Looking for Evidence of Time-Inconsistent Preferences in Asset Market Data

This study argues that strong evidence contradicting the traditional assumption of time-consistent preferences is not available. The study builds and analyzes the implications of a deterministic general equilibrium model and compares them to data from the U.S. asset market. The model implies that (1) because of dynamic arbitrage, the prices of retradable assets cannot reveal whether preferences are time-inconsistent; but (2) the prices of commitment assets, investments which must be held for their lifetime, can. These prices will be higher than the present values of their future payoffs only when preferences are timeinconsistent. And (3) when preferences are time-inconsistent, people will not hold both retradable and commitment assets. Empirical observations on two examples of commitment assets—education and individual retirement accounts— are not consistent with these model implications. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Jan is about to go out to her neighborhood bar. Before drinking anything there, Jan would like to sign a legally binding contract stating that she is allowed to drink only four beers that night. Why does she want to sign such a contract? She knows that after having four beers, she will want to have a fifth, and she wants to prevent herself from doing so. Jan is exhibiting what economists call time-inconsistent preferences: her preferences for beer, at a given date and state, change over time without the arrival of new information. An essential feature of time-inconsistency is the desire for self-commitment. People like Jan with time-inconsistent preferences are willing to pay a cost to restrict their future choices. Until recently, economists have typically assumed that the preferences of most people are consistent over time. In the last five years, however, research into the consequences of time-inconsistency has increased. Much of this has been spurred by the work of Laibson (1997) on consumption and saving. Laibson assumes that people would like to be able to commit to save more at some future date than they think they actually would otherwise save when they get to that date. He then considers the consequences of these kinds of preferences for standard macroeconomic phenomena like the covariation of household consumption and income and the level of household saving. Laibson argues that a considerable amount of introspective and experimental evidence supports his formulation of time-inconsistent preferences. However, switching from a standard modeling strategy to one with time-inconsistent preferences can dramatically change a model’s implications for economic policy. So to make such a switch, we need to have more than introspective and experimental evidence that preferences are time-inconsistent; we need supportive evidence from actual choices that affect actual outcomes. In this study, I ask, Can we see that sort of evidence in asset market data? My answer is that we cannot see this evidence in the prices of retradable assets. Rather, we need to look at the prices and holdings of what I call commitment assets. I determine this by building and analyzing the implications of a deterministic, three-period general equilibrium model. In this model economy, agents can, in the initial period, trade a one-period (short-term) bond, a two-period (long-term) bond, and a commitment asset which, as the name implies, is an investment that must be held for its lifetime. The long-term bond can be retraded in the second period; the commitment asset cannot be. Also, agents cannot borrow in the second period against the future proceeds of the commitment asset. When the agents in this model economy have timeinconsistent preferences, they have three utility discount factors. They have two in period 1, one to discount the utility of consumption between periods 1 and 2 and one to discount that utility between periods 2 and 3. And they have another in period 2, also to discount utility between periods 2 and 3. Preferences are time-inconsistent if and only if the utility discount factor between periods 2 and 3 is different in period 2 than in period 1. I follow Laibson by restricting attention to the case in which the value of this discount factor in period 1 is greater than or equal to its value in period 2; over time, that is, the discount factor may decline. In this model economy, I prove three results. First I examine the informational content of the prices of the shortand long-term bonds. Intuition might suggest that these prices are enough to tell us whether or not preferences are time-inconsistent. Again, the economy has three bond prices and three discount factors. Since we have three observable variables and three unknown parameters, it might seem plausible that we should be able to figure out whether preferences are time-consistent or time-inconsistent. As my first result, I prove that this line of reasoning is wrong: in any equilibrium, bond prices are consistent with the discount factor between periods 2 and 3 being the same in period 1 as in period 2. The mistake in the intuitive reasoning is that it ignores dynamic arbitrage. Regardless of the form of preferences, the period 1 relative price between the two bonds must equal the period 2 price of the longterm bond; otherwise, agents can make arbitrage profits. The two bond prices are thus not independent sources of information about the two discount factors. To try to learn about time-inconsistent preferences, we must turn instead to price data on the commitment asset. My second result is that we can, indeed, tell from the price of the commitment asset whether or not agents’ preferences are time-inconsistent. In particular, I prove that the price of the commitment asset is higher than the present value of its future payoffs if and only if preferences are time-inconsistent. In effect, time-inconsistent people value commitment, and this value shows up in the price of the commitment asset. My final result concerns agents’ holdings of the commitment asset and the long-term bond. I prove that if preferences are time-inconsistent, then all agents’ asset holdings are exclusive: in period 3, agents receive all income from either the commitment asset or the long-term bond; they don’t receive income from both types of assets. If some agents held both the commitment asset and the longterm bond at the end of period 2, then, on the margin, the commitment asset would provide no commitment. The agents could always reduce their consumption in period 3 by lowering their holdings of the bond. Because in this case, the commitment asset and the long-term bond would be marginally equivalent, they would have the same price. But this contradicts the second result. Therefore, in period 3, if preferences are time-inconsistent, the holdings of the commitment asset and the long-term bond must be exclusive. How do these implications of time-inconsistent preferences compare to empirical observations from the U.S. asset market? To answer that question, I examine evidence about two good examples of commitment assets: education and individual retirement accounts (IRAs). Contrary to the implications of time-inconsistent preferences, neither asset seems to have an unusually low after-tax return. And virtually all agents with education or IRAs also have highly liquid bank accounts or highly collateralizable housing. This contradicts the exclusive holdings result. I conclude that there is little evidence from data on these two commitment assets against the traditional assumption that preferences are time-consistent. A Model With Time-Inconsistent Preferences I start by developing a general equilibrium model of asset pricing inwhichpeoplehave time-inconsistentpreferences. The model has a unit measure of people, indexed by j ∈ [0,1], who all live for three periods. The world is deterministic and has a single perishable consumption good. Each agent is endowed with y1 units of consumption in period 1. Agents are each also endowed with three assets. I refer to the first two assets as bonds. Each agent is endowed with b̄2 units of a short-term bond that pays off one unit of consumption in period 2; each agent is also endowed with b̄3 units of a long-term bond that pays off one unit of consumption in period 3 but can be retraded in period 2. The last asset is a commitment asset which cannot be retraded. It pays off one unit of consumption in period 3, and each agent is endowed with b̄ c 3 om units of it. These three assets are the entire endowment of the J agents; hence, the per capita endowment in periods 2 and 3 is given by y2 = b̄2 and y3 = b̄3 + b̄ c 3 . All agents have identical preferences over future consumption streams. However, these preferences may change over time. In particular, the agents’ preferences over consumption streams (c1,c2,c3) in period 1 are representable by the utility function u(c1) + β12[u(c2) + β23u(c3)] where the β’s here represent discount factors in period 1, β12 between periods 1 and 2 and β23 between periods 2 and 3. The agents’ preferences over consumption streams (c2,c3) in period 2 are representable by this utility function: