Homogeneous production and average cost

I have discussed before that constant returns to scale (CRS) function with a single factor is of the form F(x) = cx where c is a constant. In this post, I wanted to show that if the two−factor production function has CRS, then the average cost is independent of the output level. Then I decided that such a result is too boring to deserve a post, so I am going to discuss the relationship between homogeneity of production function and average cost in full generality, following Sandmo (1970).

A function F: Rⁿ → R is homogeneous of degree k if a^kF(x) = F(ax) for all a > 0 and x in Rⁿ. A production function has constant returns to scale if it is homogeneous of degree 1. It has increasing returns to scale if homogeneous of degree k > 1, and decreasing returns to scale if homogeneous of degree k < 1. These properties can be defined locally (that is, find k as a function of x) by taking the derivative of the identity equation above with respect to a and then solving for k (assuming F is differentiable): ka^k−1F(x) = Σ F_ix_i and since a^k−1F(x) = F(ax)/a, we get k = aΣ (F_ix_i/F(ax)). Evaluating at a = 1, k = Σ (F_ix_i/F(x)).

Assume perfect competition in factor markets so that the factor prices are exogenously determined. Let r_i be the price of factor i and let r be the vector of prices. The total cost is then Σ r_ix_i. For fixed output level Q, the firm is cost minimizing with regard to x subject to its production function F(x) = Q. The Lagrangian is L = Σ r_ix_i − λ(F(x) − Q) and the first order conditions are r_i − λF_i = 0 for i = 1,..., n and F(x) − Q = 0. These n+1 equations yield the optimal x* as a function of r and Q and we can find the optimal cost C as a function of r and Q: C(r,Q) = Σ r_ix_i*.

While we are talking about cost function, let's mention that the Lagrange multiplier in cost minimization problem can be interpreted as the marginal cost. The derivative of this minimum cost function with regard to Q then is the marginal cost: dC/dQ = Σ r_i(dx_i*/dQ). By the first order conditions, r_i = λF_i. Totally differentiating F(x) = Q condition, we get Σ F_i(dx_i*/dQ) = 1. Thus, λ = dC/dQ.

Going back to the average cost problem, let A(r,Q) = C(r,Q)/Q be the average cost. To show the behavior of average cost as Q varies, take dA/dQ = Q⁻²(dC/dQ × Q − C). Thus the direction of average cost in Q depends on the sign of dC/dQ × Q − C or dC/dQ − C/Q. We showed C = Σ r_ix_i = λΣ F_ix_i and λ = dC/dQ. By the constraint, Q = F(x). Thus dC/dQ − C/Q = λ(1 − Σ (F_ix_i/F(x))). But we have also shown that k = Σ (F_ix_i/F(x)). Therefore, the average cost is increasing in Q if k > 1 (decreasing returns to scale), decreasing in Q if k < 1 (increasing returns to scale), and constant in Q if k = 1 (CRS).