Help with Econometrics and Business Forecasting (Economic Calculus stuff)

MILLS COLLEGE Name __________________ Economics 175/275 Math Modeling in Economics

PROBLEM SET #2 Due Wednesday September 23

Show and explain your work. (20 points)

1. Find ! and ! for the following functions

A. !

Evaluate these differentials at ! .

B. !

Evaluate these differentials at ! .

2. Find the local maxima and minima where they exist of the following functions:

A. !

B. !

C. !

D. !

3. Multiproduct firm:

Consider a firm that produces two products, steel (x) and aluminum (y) (with both quantities in

tons), under conditions of perfect competition. The given competitive prices are ! and

! . The firm’s revenue function is therefore

!

dz d2z

z= 3x2 + xy+5y3

(x,y)= (1,2)

z= 2lnx+5ln y

(x,y)= (3,1)

z= x2 + y2 + xy+10x+10y

z= 2x2 − y2 −4x+8y

z= 6x+3y− x2 − y2

z= x2 − y2

px =100 py = 200

R=100x+200y

Assume the firm’s total cost function is

!

A. Write down the firm’s profit function, which the shareholders wish to maximize.

B. Find the stationary points.

C. Check whether this is a maximum.

4. A firm has the production function

!

The price of its output is fixed at ! and the prices of labor and capital, respectively, are ! , ! , and there are no fixed costs. What quantities of capital and labor should it hire to maximize profit? Check to make sure necessary and sufficient conditions are satisfied.

5. A producer of low-capacity passenger planes has cost functions at two factories that produce the planes:

! and

!

where ! and ! are the amounts produced at factories 1 and 2, respectively, each month.

For these planes, the firm faces the inverse demand function:

! ,

where P is in $millions.

Find the firm’s profit-maximizing output to be produced at each factory and the selling price of a newly-produced plane. Check that the sufficiency conditions are met for a maximum. What is the profit-maximizing price per plane?

C = 4x2 + xy+2y2

Q= 6K1/3L1/2

P= 2 w= 3 r = 4

C1 = 3Q1 2 +2Q1 +6

C2 = 2Q2 2 +2Q2 +4

Q1 Q2

P= 74−6 Q1 +Q2( )

! 2

6. Consider the function ! and constrain ! . Find the unconstrained critical point. Find the boundary critical points. Evaluate the function at all of the critical points that satisfy the constraint and at the corner boundaries. What is the maximum and minimum of the constrained function?

7. Find the max/min of ! subject to the constraint ! .

8. Suppose a firm’s objective is to minimize costs from producing a product at two different plants. Let the total cost function be

!

where ! are the amounts of the good produce at plants 1 and 2, respectively. Let the constraint be

! .

Find the critical point satisfying the constraint and determine whether it is a max, min, or saddle point.

9. Suppose a firm produces output using labor and capital in amounts L and K, respectively, and the production function is given by

!

The firm wants to maximize its output subject to a cost constraint:

! ,

where 100=price of labor and 200=price of capital. How much labor and capital should it hire? What is the maximum output?

10. Suppose a consumer has the utility function ! . The price of good 1 is ! and the price of good 2 is ! , and the consumer’s budget to spend on the two goods is $84. What amounts of ! and ! maximize the consumer’s utility subject to the budget constraint? Illustrate with a graph of the level sets.

y= f (x1,x2)= 2x1 2 + x2

2 x1,x2 ∈ −1,3[ ]

f x1,x2( )= x12 + x22 x1 + x2 = 4

C x1,x2( )= 4x12 +6x22

x1,x2

x1 + x2 =100

Y = f K,L( )=16L1/3K2/3

100L +200K = 500,000

U = f x1,x2( )= 2x1x2 p1 = 5 p2 = 2

x1 x2

! 3