*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{.}

$$

- Jeffrey Wooldridge,
*Introductory econometrics: A modern approach,*Cengage Learning, 2012, p. 40. ↩