The Maxwell School
Syracuse University
Syracuse University
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.