Introduction 2 2. Profit maximizing firm with monopoly ... · Profit-maximizing firm An example Cost function C q q q( ) 10 2 Demand price function 1 pq 65 4 ... q, then the function

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



1. Introduction

Four questions…

What makes economic research so different from research in the

other social sciences (and indeed in almost all other fields)?



1. Introduction

Four questions



What are the two great pillars of economic theory?



1. Introduction

Four questions




Who are you going to learn most from at UCLA?



1. Introduction

Four questions




Who are you going to learn most from at UCLA?

What do economists do?

Discuss in 3 person groups



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.



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


11 14

65p q and 12 24

70p q

Group 2 exercise: How might you solve for the profit maximizing outputs?



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



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


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



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 .



MODEL 2

Cost function

2 2

1 2 1 1 2 2( ) 10 15 3C q q q q q q q


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



Think on the margin


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 .


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



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.


1

2

Model 2: Profit-maximizing lines.

2

1



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.



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 :


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 .


1

2



The profit is depicted below (using a spread-sheet)



MODEL 2

Suppose we alternative,

first maximizing with respect to 1q , then

2q and so on.

There are four zones.

( , )Z :


2q is increasing

( , )Z :


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



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.



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

qq

and

2

( ) 0f

qq

(3-1)



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)



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.



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 .



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 .



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.



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?



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

http://essentialmicroeconomics.com/Foundations2017/1-BasicsOfConstrainedMaximization.pdf



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 ) .



Geometry

1 2

1 2 1 1 2 20

{ ( ) | }x


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 ) .



Example with 2 commodities: 1 2

1 2 1 1 2 20

{ ( ) | }x


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.



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.



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






( ) j

j

UU x x

x

.



( )j

Ux

x

.

If the consumer has an additional E dollars then j jE p x and so j

j

Ex

p

.







( ) j

j

UU x x

x

.



( )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




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



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

*



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



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



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:


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



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



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

**




Problem

{ ( ) | ( ) ( ) 0}x

Max f x h x b g x



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)

*




Problem

{ ( ) | ( ) ( ) 0}x

Max f x h x b g x



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

.



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



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



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



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



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



The example: Cobb-Douglas utility function: 1 2

1 2 1 1 2 20

{ ( ) | }x



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.



The example: Cobb-Douglas utility function: 1 2

1 2 1 1 2 20

{ ( ) | }x



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

.




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

.

**




1 1 2 2

1 1u u

p x p x


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

*




1 1 2 2

1 1u u

p x p x


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

,



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.



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



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 .

**







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

.

*







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



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



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

.



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



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 ****




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

. ***




1 2

1 9( )U x

x x ,

2

1 1

1( )

Ux

x x

,

2

1 2

9( )

Ux

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 **




1 2

1 9( )U x

x x ,

2

1 1

1( )

Ux

x x

,

2

1 2

9( )

Ux

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

. *




1 2

1 9( )U x

x x ,

2

1 1

1( )

Ux

x x

,

2

1 2

9( )

Ux

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

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Introduction 2 2. Profit maximizing firm with monopoly ... · Profit-maximizing firm An example Cost function C q q q( ) 10 2 Demand price function 1 pq 65 4 ... q, then the function