Question

Competitive Firms - Optimal Labor and Capital

1. A competitive firm has production technology q A K La. lt can sell output at price P and hire capital and labor at competitive factor prices r and w. Write down the firm's profit maximization problem. What are the firm's first- order necessary conditions for a maximum. a. w Suppose now that α-,P-1 and -1. b. What is the firm's optimum capital labor-intensity? Why can't the optimum scale of production q be determined for this constant returns to scale technology? (hint: competitive firms also set P MC(Q)) c.

Answer 1

The production technology is basically the productino function, which is, q = A.K^{a}L^{1-a} . Capital (K) and labour (L) at competitive factor prices are r and w. The price is P.

         (a) The firm's total revenue is Pq = P[A.K^{a}L^{1-a}] , and the total cost of producing the output is rK+wL . The firm's profit will be total revenue minus total cost, ie \pi = P[A.K^{a}L^{1-a}] - (rK + wL). But, since the funtion is in the form of K and L as independent variable, the profit maximization problem will be the amount of K and L, where \pi is maximum, ie \underset{K,L}{max} [\pi] = \underset{K,L}{max}[P[A.K^{a}L^{1-a}] - (rK + wL)].

The first order condition will be attained by applying the method of Lagrange Polynomial and partial differentiation with respect to K and L (the independent variables). The first order condition is hence,

\frac{\partial }{\partial L}(P[A.K^{a}L^{1-a}]) = w and \frac{\partial }{\partial K}(P[A.K^{a}L^{1-a}]) = r

         (b) Solving the first equation, we have, (1-a)P[A.K^{a}L^{1-a-1}] = w, or, (1-a)P[A.K^{a}L^{-a}] = w,

or, (1-a)P[A.(\frac{K}{L})^{a}] = w,or, (\frac{K}{L})^{a} = \frac{w}{(1-a)P.A}

Solving the second equation, we have, (a)P[A.K^{a-1}L^{1-a}] = r, or, (a)P[A.K^{-(1-a)}L^{1-a}] = r,

or, (a)P[A.(\frac{L}{K})^{1-a}] = r,or, (\frac{L}{K})^{1-a} = \frac{r}{a.P.A}

Now, dividing equation (\frac{L}{K})^{1-a} = \frac{r}{a.P.A}, by equation (\frac{K}{L})^{a} = \frac{w}{(1-a)P.A}, we will have on the LHS,\frac{(\frac{L}{K})^{1-a}}{(\frac{K}{L})^{a}} = \frac{(\frac{K}{L})^{-(1-a)}}{(\frac{K}{L})^{a}} = \frac{(\frac{K}{L})^{a-1}}{(\frac{K}{L})^{a}} = \frac{(\frac{K}{L})^{a-1}}{(\frac{K}{L})^{a}} = \frac{(\frac{K}{L})^{a}(\frac{K}{L})^{-1}}{(\frac{K}{L})^{a}} = (\frac{K}{L})^{-1},

while on the RHS, we will have \frac{\frac{r}{a.P.A}}{\frac{w}{(1-a)P.A}} = \frac {r}{w} . \frac{(1-a).P.A}{a.P.A} = \frac {r}{w}.\frac{1-a}{a}.

Equating the LHS with RHS, we have (\frac{K}{L})^{-1} = \frac {r}{w}.\frac{1-a}{a},or,\frac{K}{L} = \frac {w}{r}.\frac{a}{1-a}.

It is the firm's optimal capital labour intensity, in order to maximize profit. Since given, a=\frac{1}{2}, and, \frac{w}{r} = 1, we have the optimal capital labour intensity is \frac{K}{L} = 1.\frac{1/2}{1-1/2} = \frac{1/2}{1/2} = 1,

As the capital labour factor prices are equal, the optimal capital labout intensity according to the given problem is 1, ie. for every unit increase in labour, capital should be increased at the same unit.

        (c) The optimum scale of production 'q' can not be determined as if we put K=L (or the opposite, ie L equals to K) in the profit maximization, we will have a profit function equal to function in terms of K and r (or L and w in the opposite case). The profit maximization problem basically equates different isoquants (q) to a constant isocost lines (rK + wL) , and we do not have a specific isocost line. For that, we need the values of given costs r and w, along with a given cost 'c', for c = (rK + wL) .