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A Theorem on the Consumer Surplus

Thursday April 03, 2014

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

Theorem. Let $m$ be a scaling factor and $y = β_{0} + β_{1} x_{1} + \dots$ 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 $y^{'} = β_{0}^{'} + β_{1}^{'} x_{1}^{'} + \dots$ such that $y^{'} = m y$ and $x_{1}^{'} = x_{1} / m$ , then,

- \frac{{\hat{y}}^{' 2}}{2 \hat{β_{1}^{'}}} = - \frac{{\hat{y}}^{2}}{2 \hat{β_{1}}} .

If $y^{'} = β_{0}^{*} + β_{1}^{*} x_{1} + \dots$ , then $β_{1}^{*} = m β_{1}$ .¹ Similarly, if $y = β_{0}^{*} + β_{1}^{*} x_{1}^{'} + \dots$ , then $β_{1}^{*} = m β_{1}$ . Therefore, if $y^{'} = β_{0}^{'} + β_{1}^{'} x_{1}^{'} + \dots$ , then $β_{1}^{'} = m^{2} β_{1}$ . Accordingly,

- \frac{{\hat{y}}^{' 2}}{2 \hat{β_{1}^{'}}} = - \frac{(m \hat{y})^{2}}{2 m^{2} \hat{β_{1}}} = - \frac{m^{2} {\hat{y}}^{2}}{2 m^{2} \hat{β_{1}}} = - \frac{{\hat{y}}^{2}}{2 \hat{β_{1}}} .

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

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