# Expenditure minimization with Leontief utility

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?

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

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 $$h_2(\textbf{p},u)=u$$.