Question

Problem 3: A firm has the following production function: ?(?1, ?2 ) = ?1 + 4?2 A)  Does this firm’s technology exhibit constant, increasing, or decreasing returns to scale? B) Suppose the firm wants to produce exactly ? units and that input 1 costs $?1 per unit and input 2 costs $?2 per unit. What are the firm’s conditional input demand functions? C) Write down the formula for the firm’s total cost function as a function of ?1, ?2, and ?. D) If ?1 = 1, ?2 = 2, what is the cost minimizing choice of ?1 and ?2 for Bob to produce 100 units of output? E) If ?1 = 1, ?2 = 2, what is the minimum cost of producing one unit of output?

Problem 4: A firm has the following production function: ?(?1, ?2 ) = ?1 1/3 ?2 4/3 A) Does this firm’s technology exhibit constant, increasing, or decreasing returns to scale? B) What is the firm’s Technical Rate of Substitution? What is the optimality condition that determines the firm’s optimal level of inputs? C) Is the marginal product of input 1 increasing, constant, or decreasing in ?1. Is the marginal product of input 2 increasing, constant, or decreasing in ?2? D) Suppose the firm wants to produce exactly ? units and that input 1 costs $?1 per unit and input 2 costs $?2 per unit. What are the firm’s conditional input demand functions? (Your solution should be functions of ?1, ?2, and ? for each input). E) Using the information from part D), write down the firm’s total cost function as a function of ?1, ?2, and ?.

Problem 5: A firm has total cost function: ?(?) = 50? 2 + 40? + 30 A) What is the total fixed cost? B) What is the average fixed cost? C) What is the total variable cost? D) What is the average variable cost? E) What is the marginal cost? F) What is the average total cost? For parts G) and H), suppose the firm is in a competitive market. G) what is the lowest price at which the firm will supply a positive quantity in long-run equilibrium? H) What price maximizes the firm’s profit? I) How much would the firm supply at the price in part H) J) At what quantity is the firm’s marginal cost equal to its average cost?

Answer 1

Problem 3:

?(?1, ?2 ) = ?1 + 4?2

A) Since this production function is the perfect substitute case. Thus, technology exhibits constant returns to scale.

B) Conditional input demands to produce exactly y units :

x1 - y w1<w2

- [0,y] w1=w2

- 0 w1>w2

The firm will demand x1 only to produce exactly y units of output if the cost of x1 (w1) is less than the cost of x2(w2). If the w1 is equal to w2 then it can demand any amount of input between 0 to y to produce y units. If w1 is greater than w2 then the firm will not demand x1 produce y units.

Similarly, for x2:

x2- 0 w1<w2

- [0,y/4] w1=w2

- y/4 w1>w2

C) total cost function:

w1x1+ w2x2 = C

D) min w1x1+ w2 x2

subject to x1+4x2 = 100

?1 = 1, ?2 = 2

Since w1< w2 firm will purchase x1 to produce 100 units of y.

It will incur the cost of 100$ to purchase 100 units of x1 to produce 100 units of y.

E) To produce one unit of y, the minimum cost is $1 .