I've been trying to figure out how the author came up with the constraints for this liquidity model in a textbook I'm reading.
$U = \pi_1u(C_1) + \pi_2u(C_2)$ where $\pi_i$ denotes the probability of an agent being of type $i$ and $C_t$ denotes consumption at time $t$. Agents of type $1$ consume at time $1$ and agents of type $2$ consume at time $2$.
In autarky, $C_1=1-I+lI$ and $C_2 = 1-I+RI$ where $0\leq I \leq 1$, $l < 1$, $R > 1$.
The setup for the unique ex ante symmetric Pareto-optimal allocation (not autarky) is
$\max \pi_1u(C_1) + \pi_2u(C_2)$
subject to the following constraints, which is what I don't understand:
$\pi_1C_1=1 - I$ and $\pi_2C_2 = RI$
There's a footnote on the word symmetric which says:
Since agents are ex ante identical, we only consider symmetric allocations $(C_1, C_2)$, where an agent's consumption profile does not depend on the agent's identity.
I think the symmetry definition might be part of what's tripping me up. It seems to me that since an agent of type $1$ can only consume at time $1$ his bundle should be $(C_1, 0)$, and the bundle for a type $2$ agent should be $(0, C_2)$, which would mean that the only symmetric bundle is $(0,0)$, but clearly this is not the author's intention.