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Habit

JOURNAL OF Economic

Journal of Economic Dynamics Control Dynamics

20 (1996) 1193 -1207 & Control

persistence, heterogeneous tastes, and imperfect competition

Gary Fethke*, Raj Jagannathan

College ofBusiness Administration, Unirsv-sity of Iowa. Iowa ci@ IA 52242-1000. IJSA

(Received January 1994; final version received May 1995)

Abstract

We examine the dynamic behavior of consumption and price in a setting where imperfectly competitive producers face consumers with various intensities of rational habit persistence. We establish that steady-state consumption and its persistence rate are lower under time-consistent monopoly than under either competition or monopoly commitment. Under both competition and monopoly commitment, the steady-state level and the transient path of consumption are unaffected by the relative mix of consumer types. In the consistent case, both the persistence rate and steady-state consumption decrease as the proportion of habitual consumers increases.

Key words: Habit persistence; Monopoly pricing; Time consistency; Adjacent com- plementarity ./EL classiJication: Dll; D42

1. Introduction

A sizable literature, initiated by the need to construct empirically plausible dynamic models of demand, has developed in which consumer tastes are represented as being temporally nonseparable. Prominent examples include labor supply and durable goods where the service or good is an intertemporal

*Corresponding author.

We are grateful to Andrew Daughety, Carol Fethke, John Kennan, Narayana Kocherlakota, Ayhan Kose, Jennifer Reinganum, anonymous referees, and participants in seminars at the University of Iowa and the University of London for helpful comments.

0165-1889/96/$1.5.00 f; 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1889(95)00895-3

1194 G. Fethke, R. Jagannathan 1 Journal of Economic Dynamics and Control 20 (1996) 1193-1207

substitute and habit persistence where goods are intertemporal complements. The initial research on habit, including that of Houthakker and Taylor (1970) Pollack (1970), Phlips (1974), and Boyer (1983), recognizes only the effects of past on current consumption. More recently, Becker and Murphy (1988), Becker, Grossman, and Murphy (1990, 1991, 1994), Constantinides (1990) and Fethke and Jagannathan (1991) consider rational choices of consumers who anticipate the effects of future variables including prices and taxes on current demand. These formulations focus primarily on the case of a representative consumer in a competitive market. None considers the implications of hetero- geneous tastes for a habit good in imperfectly competitive markets.

Many markets where goods and services are intertemporal complements display structural elements of imperfect competition if not monopoly. Com- monly recognized habit goods, such as cigarettes, alcohol, and pharmaceuticals, are produced in markets where there are elements of imperfect competition. These conditions also apply to a broad range of computer and software prod- ucts, transportation and commuting services, and investment goods with sunk costs.3 In these markets, consumers display a variety of affinities for habit goods, which makes a homogeneous-consumer formulation an unattractive structure for some problems.

In this paper, we consider noncompetitive pricing when there are habit effects and a mix of heterogeneous consumers. Two consumer types are distinguished according to the strength of their habit effects and their subjective rates of consumption depreciation. We develop optimal policies for price and consump- tion for competition, monopoly, with and without commitment, and for oligopoly. Under competition and monopoly commitment, we establish that price does not depend on the past consumption of habitual consumers. In the time-consistent case, however, price is shown to be increasing in past consump- tion, and anticipation by consumers of a rising price path leads to a lower steady-state level of consumption and a lower rate of consumption persistence than for either competition or monopoly commitment.

Intuitive and experimental notions of habit persistence, consumption rein- forcement, and addiction all require a positive relationship between present and

1 Bulow (1982), Stokey (1981), and Kahn (1986) among others, consider durable goods. Models of intertemporal substitution in the labor market are developed by Kydland and Prescott (1982), Mankiw et al. (1983 Sargent (1987), and Kennan (1988).

Becker et al. (1990) develop a two-period monopoly, and Fethke and Jagannathan (1991) develop the case for an infinite-period monopoly with h4 identical habitual consumers and N firms.

3 Recent papers by Becker, Grossman, and Murphy (1990,1994) and Chaloupka (1991) consider the empirical implications of rational habit formation in cigarette consumption. The empirical work by Ferson and Constantinides (1991) and Braun et al. (1992) demonstrates the general importance of habit effects in explaining the dynamic properties of nondurable consumption.

G. Fethke, R. Jagannnthan /Journal of Economic Dynamics and Control 20 (1996) 1193-1207 I195

future consumption. Becker and Murphy (1988, 1991) develop a condition on preferences, called adjacent complementarity, that is necessary and sufficient for habit reinforcement to occur. We demonstrate that adjacent complementarity depends not only on preferences but also on such factors as the degree of competition, time consistency of production policy, and the relative mix of consumer types. In our formulation, adjacent complementarity always holds under competition and monopoly commitment regardless of the mix of con- sumer types, but it need not hold under time-consistent monopoly or oligopoly for the same set of taste parameter values.

A main result of this paper is that under competition and monopoly commit- ment the mix of consumer types has no effect on the time path of consumption. Under time-consistent policy, however, both persistence and steady-state con- sumption of habitual consumers decreases with the proportion of habitual consumers in the market, mitigating the monopolists ability to price discrimi- nate over time.4 When generalized to the oligopoly case, the steady-state level and persistence of consumption are shown to be increasing in the number of firms, approaching the competitive values.

In Section 2, the competitive case is presented as a point of reference and comparison. Section 3 develops the monopoly commitment case. Section 4 examines the time-consistent monopoly solution. Section 5 considers the time- consistent oligopoly case. Section 6 concludes.

2. Competitive supply

The specific function we use to depict lifetime net consumer surplus is a simplified version of the learning by doing formulation described by Becker and Murphy (1988):

f- 1 aq, - fbq: + dq, c #t-s-lqs - R,q,

s=o 1

(2.1)

In the above expression, q, is new consumption (gross investment in learning); Qt = xi_, +-qs consumption capital, with the definition that q. = Qo; a, h, and d are nonnegative taste parameters; 0 < 6 < 1 is the discount factor; and 0 < C$ < 1 is the rate at which past consumption carries over to the present. In this formulation, the complementarity among present, past, and future con- sumption is captured by the parameters d and 4. Habit reinforcement implies that an increase in past use will increase the marginal utility of current consump- tion, lY%/aqaQ = d > 0, where u is per period utility. This is a sufficient condition for a myopic utility maximizer. Lifetime marginal utility is also increasing in future consumption, @U/aq, aqt + j = (Srpyd > 0, where U is life-

1196 G. Fethke, R. Jagannathan / Journal of Economic Dynamics and Control 20 (1996) 1193-1207

time utility. Concavity of the lifetime net consumer surplus function requires that b(1 - 4 ,,I%) > 2dJ_8.j The full price of one unit, R,, is the present value of current and subsequent purchases that result from buying one unit in period t, and is related to the single-period price, P,, according to R, = C,?, (61#1)jf,+~.

Heterogeneity of tastes involves assigning different values for the habit-effect parameter, d, and for the rate of depreciation, I - 4, of consumption. We consider a situation with two types of consumers: a Type 1 consumer, with time-separable utility and taste parameters d = 4 = 0, and a Type 2 consumer, with taste parameters d and 4 > 0. The ratio of Type 2 to Type 1 consumers in the market is given by 8. When 0 = cc, there are no Type 1 consumers and the formulation becomes the case of all Type 2 consumers with the same degree of habit. An individual consumer has no impact on price.

Each consumer maximizes net surplus, defined by (2.1). It is convenient to define a(1 - 64) = a, b(1 - 64) = 6, and d(1 - 264 + 6) = d( 1 - d), where a, b, and d are nonnegative, single-period taste parameters. We will assume throughout this paper that b > d to insure stability; this is a stronger condition than that required for concavity to hold.

The Euler equations are

a - bq,, = P,, t = 1,2, . . . ) (2.2)

for Type 1 consumers, and

a + &Qzl+ 1 - b,Q,, + hQzr- I = P,, t = 1,2, . . . , (2.3)

Becker et al. (1990, p. 30) assert: ... if the monopolist can engage in price discrimination, he may have an incentive to offer lower prices to persons who currently do not consume the good. This explains why cigarette companies distributed free cigarettes on college campuses in the past. In effect college students were being offered a zero current price but a positive future price once they became addicted.

5 Becker and Murphy (1988) consider two habit conditions, called reinforcement and tolerance. Reinforcement occurs when greater past consumption increases the marginal utility of current consumption and is closely associated with their notion of adjacent complementarity. Tolerance occurs when greater past consumption reduces current utility. For an excellent discussion of the various ways tastes can be described for intertemporal complements and habit goods, see Chaloupka (1991).

6The nth-order principle minors, D,, of the Hessian matrix, D = ((aU/aQlaQl)), associated with lifetime utility are given by D = (m; + ml) where mt and mt are the roots of the quadratic equation, mz + [(l + )b + 26$d]m + 6(db + d)* = 0. For mr and rnz to be real, it is required that

b (1 - 34) > 2$d. Both m, and m, are negative.

Parameter values exist where the utility function is concave but stability is not assured. The implications of these conditions for habit goods are discussed by Becker and Murphy (1988) for the case of competitive supply and by Fethke and Jagannathan (1991) for monopoly, where it is shown that monopoly reduces the range of parameter values that imply instability.

G. Fethke, R. Jagannathan /Journal of Economic Dynamic.? and Control 20 (1996) 1193-1207 1191

for Type 2 consumers, with

b

1 ~ Ml + W2) + 2w41 - 4) > o

1 -I l-264 +fi

b2 G 2!!!L l-I+

41 - 4) 1-2&p +6

> o

and Q20 given. Here, q,, and Q2, are measured on a per capita basis. When there is habit persistence, (2.3) indicates that current consumption depends upon past consumption and anticipated future consumption. Using the relationship [(l + 6)b2 - b,] = - (b - d)(l - c$), the long-run demand function for Type 2 consumers, P,, = u - (b - d)q,,, is less negatively sloped than that for Type 1 consumers, P, = a - bql m. At every price, P,, habitual consumers will pur- chase more than nonhabitual consumers, resulting in q1 x < q2m since b > d.

Under competition, P, = c and a > c. Steady-state consumption of Type 2 consumers is determined from (2.3) as

(2.4)

The long-run new consumption of Type 1 consumers is less than that of Type 2 consumers, 4: = (a - c)/b < q;. The genera1 solution to (2.3) is of the form:

Q;, = Q; + k,X, + k2R;, t = 1,2, . 1

where 0 < i,, < L2 are the roots of the characteristic equation:

bz(l +6A2)-blA=O. (2.5)

The larger root & > l/J& implying that net consumer surplus is unbounded unless k2 = 0. Using the concavity condition and the assumption that b > d, it follows that the smaller root

A

1 = bl - Cb: - 4db:11'2 < 1,

26b2 (2.6)

When P, = c, the solution for Type 1 consumers is given trivially by (2.2). The specific solution for Type 2 consumers is given by Qzt = Q; + (Qzo - Q:) Lr .

1198 G. Fethke. R. Jagannathan / Journal of Economic Dynamics and Control 20 (1996) 1193-1207

Alternatively, this solution can be described by a partial-adjustment rule:

Q:, = Q: + At (Qw , - Q;), t = 1,2, . . . . (2.7)

The coefficient 11,) 0 < A1 < 1, which reflects consumption persistence and affects the transient solution, can be shown to be increasing in 4. Thus, when the depreciation rate declines, the stock of consumption converges at a faster rate to a higher steady-state level.

Eq. (2.7) also implies that

qzf = (1 - WQ: + (4 - &Qzt-t, t = 1,2, . . . . 633)

The reinforcement effect of habit occurs whenever current consumption increases future consumption. Becker and Murphy (1988) develop a condition, termed adjacent complementarity, that implies reinforcement. In our formula- tion, the comparable condition is 1 > A1 > 4, which holds for all 4.

Under competition, the mix of consumer types, 0, does not affect the solution. Specifically, under competition, present and future prices are not affected by consumption decisions, even though these decisions are intertemporally linked for habitual consumers. As we will demonstrate below, under imperfect competi- tion, the price path may depend on the mix of consumer types. In the following sections we consider two solution concepts. In the first, a monopolist commits to a production plan for all periods. In the second, a time-consistent equilibrium is described where consumers and firms base their decisions on the current state of the system, defined by past consumption and by the optimal decisions of other agents.

3. Monopoly commitment

Commitment involves establishment of a production plan at the initial period for all future periods.* The intuition associated with the commitment solution is based on the monopolists temptation to exploit the habit effect associated with past sales and to increase price in each period. Forward-looking consumers, however, will be reluctant to buy at the initial low price if they anticipate that current consumption will precipitate higher future prices. To convince con- sumers that it will not increase future prices, the firm must commit never to base

An example of the continuous-time version of monopoly commitment with homogeneous con- sumers is provided by Clarke et al. (1982). They develop solutions for price paths that describe learning-by-doing and demand-side experience effects that include habit formation.

G. Fethke. R. Jagannathan / Journal of Economic Dynamics and Control 20 (1996) I 193-1207 I 199

future price increases on habit-induced demand effects. Credible commitment requires a supporting institut...