# Maximize profit with per unit profit of labor and capital (regulated firm)

Suppose a firm is regulated, such that it is permitted to recover the costs of its labor and capital inputs. Total revenue for the firm is therefore $$R= \mu L + \sigma K$$, where L=labor, K=capital, $$\mu$$ is the wage estimated by the regulator, and $$\sigma$$ is the cost of capital estimated by the regulator.

However, the regulator's estimation is imperfect, and the actual wage and cost of capital are w and r, respectively. Consequently, the firm can exploit this discrepancy to maximize profit, and substitute labor and capital based on whichever discrepancy is the better deal.

The firm's profit function (TR - TC) is:
$$\pi = \mu L + \sigma K - wL - rK$$
And the firm is required to meet demand Q, so it must produce Q=F(L,K), where F(*) is the production function.

The Lagrangian is
$$Lagrange= (\mu - w)L + (\sigma - r)K + \lambda(Q-F(L,K))$$

Taking the first order conditions, and subsequently the ratio of marginal products, results in the following expression:
$$\dfrac{F_k}{F_l} = \dfrac{\sigma - r}{\mu - w}$$

In the standard problem, set up as
$$Lagrange = P*F(L,K) - wL - rK$$

the profit maximization rule is
$$\dfrac{F_k}{F_l} = \dfrac{r}{w}$$
such that the marginal rate of technical substitution equals the ratio of input prices. The intuition is that the firm substitutes labor and capital based on their productivity relative to their cost.

In the regulated setup, the firm substitutes labor and capital based on their productivity relative to their per unit profit.

Does the regulated model make sense? Are there models that relate marginal productivity of labor and capital to their respective marginal profits?