JOURNAL OF ECONOMIC THEORY 38, 178184 (1986)

Satiation and Habit Persistence (or the Dieter’s Dilemma)

ROBERT F. BORDLEY

Societal Analysis Departmenr, General Mo1or.c Research Labs. Warren. Michigan 48090

Received May 1. 1984; revised July 9, 1985

Frequently individual consumption varies cyclically over time. In order to model such behavior, we have to generalize the intertemporal utility function so that it depends not only on current consumption but on total previous consumption and on the change in current consumption. Solving the resulting model requires a generalization of the Euler-Lagrange equations. Journal of Economic Li/eru/ure Classilication Numbers: 026, 213, 920. ,I” 1986 Academic Pres,. Inc.

The standard intertemporal utility consumption model [4], assumes that utility is additively separable over time, i.e.,

U(q(i, t), i = 1 -n,O<r<T)=j' exp( -ht) u(q(i, t), i= I . ..tz) df 0

where h is the discount factor and q(i, I) is the amount of good i consumed at time t. We refer to u(q( i, t ), i = 1 . n) as the instantaneous utility.

Now given unchanging prices and income, this utility function generally predicts a smoothly changing, non-cyclic, individual consumption pattern. But in practice, we often observe very cyclical behavior as individuals first consome a great deal of one good, then very little, then a great deal again. Thus consider the dilemma of some weight watchers who cyclically starve themselves, then overeat, then starve themselves again.’ Now Benhabib and Day [I] presented a model for such behavior assuming extreme myopia.2 But cyclical consumption behavior appears to occur even with non-myopic individuals. Hence getting an adequate description of such behavior requires us to generalize the standard formulation in some way.

I Of course, one could dismiss such behavior as irrational and outside the scope of economic theory. But this leads to serious questions about the relevance of economic theory. It is not at all clear that all cyclical behavior can be dismissed as irrational.

* In their model, the individual maximizes a static utility function whose parameters depend on previous consumption.

178 0022-0531/86 $3.00 Copynght (cl 1986 by Academic Press, Inc. All rights 01 reproduction in any form reserved.

SATIATION AND HABIT PERSISTENCE 179

We could generalize the individual’s utility function so that it depends not only on q(i, t) but on its integral Q(i, t) = l q(i, s) ds. Indeed if we inter- pret U as the net present value of a business and q(i, t) as capital and labor variables, this generalization leads to a model identical to that developed by Lucas [S] and Treadway [6]. As Ryder and Heal [2] showed, such a model can lead to cyclical behavior if the total amount of capital available at time t depends on how much capital is generated at previous times3

When we adapt their results to the individual context, we can conclude that:

(a) if the individual’s function depends on Q( i, t) and q(i, t), and

(b) if the individual’s income depends on how much he consumes,

then we can get cyclical behavior from the utility model. Unfortunately the weight-watcher’s dilemma arises even in situations where the individual’s income does not depend upon how much he consumes. Hence this generalization is still not adequate for our problem.

An alternate generalization is to let U depend upon q( i, t) and q’(i, t), the first derivative of q(i, t). However the mathematical formalism does not change significantly and we end up with the same result: no cyclical behavior will arise assuming standard forms for the income constraint.

What we have to do is assume that U depends on both Q(i, t) and q’(i, t). Specifically, we assume that the individual’s utility at time t depends on

U(Q(i, t)-,-tilt, q(i, t), q’(i, t)).

Thus utility depends on a satiation term, Q(j, t) - rt and a habit persistence term, q’(i, t) as well as current consumption, q(i, 2).

’ Ryder and Heal [2] showed how to deduce cyclical behavior from a model in which the utility function depends only on 9(l) and its weighted integral, z(l). Their model took the following form:

Maximize J= J

exp( -6r) u(q(c), Z(I)) d/

subject to: dk/dr= f(k)- Ik -4. O<q<fW)

dz/dt=p(c-z_). ,-(o)=z"<o. k(O)=k,>O,

where k was interpreted as the capital/labor relation. However a close examination of their model indicates that cyclical behavior can only occur if dyf/dk'#O. (Otherwise their parameters p and r become zero and cyclical behavior cannot occur. See Sects. 5 and 6 of their paper.) When we adapt their model to the individual consumption case, 0 < 9 <f(k) acts as a sort of budget constraint. With this interpretation off(k). the assumption, dff/dk2 # 0, will not generally make sense for a price-taking consumer. Hence cyclical individual consumption behavior cannot arise. But by making utility a function of q’(t) as well as q(l) and Q(t), a later section shows that we can get cyclical individual consumption behavior for a standard budget constraint.

180 ROBERT F. BORDLEY

The dependence on Q(i, t) - z(i)t gives us a satiation effect. If the individual consumed a great deal of good i at time s, then he will be sated on it for times close to s. But as time passes, that satiation eventually fades. Indeed given sufficient abstinence from good i, the individual develops a “negative satiation” or a special yearning for good i. This kind of utility function seems reasonable to describe such goods as foods.

The dependence on q'(i, t) is a habit-persistence effect. If there’s positive dependence on q'(i, t), then the individual prefers variety. If there is a negative dependence, then he prefers monotony. Again, this makes sense if we think of our goods as foods.

To study the implications of this model, we assume a quadratic utility function. We solve it with a generalized form of the Euler-Lagrange equations; hence we will not make use of Pontyagrin’s Maximum Principle. Nevertheless we will still get interpretable results.

1. THE BASIC PROBLEM

We only consider one good so that we can suppress q's dependence on the good i. If the utility function, u(Q(t), q(t), q'(t)) is an analytic function of Q(t), q(t), and q'(t), we can express it in a Taylor series expansion about Q(t) = zt, q(t) = z, and q'(t) = 0. If we then truncate all terms of order higher than two, we get the formula

4Q(f,, q(t). q’(t)) = u(O,O, 0) + uo(Q(t) - ~1) + u,(Q(t) - =[I2

+ u,(Q(f) - zf)(q(r) - --I + a,(q(t) - =)

+ dq(f) - z)’ + a,(Q(t) - ,-t) q’(f)

+ %(q(t) - z) q’(r) + a,q’(t) + +3(q’(W

Let exp( -bt) be the discount factor measuring impatience. Then the individual’s problem is to

Maximize O’(u(Q(r), q(t), q'(t)) exp( -bt) dt s

subject to: s

T

q(f) dt B 4 4(f) a 0 0

(2.1 J

where our budget constraint prevents the individual from consuming more than B units of the good. (There are more realistic forms for the budget constraint. We ignore them here for the sake of simplicity.)

Given that the utility function depends on Q(t), q(t), and q'(t), the

SATIATION AND HABIT PERSISTENCE 181

appropriate form of the Euler-Lagrange equations (see Weinstock [7]) is given by

e A’ au/aQ - d(e ~ ht au/aQ’)/dt + d2(e hr &@Q”)/dt’ = 0.

This extended form of the Euler-Lagrange equations has not previously been used in dynamic economics.

This first-order condition leads to the equation

Q”“(2aJ - Q”‘(4a, b) + Q”(2a5 -a,b - 24 + 2u,b2)

+ Q’(2b, - 2a, b + u6 6”) + Q(2a, + bu, + a5 b’)

= -(a,, + ha, + a,b’ - 2za, - x,b2) + zt(2u, + alb).

Here Q”’ and Q”” denote the third and fourth derivative, respectively, of Q. The solution of this equation can be written as4

Q(t) = c K(i) exp(k(i)r) + K(0) + KK(0) t

where (k(i), i= 1 ... 4) are the solutions of a quartic equation.5 To investigate the properties of this solution, we consider a simple case

in which the discount factor is zero. A zero discount factor, of course, means that the individual gives as much weight to future consumption as he does to present consumption. If we can deduce cyclical behavior given a zero discount factor, then it seems reasonable to expect it given a positive discount factor (where we do not give as much weight to the future as we do to the present).

Now the first-order conditions do not provide the optimum if the con- dition q(t) > 0 is violated. So we will assume they are never violated. If they were, a procedure in Kamien and Schwartz [3] describes how to get the optimum. Such a procedure would not affect our general findings.

We also need to have the second-order conditions for an optimum hold. To ensure this, we require concavity. This means that

a, <o

u4 < 0

a, < 0

‘KK(O)=z(2a, +a,b)/(2al +ba,+a,b2)

K(O) = -(a,+ ba, + a,b2)/(2a, + ha? + ash’)

-(2&z,-2a,b +a,b*)(z)(2a, +azb)/(2a, + ba, +a,b’)‘. 5k”(i)(2as) - k3(i)(4asb) + k’(i)(2a, - a,b - 2a, +2a,b’) + k(i)(2ba, ~ 2a,b + arb’)

+ (20, + ba, + a,b’)=O.

182 ROBERT F. BORDLEY

We now turn to the issue of cyclical behavior. We get cyclical solutions if and only if k(i) is a complex number. The k(i) values are the solution to

2a,k4(i)+2(a,-a,)k’(i)+2a,k(i)=O

which implies that

(2.2 1

Thus cyclical behavior occurs if k2(i) is negative or complex. Now k*(i) will be negative or complex if:

(a) we make ~1~ so negative that ~14 - a5 is positive. Since uR < 0, this makes k’(i) negative. This choice also satisfies our concavity conditions.

(b) If u4 - a5 is negative and if a, and a, are large negative numbers relative to u4 - u5, then k*(i) is complex. This choice also satisfies con- cavity.

So we do get complex values of k(i) if either u5 is very negative or ugu, is very large. It is interesting to note that a5 is the coefficient of the satiation/habit-persistence interaction while usa, is the product of the coef- ficient of the quadratic satiation term and the quadratic habit-persistence term. Thus in order to get complex values of k(i) in our model, we need both a satiation term and a habit-persistence term.

2. THE SATIATION/INERTIA INTERACTION

To give some intuition to this result, let us look at u5, the coefficient of the satiation/habit-persistence interaction. The term a5 is the coefficient of (Q(t) - it) q’(t). We have just shown that if this coefficient is sufficiently negative, k(i) could be negative giving rise to a cyclically varying q(t). We now analyze what a negative value of u5 means and why a sufficiently negative value would cause q(t) to manifest periodic behavior.

We can interpret a negative Q(t) q’(r) coefficient as a sort of second- order satiation term. If we have consumed a great deal of the good, then the desirability of increasing our consumption of the good, q’(t), may be much less than if we had not consumed very much of the good. (A hungry rat tends to increase his rate of consumption more quickly than a sated rat.)

To understand why a negative coefficient for (Q(t) - it) q’(t) leads to sinusoidal solutions, note that u5 being very negative means that whenever (Q(t) - it) is large, there is a cost to having high values of q’(t). As a result,

SATIATION AND HABIT PERSISTENCE 183

we expect the individual to have a very high q’(t) in the initial period when Q(r) is small but to have a very low q’(t) when Q(t) is large. Thus q’(r)

starts out large and then begins decreasing until it is negative. At that point, the individual begins decreasing his consumption of the good, q(t).

Eventually q(t) falls below -7 so that (Q(t) - it) begins decreasing. As (Q(t) - it) approaches zero, the interaction term u5( Q( f ) - it) q’(t) becomes smaller and smaller and the cost of positive values of q’(r) becomes less. Eventually q’(t) turns positive again and another cycle starts.

Thus we see how a strong negative interaction between satiation, (Q(t) - zt) and habit persistence can lead to cyclical behavior. We note that this model if applied to the business investment problem would lead to some sort of business cycle.

But suppose k(i) is real. Then the formula specifying the consumption path is the sum of four exponentials (until time r* when q(t) = 0 for t > t*.) To specify the constant K(i), we use the initial conditions:

Q(O)=0 q(0) = initial consumption

Q(t*)= B q’(O) = rate at which initial consumption changes

Q(r*) = 0.

This means that our solution will depend upon the initial consumption and the initial rate at which consumption is changing. This is a habit-per- sistence effect: Our solution in the time interval (0, 7) will depend upon consumption levels prior to time 0.

CONCLUSIONS

This paper generalized traditional models of individual behavior in dynamic economics by incorporating satiation and habit persistence. We found that including both factors is crucial to the production of cyclical consumption behavior. But when both effects are present and when the satiation/inertia interactions are much stronger than the importance of present consumption, we do get cyclical behavior.

ACKNOWLEDGMENT

I thank Dr. Robert M. Kleinbaum for useful comments.

184 ROBERT F. BORDLEY

REFERENCES

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2. H. RYDER AND G. M. HEAL, Optimum growth with intertemporally dependent preferences, Rev. Econ. Stud. 1 (1973). l-33.

3. M. KAMIEN AND N. SCHWARTZ. “Dynamic Optimization.” North-Holland, New York, 1981.

4. T. R. KOOPMANS, Utility and impatience, Econometricu 28 (1960). 287-309. 5. R. E. LUCAS, Adjustment cost and the theory of supply, .I. Poh. Econ. 75 (1967), 75-85. 6. A. B. TREADWAY, On rational entrepreneurial behavior and the demand for investment.

Rev. Econ. Stud. 36 (1969). 221-239. 7. R. WEINSTOCK, “Calculus of Variations,” McGraw-Hill, New York, 1952.