There are two people in the economy, A and B. Both have the utility function: U=min{x,y}. A has an initial endowment of (x,y)=(50,100) and B has (x,y)=(100,50).

I know how to draw the Edgeworth box and establish the contract curve, but how do I determine the Walrasian equilibrium ratio of Px/Py? I'm confused by the fact that I can't differentiate U, nor is there money exchanged to establish a budget constraint.

  • $\begingroup$ Welcome to econ.SE! Please consider taking the time to read the help section (economics.stackexchange.com/help) to familiarize yourself with some of our common practices. In addition, meta.math.stackexchange.com/questions/5020/… should give you a start at learning how to typeset mathematics here so that your posts say what you want them to, and also look good. Cheers! $\endgroup$ – Martin Van der Linden May 7 '15 at 1:04

You do not need money exchanged to establish a budget constraint, most of these standard micro questions work in cashless economies.

Your Px/Py is just the exchange rate between the goods. If it's 3:1, it means that you exchange 3 $y$ for one $x$ - no cash involved.

Not being able to differentiate sucks, but actually requires you to know what you're doing. The point of differentiation is finding a place at which all Pareto improving trade has been exhausted.

Here, that means that, indexing the two players by i:

$$ x_i, y_i \in \arg\max_{x,y} U^i(x, y) s.t. p_x x + p_y y = p_x x^i + p_y y^i \, , \forall i $$

Without first-order conditions, this problem is potentially hard to solve. However, most non-differentiable utility functions give you a hint on how the outcome must look like.

Here, aggregate resources add up: $A=x_1 + x_2 = y_1 + y_2$. Hence, the planner would chose some allocation where x=y for both players, as for any positive $a$ and $\epsilon$ it is inefficient to give one player $(a + \epsilon, a)$ and the other one $(A - a - \epsilon, A - a)$.

However, that leaves a continuum of distributions as potential outcomes: Those where we give players one and two $(a, a)$ and $(A-a, A-a)$ for any $0 \leq a \leq A$.

Which equilibrium we actually get depends on the relative value of the initial endowments. I will leave you with this: What does $p_x/p_y$ necessarily have to be given total endowments and preferences? What does this imply for the unique equilibrium distribution of resources among the players?


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.