Question

Consider a firm with a production function Q(L,K) = √L + 2√K, that faces an inverse demand function P (Q) = 250 - 2Q, and the labor and capital markets are also in perfect competition.

A) Derive the expression for the profit function of the firm in terms of labor and capital and express the profit maximization problem the firm faces.

B) Derive the first-order conditions, and use them to get the expressions for the optimal amounts of labor and capital, L* and K*.

C) Using the expressions for L* and K*, get the expressions for Q*, P*, R* (optimal revenue), C* , and π*.

D) Derive the expressions for the sufficient second order conditions for a maximum. Are they always satisfied, or only for a certain range of values for L and K?

Answer 1

A)let w=wage  

r= rental rate of capital

TC=L*w + r*k

TR=P*Q=(250-2Q)*Q=250Q-2*Q^2

TR=250*(√L+2√K)-2*(L+4K+4√KL)=250√L+500√K-2L-8K-8√KL

Profit=TR -TC=250√L+500√K-2L-8K-8√KL -Lw-rk

Max profit=250√L+500√K-2L-8K-8√KL -Lw-rk

B)∆Profit/∆L=150/√L-2-4√K/√L-w=0

(150-4√K)/√L=2+w

(150-4√K)=(2+w)√L

L*=[(150-4√k)/(2+w)]^2

∆profit/∆K=250/√K-8-4√L/√K-r=0

K*=[(250-4√L)/(8+r)]^2

C)Q*=[(150-4√K)/(2+w)]+2*[(250-4√L)/(8+r)]

P*=250-[(15/-4√K)/(2+w)]-2[(250-4√L)/(8+r)]

TR*=250*[(150-4√K)/(2+w)]+2*[(250-4√L)/(8+r)]-[(150-4√K)/(2+w)]+2*[(250-4√L)/(8+r)]^2

C*=w*[(150-4√k)/(2+w)]^2+r*[(250-4√L)/(8+r)]^2

Profit=250*[(150-4√K)/(2+w)]+2*[(250-4√L)/(8+r)]-{[(150-4√K)/(2+w)]+2*[(250-4√L)/(8+r)]}^2. - w*[(150-4√k)/(2+w)]^2+r*[(250-4√L)/(8+r)]^2

D)second order condition, is second degree DERIVATIVE should be negitive.

∆profit2/∆L2=75/L^3+2√K/L^3=(-75+2√k/(L^3)

∆profit 2/∆K2=-125/K^3+2√L/K^3=(-125+2√L)/k^3

So they doesn't always satisfied .

As you can see K=1406.25,

Second degree DERIVATIVE with respect to L is zero.

L=3906.25,. Second degree DERIVATIVE with respect to K is zero.

This only satisfy L<3906.25 and K<1406.25