[Profit maximizing input demand]
Consider a competitive firm with the following profit functionΠ = R-C = PQ-wL-rK,whereP = Price of the outputQ = OutputL = Labor inputK = Capital inputw = wage rate for labor inputr = rental rate for capital inputandQ(K,L) = 16K14 L14 .(a) Find the input quantities K∗and L∗that satisfy the first order conditions for the profitmaximization. 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 caseΠ∗∗ in each of the last three parts, i.e., (d)-(f).