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This is definition of Radner Equilibrium from Microeconomic Theory (Mas-Collel, Whinston and Green - Third Edition).

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I'm confused about two conditions:

$\sum_k q_k \cdot z_{ki} \le 0$ (in yellow)

The autors state that this is budget restriction. I wonder how the sum of transactions times the price could be different from zero: if Adam sells a stock for 10 dollars to Ben, the investment positions for Adam and Ben will be, respectively, -10 and 10, summing to $-10+10=0$

$\sum_i z^*_{ki} \le 0$ (in purple)

This is proposed as a feasibility condition. Why?

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    $\begingroup$ If A and B trade, it could be that A sells a stock to a central planner for 10 and B buys it from the planner for 11. This would be a waste of money, but it would be feasible. However, if A sells for 11 and B buys for 10, the planner would need to make up 1 out of nowhere, which is not feasible, the budget constraint excludes such cases. $\endgroup$ – Bayesian Jul 2 at 11:06
  • $\begingroup$ @Bayesian That's a great insight, but there's no central planner in the model $\endgroup$ – Bruno Schiavo Jul 2 at 11:28
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Condition in yellow: Each consumer $i$ is assumed to have initial wealth endowment $0$.

Condition in purple: For each asset $k$, aggregate demand $\leq$ aggregate supply, which is zero by assumption.

In practical terms, the asset $z_k$ being traded at $t = 0$ are $t = 1$ claims to commodity $k$. It is a derivative contract. In this particular case, it is a futures contract, i.e. the right and obligation to $z_k$ units of commodity $k$.

Assume the exogenous asset returns $r_{sk} = 1$ for all $k$ and $s$.

If you sell a futures contract for $z = 30$ barrels of West Texas Intermediate crude (tons of wheat/thousand pounds of coffee beans/etc.) at futures price $q = \$10$, then you pocket $z \cdot q = \$300$ at $t = 0$. At $t = 1$, you must deliver $30$ barrels of WTI crude to your counterparty. So you must purchase $z = 30$ barrels of WTI at spot price $p = \$20$, say, at a cost of $z \cdot p = \$600$.

Above is the case of a short position. Similarly, if you take a long position, it will cost $\$300$ today but tomorrow you get to have $30$ barrels of WIT, which you get to sell at the spot price $p = \$20$. This adds $\$600$ to your $t=1$ budget which can then be used to buy consumption good $x_i$'s.

Regardless of long or short position, today's cost is $z \cdot q$ and tomorrow's gain is $z \cdot p$.

If you have zero $t = 0$ wealth, the total value of contracts you buy cannot be more than the total value of contracts you sold. This is condition in yellow.

For each commodity good, if there is an aggregate of, say, $100$ contracts being sold at $t = 0$, there cannot be $101$ being bought. $$ \mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} \leq 0. $$ This is condition in purple.

Allowing for $< 0$ implicitly assumes that there are other agents unmodelled that absorbs excess supply. Alternatively, one can insist that the futures market clears, i.e. $$ \mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} = 0. $$ This would not lead to material change in the economic implications of the model.

Comments

Consumption Smoothing Under convexity and regularity assumptions on $U$, it is clear from the FOC that agents in this economy trade $t = 1$ claims at $t = 0$ in order to smooth their $t = 1$ consumption across states.

Sequential Trading In this economy, there is a futures/forward market at $t = 0$ and a spot market at $t =1$. Agents trade today subject to their budget constraint today, same for tomorrow.

Rational Expectations However, agents correctly anticipates the $t=1$ spot price at $t = 0$. Equilibrium is self-fulfilling.

Radner vs. Arrow-Debreu In an Arrow-Debreu equilibrium, agents trade $t=0$ state-contingent claims ("Arrow-Debreu securities"). There is no $t=1$ spot market. Trading only occurs once ex ante. In practical terms, AD securities are digital options, instead of futures contracts. A priori, agents' choice set of consumption profiles is more restricted in the Radner economy. A futures contract is a un-contingent claim. An important result is the following:

If asset returns ${r_{sk}}$ is complete, then an Arrow-Debreu equilibrium can be implemented by a Radner equilibrium.

This is not surprising. When the market is complete, i.e. there is enough futures contracts to span the set of states, any Arrow-Debreu payoff profile can be engineered from a suitable portfolio of futures contracts.

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  • $\begingroup$ Would you please elaborate? In yellow, why does initial zero endowment implies q.z <= 0? In purple, why is aggregate supply zero if people have assets to sell? $\endgroup$ – Bruno Schiavo Jul 4 at 20:02
  • $\begingroup$ What's your definition of an asset? A derivative contract is an asset. In any given time, the aggregate net position of any derivative contract in the financial market is zero. That is what's being traded in this economy---claims to future consumption, i.e. futures contracts. Each agent can choose to write or sell futures contracts. If you have zero initial wealth, the total value of contracts you buy cannot be more than the total value of contracts you sold. $\endgroup$ – Michael Jul 5 at 9:36
  • $\begingroup$ "Assume the exogenous r=1": what is "r" and what does it change? $\endgroup$ – Bruno Schiavo Jul 14 at 17:53
  • $\begingroup$ Who would be this unmodelled agent that absorbs excess supply? $\endgroup$ – Bruno Schiavo Jul 15 at 3:38
  • $\begingroup$ $r = [r_{sk}]$ is the exogenous asset return structure/matrix. Whether market is complete depends on whether the return matrix has full rank. $\endgroup$ – Michael Jul 15 at 16:53

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