A Theorem on the Consumer Surplus

Thursday April 03, 2014

•  demand •  economic analysis •  economics •  environmental policy •  environmental studies •  flood studies •  mathematics •  statistics • 

Theorem. Let [latex]m[/latex] be a scaling factor and [latex]y = \beta_0 + \beta_1 x_1 + \cdots[/latex] be a generalized linear model such that [latex]y[/latex] is the amount of a good or service purchased and [latex]x[/latex] is the price per unit of the good or service. If [latex]y^\prime = \beta_0^\prime + \beta_1^\prime x_1^\prime + \cdots[/latex] such that [latex]y^\prime = my[/latex] and [latex]x_1^\prime = x_1 / m[/latex], then,

[latex]-\frac{\hat{y}^{\prime\,2}}{2\hat{\beta_1^\prime}} = -\frac{\hat{y}^2}{2\hat{\beta_1}} \label{eqn:thm} \text{.}[/latex]

If [latex]y^\prime = \beta_0^\ast + \beta_1^\ast x_1 + \cdots[/latex], then [latex]\beta_1^\ast = m\beta_1[/latex].1 Similarly, if [latex]y = \beta_0^\ast + \beta_1^\ast x_1^\prime + \cdots[/latex], then [latex]\beta_1^\ast = m\beta_1[/latex]. Therefore, if [latex]y^\prime = \beta_0^\prime + \beta_1^\prime x_1^\prime + \cdots[/latex], then [latex]\beta_1^\prime = m^2\beta_1[/latex]. Accordingly,

[latex]-\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{.}[/latex]

  1. Jeffrey Wooldridge, Introductory econometrics: A modern approach,</i> _Cengage Learning, 2012, p. 40. ↩