Question: The firm's production function for gadgets, X, depends on the amounts of capital (K), and labor (L) employed:
X = 56K + 8.8L - 2K2 - 0.1L2.
(a) If the firm can sell any output at a price equal to 10, if the wage rate per unit of labor is 8 and the rental per unit of capital is 80, what is the profit maximizing level of output?
(b) At the optimum solution in (a), what is the value of the marginal rate of technical substitution in production?
(c) If the firm maximizes profit at a higher wage rate of 10 (all other parameters unchanged), what is the firm's elasticity of demand for labor over this wage range?