# Constraints on a symmetric pareto allocation under uncertainty

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.

What symmetric means, as explained in the footnote, is that the consumption profile must be identical. The consumption profile $(C_1, C_2)$ represents the consumption for the agent at date 1 (if he is type-1) $C_1$ and the consumption for the agent at date 2 (if he is type-2) $C_2$. Otherwise stated $(C_1, C_2)$ does not depend on the type and each agent has the same $C_1$ and $C_2$. Since the types coincide with the date it's a little bit trivial but it really says nothing more than that.
For the constraint, there is a mass-1 continuum of agents, each endowed with one unit of wealth, so the total wealth is 1. Then from that total wealth an amount $I$ will be invested in the productive technology whose return is $R$ per unit invested. Then it follows that :
• at date 1, an amount $1-I$ is available, and the demand will be $\pi_1C_1$, so this yields : $\pi_1C_1 = 1-I$
• at date 2, an amount $RI$ is available, and the demand will be $\pi_2C_2$, so this yields : $\pi_2C_2 = RI$