A Theorem on the Consumer Surplus

Theorem. Let m be a scaling factor and y = \beta_0 + \beta_1 x_1 + \cdots be a generalized linear model such that y is the amount of a good or service purchased and x is the price per unit of the good or service. If y^\prime = \beta_0^\prime + \beta_1^\prime x_1^\prime + \cdots such that y^\prime = my and x_1^\prime = x_1 / m, then,
$$
-\frac{\hat{y}^{\prime\,2}}{2\hat{\beta_1^\prime}} =
-\frac{\hat{y}^2}{2\hat{\beta_1}} \label{eqn:thm} \text{.}
$$

If y^\prime = \beta_0^\ast + \beta_1^\ast x_1 + \cdots, then \beta_1^\ast = m\beta_1. 1 Similarly, if y = \beta_0^\ast + \beta_1^\ast x_1^\prime + \cdots, then \beta_1^\ast = m\beta_1. Therefore, if y^\prime = \beta_0^\prime + \beta_1^\prime x_1^\prime + \cdots, then \beta_1^\prime = m^2\beta_1. Accordingly,
$$-\frac{\hat{y}^{\prime\,2}}{2\hat{\beta_1^\prime}} =
-\frac{(m\hat{y})^2}{2m^2\hat{\beta_1}} =
-\frac{m^2\hat{y}^2}{2m^2\hat{\beta_1}} =
-\frac{\hat{y}^2}{2\hat{\beta_1}} \text{.}
$$

  1. Jeffrey Wooldridge, Introductory econometrics: A modern approach, Cengage Learning, 2012, p. 40.