A Theorem on the Consumer Surplus

Thursday April 03, 2014

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Theorem. Let m be a scaling factor and y=Ξ²0+Ξ²1x1+β‹― 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 yβ€²=Ξ²0β€²+Ξ²1β€²x1β€²+β‹― such that yβ€²=my and x1β€²=x1/m, then,

βˆ’y^β€²22Ξ²1β€²^=βˆ’y^22Ξ²1^.

If yβ€²=Ξ²0βˆ—+Ξ²1βˆ—x1+β‹―, then Ξ²1βˆ—=mΞ²1.1 Similarly, if y=Ξ²0βˆ—+Ξ²1βˆ—x1β€²+β‹―, then Ξ²1βˆ—=mΞ²1. Therefore, if yβ€²=Ξ²0β€²+Ξ²1β€²x1β€²+β‹―, then Ξ²1β€²=m2Ξ²1. Accordingly,

βˆ’y^β€²22Ξ²1β€²^=βˆ’(my^)22m2Ξ²1^=βˆ’m2y^22m2Ξ²1^=βˆ’y^22Ξ²1^.
  1. Jeffrey Wooldridge, Introductory econometrics: A modern approach,</i> _Cengage Learning, 2012, p. 40. β†©

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