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  • 标题:How many paychecks? An example of a self-imposed constraint.
  • 作者:Archibald, Robert B.
  • 期刊名称:Economic Inquiry
  • 印刷版ISSN:0095-2583
  • 出版年度:1994
  • 期号:October
  • 语种:English
  • 出版社:Western Economic Association International
  • 摘要:The case of Ulysses and the sirens is one of the most interesting puzzles for the model of rational behavior.(1) Ulysses knew enough about himself to know that he would not be able to control his behavior when under the spell of the sirens, so he had himself bound to the mast, had wax put into the ears of his crew, and instructed them to completely disregard any gestures he might make. It is reasonable to ask, if Ulysses knew enough to know that he would lose control when under the spell of the sirens, why didn't he know enough to maintain control? More generally, if you are rational enough to know you are going to be irrational, why can't you stop the irrationality? Apparently things are not so simple. Psychiatrist George Ainslie [1975; 1992] argues, based upon a large body of experimental evidence in psychology, that individuals have an inherent tendency to be myopic, and this leads them to have to develop rules of behavior which limit options. Given the story of Ulysses, the process of imposing such constraints on oneself is sometimes called binding.
  • 关键词:Consumer behavior;Self control;Self-control

How many paychecks? An example of a self-imposed constraint.


Archibald, Robert B.


I. INTRODUCTION

The case of Ulysses and the sirens is one of the most interesting puzzles for the model of rational behavior.(1) Ulysses knew enough about himself to know that he would not be able to control his behavior when under the spell of the sirens, so he had himself bound to the mast, had wax put into the ears of his crew, and instructed them to completely disregard any gestures he might make. It is reasonable to ask, if Ulysses knew enough to know that he would lose control when under the spell of the sirens, why didn't he know enough to maintain control? More generally, if you are rational enough to know you are going to be irrational, why can't you stop the irrationality? Apparently things are not so simple. Psychiatrist George Ainslie [1975; 1992] argues, based upon a large body of experimental evidence in psychology, that individuals have an inherent tendency to be myopic, and this leads them to have to develop rules of behavior which limit options. Given the story of Ulysses, the process of imposing such constraints on oneself is sometimes called binding.

The Christmas Club is the traditional example of a self-imposed constraint.(2) The local credit union offers an account which has all of the attributes of the normal savings account and the added characteristic that the account holder cannot make any withdrawals until a specified date close to Christmas. Given that the lack of flexibility inherent in the Christmas Club is combined with no compensating advantage of any sort (e.g. higher interest rates), standard analysis suggests the Christmas Club would have no takers. But in fact there are many Christmas Club members.(3) Apparently a large number of individuals feel they cannot trust themselves to save money for Christmas shopping in a normal savings account. The Christmas Club gives them a way of tying their hands. It is a self-imposed constraint.

In this paper, I examine behavior which might also be an example of a self-imposed constraint. At the College of William and Mary a full-time faculty member has the option of receiving her salary in eighteen equal payments over the nine-month academic year or in twenty-four equal payments over the entire year starting in September. The twenty-four month option has the look of a self-imposed constraint.(4) The faculty member who takes this option is foregoing interest earnings from a simple plan which deposits eighteen payments, earns interest, and finances the summer. Perhaps the faculty members who elect the twenty-four payment option are like Ulysses; they know enough to know that they will not follow the plan they know enough to make.

A preference for the twenty-four payment option is not necessarily evidence of a self-imposed constraint. Taking eighteen payments, squirreling away some of each payment, and financing one's summer expenditures from what had accumulated during the academic year is a costly endeavor. Therefore a standard economic model based on transactions costs could also explain a preference for the twenty-four payment option. The purpose of this paper is to present a test of these two models.

II. THEORETICAL ANALYSES OF THE CHOICE OF THE TWENTY-FOUR PAYMENT OPTION

A Standard Economic Model

The task at hand is to explain the demand for the twenty-four payment option as compared to the eighteen payment option. Since the total salary is the same in both cases, at any nonzero interest rate the present value of the eighteen payment option dominates the present value of the twenty-four payment option. This suggests that on a strict basis of economic rationality, there would be no takers for the twenty-four payment option. At William and Mary, however, over 67 percent of the faculty choose the twenty-four payment option. There is, however, a clear difficulty with simply comparing present values in this case. Such a comparison implicitly assumes that trips to the bank are costless. A complete model should consider transactions costs.(5)

To consider transaction costs we need to specify the two options in more detail. First, since all paychecks are automatically deposited in banks, a person on the eighteen payment option would have to make twenty-four trips to the bank. On the first eighteen, she would transfer funds between accounts to build up a store of funds in a savings account, and on the last six she would make withdrawals from the savings account. These twenty-four trips to the bank take time, and time has an opportunity cost. We can assume that the opportunity cost of time is related to salary. Let z represent the per trip proportion of annual salary which is the opportunity cost of a trip.

The benefits of a trip to the bank are the excess over one twenty-fourth of annual salary which is available if one appropriately uses a savings account. Assume the faculty member has eighteen semi-monthly checks deposited in an interest-bearing account. Let X be the amount which can be withdrawn from this account twice a month such that the account has a positive balance until August 15th at which time the withdrawal of X will bring the balance to zero. Given this, the semi-monthly benefits of the eighteen-payment strategy are X - (1/24)S, where S is the annual salary.

To find X, we need to consider the balance in a savings account in the twenty-fourth period (after August 15). This balance should equal zero. At the end of the first eighteen periods in which checks are coming in, the accumulated balance will be:

(1) [B.sub.18] = [summation of] (S/18 - X) [(1 + r).sup.t-1] where t=1 to 18.

Thereafter the account will shrink in value by X each period while still accumulating interest. In the twenty-fourth period the balance would be:

(2) [B.sub.24] = [summation of] (S/18) [(1+r).sup.t-1] where t=7 to 24

- [summation of] X[(1 + r).sup.t-1] where t=1 to 24.

Setting [B.sub.24] equal to zero and solving for X yields:

(3) X = [[summation of] [(1 + r).sup.t-1] where t=7 to 24/[summation of] [(1 + r).sup.t-1]] where t=1 to 24 (S/18).

Equation 3 tells us that the benefits of the eighteen payment option depend upon the interest rate. Let b represent the expression in square brackets. If r = 0, b would be 18/24 = .75, and X is 1/24 of salary, the twenty-four payment option. As r increases from zero, b rises above .75. For example, if the interest rate is 1 percent, the value of b is .77192.

An example gives us a feel for the amount of money involved. If the semi-monthly interest rate is one quarter of a percent (with semi-monthly compounding this amounts to an annual rate of 6.176 percent), a faculty member who has an annual salary of $54,000 (the average salary in 1991-92 at William and Mary was $53,970) would have gross pay of $2,250 under the twenty-four-paycheck option and gross pay of $2,266.67 using the strategy described here. The extra $16.67 accrues twenty-four times a year. This is a considerable advantage to someone who takes the eighteen-payment option. Alternatively, we can consider the possibility that a faculty member did not follow the strategy outlined above, but rather took the eighteen-paycheck option and administered the twenty-four-paycheck option herself. Using the interest rate and salary above, this person would have $414.24 in a savings account at the end of the year. This represents an approximation of the gift the state of Virginia (William and Mary is a state school) receives from every faculty member who takes the twenty-four-paycheck option. More importantly, it is the cost of having someone else administer this policy.

The advantage of the eighteen-payment option, the benefits minus the cost, is then given by:

(4) NB = b(S/18) - (S/24) - z S.

It is important to recognize that both the benefits and costs are linear functions of salary. The trigger between NB [is greater than] 0 and NB [is less than] 0 depends upon the relationship between z, the opportunity cost of time spent shifting money between accounts and b, the interest rate term which determines the benefits of the shifting.

Self-Imposed Constraints

The demand for self-imposed constraints comes from a person recognizing that the strategy outlined in the last section is not the only feasible strategy. That strategy called for a constant deposit in the savings account and constant withdrawals. There are many other potential strategies which would also yield X dollars available for each of the semi-monthly periods in the summer. For example, there is the strategy in which one saves less than (S/18 - X) during the first half of December and makes up for it with extra savings, covering both lost principle and interest, at some later date. If one is scrupulous in such dealings, there is no loss from making these kinds of shifts. This is a big if.

I have argued elsewhere that the mismanagement of these kinds of trades is important for the analysis of self-control problems in situations in which it is the average over time of some activity which requires control.(6) Consider diet as an example. Self-control problems have a chance to start when a person convinces himself that he can eat an extra piece of chocolate cake today and compensate for it by skipping dessert tomorrow. If, in fact, he is able to complete this trade, he has no self-control problem. But if tomorrow he forgets his pledge or decides to eat dessert (and put off the no-dessert meal another day), he is on the way to a self-control problem. The diagnosis of self-control problems is that they result from people who do not regularly complete intertemporal trades they initiate, yet their bad record in this regard does not forestall them from initiating more such trades.(7) The reasons that such behavior might persist are explained in detail in the earlier paper.

Now consider the savings behavior involved in the eighteen-paycheck option. Intertemporal trades are certainly possible. In fact, they are likely, since there is no reason for the pattern of expenditures over a year to be even. There will be special purchase opportunities, e.g., the car needs to be repaired, or there is an unusually good sale at a favorite store, which trigger intertemporal trades. If one fails to complete several intertemporal trades during the first eighteen pay periods, he will have little left to finance consumption in the summer. As a consequence, such a person may build up a balance on credit cards late in the summer and incur considerable interest expense as a result. It is also possible that a person will draw down savings accumulated in earlier years, but there is considerable evidence that individuals do not appear to act as if they are willing to exercise this option.(8) And even if a person did draw down previously accumulated savings, he would have to recognize that he was giving up longer-term objectives to finance short-run mistakes.

Since the costs of incomplete intertemporal trades are visible, it is not easy for a person to put off facing the problem. The existence of credit cards does mitigate against this conclusion, and there are clearly cases in which individuals have exhibited an inability to manage the freedom to initiate intertemporal trades represented by credit cards. Many such people find their way to professional credit counselors. However, these people are exceptions, most people with potential self-control problems in this area are able to institute measures for self-control without professional help. The twenty-four-paycheck option is simply one of the techniques a person with tendencies in this direction would find appealing. A person with these kind of tendencies is typified by the person who knows that he is likely to spend money very rapidly when he has it and slow his rate of expenditure when his bank account is low. Money "burns a hole in his pocket." He is afraid that, in the words of country-western singer Billy Hill, he will have, "Too much month at the end of the money." But he is not the credit card abuser. Rather, since he is a descendant of Ulysses, he is sufficiently self-aware not to fall into that trap. He is attracted to things such as over-withholding on his income tax, so that he has a planned refund. And he is attracted to the twenty-four-paycheck option.

III. EMPIRICAL RESULTS

The reflexes of the empirical economist suggest the parameters from a demand function for the twenty-four-paycheck option should allow us to choose between the two theories presented above. Unfortunately, this is not the case. First, consider the parameter estimate for the effect of changes in interest rates on the demand for the twenty-four-paycheck option. If the level of the interest rate is significantly inversely related to the likelihood of choosing the twenty-four-paycheck option, the economic interpretation based upon transaction costs would seem to be supported. What about the self-imposed constraint interpretation? A self-imposed constraint can be thought of as a consumer good with a price. If that price, which is positively related to the interest rate, goes up, one would expect the demand for the good to go down, i.e. we would expect interest rates to be significantly negatively related to the demand for the twenty-four-payment option as a self-imposed constraint.(9) It is important to remember that Ulysses was rational about his irrationality. Since both models predict a negative sign for the coefficient for this parameter, its estimate will not help us separate them. A similar argument holds for measures of the cost of trips to the bank, the other important variable in the transactions cost model.

Since the variables from the transaction cost model would also be included in a demand function for the twenty-four-pay-check option derived from the self-imposed constraint model with the same expected sign, the only way that we can distinguish these models from one another is if there is some independent measure of the likelihood that someone is inclined to have self-control problems (and is aware of this inclination). If such a variable could be found, its significance, or lack thereof, would be a test of the self-imposed constraint model. I do not think such a variable is readily available.

Fortunately, a recent change in institutions at the College of William and Mary provided a natural experiment which gives the possibility for a clear test of the two theories. Prior to the 1990-1991 academic year, what I have described as the eighteen-paycheck option was the twenty-paycheck option.(10) In December of 1989, the Provost sent a memorandum to all faculty stating that starting the next academic year individuals could receive either eighteen paychecks or twenty-four paychecks. The two models under consideration yield opposite predictions for the shift from a twenty-paycheck option to an eighteen-paycheck option.

The economic model predicts that an eighteen-paycheck option would be more attractive than a twenty-paycheck option. The transactions costs are the same under both options, since the person needs to make twenty-four trips to the bank. However, since the salary payments are moved up in the academic year, more interest can be earned with the eighteen-paycheck option. Therefore based upon this model we would expect to see people who had chosen twenty-four paychecks, but were close to the margin under the old system (twenty paychecks), switching to the eighteen paycheck option.

The prediction from the self-imposed constraint model is not as clear. Since there is more interest earnings foregone under the eighteen-paycheck option, the price of imposing the constraint has risen. By the argument presented above, this should tend to decrease the demand for the twenty-four-paycheck option. However, the new eighteen-paycheck option requires a person to face three months without paychecks while the twenty-paycheck option only required a person to face two months without paychecks. This suggests that the self-control requirements of someone not taking the twenty-four-paycheck option have increased. In conclusion, the prediction from this model is unclear. Both the price and the efficacy of the twenty-four-paycheck option have increased. If the efficacy effect dominates, we would expect to see people who had chosen the twenty-paycheck option, but who were somewhat nervous about their ability to control urges to overspend when money is available, switching to the twenty-four-paycheck option. If, on the other hand, the price effect dominates, this model gives the same prediction as the transactions cost model.

Data from before and after the change to an eighteen-paycheck option from a twenty-paycheck option clearly are supportive of the notion that the model of the self-imposed constraint is the better explanation. There were 303 faculty members on the roster before and after the change in institutions. Prior to the change, 200 of them had elected the twenty-four-pay-check option. The transactions cost model would suggest that many of these 200 would be inclined to switch to the eighteen-paycheck option. The remaining 103 had taken the twenty-paycheck option, and the efficacy portion of the self-imposed constraint model would suggest that many of these people would be inclined to switch to the twenty-four-paycheck option. Only 4 of the 200 people who had previously selected the twenty-four-paycheck option changed to the eighteen-paycheck option, while 20 of the 103 who had previously selected the twenty-paycheck option changed to the twenty-four-paycheck option. The null hypothesis that the percentage of changers is equal is decisively rejected.(11) A statistically significantly larger percentage of individuals changed in the direction predicted by the self-imposed constraint model than the transactions cost model. It is important to recognize that this is the one result which unambiguously gives support to one of the models over the other.

IV. SUMMARY AND CONCLUSIONS

The important conclusion of the analysis is that a consumer faces two tasks when pursuing any kind of long-term goal: devising an optimal strategy and devising behavior which insures that the optimal strategy is followed. Typically, economic modeling stops with a description of the first task. The existence of self-imposed constraints is evidence that the second task should not be ignored. If people are willing to give up resources to insure that they reach some long-term goal, they must be aware of their inability to follow the optimal strategy unaided. Given that self-imposed constraints are costly, the actual strategy followed will not be the optimal one, it will be a second best. Nevertheless, this second-best strategy will be preferred to the actual behavior over time in the absence of the constraint.

The particular example analyzed in this paper, faculty members' decisions to receive pay in eighteen or twenty-four installments is not particularly significant in itself, but it illustrate the issues in this type of research nicely. Before we can claim that faculty who are choosing the twenty-four-payment option are imposing constraints on themselves, we have to face the more traditional explanation based on transactions cost. Transactions cost are a reason why someone might follow a strategy which looks like a second-best strategy. In our example, we were able to clearly demonstrate that the demand for the twenty-four-payment option followed from the desire of individuals to constrain their own behavior and not just from transaction costs. It is clear in this case that people are following a second-best strategy because they are convinced that they could not trust themselves to manage to follow the first-best one.

It will be important to see if the results here generalize to the study of other examples of self-imposed constraints which are more economically important, for example, the prevalence of over-withholding on income taxes. In addition, markets which provide assistance to those who are finding it difficult to follow an optimal life style, e.g. markets focused on not buying things like cigarettes and alcohol, are affected by the same motivations. These markets are important and cannot be correctly studied with standard economic models which focus solely on devising optimal plans.

1. Credit has to go to Richard Strotz [1956] for first talking about Ulysses and the Sirens in relationship to the standard economic model. The book by Jon Elster [1979] titled Ulysses and the Sirens is also important in focusing attention on this example.

2. Thomas Schelling [1978] leads his list of examples in "Egonomics, or the Art of Self-Management" with a discussion of the Christmas Club.

3. The Credit Union National Association [1992] estimates that 14 percent of all credit union member households have one or more "club" (there are Vacation Clubs as well as Christmas Clubs) accounts. This amounts to approximately 5.2 million households nationally.

4. Richard H. Thaler and H. M Shefrin [1981] suggest a similar interpretation for this example.

5. There are two additional models which one could consider as alternatives to the model of self-imposed constraints: a model based upon lack of information and one based upon some form of family decision-making problem. Neither of these models are, a priori, very appealing nor, as it turns out, are they consistent with the data. Regarding information problems, the members of the economics department and the business school at William and Mary, who are certainly aware of the mechanics of savings accounts, have a slightly higher preference for the twenty-four-payment option than other faculty, so it is hard to support an argument based upon lack of information. Regarding family decision-making problems, a dummy variable for married is not statistically significant in an estimated equation explaining the preference for twenty-four payments.

6. See Robert Archibald [1994].

7. George Akerlof's [1991] model of a procrastinator considers a related phenomenon.

8. Shefrin and Thaler [1988] and Thaler [1990] summarize the evidence which suggests that individuals behave as if they hold various forms of wealth in different mental accounts. Individuals use these mental accounts to separate wealth they are accumulating for different purposes, particularly funds they are trying to set aside for retirement. Typically we assume that money is fungible across various accounts, but these papers present evidence which suggests individuals do not act as if they recognize this.

9. George Stigler [1966] suggests this same interpretation regarding a Christmas Club account, "The foregone cost of putting money in a Christmas fund is the interest one could earn by putting the same money in a savings account. If interest rates on savings accounts rise, the cost of buying protection against a loss of willpower rises and less of it ought to be bought." (p. 57)

10. There were no other changes which would have influenced the decision regarding the number of paychecks at this time. In particular, the compensation for and the availability of summer school teaching were not altered.

11. The test of the equality of two proportions is described in John Freund [1984], page 338. The test is based upon a normal distribution. The z value for the test here is 5.32 which is decisively in the reject region.

REFERENCES

Akerlof, George. "Procrastination and Obedience." American Economic Review, May 1991, 1-19.

Ainslie, George. "Specious Reward: A Behavioral Theory of Impulsiveness and Impulse Control." Psychological Bulletin, July 1975, 463-509.

-----. Picoeconomics. Cambridge: Cambridge University Press, 1992.

Archibald, Robert B. "Intertemporal Trades and Problems of Self-Control." Journal of Economic Behavior and Organization, forthcoming 1994.

Credit Union National Association. The National Member Survey: People Talk About Credit Unions. Madison, Wis., 1992.

Elster, Jon. Ulysses and the Sirens. Cambridge: Cambridge University Press, 1979.

Freund, John, Modern Elementary Statistics, 6th ed. Englewood Cliffs, N.J.: Prentice Hall, 1984.

Schelling, Thomas C. "Egonomics, or the Art of Self-Management." American Economic Review, May 1978, 290-94.

Shefrin, Hersh M., and Richard H. Thaler. "The Behavioral Life-Cycle Hypothesis." Economic Inquiry, October 1988, 609-43.

Stigler, George J. The Theory of Price, 3rd ed. London: Macmillan, 1966.

Strotz, Richard H. "Myopia and Inconsistency in Dynamic Utility Maximization." Review of Economic Studies 23(3), 1956, 165-80.

Thaler, Richard. "Savings, Fungibility, and Mental Account." Journal of Economic Perspectives, Winter 1990, 199-214.

Thaler, Richard, and H. M. Shefrin. "An Economic Theory of Self-Control." Journal of Political Economy, April 1981, 392-406.

ROBERT B. ARCHIBALD, Professor of Economics, College of William and Mary.
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