Question

Apex Inc. produces gadgets. Its production function is ? = min(4?, ? − 1), for ? ≥ 0 and ? ≥ 1, where ? denotes units of capital input, ? denotes units of labor input, and ? denotes units of output. (Note that ? = 0 for all ? < 1.) The price of a unit of capital input is ? and the price of a unit of labor input is ?.

a. Find Apex’s ??? and ??? functions.

b. Find Apex’s ??? function.

c. Suppose that ? = 8 in the short run. Find Apex’s short run (contingent) demand function for labor input. (Hint: This will be a function of the target output level, ?.)

d. Suppose that ? = 8 in the short run. Find Apex’s short run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

e. Find Apex’s long run (contingent) demand functions for labor input and capital input. (Hint: These will be functions of the target output level, ?, but not the input prices, ? and ?. Why?)

f. Find Apex’s long run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

Answer 1

a) To produce at any level , inputs are needed in (4k , l-1) note that inputs are complementary.

q = min(4k, l-1) for 1 unit of q , min (4k, l-1) = 1

4k = l-1 = 1 , we get k= 1/4 , l-1 = 1 or l= 2 is needed .

If l-1 > 4k, q = min (4k, l-1) = 4k , the additional labour, adds nothing to the production pricess i.e., MP_{​​​​​l}​​​​ = 0 , but MP_{​​​​​k} = q/k = 4 .

Similarly when l-1 < 4k , q = min( 4k , l-1 ) = l-1

MP_{​​​​​L}​ = q/l = 1 , MP_{​​​​​k} = 0

When 4k = l-1 ,MP_{​​​​​k} = 4 and MP_{​​​​​l}​= 1

b )

When 4k> l-1 , RTS = MP_{​​​​​k} / MP_{​​​​​l}​​​​ = 0

when 4k<l-1 , RTS = infinity

When 4k = l-1 , RTS = 4

c) When k = 8 ,

q = min (4*8 , l-1) = min ( 32, l-1 )

At optimal level of production, 4k = l-1 = q = 32

l-1 = 32 , or l= 33

33 labour units are demanded

d) l-1 = q or l= q+1 , in short run k = 8

Cost = wl + rk = (q+1)l + 8r is the short run cost function .

e) input demand function is

q = 4k , of k = q/4

Similarly l -1 = q of l= q+1

f) Total cost function in long run C(q,w,r) = wl + rk = w(q+1) + rq/4 = q( w+ r/4) + w