The Properties Of Cost And Profit Functions

The properties of cost function \We have shown earlier that cost functions have certain properties that follow from the structure of the cost minimization problem; we have shown above that the conditional factor demand functions are simply the derivatives of the cost functions. Hence, the properties we have found concerning the cost function will translate into certain restrictions on its derivatives, the factor demand functions. These restrictions will be the same sort of restrictions we found earlier using other methods, but their development using the cost function is quite nice. 1) The cost function is nondecreasing in factor prices. It follows from this that 2) The cost function is homogeneous of degree 1 in w. Therefore, the derivatives of the cost function, the factor demands, are homogeneous of degree 0 in w. (See Chapter 26, page 482). 3) The cost function is concave in w. Therefore, the matrix of second

derivatives of the cost function-the matrix of first derivatives of the factor demand functions-is a symmetric negative semidefinite matrix. This is not an obvious outcome of cost-minimizing behavior. It has several implications. a) The cross-price effects are symmetric. That is,

b) The own-price effects are nonpositive. Roughly speaking, the conditional

factor demand curves are downward sloping. This follows since where the last inequality comes from the fact that the diagonal terms of a negative semidefinite matrix must be nonpositive.

Note that since the concavity of the cost function followed solely from the hypothesis of cost minimization, the symmetry and negative semidefiniteness of the first derivative matrix of the factor demand functions follow solely from the hypothesis of cost minimization and do not involve any restrictions on the structure of the technology. Hence, the conditional factor demand curve slopes downward.

The properties of profit function 1) Nondecmsing in output prices, nonincreasing in input prices. 2) Homogeneous of degree 1 in p. At the beginning of this chapter we proved that the profit function must satisfy certain properties. We have just seen that the net supply functions are the derivatives of the profit function. It is of interest to see what the properties of the profit function imply about the properties of the net supply functions. Let us examine the properties one by one. First, the profit function is a monotonic function of the prices. Hence, the partial derivative of ~ ( pw)i th respect to price i will be negative if good i is an input and positive if good i is an output. This is simply the sign convention for net supplies that we have adopted. Second, the profit function is homogeneous of degree 1 in the prices. We have seen that this implies that the partial derivatives of the profit function must be homogeneous of degree 0. Scaling all prices by a positive factor t won't change the optimal choice of the firm, and therefore profits will scale by the same factor t.