Consider the perfect-complements utility function u(x, y) = \min\{\alpha x, \beta y\}, with \alpha, \beta > 0.
For a target utility \mu \geq 0, find the Hicksian demands x^h(p_x, p_y, \mu) and 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\}, \quad \text{where } \alpha, \beta > 0
solve:-
\min_{x, y} \quad p_x x + p_y y \quad
\text{subject to} \quad \min\{\alpha x, \beta y\} \geq \mu
Now we will replace the inequality with the equal sign:
\min\{\alpha x, \beta y\} = \mu
Now as it is given they are complements
\implies \alpha x = \beta y
Minimum will be equal to either one -
\implies \alpha x = \mu
\therefore x = \frac{\mu}{\alpha}
similarly
\implies \beta y = \mu
\therefore y = \frac{\mu}{\beta}
and here we got the Hicksian Demand
\boxed{x^h(p_x, p_y, \mu) = \frac{\mu}{\alpha}, \quad y^h(p_x, p_y, \mu) = \frac{\mu}{\beta}}
Now , the Expenditure Function:-
e = p_x x + p_y y
Substituting the x and y:
e = p_x \left( \frac{\mu}{\alpha} \right) + p_y \left( \frac{\mu}{\beta} \right)
So, the expenditure function is:
\boxed{e(p_x, p_y, \mu) = \mu \left( \frac{p_x}{\alpha} + \frac{p_y}{\beta} \right)}