
8
7
y
6
5
U(x,y)=min{ax,by}=Const
4
3
2
1
0
0
2
4
6
8
10
x
Figure 2: Utility Function of Perfect Complements
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1 Utility Function, Deriving MRS
3
10
8
y
6
a b
U(x,y)=Ax y =Const
4
2
0
0
2
4
6
8
10
x
Figure 3: Cobb-Douglas Utility Function
10
9
8
7
y
6
5
U(x,y)=−ax+by=Const
4
3
2
1
0
0
2
4
6
8
10
x
Figure 4: Utility Function of the Situation That One Good Is Bad
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2 Budget Constraint
4
Because utility is constant along the indifference curve,
u = (x, y(x)) = C, =⇒
∂u ∂u dy
+
= 0, =⇒
∂x ∂y dx
dy
=
dx
∂u
∂x
∂u
∂y
.
M RS =
∂u
∂x
∂u
∂y
.
−
Thus,
Example (Sample utility function).
u(x, y) = xy 2 .
Two ways to derive MRS:
• Along the indifference curve
xy 2 = C.
r
c
y=
.
x
Thus,
√
dy
c
y
M RSd = −
= 3/2 =
.
dx
2x
2x
• Using the conclusion above
M RS =
2
∂u
∂x
∂u
∂y
=
y2
y
=
.
2xy
2x
Budget Constraint
The problem is about how much goods a person can buy with limited income.
Assume: no saving, with income I, only spend money on goods x and y with
the price Px and Py .
Thus the budget constraint is
Px · x + Py · y � I.
Suppose Px = 2, Py = 1, I = 8, then
2x + y � 8.
The slope of budget line is
dy
Px
=
.
dx
Py
Bundles below the line are affordable.
Budget line can shift:
−
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2 Budget Constraint
5
10
8
y
6
4
2x+y≤8
2
0
0
2
4
6
8
10
x
Figure 5: Budget Constraint
10
9
8
7
2x+y≤8
y
6
5
4
2x+y≤6
3
2
1
0
0
2
4
6
8
10
x
Figure 6: Budget Line Shifts Because of Change in Income
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3 Optimization: Interior Solution
6
10
9
8
7
2x+y≤8
y
6
5
4
x+y≤4
3
2
1
0
0
2
4
6
8
10
x
Figure 7: Budget Line Rotates Because of Change in Price
• Change in Income Assume I ′ = 6, then 2x + y = 6. The budget line shifts
right which means more income makes the affordable region larger.
• Change in Price Assume Px′ = 2, then 2x + 2y = 8. The budget line
changes which means lower price makes the affordable region larger.
3
Optimization: Interior Solution
Now the consumer’s problem is: how to be as happy as possible with limited
income. We can simplify the problem into language of mathematics:


 xPx + yPy � I 
x�0
max U (x, y) subject to
.
x,y


y�0
Since the preference has non-satiation property, only (x, y) on the budget line
can be the solution. Therefore, we can simplify the inequality to an equality:
xPx + yPy = I.
First, consider the case where the solution is interior, that is, x > 0 and y > 0.
Example solutions:
• Method 1
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3 Optimization: Interior Solution
7
10
9
8
7
y
6
5
U(x,y)=Const
4
3
2
P x+P y=I
x
y
1
0
2
4
6
8
10
x
Figure 8: Interior Solution to Consumer’s Problem
From Figure 8, the utility function reaches its maximum when the indif­
ferent curve and constraint line are tangent, namely:
Px
∂u/∂x
ux
= M RS =
=
.
Py
∂u/∂y
uy
– If
Px
ux
>
,
Py
uy
then one should consume more y, less x.
– If
Px
ux
<
,
Py
uy
then one should consume more x, less y. Intuition behind
Px
Py
= M RS:
Px
Py
is the market price of x in terms of y, and MRS is the price of x in
terms of y valued by the individual. If Px /Py > M RS, x is relatively
expensive for the individual, and hence he should consume more y.
On the other hand, if Px /Py < M RS, x is relatively cheap for the
individual, and hence he should consume more x.
• Method 2: Use Lagrange Multipliers
L(x, y, λ) = u(x, y) − λ(xPx + yPy − I).
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3 Optimization: Interior Solution
8
In order to maximize u, the following first order conditions must be satis­
fied:
∂L
= 0 =⇒
∂x
∂L
= 0 =⇒
∂y
ux
= λ,
Px
uy
= λ,
Py
∂L
= 0 =⇒ xPx + yPy − I = 0.
∂λ
Thus we have
Px
ux
=
.
Py
uy
• Method 3
Since xPx + yPy + I = 0,
y=
I − xPx
.
Py
Then the problem can be written as
max u(x, y) = u(x,
x,y
I − xPx
).
Py
At the maximum, the following first order condition must be satisfied:
ux + uy (
∂y
Px
) = ux + uy (− ) = 0.
∂x
Py
=⇒
Px
ux
=
.
Py
uy
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1 Corner Solution of Optimization
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 17, 2007
Lecture 6
Optimization, Revealed Preference, and
Deriving Individual Demand
Outline
1. Chap 3: Corner Solution of Optimization
2. Chap 3: Revealed Preference
3. Chap 4: Deriving Individual Demand, Engle Curve
1
Corner Solution of Optimization
When we have an interior solution,
Px
Ux
=
Py
Uy
must be satisfied. However, sometimes a consumer gets highest utility level
when x = 0 or y = 0. If that’s the case, we have corner solutions, and
Px
Ux
�=
,
Py
Uy
as shown in Figure 1.
In Figure 1, because people cannot consume negative amounts of goods
(bundle A), their best choice is to consume bundle B, so the quantity of y
consumed is zero. Conditions for corner solutions:
•
•
M RS =
Ux
Px
>
when y = 0.
Uy
Py
M RS =
Ux
Px
<
when x = 0.
Uy
Py
Example (An example of consumer’s problem). The parameters are
Px = 1,
Py = 1,
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1 Corner Solution of Optimization
Figure 1: Corner Solution to Consumer’s Problem.
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2
2 Revealed Preference
3
I = 2.
The utility function is
√
U (x, y) = x + 2 y.
The budget constraint is
x + y = 2.
According to the condition for an interior solution:
Px
Ux
=
.
Py
Uy
=⇒
1
1
= 1 .
√
1
y
=⇒
y = 1 =⇒ x = 1.
If the price y changes to 1:
Py = 1,
then the solution is
y = 4 =⇒ x = −3 < 0,
which is impossible.
Then we have the corner solution:
x = 0, y = 2.
x = 0 since consumer wants to consume as little as possible.
2
Revealed Preference
In the former chapters, we discussed how to decide optimal consumption from
utility function and budget constraint:
Utility
Function
=⇒ Optimal Consumption
Budget Constraint
And now we discuss how to know consumer’s preference from budget constraint
and consumption:
Budget Constraint
=⇒ Preference
Consumption
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3 Deriving Individual Demand, Engle Curve
4
10
9
8
7
6
y
X: 1.929
Y: 5.142
D
5
X: 1.478
Y: 3.761
4
C
A
3
X: 4.751
Y: 2.124
2
X: 3.949
Y: 1.101
1
0
B
0
1
2
3
4
5
x
6
7
8
9
10
Figure 2: A Contradiction of Preference. A and B are the Choices.
Example (Revealed preference). In Figure 2, two budget constraint lines inter­
sect. Assume one person’s choices are A and B respectively. Then we have
A � C,
B � D.
And Figure 2 obviously shows that
C ≻ B,
D ≻ A.
Thus,
A � C ≻ B � D ≻ A,
which is a contradiction, which means utility does not optimized and the choice
is not rational.
3
Deriving Individual Demand, Engle Curve
Use the following utility function again:
√
U (x, y) = x + 2 y,
with a budget constraint:
Px x + Py y = I.
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3 Deriving Individual Demand, Engle Curve
When
I�
5
Px2
,
Py
we have an interior solution. M RS = Px /Py . Thus,
x=
I
Px
−
,
Px
Py
y=
�
When
I�
Px
Py
�2
.
Px2
,
Py
we have a corner solution.
x = 0,
y=
I
.
Py
• Figure 3 shows a demand function of y and Py as an example. (Assume
that I, x and Px are held constant.)
• Engle Curve describes the relation between quantity and income. Figure
4 shows the relation between x and income, and Figure 5 shows that
between y and income.
Normal good. Quantity demanded of good increases with income.
Inferior good. Quantity demanded of good decreases with income.
Substitutes. Increase in price of one leads to an increase in quantity
demanded of the other.
Complements. Increase in price of one leads to an decrease in quantity
demanded of the other.
For this problem,
P2
• if I < Pxy , x and y are neither substitutes nor complements, and x is a
normal good.
• if I �
Px2
Py ,
x and y are substitutes, and y is a normal good.
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3 Deriving Individual Demand, Engle Curve
Figure 3: Demand Function for Goods ‘y’.
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3 Deriving Individual Demand, Engle Curve
7
10
9
8
7
x
6
5
4
3
2
1
P2/P
x
0
0
1
2
3
4
y
5
I
6
7
8
9
10
Figure 4: The Relation between Income and Quantity Demanded of ‘x’. Engle
curve of x.
10
9
8
7
y
6
5
4
3
2
1
P2/P
x
0
0
1
2
3
4
5
I
y
6
7
8
9
10
Figure 5: The Relation between Income and Quantity Demanded of ‘y’. Engle
curve of y.
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1 Substitution Effect, Income Effect, Giffen Goods
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 19, 2007
Lecture 7
Substitution and Income Effect, Individual and
Market Demand, Consumer Surplus
Outline
1. Chap 4: Substitution Effect, Income Effect, Giffen Goods
2. Chap 4: From Individual Demand to Market Demand
3. Chap 4: Consumer Surplus
1
Substitution Effect, Income Effect, Giffen Goods
Substitution and Income Effects
The impact of price change on quantity demanded are divided into two effects:
Substitution effect. Substitution effect is the change in an item’s consump­
tion associated with a change in the item’s price with the utility level held
constant.
Income effect. Income effect is a change in an item’s consumption associated
with a change in purchasing power with the price held constant.
Figure 1 shows the two effects: L is the old budget line. Px decreases, and
hence the new budget line is L′ . A is the optimal consumption before price
change, and C is the optimal consumption after price change. L′′ is a line that
has the same slope as L′ and is tangent with the green indifference curve that
passes through A, and B is the tangent point.
• The change from A to B is because of the substitution effect;
• The change from B to C is because of the income effect.
So the total effect is point A moving to C.
Inferior Good and Giffen Good
Now consider different positions of C (Figure 1):
• The income effect is B changing to C. In this case, an increase in income
causes an increase in the demand of x. x is a normal good.
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1 Substitution Effect, Income Effect, Giffen Goods
Figure 1: Substitution Effect and Income Effect.
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1 Substitution Effect, Income Effect, Giffen Goods
3
• The income effect is B changing to C ′ or C ′′ . In these cases, an increase
in income causes a decrease in the demand of x. x is an inferior good;
• If the total effect is A changing to C ′′ , such that a decrease in price causes
a decrease in the demand, we call x is a Giffen good.
Normal good
Inferior good
Price increases
substitution effect
income effect
substitution effect
income effect
quantity increases
quantity increases
quantity increases
quantity decreases
Table 1: Normal Good and Inferior Good
In Table 1, if x is a normal good, both substitution and income effects
increase its quantity; if x is an inferior good, discuss as follows:
1. substitution effect > income effect
→ quantity increases
2. substitution effect < income effect
→ quantity decreases. This unusual good is called a Giffen good. A Giffen
good must be an inferior good, but an inferior good is not necessarily a
Giffen good.
Giffen good. Good with an upward demand curve. (Figure 2)
Example (Giffen Good Example: Irish Potato Famine). People consumed lots of
potato but little meat (and other food) since meat was more expensive. Price of
potato rose. People had less money to consume meat, so they ate more potatoes
instead of meat.
An Example of Substitution Effects and Income Effects
Utility function Figure 3:
√
U (x, y) = x + 2 y.
Parameters:
Px = 1,
Py = 1,
I = 5.
The optimal solution is:
x = 4,
y = 1.
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1 Substitution Effect, Income Effect, Giffen Goods
4
6
5.5
5
4.5
P
4
3.5
3
2.5
2
1.5
1
0
0.5
1
1.5
2
Q
2.5
3
3.5
4
D
Figure 2: Demand Curve of Giffen Good.
i.e. the solution is at point A: (4, 1).
If price of x changes to 2, Px′ = 2, then the new optimal solution is:
x=
1
,
2
y = 4.
( 12 ,4).
i.e. the solution is at point C:
Try to find out the substitution effect, i.e.
the change from A to B.
At B, the slope of the indifference curve equals the slope of the new budget
constraint.
Thus,
1
P′
2
M RS = 1 = x′ = .
√
P
1
y
y
=⇒ y = 4.
On the other hand,
U (x, y) = x + 2 ×
√
√
4 = 4 + 2 × 1.
=⇒ x = 2.
Thus, point B is at (2,4).
Decomposition of the two effects:
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2 From Individual Demand to Market Demand
5
5
4.5
4
C (1/2,4)
B (2,4)
3.5
y
3
2.5
2
1.5
A(4,1)
1
0.5
0
0
0.5
1
1.5
2
2.5
x
3
3.5
4
4.5
5
Figure 3: Showing the Substitution effect and Income Effect.
• Substitution effect (A to B)
(4,1) =⇒ (2,4).
• Income effect (B to C)
(2,4) =⇒ ( 12 ,4).
2
From Individual Demand to Market Demand
Assume in a market there are two individuals A and B. And their demand
functions are:
QA = 1 − P,
1
QB = 1 − P.
2
When P < 1, both individuals consume, and the market demand is the sum of
the individual demands:
2
Q = QA + QB = 2 − P.
3
However, if P is larger than 1, only B consumes, so the market demand equals
the demand of B. Thus, the market demand function is
�
2 − 32 P if P � 1
Q=
.
1 − 12 P if P > 1
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3 Consumer Surplus
6
This is shown in Figure 4.
3
Consumer Surplus
Willingness to Pay. The sum of the ‘values’ of each of the units that con­
sumers consume.
Consumer Surplus. The difference between Willingness to Pay and the actual
Expenditure.
Example. Figure 5 shows the demand curve of a good. Assume now the price
is 15, then only the highest 6 individuals consume:
W ILLIN GN ESS T O P AY = 20 + 19 + 18 + 17 + 16 + 15 = 105.
On the other hand, the expenditure is
EXP EN DIT U RE = 6 × 15 = 90.
Therefore,
CON SU M ER SU RP LU S = 105 − 90 = 15.
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3 Consumer Surplus
Figure 4: Derived Market Demand from Individual Demands.
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3 Consumer Surplus
8
25
P
20
15
10
5
0
2
4
6
8
10
Q
12
14
16
18
20
Figure 5: Demand Curve for a Good. Used in consumer surplus calculation.
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1 Irish Potato Famine
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 21, 2007
Lecture 8
Irish Potato Famine, Network Externalities and
Uncertainty
Outline
1. Chap 4: Irish Potato Famine
2. Chap 4: Network Externalities
3. Chap 5: Uncertainty
1
Irish Potato Famine
Typical Giffen good. In Year 1845-1849, people consumed more potatoes
when the price increased. (Figure 1)
2
Network Externalities
Network externality. One person’s demand depends on the demands of other
people.
• [Bandwagon effect (Figure 2)] Positive network externality. When
more people buy, you will buy more.
Example. iPod: buy to be in style.
– Market demand more elastic than real demand curve.
– Seller sets lower price.
Example. Operating system: more software available.
Example. Internet telephone.
• [Snob effect (Figure 3)] Negative network externality. When others
buy, you will not buy.
– Market demand more inelastic than real demand curve.
– Seller sets Higher price.
Example. Designer clothes: want to be special.
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3 Uncertainty
2
10
9
Original Budget Constraint
8
A−>B: Substitution Effect
B−>C: Income Effect
7
Potato
6
5
4
C
3
A
2
B
New Budget Constraint
1
0
0
0.5
1
1.5
2
2.5
Other Food
3
3.5
4
4.5
5
Figure 1: Irish Potato Famine: Price Higher, Consume More
3
Uncertainty
An Outline in Uncertainty
• Preference, Decision
• Expected Value / Variability, Risk Standard Deviation
• Expected Utility
To measure risk we must know:
• All of the possible outcomes.
• The probability that each outcome will occur, the sum of the proba­
bilities that each outcome will occur = 1.
Example. Probability of Weather
• Sunny 70%.
• Rainy 5%.
• Cloudy 25%.
The sum of all the probabilities is 100%.
Objective probability. Based on observed frequency of past events.
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3 Uncertainty
3
Figure 2: Bandwagon Effect: Positive Network Externalities
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3 Uncertainty
4
Figure 3: Snob Effect: Negative Network Externalities
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3 Uncertainty
5
Subjective probability. Based on perception, theory and understanding of
outcomes.
Measures to characterize payoffs and degree of risk.
Job 1
Job 2
Example (Job).
Outcome 1
Outcome 2
$2000 with probability 50% $1000 with probability 50%
$1510 with probability 99%
$510 with probability 1%
Table 1: Compare Two Jobs, Each has Two Outcomes
Expected value.
E(x) = p1 x1 + p2 x2 + ... + pn xn ,
where x is a random variable, which has realizations x1 , x2 , ..., xn with
probability p1 , p2 , ..., pn respectively. Discuss the example. Expected val­
ues of salary from job 1 and 2 are:
E(job1) = 0.50 × 2000 + 0.50 × 1000 = 1500.
E(job2) = 0.99 × 1510 + 0.01 × 510 = 1500.
Since
E(job1) = E(job2),
we do not know which job is better.
Standard deviation.
�
σ(x) = p1 [x1 − E(x)]2 + p2 [x2 − E(x)]2 + ... + pn [xn − E(x)]2 .
We can consider the risks of those jobs from standard deviation:
�
σ1 = 0.50 × (2000 − 1500)2 + 0.50 × (1000 − 1500)2 = 500,
�
σ2 = 0.99 × (1510 − 1500)2 + 0.01 × (510 − 1500)2 = 99.5.
Since
σ1 > σ2 ,
for less risk, we will choose job 2.
Expected utility.
E[u(x)] = p1 u(x1 ) + p2 u(x2 ) + ... + pn u(xn ).
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1 Preference Toward Risk
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 26, 2007
Lecture 9
Preference Toward Risk, Risk Premium,
Indifference Curves, and Reducing Risk
Outline
1. Chap 5: Preference Toward Risk
2. Chap 5: Risk Premium
3. Chap 5: Indifference Curve
4. Chap 5: Reducing Risk: Diversification
1
Preference Toward Risk - Risk Averse / Neu­
tral / Seeking (Loving)
Three different kinds of behaviors:
Risk Averse (Figure 1)
• Facing two payoffs with the same expected value, prefer the less risky one.
• Diminishing marginal utility of income.
• Relation between the utility of expected value and expected utility
u(E(x)) > E(u(x)).
Example.
u(x) = ln x.
Risk Neutral (Figure 2)
• Facing two payoffs with the same expected value, feel indifferent.
• Linear marginal utility of income.
• Relation between the utility of expected value and expected utility
u(E(x)) = E(u(x)).
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2 Risk Premium
2
3
2.5
u(x)
2
1.5
1
0.5
0
0
1
2
3
4
5
x
6
7
8
9
10
Figure 1: The Utility Function of Risk Averse.
Example.
u(x) = x.
Risk Seeking (Figure 3)
• Facing two payoffs with the same expected value, prefer the riskier one.
• Increasing marginal utility of income.
• Relation between the utility of expected value and expected utility
u(E(x)) < E(u(x)).
Example.
u(x) = x2 .
2
Risk Premium
Risk premium. The maximum amount of money that a risk-averse person
would pay to avoid taking a risk.
Example (Job Choice). Assume that a risk-averse person whose utility function
corresponds with the curve in Figure 4 has two possible incomes.
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2 Risk Premium
3
9
8
7
6
u(x)
5
4
3
2
1
0
0
1
2
3
4
5
x
6
7
8
9
10
Figure 2: The Utility Function of Risk Neutral.
70
60
50
u(x)
40
30
20
10
0
0
1
2
3
4
5
x
6
7
8
9
10
Figure 3: The Utility Function of Risk Seeking.
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3 Indifference Curve between Expected Value and Standard
Deviation
4
20
18
16
14
u(x)
12
10
8
6
4
2
0
0
5
10
15
20
25
30
x
Figure 4: Risk Premium: A Utility Function.
• His income I might be 10 with probability 0.5 and 30 with probability 0.5.
Then the expected value of income I is:
E(1) = 10 × 0.5 + 30 × 0.5 = 20,
with an expected utility:
E(u(I)) = u(10) × 0.5 + u(30) × 0.5 = 10 × 0.5 + 18 × 0.5 = 14.
• If we offer him a fixed income I ′ , I ′ = 16, then his expected utility is:
E(u(I ′ )) = u(16) × 1 = 14 × 1 = 14.
One can see that
E(u(I)) = E(u(I ′ )).
However, E(I) − E(I ′ ) = 4. This means the person is willing to give up a value
of 4 in exchange for a riskless income. Thus, the risk premium is
Risk P remium = E(I) − E(I ′ ) = 20 − 16 = 4.
3
Indifference Curve between Expected Value
and Standard Deviation
The indifference curve we discussed before is about the quantities of two different
goods, now we consider the indifference curve about expected value and standard
deviation (Figure 5).
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3 Indifference Curve between Expected Value and Standard
Deviation
5
1330
1320
1310
1300
Ex
1290
1280
1270
1260
1250
1240
400
500
600
700
σ
800
900
1000
Figure 5: Indifference Curve between Expected Value and Standard Deviation.
Job 1
Job 2
Probability 0.5
900
625
Probability 0.5
1600
2025
Table 1: The Income and Probability of Two Jobs.
Example (Job choice). Suppose one has the following utility function
√
u(x) = x
and two job choices (see Table 1). Calculate expected utilities:
√
√
E(u(x1 )) = 0.5 × 900 + 0.5 × 1600 = 35,
√
√
E(u(x2 )) = 0.5 × 625 + 0.5 × 2025 = 35.
Thus, these two jobs give the person the same utility level, i.e. they are on a
same indifference curve.
In order to plot the indifference curve, we should calculate their expected
values and standard deviations.
E(x1 ) = 1250
σ(x1 ) = 494
E(x2 ) = 1325
σ(x2 ) = 990
Job 2 has higher expected value of income but it is riskier. (Figure 5)
Compare Figure 6 and Figure 7. The former is more risk averse since one
must compensate more for more risk.
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3 Indifference Curve between Expected Value and Standard
Deviation
6
90
80
70
60
Ex
50
40
30
20
10
0
0
1
2
3
4
5
σ
6
7
8
9
10
Figure 6: Indifference Curve between Expected Value and Standard Deviation,
Larger Slope.
20
18
16
14
Ex
12
10
8
6
4
2
0
0
1
2
3
4
5
σ
6
7
8
9
10
Figure 7: Indifference Curve between Expected Value and Standard Deviation,
Smaller Slope.
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4 Reducing Risk: Diversification
4
7
Reducing Risk: Diversification
Diversification. Reducing risk by allocating resources to different activities
whose outcomes are not closely related.
Example (Selling air conditioner and heater). Suppose that the weather
has a probability 0.5 to be hot and 0.5 to be cold. Table 2 shows the
company’s profit if all its efforts in selling air conditioners (heaters) and
the weather turns out to be hot (cold).
Weather
Air Conditioner
Heater
Hot
30,000
12,000
Cold
12,000
30,000
Table 2: Diversification: Selling Air Conditioners and Heaters.
• If one only sells air conditioners or heaters,
E(prof it) = 21, 000,
σ(prof it) = 9, 000.
• If the company puts half of its efforts in selling air conditioners and
half of its efforts in selling heaters, then the profit is always 21,000
no matter the weather is cold or hot.
E(prof it) = 21, 000,
σ(prof it) = 0.
Thus we should choose to sell both to reduce risk.
Example (Example: Stock versus mutual fund). Mutual fund may have
the same return as stock but much less risk.
Example. ”Don’t put all your eggs in one basket.”
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1 Reducing Risk: Insurance
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
September 28, 2007
Lecture 10
Insurance and Production Function
Outline
1. Chap 5: Reducing Risk: Insurance
2. Chap 6: Outline of Producer Theory
3. Chap 6: Production Function: Short Run and Long Run
1
Reducing Risk: Insurance
Reducing Risk:
• Diversification
• Insurance
Example (House insurance). Assume that one house has the proba­
bility p to catch fire, with loss l each time, i.e. the owner’s wealth
will reduce from y1 to y2 = y1 −l. If the owner pay premium k to buy
an insurance which covers the loss l when there is a fire, her wealth
will be y3 = y1 − k, for the situations listed (see Table 1).
No Fire
Fire
No Insurance
y1
y2 = y1 − l
Insurance
y3 = y1 − k
y3 = y1 − k
Table 1: Wealth of House Owner in Different Situations.
Assuming the owner is a risk-averse, the utility function is concave.
u′′ (y) < 0
If the expected wealth at both situations is equal,
y3 = (1 − p) × y1 + p × y2 .
We have
k = p × l.
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1 Reducing Risk: Insurance
Figure 1: The Utility Function of Risk Averse Person.
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2
1 Reducing Risk: Insurance
3
The insurance premium is equal to the expected payout by the in­
surance company, and we say the insurance is actuarially fair.
Since the person is risk-averse,
u(y3 ) > (1 − p) × u(y1 ) + p × u(y2 ).
she will choose to buy insurance.
If the insurance is actuarially unfair,
k > p × l.
Then
y3 < (1 − p) × y1 + p × y2 .
We do not know if the person wants to buy or not until we get
her specific utility function, but it is easy to imagine she may buy
insurance if k is close to pl.
Now we consider what is the maximum insurance premium that the
companies can charge and the costumer is still willing to buy the
insurance. In this case, let y3′ be the house owner’s wealth after
being charged the maximum premium. Then, (Figure 1)
u(y3′ ) = pu(y2 ) + (1 − p)u(y1 ).
Thus the maximum insurance premium charged is
k ′ = y1 − y3′ = y1 − E(y) + Risk P remium = p × l + Risk P remium
So are insurance companies more willing to take risk? If not, why are
they willing to sell insurance? The Law of Large Numbers can explain
this. Let L be the total loss from n customers, It is a random variable.
L
The average loss shared by each customer is L
n , and E( n ) = n × p.
The expected payout for L by the insurance company will be
E(L) = n × p × l
When
n→∞
The probability that the loss shared by each customer is equal to a
fixed number pl is almost 1. (Figure 2)
L
= p × l) →n→∞ 1.
n
Note that this argument only applies to the situation when customers’
fire accident events are independent.
Example (Illegal parking). Government has two reasonable methods
to punish illegal parking.
– Hire more police, get caught almost for sure but fine is low.
– Hire less police, get caught sometimes but the fine is high.
The latter might be more effective since people are risk averse abd
are afraid to take risk of being fined to park illegally.
P robability(
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1 Reducing Risk: Insurance
Figure 2: Distribution of
4
L
n
with Different Customer Numbers.
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2 Outline of Producer Theory
2
5
Outline of Producer Theory
• Production Function: Inputs to Outputs
• Given Quantity Produced, Choose Inputs to Minimize the Cost
• Choose Quantity to Maximize Firm’s Profit
The Production Function is
q = F (k, L).
The two inputs:
k: Capital
L: Labor
It is easier to change labor level but not to change capital in a short time.
Short run. Period of time in which quantity of one or more inputs cannot be
changed. For example, capital is fixed and labor is variable in the short
run.
Long run. Period of time need to make all production inputs variable. In the
long run, both capital and labor are variable.
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1 Short Run Production Function
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 1, 2007
Lecture 11
Production Functions
Outline
1. Chap 6: Short Run Production Function
2. Chap 6: Long Run Production Function
3. Chap 6: Returns to Scale
1
Short Run Production Function
In the short run, the capital input is fixed, so we only need to consider the
change of labor. Therefore, the production function
q = F (K, L)
has only one variable L (see Figure 1).
Average Product of Labor.
APL =
Output
q
= .
Labor Input
L
Slope from the origin to (L,q).
Marginal Product of Labor.
M PL =
∂Output
∂q
=
.
∂Labor Input
∂L
Additional output produced by an additional unit of labor.
Some properties about AP and M P (see Figure 2).
• When
M P = 0,
Output is maximized.
• When
M P > AP,
AP is increasing.
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1 Short Run Production Function
2
40
35
30
Q
25
20
15
10
5
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 1: Short Run Production Function.
10
9
8
7
6
5
AP
4
3
2
1
0
MP
0
1
2
3
4
5
L
6
7
8
9
10
Figure 2: Average Product of Labor and Marginal Product of Labor.
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2 Long Run Production Function
3
• When
M P < AP,
AP is decreasing.
• When
M P = AP,
AP is maximized.
To prove this, maximize AP by first order condition:
∂ q(L)
=0
∂L L
=⇒
∂q 1
q
− 2 =0
∂L L L
=⇒
∂q
q
=
∂L
L
=⇒
M P = AP.
Example (Chair Production.). Note that here APL and M PL are not con­
tinuous, so the condition for maximizing APL we derived above does not
apply.
Number of Workers
0
1
2
3
Number of Chairs Produced
0
2
8
9
APL
N/A
2
4
3
M PL
N/A
2
6
1
Table 1: Relation between Chair Production and Labor.
2
Long Run Production Function
Two variable inputs in long run (see Figure 3).
Isoquants. Curves showing all possible combinations of inputs that yield the
same output (see Figure 4).
Marginal Rate of Technical Substitution (M RT S). Slope of Isoquants.
dK
dL
How many units of K can be reduced to keep Q constant when we increase
L by one unit. Like M RS, we also have
M RT S = −
M RT S =
M PL
.
M Pk
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2 Long Run Production Function
4
10
9
8
7
k=3
Q
6
5
k=2
4
k=1
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
4.5
5
Figure 3: Long Run Production Function.
5
4.5
4
3.5
k
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
L
3
3.5
4
Figure 4: K vs L, Isoquant Curve.
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3 Returns to Scale
5
Proof. Since K is a function of L on the isoquant curve,
q(K(L), L) = 0
=⇒
∂q dK
∂q
+
=0
∂L dL
∂L
=⇒
−
dK
M PL
=
.
dL
M PK
Perfect Substitutes (Inputs). (see Figure 5)
10
9
8
7
k
6
5
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 5: Isoquant Curve, Perfect Substitutes.
Perfect Complements (Inputs). (see Figure 6)
3
Returns to Scale
Marginal Product of Capital.
M PK =
∂q(K, L)
∂K
Marginal Product of Labor
K constant , L ↑ → q?
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3 Returns to Scale
6
10
9
8
7
k
6
5
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 6: Isoquant Curve, Perfect Complements.
Marginal Product of Capital
L constant , K ↑ → q?
What happens to q when both inputs are increased?
K ↑ , L ↑ → q?
Increasing Returns to Scale.
• A production function is said to have increasing returns to scale if
Q(2K, 2L) > 2Q(K, L),
or
Q(aK, aL) = 2Q(K, L), a < 2.
• One big firm is more efficient than many small firms.
• Isoquants get closer as we move away from the origin (see Figure 7).
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3 Returns to Scale
7
10
9
8
Q=3
7
k
6
5
Q=2
4
3
2
Q=1
1
0
0
2
4
6
8
10
12
14
L
Figure 7: Isoquant Curves, Increasing Returns to Scale.
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1 Returns to Scale
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 3, 2007
Lecture 12
Production Functions and Cost of Production
Outline
1. Chap 6: Returns to Scale
2. Chap 6: Production Function Derivation
3. Chap 7: Cost of Production
1
Returns to Scale
Increasing Returns to Scale
(Lecture 11)
Constant Returns to Scale
• Doubling the inputs leads to double the output:
Q(2K, 2L) = 2Q(K, L).
• One big firm is as good as many small firms.
• Isoquants are equally distant apart (see Figure 1).
Decreasing Returns to Scale
• Doubling the inputs leads to an output less than twice the original output:
Q(2K, 2L) < 2Q(K, L).
• Small firms are more efficient.
• Isoquants become further apart (see Figure 2).
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1 Returns to Scale
2
12
10
k
8
6
Q=3
4
Q=2
2
Q=1
0
0
1
2
3
4
5
6
7
8
9
L
Figure 1: Isoquant Curves, Constant Returns to Scale.
10
9
8
7
k
6
5
Q=3
4
3
2
Q=2
1
0
Q=1
0
1
2
3
4
5
L
6
7
8
9
10
Figure 2: Isoquant Curves, Decreasing Returns to Scale.
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2 Production Function Derivation
3
Example (Cobb-Douglas Production Function.).
Q(K, L) = ALα K β .
We double both inputs to see what type of returns to scale the production
function has.
Q(2K, 2L) = A(2L)α (2K )β = 2α+β ALα K β = 2α+β Q(K, L).
1. If
α + β > 1,
returns to scale is increasing.
2. If
α + β = 1,
returns to scale is constant.
3. If
α + β < 1,
returns to scale is decreasing.
2
Production Function Derivation
Assume that the firm has two technologies A and B , and the corresponding
outputs are
x y
qA = min{ , },
2 1
x y
qB = min{ , },
1 2
where the inputs x and y are perfect complements (see Figure 3).
To derive production function, we must know which technology the firm chooses.
If the firm choose either A or B, but not both, the isoquant curve for the
production function is the black line (see Figure 3). This isoquant curve is not
convex. However, the firm can adopt technologies at the same time, and this
makes the isoquants convex (see Figure 4).
Thus the production function is:
⎧
y
⎨ min{ x2 , 1 }, when x > 2y.
x+y
q(x, y) =
, when 1 y � x � 2y.
⎩ 3 x y 2
min{ 1 , 2 }, when x < 12 y.
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2 Production Function Derivation
Figure 3: Deriving Production Function, Using Technology A or B.
Figure 4: Deriving Production Function, Using Technology A and B.
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4
3 Cost of Production
3
5
Cost of Production
Cost comes from factor price and how many units are used.
Accounting Cost. Actual expenses plus depreciation.
Economic Cost. Cost to a firm of using resources in production. Also called
opportunity cost, the most valuable forgone alternative.
Job 1
Job 2
Wage
150
200
Transportation Cost
0
20
Accounting Cost
0
20
Opportunity Cost
180
150
Table 1: Accounting Cost and Opportunity Cost.
Example (Two job opportunities (see Table 1)). If the person accepts Job
2, the most valuable forgone opportunity is Job 1.
Opportunity cost does not really happen but must be considered.
Sunk Cost. Expenditure that has been made and cannot be recovered.
Example (Two building choices). A firm has two building choices. For
Building 1, they have paid 500,000, and will pay 5,000,000 in the future;
for Building 2, they have not paid anything, and will pay 5,300,000 in
the future. Although Building 2 is cheaper than Building 1, the firm will
choose Building 1 because the 500,000 is sunk.
Total Cost.
Total Cost = Variable Cost + Fixed Cost.
Fixed Cost. A cost that is actually incurred, but independent of the level of
output.
Variable Cost. A cost that is actually incurred, and dependent of the level of
output.
Example (Short Run). Capital K is fixed, and Labor L is variable; hence,
the cost of K is a fixed cost, and the cost of L is a variable cost.
Here is another definition of sunk cost.
Sunk Cost. A fixed cost which is also independent of output, but whose cost
is not incurred, because of no cash outlay and no opportunity cost.
Usually fixed costs are considered sunk costs because they happen before
production begins.
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1 Short-Run Cost Function
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 15, 2007
Lecture 13
Cost Functions
Outline
1. Chap 7: Short-Run Cost Function
2. Chap 7: Long-Run Cost Function
Cost Function
Let w be the cost per unit of labor and r be the cost per unit of capital. With
the input Labor (L) and Capital (K), the production cost is
w × L + r × K.
A cost function C(q) is a function of q, which tells us what the minimum cost
is for producing q units of output. We can also split total cost into fixed cost
and variable cost as follows:
C(q) = F C + V C(q).
Fixed cost is independent of quantity, while variable cost is dependent on quan­
tity.
1
Short-Run Cost Function
In the short-run, firms cannot change capital, that is to say,
r × K = const.
Recall the production function given fixed capital level K in the short run (refer
to Lecture 11) (see Figure 1). Suppose w = 1, the variable cost curve can be
derived from Figure 1. Adding r × K to the variable cost, we obtain the total
cost curve (see Figure 2). Average total cost is
AT C =
TC
FC + V C
rK
wL(q; K)
=
=
+
.
q
q
q
q
With the definition of the average product of labor:
q
APL = ,
L
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1 Short-Run Cost Function
2
40
35
30
Q
25
20
15
10
5
0
0
1
2
3
4
5
L
6
7
8
9
10
9
10
Figure 1: Short Run Production Function.
50
45
40
TC
35
C
30
25
20
15
10
VC
5
0
0
1
2
3
4
5
q
6
7
8
Figure 2: Short Run Cost Function.
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1 Short-Run Cost Function
3
we can rewrite AT C as
AT C =
rK
w
+
,
q
APL
in which the average variable cost is
VC
wL(q; K)
w
=
=
.
q
q
APL
Likewise, we rewrite the marginal cost:
MC =
dT C
dV C
dL(q)
w
w
=
=w
= ∂q =
.
dq
dq
dq
M PL
∂L
In Lecture 11, we discussed the relation between average product of labor and
marginal product of labor (see Figure 3). We draw the curves for AV C and
10
9
8
7
6
5
AP
4
3
2
1
0
MP
0
1
2
3
4
5
L
6
7
8
9
10
Figure 3: Average Product of Labor and Marginal Product of Labor.
M C in the same way (see Figure 4). The relation between M C and AV C is:
• If
M C < AV C,
AV C decreases;
• if
M C > AV C,
AV C increases;
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2 Long-Run Cost Function
4
30
25
C
20
MC
ATC
15
10
5
0
AVC
0
1
2
3
4
5
L
6
7
8
9
10
Figure 4: Average Cost, Average Variable Cost, and Marginal Cost.
• if
M C = AV C,
AV C is minimized.
Now consider the total cost. Note that the difference between AT C and AV C
decreases with q as the average fixed cost term dies out (see Figure 4). The
relation between M C and AT C is:
• If
M C < AT C,
AT C decreases;
• if
M C > AT C,
AT C increases;
• if
M C = AT C,
AT C is minimized.
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2 Long-Run Cost Function
5
5
4.5
4
3.5
k
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
L
3
3.5
4
4.5
5
Figure 5: Isoquant Curve.
2
Long-Run Cost Function
In the long-run, both K and L are variable. The isoquant curve describes the
same output level with different combination of K and L (see Figure 5). The
slope of an isoquant curve is
−M RT S = −
M PL
.
M PK
Similarly, the isocost curve is constructed by different (K, L) with the same cost
(see Figure 6). The isocost curve equation is:
rK + wL = const,
therefore, it is linear, with a slope − wr .
Now we want to minimize the cost rK +wL subject to an output level Q(K, L) =
q. This minimum cost can be obtained when the isocost curve is tangent to the
isoquant curve (see Figure 7). Thus the slopes of these two curves are equal:
M RT S =
M PL
w
= .
M PK
r
Now consider an increase in wage (w). The slope of the isocost curve increases
(see Figure 8), and the firm use more capital and less labor. The firm’s choice
of input moves from A to B in the figure.
The expansion path shows the minimum cost combinations of labor and
capital at each level of output (see Figure 9).
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2 Long-Run Cost Function
6
7
6
5
K
4
3
2
1
0
0
1
2
3
4
5
6
7
L
Figure 6: Isocost Curve.
10
9
8
7
K
6
5
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 7: Minimize the Cost Subject to a Output Level.
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2 Long-Run Cost Function
7
10
9
8
7
K
6
5
4
B
3
A
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 8: The Change of Cost Minimized Situation.
12
11
10
9
K
8
Expansion Path
7
6
5
4
3
2
2
3
4
5
6
7
L
8
9
10
11
12
Figure 9: Expansion Path.
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2 Long-Run Cost Function
8
Example (Calculating the Cost.). Given the production function
2
2
q = L3 K 3 .
In the short run,
3
CSR (q; K) = rK + w
q2
,
K
where K is fixed.
In the long run, according to the equation
M PL
w
= ,
M PK
r
we have
K
w
= .
L
r
Then the expansion path is
w
L.
r
Substituting this result into the production function, we obtain
K=
3
r 1
L = q 4 ( )2 ,
w
3 w 1
K = q 4 ( )2 .
r
Hence, the long-run cost function is:
3
CLR (q) = wL + rK = 2q 4
√
wr.
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1 Relation Between Long Run Cost Short Short Run Cost
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 17, 2007
Lecture 14
The Cost of Production and Profit
Maximization
Outline
1. Chap 7: Relation Between Long Run Cost and Short Run Cost
2. Chap 7: Economies of Scale
3. Chap 7: Economies of Scope, Learning
1
Relation Between Long Run Cost Short Short
Run Cost
Since firms can change capital in the long run, the long run cost is always no
more than the short run cost:
CL R(q) � CSR,K (q).
Figure 1 shows three short-run total cost given different capital level. In the long
run, firms will choose the capital level which minimizes the total cost. Thus,
the long-run total cost is equal to the minimum of all possible short-run total
cost, and so long run total cost is the envelope of all short run total costs.
Likewise, long-run average cost is the envelope of all short run average cost.
From Figure 1, we know for a given product q, long run marginal cost is equal
to the corresponding short run marginal cost. Long run total cost and marginal
cost also have the following relation: (see Figure 2)
• If
LM C < LAC,
LAC is decreasing;
• if
LM C = LAC,
LAC is minimized;
• if
LM C > LAC,
LAC is increasing.
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1 Relation Between Long Run Cost Short Short Run Cost
2
10
9
8
TCSR3
7
C
6
5
TCSR2
4
TCLR
3
TCSR1
2
1
0
0
1
2
3
4
5
q
6
7
8
9
10
Figure 1: Deriving Long Run Total Cost from Short Run Total Cost.
15
SMC3
SMC2
10
LMC
C
SMC1
SAC1
SAC2
SAC3
5
LAC
0
0
5
10
15
q
Figure 2: Deriving Long Run Average Cost and Marginal Cost from Short Run
Average Cost and Marginal Cost.
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2 Economies of Scale
2
3
Economies of Scale
• Constant economies of scale:
C(aq) = aC(q), a > 1,
and in this case, AC is constant;
• Economies of scale:
C(aq) < aC(q), a > 1,
and in this case, AC is decreasing;
• Diseconomies of scale:
C(aq) > aC(q), a > 1,
and in this case, AC is increasing.
10
9
8
7
AC
C
6
5
4
economies of scale
3
diseconomies of scale
2
1
0
0
1
2
3
4
5
q
6
7
8
9
10
Figure 3: Production Dependence of Average Cost, Different Economies of
Scale.
3
Economies of Scope, Learning
Economies of Scope. When producing more than one type of product that
are closely linked, the cost is lower than when producing them separately.
Product Transformation Curve. Shows various combinations of outputs that
can be produced with a given set of inputs.
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3 Economies of Scope, Learning
4
Example (Product Transformation Curve with Economies of Scope). To produce
1 car and 1 truck, if we produce them separately, we need 2 units of K and 2
units of L; but if we produce them together, we only need 1.5 units of K and 1.5
units of L (see Figure 4). In this case, it is cheaper to produce them together;
thus the firm has economies of scope.
2
1.8
1.6
1.4
Trucks
1.2
1
K=1.5 L=1.5
(1,1)
0.8
K=1 L=1
(0.7,0,7)
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Cars
1.2
1.4
1.6
1.8
2
Figure 4: Product Transformation Curve with Economies of Scope.
In the case of economies of scope, the product transformation curve is neg­
atively sloped and concave.
The degree of economies of scope is defined as follows:
SC =
C(q1 ) + C(q2 ) − C(q1 , q2 )
.
C(q1 , q2 )
• If
SC > 0,
it is economies of scope;
• if
SC < 0,
it is diseconomies of scope.
The learning curve for a firm is shown in Figure 5, with the firm’s cumulative
output as the vertical coordinate, and amount of inputs needed to produce a
unit of output as the horizontal coordinate.
Learning causes a difference in cost between the new firm and the old firm
(see Figure 6).
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3 Economies of Scope, Learning
5
10
9
Hours of Labor Per Unit of Output
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Cumulative Number of Outputs Produced
8
9
10
9
10
Figure 5: Learning Curve of a Firm.
10
9
8
AC
7
C
6
5
4
3
AC*
2
1
0
0
1
2
3
4
5
q
6
7
8
Figure 6: Shift of Cost Curve from Learning.
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3 Economies of Scope, Learning
Figure 7: Structure of Production Theory.
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6
1 Profit Maximization
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 19, 2007
Lecture 15
Short Run and Long Run Supply
Outline
1. Chap 8: Profit Maximization
2. Chap 8: Short Run Supply
3. Chap 8: Producer Surplus
4. Chap 8: Long Run Competitive Equilibrium
1
Profit Maximization
For perfect competition in a product market, we make some assumptions:
• Price taking: either individual firms or consumers cannot affect the price.
• Product homogeneity: product of all firms are perfect substitutes.
• Free entry and exit: no special cost to enter or exit the market.
Firms choose the level of output to maximize their profits. Profit equals
total revenue minus total cost, namely
π(q) = R(q) − C(q) = P (q)q − C(q).
To maximize the profit, the following condition must hold:
dπ(q)
dR dC
=
−
= M R(q) − M C(q) = 0,
dq
dq
dq
and thus
M R(q) = M C(q).
Since
R(q) = P q,
we have
M R(q) =
dR(q)
= P,
dq
and
M R = AR,
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2 Short Run Supply
2
thus
M C(q) = P = M R = AR
is the maximization condition. Note that the condition is not sufficient. In
Figure 1), if the price is P2 , q2 and q3 both satisfy the condition, but only q3
maximizes the profit.
10
9
8
P1
7
C
6
MC
5
4
3
P2
2
1
0
q
q
2
0
1
q1
3
2
3
4
5
6
7
8
9
10
q
Figure 1: Profit Maximization.
2
Short Run Supply
Assume the firm has production costs shown in Figure 2, let us discuss its
behavior under different prices.
• When P = P1 , the firm is making profits, so it will continue to produce;
• When P = P2 , the firm has losses but still continues to produce, because if
it shuts down, the profit is −F C, and if continuing to produce, the profit
is R − T V C − F C > −F C.
• Since R < SV C, when P = P3 , the profit if the firm shuts down, −F C, is
more than the profit if it continues, R − T V C − F C, so it will shut down.
When the firm produces, it chooses the output level where M C(q) = P . There­
fore, the firm’s supply curve when it produces is just the part of M C above
T V C. When P < AV C, the firm shuts down and q = 0.
We can derive market supply from an individual firm’s supply (see Figure 3).
Define elasticity of market supply as follows:
ES =
dQ/Q
.
dP/P
Figure 4 and 5 stand for inelastic and elastic supply curves, respectively.
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2 Short Run Supply
3
10
9
8
MC
P1
7
C
6
ATC
5
4
P2
3
2
AVC
P3
1
0
0
1
2
3
4
5
q
6
7
8
9
10
Figure 2: Individual Firm’s Supply in Short Run.
10
9
8
7
MC1
P
6
Market Supply
MC2
5
4
3
2
1
0
0
1
2
3
4
5
q
6
7
8
9
10
Figure 3: Market Supply in Short Run.
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2 Short Run Supply
4
10
9
8
7
P
6
MC
5
4
3
2
1
0
0
1
2
3
4
5
q
6
7
8
9
10
9
10
Figure 4: Inelastic Market Supply Curve.
10
9
8
7
P
6
MC
5
4
3
2
1
0
0
1
2
3
4
5
q
6
7
8
Figure 5: Elastic Market Supply Curve.
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2 Short Run Supply
5
10
9
8
7
P
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
q
Figure 6: Perfectly Inelastic Market Supply Curve.
10
9
8
7
P
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
q
Figure 7: Perfectly Elastic Market Supply Curve.
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3 Producer Surplus
6
Similarly, we have perfectly inelastic market supply (see Figure 6) and perfectly
elastic market supply (see Figure 7).
Perfectly elastic market supply happens when
M C = const.
3
Producer Surplus
Producer Surplus is the difference between the firm’s revenue and the sum of
the total variable cost of producing q (see Figure 8):
P S = R − T V C = R − T V C − F C + F C = P rof it + F C.
Thus, producer surplus is the sum of profit and fixed cost.
8
7
6
MC
P
5
4
3
AVC
2
1
0
0
1
2
3
4
q
5
6
7
8
Figure 8: Producer Surplus.
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1 Long Run Competitive Equilibrium
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 19, 2007
Lecture 16
Long Run Supply and the Analysis of
Competitive Markets
Outline
1. Chap 8: Long Run Equilibrium
2. Chap 8: Long Run Market Supply
3. Chap 9: Gains and Losses from Government Policies
1
Long Run Competitive Equilibrium
In Figure1, an existing firm’s marginal cost and average total cost are SM C
and SAC. The short-run market price is 7, so existing firms are making profits.
In the long run, capital can be changed; old firms expand, new firms enter the
market, thus the supply increases, which leads to price decreasing. The price
will decrease until P = LM C = LAC, so that firms have no economic profit.
In the long run, firms earn zero profit, and in the short run, firms can have
positive profit. However, the short run profit is not always higher because firms
can also have negative profit (when P < AT C).
10
9
LMC
8
SMC
P7
6
SAC
5
LAC
4
3
2
1
0
0
1
2
3
4
5
x
6
7
8
9
10
Figure 1: Long Run Equilibrium Price.
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2 Long Run Market Supply
2
At long run competitive equilibrium:
• All firms are maximizing profit, or M R = M C.
• No firm has incentive to enter or exit earning zero economic profit (this is
the difference between short run and long run).
•
QD = QS .
In Figure 2, suppose the original price is 4. Existing firms make profit, so
new firms enter the market and the market supply curve shifts from S1 to S2 .
Now the market price is 3, existing firms make no profit, and new firms stop
entering. Thus, the equilibrium is reached.
In Figure 3, the original price 2 is lower than AC. Firms have a loss and
start leaving the market, and the market supply shifts from S1 to S2 .
5
5
4.5
4.5
S2
4
4
LAC
3.5
S1
3.5
3
3
2.5
2.5
2
2
LMC
1.5
1.5
1
1
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
D
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
Q
Q
(a) Long Run Cost.
(b) Shift of Equilibrium.
Figure 2: Long Run Equilibrium, High Price.
5
5
4.5
4.5
4
S1
3.5
3
3
2.5
2.5
2
2
LMC
1.5
1.5
1
1
0.5
0.5
0
S2
4
LAC
3.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
D
0
0.5
Q
(a) Long Run Cost.
1
1.5
2
2.5
3
3.5
4
Q
(b) Shift of Equilibrium.
Figure 3: Long Run Equilibrium, Low Price.
2
Long Run Market Supply
Assume that:
• All firms have the same technology;
• Initially firms produce at minimum long run average cost.
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2 Long Run Market Supply
3
Constant-Cost Industry
In constant-cost industry, price of inputs does not change. If the price is higher
than minimum LAC. new firms will keep entering, so the supply is perfectly
elastic at P = minimumLAC. Long run supply is a horizontal line at the price
equal to the minimum LAC (see Figure 4(b)).
5
4
4.5
3.8
4
3.6
LAC
3.5
2
3
2.8
LMC
1.5
2.6
1
2.4
0.5
0
SL
P*
3.2
2.5
P
P
3.4
P*
3
2.2
Q*
0
0.5
1
1.5
2
2.5
Q
3
3.5
4
4.5
2
5
0
0.5
1
1.5
2
2.5
Q
3
3.5
4
4.5
5
(a) Long Run Cost in Constant-Cost In- (b) Supply Curve in Constant-Cost In­
dustry.
dustry.
Figure 4: Long Run Market Equilibrium in Constant-Cost Industry.
Increasing-Cost Industry
Price of some or all inputs rises as production is expanded and demand of inputs
increases. When the price increase from P ∗ to P ′ , firms are making profit. Old
5
4.5
4
LAC
3.5
P*
3
5
P
4.5
2.5
P’
4
3.5
2
P
SL
P*
3
LMC
1.5
2.5
2
1
1.5
0.5
0
1
Q*
0
0.5
1
1.5
2
2.5
Q
0.5
3
3.5
4
4.5
5
0
0
0.5
1
1.5
2
2.5
Q
3
3.5
4
4.5
5
(a) Long Run Cost in Increasing-Cost (b) Supply Curve in Increasing-Cost InIndustry.
dustry.
Figure 5: Long Run Market Equilibrium in Increasing-Cost Industry.
firms expand and new firms enter, so the demand of inputs increase, and so do
the prices of inputs. Firm’s cost curves increase to LM C ′ and LAC ′ / Since now
firms have zero profit, new firms stop entering. The quantity supplied increases
but is still finite. Thus the supply curve is upward sloping.
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3 Gains and Losses from Government Policies
3
4
Gains and Losses from Government Policies
Consumer Surplus and Producer Surplus
Consumer Surplus. Area between demand curve and market price (see Fig­
ure 6).
Producer Surplus. Area between supply curve and market price (see Fig­
ure 6).
5
4.5
S
4
3.5
Consumer Surplus
3
P
Producer Surplus
2.5
2
D
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
Q
3
3.5
4
4.5
5
Figure 6: Consumer Surplus and Producer Surplus.
CS (Consumer Surplus) plus P S (Producer Surplus) is maximized at the quan­
tity when demand equals supply.
Price Ceiling
When there is no intervention, the equilibrium price and quantity are P ∗ and
Q∗ , respectively. Now government sets a price ceiling, namely, a maximum price
P̄ (see Figure 7). The changes in consumer surplus and producer surplus are as
follows:
ΔCS = A − B,
ΔP S = −A − C,
ΔCS + ΔP S = −B − C.
Deadweight loss, or net loss of CS + P S, is −(B + C) in this case. Government
should maximize economic efficiency: maximize CS + P S. If policies cause
deadweight loss, they impose an economy cost on the economy.
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3 Gains and Losses from Government Policies
5
5
4.5
4
P
P̄ 3.5
P *3
P
B
C
A
2.5
2
1.5
1
0
Q∗
Q′
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Q
Figure 7: Price Ceiling.
Price Floor
Government sets a price floor (price support), namely, a minimum price P (see
Figure 8). The changes in consumer surplus and producer surplus from the
competitive equilibrium (P ∗ , Q∗ ) to the new equilibrium (P , Q′ ) are as follows:
ΔCS = −A − B;
ΔP S = A − C;
ΔCS + ΔP S = −B − C.
Thus there is still a deadweight loss.
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3 Gains and Losses from Government Policies
6
5
4.5
S
4
3.5
B
C
P
3
A
2.5
D
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
Q
3
3.5
4
4.5
5
Figure 8: Price Floor.
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1 Agricultural Price Support
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 24, 2007
Lecture 17
Supply Restrictions, Tax, and Subsidy
Outline
1. Chap 9: Agricultural Price Support
2. Chap 9: Supply Restrictions
3. Chap 9: Tax and Subsidy
1
Agricultural Price Support
In this case, government sets prices higher than the free market level, and buys
excess supply (see Figure 1). The buyer’s price is shown on the y-axis in the
following graphs. The original consumer surplus equals the area between the
10
9
8
S
7
P
2
P
6
A
P
B
D
1
5
4
E
3
D
2
1
0
Q
Q
2
0
1
2
3
Q
1
4
3
5
Q
6
7
8
9
10
Figure 1: Agricultural Price Support.
demand curve and the line of price P1 ; after the price support, it equals the
area between the demand curve and the line of price P2 , thus
ΔCS = −(A + B).
The original producer surplus equals the area between the supply curve and the
line of price P1 ; after the price support, it equals the area between the supply
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2 Supply Restrictions
2
curve and the line of price P2 , thus
ΔP S = A + B + D.
Government buys quantity Q3 − Q2 at price P2 ; the cost equals the area of the
rectangular
ΔG = −(B + D + E).
The deadweight loss to the society is
DW L = −(B + E).
2
Supply Restrictions
Government restricts quantity supplied to be less than Q1 (see Figure 2). The
10
9
8
S
P
7
6
P1
5
P0
4
P2
A
B
C
D
3
2
1
0
Q
Q0
1
0
2
4
6
8
10
Q
Figure 2: Supply Restriction.
original consumer surplus equals the area between the demand curve and the
line of price P0 ; after the supply restriction, it equals the area between the
demand curve and the line of price P1 , thus
ΔCS = −(A + B).
The original producer surplus equals the area between the supply curve and the
line of price P0 ; after the supply restriction, it equals the area of the trapezoid,
with the supply curve, the line of price P1 , the line of quantity Q1 , and the price
axis as its sides, thus
ΔP S = A − C.
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2.1
Zero Quota
3
Thus, the deadweight loss is
DW L = −(B + C).
Example government measures include import quota and tariff, which benefit
domestic producers but hurt consumers.
2.1
Zero Quota
SD is the domestic supply, and DD is the domestic demand. If no import is
allowed, the domestic price is P0 . Without restriction on import, the domestic
price would be the same as the world price PW , which is lower than PD (see
Figure 3). Without import quota restriction, consumer surplus equals the area
10
9
8
S
7
D
P
6
5
P0
4
PW
A
B
C
3
DD
2
1
0
Q
Q
S
0
1
2
3
Q
0
4
D
5
Q
6
7
8
9
10
Figure 3: Zero Quota.
between the domestic demand curve and the line of price PW ; if the quota is
zero, it equals the area between the domestic demand curve and the line of price
P0 , thus
ΔCS = −(A + B + C).
Without quota restriction, producer surplus equals the area between the domes­
tic supply curve and the line of price PW ; if the quota is zero, it equals the area
between the domestic supply curve and the line of price P0 , thus
ΔP S = A.
The deadweight loss is
DW L = B + C.
2.2
Non-Zero Quota
Given the same SD , DD , and PW , now suppose the government sets non-zero
quota k. The domestic price P1 is where the difference between domestic demand
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2.3
Import Tariff
4
10
9
8
7
SD
P
6
5
4
P1
3
PW
A
B
C
D
D
D
2
1
0
QS
0
2
QS1
QD1
4
QD
6
8
10
Q
Figure 4: Non-Zero Quota.
(QD1 ) and domestic supply (QS1 ) is k (see Figure 4). Likewise, the change of
consumer surplus
ΔCS = −(A + B + C + D);
and the change of domestic producer surplus
ΔP SD = A.
The net domestic loss equals
−(ΔCS + ΔP S) = B + C + D.
The foreign producer surplus increases by excess profits, which equal the area
of rectangular C
ΔP SF = C.
The total deadweight loss is
DW L = B + D.
The domestic loss is
Domestic Loss = B + C + D.
2.3
Import Tariff
Government imposes a tariff P1 − PW on each unit imported (see Figure 5).
The change of consumer surplus and domestic producer surplus are
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3 Tax and Subsidy
5
10
9
8
7
SD
P
6
5
P
1
4
PW
3
A
B
C
D
DD
2
1
0
QS
0
QS1
2
QD1
4
QD
6
8
10
Q
Figure 5: Import Tariff.
ΔCS = −(A + B + C + D)
and
ΔP SD = A,
respectively. Foreign producers gain nothing, that is to say
ΔP SF = 0,
because C becomes the revenue of government
ΔG = C.
The deadweight loss is
DW L = B + D,
which equals to the domestic loss.
3
Tax and Subsidy
Assume that government imposes a $1 tax on each cigarette unit. Given the
market price P , if the tax is paid by
• producers, then buyers pay P and producers get P − 1;
• consumers, then buyers pay P + 1 and producers get P .
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3 Tax and Subsidy
6
Therefore, the price paid by buyers and the price received by producers always
have a difference of 1 (see Figure 6). Let PB be the buyer’s price and PS be the
seller’s price.
PD − PS = 1.
In figure 6, we put buyer’s price on the y axis. Therefore, with the tax, the
supply curve moves from S to S ′ . The equilibrium buyer’s price is PD , and
the equilibrium seller’s price is PS . Thus, the consumer surplus and producer
5
4.5
S’
4
buyer’s price
3.5
3
S
P
A
P
D
2.5
P0
2
B
D
C
P
D
S
1.5
1
0.5
0
Q1
Q
0
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 6: Tax.
surplus both decrease:
ΔCS = −(A + B),
ΔP S = −(C + D).
Government revenue
ΔG = A + C.
So, the deadweight loss is
DW L = B + D.
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1 Tax
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 26, 2007
Lecture 18
Tax, Subsidy, and General Equilibrium
Outline
1. Chap 9: Tax
2. Chap 9: Subsidy
3. Chap 16: General Equilibrium
4. Chap 16: Exchange Economy
1
Tax
Government imposes a $1 tax on every unit sold (see Figure 1), as discussed in
Lecture 17. The buyer’s price is shown on the y-axis. The consumer surplus
5
4.5
S’
4
buyer’s price
3.5
3
S
P
A
P
D
2.5
P0
2
B
D
C
P
D
S
1.5
1
0.5
0
Q
Q
1
0
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 1: Tax.
and producer surplus both decrease:
ΔCS = −(A + B),
ΔP S = −(C + D).
Government revenue
ΔG = A + C.
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2 Subsidy
2
So the deadweight loss is
DW L = B + D.
The burden of a tax is shared by consumers and producers; the relative amount
borne by consumers and producers depends on relative elasticities of demand
and supply.
• If the demand is inelastic (see Figure 2),
10
9
8
PD
7
S
P
6
P0
P
5
S
A
B
C
D
4
3
D
2
1
0
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 2: Tax Burden on Buyers, Relative Inelastic Demand Curve.
ΔCS = −(A + B),
ΔP S = −(C + D),
buyers bear most of the burden of the tax.
• If the supply is inelastic (see Figure 3),
ΔCS = −(A + B),
ΔP S = −(C + D),
producers bear most of the burden of the tax.
Pass-through fraction is the percentage of a tax borne by consumers. It tells the
fraction of tax ”passed through” to consumers through higher price. If ED = 0,
say the demand is perfectly inelastic (see Figure 4), buyers bear all of the tax
burden:
ES
= 1.
ES − ED
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2 Subsidy
3
10
9
8
S
7
P
6
P
D
P0
5
A
B
D
C
4
PS
3
D
2
1
0
0
2
4
6
8
10
Q
Figure 3: Tax Burden on Producers, Relative Inelastic Supply Curve.
Figure 4: Tax Burden on Buyers, Perfectly Inelastic Demand Curve.
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2 Subsidy
4
5
4.5
4
S
3.5
PS
P
3
P
C
D
P
A
B
0
2.5
E
B
2
1.5
D
1
0.5
Q
0
Q
1
0
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 5: Subsidy.
2
Subsidy
Government subsidizes $1 for each unit sold (see Figure 5). In this case, sellers’
price is higher than buyers’ price:
PB = PS + 1.
The consumer surplus increases by
ΔCS = A + B;
and the producer surplus increases by
ΔP S = C + D.
Government expenditure equals the whole area between PB and PS under the
quantity Q1
ΔG = −(A + B + C + D + E).
The deadweight loss is
DW L = E.
Likewise we can discuss the benefit of subsidy:
D
• if E
ES is small, namely, the demand is more inelastic, the benefit of subsidy
goes mostly to buyers;
D
• if E
ES is large, namely, the supply is more inelastic, the benefit of subsidy
goes mostly to sellers.
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3 General Equilibrium
3
5
General Equilibrium
Partial equilibrium. Ignores effects form other markets.
General equilibrium. Simultaneous determination of the prices and quanti­
ties in all relevant markets, taking into account feedback effects.
Feedback effect. The price or quantity adjustment in one market caused by
price and quantity adjustments in related markets.
Example (DVD and Movie Tickets Markets). The price of a DVD is $3, and the
price of a movie ticket is $6 at equilibrium. Now tax $1 on the movie ticket (see
Figure 6). The specific process of price change is listed as follows:
MOVIE TICKET :
′
SM → SM
,
Price change:6 → 6.35;
DVD :
The price change of movie tickets shifts the demand curve of DVD.
DV → DV′ ,
Price change:3 → 3.5;
MOVIE TICKET :
The price change of DVD shifts the demand curve of movie tickets.
′
DM → DM
,
Price change:6.35 → 6.75;
and so on. The final equilibrium prices are
P (M OV IET ICKET ) = 6.85;
P (DV D) = 3.58.
If we ignore the feedback effects, we might underestimate the price change
bought by the tax.
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3 General Equilibrium
6
10
9
SM*
8
SM
7
P
6
5
4
DM
3
2
1
0
0
2
4
6
8
10
Q
(a) Price Change of Movie Ticket.
10
9
8
7
SV
P
6
5
4
3
DV*
2
1
0
QV*
Q
V
0
1
2
3
4
5
Q
D
V
6
7
8
9
10
(b) Price Change of DVD.
Figure 6: General Equilibrium of DVD and Movie Ticket Markets.
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4 Exchange Economy
4
7
Exchange Economy
Assume that:
• there are two consumers A and B;
• there are two goods, food and clothing;
• the quantities of food and clothing are 10 and 6, and A has 7 food and 1
clothing, while B has 3 food and 5 clothing;
• they know each others’ preferences;
• transaction cost is zero.
The edgeworth box is shown in Figure 7.
6
O
B
5
Clothing
4
3
2
1
A(7F,1C)=B(3F,5C)
OA
0
0
1
2
3
4
5
Food
6
7
8
9
10
Figure 7: Edgeworth Box.
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1 Exchange Economy
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 29, 2007
Lecture 19
Efficiency in Exchange, Equity and Efficiency,
and Efficiency in Production
Outline
1. Chap 16: Exchange Economy
2. Chap 16: Contract Curve
3. Chap 16: General Equilibrium in a Competitive Market
4. Chap 16: Utility Possibilities Frontier
5. Chap 16: Production in Edgeworth Box
1
Exchange Economy
In the Edgeworth box (see Figure 1) given endowment E, the area between A’s
and B’s utility curves contains all beneficial trades, but not all are efficient; that
is to say, both A and B are better off in this area, but they will keep trading until
they cannot make both of them better. Then the possible efficient allocation
given the endowment E should satisfy that:
• there is no more room for trade,
• thus M RSA = M RSB .
2
Contract Curve
Contract curve shows all possible efficient allocations; it contains all points of
tangency between A’s and B’s indifference curves (see Figure 2).
3
General Equilibrium in a Competitive Market
Assume that consumers are price-takers. There are two consumers, A and B,
and two goods, X and Y, in the market. The total endowment of X is x units,
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3 General Equilibrium in a Competitive Market
2
10
OB
9
8
UB
7
6
Clothing
5
D
4
UA
3
C
2
E
1
0
OA
0
Food
1
2
3
4
5
6
7
8
9
10
Figure 1: Finding the Efficient Allocation in Edgeworth Box.
10
OB
9
8
7
6
5
4
3
2
1
0
OA
0
1
2
3
4
5
6
7
8
9
10
Figure 2: Contract Curve.
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4 Utility Possibilities Frontier
3
and the total endowment of Y is y units. Obviously, demand equals supply at
equilibrium. We denote the equilibrium state by
∗
∗
(Px∗ , Py∗ , (x∗A , yA
), (x∗B , yB
)).
If we suppose A has one unit of X and two units of Y initially, the budget
constraint is
xA Px + yA Py = Px + 2Py ,
and divide it by Py ,
xA
Px
Px
+ yA =
+ 2;
Py
Py
so we only care about the price ratio PPxy . For convenience, we usually set Py∗
to 1 so that the expression above has five unknowns. To find the equilibrium,
several conditions should be satisfied:
∗
∗
• (xA
, yA
) maximize A’s utility subject to the budget constraint, then we
obtain two equations;
∗
• (x∗B , yB
) maximize B’s utility subject to the budget constraint, likewise
we can obtain another two equations;
∗
∗
• the quantity is conserved, or x∗A + yA
= x and x∗B + yB
= y (actually one
of these equations is redundant because it is automatically satisfied given
the preceding four and another from these two equations).
Finally, we obtain five equations. Therefore, the problem can be solved (see
Figure 3). Assume that yA + yB > Y and xA + xB < X; it is not at equilibrium,
because Px > Px∗ , so price of X will decrease. At equilibrium, we must have
the right price ratio. For example, if the price for X is Px , Px > Px∗ , then
yA + yB > y and xA + xB < x. Y has excess demand, and X has excess supply.
4
Utility Possibilities Frontier
Utility possibilities frontier shows the utility levels when the two individuals
have reached the contract curve (see Figure 4). Choosing a point below the
frontier, for example, A, the allocation is inefficient; choosing a point above the
frontier, for example, B, the allocation is unobtainable.
5
Production in Edgeworth Box
Now we discuss the producer’s problem. There are two industries. One produces
food and the other produces clothing. The isoquant curves are shown in the
Edgeworth box (see Figure 5). E is their initial endowment of inputs. An
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5 Production in Edgeworth Box
4
10
x*
O
B
9
B
8
y*B
7
6
UB
5
4
3
1
0
A
A
2
Budget
Constraint
O
x*A
A
0
E: Initial Endowment
U
y*
1
2
3
4
5
6
7
8
9
10
Figure 3: Contract Curve.
10
. Point B
0
−10
Utility of B
Point A
.
−20
−30
Utility of A
−40
−50
0
1
2
3
4
5
6
7
8
9
10
Figure 4: Utility Possibilities Frontier.
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5 Production in Edgeworth Box
5
allocation of inputs is technically efficient if the output of one good cannot be
increased without decreasing the output of another, so
M RT SF = M RT SC .
At competitive equilibrium in the input market,
• wage and rent are equal for all industries;
• total L and K in all industries are equal to aggregate available supplies.
Similar to the consumers’ problem, the general equilibrium can be characterized
by
(w, r, (LF , KF ), (LC , KC )).
Production possibilities frontier shows various combinations of two goods that
can be produced with fixed quantity of input demanded from production con­
tract curve.
10
OClothing
9
8
7
QF
6
Capital
5
4
Q
C
3
2
1
0
E
OFood
0
1
Labor
2
3
4
5
6
7
8
9
10
Figure 5: Production in a Edgeworth Box.
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1 Production Possibilities Frontier
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 31, 2007
Lecture 20
Production Possibilities Frontier and Output
Market Efficiency
Outline
1. Chap 16: Production Possibilities Frontier
2. Chap 16: Output Market Efficiency
1
Production Possibilities Frontier
Marginal rate of transformation (M RT ):
• How much clothing must be given up to produce one additional unit of
food.
• The absolute value of the slope of the production possibilities frontier.
• If M RT increases in food, then the production possibilities frontier is
concave.
•
M RT =
M CF
.
M CC
Proof. Reducing $1 input from clothing, C decreases by
1
input to food, F increases by MC
. Thus,
F
M RT =
2
ΔC
=
ΔF
1
MCC
1
MCF
=
1
MCC
; adding $1
M CF
.
M CC
Output Market Efficiency
Suppose we have two industries, clothing and food, in the market. Consumers
have demand for the two goods. They have a representative utility U (C, F ). A
Pareto efficient result occurs when the production possibilities frontier is tangent
to the indifference curve (see Figure 3). That is to say,
M RT = M RS.
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2 Output Market Efficiency
2
10
9
OC
8
7
6
5
4
3
2
OF
1
0
0
1
2
3
4
5
6
7
8
9
10
9
10
Figure 1: Production Contract Curve.
10
9
Clothing
8
7
C
6
5
Production Possibilities Frontier
4
3
2
Food
1
0
0
1
2
3
4
5
F
6
7
8
Figure 2: Production Possibilities Frontier.
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2 Output Market Efficiency
3
10
9
8
7
C
6
Indifference Curve
5
Production Possibilities Frontier
4
3
2
1
0
0
1
2
3
4
5
F
6
7
8
9
10
Figure 3: Production Possibilities Frontier and Indifference Curve.
10
9
8
7
C
6
Indifference Curve
5
Production Possibilities Frontier
4
3
2
1
0
0
1
2
3
4
5
F
6
7
8
9
10
Figure 4: Equilibrium in the Output Market.
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2.1
General equilibrium in the output market
4
The prices are PF for food, and PC for clothing. When the market reaches
its equilibrium, industries are maximizing their profits, so
M CF (q) = PF ;
M CC (q) = PC .
Thus,
M RT =
M CF
PF
=
.
M CC
PC
Consumers maximize their utility, so
M RS =
PF
.
PC
Combining the equations together, we obtain (see Figure 4)
M RT =
PF
= M RS.
PC
Consider non-equilibrium prices PF′ and PC′ ,
PF′
PF
<
.
′
PC
PC
Given the prices, food has a shortage and clothing has an excess (see Figure 5).
The prices will change to adjust to the equilibrium state, namely, PF′ increases
and PC′ decreases.
2.1
General equilibrium in the output market
Example (Gains from Free Trade). Assume that Holland and Italy both produce
cheese and wine, unit of labor required is provided in Table 2.1). If these
Holland
Italy
Cheese
1
6
Wine
2
3
Table 1: Unit of Labor Required in Cheese and Wine Production.
two countries cannot trade cheese or wine, we consider the domestic markets
separately. The price ratio is not the same:
H
I
PW
PW
<
.
PCH
PCI
Consumer utility levels are UH and UI , respectively. However, if they can trade,
Holland exports cheese and imports wine, and Italy exports wine and imports
cheese. The prices ratio will adjust to agree, and people in both countries are
better off because both indifference curves move upwards (see Figure 6). The
′
new utility levels are UH
and UI′ .
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2.1
General equilibrium in the output market
5
10
9
8
7
C
6
Supply
5
Demand
4
3
2
1
0
0
1
2
3
4
5
F
6
7
8
9
10
Figure 5: Non-equilibrium Consumption and Production.
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2.1
General equilibrium in the output market
6
14
Holland
12
10
8
UH’
6
4
UH
2
0
0
1
2
3
4
5
6
7
8
9
10
8
9
10
(a) Trade in Holland.
10
Italy
9
8
7
C
6
U’
5
I
4
3
U
I
2
1
0
0
1
2
3
4
5
W
6
7
(b) Trade in Italy.
Figure 6: Gains from Free Trade.
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1 Why Markets Fail
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 2, 2007
Lecture 21
Why Markets Fail
Outline
1. Chap 16: Why Markets Fail
1
1.1
Why Markets Fail
Market Power
Inefficiency arises when a producer or supplier of a factor input has market
power, for example, monopoly power, that can profitably charge a price greater
than marginal cost.
1.2
Incomplete Information
For example, in the second-hand car market, sellers know more about the cars
than buyers. Final allocation might be inefficient when there is incomplete
information.
1.3
Externalities
Consumption or production has indirect effect on other consumption or pro­
duction, which is not reflected in market prices. An example is air and water
pollution by a factory.
1.4
Public Goods
For one firm’s new technology, others may copy it if there is no patent law; all
firms are thus waiting for others to invent.
Examination 2 Review
Examination: Chapter 6, 7, 8, 9, and 16. (Review Lectures 10–20.)
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1 Monopoly
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 7, 2007
Lecture 22
Monopoly
Outline
1. Chap 10: Monopoly
2. Chap 10: Shift in Demand and Effect of Tax
1
Monopoly
The monopolist is the single supply-side of the market and has complete control
over the amount offered for sale; the monopolist controls price but must operate
along consumer demand.
1.1
Revenue in Monopoly
Review the revenue in perfect competition:
R = PQ
AR = M R = P.
(1.1)
(1.2)
Revenue of monopolist is also
R = P (Q)Q,
but P changes with Q because the monopolist faces the whole market demand
and his quantity supplied affects the market price. Then the average revenue is
AR =
R
= P (Q);
Q
and the marginal revenue is
MR =
dR
d(P Q)
dP
=
= P (Q) + Q
.
dQ
dQ
dQ
The relation between P and Q is determined by the demand curve (see Figure 1).
Since
dP
< 0,
dQ
M R < P (Q).
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1.2
Output Decision in Monopoly
2
Example (A Demand Function). Suppose the price is
P = 10 − QD ,
where QD is the quantity demanded. Calculate the average revenue and the
marginal revenue:
AR = P = 10 − Q;
dP
MR = p + Q
= 10 − 2Q.
dQ
15
10
Demand Curve
P1
5
P2
0
0
Q1
5
Q2
10
15
Figure 1: Demand and Supply of Monopolist.
1.2
Output Decision in Monopoly
The monopolist will maximize its profit
π(Q) = R(Q) − C(Q),
which is the difference of revenue and cost. When maximized,
dπ
dR dC
=
−
= 0,
dQ
dQ dQ
namely,
M R = M C,
so the monopolist would choose this point to produce; because
P > M R,
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1.3
Lerner’s Index
3
P > M C.
The profit equals to
(AR − AC)Q = (P − AC)Q
(see Figure 2).
15
MC
Demand Curve
10
P
AR
AC
P*
5
0
MR
Q*
0
5
10
15
Q
Figure 2: Output Decision of Monopolist.
1.3
Lerner’s Index
Rewrite the marginal revenue:
MR = P + Q
dP
Q dP
1
= P + P(
)=P +P
.
dQ
P dQ
ED
The monopolist chooses to produce the quantity where
MC = MR = P + P
Thus,
1
.
ED
1
P − MC
=
,
|ED |
P
(1.3)
which is the makeup over M C as a percentage of price; this fraction is less than
1. L = P −PMC measures the monopoly power of a firm and is called Lerner’s
index.
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2 Shift in Demand and Effect of Tax
4
• In a competitive market,
M C = P,
and the makeup is zero.
• In a monopolistic market,
M C < P,
and the makeup is larger than zero.
Comments:
1. The makeup increases with the inverse of demand elasticity.
2. The larger the demand elasticity, the less profitable it is to be a monopolist
(see Figure 3 and 4).
3. A monopolist never produces a quantity at the inelastic portion of demand
curve, since the makeup right hand side of Equation 1.3 is less than one.
10
9
D
8
P*
7
MR
6
5
4
MC
3
2
1
0
Q*
0
1
2
3
4
5
6
7
8
9
10
Figure 3: Inelastic Demand.
2
Shift in Demand and Effect of Tax
Compare the competitive market and the monopolistic markets.
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2.1 Supply Curve of Competitive Market and Monopolistic
Markets
5
10
9
8
P*
7
MR D
P
6
5
4
MC
3
2
1
0
Q*
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 4: Elastic Demand.
2.1
Supply Curve of Competitive Market and Monopolis­
tic Markets
The supply curve in competitive markets is determined by M C, and there is no
supply curve for monopolistic markets.
2.2
Shift in Demand
In competitive markets, when demand shifts, the changes in price and quantity
has a positive relation, namely, if the price raises, the quantity increases. In
monopolistic markets, when the demand shifts, it may be the case that only
price changes (see Figure 5), only quantity changes (see Figure 6), or both
change.
2.3
Effect of Tax
In competitive marketes, buyer’s prices raise less than the tax, and the burden is
shared by Producers and Consumers; in monopolistic markets, the price might
raise more than tax (see Figure 7).
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2.3
Effect of Tax
6
15
AR2
MR2
10
MC
P
P2
P1
5
0
MR1
AR1
Q1=Q2
0
5
10
15
Q
Figure 5: Only Price Change in Monopoly.
15
AR2
10
MC
P
MR2
5
P1=P2
MR1
0
AR1
Q1
0
Q2
5
10
15
Q
Figure 6: Only Quantity Change in Monopoly.
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2.3
Effect of Tax
7
10
9
8
D
7
MR
P
6
P2
5
MC2=MC1+T
4
P1
3
MC1
2
1
0
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 7: Price Might Raise More than Tax.
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1 Multi-Plant Firm
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 9, 2007
Lecture 23
Monopoly and Monopsony
Outline
1. Chap 10: Multi-Plant Firm
2. Chap 10: Social Cost of Monopoly Power
3. Chap 10: Price Regulation
4. Chap 10: Monopsony
1
Multi-Plant Firm
How does a monopolist allocate production between plants?
Suppose the firm produces quantity Q1 with cost C1 (Q1 ) for plant 1, and quan­
tity Q2 with cost C2 (Q2 ) for plant 2. The total quantity is
QT = Q1 + Q2 .
And the profit is
π = QT P (QT ) − C1 (Q1 ) − C2 (Q2 ) = (Q1 + Q2 )P (Q1 + Q2 ) − C1 (Q1 ) − C2 (Q2 ).
To solve, use the first order constraint:
dπ
dP (Q1 + Q2 )
dC1
= P (Q1 + Q2 ) + (Q1 + Q2 )
−
= 0,
dQ1
dQ1
dQ1
Since
P (QT ) + QT
dP (QT )
dP (QT )
= P (QT ) + QT
= M R(QT ),
dQ1
dQT
M R(QT ) = M C1 (Q1 ).
Similarly,
M R(QT ) = M C2 (Q2 ).
Thus,
M R(QT ) = M C1 (Q1 ) = M C2 (Q2 ).
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2 Social Cost of Monopoly Power
2
2
Social Cost of Monopoly Power
Firstly, compare the producer and consumer surplus in a competitive market and
a monopolistic market. In the competitive market, the quantity is determined
by
M C = AR,
while in the monopolistic market, the quantity is determined by
MC = MR
(see Figure 1). Therefore, in going from a perfectly competitive market to a
10
9
8
7
P
6
PM
5
4
A
PC
MR
3
D=AR
MR
2
1
0
MC
B
C
0
1
2
QM Q
C
3
4
5
Q
6
7
8
9
10
Figure 1: Consumer and Producer Surplus in Monopolist Market.
monopolistic market, the change of consumer surplus and producer surplus are,
respectively,
ΔCS = −(A + B),
and
ΔP S = A − C.
The deadweight loss is
DW L = B + C.
In fact, social cost should not only include the deadweight loss but also rent seek­
ing. The firm might spend to gain monopoly power by lobbying the government
and building excess capacity to threaten opponents.
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3 Price Regulation
3
3
Price Regulation
In perfectly competitive markets, price regulation causes deadweight loss, but
in monopoly, price regulation might improve efficiently. Now we discuss four
possible price regulations in monopolistic markets. P1 , P2 , P3 , P4 are:
•
P1 ∈ (PC , PM );
•
P2 = PC ;
•
P3 ∈ (P0 , PC );
•
P4 < P0 .
10
9
8
7
MC
P
6
5
4
3
PM
AC
PC
P0
MR
2
D=AR
1
0
0
1
2
QM Q
C
3
4
5
Q
6
7
8
9
10
Figure 2: Comparing Competitive and Monopolist Market.
Price between the competitive market price and monopolist market price.
Suppose the price ceiling is P1 . The new corresponding AR and M R
curves are shown in Figure 3. Given the new M R curve, the equilibrium
quantity will be Q1 .
Q1 ∈ (QM , QC ).
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3 Price Regulation
4
10
9
8
7
MC
P
6
5
4
P
MP
P 1
C
MR
3
2
1
0
0
1
2
Q Q1Q
M
C
3
4
AR
5
Q
6
7
8
9
10
Figure 3: Price between the Competitive Market Price and Monopolist Market
Price.
Price equal to the competitive market price. The new corresponding M R
and AR curves are shown in Figure 4. In this case the equlibrium price
and quantity are as same as those of the competitive market.
Price between the competitive market price and the lowest average cost.
Suppose the price ceiling is P3 . The new corresponding M R and AR
curves are shown in Figure 5. The equilibrium quantity will be Q3 .
Q3 ∈ (QC , Q0 ).
The new bold line describes the relation between price and quantity.
Price lower than the lowest average cost. The firm’s revenue is not enough
for the cost, thus it will quit the market. There is no production.
The analysis shows that if the government sets the price ceiling equal to P2 , the
outcome is the same as in a competitive market, and there is no deadweight
loss.
Natural monopoly. In a natural monopoly, a firm can produce the entire
output of the industry and the cost is lower than what it would be if there
were other firms. Natural monopoly arises when there are large economies
of scale (see Figure 6). If the market is unregulated, the price will be PM
and the quantity will be QM . To improve efficiency, the government can
regulate the price. If the price is regulated to be PC , the firm cannot cover
the average cost and will go out of business. PR is the lowest price that
the government can set so that the monopolist will stay in the market.
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3 Price Regulation
5
10
9
8
7
MC
P
6
5
4
PC
MR
3
2
AR
1
0
0
1
2
3
QC
4
5
Q
6
7
8
9
10
Figure 4: Price Equal to the Competitive Market Price.
10
9
8
7
MC
P
6
5
4
3
AC
PCP
P0 3 MR
2
1
0
0
1
2
3
Q AR
QC 3
4
5
Q
6
7
8
9
10
Figure 5: Price between the Competitive Market Price and the lowest Average
Cost.
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4 Monopsony
6
10
9
8
7
P
6
PM
5
P
4
AC
R
3
MC
P
2
C
1
0
0
1
2
Q MRQR
M
3
4
Q AR
C
5
Q
6
7
8
9
10
Figure 6: Regulating the Price of a Natural Monopoly.
4
Monopsony
Monopsony refers to a market with only one buyer. In this market, the buyer
will maximize its profit, which is the difference of value and expenditure:
max Π(Q) = V (Q) − E(Q).
When the profit is maximized,
d
(V (Q) − E(Q) = 0.
dQ
Thus
M V = M E,
namely, the marginal value (additional benefit form buying one more unit of
goods) is equal to the marginal expenditure (addtional cost of buying one more
unit of goods).
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1 Monopsony
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 14, 2007
Lecture 24
Monopoly and Monopsony
Outline
1. Chap 10: Monopsony
2. Chap 10: Monopoly Power
3. Chap 11: Price Discrimination
1
Monopsony
A monopsony is a market in which there is a single buyer. Typically, a monop­
sonist chooses to maximize the total value derived from buying the goods minus
the total expenditure on the goods: V (Q) − E(Q).
Marginal value is the additional benefit derived from purchasing one more unit
of a good; since the demand curve shows the buyer’s additional willingness to
pay for an additional unit, marginal value and the demand curve coincide.
Marginal expenditure is the additional cost of buying one more unit of a
good. Average expenditure is the market price paid for each unit, which is
determined by the market supply (see Figure 1). Now compare the competitive
and monopsony market.
• Competitive buying firms are price takers: The price P ∗ is fixed; therefore,
E = P ∗ × Q.
And then
AE = M E = P ∗
(see Figure 2).
• Monopsonist:
E = PS∗ (Q) × Q.
By definition,
AE =
and
ME =
E
= PS (Q);
Q
dE
dPS (Q)
= PS (Q) + Q∗ ×
.
dQ
dQ
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2 Monopoly Power
2
10
9
8
ME
7
S=AE
Price
6
5
PC
4
PM
B
A
C
3
D=MV
2
1
0
QM
0
1
2
QC
3
4
5
Quantity
6
7
8
9
10
Figure 1: Monopsony Market.
Since the supply curve is upward sloping,
M E > PS (Q) = AE.
To maximize
V (Q) − E(Q),
we have
M V (Q) = M E(Q).
Buyers gain A−B from monopsony power, while sellers lose A+C (see Figure 1);
the deadweight loss is B + C.
2
Monopoly Power
There usually is more than one firm in the market, and they have similar but
different goods. The Lerner’s index is
L=
P − MC
1
=
,
P
|Ed |
in which |Ed | is the elasticity of demand for a firm, as oppose to market demand
elasticity.
There are several factors that affect monopoly power.
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3 Price Discrimination
3
10
9
8
7
S
Price
6
PC
5
4
D
3
2
1
0
QC
0
1
2
3
4
5
Quantity
6
7
8
9
10
Figure 2: Competitive Buying Market.
• Elasticity of Market Demand: If the market demand is more elastic, the
firm’s demand is also more elastic. In a competitive market, elasticity of
demand for a firm is infinite. With more than one firm, a single firm’s
demand is more elastic than market demand.
• Number of Firms in Market: With more firms, the firm’s demand elasticity
is higher, namely, the market power is less.
• Interaction among Firms: If competitors are more aggressive, firms have
less market power; if firms collude, they thus have more market power.
3
Price Discrimination
Without market power, the producer would focus on managing production; with
market power, the producer not only manages production, but also sets price
to capture consumer surplus.
First Degree Price Discrimination
Knowing each consumer’s identity and willingness to pay, the producer charges
a separate price to each customer.
•
M R(Q) = PD (Q).
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3 Price Discrimination
4
10
9
MC
8
7
6
5
4
D=AR
3
2
MR
1
0
0
1
2
3
4
5
6
7
8
9
10
Figure 3: First Degree Price Discrimination.
• Choose Q∗ such that
M R(Q∗ ) = M C(Q∗ )
Q∗ is efficient.
• When the consumer surplus is zero, the producer surplus is maximized.
This kind of price discrimination is not usually encountered in real world.
Second Degree Price Discrimination
The producer charges different unit prices for different quantity purchased. It
applies to the situation when consumers are heterogeneous and the seller cannot
tell their identity, and consumers have multiple unit demand.
Third Degree Price Discrimination
Refer to next lecture.
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1 Third Degree Price Discrimination
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 16, 2007
Lecture 25
Pricing with Market Power
Outline
1. Chap 11: Third Degree Price Discrimination
2. Chap 11: Peak-Load Pricing
3. Chap 11: Two-Part Tariff
1
Third Degree Price Discrimination
Third degree price discrimination is the practice of dividing consumers into two
or more groups with separate demand curves and charging different prices to
each group (see Figure 1). Now maximize the profit:
10
10
9
9
9
8
8
7
7
7
6
6
6
P1
5
4
4
3
3
2
2
Q1 MR1
1
0
D1
0
1
2
3
4
8
Price
5
Price
Price
10
4
P2
5
Quantity
6
7
8
(a) Group 1.
9
10
3
2
1
0
MC
5
0
1
D2
MR2
Q2
2
3
4
5
Quantity
6
7
1
8
9
10
(b) Group 2.
0
MR(QT)
QT
0
1
2
3
4
5
Quantity
6
7
8
9
10
(c) Total Market.
Figure 1: Third Degree Price Discrimination.
π(Q1 , Q2 ) = P1 (Q1 )Q1 + P2 (Q2 )Q2 − C(Q1 + Q2 );
first order conditions
and
∂π
=0
∂Q1
∂π
=0
∂Q2
give
M R1 (Q1 ) = M C(Q1 + Q2 ),
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2 Peak-Load Pricing
2
and
M R2 (Q2 ) = M C(Q1 + Q2 );
finally,
M R1 (Q1 ) = M R2 (Q2 ) = M C(Q1 + Q2 ).
Because
M R1 = P1 (1 −
1
),
|E1 |
M R2 = P2 (1 −
1
),
|E2 |
and
we have
P1
1 − 1/|E1 |
=
;
P2
1 − 1/|E2 |
since
|E1 | < |E2 |,
P1 > P2 .
10
10
10
9
9
9
8
8
7
7
7
6
6
6
5
5
4
4
3
3
2
2
1
0
MR1
0
1
D1
2
8
P2
3
4
5
Quantity
6
7
8
9
10
(a) Group 1.
MC(QT)
5
4
3
2
D2
Q2 MR2
1
0
Price
Price
Price
Sometimes a small group might not be served (see Figure 2). The producer only
0
1
2
3
4
1
5
Quantity
6
7
(b) Group 2.
8
9
10
0
MR(QT)
QT
0
1
2
3
4
5
Quantity
6
7
8
9
10
(c) Total Market.
Figure 2: Third Degree Price Discrimination with a Small Group.
serves the second group, because the willingness to pay of the first group is too
low.
2
Peak-Load Pricing
Producers charge higher prices during peak periods when capacity constraints
cause higher M C.
Example (Movie Ticket). Movie ticket is more expensive in the evenings.
Example (Electricity). Price is higher during summer afternoons.
For each time period,
MC = MR
(see Figure 3).
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3 Two-Part Tariff
3
10
10
9
9
8
MC
7
7
6
6
Price
Price
8
5
4
3
5
PM
4
3
PL
MR1
1
0
1
D1
QL
2
3
4
D2
MR2
2
2
0
MC
1
5
Quantity
6
7
8
9
0
10
QM
0
1
2
(a) Period 1.
3
4
5
Quantity
6
7
8
9
10
(b) Period 2.
Figure 3: Peak-Load Pricing.
3
Two-Part Tariff
The consumers are charged both an entry (T ) and usage (P ) fee, that is to say,
a fee is charged upfront for right to use/buy the product, and an additional fee
is charged for each unit that the consumer wishes to consume. Assume that the
firm knows consumer’s demand and sets same price for each unit purchased.
Example (Telephone Service, Amusement Park.).
When there is only one consumer. If the firm sets usage fee
P = M C,
consumer consumes Q∗ units (see Figure 4), and the firm can set entry fee
T = A,
and extract all the consumer surplus.
• If setting
P1 > M C,
total revenue is
R1 = A1 + P1 × Q1 ,
and cost is
C1 = M C × Q1 ,
then the profit is
π1 = A − B1 .
• If setting
P2 < M C,
total revenue is
R2 = A2 + P2 × Q2 ,
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3 Two-Part Tariff
4
10
9
8
7
CS
Price
6
A
5
MC
4
3
D
2
1
Q*
0
0
1
2
3
4
5
Quantity
6
7
8
9
10
Figure 4: Entry Fee of One Consumer.
10
10
9
9
8
8
CS
7
4
B2
5
3
D
2
1
1
2
3
P2
2
D
1
Q1
0
MC
A2
4
P1
3
0
CS
6
MC
B1
5
Price
6
Price
7
A1
Q*
4
5
Quantity
Q2
Q*
6
7
8
9
10
(a) Price Higher than Marginal Cost.
0
0
1
2
3
4
5
Quantity
6
7
8
9
10
(b) Price Lower than Marginal Cost.
Figure 5: Two-Part Tariff.
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3 Two-Part Tariff
5
and cost is
C2 = M C × Q2 ,
then the profit is
π2 = A − B2 .
Either B1 or B2 is positive, so the best unit price that maximized the producer
surplus is exactly M C.
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1 Two-Part Tariff
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 19, 2007
Lecture 26
Pricing and Monopolistic Competition
Outline
1. Chap 11: Two-Part Tariff
2. Chap 11: Bundling
3. Chap 12: Monopolistic Competition
1
Two-Part Tariff
When there are two consumers. Consumer 1 has higher demand than consumer
2. If setting P = M C, consumer 1 consumes Q1 units and consumer 2 consumer
Q2 units. A1 is consumer 1’s consumer surplus, and A2 is consumer 2’s consumer
surplus. Assume that 2A2 > A1 . Then the maximum entry fee the firm can
charge is A2 . If more than A2 is charged, consumer 2 would not consume.
10
9
8
A1
7
Price
6
A2
5
MC
4
3
D1
2
Q2
1
0
0
1
2
3
D2
Q1
4
5
Quantity
6
7
8
9
10
Figure 1: Entry Fee of Two Consumers.
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1 Two-Part Tariff
2
Now consider the case that price is higher or lower than the marginal cost.
• If setting
P > M C, T = A′2 ,
we have
π1 = A′2 + Q′1 × (P − M C) = A2 + C,
and
π2 = A′2 + Q′2 × (P − M C) = A2 − B,
thus
π = π1 + π2 = 2A2 + C − B.
Because
C>B
(see Figure 2),
π > 2A2 .
• If setting
P < M C, T = A′′2
we have
π1 = A′′2 − Q′′1 × (M C − P ) = A2 − D,
and
π2 = A′′2 − Q′′2 × (M C − P ) = A2 − E,
thus
π = π1 + π2 = 2A2 − D − E.
Always
π < 2A2 .
Summary: the firm should set
• usage fee
P > M C,
namely, larger than the marginal cost;
• entry fee
T = A2 ,
namely, equal to the remaining consumer surplus of the consumer with
the smaller demand.
Summary: If the demands of two consumers are more similar, the firm should
set usage fee close to M C and higher entry fee; if the demands of two consumers
are less similar, the firm should set higher usage fee and lower entry fee.
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1 Two-Part Tariff
3
10
9
8
7
Price
6
A’1
5
A’2
4
3
P
C
B
2
MC
1
0
Q’2
0
1
2
Q’1
3
4
5
Quantity
6
7
8
9
10
Figure 2: Two-Part Tariff: Price Higher than Marginal Cost
10
9
8
7
Price
6
5
4
A’’1
A’’2
3
MC
2
D
1
0
P
0
Q’’2
1
2
3
4
5
Quantity
F
E
Q’’1
6
7
8
9
10
Figure 3: Two-Part Tariff: Price Lower than Marginal Cost
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2 Bundling
2
4
Bundling
Bundling means packaging two or more products, for example, vacation travel
usually has a packaging of hotel, airfare, car rental, etc.
Assume there are two goods and many consumers in the market, and the con­
sumers have different reservation prices (willingness to pay).
See Figure 4 and 5. The coordinates are the reservation prices of the two goods
respectively.
If the firm sells the goods separately with prices P1 and P2 (see Figure 4),
• when
r1 > P1 ,
and
r2 > P2 ,
the consumer will buy both good 1 and 2;
• when
r1 > P1 ,
but
r2 < P2 ,
the consumer will only buy good 1;
• when
r2 > P2 ,
but
r1 < P1 ,
the consumer will only buy good 2;
• when
r1 < P < 1,
and
r2 < P < 2,
the consumer will buy neither good 1 nor 2.
If the firm sells the two goods in a bundle and charges price PB ,
• if
r1 + r2 > PB ,
the consumer will buy the bundle;
• if
r1 + r2 < PB ,
the consumer will not buy the bundle.
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2 Bundling
5
10
r2
9
8
7
6
5
4
3
2
r1
1
0
0
1
2
3
4
5
6
7
8
9
10
9
10
Figure 4: Price without Packaging.
10
r2
9
8
7
6
5
4
3
2
r1
1
0
0
1
2
3
4
5
6
7
8
Figure 5: Price with Packaging.
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2 Bundling
6
10
r2
9
8
7
6
(5,5)
.
.(4,4)
5
4
3
2
.(1,1)
1
0
0
1
.(2,2)
r1
2
3
4
5
6
7
8
9
10
Figure 6: Bundling Example 1.
Bundling Example 1: the four points in Figure 6 represent the four con­
sumers’ reservation values. Consider two pricing strategies – one is that the two
goods are sold separately with prices P1 = 3 and P2 = 3, and the other is that
the two goods are sold in a bundle with price PB = 6. Without bundling, the
revenue is
R = 12,
and with bundling, the revenue is
R = 12;
bundling does not do better.
Bundling Example 2: Consider the other four consumers shown in Figure 7
and the firm chooses between the two pricing strategies mentioned before. With­
out bundling, the revenue is
R = 12,
and with bundling, the revenue is
R = 24;
obviously, bundling strategy benefits the producer in this case
Conclusion: bundling works well when
• the consumers are heterogeneous;
• price discrimination is not possible;
• the demand for different goods are negatively correlated.
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3 Monopolistic Competition
7
10
r2
9
8
7
6
.
(1,5)
5
.
(2,4)
4
3
.(4,2)
2
.(5,1)
1
0
0
1
2
3
4
5
r1
6
7
8
9
10
Figure 7: Bundling Example 2.
3
Monopolistic Competition
In monopolistic competition,
• there are many firms;
• there is free entry and exit;
• products are differentiated but close substitutes.
Thus
• each firm faces a distinct demand, which is downward sloping and elastic;
• there is no profit in long run (see Figure 8 and 9);
• price is higher than marginal cost because firms have some monopoly
power, and thus there is some deadweight loss.
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3 Monopolistic Competition
8
10
9
8
7
MC
6
P
P
S
5
PROFIT
AC
4
3
2
1
0
0
1
2
DS=AR
MR
QS
S
3
4
5
Q
6
7
8
9
10
Figure 8: Short Run in Monopolistic Competition Market.
10
9
8
7
MC
P
6
5
AC
4
PL
3
2
1
0
DL=AR
QL MRL
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 9: Long Run in Monopolistic Competition Market.
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1 Game Theory
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 21, 2007
Lecture 27
Game Theory and Oligopoly
Outline
1. Chap 12, 13: Game Theory
2. Chap 12, 13: Oligopoly
1
Game Theory
In monopolistic competition market, there are many sellers, and the sellers do
not consider their opponents’ strategies; nonetheless, in oligopoly market, there
are a few sellers, and the sellers must consider their opponents’ strategies. The
tool to analyze the strategies is game theory.
Game theory includes the discussion of noncooperative game and coopera­
tive game. The former refers to a game in which negotiation and enforcement of
binding contracts between players is not possible; the latter refers to a game in
which players negotiate binding contracts that allow them to plan joint strate­
gies.
A game consists of players, strategies, and payoffs.
Now assume that in a game, there are two players, firm A and firm B; their
strategies are whether to advertise or not; consequently, their payoffs can be
written as
πA (A� s strategy, B � s strategy)
and
πB (A� s strategy, B � s strategy)
respectively.
Now let’s represent the game with a matrix (see Table 1). The first row is the
situation that A advertises, and the second row is the situation that A does not
advertise; the first column is the situation that B advertises, and the second
column is the situation that B does not advertise. The cells provide the payoffs
under each situation. The first number in a cell is firm A’s payoff, and the
second number is firm B’s payoff.
Dominant strategy is the optimal strategy no matter what the opponent
does. If we change the element (20, 2) to (10, 2), no matter what the other firm
does, advertising is always better for firm A (and firm B). Therefore, both firms
have a dominant strategy.
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2 Oligopoly
Firm A
2
Advertise
Not Advertise
Firm B
Advertise Not Advertise
10,5
15,0
6,8
20,2
Table 1: Payoffs of Firm A and B.
When all players play dominant strategies, we call it equilibrium in dominant
strategy.
Now back to original case, B has dominant strategy, but A does not, because
• if B advertises, A had better advertise;
• if B does not advertise, A had better not advertise.
So we see that not all games have dominant strategy. However, since B has
dominant strategy and would always advertise, A would choose to advertise in
this case.
Now consider another example. Two firms, firm 1 and firm 2, can produce
crispy or sweet. If they both produce crispy or sweet, the payoffs are (−5, −5);
if one of them produces crispy while the other produces sweet, the payoffs are
(10, 10).
Firm 1
Crispy
Sweet
Firm 2
Crispy Sweet
-5,-5
10,10
10,10
-5,-5
Table 2: Payoffs of Firm 1 and 2.
There is no dominant strategy for both firms. We define another equilibrium
concept – Nash equilibrium.
Nash equilibrium is a set of strategies such that each player is doing the best
given the actions of its opponents.
In this case, there are two Nash equilibriums, (sweet, crispy) and (crispy, sweet).
2
Oligopoly
Small number of firms, and production differentiation may exist.
Different Oligopoly Models
1. Cournot Model: firms produce the same good, and they choose the pro­
duction quantity simultaneously.
2. Stackelberg Model: firms produce the same
3. Bertrand Model: firms produce the same good, and they choose the price.
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2.1
2.1
Cournot Model
3
Cournot Model
Example. Market has demand
P = 30 − Q,
with two firms, so
Q = Q1 + Q2 ,
and assume that there is no fixed cost and marginal cost,
M C1 = M C2 = 0.
Firm 1 would like to maximize its profit
P × Q1 ,
or
(30 − Q1 − Q2 ) × Q1 ;
from the
d
((30 − Q1 − Q2 ) × Q1 ) = 0,
dQ1
we have firm 1’s reaction function
Q1 = 15 −
Q2
,
2
in which the Q2 is the estimation of firm 2’s production by firm 1.
In the same way, firm 2’s reaction function is
Q2 = 15 −
Q1
,
2
in which the Q1 is the expectation of firm 1’s production by firm 2.
At equilibrium, firm 1 and firm 2 have correct expectation about the other’s
production, that is,
Q1 = Q1 ,
Q2 = Q2 .
Thus, at equilibrium,
Q1 = 10,
and
Q2 = 10.
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1 Stackelberg
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 26, 2007
Lecture 28
Oligopoly
Outline
1. Chap 12, 13: Stackelberg
2. Chap 12, 13: Bertrand
3. Chap 12, 13: Prisoner’s Dilemma
In the discussion that follows, all of the games are played only once.
and
1
Stackelberg
Stackelberg model is an oligopoly model in which firms choose quantities se­
quentially.
Now change the example discussed in last lecture as follows: if firm 1 pro­
duces crispy and firm 2 produces sweet, the payoff is (10, 20); if firm 1 produces
sweet and firm 2 produces crispy, the payoff is (20, 10) (see Table 1).
Firm 1
Crispy
Sweet
Firm 2
Crispy Sweet
-5,-5
10,20
20,10
-5,-5
Table 1: Payoffs of Firm 1 and 2.
�
�
−5, −5 10, 20
20, 10 −5, −5
This is an extensive form game; we use a tree structure to describe it.
Firm 1
Crispy
Sweet
Firm 2
Firm 2
Crispy
Sweet
Crispy
Sweet
(-5,-5)
(10,20)
(20,10)
(-5,-5)
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2 Bertrand
2
Start from the bottom using backward induction, namely, solve firm 2’s
decision problem first, and then firm 1’s. If firm 1 chooses crispy, firm 2 will
choose sweet to get a higher payoff. If firm 2 chooses sweet, firm 2 will choose
crispy. Knowing this, firm 1 will choose sweet in the first place. In this case,
going first gives firm 1 the advantage. Now consider the case we discussed for
the Cournot model, but firm 1 chooses Q1 first, and firm 2 choose Q2 later. For
firm 2, the first order condition
d
(30 − Q1 − Q2 ) × Q2 = 0
dQ2
gives that
Q2 (Q1 ) = 15 −
Q1
.
2
For firm 1,
d
(30 − Q1 − Q2 (Q1 ) × Q1 = 0
dQ1
gives that
Q1 = 15.
Thus, the result will be
Q1 = 15,
π1 = 112.5;
Q2 = 7.5,
π2 = 56.25.
In this case, firm 1 also has advantage to go first.
2
Bertrand
The Bertrand model is the oligopoly model in which firms compete in price.
First assume that two firms produce homogeneous goods and choose the prices
simultaneously. Assume two firms have the same marginal cost
M C1 = M C2 = 3;
consumers buy goods from the firm with lower price. If
P1 = P2 = 4,
the two firms share the market equally, but this is not the equilibrium. The
reason is that one firm can get whole demand by lowering the price a little;
therefore, the equilibrium will be
P1 = P2 = 3,
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2 Bertrand
3
when the price is equal to the marginal cost. Now we check if
P1 = 3
is the best choice for firm 1 given
P2 = 3.
When
P1 = 3,
π1 = 0;
if
P1 > 3,
consumers will not buy firm 1’s goods, thus
π1 = 0;
if
P1 < 3,
the price is lower than the marginal cost, thus
π1 < 0.
It follows that
P1 = 3
is optimal for firm 1; by analogy, we can get the same conclusion for firm 2.
Therefore,
P1 = P2 = 3 = M C
in a Bertrand game with homogeneous goods. This is like the competitive
market.
Suppose the goods from the two firms are heterogeneous, but substitutes.
Firm 1 and firm 2 face the following demands:
Q1 = 12 − 2P1 + P2 ,
and
Q2 = 12 − 2P2 + P1 .
Firm 1’s and firm 2’s reaction functions are
P1 = 3 +
P2
,
4
P2 = 3 +
P1
.
4
and
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3 Prisoner’s Dilemma
4
At equilibrium,
P1 = P 1 ,
and
P2 = P 2 ;
so
P1 = P2 = 4,
Q1 = Q2 = 8,
and
π1 = π2 = 32.
Consider the case when the firms choose prices sequentially. Supposing firm 2’s
first order condition
d
(12 − P2 + P1 ) × P2 = 0
dQ2
and firm 1’s first order condition
d
(12 − 2P1 + P2 (P1 )) × P1 = 0.
dQ1
From the first equation
P1
,
4
and then substitute it into the second equation, we obtain
P2 (P1 ) = 3 +
2
P1 = 4 .
7
Therefore,
1
π1 = 32 ;
4
1
P2 = 4 ,
14
and
15
.
98
In this case, we can see that the firm who goes first has disadvantage, when
competing in price.
π2 = 33
3
Prisoner’s Dilemma
Criminals A and B cooperated, and then got caught. However, the police have
no evidence; so they have to interrogate A and B separately, trying to make
them tell the truth.
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3 Prisoner’s Dilemma
Firm A
5
Betray
Silent
Firm B
Betray Silent
-3,-3
0,6
-6,0
-1,-1
Table 2: Payoffs of Firm A and B.
The above matrix shows A and B’s payoffs. Given the payoffs, A and B
choose to tell the truth (betray) or keep silent. We can see that, if they both
keep silence, the result (−1, −1) is best for them; nonetheless, if one of them
betrays another, he will be free but his companion will have payoff -6; moreover,
if both of them betray, they will face the result (−3, −3).
Consider what A thinks. Whether B keeps silence or betrays him, A will
always be better off if he betrays; so will B. Therefore, the result of this problem
is (−3, −3), namely, both prisoners betray.
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1 Collusion – Prisoners’ Dilemma
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 28, 2007
Lecture 29
Strategic Games
Outline
1. Chap 12, 13: Collusion – Prisoners’ Dilemma
2. Chap 12, 13: Repeated Games
3. Chap 12, 13: Threat, Credibility, Commitment
4. Chap 14: Maximin Strategy
1
Collusion – Prisoners’ Dilemma
Last time we talked about the prisoners’ dilemma. The conclusion is that they
will betray the other.
Now apply it to the cases of Cournot and Bertrand models.
In the Cournot model, the demand is
P = 30 − Q1 − Q2 .
The equilibrium will be
Q1 = Q2 = 10,
with
π1 = π2 = 100.
However, to maximize their total profits, they should choose a total quantity
Q so that
d
(Q(30 − Q)) = 0,
dQ
which follows that
Q = 15.
If they share profit equally,
Q1 = Q2 = 7.5,
and
π1 = π2 = 112.5.
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1 Collusion – Prisoners’ Dilemma
2
10
9
8
Q2(Q1)
7
6
5
4
3
Q1(Q2)
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Figure 1: Reaction Curves in Cournot Model.
Obviously, the latter case will make both of them better off. But given the
opponent produces 7.5, each of them can increase the profit by producing more
(see Figure 1).
In Bertrand model, demand functions for firm 1 and firm 2 are
Q1 = 12 − 2P1 + P2 ,
and
Q2 = 12 − 2P2 + P1 .
Equilibrium is
P1 = P2 = 4,
with
π1 = π2 = 32.
However, firms can choose P1 and P2 together to maximize the total revenue
π = P1 (12 − 2P1 + P2 ) + P2 (12 − 2P2 + P1 ).
By first order condition, we obtain
12 − 4P1 + 2P2 = 0,
and
12 − 4P2 + 2P1 = 0.
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2 Repeated Games
3
Thus
P1 = P2 = 6,
with
π1 = π2 = 36.
But in this case, each firm has incentive to lower its price given the other
firm’s price (see Figure 2).
10
9
8
P2(P1)
7
6
5
P1(P2)
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Figure 2: Reaction Curves in Bertrand Model.
2
Repeated Games
Back to the prisoners’ problem. If suspect A and B will cooperate for infinite
periods, and they are both patient, they care about future payoffs. Because if
one of them betrays this time, the opponent will lose the trust and betray in
the future; the payoff changes from −1 to −3 for each time. Therefore, both A
and B would like to keep silence. But if they are impatient, and only consider
today’s payoff, they will still betray. Now move on to the case that A and B
will cooperate for finite number times which is fairly large. We deduce from the
last time they cooperate; the answer is that they will betray for the last time,
so will they for other opportunities. Therefore, the collusion between A and B
succeed only if they will be cooperative forever and are patient.
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3 Threat, Credibility, Commitment
3
4
Threat, Credibility, Commitment
Back to the crispy-sweet question.
Firm 1
Crispy
Sweet
Firm 2
Crispy Sweet
-5,-5
10,20
20,10
-5,-5
Table 1: Payoffs of Firm 1 and 2.
Firm 1
Crispy
Sweet
Firm 2
Crispy
(-5,-5)
Firm 2
Sweet
Crispy
Sweet
(10,20)
(20,10)
(-5,-5)
In order to get the largest 20 by producing sweet, firm 2 tries to make firm
1 believe that firm 1 should choose crispy by claiming that it always produces
sweet no matter what firm 1 produces. However, firm 1 can ignore firm 2’s
announcement because once firm 1 choose sweet, firm 2 will produce crispy.
Suppose that firm 2 will advertise and so change the payoffs.
Firm 1
Crispy
Sweet
Firm 2
Crispy Sweet
-5,-5
10,35
20,10
-5,10
Table 2: Payoffs of Firm 1 and 2.
Firm 1
Crispy
Sweet
Firm 2
Firm 2
Crispy
Sweet
Crispy
Sweet
(-5,-5)
(10,35)
(20,10)
(-5,10)
In this case, firm 2 feels indifferent between choosing crispy or sweet when
firm 1 produces sweet, and chooses sweet when firm 1 produces crispy. So it is
credible if firm 2 claims that it always chooses sweet, and then firm 1 had better
choose crispy. This example tells us that firm 2 had to do something to make
the announcement credible.
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4 Maximin Strategy
4
5
Maximin Strategy
See Table 3. Firm B has dominant strategy: advertise.
Therefore, the equilibrium should be both A and B advertise.
However, if firm B does not choose the rational option, the minimum payoff
of A is 5 if A advertises, and 8 if A does not advertise.
The maximin strategy is the strategy that renders the highest minimum
payoff.
When A cannot tell whether B is rational or not, A might use maximin
strategy. In this case, the maximin strategy of A is:
Firm A
Advertise
Not Advertise
Firm B
Advertise Not Advertise
10,5
5,0
8,8
15,2
Table 3: Payoffs of Firm A and B.
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1 Dominant Firm Model
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
November 30, 2007
Lecture 30
Dominant Firm Model and Factor Market
Outline
1. Chap 12, 13: Dominant Firm Model
2. Chap 14: Factor Market
1
Dominant Firm Model
The dominant firm model is the model that in some oligopolistic markets, one
large firm has a major share of total sales, and a group of smaller firms supplies
the remainder of the market. The large firm has power to set a price that
maximizes its own profits. A dominant firm exists because it has lower marginal
cost than the other fringe firms.
Assume the fringe firms’ total supply is SF , the market demand is DM , then
the dominant firm’s demand is (see Figure 1)
D D = D M − SF .
Knowing DD , we can derive M RD . The dominant firm produces at a quantity
QD that satisfies
M RD = M CD .
Correspondingly, the price is P ∗ . The fringe firm’s supply curve thus shows QF .
Furthermore, the total quantity is
QT = QF + QD .
Example (OPEC). OPEC is an example of a successful cartel, which can be
regarded as a dominant firm.
Cartels are more likely to succeed if
• demand is inelastic, and
• supply of non-Cartel producers is inelastic.
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2 Factor Market
2
10
9
P*
8
MCD
DM
7
DD
P
6
SF
5
4
3
MRD
2
1
0
QF
0
1
2
QD
3
QT
4
5
Q
6
7
8
9
10
Figure 1: Dominant Firm Model.
2
Factor Market
The last chapters were about product market, or output market, in which
• individuals are buyers, and
• firms are producers;
we start to discuss factor markets, or input markets, in which
• individuals are producers, and
• firms are buyers.
Firms need labor and capital to produce.
Outline
1. Demand of Labor
2. Supply of Labor
2.1
Demand of Labor
Demands of labor are different in short run and long run markets, and condi­
tional and unconditional market (see Table 1). Firms use labor and capital as
input.
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2.1
Demand of Labor
Conditional
Unconditional
3
Short Run
Output price fixed
Other factors fixed
Output Price varies
Other input factors fixed
Long Run
Output price fixed
Other input factors vary
Output Price varies
Other Inputs vary
Table 1: Demand of Labor.
Short Run Demand of Labor. Only labor is variable.
The prices for L and K are w and r respectively.
Define marginal revenue product of labor M RPL to be additional revenue
from an additional unit of labor.
M PL is the additional output obtained from an additional unit of labor;
M R is the additional revenue from an additional unit of output. Therefore,
M RPL =
dR
dR ∂Q
=
= M R × M PL .
dL
dQ ∂L
Firm chooses Q such that
w = M RPL (L),
so the marginal revenue and marginal cost at hiring one more unit of labor
are the same.
• If output market is competitive,
MR = P;
if it is not competitive,
MR < P
(see Figure 2 and 3).
• Given w, we derive the firm’s demand for labor from
w = M RPL (L).
M RPL decreases in L; therefore, M RPL is the firm’s short run de­
mand curve.
Long Run Demand of Labor. Both K and L are variable.
w decreases then M C decreases, Q increases, and L increases. With higher
L, M PK increases, so the firm uses more K, and then M PL increases
further, and the firm hires more labor. Thus, the demand of labor is more
elastic than that in short run (see Figure 5).
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2.1
Demand of Labor
4
10
9
8
7
P
6
5
MPL× P
4
MPL× MR
3
Competitive
Not Competitive
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 2: Marginal Revenue Product of Labor.
10
9
8
7
W
6
5
DL=MPL× MR
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 3: Marginal Revenue Product of Labor in Competitive Market.
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2.1
Demand of Labor
5
10
9
8
7
W
6
DL
5
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 4: Marginal Revenue Product of Labor Increases in Price.
10
9
MRP,L
8
7
W
6
MRPL
5
DL
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 5: Marginal Revenue Product of Labor in Long Run.
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2.1
Demand of Labor
6
10
9
8
7
W
6
MRP,L
5
DL
MRPL
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 6: Unconditional on Output Market Price.
Unconditional on Output Market Price. The discussion before was based
on the assumption that the output price is fixed. Now consider the case
when the output price is unconditional so that it is not fixed.
If w decreases, L increases and Q increases, and so P decreases; with
M RPL decreases, Q and L decrease.
The demand is less elastic than when output P is fixed (see Figure 6).
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1 Supply of Labor
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
December 3, 2007
Lecture 31
Factor Market
Outline
1. Chap 14: Supply of Labor
2. Chap 14: Demand of Labor
1
Supply of Labor
We derive the supply of labor by solving consumers’ utility maximization prob­
lems.
Two variables determining the utility are leisure (L), which is measured by
hours, and income (Y ); the prices are w and 1 respectively.
To maximize u(L, Y ), we have
∂u
∂L
∂u
∂Y
= w.
If w increases, on one hand, higher wages encourage people to work more (point
A to point B), which is a substitution effect; on the other hand, higher wages
allow the worker to purchase more goods, including leisure, which reduces work
hours (point B to point C), which is an income effect (see Figure 1).
When the wage is higher, if the substitution effect exceeds the income effect,
labor supply increases, and leisure decreases; if the income effect exceeds the
substitution effect, labor supply decreases, and leisure increases (see Figure 2).
Like product markets, competitive, monopolistic, and monopsonistic mar­
kets are types of factor markets.
In a competitive factor market, if the product market is also competitive,
M RPL = P × M PL .
If the product market is monopolistic,
M RPL = M R × M PL = P (1 −
1
) × M PL .
|ed |
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1 Supply of Labor
2
10
9
8
7
C
Y
6
B
5
4
A
Income Effect
3
2
1
Substitution Effect
0
2
3
4
5
6
7
L
8
9
10
11
12
Figure 1: Substitution Effect and Income Effect of Labor Supply.
10
9
8
Supply of Labor
7
Income Effect
> Substitution Effect
Wage
6
5
Income Effect
< Substitution Effect
4
3
2
1
0
3
3.5
4
4.5
5
5.5
6
Hours of Work per Day
6.5
7
7.5
8
Figure 2: Backward-Bending Supply of Labor.
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1.1
Factor Competitive
3
10
9
8
SL
7
w
6
w*
5
4
3
DL=P× MPL
2
1
0
L*
0
1
2
3
4
5
L
6
7
8
9
10
Figure 3: Competitive Factor Market.
1.1
Factor Competitive
Competitive market is most efficient, and there is no deadweight loss (see Fig­
ure 3).
When M R < P , both w and L decrease; the market is then not as efficient
as competitive market, and has deadweight loss (see Figure 4).
1.2
Factor Monopsony
Marginal Value equals the demand. Marginal Expenditure
ME =
∂PS (Q)Q
∂PS
=
Q + PS > PS .
∂Q
∂Q
Because L is determined by
M E = M V,
we can see that
′
w < w∗ ,
and
′
L < L∗
(see Figure 5).
One example of factor monopsonist is the government hiring soldiers.
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1.2
Factor Monopsony
4
10
9
8
SL
7
w
6
w*
5
P× MPL
w,
4
3
2
1
0
L, L*
0
1
2
3
4
DL=MR× MPL
5
L
6
7
8
9
10
9
10
Figure 4: Noncompetitive Factor Market.
10
9
ME
8
S=AE
7
w
6
w*
5
w,
4
3
D=MV
2
1
0
L,
0
1
2
3
L*
4
5
L
6
7
8
Figure 5: Monopsonistic Factor Market.
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1.3
1.3
Factor Monopoly
5
Factor Monopoly
An example of monopoly power in factor markets involves labor unions.
Economic rent is the difference between payments to a factor of production
and the minimum payment that must be spent to obtain the factor; it is like
producer surplus in a product market (see Figure 6).
10
9
8
SL
7
6
w
w
5
Economic Rent
4
3
2
1
0
0
0.5
1
1.5
2
L
2.5
3
3.5
4
Figure 6: Economic Rent.
When some workers lose their jobs, remaining workers have higher wages.
If the union tries to maximize the number of workers hired, it should set the
wage and labor employed w∗ and L∗ ; if the union tries to maximize economic
rent, it should set the wage and labor employed w1 and L1 .
w1 > w∗ ,
and
L1 < L∗
(see Figure 7).
It is hard to say which one is better for the workers.
Now consider a model of union workers and non-union workers. Assume
the demand for union workers is DU , and the demand for non-union workers is
DN U . The total market demand
DL = DU + DN U
is fixed.
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1.3
Factor Monopoly
6
10
9
8
SL
w,
7
w
6
w*Economic
5
Rent
4
3
DL
MR
2
1
0
L,
0
1
2
3
L*
4
5
L
6
7
8
9
10
Figure 7: Monopoly Power of Sellers of Labor.
When a monopolistic union raises the wage rate in the unionized sector of
the economy from w∗ to wU , employment in that sector falls; for the total supply
of labor to remain unchanged, the number of non-union workers increases and
the wage in the non-unionized sector must fall from w∗ to wN U (see Figure 8).
Assume the total supply of workers is 60; the demands for nonunion and
union workers are
1
wN U = 30 − LN U ,
2
wU = 30 − LU .
• When the union does not intervene,
wN U = wU = w.
Thus
LN U = 60 − 2w,
and
LU = 30 − w.
Then
L = 90 − 3w = 60,
which gives
w = 10,
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1.3
Factor Monopoly
7
10
9
wU
8
SL
7
DU
w
6
DNU
w*
5
4
3
wNU
2
DL
1
0
0
1
2
3
4
5
6
Number of Workers
7
8
9
10
Figure 8: Wage Discrimination in Labor Market.
and therefore
LU = 20,
LN U = 40.
• When the union maximizes the total wage of union workers as a monop­
olist, the first order condition is
d
d
(wU × LU ) =
(30 − LU ) × LU = 0.
dLU
dLU
Then
30 − 2LU = 0,
LU = 15;
thus
wU = 15.
For the nonunion workers,
LN U = 45,
wN U = 7.5.
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2 Demand of Supply
2
8
Demand of Supply
In competitive factor market, assume
Q = 10L − L2 ,
and
P = 1.
M RPL = M PL × M R = 10 − 2L.
w is marginal cost of hiring labor, thus
w = 10 − 2L,
then
LD =
10 − w
.
2
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1 Present Discount Value
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
December 5, 2007
Lecture 32
Investment, Savings, Time and Capital Markets
Outline
1. Chap 15: Present Discount Value
2. Chap 15: Bond
3. Chap 15: Effective Yield
4. Chap 15: Determine Interest Rate
1
Present Discount Value
Present discount value (PDV) determines the value today of a future flow of
income.
Payment A
Payment B
Today
100
20
1 year
100
100
2 year
0
100
Table 1: Two Payments.
Consider the two payments, A and B, in Table 1.
Because the present value of 1 dollar in n years is
1
,
(1 + r)n
where r is the interest rate, the present values of A and B are
100 +
and
20 +
100
,
1+r
100
100
+
,
1 + r (1 + r)2
respectively.
• If r is low, P V of B is larger than P V of A.
• If r is high, P V of A is larger than P V of B.
Several examples are provided in Table 2.
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2 Bond
2
Value of r
P V of A
P V of B
0.05
195.24
205.94
0.10
190.90
193.54
0.15
186.96
182.57
Table 2: Present Values.
2
Bond
A bond is a contract in which a borrower (issuer) agrees to pay the bondholder
(the lender) a stream of money.
For instance, a payment consists of a coupon payment of 100 dollars per year
for 10 years, and a principal payment of 1000 dollars in 10 years.
P V of the bond is
PV =
100
100
100
1000
+
+ ... +
+
.
1 + r (1 + r)2
(1 + r)10
(1 + r)10
With a higher interest rate, the present discount value is lower (see Figure 1).
10
9
8
7
PDV
6
5
4
3
2
1
0
2
3
4
5
6
7
r
8
9
10
11
12
Figure 1: Present Discount Value and Interest Rate.
Perpetuity is a bond that pays a fixed amount of money each year forward:
PV =
100
100
100
+
+ ... =
.
2
1 + r (1 + r)
r
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3 Effective Yield
3
3
Effective Yield
Effective yield is the interest rate that equates the present value of a bond’s
payment stream with the bond’s market price.
Riskier bonds have higher yields. An effective yield equals risk-free interest
rate plus risk premium.
When we choose between projects, we can compare the present value, or
compare the yield rate, and choose the higher one.
Time (Year)
Project A (Dollar)
Project B (Dollar)
0
-50
-20
1
5
4
2
55
24
Table 3: Two Projects.
Assume
r = 15%.
P V of A is -4, and P V of B is 11. P V of B is higher; thus firm should invest
in B.
Now calculate the yield rates.
For project A,
5
55
50 =
+
,
1 + rA
(1 + rA )2
rA = 10%.
For project B,
20 =
4
24
+
,
1 + rB
(1 + rB )2
rB = 20%.
Here yield rate of B is higher. Firm should invest in B again.
In this case, the results of both criteria are consistent; however, they are not
always consistent.
4
Determine Interest Rate
The interest rate is the price that borrowers pay lenders to use their funds. It
is determined by supply and demand for loanable funds. Demand for loanable
funds comes from firms and governments that want to make capital investments.
Supply of loanable funds comes from household savings (see Figure 2).
Suppose a consumer only lives for two periods, intertemporal utility function
u(C1 , C2 ) will be maximized, under the budget constraint
P V = Y1 +
Y2
C2
= C1 +
,
1+r
1+r
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4 Determine Interest Rate
4
10
9
8
Savings
7
Interest Rate
6
5
4
3
Investment
2
1
0
0
1
2
3
4
5
6
Quantity of Loanable Funds
7
8
9
10
Figure 2: Supply and Demand of Funds.
in which C1 and C2 stand for consumptions in period 1 and 2, and Y1 and Y2
are incomes in period 1 and 2 respectively.
When the consumer’s utility is maximized
∂u
∂C1
∂u
∂C2
= 1 + r.
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1 Adverse Selection
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
December 7, 2007
Lecture 33
Asymmetric Information
Outline
1. Chap 17: Adverse Selection
2. Chap 17: Moral Hazard
1
1.1
Adverse Selection
Used Car Market
Buyers do not know the quality of each car but know quality distribution.
Assume there are three cars, and their prices are 0, 5, and 10, respectively.
The consumer’s willingness to pay is 5, so the seller of 10 will leave the
market.
As a result, the consumer’s willingness to pay decreases to 2.5; thus the seller
of 5 will leave the market.
Finally, the willingness to pay decreases to 0; market fails, and only car stays
is the worst one. This is called the Lemon Problem.
1.2
Insurance Market
Insurance companies do not know how healthy each person is.
For instance, the probabilities of getting sick of A and B are shown in Table 1.
When one is sick, the insurance company gives him 10 dollars to cover medical
expense.
A
B
Sick
0.1
0.5
Healthy
0.9
0.5
Table 1: Probability of Health.
Thus the expected expense for A is 1, and that for B is 5.
Since the company cannot tell who is healthy, it sets a premium of 3.
Those healthy people who are risk-averse enough would accept the $3 pre­
mium; those who are not risk-averse enough would reject the $3 premium. If
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2 Moral Hazard
2
only unhealthy people accept the insurance contract, the insurance company
has to adjust the premium to $5.
Solve this problem by requiring people to do a physical examination before
buying insurance – the examination works as a certificate, like credit history for
banks.
2
Moral Hazard
Moral hazard occurs when the insured party whose actions are unobserved by
the insurer can affect the probability or magnitude of a payment associated with
an event. For example, it often occurs in insurance: if my home is insured, I
might be less likely to lock my doors or install a security system.
Assume jewelry is worth $10. The probability to be stolen is 0.5. If the
owner spend $2 to hire a guard, the probability decreases to be 0.1. Because
10 × 0.9 + 0 × 0.1 − 2 = 7,
10 × 0.5 + 0 × 0.5 = 5,
one will hire a guard.
If the owner asks for insurance, and the insurance will pay $10 if the jewelry
is stolen. If the owner hires a guard, the actuarially fair insurance premium is
p = 10 × 0.1 = 1.
However, the owner buys the insurance, he will not hire a guard.
If the insurance company only cover $4.9 when stolen; and the insurance
premium is P :
Hiring a guard, the owner’s payoff is
10 × 0.9 + 4.9 × 0.1 − 2 − P = 7.49 − P ;
not hiring a guard, the owner’s payoff is
10 × 0.5 + 4.9 × 0.5 − P = 7.45 − P.
Thus, the owner will hire a guard, and the actuarially fair insurance premium
is
P = 4.9 × 0.1 = 0.49.
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1 Efficient Wage Theory
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
December 10, 2007
Lecture 34
Externalities, Market Failure and Government
Outline
1. Chap 17: Efficient Wage Theory
2. Chap 18: Externalities
3. Chap 18: Property Rights
4. Chap 18: Common Property Resources
1
Efficient Wage Theory
Use the efficient wage theory to explain the presence of unemployment.
Suppose the wage is w, and workers can choose to work or shirk provided a
benefit of S. The unemployment rate is u, and the workers get caught and fired
with a probability p. If a worker shirks, he can get
S + (1 − p)w + p(1 − u)w = S + w(1 − pu),
if a worker does not shirk, he gets w.
Therefore, a worker will work if
w � S + w(1 − pu),
that is,
w�
S
.
pu
This is called nonshirking constraint.
Without information asymmetry, the market wage is wC , and full employ­
ment exists at LC . With information asymmetry, the nonshirking constraint
and the demand of labor determine the wage w∗ and labor L∗ (see Figure 1).
With greater asymmetric information, the probability that shirking is de­
tected, P decreases, and thus the nonshirking constraint rises. The wage and
′
labor are w and L′ respectively (see Figure 1).
Thereby
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2 Externalities
2
10
9
Nonshirking Constraint
SL
8
w’
7
w
6
w*
5
4
wC
3
2
DL
1
L’
0
0
1
2
3
LC
L*
4
5
L
6
7
8
9
10
Figure 1: Unemployment in a Shirking Model.
•
w∗ > wC ,
L∗ < LC ;
•
′
w > w∗ ,
′
L < L∗ .
2
Externalities
Externalities are the effects of production and consumption activities not di­
rectly reflected in the market.
They can be negative or positive.
Negative Externalities. Action by one party imposes a cost on another party.
Example (Pollution). Pollution is not reflected in market because at mar­
ket, residents do not demand firm pay for that cost.
Positive Externalities. Action by one party benefits another party.
Example (Beautiful Garden). If your neighbor has a beautiful garden, you
are happier, but you do not pay your neighbor.
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2 Externalities
3
Negative Externality
An example is steel plant dumping waste in the river as it makes steel.
That imposes cost on fisherman downstream. Marginal external cost (M EC)
is the increase in this cost for each additional unit of steel production.
Marginal social cost (M SC) is M C plus M EC.
Given the market price P , a firm chooses to produce q1 , but if taking external
cost into account, a firm should produce at q ∗ (see Figure 2).
10
9
MSC
8
7
MC
P
6
5
4
3
2
0
q1
q*
1
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 2: A Firm with Negative Externality.
In a competitive market, the equilibrium price and quantity are P1 and q1 ,
but the efficient outcome should be P ∗ and q ∗ (see Figure 3). The failure to
incorporate external cost creates deadweight loss.
Positive Externality
Landscaping generates external benefits to the neighbors.
Like the example above, the marginal social benefit (M SB) is the sum of
private benefit (which is the demand) and the marginal external benefit (M EB).
The quantity q1 consumed in the market is less than the efficient level q ∗ (see
Figure 4).
Solution to Externality
Here are some solutions with government intervenes.
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2 Externalities
4
10
9
MSC
8
DWL
7
MC
6
P
P*
5
4
P1
3
D
2
1
0
q1
q*
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 3: The Whole Industry with Negative Externality.
10
9
8
7
P
6
P1
5
MC
4
3
2
MSB
1
0
q1
0
1
2
D
q*
3
4
5
Q
6
7
8
9
10
Figure 4: External Benefits.
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3 Property Rights
5
• Tax each unit produced by M EC. The marginal cost of the firm is
M C + T = M C + M EC = M SC,
then the firm will choose efficient output.
• Create a standard and monitor pollution. Control the quantity produced
or pollution emission.
3
Property Rights
When property rights are well-specified, economic efficiency may be achieved
without government intervention.
• Factory can install a filter.
• Fishermen can pay for a treatment plant to intercept and clean up factory
waste.
Factory
No Filter
Filter
No Filter
Filter
Fishermen
No Treatment
No Treatment
Treatment
Treatment
Factory’s Profit
500
300
500
300
Fishermen’s Profit
100
500
200
300
Total Profit
600
800
700
600
Table 1: Profits Under Alternative Emissions Choices.
In this case (see Table 1), the most efficient result is that factory installs
filter and fishermen do not pay for treatment.
• If fishermen own the river, they can sue the plant for damages $400. The
factory has two options.
– The factory do not install the filter and pay damages. Profit
500 − 400 = 100.
– The factory install filter. Profit is 300.
Thus the factory will install the filter.
• If factory owns the river, fishermen have three options.
– Fishermen put in treatment plant. Profit is 200.
– Fishermen pay the cost of filter installation to the factory. Profit
500 − 200 = 300.
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4 Common Property Resources
6
– No plant, no filter. Profit is 100.
A payment to the factory by the fisherman results in an efficient outcome
and is in their own interest.
Theorem (Coase Theorem). When parties can bargain without cost and to their
mutual advantage, the outcome will be efficient, regardless of how the property
rights are specified.
4
Common Property Resources
Everyone has free access to a renewable resource, for example, lake, forest, and
so on.
Without control, the quantity consumed is q1 where private cost is equal to
marginal benefit (demand). However, the efficient level of quantity is q ∗ where
M SC = M B(D) (see Figure 5).
10
9
MSC
8
Private Cost
7
P
6
5
P1
4
3
D (MB)
2
1
0
q1
q*
0
1
2
3
4
5
Q
6
7
8
9
10
Figure 5: Common Property Resources.
Some measures to prevent from consuming too much:
• Government puts restrictions on production quantity.
• Set private ownership and the owner sets fees for use of resources.
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1 Public Goods
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
December 12, 2007
Lecture 35
Public Goods
Outline
1. Chap 18: Public Goods
1
Public Goods
Characteristics of public goods:
Nonrival. For any given level of production, the marginal cost of providing it
to an additional consumer is zero: enjoy it rather than use it up.
Nonexclusive. People cannot be excluded from consuming the good. Difficult
to charge for its enjoyment.
Example (Roads).
Example (Streetlight). Once streetlight is setup, everyone can see the light
(nonexclusive).
It is more efficient to have government provide public goods. Government
provides and imposes tax.
Free-rider problem. Consumer need to pay for public goods, and they tend
to understate the value.
Example (Streetlight). Assume cost of building a street light is 1. A’s
and B’s reservation values for the street light are both 1, but they are not
known by government. Then government asks A and B to announce their
value. The street light will be built if
VA + VB � 1,
and people who have value share the cost. Their payoffs are shown in
Table 1). Therefore, A and B will both announce the value as 0. The
result is inefficient: consumers wait for others to pay for the public goods.
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1 Public Goods
2
B
A
Announce Value as 0
Announce Value as 1
Announce Value as 0
0,0
0,1
Announce Value as 1
1,0
1/2,1/2
Table 1: Payoffs of A and B.
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