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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.

details: http://imgur.com/TpVjg4w

$U = \pi_1u(C_1) + \pi_2u(C_2)$ 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.

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The idea is that since agents are ex-ante identical, they should have the same (ex-ante) expected utility.

Ex-ante identical means that even the agent does not know what's his type until resolution of type at date-1. Because they have the same preferences each agent's consumption profile must be identical. And the analysis can then focus on symmetric equilibrium.

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 : $\pi_2C_2 = RI$
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