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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?

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    $\begingroup$ 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. $\endgroup$ – Giskard Oct 15 '15 at 6:45
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This answer is in response to the edited (2015.10.16 ca. 12:30 CET) question.

I think the concept of equilibrium means that given the circumstances you have no incentive to alter your behavior. (This is pretty much based on the Nash-equilibrium.)
In this situation equilibrium consists of a price and an allocation of goods. Given the price everyone makes a trade decision. The decision has to be feasible both with respect to the budget and the physical availability of the goods, i.e. you cannot buy goods that do not exist. We are in equilibrium if everyone's decision was optimal (for themselves).
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.

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  • $\begingroup$ As far as I understand, the definition of the "demand" of an agent is: the optimal bundle that is in the agent's budget set. Hence, there cannot be infinite demand because the budget is finite. $\endgroup$ – Erel Segal-Halevi Oct 16 '15 at 11:52
  • $\begingroup$ @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 $x$. $\endgroup$ – Giskard Oct 16 '15 at 14:23
  • $\begingroup$ As far as I understand, the whole idea of price theory is that each agent has to know only his own bundle and the price vector; he does NOT have to know the bundles of other players, since all the information about the bundles of other players is already conveyed by the price vector. The price is determined in such a way that all demand exactly matches the supply. $\endgroup$ – Erel Segal-Halevi Oct 17 '15 at 18:09
  • $\begingroup$ @ErelSegal-Halevi I see what you mean. After having thought this about I see no contradiction, perhaps such a price always exists. Wikipedia says general equilibrium is only unique with some conditions: en.wikipedia.org/wiki/General_equilibrium_theory#Uniqueness $\endgroup$ – Giskard Oct 17 '15 at 21:30

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