# Does the Lucas (1978) asset pricing model feature complete markets?

The Lucas (1978) asset pricing model seems to be one of the workhorse models in finance / asset pricing models. It also seems to be the case that the environment, with claims to $$n$$ (exogenous) productive units traded, features complete markets. I have also seen another version of this model with the agent trading a single stock and a single bond that the instructor told me had complete markets.

To me, this is not obvious at all -- the agent does not have access to Arrow securities for each state, or something of that sort. In fact, in the original paper, the states are continuous, and this makes it even harder for me think about market completeness.

Can anyone tell me why (or perhaps why not) this model features market completeness?

Any help would be greatly appreciated (or pointers to relevant links would work). Thanks!

Edit #1: ----------------------------------------

Just to clarify, while I understand that in equilibrium the net holdings (of all the identical agents) must be zero, my concern was that having access to AD securities vs. not having access may have asset pricing implications. What I mean by this is that although the outcome in terms of asset holdings must (almost by construction) be equal in any case, but perhaps the presence of AD securities may distort the prices of other assets (e.g. equity claims "trees").

Intuitively, perhaps the price of trees must be different in the presence of AD securities to offset the agent's desire to hold the AD securities in equilibrium. My next logical step in determining whether or not this is the case, was to think about whether the markets were complete or not in the first place -- if markets were already complete, then the additional AD securities are "redundant" is some sense, and prices should not change. If markets are incomplete, my initial intuition was that this may possibly have consequences (not in asset positions, but in prices).

To me, this is not obvious at all, and something I would really like to see a proof of.

• Thanks for the edit. My thoughts are that the proof is trivial. Consider one model where the agent can only invest in the trees. Trying to write notation compactly (loosely speaking), let this first model be $\max_{c, x} U(\{c\})$ st $\{c,x\} \in \Gamma$ . Then consider another model where the agent invests in trees and AD securities that are in zero net supply. This is $\max_{c,x,a} U(\{c\})$ st $\{c,x\} \in \Gamma$ and st $a_{i,t} = 0$ $\forall i,t$, where $a_{it}$ are the AD security holdings for state $i$ at time $t$. The models should result in the exact same outcomes, right? – jmbejara Mar 21 at 20:51
• Thanks for the continuous update! This has been very helpful, and I have accepted your answer. – mathsquestions1 Mar 25 at 22:52
• No problem! Good question (and feedback)! – jmbejara Mar 26 at 1:57

It seems to me that we might as well say that markets are complete. It seems to me to be somewhat inconsequential since this is a representative agent model and that market clearing requires that the only assets that the agent may hold in positive net supply are the $$n$$ trees. The prices of all Arrow-Debreu securities may be inferred from shadow prices. Suppose the optimization problem is rewritten such that the agent may invest in the $$n$$ trees plus the set of all Arrow-Debreu securities. (There will be redundancies---I'm hand-waving a little.) The market clearing conditions say that, in equilibrium, the agent must have a unit holding in each of the $$n$$ trees and zero holdings in the rest of the assets and the price of each of these securities in equilibrium must be set to support these holdings. Since market completeness is a matter of whether or not the agent may trade in all of these securities, we may interpret the situation a two different ways: either the agent can invest in the Arrow-Debreu securities (as well as the trees) but the AD securities must be in zero net supply or the agent cannot invest in the AD securities (only in the trees). In either case, the result is the same.

On the other hand, if you had a model with more than one agent and they had different endowment processes, then the question of market completeness would matter for equilibrium outcomes.

Reference

Take a look at the notes here.

We have used the equilibrium of the model to determine a single price, the price of a tree. With trading in trees, markets are complete. However we can introduce additional (redundant) assets and use the same approach to determine their prices.

Your question is somewhat related to the question asked here: "Complete Markets in Continuous Time". See the solutions given there.

In addition, you can also find a discussion in these notes by Markus Brunnermeier:

If the horizon is infinite, the number of events is also infinite. Does that imply that we need an infinite number of assets to make the market complete?

• List item
• Do we need assets with all possible times to maturity and events to have a complete market?
• No. Dynamic completion.
• Arrow (1953) and Guesnerie and Jaffray (1974)

The conclusion, which is in the slide titled "How many assets to complete market?", is that, in discrete time, it depends on the "branching number" (see the slides) and that, in continuous time, you can get complete markets simply with a risk-free bond and $$n$$ independent risky assets where $$n$$ is the number of independent Brownian motion underlying the economy.

• Thank you for the helpful answer. One last question -- the original paper is in discrete time, continuous state (not vice versa). Does the argument still go through in a similar way in that particular setting as well? – mathsquestions1 Mar 19 at 19:38
• You're right. My apologies. I've rewritten the answer the best I could. It may still have errors, though. Let me know what you think. – jmbejara Mar 20 at 6:45