程序代写代做代考 1. Sets and Sequences

1. Sets and Sequences

Microeconomic Theory -1- Introduction and maximization

Introduction 2

Maximization

1. Profit maximizing firm with monopoly power 6

2. General results on maximizing with two variables 22

3. Non-negativity constraints 25

4. First laws of supply and input demand 27

5. Resource constrained maximization – an economic approach 30

41 pages

Microeconomic Theory -2- Introduction and maximization

Introduction

Four questions…

What makes economic research so different from research in the

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

Microeconomic Theory -3- Introduction and maximization

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?

Microeconomic Theory -4- Introduction and maximization

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?

Microeconomic Theory -5- Introduction and maximization

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

Microeconomic Theory -6- Introduction and maximization

Maximization

1. Profit-maximizing firm

Example 1:

Cost function

2( ) 5 12 3C q q q  

Demand price function

( ) 20p q q 

Group exercise: Solve for the profit maximizing output and price.

Microeconomic Theory -7- Introduction and maximization

Example 2: 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

1
1 14

85p q  and 12 2490p 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

1
1 14

65p q  and 12 2470p q 

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

Microeconomic Theory -8- Introduction and maximization

MODEL 1:

Revenue

21 1
1 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 24 485 90 (10 15 2 3 2 )q q q q q q q q q q        

2 29 91 2 1 2 1 24 475 75 3q q q q q q    

Microeconomic Theory -9- Introduction and maximization

Think on the margin

Marginal profit of increasing 1q

9
1 22

75 3q q
q


  


.

Therefore the profit-maximizing choice is

2 2
1 1 2 2 29 3

( ) (75 3 ) (25 )q m q q q     .

2 29 91 2 1 2 1 24 475 75 3q q q q q q     

Marginal profit of increasing 2q

9
1 22

75 3q q
q


  


.

Therefore the profit-maximizing choice is

2 2
2 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

Microeconomic Theory -10- Introduction and maximization

2
1 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  .

Microeconomic Theory -11- Introduction and maximization

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

1
1 14

65p q  and 12 2470p q 

Revenue

21 1
1 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 24 465 70 (10 15 3 )q q q q q q q q q q        

2 25 51 2 1 2 1 24 455 55 3q q q q q q    

Think on the margin

Microeconomic Theory -12- Introduction and maximization

Marginal profit of increasing 1q

5
1 22

55 3q q
q


  


.

Therefore, for any 2q the profit-maximizing 1q is

2
1 1 2 25

( ) (55 3 )q m q q   .

Marginal profit of increasing 2q

5
1 22

55 3q q
q


  


.

Therefore, for any 1q the profit-maximizing 2q is

2
2 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.

Microeconomic Theory -13- Introduction and maximization

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

Model 2: Profit-maximizing lines.

Microeconomic Theory -14- Introduction and maximization

Model 1:

Is 1 2( , )q q q the profit-maximizing output vector?

The profit-maximizing lines divide the

positive quadrant into four zones.

The arrows indicate the directions of

in which 1 2( , )q q increases.

Consider the point
0q .

Output is higher in the diagonally shaded region

and lower in the dotted region.

Thus the level set through
0q must have a negative slope.

A similar argument can be used in the other three quadrants.

Model 1: Profit-maximizing lines

Microeconomic Theory -15- Introduction and maximization

The level set
0( ) ( )q q  in the ( , )Z   region

Profit is higher in the shaded region

Note that the level set is parallel to the 2q axis

at the point of intersection with the maximizing

line for 2q and is parallel to the horizontal

axis at the point of intersection with the

maximizing line for 1q

Model 1: Level set for profit

Microeconomic Theory -16- Introduction and maximization

The level set
0( ) ( )q q 

and superlevel set
0( ) ( )q q 

are depicted opposite.

Model 1: Level set for profit

Microeconomic Theory -17- Introduction and maximization

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

Microeconomic Theory -18- Introduction and maximization

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

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.

Microeconomic Theory -19- Introduction and maximization

MODEL 2

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

never leads to the intersection point (10,10)q  .

Microeconomic Theory -20- Introduction and maximization

Local maximum 1( ,0)q q on the 1q axis

By an essentially identical argument, there is a second local maximum q on the 2q axis.

Microeconomic Theory -21- Introduction and maximization

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.

Microeconomic Theory -22- Introduction and maximization

2. 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

( ) 0
f

q
q





and

( ) 0
f

q
q





(3-1)

Microeconomic Theory -23- Introduction and maximization

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 2

1 1

( , ) 0
f

q q
q q

 


 
and

1 2

2 2

( , ) 0
f

q q
q q

 


 
(3-2)

Microeconomic Theory -24- Introduction and maximization

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 the first and second order necessary conditions hold at q and the level sets are closed loops

around q , then the function ( )f q has a local maximum at q

Proposition: Sufficient conditions for a global maximum

If the first and second order necessary conditions hold at q and the level sets are closed loops

around q and the FOC hold only at q , then this is the global maximizer.

Microeconomic Theory -25- Introduction and maximization

3. Non-negativity constraints

Many economic variables cannot be negative. Suppose this is true for all variables

Let 1( ,…, )nx x x solve
0

{ ( )}
x

Max f x


We will consider the first variable.

It is helpful to write the optimal value

of all the other variables as 1x . Then

1x solves
1

1 1
0

{ ( , )}
x

Max f x x


Case (i) 1 0x 

This is depicted opposite.

For 1x to be the maximizer,

the graph of 1 1( , )f x x must be zero at 1x .

Case (i): Necessary condition for a maximum

Microeconomic Theory -26- Introduction and maximization

Case (ii) 1 0x 

This is depicted opposite.

For 1x to be the maximizer,

the graph of 1 1( , )f x x cannot be strictly

positive at 1x .

Taking the two cases together,

( ) 0
f

x
x





, with equality if 1 0x 

An identical argument holds for all of the

variables.

Necessary conditions

( ) 0
f

x
x





, with equality if 1 0x 

Case (ii): Necessary condition for a maximum

Microeconomic Theory -27- Introduction and maximization

4. Laws of supply and input demand

The first law of firm supply

As an output price p rises, the maximizing output ( )q p increases (at least weakly).

Case (i) (0)p MC Case (ii) (0)p MC

As the output price rises, the profit-maximizing output increases (at least weakly).

Fig. 1: Profit-maximizing output

Microeconomic Theory -28- Introduction and maximization

The firm’s supply curve

For prices below (0)MC , supply is zero. For higher prices the graph of marginal cost ( )MC q is the

supply curve.

Fig. 2: Firm’s supply curve

Microeconomic Theory -29- Introduction and maximization

The first law of input demand

As an input price r rises, the maximizing input ( )z r decreases (at least weakly).

The rate at which revenue rises as the input (and hence output) rises is called the Marginal Revenue

Product (MRP).

Fig. 3: Firm’s input demand curve

Microeconomic Theory -30- Introduction and maximization

5. Resource constrained maximization – – an economic approach

Problem:
0

{ ( ) | ( ) }
x

Max f x g x b




Let x be the solution to this problem.

Interpretation, if the firm chooses x it requires ( )g x units of a resource that is fixed in supply (.e.

Floor space of plant, highly skilled workers)

Assumption 1: No solution,
*x , to the maximization problem

0
{ ( )}

x
Max f x


satisfies the resource

constraint. Therefore, at x , this constraint is binding.

We interpret ( )f x as the profit of the firm.

Microeconomic Theory -31- Introduction and maximization

To solve this problem, we consider the “relaxed problem” in which the firm can purchase additional

units at the price  . Since this is a hypothetical opportunity, economists refer to the price as the

“shadow price” of the resource rather than a market price.

Suppose that the firm purchases ( )g x b additional units. Its profit is then

( ) ( ( ) ) ( ) ( ( ))f x g x b f x b g x      L

The relaxed problem is then

{ ( ) ( ( ))}
x

Max f x b g x


  L

First Order Necessary Conditions:

Necessary conditions for ( )x  to solve
0

{ ( , )}
x

Max x 

L

( , ) ( ) ( ) 0
j j j

f g
x x x

x x x
 

  
  

  

L
, with equality if 0jx  , 1,2j 

Microeconomic Theory -32- Introduction and maximization

Let ( )z g x be demand for the resource.

In Section 4 it was argued that

Demand, ( )z r declines as the price rises.

If the resource price is sufficiently high it is

more profitable to sell all of the resource.

Case (i) (0)z b

Supply exceeds demand art every price

So the market clearing price 0  .

Demand for the resource in the relaxed problem

Microeconomic Theory -33- Introduction and maximization

Case (ii): (0)z b

At the price  , demand for the resource is equal to b.

Suppose we find such a price  .

SInce x is profit-maximizing,

( ) ( ( ) ) ( ) ( ( ))f x g x b f x b g x      L

Since demand for the resource equals supply

At the price  , it follows that

( ) ( ) ( ( ))f x f x b g x   L (*)

Now consider the original problem,
0

{ ( ) | ( ) }
x

Max f x g x b


 .

For any feasible x it follows that ( ) 0b g x  . Appealing to (*),

( ) ( ) ( ( )) ( )f x f x b g x f x    L

Thus x solves the original problem.

Demand for the resource equals supply at price

Microeconomic Theory -34- Introduction and maximization

Summary: Necessary conditions for a maximum with a resource constraint, i.e.

0
{ ( ) | ( ) 0}

x
Max f x b g x


 

Consider the relaxed problem in which there is a market for the resource and the firm owns b units

of the resource. If the price of the resource is  , then profit in the relaxed problem is

( ) ( ( ) ) ( ) ( ( ))f x g x b f x b f x      L .

Since this market is a theoretical rather than an actual market we call the price a shadow price.

Suppose we find a shadow price 0  and x such that the Necessary First Order Conditions for the

relaxed problem are satisfied and in addition,

(i) ( ) 0 0b g x     (ii) 0 ( ) 0b g x     .

Then these conditions are the necessary conditions for the resource constrained problem.

Microeconomic Theory -35- Introduction and maximization

Solving for the maximum

Example 1: Output maximization with a budget constraint

31 2

1 2 3{ ( ) }
x X

Max q f x x x
 


  where { 0| }X x p x b    and 0p

Preliminary analysis

If q takes on its maximum at x , then, for any strictly increasing function ( )g q ,

( ) ( ( ))h x g f x also takes on its maximum at x .

In this case the function ( ) lng q q simplifies the problem since

1 1 2 2 3 3( ) ln ( ) ln ln lnh x f x x x x      , where
3

1j
j






The derivatives of lnq are very simple since each term has only one variable. The new problem is

0
1

{ ( ) ln | 0}j j
x

Max h x x b p x




    .

Microeconomic Theory -36- Introduction and maximization

0
1

{ ( ) ln | 0}j j
x

Max h x x b p x




   

We write down the profit in the relaxed problem in which there is a market price  for the resource.

Mathematicians call this the Lagrangian.

If the firm sells b p x  units of the resource, then the profit of the firm is

3 3

1 1

ln ( )j j j j
j j

x b p x 
 

   L

Necessary conditions for profit maximization

0
j

j j

p
x x





  



L
, with equality if 0jx  , 1,2,3j  .

Note that as 0jx  the first term on the right hand side increases without bound. Therefore the right

hand side cannot be negative. Then

0,
j

j j

p
x x





  



L
1,2,3j  . Therefore

j jp x



 , 1,2,3j 

Microeconomic Theory -37- Introduction and maximization

We have shown that

j jp x



 , 1,2,3j  (2-1)

Summing over the commodities,

3 3

1 1

1j
j j

j j

b p x


  
    , since

1j
j






Appealing to (2-1) it follows that

b
x


 , 1,2,3j 

Microeconomic Theory -38- Introduction and maximization

Example 2: Utility maximization

A consumer’s preferences are represented by a strictly increasing utility function ( )U x , where

( ) 0U x  if and only if 0x . The consumer’s budget constraint is 1 1 … n np x p x p x I     where

the price vector 0p .

The consumer chooses x to solve
0

{ ( ) | }
x

Max U x p x I


  .

Group Exercise:

(1) Explain why 0x  and p x I 

(ii) Show that the FOC can be written as follows:

1 1

… n

p p




  .

(iii) Provide the intuition behind these conditions.

Microeconomic Theory -39- Introduction and maximization

A graphical approach

Suppose x solves
0

{ ( ) | }
x

Max U x p x I


  . Define 3( ,…, )nz x x . Then

1 2( , )x x solves 1 2 1 1 2 2
0

{ ( , , ) | }
x

Max U x x z p x p x p z I


    .

Hence

1 2( , )x x solves 1 2 1 1 2 2
0

{ ( , , ) | }
x

Max U x x z p x p x I I p z


     .

We can illustrate this two variable problem

in a figure showing the 2 commodity budget

constraint and level sets of the function

1 2( , , )U x x z .

Choosing commodities 1 and 2

Microeconomic Theory -40- Introduction and maximization

The slope of the budget line is 1

p


But what is the slope of the level set?

Note that the level set implicitly defines

a function 2 1( )x x . That is

1 1 1 2( , ( ), ) ( , ,, )U x x z U x x z 

Differentiate with respect to 1x

1 1 1

1 1 2

( , ( ), ) ( ) 0
d U U

U x x z x
dx x x

 
 

  
 

Therefore the slope of the level set is

1 2

( ) /
U U

x
x x


 

 
 

Choosing commodities 1 and 2

Microeconomic Theory -41- Introduction and maximization

At the maximum the slopes are equal.

Therefore

1
1 2 1 2

2 1 2

( , , ) / ( , , )
p U U

x x z x x z
p x x

 

 

i.e.

2 1 2

( ) / ( )
p U U

x x
p x x

 

 

Exactly the same argument holds for every

pair of commodities.

Therefore

( ) / ( )i

j i j

p U U
x x

p x x

 

 

for all ,i j

Rearranging this equation,

( )( )
ji

i j

UU
xx

p p




 .

Choosing commodities 1 and 2

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