# General Equilibrium with Perfect Substitutes

I came across the following problem:

The quantities of an economy’s only two goods are denoted by $$X$$ and $$Y$$; no production is possible. Ann’s and Ben’s preferences are described by the utility functions $$u_A(x,y) = X+Y$$ and $$u_B(x,y) = XY$$.

Ann owns the bundle $$(0,5)$$ and Ben owns the bundle $$(30,5)$$. Determine the Walrasian equilibrium price(s) and allocation(s).

It is quite easy to determine B's offer curve (with the price of $$Y$$ ($$p_2$$) normalised to 1): $$OC_B = \left(\frac{30p_1 + 5}{2p_1}; \frac{30p_1 + 5}{2}\right)$$

However, I don't know how to proceed now, since consumer A is indifferent between good X and good Y. The general Marshallian demands for a function of the form $$u_A(x,y) = \alpha X+ \beta Y$$ are (if we assume that the consumer consumes the same quantities in case we have $$p_1 = \frac{\alpha}{\beta}p_2$$) are the following:

$$X^{M} = \left\{ \begin{array}{ll} m/p_1 & \mbox{if } p_1 < \frac{\alpha}{\beta}p_2 \\ m/2p_1 & \mbox{if } p_1 = \frac{\alpha}{\beta}p_2 \\ \ 0 & \mbox{else} \end{array} \right.$$

and

$$Y^{M} = \left\{ \begin{array}{ll} m/p_2 & \mbox{if } p_1 > \frac{\alpha}{\beta}p_2 \\ m/2p_2 & \mbox{if } p_1 = \frac{\alpha}{\beta}p_2 \\ \ 0 & \mbox{else} \end{array} \right.$$

So if we replace exogenous income with the endowments $$\omega = e_1p_1 + e_2p_2$$, and normalize the price of Y to 1 again, our offer curve would look exactly the same as the one of consumer B (since $$\alpha = \beta = 1$$), because we are in the case where consumer A splits his consumption equally (by assumption), since we have that $$MRS_A = \frac{1}{1} = 1$$. And since $$MRS = \frac{p_1}{p_2}$$ in equilibrium, we must have that $$p_1 = p_2 = 1$$ if I'm not mistaken.

$$OC_A = \left(\frac{30p_1 + 5}{2p_1}; \frac{30p_1 + 5}{2}\right)$$

Solving for the Walrasian Equilibrium, we would get that $$p_1 = 1/6$$, which corresponds to our price ratio, since $$p_2 = 1$$.

This somehow doesn't seem right to me, because the fact that consumer A wants to consumer both goods equally if $$p_1 = \frac{\alpha}{\beta}p_2$$ is just an assumption that we have made to be able to define that case. In reality, however, consumer A is totally indifferent between goods X and Y, so he can freely switch between the two and does not have to consume them $$50:50$$. Also, what makes this approach very confusing to me as well is the fact that the prices are endogenous within this framework. So the prices should be able to adjust freely according to demand, which makes the above definition of the Marshallian demands redundant.

My second question is that given this logic, would it even make sense to look at what happens if $$p_1 \neq \frac{\alpha}{\beta}p_2$$? As I said, the prices are endogenous, so we cannot impose certain prices in the beginning already. How do you approach such a problem with perfect substitutes in general then? Say we have for example $$u_A(x,y) = 2X+3Y$$ and $$u_B(x,y) = XY$$. Does the result change?

To solve for competitive equilibrium, we can first find the demand : Demand for commodity $$X$$ by A is $$x_A = \frac{5}{p_x}$$ if $$p_x < 1$$, $$x_A \in [0,5]$$ if $$p_x = 1$$, $$x_A = 0$$ otherwise. Demand for commodity $$X$$ by B is $$x_B = \frac{(30p_x+5)}{2p_x}$$. Now we can equate demand and supply and solve for $$p_x$$.
$$x_A + x_B = 30$$ yields $$p_x = \frac{1}{2}$$.