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.
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Akerlof, George. "Procrastination and Obedience." American
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-----. Picoeconomics. Cambridge: Cambridge University Press, 1992.
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forthcoming 1994.
Credit Union National Association. The National Member Survey: People
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Press, 1979.
Freund, John, Modern Elementary Statistics, 6th ed. Englewood Cliffs,
N.J.: Prentice Hall, 1984.
Schelling, Thomas C. "Egonomics, or the Art of
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Shefrin, Hersh M., and Richard H. Thaler. "The Behavioral
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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
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ROBERT B. ARCHIBALD, Professor of Economics, College of William and
Mary.