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The Stone-Geary function is a widely-used generalization of Cobb-Douglas. For two goods, the form of the function is:

`U=(X-\beta_X)^\alpha \cdot (Y-\beta_Y)^{1-\alpha}`

The corresponding demand equations are:

`X=\beta_X+(\alpha (M-\beta_X P_X-\beta_Y P_Y))/P_X`

`Y=\beta_Y+((1-\alpha) (M-\beta_X P_X-\beta_Y P_Y))/P_Y`

Stone-Geary is a generalization because it contains the Cobb-Douglas function as a special case when both `\beta` terms are 0.

What makes the Stone-Geary particularly useful is that it can accommodate goods that have different income elasticities, such as 0.5 for a necessity or 1.8 for a luxury good. Cobb-Douglas demands, in contrast, always have an income elasticity of exactly 1.

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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 04/20/2019