2
$\begingroup$

The maximization problem is:

Maximize $u(x_1, x_2) = \min[a_1x_1, a_2x_2]\; \ \text{s.t.}\;\; p_1x_1 + p_2x_2 \leq$ $w$, in which $x_i, p_i$ is the amount and price of good $i$, $w$ is the total budget available.

What I have been told to deal with this $\min[.,.]$ function is to solve it graphically. It's very easy to see on the graph that the maximization happens when $a_1x_1 = a_2x_2$. But I wonder if there is a way to solve this algebraically? I'm stumped at the first step, which is to derive $\dfrac{\partial u(x_i)}{\partial x_i}$.

$\endgroup$
2
$\begingroup$

Since the utility function has the Leontief form, then the two goods are perfect complements. Therefore the consumer will always choose the kink point where $a_1x_1 = a_2x_2$, i.e. the maxima $(x_1^*,x_2^*)$ satises $a_1x_1^* = a_2x_2^*$.

Also since the consumer will spend all his/her income, we can have two variables $(x_1^*,x_2^*)$ and two equalities: $$\left\{ \begin{align} a_1x_1^* &= a_2x_2^*\\ p_1x_1^* +p_2x_2^*&=w\\ \end{align}\right. $$ Therefore by solving the two equations, we can nd the demand function for each good: $$\left\{ \begin{align} x_1^* &= \frac{a_2 w}{a_2p_1+a_1p_2}\\ x_2^*&=\frac{a_1 w}{a_2p_1+a_1p_2}\\ \end{align}\right. $$

Note that we cannot equate the MRS with the slope of the budget line here, because the MRS is not defined at the point where $a_1x_1 = a_2x_2$.

$\endgroup$
  • $\begingroup$ Thanks for your help. As indicated in my question, I already knew how to this. The part that troubles me is Therefore the consumer will always choose the kink point. I can very easily see this on graph and by tuition. But is there a way to show this algebraically, i.e. via partial derivative of the $min[.,.]$ function? $\endgroup$ – Heisenberg Dec 13 '13 at 22:03
  • $\begingroup$ The answer at this is "Note that we cannot equate the MRS with the slope of the budget line here, because the MRS is not defined at the point where $a_1x_1=a_2x_2$. So you cannot use the method with partial derivatives... $\endgroup$ – alexjo Dec 13 '13 at 22:13
  • $\begingroup$ @alexjo Is there any way we can apply this reasoning to this slightly more complicated problem? economics.stackexchange.com/questions/9012/… $\endgroup$ – Mathemanic Nov 5 '15 at 5:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.