Averch–Johnson effect

Suppose that a regulated firm wishes to maximize its profit:

$${\displaystyle \pi =R(K,L)-wL-rK}$$

where ${\displaystyle R(K,L)}$ is the revenue function, ${\displaystyle K}$ is the firm's capital stock, ${\displaystyle L}$ is the firm's labor stock, ${\displaystyle w}$ is the wage rate, and ${\displaystyle r}$ is the cost of capital. The firm's profit is constrained such that:

$${\displaystyle \sigma ={R-wL \over {K}}}$$

where ${\displaystyle \sigma }$ is the allowable rate of return. Assume that ${\displaystyle \sigma >r}$. We may then form a functional to find the firm's optimal action:

$${\displaystyle J=R(K,L)-wL-rK-\lambda [R(K,L)-wL-\sigma K]}$$

where ${\displaystyle \lambda }$ is the Lagrange multiplier (also known as the shadow price). The derivatives of this functional are:

$${\displaystyle {\begin{aligned}{\partial J \over {\partial K}}&=(1-\lambda )R_{K}-r+\lambda \sigma \\{\partial J \over {\partial L}}&=(1-\lambda )R_{L}-(1-\lambda )w\end{aligned}}}$$

Taken together, this implies that:

$${\displaystyle R_{K}={r-\lambda \sigma \over {1-\lambda }},\quad R_{L}=w}$$

The ratio of the marginal product of capital and the marginal product of labor is:

$${\displaystyle {R_{K} \over {R_{L}}}={r-\alpha \over {w}},\quad \alpha ={\lambda \over {1-\lambda }}(\sigma -r)}$$

Since this new cost of capital is perceived to be less than the market cost of capital, the firm will tend to overinvest in capital.[4]

• Law and economics
• Public utilities commission
• Rate-of-return regulation

• Greer, Monica (2012). Electricity Marginal Cost Pricing: Applications in Eliciting Demand Responses. Waltham, MA: Butterworth-Heinemann.
• Lesser, Jonathan A.; Giacchino, Leonardo R. (2013). Fundamentals of Energy Regulation (2nd ed.). Public Utilities Reports, Inc.
• Willis, H. Lee; Philipson, Lorrin (2019). Understanding Electric Utilities and De-Regulation. Power Engineering. Boca Raton, FL: CRC Press.

• Body of Knowledge on Infrastructure Regulation: Incentive Features and Other Properties
• The Averch Johnson Effect