Please show all work Suppose we have the following production function q = f(K,L

Question

Please show all work

Suppose we have the following production function q = f(K,L) = L^1/2 +K^1/2 Where q=output, L= labor and K=capital The profit function for a competitive firm is Profit = Pq - wL - rK Where P=market price, w= wage rate and r=rental rate. Solve the first-order conditions for K and L. Find the second-order sufficient condition and create the Hessian Matrix. Do we have profit maximization? If P= 1000, w=20 and r= 10, find the optimal K, L and Profit. Also check the Second Order Conditions.

Explanation / Answer

Profit = P*q - wL -rk

Putting q = L^1/2 + k^1/2

Profit = P*(L^1/2 + k^1/2) - wL - rk

dProfit/dL = 1/2(P/L^1/2) - w

dProfit/dk = 1/2(P/k^1/2) - r

For first order condition

Putting dprofit/dl = 0 , dprofit/dk = 0

dProfit/dL = 1/2(P/L^1/2) - w = 0

dProfit/dk = 1/2(P/k^1/2) - r = 0

P = 2*w*L^1/2

P = 2*r*k^1/2

P = 2*w*L^1/2 = 2*r*k^1/2

w/r = k^1/2/L^1/2

For Second order Condition

FLL = d2Profit/dL2 = -1/4(P/L^3/2)

FLk = d2profit/dLdK = 0

Fkk = d2Profit/dK2 = -1/4(P/k^3/2)

FkL = d2Profit/dkdL = 0

Hessian Matrix = -1/4(P/L^3/2) 0

0 -1/4(P/K^3/2)

iii.    w/r = k^1/2/L^1/2

20/10 =  k^1/2/L^1/2

400/100 = K/L

400*L = 100*k

P = 2*w*L^1/2

1000 = 2*w*L^1/2

1000 = 2*20*L^1/2

L = 10000//16

K = 10000/4

Profits =

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