程序代写代做代考 Excel John Riley MAE Homework 1 9 October 2018

John Riley MAE Homework 1 9 October 2018

Due next Thursday before class.

Based on questions today I have made some minor changes. These are marked in red. Do not

submit your spread-sheet. I have added a question that can be quickly answered using Solver.

Just submit the results of the numerical maximization.

Note that question 3 is very similar to the question you considered in the TA session. I have

tried to organize the question so that the method of approach is clear.

1. Profit maximization

A firm has a cost function 2 2
1 2 1 2 2

( ) 40 60 (2 5 )C q q q q q q     .

The firm is a price take in output markets. The price vector is (168,372)p  .

(a) Solve for the unique critical point 0q  of the profit function.

(b) Explain why this is not the profit-maximizing output vector.

(c) Is q a local maximum?

(d) What output vector does maximize the profit of the firm.

(e) Use Solver in Excel to solve the maximization problem numerically. Then choose some

number b between 2 and 10 (not necessarily an integer and solve numerically if the cost

function is

2 2
1 2 1 2 2

( ) 40 60 (2 5 )C q q q q q bq    

How does your numerical answer change when b rises?

Note: You are not expected to do part (e) analytically. Simply submit two values of b and the

associated outputs.

2. Output maximization

A firm has production function 1

1 2

1 1
( ) ( )q F z

z z


   .

(a) Show that if inputs are scaled up by  , then output is also scaled up by  , i.e.

( ) ( )F z F z  .

(b) The input price vector is 0p  . The manager is asked to maximize output given a budget

of B dollars. Explain why the solution to this maximization problem is also the solution to the

following simpler maximization problem

John Riley MAE Homework 1 9 October 2018

0

1 2

1 1
{ | 0}

z
Max B p z

z z
    

(c) Use the Lagrange method for the simpler problem and show that
1/2

1
1 1 1/2

p
p z


 . (By an

identical argument it follows that
1/2

2
2 2 1/2

p
p z


 .)

(d) Solve for
1/2

 and hence show that
1/2 1/2 1/2

1 1 2

1

( )1 p p p

z B


(e) Solve also for
2

1

z
. Hence obtain an expression for maximized output

3. Walrasian equilibrium

Alex has an endowment (8,8)
A

  and Bev has an endowment of (7,12)
B

  . Alex likes each

unit of commodity 1 twice as much as each unit of commodity 2. Bev likes each unit of

commodity 1 four times as much as each unit of commodity 2.

(a) Explain why the optimal choice for each consumer is not affected if the price vector changes

from
1 2

( , )p p p to
1 2

( , )p p p   for any 0  . Thus we can always “normalize” by

choosing
1

j
p

  so that the price of commodity j is equal to 1.

(b) Suppose that
2

1p  . Explain why the market supply of commodity 1,
1 1
( ,1)S p is as

depicted below.

John Riley MAE Homework 1 9 October 2018

(c) For what price of commodity 1 will (i) neither (ii) one (iii) two consumers have a strictly

positive demand for commodity 1.

(d) Fully determine the market demand,
1 1
( ,1)D p for every price

1
p and depict it in a separate

figure. Explain any horizontal segments.

(e) What is the equilibrium price of commodity 1.

(f) Let ( ) ( ) ( )A B
j j j

x p x p x p  be total demand for commodity j given price vector p and let

A B

    be the total endowment vector. Write down the budget constraints and hence

show that

1 1 1 2 2 2
( ( ) ) ( ( ) ) 0p x p p x p    

Use this result to show that if supply equals demand in the market for commodity 1, then

supply must also equal demand in the market for commodity 2.

4. Equilibrium trades in an exchange economy

Alex has a utility function
1 2

( )
A a a a

U x x x and Bev has a utility function
1 2

1 1
( )

B b

b b
U x

x x
   .

(a) Explain why the slope of a level set at x is
1 2

( ) / ( )
h h

U U
x x

x x

 

 

. Why is
1 2

( ) / ( )
h h

U U
x x

x x

 

 

called the consumer’s marginal rate of substitution?

(b) Suppose both consumers have the same consumption bundle x .

Show that ( ) ( )
a b

M xRS MRS x if
2 1

x x . i.e. the consumption bundle is on the 45 line.

Who has the higher MRS at if (i)
2 1

x x (ii)
2 1

x x ?

You might try drawing the level sets of Alex and Bev through (2,1)x  in a single diagram.

(c) Suppose ( , )
a b

     so that the endowment is on the 45 line.

Explain why demand cannot be equal to supply if
1 2

p p . What if
1 2

p p ?

John Riley MAE Homework 1 9 October 2018

(d) Suppose
a b

  and the endowment is above the 45 line (more endowment of

commodity 2.)

In the Walrasian Equilibrium, which consumer will sell commodity 1?

(e) Suppose
2 1 2 1

/ /
a a b b

    . In the Walrasian Equilibrium, is it clear which consumer will sell

commodity 1?

Remark: In parts (d) and (e) you are not expected to solve for the WE price vector.