1. (9 points) Consider a firm that uses capital (K) and labor (L) to generate un
ID: 1119744 • Letter: 1
Question
1. (9 points) Consider a firm that uses capital (K) and labor (L) to generate units of output (). Output is sold in a purely competitive market at price p. The firm's production function is f(K, L)-K + L2. The input cost of capital is r and the input cost of labor isw. (a) (2 points) Write down the firm's profit equation (b) (4 points) Find the firm's profit-maximizing quantities of K and L as functions of p, r, and w. (c) (3 points) Plug these back into the profit equation to find the firm's profits as functions of p, r, and w.Explanation / Answer
Solution to a)
The firm is in perfectly competitve market. So MR = AR in this market. AR is p. The production function is
f(K,L)= K2+L2
The cost of production is found by multiplying cost of factors to production function, here C = rK2 +wL2
The profit function is the difference of cost and revenue. P = R-C ie., Profit = (P* no of units) - (rK 2+ wL2)
Solution to b
The profit will be maximised when MR = MC. Marginal cost and revenue should meet. To find the MR and MC, take first order differentials.
MR = MC
dR/dx = dC/dx
d/dx(TR) = d/dx(rK 2 +wL2)
p = 2rK + 2wL - Profits will be maximised when 2 units of K and L are used.
Solution to c)
Profits = px - rK 2 + wL2
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