# Unable to determine price ratio for a pure exchange economy

The problem is simple. However, I'm having trouble deriving the price vector for the system. What exactly am I doing wrong/missing out?

Two consumers.

$$U_A = \min(x,y), \hskip 20pt U_B = x+y.$$

Initial endowments $$(x_A,y_A)=(100,100), \hskip 20pt (x_B,y_B)=(50,0).$$

Putting A on the bottom left, the contract curve will be a line with slope 1 from A, from point (0,0) to (100,100), A's endowment point. [Seen from A's perspective.] All points on this contract curve are points of 'tangency' between A's kinked indifference curves, and B's linear indifference map.

$x_A= 100(p_x+p_y)/p_x$

$x_B = 0$ (if $p_x>p_y$)

$x_B = 50$ (if $p_x<p_y$)

A cannot benefit from more than (100,100), while B will benefit from as much of either good as she can get: i.e. endowment point lies on contract. Thus, the endowment point can be the competitive equilibrium allocation.

The question is, what will be the equilibrium price vector.

If we keep the price vector as (1,1), X can sell and buy back endowment, and so can B. So, (1,1) makes the endowment point affordable. However, so does any price vector where $p_x<p_y$ (ensuring 0 demand for Y from B.)

Does a unique price vector (ratio) exist for this system? How do we derive it? And what am I doing wrong?

Any help (resources on CE problems of this kind) will be wonderful. Thanks.

Your analysis seems to be spot on, except for some minor typos: $$x_A^* = 100 \frac{p_x + p_y}{p_x + p_y} = 100,$$ because the value of A's endowment is $100 \cdot (p_x + p_y)$. And you could include the case of $$x_B^* \in [0,50] \mbox{ if } p_x = p_y.$$

As you state at the end the equilibrium price vector is not unique, there is equilibrium for all $\frac{p_x}{p_y} < 1$, because with this price vector we have $$(x_A^*,y_A^*) = (100,100), \hskip 20pt (x_B^*,y_B^*) = (50,0)$$ and the market is then indeed in equilibrium as $$x_A^* + x_B^* = x_A + x_B, \hskip 20pt y_A^* + y_B^* = y_A + y_B.$$ The price ratio $\frac{p_x}{p_y} = 1$ also results in equilibrium, because one of the optimal consumptions of B is still $(x_B^*,y_B^*) = (50,0)$.

• I denote the optimal demand of good $x$ by consumer $i$ with $x_i^*$, and I denoted her initial endowment with $x_i$. Alternately you could denote the initial endowment with $\omega_i^x$ to make a more striking difference. This frees up $x_i$ for other uses, e.g. in the utility function. – Giskard Jun 20 '15 at 17:36

Equilibrium price vector $(p_x, p_y=1)$ and allocation $((x_A, y_A), (x_B, y_B))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the two consumers, i.e. it must lie on the demand functions)

• $(x_A, y_A) = \left(\frac{100p_x + 100}{p_x + 1}, \frac{100p_x + 100}{p_x+1}\right) = (100, 100)$
• $(x_B, y_B) = \begin{cases} \left(\frac{50p_x}{p_x}, 0\right) = (50,0) & \text{if } p_x \leq 1 \\ \left(0, 50p_x\right) & \text{if } p_x > 1 \end{cases}$

Feasibility Conditions

• $x_A + x_B = 150$
• $y_A + y_B = 100$

Clearly, any price vector $(p_x, p_y)$ satisfying $p_x \leq 1$ and $p_y = 1$, and the allocation $((x_A, y_A), (x_B, y_B)) = ((100, 100), (50, 0))$ is the equilibrium.

• How is this different from my answer? – Giskard Mar 1 '17 at 8:35
• I guess its the same. I didn't read your answer when I posted it. I will be careful in future to avoid duplication. – Amit Mar 1 '17 at 8:49