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Consider an exchange economy with $2$ goods and $2m$ identical Households, but in this case each household has utility function $u(x_1,x_2)=x_{1}^2+x_{2}^2$, and endowments $w_1=w_2$. Show that there exist prices at which agents can engage in mutually beneficial trades.

My thinking: enter image description here

Here $MRS_{x_1,x_2}$ is not equal to $\frac{p_1}{p_2}$ because the utility is not convex.

$MRS_{x_1,x_2} = \frac{2x_1}{2x_2} = \frac{w_1}{w_2} = 1$.

Any suggestions on where I can go from here?

Sergio Parreiras
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OGC
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1 Answers1

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As you can see the $MRS$ increases as you move along an indifference curve, $x_1\nearrow$ and $x_2\searrow$. That means the consumer likes specialization, the more he or she gets of $x_1$ the more she is willing to give-up of good $x_2$ to get more of good $x_1$ and also the other way around.

Consumers have the same amount of $x_1$ and $x_2$ but they like specialization (without any particular preference of good $x_1$ over good $x_2$).

This suggests than an optimal trade will involve one consumer giving all of her $x_1$ in exchange for all the $x_2$ of another consumer. As the number of consumers is even, this can work. The (implicit) prices in this trade are $p_1=p_2$.

Now, you have to prove formally that the suggested trade does in fact make everyone better off.

Sergio Parreiras
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  • I don't think the instructor is looking for a formal proof of existing prices so that agents can engage in mutually beneficial trades. An explanation would be fine. – OGC Feb 19 '14 at 22:30
  • Why does the the number of consumers need to be even for this to work? – OGC Feb 19 '14 at 22:31
  • @user36829: with the same number of people is easy to see that half of them will consume only x1 and the other half only x2. Think about the case with 3 people and try to see how things would change. – Sergio Parreiras Feb 20 '14 at 15:37