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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 of‘Business Administration, Unirsv-sity of Iowa. Iowa c’i@ 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 complementarity ./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 heterogeneous 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 consumption 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 consumption, 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 reinforcement, 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 consumer 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 commitment the mix of consumer types has no effect on the time path of consumption. Under time-consistent policy, however, both persistence and steady-state consumption of habitual consumers decreases with the proportion of habitual consumers in the market, mitigating the monopolist’s 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

a’q, - fb’q: + d’q, 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 consumption is captured by the parameters d’ and 4.” Habit reinforcement implies that an increase in past use will increase the marginal utility of current consumption, 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 = (Srpy’d’ > 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%) > 2d’J_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:) L’r .

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 competition, 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 monopolist’s 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 consumers 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 institutional arrangement, for example, firm reputation or

long-term contracting. A monopolist who can commit to a production path will maximize profit:

T( = j& g is-’ (Pt - 4 hr + OQdr f-l

(3.1)

subject to

p, = a - bq,, = a + SbzQa+ 1 - b,Q,, + bZQzt- 1, (3.2)

for t = 1,2, . . . and Qzo given. The constraint (3.2) equates the Euler equations of the two consumer types, (2.2) and (2.3), and represents the requirement that all consumers must pay the same single-period price. Per capita consumption in period t, Qt = (qll + tIQZ1)/(l + O), is a weighted average of Type 1 and Type 2 consumption.

The Euler equations associated with monopoly commitment are

-d’b:Q,,+, + &(2b, +WQzt+l 42% +b,(b, +WlQ,,

+ bz(2b, + WQ,,-, - b:Qzt-2

= +(a -c) [h,(l + 6) - bl - Ob], t = 2, 3, .

The steady state solution is

a-c (1 -4,Q: = q:=2(b_d).

(3.3)

(3.4)

Long-run new consumption of Type 1 consumers is less than that of Type 2 consumers, q: = +(a - c)/b < qf. Just as in the competitive case, the mix of consumer types, 0, does not affect the steady-state consumption of either type. The reason for this result will become apparent by examining the commitment price path.

When the trial solution Q2, = ki,’ is substituted into the homogeneous equations associated with (3.3), we obtain the characteristic equation:

[b, ,I - b2(1 + SA’)] [(b, + bt))l, - b2(l + an’)] = 0. (3.5)

1200 G. Fethke, R. Jagannathan / Journal of Economic Dynamics and Control 20 (1996) 1193-1207

The general solution for all t is of the form Qz, = Q: + kll; + k2& +

kJ& + k&. Since the larger root in each factor, ,I2 and L,, exceeds f/J%, it follows that a finite utility function requires setting kl = k4 = 0. The smaller roots, II and ,Is, are such that 0 < 12s < A1 < 1 since b > d. The root A1 is identical to that for the competitive case, (2.6). Using the general solution, the initial value, QZ,,, and the Euler equation at t = 1, we determine the unique values for k, and k3. The resulting solution for consumption of Type 2 consumers is

Qzt = Q: + (fQm - Q%I + tQzo&

= $Q’i‘, + +Q,d:, t = 1,2, . . . . (3.6)

The commitment solution, apart from the effect of initial consumption Q2,,, is one half of the competitive solution, Q$.

Substituting (3.6) into (3.2) provides the price path:

I’, = +(a + c) + +ObQ&. (3.7)

Apart from the decreasing effect of initial stock, price in each period is the single-period monopoly price. When there is a mix of consumer types, it is optimal to spread the price adjustment to Q 20 smoothly over the horizon to a steady-state price, PR = $(a + c).

In the absence of a credible mechanism to enforce commitment, the temptation to revise the original plan presents itself each period. If plan revision by the monopolist occurs, consumers will expect similar changes in the future. In the current situation, if the monopolist does not adhere to the original plan, consumers will anticipate that past consumption affects current price and rationally take this fact into consideration when forming expectations. The monopolist, in turn, will be constrained by the fact that consumers realize that current consumption will affect future prices. The result is a breakdown in commitment, which motivates our examination of a time-consistent equilibrium.

4. Time-consistent monopoly

The time-consistent equilibrium is a symmetric Markovian equilibrium where the optimal policies for consumption and price are linear rules whose coefficients depend on taste and technology parameters and on the structure of the product market. In the infinite-horizon model, the form of the problems

G. Fethke, R. Jagannathan 1 Journal of Economic Dynamics and Control 20 (1996) 1193-3207 1201

confronting consumers and the monopolist are identical in each period, imply-

ing stationary rules for consumption and price.’

Each period, the firm takes the consumer demand functions as given and determines current output. Consumers, in turn, take the corresponding price and previous consumption as given and determine current consumption. All agents follow optimal policies in all future periods. With heterogeneous tastes and the two consumer types, we will establish that policies that reflect this sequence of optimal decisions for price, consumption of Type 1 and Type 2 consumers are

f’, = a - bq,, = PT +B(Qzt-, - Q,',, (4.1)

411 = d + &,(QwI -Q;,, (4.2)

Qzr = Q; + &z(Qzt-~ - Q;,. t = 1,2, . . . (4.3)

Here, price and consumption are linear functions of the past consumption of Type 2 consumers. We will also determine the unique values and properties of the six defining coefficients, PT,J, q;, LIZ, QT, and ILZ2.

Consumers anticipate that optimal output policy, (4.3) will be followed in all future periods. Imposing this additional restriction on consumer Euler equa-

tions (2.2) and (2.3) implies that the firm will select output in each period to maximize profit, (3.1), subject to the condition that all consumers pay the same price, P,, (3.2), and that the optimal policy, (4.3) holds for allfuture periods, that

is, Qztfl = Q2' + L (Qlt - QT) for t = 1, 2, We will demonstrate that this rule, with uniquely determined values for QT and LZ2, also holds for the current period, thereby establishing the optimality of the policies, (4.1)-(4.3) for all periods.

The Euler equations associated with the firm’s problem, maximizing (3.1) subject to (3.2) and (4.3) are

‘This approach for solving infinite-period dominant player models 1s similar to those used by

Kydland (1977) Stokey (198 l), and Kahn (1986) to solve finite-period problems. When the horizon is

extended in these models, the solutions to the finite-horizon models approach a unique stationary

linear feedback rule. The homogeneous taste case with M habitual consumers and N Cournot

producers is developed by Fethke and Jagannathan (1991).

1202 G. Fethke, R. Jagannafhan / Journal of Economic Dynamics and Control 20 (1996) 1193-1207

= (c - U)[bl + be - &(l + &,)I

- &[2bl + Bb - 2662(1 + &,)I (1 - &z) Q,‘, t-1,2, . . . . (4.4)

When the trial solution Qzr = kl’ is substituted into the homogeneous equations associated with (4.4), we obtain the characteristic equation:

(4.5)

This expression is positive for AZ2 = 0, negative for AZ2 = il, the competitive value of the persistence coefficient, and strictly convex over the interval 0 I AI2 I 1,. These conditions imply that there is a unique root, L,,(e), such that 0 < &( co ) < &,(e) < J.,,(O) = ;1i, where &( cc ) is consumption persistence for the case of homogeneous Type 2 consumers. Since (4.5) is linear in 8, it is clear that &(e) is decreasing in 8.

The steady-state value for consumption stock, derived from (4.4) is

Q; = (a - c)(Ob + b, - 6bz(l + A,,))

(b -d)(l - q5)(Ob + b, - 6b2(1 + i,,)) + ((b - d)(l - 4) + Bb)(b, - 66,(1 + &))’

(4.6)

A direct comparison with the commitment steady-state, Qf in (3.4), which is independent of 8, reveals that QT < Qf for all 8 because AX2 < l/6. Specifically, Q:(e) = Qf when 8 = 0 and is decreasing in 8. The steady-state prices are such that PT > PR > c, with P’(0) = PR and PT(0) increasing in 0.” In addition to the usual monopoly inefficiency, there exists an inefficiency associated with the firm’s inability to commit, since both consumer types and the firm are better off under commitment than under time-consistent policy.

The optimal consumption policy of Type 2 agents is thus shown to be the partial-adjustment expression, (4.3) with 2 22 uniquely determined by (4.5) and steady-state consumption, QT, given by (4.6). Eq. (4.3) also implies that

q2t = (1 - ~22)Q; + (A22 - 4)Qzt- 1, t = 1,2, . . . . (4.7)

In the competitive case, the adjacent complementarity condition, A1 > 4, holds for all 4. Here, 1222 > 4, holds for 0 < C$ I d*(e) < 1, where 4*(e) is decreasing in 0. Thus, there exist values for the depreciation rate, 1 - 4, where adjacent

lo We have established that PR > c. The result Qr < Qf, combined with the time-consistent steady-state price derived from (2.3). implies that PT z PR.

G. Fethke. R. Jagannathan / Journal of Economic Dynamics and Control 20 (I 996) 1193-1207 1203

complementarity fails to hold. Past consumption may be unrelated, if AZ2 = 4, or even negatively related, if AZ2 < 4, to current consumption even though an increase in past consumption raises the marginal utility of current consumption. The pricing policy of the firm can fully offset the habit forming preferences for positive demand reinforcement, which is implied by d’ > 0.

The basic notion of adjacent complementarity, which supports the concept of habit reinforcement, depends on pricing policy. Substitution of (4.3), with Qf determined by (4.6) and AZ2 by (4.5) into (2.3) provides the optimal price policy,

Pt = a - (b - W - 4,Q; + (b,(l + 6&J - b,&,)(Q,,- 1 - Q;)

= PT + B(Qzt-I - Q;,, t = 1,2, . . . (4.8)

Consumers anticipate that the firm will set price according to (4.1), where J? = [b,(l + 6&) - b13,22] > 0, so that price, P,, is increasing with respect to QX1_ 1. To establish this result, note that JI(Lz2) is decreasing in AZ2 on the interval 0 5 i b22 I RI. SinceJ(ir) = 0, it follows thatJ(Az2) > 0 for AZ2 < i,, Becker et al. (1990, p. 30) argue in the context of a two-period model where price increases from period 1 to period 2, that ‘. . a monopolist may lower price to get more consumers ‘hooked’ on the addictive good’. While it is true that starting from a low initial consumption that time-consistent price will increase over time, price policy is not under the complete control of the firm. Habit persistence and the subsequent reinforced demand lead the monopolist to increase price each period, but higher current and expected future prices prompt consumers to reduce current consumption. The price policy (4.8) reflects both of these forces.

Finally, substitution of (4.8) into (2.2) provides the consumption policy for Type 1 consumers,

q1 f = (b - d)(l - 4) Q’ _ C&U + h&J - b1&21 b

2 b

(Q2,-1 - Q;,

= d + 42 (Q2r- I - Q;), t= 1,2, . . . , (4.9)

as given in (4.2), with qT < qr and AI2 = -&l/b < 0. The consumption of Type 1 consumers is indirectly affected by the past consumption of Type 2 because habit-reinforced demand of Type 2 consumers increases the price level each period and reduces the consumption of all consumers in the market.

For the competitive case and monopoly commitment, we established that the fraction of Type 2 consumers in the market, 8, has no effect on either steady-state consumption, Q$ and Qf or consumption persistence, AI. Under time-consistent policy, steady-state consumption, Q:(e), and consumption

1204 G. Fethke, R. Jagannathan 1 Journal of Economic Dynamics and Control 20 (1996) 1193-1207

persistence, LZ2(0), are decreasing in 8, while the responsiveness coefficient,J (H), is increasing in 8. Consequently, the smallest steady-state consumption, the lowest persistence value of 1222 and the highest value of the price responsiveness coefficientJ occur when consumers are all Type 2.

5. Time-consistent policy with oligopoly supply

It is possible to extend the formulation to include oligopoly. Here, we consider the time-consistent Cournot equilibrium with N firms.” As before, there are two consumer types, Type 1, with time-separable tastes, and Type 2, who display a habit effect, 4, d’ > 0. Each firm selects a strategy that specifies the number of units it will sell given sales of all other firms up to that date, and expected future output, as given by (4.3). These expectations must be consistent with consumer expectations of future sales.

Thejth producer seeks to maximize

(5.1)

subject to

+ b2 5 Qxjt- 1 + b f 41jt = 0, t = 1,2, . . (5.2) j=l j=l

Here iijr and Q”zjr are the Type 1 consumption and Type 2 consumption produced by the jth firm. The constraint (5.2) is the N-firm version of the equations that result when (4.3) is substituted into (3.2).

After aggregation, the Euler equations depicting the consumption path of Type 2 consumers are

&Ceb + (N + l)(b, - SWn)l Qzt+ I

- (N + l)C6b: + (b, - &b,)(Qb + bl - &Ml Qzr

+ b2Ceb + (N + Mb, - 6b2~22)l Q21- I

‘I With a homogeneous product and firms competing in prices, rather than quantities, consumers

will buy from the producer offering the lowest price each period. Bertrand competition will drive

price to marginal cost, thus replicating the competitive supply case described in Section 2.

G. Fethke. R. Jagannathan /Journal of Economic Dynamics and Control 20 (1996) 1193-1207 1205

= N(c - a)(Uh + bl - 6b2&) - 6b2 [bfl + (N + l)(b, - bz(l + &,))I

x (1 - &IQ:, t = 1,2, . . . (5.3)

The steady-state value for consumption is

Q;(N) =

N(a - c)(Ub + b, - 6b2(1 + in))

N(b - d)(l - +)(Ob + b, - 6bz(l + Jzz)) + ((b - 6)(1 - 4) + Ob)(b, - 6b2(1 + 122))

(5.4)

When N = 1, this expression reduces to the monopoly case, (4.6). As N + crj, steady-state consumption, Q;(N), converges to the competitive level, (a - c)/(b - d).

The characteristic equation associated with (5.3) is

-(N + l)[b,(l + 61:,) - b,&2][2db2&2 - b,]

+ bU[b,(N + 6(N + 2)&) - (N + l)bl,%,,] = 0. (5.5)

This condition for AZ2 is a generalization of the monopoly case, (4.5), to the case of N firms. In particular, the expression is positive at AZ2 = 0, negative at the competitive value of the persistence coefficient, Al, given by (2.6), and strictly convex over the interval 0 5 AZ2 I L1. These conditions imply that there is a unique stable value of the persistence coefficient, L,,(N), that is increasing in N and approaches 1, as N --* 00. ” As the number of producers increases, the

expected impact on future price of past consumption decreases, stimulating

consumption at each point in time. For any N, consumption persistence of Type 2 agents, I,,(N), is decreasing with respect to the fraction, 0, of Type 2 agents in the market.13 Also, the coefficient Rlz that links consumption of Type 1 agents

“Let A,,(N) be the solution of (5.5). Increasing N to N + 1 and lettmg A,, = Az2(N) results in

a positive value for (5.5):

[b,(l + 6&) - blL,,] [b, + Ob - 26b2&2] > 0.

The first factor is positive because we have shown that 1,,(N) < 1, for any N. The second factor is also positive, which together with the other properties of (5.5) implies that i,,(N) is increasing in N.

l3 Eq. (5.5) is linear in 0. Denoting the solution to (5.5) as Az2(N,0), we note that A2Z(N.0) = ,I1 and

J.,,(N, cu) < I,; thus, it follows that 1,,(N,B) is decreasing in 0.

1206 G. Fethke, R. Jagannathan /Journal of Economic Dynamics and Control 20 (1996) 1193-1207

to past consumption of Type 2 consumers is nonpositive, decreasing in the fraction of Type 2 consumers, and approaches zero as N + co.

6. Conclusions

We develop a model in which the market contains a mix of consumer types, namely, those with and without habit formation. We consider the cases of competitive supply, commitment and time-consistent monopoly, and time-consistent oligopoly. For competition and monopoly commitment, the persistence and steady-state consumption of consumers with habit formation are unaffected by the mix of consumer types. Consumption of habitual consumers follows a partial-adjustment rule, while the consumption of nonhabitual consumers is a constant. When marginal cost is constant and a mechanism exists that allows a monopolist to commit to a production path, then price (apart from an assignment associated with initial consumption) is independent of past consumption.

When commitment is not possible, we analyze a time-consistent equilibrium. Here, the past consumption of habitual consumers will affect the consumption paths of all consumers. This effect, which operates through the price level, reduces the steady-state consumption and the rate of consumption persistence of habitual consumers below that associated with commitment and competition and also reduces the consumption of nonhabitual consumers. As the fraction of habitual consumers increases, both steady-state price and the responsiveness of price to past consumption increase.

Finally, we consider the case of time-consistent oligopoly. This extension reveals that the consumption persistence and steady-state consumption of habitual consumers is increasing in the number of producers, approaching the competitive case. Consumption persistence under oligopoly is decreasing in the fraction of habitual consumers for any number of firms.

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