Consider the hybrid utility function u(x, y) = \min\{\alpha x+\beta y,\beta x+\alpha y\}, with 0 < \alpha < \beta.
For a target utility \mu \geq 0, find the Hicksian demand correspondence (X^h,Y^h)(p_x, p_y, \mu), and the expenditure function e(p_x, p_y, \mu).
Utility Function:
u(x,y) = \min\{\alpha x + \beta y, \beta x + \alpha y\}
where
0 < \alpha < \beta
solve
\min p_x x + p_y y
S.t.
\min\{\alpha x + \beta y, \beta x + \alpha y\} \geq \mu
Replace ≥ by =
\min\{\alpha x + \beta y, \beta x + \alpha y\} = \mu
now bringing terms together :
\alpha x + \beta y = \beta x + \alpha y
(\alpha - \beta)x = (\alpha - \beta)y
since
\alpha \neq \beta
so
x = y
Substituting
\alpha x + \beta x = \mu
(\alpha + \beta)x = \mu
\therefore \boxed{x = \frac{\mu}{\alpha + \beta}, \quad y = \frac{\mu}{\alpha + \beta}}
Now lets suppose
\alpha x + \beta y < \beta x + \alpha y
then
\alpha x + \beta y = \mu
Cost per unit of utility
\text{Good } x:\text{cost per unit utility} = \frac{p_x}{\alpha}
\text{Good } y:\text{cost per unit utility} = \frac{p_y}{\beta}
So,
If \frac{p_x}{\alpha} < \frac{p_y}{\beta}, buy only x:
(x, y) = \left( \frac{\mu}{\alpha}, 0 \right), \quad E = \frac{p_x \mu}{\alpha}
If \frac{p_y}{\beta} < \frac{p_x}{\alpha}, buy only y:
(x, y) = \left( 0, \frac{\mu}{\beta} \right), \quad E = \frac{p_y \mu}{\beta}
Similarly
Only x is purchased:
\left( \frac{\mu}{\beta}, 0 \right), \quad E = \frac{p_x \mu}{\beta}
Only y is purchased:
\left( 0, \frac{\mu}{\alpha} \right), \quad E = \frac{p_y \mu}{\alpha}
From this Hicksian Demand -
\boxed{
(X^h, Y^h) (px,py,{\mu})=
\begin{cases}
\left( \dfrac{\mu}{\alpha}, 0 \right) & \text{if } \dfrac{p_x}{\alpha} < \dfrac{p_y}{\beta} \\[12pt]
\left( 0, \dfrac{\mu}{\beta} \right) & \text{if } \dfrac{p_y}{\beta} < \dfrac{p_x}{\alpha}\\[12pt]
\left( \dfrac{\mu}{\beta}, 0 \right) & \text{if } \dfrac{p_x}{\beta} < \dfrac{p_y}{\alpha} \\[12pt]
\left( 0, \dfrac{\mu}{\alpha} \right) & \text{if } \dfrac{p_y}{\alpha} < \dfrac{p_x}{\beta}\\[12pt]
\left( \dfrac{\mu}{\alpha + \beta}, \dfrac{\mu}{\alpha + \beta} \right) & \text{if } \dfrac{\alpha}{\beta} \leq \dfrac{p_x}{p_y} \leq \dfrac{\beta}{\alpha} \\[12pt]
\end{cases}
}
Now Expenditure Function:-
e = p_x x + p_y y
Substitute x and y
e = p_x \left( \frac{\mu}{\alpha + \beta} \right) + p_y \left( \frac{\mu}{\alpha + \beta} \right)
\therefore \boxed{e(p_x, p_y, \mu) = \mu \left( \frac{p_x}{\alpha} + \frac{p_y}{\beta} \right)}