Golden Eggs and Hyperbolic Discounting

I. Introduction

"Use whatever means possible to remove a set amount of money from your bank account each month before you have a chance to spend it" —New York Times "Your Money" column [1993]

Many people place a premium on the attribute of self-control. Individuals who have this capacity are able to stay on diets, carry through exercise regimens, show up to work on time, and live within their means. Self-control is so desirable that most of us complain that we do not have enough of it. Fortunately, there are ways to compensate for this shortfall. One of the most widely used techniques is commitment. For example, signing up to give a seminar is an easy way to commit oneself to write a paper. Such commitments matter since they create constraints (e.g., deadlines) that generally end up being binding.

Strotz [1956] was the first economist to formalize a theory of commitment and to show that commitment mechanisms could be potentially important determinants of economic outcomes. He showed that when individuals' discount functions are nonexponential, they will prefer to constrain their own future choices. Strotz noted that costly commitment decisions are commonly observed:

… we are often willing even to pay a price to precommit future actions (and to avoid temptation). Evidence of this in economic and other social behaviour is not difficult to find. It varies from the gratuitous promise, from the familiar phrase "Give me a good kick if I don't do such and such" to savings plans such as insurance policies and Christmas Clubs which may often be hard to justify in view of the low rates of return… Joining the army is perhaps the supreme device open to most people, unless it be marriage for the sake of "settling down."

Strotz's list is clearly not exhaustive. In general, all illiquid assets provide a form of commitment, though there are sometimes additional reasons that consumers might hold such assets (e.g., high expected returns and diversification). A pension or retirement plan is the clearest example of such an asset.

There exists a class of assets that provide a store of illiquid value, like savings bonds, and certificates of deposit. All of the illiquid assets discussed above have the same property as the goose that laid golden eggs. The asset promises to generate substantial benefits in the long run, but these benefits are difficult, if not impossible, to realize immediately. Trying to do so will result in a substantial capital loss.

Instruments with these golden eggs properties make up the overwhelming majority of assets held by the U.S. household sector. For example, the Federal Reserve System publication Balance Sheets for the U.S. Economy 1945–94 reports that the household sector held domestic assets of $28.5 trillion at year-end 1994. Over two-thirds of these assets were illiquid.

Despite the abundance of commitment mechanisms, and Strotz's well-known theoretical work, intrapersonal commitment phenomena have generally received little attention from economists. This deficit is probably explained by the fact that commitment will only be chosen by decision-makers whose preferences are dynamically inconsistent, and most economists have avoided studying such problematic preferences. However, there is a substantial body of evidence that preferences are dynamically inconsistent.

Research on animal and human behavior has led psychologists to conclude that discount functions are approximately hyperbolic [Ainslie 1992]. Hyperbolic discount functions are characterized by a relatively high discount rate over short horizons and a relatively low discount rate over long horizons. This discount structure sets up a conflict between today's preferences, and the preferences that will be held in the future.

II. The Consumption Decision

I consider a highly stylized commitment technology that is amenable to an analytic treatment. Specifically, I assume that consumers may invest in two instruments: a liquid asset x and an illiquid asset z. Instrument z is illiquid in the sense that a sale of this asset has to be initiated one period before the actual proceeds are received. So a current decision to liquidate part or all of an individual's z holding will generate cash flow that can be consumed no earlier than next period. By contrast, agents can always immediately consume their x holdings.

I adopt the following discount structure to capture the qualitative properties of a generalized hyperbolic discount function: events t periods away are discounted with factor (1 + at)^−γ/a.

I call the discount structure in equation (1) "quasi-hyperbolic." Note that the quasi-hyperbolic discount function is a discrete time function with values {1, βδ, βδ², βδ³, …}. When 0 < β < 1, the discount structure mimics the qualitative property of the hyperbolic discount function, while maintaining most of the analytical tractability of the exponential discount function.

The constraint x_t ≥ 0 rules out forced savings contracts. If the consumer could set x_t to any negative value, then she could perfectly commit her future savings behavior. Two arguments support this implicit assumption against forced savings contracts.

First, such contracts are susceptible to renegotiation by tomorrow's self, and in any finite-horizon environment, the contract would unwind. Second, such contracts are generally unenforceable in the United States. U.S. courts will generally not enforce contracts with a penalty of this kind — courts allow "liquidated damages" which reflect losses likely to be experienced by the promisee, but courts do not allow "penalties" which do not reflect such losses [Goetz and Scott 1977].

III. Equilibrium Strategies

This section characterizes the equilibrium strategies of the game described above. To analyze equilibrium behavior when preferences are dynamically inconsistent, it is standard practice to formally model a consumer as a sequence of temporal selves making choices in a dynamic game (e.g., Pollak [1968], Peleg and Yaari [1973], and Goldman [1980]).

Hence, a T-period consumption problem translates into a T-period game, with T players, or "selves," indexed by their respective periods of control over the consumption decision. I look for subgame perfect equilibrium (SPE) strategies of this game.

THEOREM 1. Fix any T-period consumption game with exogenous variables satisfying A1. There exists a unique resource-exhausting joint strategy, s* ∈ S, that satisfies P1–P4, and this strategy is the unique subgame perfect equilibrium strategy of this game.

IV-A. Comovement of Consumption and Income

There is a growing body of evidence that household consumption flows track corresponding household income flows "too" closely, generating violations of the life-cycle/permanent-income consumption model. In particular, household consumption is sensitive to expected movements in household income (e.g., Hall and Mishkin [1982], Zeldes [1989], Carroll and Summers [1991], etc.). Many of these authors find that consumption tracks expected income changes even when consumers have large stocks of accumulated assets.

Several models have been proposed to explain the consumption-income comovement. Carroll [1992] proposes a buffer-stock theory... Attanasio and Weber [1993] argue that demographic dynamics explain much of it.

The golden eggs model provides a new explanation for the observed comovement in consumption and income. In the model, self t−1 chooses x_t−1 to constrain the consumption of self t. In this way "early" selves manipulate the cash flow process by keeping most assets in the illiquid instrument. In equilibrium consumption is exactly equal to the current level of cash flow: c_t = y_t + R_tx_t−1.

An example may help to make this more concrete. Let the horizon be infinite. Assume that labor income follows a trending high-low process: y_t = ȳ·e^gt when t is odd, y_t = y·e^gt when t is even. Assume that the interest rate is constant and exp(ρg) = δR.

A regression of Δln c_t on Δln y_t yields a coefficient of .40. Since the income process is completely deterministic, this implies that predictable changes in income are associated with changes in consumption. Hence, consumption tracks income despite the fact that the consumer in this example controls a substantial asset stock (K/Y ≈ 3).

IV-B. Aggregate Saving

In most intertemporal rational choice models, high discount rates are a necessary condition for consumption-income comovement. Such relatively high discount rates, however, tend to imply relatively low levels of capital accumulation in general equilibrium (see Aiyagari [1992]).

The golden eggs model generates consumption-income comovement even when actors are wealthy. This is because in equilibrium decisions to dissave out of the illiquid asset stock do not depend on β. Self t is not able to consume the illiquid asset immediately, so self t does not consider trade-offs between consumption today and consumption tomorrow when dissaving from the illiquid instrument. The value of β is superfluous for such a decision—from self t's perspective—and hence the steady state capital stock is independent of β.

PROPOSITION 1. In the economy described above there exists a unique steady state that satisfies A1. In that steady state

exp(ρg) = δR

IV-C. Asset-Specific MPCs

Thaler [1990] argues that consumers have different marginal propensities to consume for different categories of assets. He presents evidence that an unexpected increase in the value of an equity portfolio will have a very small effect on consumption, while an unexpected job-related bonus will be immediately consumed. He cites a wide body of evidence which suggests that "the MPC from [current income] is close to unity, the MPC from [future income] is close to zero, and the MPC from [net assets] is somewhere in between."

Thaler explains this behavior by postulating that consumers use a system of nonfungible mental accounts to guide rule-of-thumb decision-making. By contrast, the golden eggs model predicts that even fully rational consumers will exhibit asset-specific MPCs.

IV-E. Declining Savings Rates in the 1980s

The golden eggs model may help to explain the decline in U.S. savings rates during the 1980s. The first explanation is driven by the fact that during the 1980s a relatively large proportion of national income was realized as cash flow to consumers. Hatsopoulos, Krugman, and Poterba [1989] document the observation that cash flow to consumers (as a percentage of NNP) was high during the 1980s relative to the 1970s.

However, I am unsatisfied with this first story for reasons that I describe below. The golden eggs model can only explain the high MPC out of cash flow; the model cannot explain why the cash flow was high in the first place. Hence, application of the golden eggs model may only relabel the puzzle...

The golden eggs model suggests a second explanation for the low level of savings during the past decade. The 1980s was a period of rapid expansion in the U.S. consumer credit market. Increasing access to instantaneous credit has reduced the effectiveness of commitment devices like illiquid assets.

In 1970 only 16 percent of all U.S. families had a third party credit card... By 1989 54 percent had one. From 1970 to 1995, revolving credit grew from 3.7 percent to 36.3 percent of total consumer credit.

PROPOSITION 5. Consider the general equilibrium economy analyzed above, but now assume that consumers can instantaneously borrow against their illiquid asset. This economy is equivalent to one in which there is no illiquid asset (i.e., x is the only asset).

COROLLARY. In the steady state characterized in Proposition 5 the capital-output ratio is less than the steady state capital-output ratio in the economy with the commitment technology.

Table I shows that for a β value of 0.6, elimination of the commitment technology raises the steady state real interest rate 1.3 percentage points. This corresponds to a reduction in the capital-output ratio of 0.3. The ratio of national net worth to gross national product fell from an average of 3.2 from 1946–1984, to 2.8 in 1994.

IV-F. Welfare Analysis of Financial Innovation

The introduction of instantaneous credit increases consumers' choice sets. Standard economic models imply that this development might lower levels of capital accumulation, but would raise consumer welfare. Yet, in the United States, policy-makers and pundits are concerned that instantaneous credit is somehow bad for consumers.

The golden eggs framework provides a formal model of the costs of financial innovation. By enabling the consumer to instantaneously borrow against illiquid assets, financial innovation eliminates the possibility for partial commitment. This has two effects on the welfare of the current self. First, the current self no longer faces a self-imposed liquidity constraint and can therefore consume more in its period of control. Second, future selves are also no longer liquidity constrained and may also consume at a higher rate out of the wealth stock that they inherit. The first effect makes the current self better off. The second effect makes the current self worse off ... Under most parameterizations the impact of the second effect dominates, and the welfare of the current self is reduced.

V. Evaluation and Extensions

The model helps to explain many of the empirical puzzles in the consumption literature, notably consumption-income tracking and asset-specific MPCs.

However, the model has several drawbacks that suggest four important areas to pursue extensions:

First, the golden eggs model does not explain how consumers accumulate assets in the first place.

A second problem associated with the model is the anomalous prediction that consumers will always face a binding self-imposed liquidity constraint... predicting that consumers should have no liquid funds left.

A third problem is that some consumers may not need to use external commitment devices to achieve self-control. Consumers may have internal self-control mechanisms, like "will power".

The fourth problem is that some consumers may have access to an array of "social" commitment devices that are far richer than the simple illiquid asset... like marriage, work, and friendship.