<p><em>Condition in yellow</em>: Each consumer <span class="math-container">$i$</span> is assumed to have initial wealth endowment <span class="math-container">$0$</span>. </p>

<p><em>Condition in purple</em>: For each asset <span class="math-container">$k$</span>, aggregate demand <span class="math-container">$\leq$</span> aggregate supply, which is zero by assumption. </p>

<p>In practical terms, the asset <span class="math-container">$z_k$</span> being traded at <span class="math-container">$t = 0$</span> are <span class="math-container">$t = 1$</span> claims to commodity <span class="math-container">$k$</span>. It is a derivative contract. In this particular case, it is a futures contract, i.e. the right and obligation to <span class="math-container">$z_k$</span> units of commodity <span class="math-container">$k$</span>.</p>

<p>Assume the exogenous asset returns <span class="math-container">$r_{sk} = 1$</span> for all <span class="math-container">$k$</span> and <span class="math-container">$s$</span>.</p>

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

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

<p>Regardless of long or short position, today's cost is <span class="math-container">$z \cdot q$</span> and tomorrow's gain is <span class="math-container">$z \cdot p$</span>.</p>

<p>If you have zero <span class="math-container">$t = 0$</span> wealth, the total value of contracts you buy cannot be more than the total value of contracts you sold. This is condition in yellow.</p>

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

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

<p><strong>Comments</strong></p>

<p><strong><em>Consumption Smoothing</em></strong> 
Under convexity and regularity assumptions on <span class="math-container">$U$</span>, it is clear from the FOC that agents in this economy trade <span class="math-container">$t = 1$</span> claims at <span class="math-container">$t = 0$</span> in order to smooth their <span class="math-container">$t = 1$</span> consumption across states. </p>

<p><strong><em>Sequential Trading</em></strong> 
In this economy, there is a futures/forward market
at <span class="math-container">$t = 0$</span> and a spot market at <span class="math-container">$t =1$</span>. Agents trade today subject to their budget constraint today, same for tomorrow.</p>

<p><strong><em>Rational Expectations</em></strong> 
However, agents correctly anticipates the <span class="math-container">$t=1$</span> spot price at <span class="math-container">$t = 0$</span>. Equilibrium is self-fulfilling.</p>

<p><strong><em>Radner vs. Arrow-Debreu</em></strong> 
In an <em>Arrow-Debreu equilibrium</em>, agents trade <span class="math-container">$t=0$</span> state-contingent claims ("Arrow-Debreu securities"). There is no <span class="math-container">$t=1$</span> 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:</p>

<p><em>If asset returns <span class="math-container">${r_{sk}}$</span> is complete, then an Arrow-Debreu equilibrium can be implemented by a Radner equilibrium.</em></p>

<p>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. </p>

CC BY-SA 4.0
2019-07-04T20:02:27.557Z
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?
Incognito
CC BY-SA 4.0
2019-07-05T09:36:42.697Z
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.
Michael
CC BY-SA 4.0
2019-07-14T17:53:28.937Z
"Assume the exogenous r=1": what is "r" and what does it change?
Incognito
CC BY-SA 4.0
2019-07-15T03:38:05.867Z
Who would be this unmodelled agent that absorbs excess supply?
Incognito
CC BY-SA 4.0
2019-07-15T16:53:23.723Z
$r = [r_{sk}]$ is the exogenous asset return structure/matrix. Whether market is complete depends on whether the return matrix has full rank.
Michael
CC BY-SA 4.0
2019-07-15T16:56:25.840Z
"Unmodelled" implies that, by definition, you don't know who they are nor do you really care as a modeler---"noise traders", in the finance literature. One could easily insist that the futures/forward market clears and remove that consideration.
Michael
CC BY-SA 4.0
2019-07-15T16:56:48.323Z
OK. r= 1 means that a 1 dollar asset yields 1 dollar in the following period, right? Also, what does r=1 or r=0.5 changes in your example?
Incognito
CC BY-SA 4.0
2019-07-15T16:59:49.017Z
$r = 1$ means if a futures contract is $z$ dollars today, it's also $z$ dollars tomorrow.
Michael
CC BY-SA 4.0
2019-07-04T10:36:37.630Z
2019-07-15T16:50:42.493Z
2019-07-15T16:50:42.493Z
Michael
<p>This is definition of Radner Equilibrium from Microeconomic Theory (Mas-Collel, Whinston and Green - Third Edition).</p>
<p><a href="https://i.sstatic.net/7XKJV.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7XKJV.jpg" alt="enter image description here" /></a></p>
<p>I'm confused about two conditions:</p>
<p><span class="math-container">$\sum_k q_k \cdot z_{ki} \le 0$</span> (in yellow)</p>
<p>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 <span class="math-container">$-10+10=0$</span></p>
<p><span class="math-container">$\sum_i z^*_{ki} \le 0$</span> (in purple)</p>
<p>This is proposed as a feasibility condition. Why?</p>

CC BY-SA 4.0
2019-07-02T11:06:52.210Z
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.
Bayesian
CC BY-SA 4.0
2019-07-02T11:28:37.563Z
@Bayesian That's a great insight, but there's no central planner in the model
Incognito
<microeconomics><general-equilibrium><uncertainty><radner-equilibrium>
Conditions for Radner Equilibrium
Incognito
# Conditions for Radner Equilibrium

(Asked by: username_0 on 2019-07-01)

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

[![enter image description here](https://i.sstatic.net/7XKJV.jpg)](https://i.sstatic.net/7XKJV.jpg)

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?

**Tags:** microeconomics, general-equilibrium, uncertainty, radner-equilibrium

> **Comments:**
> **username_1**: 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.  
> **username_0**: @username_1 That's a great insight, but there's no central planner in the model  

---

## 1 answers

### Answer by username_2 on 2019-07-04. Score: 2

*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.

> **Comments:**
> **username_0**: 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?  
> **username_2**: 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.  
> **username_0**: "Assume the exogenous r=1": what is "r" and what does it change?  
> **username_0**: Who would be this unmodelled agent that absorbs excess supply?  
> **username_2**: $r = [r_{sk}]$ is the exogenous asset return structure/matrix. Whether market is complete depends on whether the return matrix has full rank.  
> **username_2**: "Unmodelled" implies that, by definition, you don't know who they are nor do you really care as a modeler---"noise traders", in the finance literature. One could easily insist that the futures/forward market clears and remove that consideration.  
> **username_0**: OK. r= 1 means that a 1 dollar asset yields 1 dollar in the following period, right? Also, what does r=1 or r=0.5 changes in your example?  
> **username_2**: $r = 1$ means if a futures contract is $z$ dollars today, it's also $z$ dollars tomorrow.  

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