for all $s^t$ for all $t \geq 0$.

To design a socially optimal allocation, the social planner wants to know what agent $1$ believes about the endowment sequence and how they feel about bearing risks.

To design a socially optimal allocation, the social planner wants to know what each agent $i$ believes about the endowment sequence and how they feel about bearing risks.

As for the endowment sequences, agent $i$ believes that nature draws i.i.d. sequences from joint densities

* each agent's one period utility function $u(\cdot) = \ln(\cdot)$

* each agent $i$'s probability model $\{\pi_t^i(s^t)\}_{t=0}^\infty$

Consequently, we anticipate that these objects will appear in the social planner's rule for allocating the aggregate endowment each period.

Consequently, it is natural to anticipate that these objects will appear in the social planner's rule for allocating the aggregate endowment each period.

First-order necessary conditions for maximizing welfare criterion {eq}`eq:welfareW` subject to the feasibility constraint {eq}`eq:feasibility` are

then we can represent the social planner's allocation rule as

$$

c_t^1(s^t) = \lambda_t(s^t) .

c_t^1(s^t) = \lambda_t(s^t)

$$

and of course

$$

c_t^2(s^t) = 1- \lambda_t(s^t) .

$$

## If you're so smart, $\ldots$

Let's compute some values of limiting allocations {eq}`eq:allocationrule1` for some interesting possible limiting

values of the likelihood ratio process $l_t(s^t)$:

values of the likelihood ratio process $l_t(s^t)$.

As our first case, let's suppose that

$$l_\infty (s^\infty)= 1; \quad c_\infty^1 = \lambda$$

* In the above case, both agents are equally smart (or equally not smart) and the consumption allocation stays put at a $\lambda, 1 - \lambda$ split between the two agents.

* In this case, both agents are equally smart (or equally not smart) and the consumption allocation stays put at a $\lambda, 1 - \lambda$ split between the two agents.

$$l_\infty (s^\infty) = 0; \quad c_\infty^1 = 0$$

As our second case, let suppose that

* In the above case, agent 2 is "smarter" than agent 1, and agent 1's share of the aggregate endowment converges to zero.

$$l_\infty (s^\infty) = 0; \quad c_\infty^1 = 0$$

* In this case, agent 2 is "smarter" than agent 1, and agent 1's share of the aggregate endowment converges to zero.

As our third case, let's suppose that

$$l_\infty (s^\infty)= \infty; \quad c_\infty^1 = 1$$

* In the above case, agent 1 is smarter than agent 2, and agent 1's share of the aggregate endowment converges to 1.

* In this case, agent 1 is smarter than agent 2, and agent 1's share of the aggregate endowment converges to 1.

```{note}

These three cases are somehow telling us about how relative **wealths** of the agents evolve as time passes.

* when the two agents are equally smart and $\lambda \in (0,1)$, agent 1's wealth share stays at $\lambda$ perpetually.

* when agent 1 is smarter and $\lambda \in (0,1)$, agent 1 eventually "owns" the entire continuation endowment and agent 2 eventually "owns" nothing.

* when agent 2 is smarter and $\lambda \in (0,1)$, agent 2 eventually "owns" the entire continuation endowment and agent 1 eventually "owns" nothing.

Continuation wealths can be defined precisely after we introduce a competitive equilibrium **price** system below.

We can define continuation wealths precisely after we construct a competitive equilibrium **price** system below.

```

Soon we'll do some simulations that will shed further light on possible outcomes.

But before we do that, let's take a detour and study some "shadow prices" for the social planning problem that can readily be

converted to "equilibrium prices" for a competitive equilibrium.

But before we do that, let's take a detour and study some "shadow prices" for the social planning problem.

We shall soon see that these shadow prices can readily be converted to "equilibrium prices" for a competitive equilibrium.

Doing this will allow us to connect our analysis with an argument of {cite}`alchian1950uncertainty` and {cite}`friedman1953essays` that competitive market processes can make prices of risky assets better reflect realistic probability assessments.

Doing this will allow us to connect our analysis with claims by {cite}`alchian1950uncertainty` and {cite}`friedman1953essays` that transfers of wealth coming from trades in competitive asset markets eventually make prices of risky assets reflect realistic probability assessments.

## Competitive equilibrium prices

Two fundamental welfare theorems for general equilibrium models lead us to anticipate that there is a connection between the allocation that solves the social planning problem we have been studying and the allocation in a **competitive equilibrium** with complete markets in history-contingent commodities.

Two fundamental welfare theorems for general equilibrium models lead us to anticipate that there is a connection between the allocation that solves the social planning problem we have been studying and the allocation in a **competitive equilibrium** with complete markets in contingent claims to time- and history-dependent consumption goods.

```{note}

For the two welfare theorems and their history, see <https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics>.

We'll sketch it now.

In a competitive equilibrium, there is no social planner that dictatorially collects everybody's endowments and then reallocates them.

In a competitive equilibrium, no social planner dictatorially collects everybody's endowments and then reallocates them.

Instead, there is a comprehensive centralized market that meets at one point in time.

There are **prices** at which price-taking agents can buy or sell whatever goods that they want.

Trade is multilateral in the sense that that there is a "Walrasian auctioneer" who lives outside the model and whose job is to verify that

each agent's budget constraint is satisfied.

Trade is multilateral in the sense that that there is a "Walrasian auctioneer" who lives outside the model and whose job it is to verify that each agent's budget constraint is satisfied.

That budget constraint involves the total value of the agent's endowment stream and the total value of its consumption stream.

That budget constraint requires that the total value of the agent's endowment stream be at least as the total value of its consumption stream.

These values are computed at price vectors that the agents take as given -- they are "price-takers" who assume that they can buy or sell

whatever quantities that they want at those prices.

These values are computed at price vectors that the agents take as given -- the agents are "price-takers" who assume that they can buy or sell whatever quantities that they want at those prices.

Suppose that at time $-1$, before time $0$ starts, agent $i$ can purchase one unit $c_t(s^t)$ of consumption at time $t$ after history

$s^t$ at price $p_t(s^t)$.

Suppose that at time $-1$, before time $0$ starts, agent $i$ can purchase one unit $c_t(s^t)$ of consumption at time $t$ after history $s^t$ at price $p_t(s^t)$.

Notice that there is (very long) **vector** of prices.

The market meets once at time $-1$.

At times $t =0, 1, 2, \ldots$ trades made at time $-1$ are executed.

At times $t =0, 1, 2, \ldots$ trades made at time $-1$ are simply executed, i.e., the promised deliveries are made.

* in the background, there is an "enforcement" procedure that forces agents to carry out the exchanges or "deliveries" that they agreed to at time $-1$.

We want to study how agents' beliefs influence equilibrium prices.

We want to study how agents' probability models influence equilibrium prices.

Agent $i$ faces a **single** intertemporal budget constraint

According to budget constraint {eq}`eq:budgetI`, trade is **multilateral** in the following sense

* we can imagine that agent $i$ first sells his random endowment stream $\{y_t^i (s^t)\}$ and then uses the proceeds (i.e., his "wealth") to purchase a random consumption stream $\{c_t^i (s^t)\}$.

* imagine that agent $i$ first sells his random endowment stream $\{y_t^i (s^t)\}$ and then uses the proceeds (i.e., his "wealth") to purchase a random consumption stream $\{c_t^i (s^t)\}$.

Agent $i$ puts a Lagrange multiplier $\mu_i$ on {eq}`eq:budgetI` and once-and-for-all chooses a consumption plan $\{c^i_t(s^t)\}_{t=0}^\infty$

to maximize criterion {eq}`eq:objectiveagenti` subject to budget constraint {eq}`eq:budgetI`.

Agent $i$ attaches a Lagrange multiplier $\mu_i$ to budget constraint {eq}`eq:budgetI` and once-and-for-all chooses a consumption plan $\{c^i_t(s^t)\}_{t=0}^\infty$

that maximizes criterion {eq}`eq:objectiveagenti` subject to budget constraint {eq}`eq:budgetI`.

This means that the agent $i$ chooses many objects, namely, $c_t^i(s^t)$ for all $s^t$ for $t = 0, 1, 2, \ldots$.

for $i=1,2$.

If we divide equation {eq}`eq:priceequation1` for agent $1$ by the appropriate version of equation {eq}`eq:priceequation1` for agent 2, use

If we divide equation {eq}`eq:priceequation1` for agent $1$ by the appropriate version of equation {eq}`eq:priceequation1` for agent 2, impose the feasibility condition

$c^2_t(s^t) = 1 - c^1_t(s^t)$, and do some algebra, we'll obtain

$$

c_t^1(s^t) = \frac{\mu_1 l_t(s^t)}{\mu_2 + \mu_1 l_t(s^t)} .

$$ (eq:allocationce)

We now engage in an extended "guess-and-verify" exercise that involves matching objects in our competitive equilibrium with objects in

our social planning problem.

We now embark on an extended "guess-and-verify" exercise that involves matching objects in our competitive equilibrium with objects in our social planning problem.

* we'll match consumption allocations in the planning problem with equilibrium consumption allocations in the competitive equilibrium

* we'll match "shadow" prices in the planning problem with competitive equilibrium prices.

Notice that if we set $\mu_1 = 1-\lambda$ and $\mu_2 = \lambda$, then formula {eq}`eq:allocationce` agrees with formula

{eq}`eq:allocationrule1`.

Notice that if we set $\mu_1 = 1-\lambda$ and $\mu_2 = \lambda$, then formula {eq}`eq:allocationce` agrees with formula {eq}`eq:allocationrule1`.

* doing this amounts to choosing a **numeraire** or normalization for the price system $\{p_t(s^t)\}_{t=0}^\infty$

```{note}

For information about how a numeraire must be chosen to pin down the absolute price level in a model like ours that determines only

relative prices, see <https://en.wikipedia.org/wiki/Num%C3%A9raire>.

For information about how a numeraire must be chosen to pin down the absolute price level in a model like ours that determines only relative prices, see <https://en.wikipedia.org/wiki/Num%C3%A9raire>.

```

If we substitute formula {eq}`eq:allocationce` for $c_t^1(s^t)$ into formula {eq}`eq:priceequation1` and rearrange, we obtain

Complete markets models with homogeneous beliefs, a kind often used in macroeconomics and finance, are studied in this quantecon lecture {doc}`ge_arrow`.

{cite}`blume2018case` discuss a paternalistic case against complete markets. Their analysis assumes that a social planner should disregard individuals preferences in the sense that it should disregard the subjective belief components of their preferences.

{cite}`blume2018case` discuss a paternalistic case against complete markets. They study the consequences of assuming that a social planner disregards individuals preferences in the sense that it ignores the subjective belief components of their preferences and replaces it with the social planner's beliefs about probabilities.

Likelihood processes play an important role in Bayesian learning, as described in {doc}`likelihood_bayes` and as applied in {doc}`odu`.