The profit-maximizing firm's demand for labor, L, and capital, K, is contributed to by the wage paid to labor, w, the rental rate of capital, c, the price of the output, P, and the level of output, Q, which is a function of labor and capital. Specifically:

Q(L, K) = L K

P(Q) = a bQ

TC(L, K) = wL + cK

Where a > 0, b > 0, w > 0, and c > 0

(a) from the firm's profit function, derive the firm's demand for labor

(b) derive the elasticity of labor with respect to w

(c) and the cross-price elasticity of labor with respect to c

A firm mandates that every executive-level employee will now donate $500 to local charities. Demonstrate what you expect to happen using the graph of consumption-leisure below.

*The y-axis in terms of ($) and the x-axis in terms of (leisure)