Perfect risk sharing in Arrow-Debreu w/ the same subjective beliefs over states So I'm looking at a 2-agent Arrow-Debreu economy with one good. Consumption and endowments are zero in t=0, and 2 states are possible in t=1 with aggregate endowment in both states equal to 1. 

We assume utility is strictly increasing and strictly quasiconcave. My question is this:

My professor says by strict monotonicity 

$\dfrac{v^{'}_{1}\left(x^1_1\right)}{v^{'}_{1}\left(x^2_1\right)} = \dfrac{v^{'}_{2}\left(1 - x^1_1\right)}{v^{'}_{2}\left(1 - x^2_1\right)} \Rightarrow x^1_1 = x^2_1$

I can see this is obviously true if $v(\cdot)$ is concave, but we only have strict quasi-concavity. For example $f(x) = x^2$ is strictly convex yet strictly quasiconcave. Since agent 1 and agent 2 are not required to have the same utility function, it is possible for agent 1 to have a convex utility and agent 2 to have a concave one. In short, we cannot say the second derivatives are the same sign without the concavity assumption. 

Moreover, wouldn't a counter example be aif $v_i(x) = x$ for $i = 1,2$. Then $x_1^1 \neq x_1^2$ would still imply the ratio holds. Is it then something about an interior solution?
In two details you seem to be mistaken:


You need strict concavity of $v(\cdot)$, not concavity.
According to the definition of quasiconcavity, the function $x^2$ is not quasiconcave, it is quasiconvex.


Your main point is correct. If both functions are linear, then $x_i^1 = x_i^2$ no longer needs to hold, as the goods $x_i^1$ and $x_i^2$
are perfect substitutes for consumer $i$. Hence all consumption decisions are correct as long as $i$ trades at a price ratio of 1. This is indeed the equilibrium price ratio. The same can be said for the other consumer.