# Calculating price in a pure exchange economy

The problem is simple, I'm not really sure of my answer though.

Consider an economy where there are two consumers and two goods: $$U_1(x_{11}, x_{21}) = x_{11}$$ $$U_2(x_{12}, x_{22}) = x_{22}$$ $X_1 = X_2 = R_2^+$ ; $e_i ≥ 0, i = 1, 2.$

For what values of endowments does an equilibrium exist? What are the prices in the cases where an equilibrium does exist?

Now we cant really use the tangency condition here to calculate the competitive allocation. Intuitively, it seems like there'd be an exchange of commodities. To me, it seems like there'd be different cases.

Case 1: Individual 1 has both $x_1$ and $x_2$, whereas individual 2 has only $x_2$. Since individual 1 only values $x_1$, the price of $x_1$ in this case would be 0, since he doesn't really need to trade. I'm not really sure how to calculate the price of $x_2$ though.

Case 2: Similarly, Individual 1 has some $x_1$ and no $x_2$. Individual 2 has some of both. In this case, price of $x_2$ should be zero, but again, not sure about the price for $x_1$. (I'm not really sure it matters though, since we usually always calculate relative prices)

Case 3: Both individuals have some quantity of both $x_1$ and $x_2$. In this case, the income from the excess of the commodity not required by the individual must be equal to the price of the commodity he needs to buy. The relative price in this case comes out to be $\frac{X-x_{11}}{x_{21}}$.

• You use both $y$ and $x_2$. Perhaps you ought to pick one. – Giskard Feb 9 '16 at 17:39
• Have you considered using a market equilibrium approach? Let $p$ denote the price ratio. Then given the initial endowments some very simple functions $D_{11}(p), D_{12}(p), D_{21}(p), D_{22}(p)$ exist and for any good total demand equals total supply. – Giskard Feb 9 '16 at 18:13
• Either no, or I don't understand what you are saying. An example: $$D_{11}(p) = \frac{e_{11} \cdot p + e_{21}}{p}.$$ – Giskard Feb 9 '16 at 18:50