# Non-unique utilities in a competitive equilibrium

In continuation to this question: Uniqueness of utilities in competitive equilibrium I think I found a simple example in which the utilities in equilibrium are not unique, and wanted to check if it is true.

This is an exchange economy with two goods and two agents with utilities:

$$u_1(x,y)=2x+y$$ $$u_2(x,y)=3x+y$$

Initially, each agent has 1 unit of $x$ and 5 units of $y$.

Suppose $p_y=1$. Then, whenever $p_x\in[2,3]$, the economy is in equilibrium, since agent 1 wants to sell all his $x$ endowment and agent 2 wants to buy all $x$. So the equilibrium utilities will be:

$$u_1 = 5+p_x$$ $$u_2 = 11-p_x$$

This shows that utilities are not unique.

A. Are my calculations correct?

B. If they are correct, what is going to happen in a real competitive economy? Since the equilibria are essentially different, which of them will prevail?

EDIT: After the price theory book again, I am not sure about my calculations. The book defines a competitive equilibirum as a situation in which the sum of demands exactly equals the sum of endowments. The demand of an agent is defined as the optimal bundle he can afford using his initial endowment. Now:

• Whenever $p_x<3$, the demand of agent 2 is $(1+5/p_x,0)$ - he wants to buy $x$ with all his endowment of $y$. Since the total amount of $x$ is 2, this implies $p_x\geq 5$ - a contradiction.
• Whenever $p_x>3$, the demand of both agents is $(0,5+p_x)$ - both want to buy only $y$, which is obviously not an equilibrium.
• Hence we must have $p_x=3$. Now agent 2 is indifferent between buying or selling $x$, while agent 1 wants to buy only $y$. Hence, the demand of agent 1 is $(0,5+p_x)=(0,8)$.
• So, in this case there is a unique eqilibrium in which $p_x=3$, agent 1 gets $(0,8)$ and agent 2 gets $(2,2)$; the utilities are 8 and 8.

Now I am confused: is the utility vector unique or not?

• A. Your calculation seems to be correct. B. Just like in game theory if you have multiple equilibria and wish to select one you need to come up with additional assumptions. – Giskard Oct 15 '15 at 6:45

In your situation for any $p_x \in [0,2]$ there exists an equilibrium allocation, where consumer 2 bought the good $x$ from consumer 1 for $p_x$ units of good $y$. There is no better feasible decision for either of them.
If you were to adhere to the strict/dogmatic interpretation there would be no equilibrium at all, because for any $p_x <3$ consumer 2 wants to buy an infinite amount of good $x$ and for any $p_x > 2$ consumer 1 wants to sell an infinite amount of good $x$. Since there is no $p_x$ such that $p_x \leq 2, \ p_x \geq 3$ you could not have an equilibrium with finite amounts.
• @ErelSegal-Halevi Yes, exactly. That is why the demand is not always $(1+5/p_x,0)$, only if this is in the budget set. Which is limited by physical contraints, such as non negativity and in this case an upper limit on the number of goods $x$. – Giskard Oct 16 '15 at 14:23