Question

Given a perfectly competitive firm in the input and output markets where: P0= exogenous price, Q = f(L, K0) where dQ/dL > 0 and d2Q/dL2< 0, the cost function where: C(L, K0) = r0K0+ w0L; r0= exogenous rental rate of capital, K0= exogenous capital stock, and w0= exogenous wage.

a)State the firm’s profit function in terms of L.

b)Find the F.O.C. that maximizes profit at L*.

c)Interpret the F.O.C.

d)Find the S.O.C. that maximizes profit at L*.

e)Interpret the S.O.C.

f)Find dL*/dP0

g)Interpret the derivative in (f).

Answer 1

A). P0,K0,w0,r0 is exogenous variable then profit function in terms of L will be,

Profit (u)= total revenue - total cost

u(L)= p0*f(L,K0) -( r0K0 + w0L)

B). The firm profit will be maximum at where first degree derivative of profit function with respect to labour is zero and the second degree derivative of profit function is negitive.

First order condition:

du/dL=p0*df/dL-w0(first degree derivative)

du/dL=0

p0*df/dL-w0=0

p0*df/dL=w0.

C). The first order condition tells that firms profit would be at that quantity where value of marginal product of additional unit of labour is equal to additional cost of unit labour or wage.

D). The second order condition is second degree derivative of profit function should be negitive.

the second order condition:

d^2u/dL^w=p0*d^2f/dL^2

p0*d^2f/dL<0

E).the second order condition tells that after profit maximization quantity , hiring additional unit of labour cost should be more than its value of marginal product.

F).L*=(p0f-u-r0K0)/w

L* = equilibrium quantity of labour

dL*/dp0=f(L,K0)/w

G).The above derivative tells us that given wage , interest, quantity of capital unchanged, change in one additional unit of market price of good will result in change in equilibrium labour quantity by output-wage ratio.