Stone–Geary utility function

The Stone–Geary utility function takes the form

${\displaystyle U=\prod _{i}(q_{i}-\gamma _{i})^{\beta _{i}}}$

where ${\displaystyle U}$ is utility, ${\displaystyle q_{i}}$ is consumption of good ${\displaystyle i}$, and ${\displaystyle \beta }$ and ${\displaystyle \gamma }$ are parameters.

For ${\displaystyle \gamma _{i}=0}$, the Stone–Geary function reduces to the generalised Cobb–Douglas function.

The Stone–Geary utility function gives rise to the Linear Expenditure System.^[1] In case of ${\displaystyle \sum _{i}\beta _{i}=1}$ the demand function equals

${\displaystyle q_{i}=\gamma _{i}+{\frac {\beta _{i}}{p_{i}}}(y-\sum _{j}\gamma _{j}p_{j})}$

where ${\displaystyle y}$ is total expenditure, and ${\displaystyle p_{i}}$ is the price of good ${\displaystyle i}$.

The Stone–Geary utility function was first derived by Roy C. Geary,^[2] in a comment on earlier work by Lawrence Klein and Herman Rubin.^[3] Richard Stone was the first to estimate the Linear Expenditure System.^[4]