Question: Consider a competitive firm with the following profit function
Π = R-C = PQ-wL-rK
where P = Price of the output
Q = Output
L = Labor input
K = Capital input
w = wage rate for labor input
r = rental rate for capital input
and Q(K,L) = 16K^1/4 L^1/4
(a) Find the input quantities K*and L*that satisfy the first order conditions for the profit maximization. Assume the parameter values P = 4, w = 1, and r = 1.
(b) Compute the profit maximizing output Q*.
(c) Compute the maximum profit Π*.
(d) If the wage rate w changes to w = 2, determine the new L**.
(e) If the rental rate r changes to r = 2, determine the new L**.
(f) If the output price P changes to P = 16, determine the new L**.Hint: All other parameters remain as in the original case in last three parts.
(g) Also compute the profit maximizing output Q** and the maximum profit in each case Π** in each of the last three parts, i.e., (d)-(f).