A Theorem on the Consumer Surplus | James Howard A Theorem on the Consumer Surplus | James Howard

# 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,</i> _Cengage Learning, 2012, p. 40.