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?

  • $\begingroup$ Downvoter, what's the problem with the question? $\endgroup$ – PhDing Oct 5 '18 at 21:20
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    $\begingroup$ Downvote by me. What have you tried so far? This question seems to be very solvable with a medium amount of research effort. $\endgroup$ – Giskard Oct 6 '18 at 8:04

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)

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    $\begingroup$ The two methods you present seem correct (though I would like to see details on the second one), but both are also really inefficient ways of solving the expenditure minimisation problem. Why not solve it directly with the given utility function? $\endgroup$ – Theoretical Economist Oct 5 '18 at 1:36
  • $\begingroup$ @TheoreticalEconomist What do you mean by "directly"? How would you set the constraint in this case? $\endgroup$ – PhDing Oct 5 '18 at 21:21

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