Question

Conditional/Unconditional demand for an input factor A firm produces an output using production function Q = F(L, K):= L1/2K1/3. The price of the output is $3, and the input factors are priced at pL 1 and pK-6 (a) Find the cost function (as a function of output Q). Then find the optimal amount of inputs i.e., L and K) to maximize the profit (b) Suppose w changes. F'ind the conditional labor deand funtionL.Px G) whene function L(PL.PK for Q is the output in part (a). (c) Find the unconditional labor demand function L(PL.PK 6) (d) Draw the conditional and the unconditional demand functions in one graph

Answer 1

1) Cost = p_L * L + p_K *K = 1L+6K

Profit = PQ - Cost = 3L^1/2K^1/3 - L -6K

for maximum profit dP/dL = 0 & dP/dK = 0

dP/dL = 1.5K^1/3 / L^1/2 -1 =0 => (3/2)K^1/3 = L^1/2

dP/dK = L^1/2 / K^2/3 -6 = 0 => L^1/2 = 6K^2/3

solving both:

6K^2/3 = (3/2)K^1/3

K^1/3 = 1/4

=> K = 1/64

=> L = 36K^4/3 = 36(1/64)^4/3 = 36/256 = 9/64

Q* = L^1/2K^1/3 = 3/8 * 1/4 = 3/32

2. Q=Q*(output at profit maximizing values of input)

MRTS = (MP_L / MP_K) = (0.5* K^1/3 / L^1/2 ) / (L^1/2 / 3K^2/3) = 1.5K/L

this should be equal to w/r = p_L / p_K = 6/6 =1

=> 1.5K/L = 1

=> L/1.5 = K

putting in production function:

Q = L^1/2K^1/3

Q = L^1/2L^1/3 / (1.5)^1/3 = L^5/6 / (1.5)^1/3

L^5/6 = (Q)*(1.5)^1/3

L = 1.176* Q^6/5 [ unconditional Labor Demand function] (part C)

as Q = Q* ( profit maximal output) = 3/32

L = 1.176* (3/32)^6/5

L = 0.06867 [ conditional labor demand function]

d) Unconditional functional will be upward sloping monotonically increasing function while conditional functional will be a straight horizontal line.