Expenditure minimization with Leontief utility

Question

I need to solve the expenditure minimization in a context where $u(x,y) = min\{x,y\}$, i.e. where utility is Leontief.

The minimization problem is

$$\text{min}_{x,y}\,\,p_xx+p_yy \\ \text{subject}\,\,\text{to}\,\,\text{min}\{x,y\} \geq u$$

I know that if I had to maximize the sam utility function I had to impose the equality between the content of the curly brackets. But I am stuck in getting how should I behave in this context. Should I proceed by cases?

Answer 1

The Lagrangian method wouldn't be of any use, because Leontief function is not differentiable at the point of optimality/kink. However, you can consider the following approach.

We know that for $U(x,y)=min\{x,y\}$, optimalilty occurs at the point where $x=y$. Let the budget correspondence be $p_1x+p_2y\leq w$, where $w$ is the income level. Optimal consumption bundle occurs when all the income is used up. Thus, we have $p_1x+p_2x= w$ (since $x=y$ at point of optimality). This gives us the demand correspondences as $x(\textbf{p},w)=y(\textbf{p},w)=\frac{w}{p_1+p_2}$. Using the demands, we can find out the indirect utility as $v(\textbf{p},w)=U(x(\textbf{p},w),y(\textbf{p},w))=min\{\frac{w}{p_1+p_2},\frac{w}{p_1+p_2}\}=\frac{w}{p_1+p_2}$.

Next, we turn towards the duality principle,i.e; for a given level of utility, we have $v(\textbf{p},e(\textbf{p},u))=u$. Thus, we have $\frac{e(\textbf{p},u)}{p_1+p_2}=u$, or $e(\textbf{p},u)=u(p_1+p_2)$. Next, we appeal to Shephard's Lemma,i.e; $\frac{\partial e(\textbf{p},u)}{\partial p_i}=h_i(\textbf{p},u)$. Notice that the expenditure function is differentiable in prices. Thus, we get $h_1(\textbf{p},u)=u$ and $h_2(\textbf{p},u)=u$.

Another approach is possible. You can also solve the EMP for the standard CES utility function, and then derive the respective hicksian demands for Leontief function by taking suitable limits on the elasticity.(I don't know of any material where they've used this method, but it is possible according to theory)