Calculating price in a pure exchange economy

Question

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 $\frac{X-x_{11}}{x_{21}}$.

Does the answer make sense?

Answer 1

Let $\omega_1 = (\omega_{1}^X, \omega_{1}^Y)$, and $\omega_2 = (\omega_{2}^X, \omega_{2}^Y)$ be the endowment of the two consumers, respectively. Also, their utility functions are $u_1(x_1, y_1) = x_1$ and $u_2(x_2, y_2) = y_2$, respectively.

We assume that there is positive quantity of both the goods in the economy i.e. $\omega_{1}^X + \omega_{2}^X >0$ and $\omega_{1}^Y + \omega_{2}^Y > 0$.

We'll show that the competitive equilibrium exist if and only if either ($\omega_1^Y>0$ and $\omega_2^X>0$) or ($\omega_1^Y=0$ and $\omega_2^X=0$)

Proof: We'll first show that the competitive equilibrium exist if ($\omega_1^Y>0$ and $\omega_2^X>0$) holds. Consider $(p_X, p_Y=1)$, value of endowment of individual $1$ is $p_X\omega_1^X + \omega_1^Y$ and he'll demand only good $X$:

$x_1^d = \dfrac{p_X\omega_1^X + \omega_1^Y}{p_X} $.

Clearly,

$x_2^d = 0 $

To find the equilibrium price $p_X$, we just need to solve: $\dfrac{p_X\omega_1^X + \omega_1^Y}{p_X} = \omega_1^X + \omega_2^X $ and we'll get $p_X = \dfrac{\omega_1^Y}{\omega_2^X}$. So the competitive equilibrium price is $(p_X = \dfrac{\omega_1^Y}{\omega_2^X}, p_Y=1)$ and the allocation is $(x_1, y_1) = (\omega_1^X + \omega_2^X, 0)$ and $(x_2, y_2) = (0,\omega_1^Y + \omega_2^Y)$.

We can now show that the competitive equilibrium exist if ($\omega_1^Y=0$ and $\omega_2^X=0$) holds. In this case the endowment allocation $(x_1, y_1) = (\omega_1^X, 0)$ and $(x_2, y_2) = (0, \omega_2^Y)$ is the competitive equilibrium supported by any prices of the form $(p_X>0, p_Y=1)$.

Now we can show that the competitive equilibrium does not exist if either ($\omega_1^Y=0$ and $\omega_2^X>0$) or ($\omega_1^Y>0$ and $\omega_2^X=0$).

We'll show this for the case of ($\omega_1^Y=0$ and $\omega_2^X>0$), the proof is analogous for the other case. First observe that none of the prices can be $0$ in a competitive equilibrium because $u_1$ is strictly increasing in $x_1$ and $u_2$ is strictly increasing in $y_2$. So, proceeding in the same way as above, we get the total demand for $X$ as $x_1^d = \dfrac{p_X\omega_1^X}{p_X} = \omega_1^X$ because only 1 demands $X$. Clearly, $x_1^d = \omega_1^X< \omega_1^X + \omega_2^X$. Therefore, competitive equilibrium does not exist in this case. Likewise we can also show this for the case of ($\omega_1^Y>0$ and $\omega_2^X=0$).