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 $0 + 3 + 3p = 4$).
