Question

Consider the case of a firm that produces output x (sold at price p) using a production function x = A*l^(α)*k^(1‐α‐β)*e^β, where l is labor, k is capital, and e is energy (for example, oil or electricity).

a) What is the interpretation of A?

b) Under what condition(s) does the production function exhibit constant returns to scale? Is it homogeneous? Are the marginal products of inputs increasing, constant, or decreasing?

c) Set up the profit maximization problem for the firm.

d) Find the optimality conditions for all the inputs in the production function.

e) Find the input demand functions for labor, capital, and energy.

f) Find the optimal supply function for x.

g) Set up the cost minimization problem. Show the relevant Technical Rates of Substitution.

h) Assume that the market is populated by N consumers and M firms and that the production function is the one given above. Do you expect the market to provide the most efficient allocation? Under what conditions would the allocation be inferior to the one provided by a benevolent social planner?

Answer 1

a) A represents the total factor productivity and it captures the growth in output unexplained by the factors of production in the production function. it captures the efficiency and technology in the production of a good or service.

b) A production function exhibits Costant returns to scale if all factors change by the same factor, a and the output also changes by a factor of a.

i.e.

if all input increase by a,
fal, ak, ae
, then if , we say that function exhibits constant return to scale

here, X = l^αk^{(1-α - β)}e^β => (al)^α(ak^{(1-α - β)})(ae)^β = (a)^{α+ 1-α - β + β}l^αk^{(1-α - β)}e^β = aX

Hence, given production function exhibits constnt returns to scale

Alternatively, a cobb Douglas function exhibits constant returns to scale if sum of exponents of all factors of production is equal to 1, i.e. $\alpha + (1-\alpha -\beta ) +\beta = 1$ . therefore, the given production function exhibits Constant return to scale.

A function is said to be homogeneous of degree n if the multipli­cation of all the independent variables by the same constant, say a, results in the multiplication of the dependent variable by aⁿ.

As shown above, the given function is homogenous of degree 1.

Marginal Product of labor, MP_l = $\frac{\partial X}{\partial l} =\alpha (X/l)$

Marginal Product of Capital, MP_k = $\frac{\partial X}{\partial k} =(1-\alpha -\beta ) (X/k)$

Marginal Product of Energy, MP_e = $\frac{\partial X}{\partial e} =(\beta ) (X/e)$

As every unit of factor of protection increases, Marginal product will fall as shown by the above equation. therefore, production function exhibits decreasing marginal product of all factors.

c) Price = p

Quantity = X

Total revenue = pX

Let wage per labor = w, rental rate of capital = r and rental rate of energy = r_e

Therefore, Total cost = wl + rk + r_ee

=> Profit = pX - wl + rk + r_ee

d) Optimality condition for any input of production: Marginal Product of factor = Marginal COst of hiring that factor

therefore,

Optimal Condition for labor is $\mathbf{w =\alpha (X/l)}$

Optimal Condition for Capital is $\mathbf{r =(1- \alpha-\beta ) (X/k)}$

Optimal Condition for labor is $\mathbf{re =\beta (X/e)}$