程序代写代做代考 1. Sets and Sequences

1. Sets and Sequences

Microeconomic Theory -1- WE with production

Walrasian Equilibrium with production

1. Convex sets and concave functions 2

2. Production sets 14

3. WE in a constant returns to scale economy 28

4. WE with diminishing returns to scale 35

First two sections recently edited.

Microeconomic Theory -2- WE with production

Convex sets and concave functions

Convex combination of two vectors

Consider any two vectors 0z and 1z . A weighted average of these two vectors is

0 1(1 )z z z     , 0 1 

Such averages where the weights are both strictly positive and add to 1 are called the convex

combinations of 0z and 1z .

Convex sets and concave functions

Microeconomic Theory -3- WE with production

Convex combination of two vectors

Consider any two vectors 0z and 1z . The set of weighted average of these two vectors can be

written as follows.

0 1(1 )z z z     , 0 1 

Such averages where the weighs are both strictly positive and add to 1 are called the convex

combinations of 0z and 1z .

Convex set

The set
nS  is convex if for any 0z and 1z in S,

every convex combination is also in S

A convex set

Convex combination of two vectors

Microeconomic Theory -4- WE with production

– – another view

Consider any two vectors 0z and 1z .

The set of weighted average of these

two vectors can be written as follows.

0 1(1 )z z z     , 0 1 

Rewrite the convex combination is follows:

0 1 0( )z z z z   

The vector z is a fraction 

of the way along the line joining 0z and 1z

Microeconomic Theory -5- WE with production

Concave functions of 1 variable

Definition 1: A function is concave if, for every 0x and 1x ,

the graph of the function is above the line

joining
0 0( , ( ))x f x and

1 1( , ( ))x f x , i.e.

0 1( ) (1 ) ( ) ( )f x f x f x    

for every convex combination

0 1(1 )x x x    

Note that as the distance between
1x and 0x

approaches zero, the line passing through

two blue markers becomes the tangent line.

Microeconomic Theory -6- WE with production

Tangent line is the linear approximation of the function f at 0x

0 0 0( ) ( ) ( )( )Lf x f x f x x x   .

Note that the linear approximation has the same value at
0x and the same first derivative (the slope.)

In the figure ( )Lf x is a line tangent to the graph of the function.

Definition 2: Differentiable concave function

A differentiable function is concave if every tangent line is above the graph of the function. i.e.,

0 0 1 0( ) ( ) ( )( )f x f x f x x x  

Microeconomic Theory -7- WE with production

Definition 3: Concave Function

A differentiable function f defined on an interval X is concave if ( )f x , the derivative of ( )f x is

decreasing.

The three types of differentiable concave function are depicted below.

Note that in each case the linear approximations at any point
0x lie above the graph of the function.

Microeconomic Theory -8- WE with production

Concave function of n variables

Definition 1: A function is concave if, for every 0x and 1x ,

0 1( ) (1 ) ( ) ( )f x f x f x     for every convex combination 0 1(1 )x x x     , 0 1 

(Exactly the same as the definition when 1n )

Microeconomic Theory -9- WE with production

Linear approximation of the function f at 0x

0 0 0

( ) ( ) ( )( )
n

L j j
j j

f
f x f x x x x

x


  


 .

Note that for each jx the linear approximation has the same value at
0x and the same first derivative

(the slope.)

Definition 2: Differentiable Concave function

For any
0x and 1x

1 0 0 0

( ) ( ) ( )( )
n

j j
j j

f
f x f x x x x

x


  




Microeconomic Theory -10- WE with production

Group exercise: Appeal to one of these definitions to prove the first of the following important

propositions.

Proposition

If ( )f x is concave, and x satisfies the necessary conditions for the maximization problem

0
{ ( )}

x
Max f x


then x solves the maximization problem.

Proposition

If ( )f x and ( )h x are concave, and x satisfies the necessary conditions for the maximization problem

0
{ ( ) | ( ) 0}

x
Max f x h x




then x is a solution of the maximization problem

Remark: This result continues to hold if there are multiple constraints ( ) 0ih x  and each function

( )ih x is concave.

Microeconomic Theory -11- WE with production

Concave functions of n variables

Proposition

1. The sum of concave functions is concave

2. If f is linear (i.e. 0( )f x a b x   ) and g is concave then ( ) ( ( ))h x g f x is concave.

3. An increasing concave function of a concave function is concave.

4. If ( )f x is homogeneous of degree 1 (i.e. ( ) ( )f x f x  for all 0  )

and for some increasing function ( )g  , ( ) ( ( ))h x g f x is concave, then ( )f x is concave.

5. If ( )f x is homogeneous of degree 1 (i.e. ( ) ( )f x f x  for all 0  ) and the superlevel sets of

( )f x are convex, then ( )f x is concave.

Remark: The proof of 1-3 follows directly from the definition of a concave function. The proofs of 4

and 5 are very similar and more subtle.

Microeconomic Theory -12- WE with production

Group exercise: Prove that the sum of concave functions is concave.

Group Exercise: Prove the following result

Proposition: Concave functions have convex superlevel sets

If ( )f x is a concave function then the superlevel sets of ( )f x are convex sets. i.e.,

If
0 1,x x are in the superlevel set { | ( ) }S x f x k  then every convex combination is in S .

Group Exercise: Output maximization with a fixed budget

A plant has the CES production function

1/2 1/2 2
1 2( ) ( )F z z z  .

The CEO gives the plant manager a budget B and instructs her to maximize output. The input price

vector is 1 2( , )r r r . Solve for the maximum output ( , )q r B .

Class Exercise: What is the firm’s cost function

Microeconomic Theory -13- WE with production

2. Production sets and returns to scale (first 3 pages are a review)

Feasible plan

If an input-output vector ( , )z q where 1( ,…., )mz z z and 1( ,…, )nq q q is a feasible plan if q can be

produced using z .

Production set

The set of all feasible plans is called the firm’s production set.

Microeconomic Theory -14- WE with production

Production sets

Feasible plan

If an input-output vector ( , )z q where 1( ,…., )mz z z and 1( ,…, )nq q q is a feasible plan if q can be

produced using z .

Production set

The set of all feasible plans is called the firm’s production set.

Production function

If a firm produces one commodity the maximum output for some input vector z ,

( )q G z

is called the firm the firm’s production function

Microeconomic Theory -15- WE with production

Production sets

Feasible plan

If an input-output vector ( , )z q where 1( ,…., )nz z z and 1( ,…, )nq q q is a feasible plan if q can be

produced using z .

Production set

The set of all feasible plans is called the firm’s production set.

Production function

If a firm produces one commodity the maximum output for some input vector z ,

( )q G z

is called the firm the firm’s production function

Example 1: One output and one input

{( , )}| 0 2 }f f f ffS z q q z  

Microeconomic Theory -16- WE with production

Example 1: One output and one input

{( , )}| 0 2 }f f f ffS z q q z  

Note that the production function

( ) 2f f f fq G z z 

Is homogeneous of degree one

( ) ( )f f f fG z G z  .

Such a firm is said to exhibit constant returns to scale

Microeconomic Theory -17- WE with production

Example 1: One output and one input

{( , )}| 0 2 }f f f ffS z q q z  

Note that the production function

( ) 2f f f fq G z z 

Is homogeneous of degree one

( ) ( )f f f fG z G z  .

Such a firm is said to exhibit constant returns to scale

Example 2: One output and one input

2{( , ) 0| }f f f ffS z q q z  

Equivalently

1/2{( , ) 0| ( ) }f f f ffS z q q z  

Group Exercise: Show that
fS is convex

Microeconomic Theory -18- WE with production

Example 3: two inputs and one output

1/3 2/3
1 2{( , ) 0| ( , ) ( ) ( ) 0}

f fS z q h z q A z z q    

Note that the production function is concave (why?)

Hence ( , )h z q is concave. (why)

Remark: We have proved that the superlevel set of a concave function are convex so fS is a convex

set

Example 4: one input and two outputs

2 2 2
1 1 2 2{( , ) 0| ( , ) 0}

f fS z q h z q z a q a q     

Equivalently

2 2 1/2
1 1 2 2{( , ) 0| ( , ) ( ) 0}

f fS z q h z q z a q a q     

Class Exercise: Explain why fS is a convex set

Microeconomic Theory -19- WE with production

Aggregate production set

Let
1{ }

f F
fS  be the production sets of the F firms in the economy.

The aggregate production set is

1 … FS S S  

That is

( , )z q S if there exist feasible plans
1{( , )}

f f F
fz q  such that

( , ) ( , )
F

f f

z q z q


 .

***

Microeconomic Theory -20- WE with production

Aggregate production set

Let
1{ }

f F
fS  be the production sets of the F firms in the economy.

The aggregate production set is

1 … FS S S  

That is

( , )z q S if there exist feasible plans
1{( , )}

f f F
fz q  such that

( , ) ( , )
F

f f

z q z q


 .

Example 1: {( , ) 0| 2 0}
f f f f fS z q z q   

Exercise: Prove this using the methods from Example 2.

Microeconomic Theory -21- WE with production

Example 2:
2{( , ) | ( ) }f f f f fS z q q z 

(a) Show that with two firms the aggregate production set is
2{( , ) | 2 }S z q q z 

(b) What is the industry production set if there are 4 firms?

Group Exercise

HINT: The maximum output of the two firms is

1 2 1 2 1 2 2 2 1 2{ | ( ) , ( ) , }q Max q q q z q z z z z      .

Rather than use the Lagrange method with 3 constraints, note that for any 1z and 2z output is

maximized by choosing 1q and 2q so that the first two inequalities are binding.

Method 1: Then
2( )f fz q and so the problem is reduced to a one constraint problem.

1 2

1 2 1 2 2 2

,
{ | ( ) ( ) }

q q
q Max q q q q z   

Method 2: Since
1/2( )f fq z it follows that maximized output is

1 2

1 2 1 1/2 2 1/2 1 2

,
{ ( ) ( ) | }

q q
q Max q q z z z z z     

Remark: You might switch to subscripts to avoid confusion.

Microeconomic Theory -22- WE with production

Aggregation Theorem for price taking firms

Proposition: If there are 2 firms in an industry, prices are fixed and ( , )
f fz q is profit maximizing for

firm , 1,2f f  then 1 2 1 2( , ) ( , )z q z z q q   is industry profit-maximizing.

Microeconomic Theory -23- WE with production

Aggregation Theorem for price taking firms

Proposition: If there are 2 firms in an industry, prices are fixed and ( , )
f fz q is profit maximizing for

firm , 1,2f f  then 1 2 1 2( , ) ( , )z q z z q q   is industry profit-maximizing.

Proof: Let f be maximized profit of firm f Since the industry can mimic the two firms, industry

profit cannot be lower. Suppose it is higher. Then for some feasible ˆˆ( , )
f fz q , 1,2f  ,

1 2 1 2 1 2ˆ ˆ ˆ ˆ( ) ( )p q q r z z       .

Microeconomic Theory -24- WE with production

Aggregation Theorem for price taking firms

Proposition: If there are 2 firms in an industry, prices are fixed and ( , )
f fz q is profit maximizing for

firm , 1,2f f  then 1 2 1 2( , ) ( , )z q z z q q   is industry profit-maximizing.

Proof: Let f be maximized profit of firm f Since the industry can mimic the two firms, industry

profit cannot be lower. Suppose it is higher. Then for some feasible ˆˆ( , )
f fz q , 1,2f  ,

1 2 1 2 1 2ˆ ˆ ˆ ˆ( ) ( )p q q r z z       .

Rearranging the terms,

1 1 1 2 2 2ˆ ˆˆ ˆ( ) ( ) 0p q r z p q r z         

Then either

1 1 1 2 2 2ˆ ˆˆ ˆ or p q r z p q r z       

But then
1 1( , )z q and

1 1( , )z q cannot both be profit-maximizing.

QED

Remark: Arguing in this way we can aggregate to the entire economy.

Microeconomic Theory -25- WE with production

7. Walrasian equilibrium (WE) with Identical homothetic preferences & constant returns to scale

Consumer h has utility function 1 2 1 2( , )
h h h hU x x x x . The aggregate endowment is ( ,1)a . All firms

have the same linear technology. Firm f can produce 2 units of commodity 2 for every unit of

commodity 1. That is the production function of firm f is 2
f fq z

Then the aggregate production function is 2q z .

Microeconomic Theory -26- WE with production

Walrasian equilibrium (WE) with Identical homothetic preferences and constant returns to scale

Consumer h has utility function 1 2 1 2( , )
h h h hU x x x x . The aggregate endowment is ( ,1)a . All firms

have the same linear technology. Firm f can produce 2 units of commodity 2 for every unit of

commodity 1. That is the production function of firm f is 2
f fq z

Then the aggregate production function is 2q z .

Aggregate production set

If the industry purchases z units of commodity 1

it can produce 2q z units of commodity 2.

Then total supply of each commodity is

( ,1 2 )a z z  .

This is depicted opposite.

Microeconomic Theory -27- WE with production

Maximizing the utility of the representative consumer

Simply solve for the utility maximizing point

In the aggregate production set.

Slope = -2

Microeconomic Theory -28- WE with production

Walrasian Equilibrium

First consider profit maximization

The profit of firm f is

2 2 1 1 2 1 1 1 1 2 12 (2 )
f f f f f fp q p z p z p z z p p       .

Microeconomic Theory -29- WE with production

Walrasian Equilibrium

First consider profit maximization

The profit of firm f is

2 2 1 1 2 1 1 1 1 2 12 (2 )
f f f f f fp q p z p z p z z p p       .

Case (i) 1

2
p

p
 : the profit maximizing firm will purchase no inputs and so produce no output.

Case (ii) 1

2
p

p
 : No profit maximizing plan

Case (iii) 1

2
p

p
 : any input-output vector 1 2 1 1( , ) ( ,2 )z q z z is profit maximizing.

Note that in case (iii) the profit is zero.

Thus the is a WE with production the price ratio is 1

2
p

p
 and maximized profit is zero.

Microeconomic Theory -30- WE with production

WE with no production

When is this the case?

If so the representative consumer

does not trade

FOC

1 2

( ) ( )
U U

x x

p p

 
 

 


Therefore

1 1 2

2 1

( )
1

( )

p x

Up a
x









  




slope = -2

Microeconomic Theory -31- WE with production

There are two cases. Case (i)
1

2
a  Case (ii):

2
a 

The indifference curve through  is depicted below.

In the second case the no trade price ratio 1

p a
 exceeds 2. As we have seen, production is not

profitable at such a price ratio. Thus the right hand diagram depicts a WE.

Case (ii)

Slope = -2

Case (i)

Microeconomic Theory -32- WE with production

Case (i)

Demand

Consider the representative consumer.

The optimal consumption is depicted.

We know from the analysis of the

Firm that the price ratio is

2
p

p


Also the profit is zero (no dividends)

Therefore the boundary of the feasible set

of outcomes is the budget line of the

representative consumer.

p x p    .

The representative consumer’s best choice is then *x

Slope = -2

Microeconomic Theory -33- WE with production

Second example:

One output and one input

1/2{( , ) 0| ( ) }f f f f f fS z q q a z  

There are two firms
1 2( , ) (3,4)a a 

The aggregate endowment is (12,0)

Consumer preferences are as in the

previous example.

Exercise

(a) Show that the aggregate production set

Can be written as follows:

1/2{( , ) 0| 5 }S z q q z  

(b) What is the best consumption vector *x

Microeconomic Theory -34- WE with production

Profit maximizing firm

Note that 1 1z x  so the level set for maximized profit can be rewritten as follows:

2 1 2 2 1 2 2 1 1 1 1 1( )p q p z p x p z p x p x p x p          

Rearranging terms, the maximum profit level set is

1 1p x p  

Microeconomic Theory -35- WE with production

Note that the maximum profit level set

1 1p x p  

Is also the consumer’s budget set.

Adding the indifference curves, for any

price vector we can solve for the demands.

For the price vector shown there is

Excess demand for commodity 2 and

So excess supply of commodity 1.

Prices adjust.

Microeconomic Theory -36- WE with production

Walrasian Equilibrium

Class Exercise:

For the following economy,

solve for the WE allocation and prices.

Aggregate production set

1/2

1 2 2 1{( , ) 0 | 5 }S z x x z  

Utility function

1 2 1 2( , )
h h h hU x x x x

Homothetic hence we consider the

Representative consumer

1 2 1 2( , )U x x x x

Aggregate endowment (12,0) .

Microeconomic Theory -37- WE with production

Answer to exercise

Example 1: {( , ) 0| 2 0}
f f f f fS z q z q   

Exercise: Prove this using the methods from Example 2.

Proof:

With firm inputs 1z and 2z , maximized outputs of the two firms are
1 12q z and

2 22q z . Therefore

maximized to total output is
1 22( )q z z  . If the total input available is z then 1 2z z z  and so

1 22( ) 2q z z z   .

Therefore the aggregate production set is {( , ) | 2 0}S z q z q  

Microeconomic Theory -38- WE with production

Answer to exercise:

One output and one input

1/2{( , ) 0| ( ) }f f f f f fS z q q a z  

There are two firms
1 2( , ) (3,4)a a 

(a) Show that the aggregate production set

can be written as follows:

1/2{( , ) 0| 5 }S z q q z  

If the allocation of the input to firm 1 is 1z , then maximized output is
1 1 1/23( )q z . Similarly

2 2 1/24( )q z and so

1 2 1 1/2 2 1/23( ) 4( )q q z z  

Maximized industry output is therefore

1 2 1 1/2 2 1/2 1 2{ 3( ) 4( ) | }q Max q q z z z z z     

The problem is concave so the necessary condition are sufficient. We look for a solution with

1 2( , ) 0z z  . The Lagrangian is

Microeconomic Theory -39- WE with production

1 1/2 2 1/2 1 23( ) 4( ) ( )L z z z z z    

FOC

1 1/2
1

3
( ) 0

L
z

q



  


, 1 1/2

4
( ) 0

L
z

q



  



Therefore

1 1/2 2 1/2

3 4

( ) ( )z z


Squaring and appealing to the Ratio Rule,

1 2 1 2

9 16 25 25

z z z z z
  



Therefore

1 9

25
z z and

2 16

25
z z and so

1 1 1/2 1/293( )
5

q z z  and
2 2 1/2 1/2164( )

5
q z z  .

So
1 2 1/25q q q z  

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