<p>I am faced with defining an equilibrium for a two person (infinitely lived) market with borrowing and lending of one period bonds. Their utility of consumption is just <span class="math-container">$\sum^\infty_{t=0} \beta^t ln(c^i_t)$</span> with endowments <span class="math-container">$e^1_t = (8,2,8,2,8,...)$</span> and <span class="math-container">$e^2_t = (2,8,2,8,...)$</span></p>
<p>Additionally these individuals have some share of &quot;trees&quot; that give a fixed amount of some good and can be traded. <strong>There is no uncertainty in this model.</strong> If this were a regular  i.e complete market, I would know how to define the ADE but I am told that the there is limited enforcement, i.e. contracts must satisfy a no-default constraint. In this context default means that at any point in time agents can walk away from the contract and the punishment is that their trees are seized and they are prevented from borrowing-lending in the future.</p>
<p>I do not know how this would fit/ change the definition of the AD equilibrium. My thinking is that agents would never renege on their contracts because this would lead to a decrease in total consumption over all periods. So when they maximize they wouldn't even consider defaulting. I can't find anything in my notes about this nor in Dick Kreuger's macro theory notes.</p>
<p>Any help would be appreciated!</p>

CC BY-SA 4.0
2023-10-15T22:38:34.743Z
Are you sure you are supposed to define a version of an AD equilibrium? There are equilibrium notions for such models that have nothing to do with prices. See for example http://www.sfu.ca/~kkasa/kocherla_96.pdf
Michael Greinecker
CC BY-SA 4.0
2023-10-16T00:23:46.597Z
I am sure. The question specifically says "Define an Arrow Debreu Equilibrium in this case". I thought it was odd which is why I think its kind of a trick question?
KeithRoberts
CC BY-SA 4.0
2023-10-16T07:03:48.037Z
One thing you could do is restrict the budget set to consumption streams such that your continuation value at each period is at least what you would get from deviating, getting the trees seized and being forced to consume the endowment.
Michael Greinecker
<macroeconomics><academic-graduate>
Defining an Arrow Debreu Equillibrium with a No Default Constraint
KeithRoberts
# Defining an Arrow Debreu Equillibrium with a No Default Constraint

(Asked by: username_0 on 2023-10-15)

I am faced with defining an equilibrium for a two person (infinitely lived) market with borrowing and lending of one period bonds. Their utility of consumption is just $\sum^\infty\_{t=0} \beta^t ln(c^i\_t)$ with endowments $e^1\_t = (8,2,8,2,8,...)$ and $e^2\_t = (2,8,2,8,...)$

Additionally these individuals have some share of "trees" that give a fixed amount of some good and can be traded. **There is no uncertainty in this model.** If this were a regular i.e complete market, I would know how to define the ADE but I am told that the there is limited enforcement, i.e. contracts must satisfy a no-default constraint. In this context default means that at any point in time agents can walk away from the contract and the punishment is that their trees are seized and they are prevented from borrowing-lending in the future.

I do not know how this would fit/ change the definition of the AD equilibrium. My thinking is that agents would never renege on their contracts because this would lead to a decrease in total consumption over all periods. So when they maximize they wouldn't even consider defaulting. I can't find anything in my notes about this nor in Dick Kreuger's macro theory notes.

Any help would be appreciated!

**Tags:** macroeconomics, academic-graduate
> **Comments:**
> **username_1**: Are you sure you are supposed to define a version of an AD equilibrium? There are equilibrium notions for such models that have nothing to do with prices. See for example http://www.sfu.ca/~kkasa/kocherla_96.pdf  
> **username_0**: I am sure. The question specifically says "Define an Arrow Debreu Equilibrium in this case". I thought it was odd which is why I think its kind of a trick question?  
> **username_1**: One thing you could do is restrict the budget set to consumption streams such that your continuation value at each period is at least what you would get from deviating, getting the trees seized and being forced to consume the endowment.  

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