# Non-existence of general equilibrium with quasi-linear utility

I recently came across an interesting example of why general equilibrium need not exist with quasi-linear utility (in case anyone is interested, I'm posting this at the end). To make the example work, you need to assume that one person's endowment is sufficiently low to rule out an equilibrium in which all agents choose an interior bundle.

I was wondering which assumption of standard existence theorems this violates. I guess this is the assumption that preferences are convex?

I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium?

The example. Suppose that there are two individuals $$A, B$$, each with the same utility function $$u = x + \ln(y)$$. Endowments are $$w^a = (0, 1)$$, $$w^b = (4, 3)$$ where the first component denotes possession of good $$x$$. Normalise $$p_x \equiv 1$$ and define $$p \equiv p_y$$. Solving $$A$$'s optimisation problem reveals that they demand $$x^a = (p-1, 1/p)$$ if $$p \geq 1$$; but otherwise demand their endowment so $$x^a = (0, 1)$$. Meanwhile, solving $$B's$$ problem reveals that $$x^b = (3 + 3p, 1/p)$$.

To begin, let's check if there is an equilibrium with $$p \geq 1$$. Since there are $$4$$ units of $$y$$ in total, market clearing would then require that $$\frac{1}{y} + \frac{1}{y} = 4 \iff p = \frac{1}{2}$$ which contradicts $$p \geq 1$$. So there is no such equilibrium.

Next, let's look for any equilibrium with $$p < 1$$. Since there are $$4$$ units of $$x$$, market clearing would require $$0 + 3 + 3p = 4 \iff p = 4/3$$ which contradicts $$p < 1$$. So there is no equilibrium with $$p < 1$$ either.

UPDATE: So it turns out I just made a very basic error, and the equilibrium is $$p = 1/3$$ (the solution to the equation $$0 + 3 + 3p = 4$$).

I was wondering which assumption of standard existence theorems this violates.

None. An equilibrium exists (defined by the starting allocation), it is merely not in the interior of the set. The existence theorems I know make no such guarantees.

I guess this is the assumption that preferences are convex?

The preferences in the example are convex.

Suggestion:
Heavily edit this question or post a new one based on your third question:

I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium. Does this illustrate anything general and interesting?

I recommend rephrasing this as well, as the final line seems clearly opinion-based. Suggested text:
"In general equilibrium are there any existence theorems that guarantee the existence of an interior equilibrium? By interior I mean [explain if you mean only the good space or if you also wish to restrict the price vector.]"

• Can you explain why an equilibrium exists? In the question, I provide what seems to be a valid proof that there is no equilibrium price. (Though I also understand your point that, in some sense, we would expect individuals to just consume their endowment -- for example, there are no mutually beneficial trades from this starting point.) Oct 20 '21 at 11:46
• @afreelunch I did not read your detailed calculations because I trust the theorems (: But it seems like $p=1/3$ would do the trick; everyone's inital endowment is an optimal choice given this price of good $y$. So I guess the error is in your final paragraph. Oct 20 '21 at 11:51
• Seems to like I have made a very stupid error! Indeed $p = 1/3$ does work, alas I failed to correctly solve the equation $3 + 3p = 4$.... Oct 20 '21 at 11:53
• I guess I should delete the question? (Thanks anyway for clearing this up for me!) Oct 20 '21 at 11:54
• @afreelunch Or you could accept the answer (: Oct 20 '21 at 11:54