The Journal of Socio-Economics 38 (2009) 415–420

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The Journal of Socio-Economics

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Consumption theory with reference dependent utility

Fredrik W. Andersson ∗

Göteborg University, Department of Economics, Vasagatan 1, P.O. Box 640, 405 30 Göteborg, Sweden

a r t i c l e i n f o

Article history:Received 3 April 2007Received in revised form 13 December 2008Accepted 7 February 2009

JEL classification:D91E21

Keywords:

a b s t r a c t

This paper presents a closed form consumption function for an individual when his utility depends bothon his own current and previous consumption and on the consumption by his relevant others. Given thismodel, I argue that we can introduce an alternative definition of marginal propensity to consume (MPC)in addition to the traditional definition. This alternative definition can be called the individual’s total MPC,which I show is smaller than the traditional MPC.

© 2009 Elsevier Inc. All rights reserved.

Consumption decisionConsumption by relevant othersHM

1

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tion of the individual’s MPC given different notions of what the

1d

abit-formation behaviorarginal propensity to consume

. Introduction

Empirical evidence suggests that individuals evaluate own con-umption (income) by comparing it to the consumption (income)evels of others; see e.g. Solnick and Hemenway (1998), Johansson-tenman et al. (2002), Alpizar et al. (2005), and Andersson (2008).his paper presents a general consumption model that is anxtended version of Alessie and Lusardi’s (1997) consumptionodel. In Alessie and Lusardi (1997), individuals merely care about

heir own current and previous consumption. I add the assump-ion that individuals also compare own consumption with that seenmong relevant others, and derive a closed form consumption func-ion for an arbitrary individual. Since an individual’s consumptionlso depends on the consumption by his relevant others, I intro-uce the individual’s total marginal propensity to consume (totalPC). Earlier theories like Hall’s permanent income hypothesis

PIH) (Hall, 1978), and a pure habit formation behavior model, suchs Alessie and Lusardi (1997), imply larger marginal propensities toonsume than found in this model.

Is it realistic that individuals only have their own previousonsumption levels as reference? Probably not. From a psycholog-

cal perspective, individuals compare own consumption also withhe consumption levels of relevant others. Duesenberry (1949, p.8) argues that “Any particular consumer will be influenced byonsumption of people with whom he has social contacts. . .”;

∗ Tel.: +46 31 773 2679.E-mail address: fredrik.andersson@economics.gu.se.

053-5357/$ – see front matter © 2009 Elsevier Inc. All rights reserved.oi:10.1016/j.socec.2009.02.006

he coins this concept “the demonstration effect.” Duesenberry’snotion has long been overlooked in economics models, although hehas advocators within psychology. For example, Runciman (1966)argues that individuals have both a space and time dimension ofcomparison. Frank (1985, p. 146) presents an explanation to whyeconomists are not keen on adopting the space dimension: “Tomany economists, the notion of consumers being strongly influ-enced by demonstration effects must have seemed troublinglyinconsistent with the reasoned pursuit of self-interest, if not com-pletely irrational.” It seems reasonable to extend Alessie andLusardi’s (1997) model by including Duesenberry’s demonstrationeffect. For example Frank (1985, p. 150) supports this by arguing:“. . . concerns about relative standing are perfectly compatible withthe economist’s view that people pursue their own interest in arational way.” I believe this extended consumption model addsmore knowledge about individuals’ actual consumption decision.1

This paper has the following structure: Section 2 describes anindividual’s utility maximization problem. In Section 3, I derivethe individual’s closed form consumption function in addition toa recursive consumption function. Section 4 discusses the defini-

individual utility depends on, and finally, Section 5, presents someconcluding remarks.

1 I use an additive comparison function since Wendner (2002, p. 16) argues that“the multiplicative [i.e., ratio] specification is not in line with elementary propertiesof habit persistence.” Ratio comparisons are used by, e.g. Abel (1990), Carroll et al.(1997), Carroll (2000), and Aronsson and Johansson-Stenman (2008).

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. The individual’s utility

.1. The individual’s utility function arguments

In order to emphasize how important individuals’ social inter-ctions with each other are, Aristotle referred to human beings asocial animals. By looking at psychological and sociological motives,.g. Duesenberry (1949), Runciman (1966), Frank (1985), and Elsternd Loewenstein (1992) argue that individuals have both a spacend a time dimension of comparison. I.e., individuals compare theirwn current consumption with a reference level that is a func-ion of both the consumption by relevant others and their ownrevious consumption. Compared to the two consumption modelsentioned in Section 1 this adds more realism to what individ-

als’ utility depends on. Put differently, Scitovsky (1992) argueshat people wish to keep their status in relation to their referenceevel, since losing status may be painful. Here I extend Alessie andusardi’s (1997) model by assuming that people also care about theonsumption among relevant others. Then, the “psychological” con-umption amount that utility depends on at time �, for an arbitraryndividual, is:

∗� = c� − �c�−1 − �c̄�, (1)

here � ∈ [0, 1] controls how much the individual cares about theonsumption among his relevant others,2c̄� .3 The higher the �, theore the individual cares. The other parameter, � ∈ [0, 1], controls

ow much the individual cares about his own previous consump-ion, and � > 0 implies that the individual has a habit-formationehavior. The higher the � , the more the individual cares aboutis previous consumption. The formulation in Eq. (1) will then boilown to the one used by Alessie and Lusardi (1997) for � > 0 and= 0, and when � = � = 0 it will reflect the conventional model assed by, e.g. Hall (1978).

.2. The individual’s utility maximization problem

By assumption, the individual’s utility, u(c∗�), is concave, contin-

ous, and twice differentiable over the interior of the individual’s∗�set, and moreover I restrict the individual’s consumption amount,� , to always be non-negative.

In order for the individual to optimize his consumption profile,e needs to predict at time � his stock of human wealth, which ishe present discounted value of his expected future labor incomend the current value of his non-human wealth (a). I assume that thendividual has a finite life, gives no bequests at period T, dies withoutny debt, and lives in a world with a perfect capital market (i.e.,ndividuals can borrow and lend at the same constant4 interest rate)n addition he is not liquidity constrained. Furthermore, I assumehat the individual has perfect foresight about his own future laborncome and the future consumption among his relevant others; i.e.,he information is complete and there is no uncertainty.

Then the individual’s intertemporal maximization problem cane specified as

T∑

ax{c� }T�=t

U� =�=t

ˇ�u(c∗�(c�, c�−1, c̄�)), (2)

2 Relevant others refers to, e.g. neighbors, co-workers, and friends.3 This is similar to the psychological consumption that Alonso-Carrera et al. (2004)

se in a paper that analyzes the circumstances under which consumption by relevantthers is a source of inefficiency. They also included a third reference argument,hich is the previous consumption, c̄t−1, of relevant others.4 The interest rate is independent of the capital stock in the economy.

Economics 38 (2009) 415–420

subject to his intertemporal budget constraint

T∑�=t

(1

1 + r

)�

c� +(

11 + r

)T+1aT+1 = a� +

T∑�=t

(1

1 + r

)�

y�, (3)

and(1

1 + r

)T+1aT+1 ≥ 0, (4)

where a� and c�−1 are given. Since the individual cannot haveunpaid debts at period T, aT+1 cannot be less than zero. Moreover,from the individual intertemporal utility maximization problem, itis not optimal for the individual to have unused resources when hedies, hence aT+1 = 0 will always hold. Constraints (3) and (4) cantherefore be combined into:

a� +T∑

�=t

(1

1 + r

)�

y� −T∑

�=t

(1

1 + r

)�

c� = 0. (5)

When the interest rate, r, is constant over time, the intertempo-ral budget constraint implies that the present discounted value ofconsumption is equal to the individual’s initial wealth (a) plus hispresent discounted labor income (y).

Furthermore, I assume that the consumption among relevantothers is not affected by the individual’s consumption; i.e., c̄� isexogenously given.

The individual’s discount factor, ˇ = 1/(1 + �), is constant overtime, where � > 0, and is the individual’s pure time preference. Thisrules out any possibility of discontinuity of U� (i.e., assures that U�

does not diverge to infinity).The individual’s intertemporal maximization problem is then

solved by maximizing his lifetime utility (2) subject to his intertem-poral budget constraint (5). The Lagrangian function for thisproblem is:

max{c� }T�=tL(c�, c�+1, . . . ; �) =

T∑�=t

ˇ�u(c∗�(c�, c�−1, c̄�))

+ �

(a� +

T∑�=t

(1

1 + r

)�

y� −T∑

�=t

(1

1 + r

)�

c�

), (6)

where � is the constant Lagrange multiplier. The first order condi-tion for an interior solution at an arbitrary period t is:

∂L(·)∂ct

= ˇt ∂u(c∗t )

∂c∗t

∂c∗t

∂ct+ ˇt+1

∂u(c∗t+1)

∂c∗t+1

∂c∗t+1

∂ct− �(

11 + r

)t

= 0. (7)

Since this expression holds for all t, it is obvious that it also holdsfor t + 1:

∂L(·)∂ct+1

= ˇt+1∂u(c∗

t+1)

∂c∗t+1

∂c∗t+1

∂ct+1+ ˇt+2

∂u(c∗t+2)

∂c∗t+2

∂c∗t+2

∂ct+1− �(

11 + r

)t+1

= 0. (8)

Then solving for the individual’s marginal rate of substitution (MRS)by combining (7) and (8), we have (after some manipulation):

(∂u(c∗t+1)/∂c∗

t+1)(∂c∗t+1/∂ct+1)+ˇ(∂u(c∗

t+2)/∂c∗t+2)(∂c∗

t+2/∂ct+1)

(∂u(c∗t )/∂c∗

t )(∂c∗t /∂ct) + ˇ(∂u(c∗

t+1)/∂c∗t+1)(∂c∗

t+1/∂ct)

1 + �

=1 + r

. (9)

Up to this point, the individual’s MRS is valid for both a ratio andan additive comparison function. Let us continue the derivationof the individual’s MRS with the additive comparison function as

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n (1). Thus, Eq. (1) has the following properties for the additiveomparison function:

∂c∗t

∂ct=

∂c∗t+1

∂ct+1= 1, (10)

∂c∗t+1

∂ct=

∂c∗t+2

∂ct+1= −�. (11)

ssuming that the individual’s pure value of time preference isqual to the interest rate (� = r), the individual’s MRS (9) may beewritten with the additional properties in (10) and (11) as

(∂u(c∗t+1)/∂c∗

t+1) − �ˇ(∂u(c∗t+2)/∂c∗

t+2)

(∂u(c∗t )/∂c∗

t ) − �ˇ(∂u(c∗t+1)/∂c∗

t+1)= 1. (12)

q. (12) is satisfied if and only if:

∂u(c∗t )

∂c∗t

=∂u(c∗

t+1)

∂c∗t+1

=∂u(c∗

t+2)

∂c∗t+2

= �, ∀ t, (13)

here ˝ is a constant (see Appendix A for the proof), i.e., thearginal utility of psychological consumption must be constant

ver time if the MRS between any two periods equals 1. Any con-ave utility function implies that when the marginal utility ofsychological consumption is constant, the level of psychologicalonsumption is also constant ⇒ c∗

� = constant. (see Lemma 1).

emma 1. Eq. (13) implies that the path of psychological consump-ion {c∗

t }Tt=0 is constant over time.

Hence, if the consumption by relevant others increases ineriod t + 1, the individual’s consumption in period t + 1 must also

ncrease in order to keep the marginal utility of psychological con-umption constant. By utilizing this knowledge, it is possible toerive the individual’s consumption change:

ct+1 = �ct + �c̄t+1, (14)

here ct+1 = ct+1 − ct . This shows that the individual’s con-umption change in period t + 1 depends on his own previousonsumption change and the current consumption changes amongis relevant others.5 This is a general Euler equation that boils downlessie and Lusardi’s (1997) consumption model when � > 0 and= 0 and Hall’s PIH (1978) when � = � = 0.

. The individual’s closed form consumption function

The Euler equation is a recursive consumption function, which isere possible to rewrite as a function of the individual’s consump-ions in the first period as follows:6

t = (1 + g − �) − g�t

1 − �c0 + �

⎡⎣T−2−j∑

t=2

T∑j=0

�jc̄t

⎤⎦ , ∀ t ≥ 2. (15)

use the assumption that c1 = (1 + g)c0 (the individual’s consump-ion changed with rate g between period t = 0 and t = 1) in order to

eceive a simple expression for the consumption in the first period.7

consumption path derived from Eq. (15) will satisfy the Eulerquation. When I consider the individual’s budget constraint it is

5 My only interest here is the interior solution to the intertemporal maximizationroblem.6 In order to derive this equation I use the Euler equation. I first define the indi-

idual’s consumption in period t = 1. Then I lead the Euler equation one period andse the information from the consumption in period t = 1, and so forth.7 This corresponds to the case when the individual’s utility in the first period

epends merely on past consumption. It is not until the second period that thendividual’s utility depends on the consumption by his relevant others.

Economics 38 (2009) 415–420 417

possible to derive his closed form consumption function. By plac-ing (15) in (5), it is possible to write the individual’s first periodconsumption as

c0 =

(1 + g)+˚ − �

⎡⎣at +

T∑t=0

(1

1 + r

)t

yt −�

T−2−j∑t=2

T∑j=0

�jc̄t

⎤⎦ ,

(16)

where

= (1 + �)(1 + r − �)r(1 + r)T , (17)

˚ = (1 + g − �)(1 + r − �)r((1 + r)T − 1), (18)

� = g r�((1 + r)T − �T ). (19)

Eq. (15) together with Eq. (16) give a unique consumption paththat satisfies both the Euler equation and the individual’s budgetconstraint.

In order to discuss one of the aims in this paper, I need to derivea consumption function for the individual that depends on the indi-vidual’s previous consumption. Hence, I rewrite the intertemporalbudget constraint (5) by substituting in c� from (1), and after somemanipulation I solve for the present discounted value of c∗

� :8

T∑�=t

(1

1 + r

)�

c∗� = −�c�−1 − �

T∑�=t

(1

1 + r

)�

c̄�

+[

1 − �

1 + r

](a� +

T∑�=t

(1

1 + r

)�

y�

). (20)

Using Lemma 1, I can derive from (20) a recursive consumptionfunction, which satisfies the individual’s budget constraint, for theindividual:9

c� = (1 + r)T − 1

(1 + r)T+1 − 1�c�−1︸ ︷︷ ︸

i

+ (1 + r)T − 1

(1 + r)T+1 − 1�c̄�︸ ︷︷ ︸

ii

− �r(1 + r)T

(1 + r)T+1 − 1

T∑�=t+1

(1

1 + r

)�

c̄�

︸ ︷︷ ︸iii

+(

1 − �

1 + r

)r(1 + r)T

(1 + r)T+1 − 1

[a� +

T∑�=t

(1

1 + r

)�

y�

]︸ ︷︷ ︸

iv

. (21)

From Eq. (21) it follows that the individual’s consumptionlevel at time � depends on four features: (i) a habit level ofconsumption—the individual’s own previous consumption, (ii) akeeping-up effect —current consumption by relevant others, (iii) afear (future potential disutility) of falling behind effect —the futureconsumption by relevant others, and (iv) the wealth effect, which is

also present in the PIH, although it is reduced when the individ-ual has a habit-formation behavior (� > 0).10 The general recursiveconsumption function in Eq. (21) boils down to two other con-sumption functions found in the literature: Hall’s (1978) PIH when

8 We can easily verify that if � = 0, the intertemporal budget constraint collapsesto an ordinary “textbook” intertemporal budget constraint, and hence, if ˛ = 0, itcollapses to Alessie and Lusardi’s (1997) intertemporal budget constraint.

9 We know that Lemma 1 implies∑T

�=t(1/(1 + r))� c∗

� = ((1 + r)T+1 −1)/(r(1 + r)T )c∗

� .10 The number of lagged variables depends on how many lagged variables are

included in the c∗� measure.

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18 F.W. Andersson / The Journal of

= � = 0, which shows that an individual’s consumption at time �s equal to the annuity value of his lifetime resources and is constantver time, and the one in Alessie and Lusardi (1997), when � > 0nd � = 0. The individual’s consumption at time � then dependsartly on his previous consumption and partly on his permanent

ncome.

.1. The impact of the individual’s permanent income

It is possible to rewrite Eq. (21) by using concepts such as thendividual’s permanent income and the permanent consumption byis relevant others,11 where the individual’s permanent income (yp

�)s the annuity value of the sum of the current non-human wealth (a�)nd h uman wealth (present discounted value of future income),nd where the permanent consumption by relevant others (c̄p

�+1)s the annuity value of the present discounted value of future con-umption. Thus, the individual’s consumption in period � is:

� = ��c�−1 + ��c̄� − �c̄p�+1 +

(1 − �

1 + r

)yp

�, (22)

here � = ((1 + r)T − 1)/((1 + r)T+1 − 1), yp� = r(1 + r)T /

(1 + r)T+1 − 1)(a� +∑T

�=t(1/(1 + r))�y�), and c̄p�+1 = r(1 + r)T /

(1 + r)T+1 − 1)∑T

�=t+1(1/(1 + r))� c̄� .We can see from (22) that a change in permanent income has the

ame effect on the individual’s consumption as if the psychologi-al consumption measure would merely include a habit formationehavior; i.e., the individual’s utility depends on c∗

� = c� − �c�−1.hus, the individual’s consumption changes with his permanentncome as

∂c�

∂yp�

= 1 − �

1 + r. (23)

rom Eq. (23) we can see that if the individual increases his concernbout his previous consumption, i.e., his habits, then a change in hisermanent income changes his consumption to a lesser extent.

∂2c�

∂yp�∂�

= − 11 + r

< 0. (24)

hen the individual has a negative change in his permanentncome, stronger habits (higher �) implies that the individual’s con-umption decreases by a smaller amount than with weaker habits.his smoother reduction comes from the fact that it takes time forn individual to alter his consumption habits.

.2. The impact of the consumption by relevant others

How does the consumption at time � depend on a change in theegree of concern about the consumption by relevant others, i.e.,ith �?

∂c�

∂�= c̄�︸︷︷︸

i

− c̄p�+1︸︷︷︸ii

. (25)

he individual’s consumption is affected in two ways: (i) The firstffect stems from the individual’s wish to keep up with the con-umption levels of his relevant others at time �; this is captured byhe first term in (25). This implies that the individual’s consumption

evel adjusts upward by a fraction of the current consumption byelevant others, i.e., the individual’s consumption to some degreeracks the consumption levels of his relevant others. (ii) The sec-nd effect arises if the individual perceives the future consumption

11 It would also be possible to rewrite the permanent consumption by relevantthers as permanent income if we assume a constant saving rate.

Economics 38 (2009) 415–420

path of the relevant others as painful—i.e., it reduces the individ-ual’s utility. This is captured by the last term in (25), where we cansee how the individual’s consumption is negatively affected by hisrelevant others’ future consumption. Hence, the higher the futurepermanent consumption among his relevant others, the more theindividual’s consumption at time � is reduced.

3.2.1. A temporary increase in the consumption by relevant othersConsider a temporary increase in the consumption by relevant

others at current time, �. This may be for example a bonus, i.e.,an extra amount of money. This increase boosts the individual’scurrent consumption, since he wishes to keep up with them:

∂c�

∂c̄�= � > 0. (26)

This effect may impact the individual’s consumption growth, andhence may provide some insight into why consumption growshigher than income.

“. . . in each of the past three years, real consumer outlays havegrown faster than real aftertax income.”—Business Week 17/4 2006 (U.S.: It’s Way Too Early to CountConsumers Out)

The individual knows that the consumption by relevant othersincreases at time � but not at time � + 1. He will therefore increasehis consumption at time � to not lose status.

3.2.2. A permanent increase in the consumption by relevantothers

Veblen (1934) states that it is the best-off members in a societywho establish the consumption standard for the rest, and then peo-ple below wish to emulate their consumption. Duesenberry (1949,p. 101) claims that “Low-income groups are affected by consump-tion of high-income groups but not vice versa.”, i.e., individualsmake upward comparisons when they evaluate their consumptionlevel. Similar thoughts, i.e., are voiced by, e.g. Schor (1998, p. 4) whoargues that individuals “make comparison with, or choose, a ‘refer-ence group,’ people whose income are three, four, or five times hisor her own.” She finds that individuals with lower financial statusthan their reference groups save significantly less than individualswith better financial status than their reference groups.

If the increase in the consumption by relevant others is perma-nent, i.e., also c̄p

�+1 changes and not just c̄t , then the individual’sconsumption changes as

∂c�

∂c̄�+ ∂c�

∂c̄p�+1

= �� − � = �(� − 1) < 0, since � < 1. (27)

Hence, the individual’s consumption is negatively affected by a per-manent consumption increase among relevant others.12

4. Marginal propensity to consume

4.1. A review of marginal propensity to consume

Keynes (1936, p. 36) argues that “The fundamental psycholog-ical law. . . is that men are disposed, as a rule and on the average,to increase their consumption as their income increases, but not

consumption function hypothesizes that if an individual’s currentincome rises/falls by one unit, then his consumption should rise/fallproportionally with the MPC, which is less than 1. This is the absolute

12 This effect is a result of the individual’s intertemporal budget constraint, whichimplies that his present discounted value of consumption cannot be larger than thepresent discounted value of his human and non-human wealth.

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ncome hypothesis. Another venue is Friedman’s (1957) permanentncome hypothesis, where MPC is determined by the relative vari-tions in permanent and transitory incomes. When the variationn permanent income is much greater than the variation in transi-ory income, consumption rises almost one-for-one with currentncome. Intuitively, an individual’s consumption increases whenis permanent income increases. Hall (1978) derives his versionf the permanent income hypothesis within a framework of explicitntertemporal utility maximization, rather than merely assertingproportional dependence between consumption and permanent

ncome, and predicts that the MPC is 1. Thus, any change in thendividual’s permanent income affects his consumption one-for-ne. Duesenberry (1949) rejects this symmetric notion, claiminghat once consumption habits and status are acquired, it is hardor an individual to alter them. He therefore postulates the relativencome hypothesis. At a certain income level, certain status and cer-ain consumption habits are formed, and these are not completelybandoned if income falls; it takes time for individuals to adjustonsumption downward. Alessie and Lusardi (1997) derive a recur-ive consumption function from Duesenberry’s (1949) notion. Themplication is that when an individual cares both about his currentnd previous consumption, his MPC is smaller than in Hall’s (1978)odel of PIH. Although I use perfect foresight about future flows,

he MPC is the same as if the individual would have a stochasticabor income process.

We are now ready to compare the different marginal propensi-ies to consume from Hall’s (1978) model of PIH, from Alessie andusardi (1997), and from Eq. (22).

.2. The individual’s marginal propensity to consume

After deriving the individual’s recursive consumption function,t is possible to derive and study the individual’s MPC. The tradi-ional definition of an individual’s MPC is:

MPC defines how much an individual’s consumption changes whenhis permanent income is changed by one unit. 13

This definition is used by, e.g. Keneys. However, when we nowllow the individual’s utility to depend on his psychological con-umption as in Eq. (1), instead of on his absolute consumption, it isossible to add a new definition of the individual’s MPC.

.2.1. An increased permanent income affects MPCThe individual’s recursive consumption function is the same as

q. (22):

� = ��c�−1 + ��c̄� − �c̄p�+1 +

(1 − �

1 + r

)yp

�. (28)

ifferentiating Eq. (28) with respect to yp� yields the traditional mea-

ure of an individual’s MPC (we did the exact same procedure inection 3.1 but never introduced the concept of MPC):

pc = ∂c�

∂yp�

= 1 − �

1 + r. (29)

ence, if the individual would have had one unit higher permanentncome, his consumption at time � would change by 1 − (�/(1 + r))nits. We can see that the individual’s MPC depends merely on howuch he cares about his previous consumption (�). This is also an

ndividual’s MPC in the model by Alessie and Lusardi (1997), hencepc ≡ mpcA&L .The other extreme case is when � = 0, which corresponds to

all’s (1978) model of PIH. Then, it is easy to see that the individual’s

13 One unit higher permanent income implies that the income in each periodncreases by one unit.

Economics 38 (2009) 415–420 419

MPC is:

mpcPIH = ∂c�

∂yp�

= 1. (30)

This implies that if the individual’s permanent income is one unithigher, then consumption increases under Hall’s (1978) model ofPIH by the corresponding amount.

However, when we now allow psychological and sociologicalaspects to enter the economic models, we need to have a broaderdefinition of the individual’s MPC. I argue that we should alsoinclude the effect of the consumption of relevant others, both thecurrent (c̄�+1) and future permanent consumption (c̄p

�+1), in thedefinition of the MPC, since the effect of the individual’s own per-manent income on his consumption is only one of several effects.I therefore put forward another definition of the individual’s MPC,which takes into account the consumption of the individual’s rele-vant others. I introduce the individual’s total MPC as

The total MPC defines by how much an individual’s consumptionchanges when his permanent income is changed by one unit, andat the same time the permanent consumption among his relevantothers increases by one unit.

4.2.2. The consumption by relevant others affects MPCWe assume that structural change in the economy has a per-

manent effect on the consumption by relevant others; both c� andc̄p

� change in the same direction. The individual’s total MPC, withrespect to a permanent increase in the consumption by his relevantothers is hence:

mpctotal = ∂c�

∂c̄�+ ∂c�

∂c̄p�+1

+ ∂c�

∂yp�

= 1 + �� − � − �

1 + r, (31)

where � < 1. With the assumptions that � > 0 and 0 < � < 1, it ispossible to formulate the following proposition:

Proposition 1. When an individual’s utility depends on both his owncurrent and previous consumption in addition to the current consump-tion by his relevant others, his MPC is lower than if he had merely hada habit formation behavior, as in Alessie and Lusardi (1997), while theMPC is the highest in Hall’s (1978) PIH model.

Proof 1. The inequality 1 = mpcPIH > mpcA&L = 1 − (�/(1 + r))holds when � ∈ (0, 1], since we have that �/(1 + r) > 0. Fur-thermore, when � ∈ (0, 1] and � ∈ (0, 1), we have that 1 +�(� − 1) − (�/(1 + r)) = mpctotal < mpcA&L = 1 − (�/(1 + r)) holdssince �(� − 1) < 0. Hence, it is obvious that mpcPIH > mpcA&L >mpctotal. �

Hence, when � < 1, � > 0, and 0 < � < 1, we can conclude thatthe total MPC is smallest: mpcPIH > mpcA&L > mpctotal.

Thus, if the individual has both a habit-formation behavior andcares to some degree about the consumption among his relevantothers, then his consumption is less sensitive to an increase in hispermanent income and to the permanent consumption levels ofhis relevant others compared to if he merely has a habit-formationbehavior and obviously if only his absolute consumption drives util-ity. This “smoothness” depends on the fact that the individual doesnot wish to fall behind the consumption among his relevant othersin the future, and therefore he adjusts his consumption level less.

Hence, this model may be better at explaining the “excess smooth-ness” phenomenon14 found in the consumption data compared toearlier theories, which opens up for further research.

14 See, e.g. Deaton (1992).

4 Socio-

5

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ccswtFdtAtlmu

tat(TwiC

A

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Ps

a

Pc

Oxford University Press, New York, Oxford.

20 F.W. Andersson / The Journal of

. Concluding remarks

By looking at psychological and sociological arguments of whatndividuals’ utility depends on, I extend Alessie and Lusardi’s1997) consumption model, in which individuals merely have habit-ormation behaviors, by adding the notion that individuals also carebout the consumption among their relevant others. I also derivegeneral closed form consumption function that boils down to

he closed form consumption functions of Hall’s (1978) PIH, or aabit-formation behavior such as in Alessie and Lusardi’s (1997)onsumption models.

The extension of Alessie and Lusardi’s (1997) psychologicalonsumption measures (that utility depends on) implies that theonsumption among relevant others affects an individual’s con-umption in two ways: The individual’s consumption (i) increaseshen the current consumption increases and (ii) decreases when

he future permanent consumption by relevant others increases.urthermore, a change in the consumption among relevant othersoes not affect the individual’s traditional MPC. However, I showhat the total MPC is lower compared to both Hall’s (1978) andlessie and Lusardi’s (1997) MPC. This is a consequence of the fact

hat the individual does not wish to fall behind the consumptionevel of his relevant others in the future. Furthermore, this model

ay be better at explaining the “excess smoothness” which opensp for further research.

It should be possible to test whether an individual’s consump-ion depends on both his own previous consumption and that seenmong his relevant others, and whether new information regardinghe individual’s own future income and that among relevant othersrelaxing the assumption of perfect foresight) affects consumption.his is in a sense analogous with papers in the happiness literature,here empirical evidence suggests that happiness of individuals

s affected by the income level of relevant others; see, e.g. Ferrer-iarbonell (2005).

cknowledgements

I appreciate the thoughtful comments of Katarina Nordblom,lof Johansson-Stenman, Håkan Locking, Wlodek Bursztyn, Johnassler, Markus Knell, Debbie Axlid for editorial advices, and twononymous referees for valuable comments on an earlier version ofhis paper.

ppendix A

We know that an individual’s MRS under certain assumptions isqual to (the same as Eq. (12)):

(∂u(c∗t+1)/∂c∗

t+1) − �ˇ(∂u(c∗t+2)/∂c∗

t+2)

(∂u(c∗t )/∂c∗

t ) − �ˇ(∂u(c∗t+1)/∂c∗

t+1)= 1. (32)

roposition 2. The only admissible time series of psychological con-umption, {c∗

� }T�=t , that satisfies Eq. (32) when T → ∞ is:

∂u(c∗t )

∂c∗ =∂u(c∗

t+1)

∂c∗ =∂u(c∗

t+2)

∂c∗ = ˝, ∀ t, (33)

t t+1 t+2

nd where ˝ is a constant.

roof 2. If the marginal utility of psychological consumption is notonstant over time and (∂u(c∗

t+1)/∂c∗t+1) /= (∂u(c∗

t )/∂c∗t ), it is possible

Economics 38 (2009) 415–420

to calculate (∂u(c∗t+2)/∂c∗

t+2). In order for Eq. (32) to be satisfied,

∂u(c∗t+2)

∂c∗t+2

= 1 + �ˇ

�ˇ

∂u(c∗t+1)

∂c∗t+1

− 1�ˇ

∂u(c∗t )

∂c∗t

. (34)

Eq. (34) is a second order difference equation whose general solu-tion is:

∂u(c∗t+n)

∂c∗t+n

= A + B(

1�ˇ

)n

, (35)

where A and B are arbitrary constants. When an individual has ahabit-formation behavior, then 0 < �ˇ < 1, and then Eq. (35) showsthat the individual’s marginal utility of psychological consumptionis growing over time. When an individual’s utility function is con-cave, then his psychological consumption, c∗

t+n, decreases over time,which implies that his absolute consumption also decreases overtime. This is not a utility maximization, i.e., (2) is not maximized.Hence, Proposition 2 is true, and we can formulate Lemma 1. �

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