The Solow Model [The first order conditions for profit maximization] - Repkine , Johan Lindén

The fact that the Solow condition follows from cost minimization, and so does not require profit maximization, means that it can be expected to hold not only for firms, but equally well for other employers such as government agencies and household organizations.

Obviously profit maximization entails cost minimization. Nevertheless it may be instructive to see how the Solow condition is derived from the first order conditions for profit maximization.

Suppose that a firm produces its output q according to the production function

q = F(K, e(w)L) 

and sells it on a product market at a price p. If the product market is competitive, the price is constant, but in order to allow for the possibility that the firm wields some market power on its product market, assume that it faces the inverse product demand curve p = p(q). It buys or rents its capital K on a competitive market for physical capital at a rental cost r, and can hire any amount of labor L, which provides effective labor services depending on the wage w according to the efficiency function e = e(w).

The firm chooses its amount of capital K and labor L, as well as the wage w that it pays its workers, to maximize its profits:

π = p(q)q − rK − wL 

The first order conditions with respect to K, L, and w set the derivatives of profits with respect to these three variables equal to zero:

∂π/∂K = (p(q) + q dp/ dq ) ∂F /∂K (K, e(w)L) − r = 0 

∂π ∂L = (p(q) + q dp/ dq )e(w) ∂F/ ∂L(K, e(w)L) − w = 0

∂π ∂w = (p(q) + q dp/ dq )Le′ (w) ∂F/ ∂L(K, e(w)L) − L = 0

The first of these conditions simply states the usual condition for an optimal capital stock, that the marginal revenue product of capital is equal to the rental cost of capital:

MRPK = (p + q dp /dq ) ∂F /∂K = r 

The second condition requires that the marginal revenue product of effective labor input be equal to the cost per unit of effective labor:

MRPL = (p + q dp/ dq ) ∂F/ ∂L = w e(w)  

Substituting the second condition into the third, results in the condition that the wage w is chosen so that these costs of effective labor are minimized:

w / e(w) e ′ (w) = 1