introduction 2 2. profit maximizing firm with monopoly ... · profit-maximizing firm an example...
TRANSCRIPT
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Microeconomic Theory -1- Introduction
© John Riley October 4, 2017
1. Introduction 2
2. Profit maximizing firm with monopoly power 6
3. General results on maximizing with two variables 18
4. Model of a private ownership economy 22
5. Consumer choice (constrained maximization) 26
6. Market demand and equilibrium in an exchange economy 55
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Microeconomic Theory -2- Introduction
© John Riley October 4, 2017
1. Introduction
Four questions…
What makes economic research so different from research in the
other social sciences (and indeed in almost all other fields)?
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Microeconomic Theory -3- Introduction
© John Riley October 4, 2017
1. Introduction
Four questions
What makes economic research so different from research in the
other social sciences (and indeed in almost all other fields)?
What are the two great pillars of economic theory?
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Microeconomic Theory -4- Introduction
© John Riley October 4, 2017
1. Introduction
Four questions
What makes economic research so different from research in the
other social sciences (and indeed in almost all other fields)?
What are the two great pillars of economic theory?
Who are you going to learn most from at UCLA?
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Microeconomic Theory -5- Introduction
© John Riley October 4, 2017
1. Introduction
Four questions
What makes economic research so different from research in the
other social sciences (and indeed in almost all other fields)?
What are the two great pillars of economic theory?
Who are you going to learn most from at UCLA?
What do economists do?
Discuss in 3 person groups
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Microeconomic Theory -6- Introduction
© John Riley October 4, 2017
2. Profit-maximizing firm
An example
Cost function
2( ) 10C q q q
Demand price function
14
65p q
Group exercise: Solve for the profit maximizing output and price.
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Microeconomic Theory -7- Introduction
© John Riley October 4, 2017
Two products
MODEL 1
Cost function
2 2
1 2 1 1 2 2( ) 10 15 2 3 2C q q q q q q q
Demand price functions
11 14
85p q and 12 24
90p q
Group 1 exercise: How might you solve for the profit maximizing outputs?
MODEL 2
Cost function
2 2
1 2 1 1 2 2( ) 10 15 3C q q q q q q q
Demand price functions
11 14
65p q and 12 24
70p q
Group 2 exercise: How might you solve for the profit maximizing outputs?
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Microeconomic Theory -8- Introduction
© John Riley October 4, 2017
MODEL 1:
Revenue
21 11 1 1 1 1 1 14 4
(85 ) 85R p q q q q q , 21 12 2 2 2 2 2 24 4
(90 ) 90R p q q q q q
Profit
1 2R R C
2 2 2 21 11 1 2 2 1 2 1 1 2 24 4
85 90 (10 15 2 3 2 )q q q q q q q q q q
2 29 91 2 1 2 1 24 4
75 75 3q q q q q q
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Microeconomic Theory -9- Introduction
© John Riley October 4, 2017
Think on the margin
Marginal profit of increasing 1q
91 22
1
75 3q qq
.
Therefore the profit-maximizing choice is
2 21 1 2 2 29 3
( ) (75 3 ) (25 )q m q q q .
2 29 91 2 1 2 1 24 4
75 75 3q q q q q q
Marginal profit of increasing 2q
91 22
2
75 3q qq
.
Therefore the profit-maximizing choice is
2 22 2 1 1 19 3
( ) (75 3 ) (25 )q m q q q .
The two profit-maximizing lines are depicted.
Model 1: Profit-maximizing lines
1
2
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Microeconomic Theory -10- Introduction
© John Riley October 4, 2017
21 1 2 23
( ) (25 )q m q q , 22 2 1 13
( ) (25 )q m q q
If you solve for q satisfying both equations you will find that the unique solution is
1 2( , ) (10,10)q q q .
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Microeconomic Theory -11- Introduction
© John Riley October 4, 2017
MODEL 2
Cost function
2 2
1 2 1 1 2 2( ) 10 15 3C q q q q q q q
Demand price functions
11 14
65p q and 12 24
70p q
Revenue
21 11 1 1 1 1 1 14 4
(65 ) 65R p q q q q q , 21 12 2 2 2 2 2 24 4
(70 ) 70R p q q q q q
Profit
1 2R R C
2 2 2 21 11 1 2 2 1 2 1 1 2 24 4
65 70 (10 15 3 )q q q q q q q q q q
2 25 51 2 1 2 1 24 4
55 55 3q q q q q q
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Microeconomic Theory -12- Introduction
© John Riley October 4, 2017
Think on the margin
Marginal profit of increasing 1q
51 22
1
55 3q qq
.
Therefore, for any 2q the profit-maximizing
1q is
21 1 2 25
( ) (55 3 )q m q q .
Marginal profit of increasing 2q
51 22
2
55 3q qq
.
Therefore, for any 1q the profit-maximizing
2q is
22 2 1 15
( ) (55 3 )q m q q
The two profit-maximizing lines are depicted.
If you solve for q satisfying both equations you will find that the unique solution is
1 2( , ) (10,10)q q q .
Model 2: Profit-maximizing lines.
2
1
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Microeconomic Theory -13- Introduction
© John Riley October 4, 2017
These look very similar to the profit-maximizing lines in Model 1. However now the profit-maximizing
line for 2q is steeper (i.e. has a more negative slope).
As we shall see, this makes a critical difference.
Model 1: Profit-maximizing lines
1
2
Model 2: Profit-maximizing lines.
2
1
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Microeconomic Theory -14- Introduction
© John Riley October 4, 2017
Is q the profit-maximizing output bundle (output vector)?
Group Exercise: For model 2 solve for maximized profit if only one commodity is produced.
Compare this with the profit if (10,10)q is produced.
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Microeconomic Theory -15- Introduction
© John Riley October 4, 2017
Model 1
Suppose we alternate, first maximizing
with respect to 1q , then
2q and so on.
There are four zones.
( , )Z :
The zone in which 1q is increasing and
2q is increasing
( , )Z :
The zone in which 1q is increasing and
2q is decreasing
and so on…
If you pick any starting point you will find this process leads to the intersection point (10,10)q .
Model 1: Profit-maximizing lines
1
2
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Microeconomic Theory -16- Introduction
© John Riley October 4, 2017
The profit is depicted below (using a spread-sheet)
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Microeconomic Theory -17- Introduction
© John Riley October 4, 2017
MODEL 2
Suppose we alternative,
first maximizing with respect to 1q , then
2q and so on.
There are four zones.
( , )Z :
The zone in which 1q is increasing and
2q is increasing
( , )Z :
The zone in which 1q is increasing and
2q is decreasing
and so on…
If you pick any starting point you will find this process
never leads to the intersection point (10,10)q .
2
1
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Microeconomic Theory -18- Introduction
© John Riley October 4, 2017
The profit function has the shape of a saddle. The output vector q where the slope in the direction of
each axis is zero is called a saddle-point.
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Microeconomic Theory -19- Introduction
© John Riley October 4, 2017
3. General results
Consider the two variable problem
1 2{ ( , )}q
Max f q q
Necessary conditions
Consider any 0q . If the slope in the cross section parallel to the 1q -axis is not zero, then by
standard one variable analysis, the function is not maximized. The same holds for the cross section
parallel to the 2q -axis. Thus for q to be a maximizer, the slope of both cross sections must be zero.
First order necessary conditions for a maximum
For 0q to be a maximizer the following two conditions must hold
1
( ) 0f
and
2
( ) 0f
(3-1)
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Microeconomic Theory -20- Introduction
© John Riley October 4, 2017
Suppose that the first order necessary conditions hold at q . Also, if the slope of the cross section
parallel to the 1q -axis is strictly increasing in
1q at q , then 1q is not a maximizer. Thus a necessary
condition for a maximum is that the slope must be decreasing. Exactly the same argument holds for
2q .
We therefore have a second set of necessary conditions for a maximum. Since they are
restrictions on second derivatives they are called the second order conditions.
Second order necessary conditions for a maximum
If 0q is a maximizer of ( )f q , then
1 1
( ) 0f
qq q
and
2 2
( ) 0f
qq q
(3-2)
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Microeconomic Theory -21- Introduction
© John Riley October 4, 2017
As we have seen, these conditions are necessary for a maximum but they do not, by themselves
guarantee that q satisfying these conditions is the maximum.
However, if the step by step approach does lead to q then this point is a least a local maximizer.
Proposition: Sufficient conditions for a local maximum
If in the neighborhood of 0q satisfying the first order conditions (FOC) the step by step approach
leads to q , then the function ( )f q has a local maximum at q
Proposition: Sufficient conditions for a global maximum
If the conditions above hold at q and the FOC hold only at q , then this is the global maximizer.
If there is a unique q it is the global maximizer.
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Microeconomic Theory -22- Introduction
© John Riley October 4, 2017
4. Model of a private ownership economy
Commodities:
The set of commodities is {1,..., }nN
Endowments:
0j the initial endowment of commodity j (land, labor, coconuts…)
Consumers:
The set of consumers is {1,..., }HH .
Each consumer’s preferences can be represented by a continuously differentiable strictly increasing
function ( ), 1,...,hU x h H where 1( ,..., )h h h
nx x x
Notation: If i ix x for 1,...,i n and the inequality is strict for some i we write x x .
Then utility is increasing if x x implies that ( ) ( )h hU x U x .
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Microeconomic Theory -23- Introduction
© John Riley October 4, 2017
Firms
The set of firms is {1,..., }FF .
Transformers of inputs into outputs:
Let 1( ,..., )f f f
nz z z be a vector of firm f ’s inputs.
Each component is a quantity of one of the commodities.
Let fq be a vector of the outputs.
Production vector
( , )f f fy z q is an ordered list of all the inputs and outputs.
Each firm must choose among the production vectors that are feasible.
We write this set of feasible plans as fY .
Private ownership:
Shareholdings: hfk is the share-holding of consumer h in firm f .
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Microeconomic Theory -24- Introduction
© John Riley October 4, 2017
Feasible consumption vectors
1 1 1 1
h h F Fh h f f
j j j j
h h f f
x z q
, where ( , )f f f fy z q Y (production plans are feasible)
total consumption total endowment - total inputs + total outputs
Price taking equilibrium allocation
An allocation to consumers, 1{ }h H
hx , feasible production plans 1{ }f F
fy and a price vector 0p
Such that
(i) No consumer has a strictly preferred point in his/her budget set.
(ii) There is no strictly more profitable feasible plan for any firm.
(iii) All markets clear.
1 1 1 1
h h F Fh h f f
j j j j
h h f f
x z q
and 0jp if the inequality is strict.
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Microeconomic Theory -25- Introduction
© John Riley October 4, 2017
We will look at simple economies to gain insights into how commodities are allocated via markets.
For now consider one example.
Alex likes only bananas. Bev likes only coconuts.
Alex has an endowment of 4 bananas and 6 coconuts (4,6)A .
Bev has an endowment of 10 bananas and 7 coconuts (10,7)A .
Group exercise:
What are market clearing prices in this economy?
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Microeconomic Theory -26- Introduction
© John Riley October 4, 2017
5. Consumers
We begin by considering a consumer maximization problem.
1 10
{ ( ) | ... }h
n nx
Max U x p x p x I
Using the sumproduct notation 1 1 ... n na b a b a b , we can write this as follows:
0
{ ( ) | }h
xMax U x p x I
.
We will assume that ( )hU x is a strictly increasing function.
This is an example of a maximization with a single linear resource constraint.
As you will see by looking at sections A-C in the following slides, an almost identical argument holds
for any maximization problem with resource constraints.
http://essentialmicroeconomics.com/Foundations2017/1-BasicsOfConstrainedMaximization.pdf
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Microeconomic Theory -27- Introduction
© John Riley October 4, 2017
0
{ ( ) | }h
xMax U x p x I
.
Let x be a solution. Note that, since ( )U x is strictly increasing, p x I
Example with 2 commodities: 1 2
1 2 1 1 2 20
{ ( ) | }x
Max U x x x p x p x I
Note that utility is zero if consumption of either commodity is zero. Therefore every component of
the solution x is strictly positive. (We write 0x ) .
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Microeconomic Theory -28- Introduction
© John Riley October 4, 2017
Geometry
1 2
1 2 1 1 2 20
{ ( ) | }x
Max U x x x p x p x I
In the 2 commodity case we can represent
preferences by depicting points for which utility
has the same value.
Such a set of points is called a level set. In the figure
the level sets are the boundaries of the blue, red and
green shaded regions.
Note that utility is zero if consumption of either commodity is zero. Therefore every component
of the solution x is strictly positive.(We write 0x ) .
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Microeconomic Theory -29- Introduction
© John Riley October 4, 2017
Example with 2 commodities: 1 2
1 2 1 1 2 20
{ ( ) | }x
Max U x x x p x p x I
Level sets
In mathematical notation the 4 level sets are
{ | ( ) 0}x U x ,{ | ( ) 1}x U x , { | ( ) 2}x U x ,{ | ( ) 3}x U x .
Three of them are what economists
often call indifference curves.
Superlevel sets
The set of points on or above a level set
{ | ( ) }x U x k
is called a superlevel set.
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Microeconomic Theory -30- Introduction
© John Riley October 4, 2017
The set of points satisfying
1 1 2 2p x p x I
Can be rewritten as 12 1
2 2
pIx x
p p
This is a line of slope 1
2
p
p
In the figure both components of the solution
1 2( , )x x x are strictly positive.
(Mathematical shorthand 0x .)
So the slope of the budget line is
equal to the slope of the level set.
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Microeconomic Theory -31- Introduction
© John Riley October 4, 2017
Necessary conditions for a maximum
To a first approximation, if a consumer currently, choosing x can increase consumption of
commodity j by jx , the change in utility is
( ) j
j
UU x x
x
.
This is depicted in the figure..
The slope of the tangent line at x is
( )j
Ux
x
.
Fig. 4-1: Utility as a function of
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Microeconomic Theory -32- Introduction
© John Riley October 4, 2017
Necessary conditions for a maximum
To a first approximation, if a consumer currently, choosing x can increase consumption of
commodity j by jx , the change in utility is
( ) j
j
UU x x
x
.
This is depicted in the figure..
The slope of the tangent line at x is
( )j
Ux
x
.
If the consumer has an additional E dollars then j jE p x and so j
j
Ex
p
.
Fig. 2-3: Utility as a function of
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Microeconomic Theory -33- Introduction
© John Riley October 4, 2017
Necessary conditions for a maximum
To a first approximation, if a consumer currently, choosing x can increase consumption of
commodity j by jx , the change in utility is
( ) j
j
UU x x
x
.
This is depicted in the figure..
The slope of the tangent line at x is
( )j
Ux
x
.
If the consumer has an additional E dollars then j jE p x and so j
j
Ex
p
.
The increase in utility is therefore 1
( ) ( ) ( )j
j j j j j
U U E UU x x x x E
x x p p x
Fig. 4-1: Utility as a function of
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Microeconomic Theory -34- Introduction
© John Riley October 4, 2017
We have seen that
1( )
j j
UU x E
p x
Therefore
1
( )j j
U Ux
E p x
In the limit as E approaches zero, this becomes the rate at which utility rises as expenditure on
commodity j rises.
1( )
j j
Ux
p x
is the marginal utility per dollar as expenditure on commodity j rises
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Microeconomic Theory -35- Introduction
© John Riley October 4, 2017
Suppose that the consumer spends 1 dollar less on commodity j . His change in utility is
1( )
j j
Ux
p x
. He then spends the dollar on commodity i .
The change in utility is 1
( )i i
Ux
p x
. The net change in utility is therefore
1 1
( ) ( )i i j j
U Ux x
p x p x
*
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Microeconomic Theory -36- Introduction
© John Riley October 4, 2017
Suppose that the consumer spends 1 dollar less on commodity j . His change in utility is
1( )
j j
Ux
p x
. He then spends the dollar on commodity i .
The change in utility is 1
( )i i
Ux
p x
. The net change in utility is therefore
1 1
( ) ( )i i j j
U Ux x
p x p x
Case (i) , 0i jx x
If the change in utility is strictly positive the current utility can be increased by consuming more of
commodity i and less of commodity j . If it is negative, utility can be increased by spending less
commodity j and more on commodity i . Thus a necessary condition for x to be utility maximizing is
that
1 1
( ) ( )i i j j
U Ux x
p x p x
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Microeconomic Theory -37- Introduction
© John Riley October 4, 2017
Case (ii) 0j ix x
If the difference in marginal utilities is positive current ( )U x can be increased by spending a positive
amount on commodity j . Thus a necessary condition for x to be utility maximizing is that
1 1
( ) ( )i i j j
U Ux x
p x p x
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Microeconomic Theory -38- Introduction
© John Riley October 4, 2017
Case (ii) 0j ix x
If the difference in marginal utilities is positive current ( )U x can be increased by spending a positive
amount on commodity j . Thus a necessary condition for x to be utility maximizing is that
1 1
( ) ( )i i j j
U Ux x
p x p x
Let be the common marginal utility per dollar for all those commodities that are consumed in
strictly positive amounts. We can therefore summarize the necessary conditions as follows:
Necessary conditions for a maximum
If 0jx then 1
( )j j
Ux
p x
If 0jx then 1
( )j j
Ux
p x
Note: Since is the rate at which utility rises with income it is called the marginal utility of income
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Microeconomic Theory -39- Introduction
© John Riley October 4, 2017
For the general problem,
{ ( ) | ( ) ( ) 0}x
Max f x h x b g x
j
j
gb x
x
is the extra resources needed to increase commodity j by jx .
Therefore 1
j
j
x bg
x
is the extra output you can get using an extra b of the resource.
An almost identical argument then yields the following necessary conditions.
If 0jx then ( )j
j
f
xx
g
x
. If 0jx then ( )j
j
f
xx
g
x
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Microeconomic Theory -40- Introduction
© John Riley October 4, 2017
Use the Lagrangian to write down the necessary conditions
Problem
{ ( ) | ( ) ( ) 0}x
Max f x h x b g x
The Lagrangian for the maximization problem is defined as follows:
( , ) ( ) ( ) ( ) ( ( ))x f x h x f x b g x L , where 0
**
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Microeconomic Theory -41- Introduction
© John Riley October 4, 2017
Use the Lagrangian to write down the necessary conditions
Problem
{ ( ) | ( ) ( ) 0}x
Max f x h x b g x
The Lagrangian for the maximization problem is defined as follows:
( , ) ( ) ( ) ( ) ( ( ))x f x h x f x b g x L , where 0
Necessary conditions for x to be a solution to this maximization problem.
( , ) ( ) ( ) 0j j j j j
f h f gx x x
x x x x x
L, with equality if 0jx . 1,...,j n (1)
( , ) ( ) ( ) 0x h x b g x
L, with equality if 0 . (2)
*
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Microeconomic Theory -42- Introduction
© John Riley October 4, 2017
Use the Lagrangian to write down the necessary conditions
Problem
{ ( ) | ( ) ( ) 0}x
Max f x h x b g x
The Lagrangian for the maximization problem is defined as follows:
( , ) ( ) ( ) ( ) ( ( ))x f x h x f x b g x L , where 0
Necessary conditions for x to be a solution to this maximization problem.
( , ) ( ) ( ) 0j j j j j
f h f gx x x
x x x x x
L, with equality if 0jx . 1,...,j n (1)
( , ) ( ) ( ) 0x h x b g x
L, with equality if 0 . (2)
From (1), note that
if ix and 0jx , then j i
j i
f f
x x
g g
x x
. And if 0j ix x , then j i
j i
f f
x x
g g
x x
.
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Microeconomic Theory -43- Introduction
© John Riley October 4, 2017
An alternative approach
From the argument above, if both commodity i and commodity j are consumed, then the ratio
of their marginal utilities must be equal to the price ratio.
To understand this consider a change in ix and jx that leaves the consumer on the same level set. i.e.
1 1 2 2 1 2( , ) ( , )U x x x x U x x
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Microeconomic Theory -44- Introduction
© John Riley October 4, 2017
An alternative approach
From the argument above, if both commodity i and commodity j are consumed, then the ratio
of their marginal utilities must be equal to the price ratio.
To understand this consider a change in ix and jx that leaves the consumer on the same level set. i.e.
1 1 2 2 1 2( , ) ( , )U x x x x U x x
Above we showed that, to a first approximation,
( ) j
j
UU x x
x
.
If we increase the quantity of commodity j and reduce the quantity of commodity i , then the net
change in utility is
( ) ( )j i
j i
U UU x x x x
x x
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Microeconomic Theory -45- Introduction
© John Riley October 4, 2017
We have argued that ( ) ( )j i
j i
U UU x x x x
x x
For this net change to be zero,
j i
i
j
U
x x
Ux
x
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Microeconomic Theory -46- Introduction
© John Riley October 4, 2017
We have argued that ( ) ( )j i
j i
U UU x x x x
x x
For this net change to be zero,
j i
i
j
U
x x
Ux
x
In the figure, - 2
1
x
x
is the slope of the level set at x .
The ratio is the rate at which 1x must be substituted into
the consumption bundle to compensate for a reduction in 2x
Hence we call it the marginal rate of substitution of 1x for
2x .
Definition: Marginal rate of substitution
( , ) ii j
j
U
xMRS x x
U
x
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Microeconomic Theory -47- Introduction
© John Riley October 4, 2017
For x to be the maximizer the rate at which 1x
can be substituted into the budget as 2x is
reduced must leave total expenditure
on the two commodities constant, i.e.,
0i i j jp x p x
Then along the budget line
j i
i j
x p
x p
Graphically, the slope of the budget line
must be equal to the slope of the indifference curve at x ie.
( , ) i ii j
j
j
U
x pMRS x x
U p
x
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Microeconomic Theory -48- Introduction
© John Riley October 4, 2017
The example: Cobb-Douglas utility function: 1 2
1 2 1 1 2 20
{ ( ) | }x
Max U x x x p x p x I
Necessary conditions for a maximum
Method 1: Equalize the marginal utility per dollar
To make differentiation simple, try to find an increasing function of the utility function that is simple.
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Microeconomic Theory -49- Introduction
© John Riley October 4, 2017
The example: Cobb-Douglas utility function: 1 2
1 2 1 1 2 20
{ ( ) | }x
Max U x x x p x p x I
Necessary conditions for a maximum
Method 1: Equalize the marginal utility per dollar
To make differentiation simple, try to find an increasing function of the utility function that is simple.
Define the new utility function ( ) ln ( )u x U x
The new maximization problem is
1 1 2 20
{ ( ) ln ( ) | }x
Max u x U x p x p x I
That is
1 1 2 2 1 1 2 2
0{ ln ln | }
xMax x x p x p x I
Note that
j
j j
u
x x
.
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Microeconomic Theory -50- Introduction
© John Riley October 4, 2017
Necessary conditions
1 1 2 2
1 1u u
p x p x
. For the example it follows that 1 2
1 1 2 2p x p x
.
Also 1 1 2 2p x p x I .
Technical tip
Ratio Rule:
If 1 2
1 2
a a
b b and
1 2 0b b then 1 2 1 2
1 2 1 2
a a a a
b b b b
.
**
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Microeconomic Theory -51- Introduction
© John Riley October 4, 2017
Necessary conditions
1 1 2 2
1 1u u
p x p x
. For the example it follows that 1 2
1 1 2 2p x p x
.
Also 1 1 2 2p x p x I .
Technical tip
Ratio Rule:
If 1 2
1 2
a a
b b and
1 2 0b b then 1 2 1 2
1 2 1 2
a a a a
b b b b
.
Therefore
1 2 1 2 1 2
1 1 2 2 1 1 2 2p x p x p x p x I
*
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Microeconomic Theory -52- Introduction
© John Riley October 4, 2017
Necessary conditions
1 1 2 2
1 1u u
p x p x
. For the example it follows that 1 2
1 1 2 2p x p x
.
Also 1 1 2 2p x p x I .
Technical tip
Ratio Rule:
If 1 2
1 2
a a
b b and
1 2 0b b then 1 2 1 2
1 2 1 2
a a a a
b b b b
.
Therefore
1 2 1 2 1 2
1 1 2 2 1 1 2 2p x p x p x p x I
We can then solve for 1x and
2x
Cobb-Douglas demands
1 2
j
j
j
Ix
p
,
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Microeconomic Theory -53- Introduction
© John Riley October 4, 2017
Method 2: Equate the MRS and price ratio
1 2
1 2( )U x x x
. Then 1 21
1 1 2
1
Ux x
x
and 1 2 1
2 1 2
2
Ux x
x
1 2
1 2
1
1 1 1 2 1 21 2 1
2 1 2 2 1
2
( , )
U
x x x xMRS x x
U x x x
x
.
Then to be the maximizer,
1 2 11 2
2 1 2
( , )x p
MRS x xx p
As we have seen, it is helpful to rewrite this as follows:
1 1 2 2
1 2
p x p x
.
Then proceed as before.
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Microeconomic Theory -54- Introduction
© John Riley October 4, 2017
Data Analytics (Taking the model to the data)
11
1 2
( , )j
Ix p I
p
.
Take the logarithm
1
1 2
ln ln( ) ln lnj jx I p
The model is now linear. We can then use least squares estimation
0 1ln (ln ln )j jx a a I p
or
0 1 2ln ln lnj jx a a I a p
Exercise: If 1 2
1 1 2 2( ) ( ) ( )U x a x a x
, solve for the demand function 1( , )x p I
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Microeconomic Theory -55- Introduction
© John Riley October 4, 2017
6. Market demand in a Cobb-Douglas economy with no production
Consumer hH has some initial endowment 1( ,...., )n . With a price vector p , the market value
of this endowment is h hI p . If the consumer sells his entire endowment he can then purchase
any consumption bundle hx satisfying
h h hp x I p .
**
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Microeconomic Theory -56- Introduction
© John Riley October 4, 2017
6. Market demand in a Cobb-Douglas economy with no production
Consumer hH has some initial endowment 1( ,...., )n . With a price vector p , the market value
of this endowment is h hI p . If the consumer sells his entire endowment he can then purchase
any consumption bundle hx satisfying
h h hp x I p .
Consider a 2-commodity economy in which each consumer has the same Cobb-Douglas preferences.
1 2
1 2( ) ( ) ( )h h hU x x x
.
*
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Microeconomic Theory -57- Introduction
© John Riley October 4, 2017
5. Market demand in a Cobb-Douglas economy with no production
Consumer hH has some initial endowment 1( ,...., )n . With a price vector p , the market value
of this endowment is h hI p . If the consumer sells his entire endowment he can then purchase
any consumption bundle hx satisfying
h h hp x I p .
Consider a 2-commodity economy in which each consumer has the same Cobb-Douglas preferences.
1 2
1 2( ) ( ) ( )h h hU x x x
.
From section 4, demand for commodity 1 is
1 1
1
( , )h
h h Ix p I
p where 1
1
1 2
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Microeconomic Theory -58- Introduction
© John Riley October 4, 2017
Suppose that all consumers have the same preferences
Then a consumer with a budget of 1 has a consumption of 11
1 2 1
1( ,1)x p
p
. Therefore consumer
h ’s demand is simply scaled up by her income.
1 ( , ) ( ,1)h h hx p I I x p
11
1
( ,1)x pp
where 1
1
1 2
1 ( , ) ( ,1)h h hx p I I x p
Summing over consumers, Cobb-Douglas market demand for two consumers is
1 1 1 2 1 2
1 1 1 1 1 1( ) ( , ) ( , ) ( ,1) ( ,1) ( ,1)( )h hx p x p I x p I x p I x p I x p I I
For H consumers, market demand is
1 1
1 1 1 1 1
1
( ) ( , ) ( ,1) ... | ( ,1) ( ,1)( ... )H
h h H H
h
x p x p I x p I x p I x p I I
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Microeconomic Theory -59- Introduction
© John Riley October 4, 2017
Consider a very simple exchange economy
No production. On one end of the island there are banana plantations. At the other end there are
coconut plantations.
Given a price vector p , if consumer h has an endowment of 1 2( , )h h h , this has market value
1 1 2 2
h h h hI p p p .
She can trade this for any other bundle of bananas and coconuts costing less. So her maximiaation
problem is
0{ ( | }
h
h h h h h
xMax U x p x p I
.
The vale of all the endowment is 1 ... hI I p where 1
Hh
h
.
Therefore if all consumers have the same Cobb-Douglas preferences
11
1
( )x p pp
.
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Microeconomic Theory -60- Introduction
© John Riley October 4, 2017
Equilibrium in an exchange economy with Cobb-Douglas utility 1 2
1 2( ) ( )h h hU x x
The total supply of commodity 1 is 1 . The market commodity 1 clears if
11 1 1 1 1 1
1 1
1( ) ( ) 0x p p p p
p p
Solving
1 1 1 1 1 2 2( )p p p
1 1 2 1 2
2 1 1 2 1
1p
p
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Microeconomic Theory -61- Introduction
© John Riley October 4, 2017
Additional example: (A bit technical for a closed book exam)
1 2
1 9( )U x
x x ,
2
1 1
1( )
Ux
x x
,
2
1 2
9( )
Ux
x x
.
Solve the budget problem: 1 2{ ( ) | 4 21Max U x x x ****
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Microeconomic Theory -62- Introduction
© John Riley October 4, 2017
Additional example: (A bit technical for a closed book exam)
1 2
1 9( )U x
x x ,
2
1 1
1( )
Ux
x x
,
2
1 2
9( )
Ux
x x
.
Solve the budget problem: 1 2{ ( ) | 4 21Max U x x x
Step 1: 0x (why?)
1 2
2 2
1 2 1 2
1 9
4
U U
x x
p p x x
. ***
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Microeconomic Theory -63- Introduction
© John Riley October 4, 2017
Additional example: (A bit technical for a closed book exam)
1 2
1 9( )U x
x x ,
2
1 1
1( )
Ux
x x
,
2
1 2
9( )
Ux
x x
.
Solve the budget problem: 1 2{ ( ) | 4 21Max U x x x
Step 1: If 0x
1 2
2 2
1 2 1 2
1 9
4
U U
x x
p p x x
.
Step 2: Make the denominators linear in each commodity by taking the square root.
1 2
1 3
2x x **
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Microeconomic Theory -64- Introduction
© John Riley October 4, 2017
Additional example: (A bit technical for a closed book exam)
1 2
1 9( )U x
x x ,
2
1 1
1( )
Ux
x x
,
2
1 2
9( )
Ux
x x
.
Solve the budget problem: 1 2{ ( ) | 4 21Max U x x x
Step 1: If 0x
1 2
2 2
1 2 1 2
1 9
4
U U
x x
p p x x
.
Step 2: Make the denominators linear in jx by taking the square root.
1 2
1 3
2x x
Step 3: Convert the denominators into expenditure and appeal to the Ratio Rule
1 2 1 2
1 6 7 7
4 4 21x x x x
. *
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Microeconomic Theory -65- Introduction
© John Riley October 4, 2017
Additional example: (A bit technical for a closed book exam)
1 2
1 9( )U x
x x ,
2
1 1
1( )
Ux
x x
,
2
1 2
9( )
Ux
x x
.
Solve the budget problem: 1 2{ ( ) | 4 21Max U x x x
Step 1: If 0x
1 2
2 2
1 2 1 2
1 9
4
U U
x x
p p x x
.
Step 2: Make the denominators linear in jx by taking the square root.
1 2
1 3
2x x
Step 3: Convert the denominators into expenditure and appeal to the Ratio Rule
1 2 1 2
1 6 7 7
4 4 21x x x x
.
Step 4: Solve for the demands
1 3x 2 4.5x