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IOWA STATE UNIVERSITY

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									                           IOWA STATE UNIVERSITY
                                             Are the Joneses making you financially vulnerable?

                                                   Richard C Barnett, Joydeep Bhattacharya, Helle Bunzel

                                                                            April 2008

                                                                 Working Paper # 08011


                                                   Department of Economics
                                                    Working Papers Series




                                                                      Ames, Iowa 50011

Iowa State University does not discriminate on the basis of race, color, age, religion, national origin, sexual orientation, gender identity, sex, marital status,
disability, or status as a U.S. veteran. Inquiries can be directed to the Director of Equal Opportunity and Diversity, 3680 Beardshear Hall, (515) 294-7612.
            Are the Joneses making you financially vulnerable?




                                              Richard Barnett∗

                                            Villanova University



                                          Joydeep Bhattacharya†

                                           Iowa State University



                                                 Helle Bunzel‡

                                           Iowa State University




                                                April 30, 2008§




  ∗ Corresponding  author: Richard C. Barnett, Department of Economics, Bartley Hall Rm 3008, 800 Lancaster Avenue,
Villanova, PA 19085; Ph: (610) 519-6321; Email address: Richard.Barnett@villanova.edu
   † Department of Economics, Heady Hall, Iowa State University, Ames IA 50011; E-mail: joydeep@iastate.edu

   ‡ Department of Economics, Heady Hall, Iowa State University, Ames IA 50011; E-mail: hbunzel@iastate.edu

   § We thank Saqib Jaffrey, Rajesh Singh, and especially Subir Bose, for highly valuable input.




                                                        1
                     Are the Joneses making you financially vulnerable?

   Abstract: This note studies a model in which heterogeneous income agents get a utility boost only

when their consumption catches up with the Joneses’. The resulting utility function is non-concave. In

this setup, participation in a fair consumption lottery has the potential to make some agents ex-ante

better off but more financially vulnerable. More income-diverse people join the lottery pool when the

‘kick’ from catching up increases. Worsening income inequality may increase the number of financially

vulnerable people. The analysis sheds light on some aspects of the ongoing sub-prime mortgage crisis.




JEL Classifications: D 01, R 21

Keywords: catching up with the Joneses, consumption externalities, non-concave utility, lotteries,

inequality




                                                  2
1     Introduction

There is a fair bit of evidence to suggest that people care not just about their absolute level of con-

sumption (or income) but also about how it compares to that of their neighbors.1 If Mr. X’s rich

neighbor builds a bigger/fancier house than his, it negatively affects his happiness. If this “neighbor

effect” is strong enough, Mr. X might be compelled to respond. He could, for example, take on a bigger

mortgage/home-equity loan, and use it to finance two extra bedrooms or a deck. Such an action may

make him more “financially vulnerable”, for if interest rates on the loan rises or house prices plummet,

he may cease to be able to afford the extra rooms or the deck. This paper models this exact scenario,

albeit in a fairly stylized fashion.

    To keep things as simple as possible, the model economy is static and populated by a large number

of agents with varying incomes. Utility is reference-dependent: only if consumption exceeds a reference

or benchmark (identified here with the consumption of the Joneses) does the agent get a fixed utility

kick from catching up with the Joneses. This feature introduces a jump or discontinuity in utility at

this benchmark. In technical terms, even though utility is smooth and concave below and above the

benchmark, the overall utility function is rendered non-concave by this discontinuity.

    In such a setting, as originally discussed by Friedman and Savage (1948) and more recently by Hartley

and Farell (2002), some agents may do better in an ex-ante sense were they to purchase consumption

lotteries. In our case, the lottery is of the following form: if you pledge your entire income to a lottery

(i.e., you are willing to randomize your income), the lottery pays out a low level of consumption with

some probability and with the remaining probability it pays just enough to allow you to catch up with

the Joneses. The odds of winning the lottery, getting an income enough to support the benchmark level

of consumption, depend positively on the income pledged. The upshot is that rational and risk-averse

individuals with a desire to catch up with the Joneses take on more risk than they would if they didn’t

covet their neighbors’ consumption.

    Who does not participate in such a lottery? Those whose certainty income is high enough to afford

the aforementioned benchmark level of consumption. However, not everyone else participates; after all,

those with sufficiently low incomes face zero odds of winning such a lottery (and hence, are happier

receiving their sure income). In fact, as we show, there are people in a certain ‘middle’ range of incomes

who enter this lottery; in the process, they make themselves “financially vulnerable”. These are people

who are able to catch up with the Joneses only if they win the lottery. We go on to show that ceteris
   1 Luttmer (2005), for example, finds convincing evidence that “an increase in neighbors’ earnings and a similarly sized

decrease in own income each lead to a reduction in happiness of about the same order of magnitude.” See Clark, Frijters,
and Shields (forthcoming) for a insightful review of the issues.




                                                           3
paribus, a mean-preserving increase in the spread of the income distribution may increase the number

of financially vulnerable people.

    In light of the above discussion, one can reinterpret the aforementioned fable of the jealous neighbors

and their houses as follows. Mr. X’s neighbor has a bigger/fancier house than Mr. X and he routinely

fret over this.2 He, however, does not have the resources to buy a house similar to his neighbor’s. In

fact, he cannot even afford the 20% downpayment the bank needs to see before they offer a standard

mortgage, one that is large enough for him to buy a house like his neighbors’. The bank may, however,

be willing to loan him more than the standard 80% or issue a variable rate home-equity line of credit. If

Mr. X accepts such contracts, he becomes more “financially vulnerable”; for if the variable rate spikes

or house prices crash precluding a refinance of his mortgage, he is in financial trouble. He may lose the

house. It is in this sense that our analysis sheds light on certain aspects of the sub-prime mortgage crisis

(“housing crisis”) currently plaguing the U.S.



2     The model

2.1     Preliminaries

Consider a simple static model in which people (indexed by i) have income yi . Agent i draws yi from

a continuous distribution with cdf G (yi ; σ) and density g (yi ; σ) with support [0, yU ] and σ denotes the

                                ¯
spread of the distribution. Let y denote the mean value of y. There is a single consumption good with

price normalized to 1 implying consumption equals the income.

    Agents’ preferences over consumption (c) is summarized by the utility function U (c) as follows:
                                           ⎧
                                           ⎪
                                           ⎨ u (c) + μ if c ≥ κ
                                   U (c) ≡                       .
                                           ⎪
                                           ⎩ u (c)      if c < κ

Here μ > 0 is a parameter capturing the ‘utility boost’ one gets when one’s consumption beats κ > 0

— the to-be-specified level of consumption the agent aspires to match. More generally, we think of μ as

the utility boost one gets from the realization that one has “made it in life”. We refer to the Joneses,

collectively, as those whose incomes meet or exceed κ. We assume the function u is strictly increasing

and strictly concave; also u (0) can be zero, positive, or negative. These preferences include an extremely

simple characterization of the notion of jealousy alluded to in the introduction: if Mr. X’s neighbor

builds a house of size bigger than κ, then relative to him, Mr. X gets a reduced utility of just u (c) .3
   2 Websites, such as Zillow.com, contain data on 52 million house valuations and get four million hits a month. As

O’Brien (2007) argues, “you type in your address to check out the Zestimate, an approximation of your home’s market
value. It appears in a little pop-up superimposed on a photographic map of your neighborhood. The number might make
you smile; it could make you angry....Next, you check your neighbors’ Zestimates.”
   3 Also, notice that utility is defined only on consumption. To keep things simple, we have ignored things like leisure




                                                           4
                                      utility


                                                  μ




                                                  κ                              c

                                           Figure 1: The utility function




    The agent has a two-piece utility function, the pieces defined over non-overlapping portions of the

consumption domain. For values of consumption that exceed κ, the agent receives utility not just from

consumption but also something extra (μ) from the very act of successfully catching up with the Joneses.

In terms of a picture, there is a discrete jump (discontinuity in U (c)) exactly at c = κ. Notice that

the marginal utility from consumption is the same irrespective of whether c ≤ κ or c > κ. Note that

the exact functional form for the utility function is not crucial; the results are more general and apply

whenever reference point (or aspiration level) effects create a two-piece utility function.4


2.2     Lotteries

We consider a setting in which consumption lotteries are accessible freely by all agents. These are fair

in the sense that the expected payout from the lottery equals the amount invested in it. Suppose agent

i invests his entire income yi in such a lottery. The lottery pays κ if agent i wins the lottery; otherwise

it pays cl < yi . Assume κ > cl . Let αi be the probability of agent i winning the lottery. Then, since the

lottery is fair,
                                                                          yi − cl
                                      αi κ + (1 − αi ) cl = yi ⇒ αi =             .
                                                                          κ − cl
although we are confident that the main thrust of our results will survive upon inclusion of leisure.
   4 Indeed, the jump at κ is not crucial to the analysis. One can redefine preferences as
                                               ⎧
                                               ⎪
                                               ⎨ u (c + θ (c − κ)) if c > κ
                                      U (c) ≡                                ; θ>0
                                               ⎪
                                               ⎩        u (c)       if c ≤ κ

In this case, there is no discontinuity but the utility function is not differentiable at κ (because the marginal utility
as c −→ κ− is different from the same as c −→ κ+ ). The main results, suitably amended, are preserved under these
preferences.




                                                            5
Since αi ∈ (0, 1), it follows that agent i participates in such a lottery if κ > yi > cl . In other words,

the subset of agents that potentially participate in such a lottery are those with income yi ∈ (cl , κ) .

In passing, notice that those who invest more in the lottery (i.e., those with high y) face better odds

of winning; in the limit, anyone investing y = κ has a 100% chance of winning back what is put in.

Similarly, if only cl is put in, there is no shot at winning the lottery and receiving the prize κ. Those

whose incomes are below cl don’t participate because they prefer the sure thing (their income) to the

lottery. The same is true of those people whose incomes are above κ.

   The issue at hand is: how is cl selected? We posit a perfectly competitive lottery industry where

individual lottery firms are Nash competitors. In other words, cl is chosen by maximizing the expected

utility of any lottery participant. We start by writing down the indirect expected utility to agent i from

participation in the lottery:

                                      Ui = αi (u (κ) + μ) + (1 − αi ) u (cl )

which may be rewritten as
                                          µ             ¶
                                              yi − cl
                                   Ui =                     [u (κ) + μ − u (cl )] + u (cl ) .
                                              κ − cl

Then, we compute a value of cl which maximizes Ui , taking as given κ. It is easy to check that cl solves:

                                                             u (κ) + μ − u (cl )
                                              u0 (cl ) =                         .                                           (1)
                                                                   κ − cl

Notice cl is the same for all i.

   Different people put in different amounts all hoping to win the lottery and receive κ as the prize.
                                                                           κ−yi
Person i puts yi into the lottery and faces a probability                  κ−cl   of losing the lottery. In that case, the
                                                                                     κ−yi
amount he loses is (yi − cl ) ; therefore his expected loss is l (yi ) ≡             κ−cl   (yi − cl ) . It is easy to check that

                                ∂l (yi )   (κ + cl − 2yi )             κ + cl
                                         =                 ≷ 0 ⇐⇒ yi ≶
                                 ∂yi           κ − cl                    2

This means that expectation of loss reaches its maximum for the income at the midpoint, κ+cl .
                                                                                         2




                                                                  6
                 U(c)




                                                               μ




                                cl           cla          κ                             c

                               Figure 2: The lottery (the heavy blue line)




   Why can’t a competitor offer a lottery that delivers κ in the good state and, say cla ∈ (cl , κ) in the

bad state? As Figure 2, illustrates, such a lottery would generate an expected utility on the thin red

line which forever lies below the heavier blue line (the lottery between κ and cl ).

   For future reference, the “envelope” expected utility function is given by
                           ⎧
                           ⎪
                           ⎪
                           ⎪
                           ⎪ µ                u (c) + μ                      if c ≥ κ
                           ⎪
                           ⎨            ¶
                                yi − cl
                   U (c) ≡                [u (κ) + μ − u (cl )] + u (cl ) if cl ≤ c < κ .
                           ⎪ κ − cl
                           ⎪
                           ⎪
                           ⎪
                           ⎪
                           ⎩                    u (c)                        if c < cl

2.3    Who participates in the lottery?

Under certain conditions, holding κ fixed, (1) has a unique solution which we denote by c∗ ≡ c∗ (μ) . To
                                                                                        l    l

see this, rewrite (1) as

                                     u0 (cl ) (κ − cl ) + u (cl ) = u (κ) + μ.                          (2)

Using κ > cl , it is easily verified that the slope of the left hand side of (2) is negative. Suppose we make

the following assumption:


Assumption 1




                                        u0 (0) κ + u (0) > u (κ) + μ .


                                                        7
Then, it follows that the left hand side of (2) at cl = 0 starts off above the right hand side; at cl = κ, the

right hand side exceeds the left hand side. Therefore, if Assumption 1 holds, there is a unique, positive

c∗ ; otherwise, c∗ = 0.
 l               l

   For future use, also note that differentiating (2) yields

                                       ∂c∗ (μ)
                                         l         1        1
                                               =      ∗ u00 (c∗ ) < 0.
                                         ∂μ      κ − cl       l

   If μ > 0, people with incomes in the range (c∗ , κ) will optimally choose to become financially
                                                l

vulnerable (i.e., they would prefer randomization) and take on risks they would otherwise not take (in

the absence of catching up). The implication of ∂c∗ (μ) /∂μ < 0 is that an increase in the utility kick
                                                  l

from catching up draws more income-diverse participants into the lottery pool.

   This does not necessarily imply, when μ rises, the number of people participating in the lottery goes

up. That depends on the probability mass on various income ranges. For future reference, note for given
                                                                Rκ
κ, the number of people participating in the lottery is given by c∗ g (yi , σ) dyi .
                                                                     l


   One can map the story of people buying lotteries into a story of people buying house mortgages.

Consider a setting in which income-diverse people are seeking to mortgage their houses. Roughly

speaking, banks offer some home buyers [in the income range, (c∗ , κ)] the following kind of deal: “Since
                                                              l

you do not qualify for a conventional type of mortgage, or at least one of size κ, we can offer you an

alternative type of mortgage, one with ‘special’ conditions (such as, a variable interest rate) attached.

If you meet these conditions, you get to stay in the house; otherwise not. In the latter case, you will

end up in a house that is significantly smaller than what you can if you do not enter this deal.” One

can think of c∗ as the house one would end up with, once fees, bankruptcy costs, etc. have been paid.
              l

A home buyer wishing to catch up with his neighbors decides to take on this bet, and in the process,

makes himself financially vulnerable. That same buyer could purchase a smaller not-as-fancy a house

with a safer conventional mortgage but would have to forego the utility boost of living in a larger house

and the associated feeling of ‘making it’ in life.


2.4    Determination of the consumption standard

The final issue that needs to be resolved is: how is κ determined? While there is no uniquely acceptable

way of defining κ, it may be insightful to consider a setting in which κ is computed to be a positive

fraction, λ, of the average consumption of those with consumption exceeding κ, i.e.,
                                              R yU
                                               κ
                                                   yi g (yi , σ) dyi
                                       κ = λ R yU                    .                                   (3)
                                                 κ
                                                    g (yi , σ) dyi

Our definition of the consumption standard embodies the notion that “..a barrage of magazines and

television shows celebrating the toys and totems of the rich has fostered a whole new level of desire

                                                     8
across class groups....a "horizontal desire", coveting a neighbor’s goods, has been replaced by a "vertical

desire," coveting the goods of the rich and the powerful seen on television”. (Steinhauer, 2005).

    Integrating by parts,
          Z   yU                                             Z   yU                                        Z   yU
                   yi g (yi , σ) dyi = [yi G (yi , σ)]yU −
                                                      κ               G (yi , σ) dyi = yU − κG (κ, σ) −             G (yi , σ) dyi .
              κ                                                  κ                                             κ


It follows that (3) can be written as
                                                                µ     Z            yU                 ¶
                                    κ [1 − (1 − λ) G (κ, σ)] = λ yU −                   G (yi , σ) dyi ,                               (4)
                                                                                  κ


implying that κ may be computed from (4). Notice that the consumption benchmark (κ) is endogenous;

in particular, it varies with the distribution of income.5

    It is easy to establish a unique solution to (3); in fact, there exists a unique κ for any λ ∈ (0, 1).6

The appendix contains a proof.


2.5       Change in the spread

How does a change in the spread of the underlying income distribution affect κ — the consumption

standard? To answer this, we introduce a mean-preserving increase in the spread — a redistribution

of mass away from the center and towards the tails. More formally, if a new unimodal distribution

of a variable x, F (x, σ 2 ) , is obtained from an old one, F (x, σ 1 ) , via a mean-preserving increase in

spread from σ 1 to σ2 , it exhibits the single crossing property. That is, there exists an x∗ such that

F (x, σ 2 ) R F (x, σ 1 ) whenever x Q x∗ . Also, when the mean is preserved, and F (.) is a symmetric and

unimodal distribution, the single crossing takes place at the mean, i.e., x∗ = x.
                                                                               ¯

    From (4), we have
                                               Ry
                                       ∂κ   −λ κ U G2 (yi , σ) dyi + κ (1 − λ) G2 (κ, σ)
                                          =                                              .                                             (5)
                                       ∂σ   (1 + λ) (1 − G (κ, σ)) − κ (1 − λ) g (κ, σ)
                    ∂κ
The sign of         ∂σ   is ambiguous.7

    To get a sense of the intuition for what happens when the spread changes, focus attention on Figure

3 which is drawn assuming g to be a unimodal symmetric distribution. In this figure, the heavier green

curve represents a mean-preserving increase in spread of the original distribution (shown as the thin

black line). For ease of presentation, we start by considering a fixed κ (the bold red vertical line drawn
   5 To  foreshadow, below we consider mean-preserving changes in spread. Had we computed κ to be average consumption,
it would have remained invariant to mean-preserving changes in the spread. In that case, changes in the income distribution
would have had no effect. Our way of defining κ gets around this issue.
    6 When λ = 1, the only solution to (3) is κ = y .
                                                      U
    7 Suppose G (.) is a symmetric and unimodal distribution. In that case, when σ increases in a mean preserving manner,

G (κ; σ) rises (falls) for all κ below (above) the single-crossing point (which, in this case, is the mean c = y). In short, for
                                                                                                           ¯ ¯
a symmetric distribution, G2 (κ, σ) R 0 as κ S c. While this allows us to sign some of the terms in (5), the sign of ∂σ
                                                    ¯                                                                        ∂κ

remains ambiguous.




                                                                        9
here to the left of the mean). The cl line is a bold yellow line to the left of κ. Symmetry ensures that

C + H = 0.5 = H + G implying C = G; similarly, B + P = E + F. Under the initial distribution,

two groups of people do not participate in the lottery: i) those who cannot afford to participate in the

lottery (mass Q), and ii) those who can catch up with the Joneses without participating in the lotteries

— the Joneses with mass B + C + D + H. The mass of people who participate in the lottery is given by

A + P.




                                 g(c)



                                                                       B   C

                                                               P
                                                          m

                                                                       D
                                                     E                         H

                                             F             A                       G
                                                 Q
                                                     cl            κ                     c


                            Figure 3: A mean-preserving increase in spread




   Under the new distribution, the mass of people who cannot afford to participate in the lottery has

now increased to Q + F. The mass of those who can keep up without participating is now lower: this is

because D + H + G < B + C + D + H ⇔ G < B + C which is true because C = G. Finally, the mass of

people who do participate in the lottery is now given by A + E which is more than the original A + P

if E > P (which the figure assumes to be true).

   The above diagrammatic description of the effects assumed a fixed κ (that is, it assumed the position

of the heavy red line remained unchanged when the spread increased). From (5), it is clear that the

direction of movement of the red line (subsequent to a increase in spread) is not known. Hence, the

impact on the mass of people entering the lottery remains, in general, unknown.




                                                          10
2.6    An example with skewed distributions

Thus far, we had restricted the discussion to symmetric distributions, purely for analytical ease. Be-

low, we numerically explore various interesting facets of the equilibrium for more general distributions.

Assume yi is drawn from a Beta distribution with support [0, yU ] = [0, 1] with mean held fixed at 0.48.

   We assume the quadratic form for utility:


                                                    B 2
                                     u (c) = Ac −     c ;        A > 0, B > 0
                                                    2

For this specification, u0 (c) > 0 ⇔ A − Bc > 0 ⇔ c < (A/B) . Since c ≤ yU , a sufficient condition for

u0 (c) > 0 ∀c is yU < (A/B) . It is easy to verify that (1) reduces to the quadratic form
                                 µ       ¶                   µ ¶
                                     B                  B 2
                                             c2
                                          − (Bκ) cl +
                                              l           κ − μ = 0,
                                     2                  2
                                                           q
whose sole economically valid solution is given by cl = κ − 2μ (assumed to be non-negative).
                                                             B

   We pick the following parameter configuration: A = 10, B = 5, λ = 0.8, and μ = 0.3. Figures 4a

and 4b below respectively document changes in κ and the fraction (%) of lottery participants in the

economy as the spread of the distribution increases (from 0.05 to 0.5) in a mean-preserving manner.


        0.54
        0.52
          0.5
        0.48
        0.46
        0.44
        0.42

                             0.1                  0.2               0.3         0.4         0.5

                                         Figure 4a: κ against spread




                                                        11
            55

            50

            45

            40

            35

            30


                               0.1               0.2                0.3                0.4               0.5
                 Figure 4b: Fraction of lottery participants in the economy against spread




     Clearly, in this example, κ rises with the spread. Also, there is a range of the spread for which

the proportion of lottery participants in the economy is steadily increasing. In this range, as income

inequality rises, more and more people, urged on by their desire to stay ahead of their neighbors, enter

the lottery, making themselves financially vulnerable in the process.8



3          Concluding remarks: Implications for the housing crisis

Somewhere around the late 1990s, the U.S. Congress brought pressure on banks and other financial

institutions to “democratize credit” — find ways to make home mortgages accessible to a wider spectrum

of people including those that had checkered credit histories and could not qualify for conventional

80%-down mortgages. Money, at that time, was fairly cheap, and financial institutions took it upon

themselves to find new and creative ways — very low or zero down, and interest-only — loans to prospective

homeowners, also known as sub-prime mortgages. As Lowenstein (2008) argues, “in an earlier era, such

people would have been restricted from borrowing more than 75 percent or so of the value of their

homes, but during the great bubble, no such limits applied.”
    8 In
       this example, the fraction of lottery participants in the economy declines beyond a level of income inequality. This
happens because at sufficiently high levels of the spread, a lot of mass is shifted away from the middle incomes on to the
really poor and the really rich — income groups that do not participate in the lottery, albeit for very different reasons.


                                                            12
    The current housing crisis for the most part started around 2001 when such people with less-than-

perfect credit bought houses at so-called teaser rates – “interest rates that start out low but quickly

reset to become significantly higher.” Since subprime borrowers cannot afford to make payments once

the rates rise, they would need to refinance and soon. As James Surowiecki (2007) wrote in the New

Yorker, “if the credit and housing markets were more buoyant, these people might [have been] able to

refinance their loans or sell their homes. But the sharp decline in housing prices and the tightening

of credit standards [had] closed off those options for most sub-prime borrowers, which pretty much

guarantee[d] a big increase in the number of foreclosures.” In Surowiecki’s opinion, “too many people

spent far too much borrowed money on houses with prices that were far too high, and that they are

now stuck in homes that they can’t really afford and can’t sell.”

    Our analysis offers some qualitative insights on some aspects of this ongoing crisis.9                      It is well

known that between 1978 and 2006, the median square footage in new single-family houses has gone

up by 36% (from 1650 sq. ft to 2237 sq. ft) while average square footage has gone up from 1750 to

2465 (a 40% increase). Inequality in incomes has also surged. Lahart and Evans (2008) report that

“adjusted for inflation, income of the top 1% of earners grew at an annual rate of 11% from 2002 to 2006,

...[while] incomes of the bottom 99% grew at less than 1% annually.” Very loosely, one can interpret

these “factoids” as evidence that the proverbial Joneses are richer and live in bigger/fancier houses than

before — as a consequence, it has become harder than ever to catch up with them.10 As Adler (2006)

observes, “in the United States, there are now millions of people with lots of money, and their wealth

shifts the frame of reference for those just below them.” In the context of our simple framework, when

κ rises (the Joneses become richer), more people may participate in the lottery, hoping to win the prize

κ and successfully catch up. This is similar in spirit to the idea that Surowiecki advances. In wanting

to catch up with the Joneses, more and more people took a bet with their mortgages, promising to pay

interest rates beyond their means, hoping that house prices would rise to eventually cover the bet.

    Three additional remarks concluding are in order. First, the analysis presented is fairly general and,

suitably amended, applies to many other situations, such as, people “maxing out” on their credit cards

to buy the consumer durables their neighbors own. Second, our analysis is necessarily qualitative and the

“evidence” we offer in support of the suggested mechanism is anecdotal. Exploration of a quantitatively-

tight link between the model and the data is well beyond the scope of the current paper. Finally, we

do not wish to advance any policy implications (say, on regulation of lending practices of banks) from
     9 Of course, the housing crisis also involved significant problems with the banking and financial sectors — the way they

were regulated, their lending practices, the ratings of mortgage-backed securities, and so on. Lowenstein (2008) contains
a useful discussion.
   1 0 Steinhauer (2005) remarks: “While the rest of the United States may appear to be catching up with the Joneses, the

richest Joneses have already moved on.”



                                                            13
our analysis. In the narrow context of the model, the banks break even and all lottery participants

are clearly better off in an ex-ante sense. Ex-post, however, many of them are hurt. Issues relating to

taxation of the consumption externality imposed by the Joneses — such as ending the mortgage interest

tax deduction on large homes — are also left for future research.




                                                   14
A      Appendix

This appendix establishes a unique solution for κ exists for the equation
                                                     R yU
                                                      κ
                                                          yi g (yi , σ) dyi
                                                κ = λ R yU
                                                        κ
                                                           g (yi , σ) dyi
                        yU
                            yi g(yi ,σ)dyi
Define T (κ) ≡ κ − λ     κ
                          yU
                             g(yi ,σ)dyi
                                             . The function T (κ) is monotonically increasing over the interval
                         κ


(0, yU ) since
                                                   R yU                                   R yU
                  0                   κg (κ, σ)     κ
                                                          g (yi , σ) dyi − g (κ, σ)        κ
                                                                                                 yi g (yi , σ) dyi
                 T (κ) = 1 − λ                                   ¡R yU               ¢2
                                                                   κ
                                                                       g (yi , σ) dy
                                       Ry
                               g (κ, σ) κ U (κ − yi ) g (yi , σ) dyi
                         = 1−λ        ¡R yU               ¢2         >0
                                        κ
                                            g (yi , σ) dy

and (κ − yi ) < 0 over the interval of intergration.

    In addition, using Leibnitz’s Rule, we have lim T (κ) = (1 − λ) yU > 0, when λ < 1. On the other
                                                           κ→yU

hand, lim T (κ) = 0 − λ¯ < 0 for any λ ∈ (0, 1) . The rest is immediate.¥
                       y
       κ→yL




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References

 [1] Adler, Margot. 2006. “Behind the Ever-Expanding American Dream House”, All Things Considered,

    NPR July 4

 [2] Clark, Andrew. E, Paul Frijters, and Michael A. Shields “Relative Income, Happiness and Utility:

    An Explanation for the Easterlin Paradox and Other Puzzles”, forthcoming Journal of Economic

    Literature.

 [3] Friedman, Milton, and Savage, Leonard J., 1948, “The Utility Analysis of Choices Involving Risk,”

    Journal of Political Economy, 56 (August): 279-304.

 [4] Hartley, Roger, and Farrell, Lisa, 2002, “Can Expected Utility Theory Explain Gambling?,” Amer-

    ican Economic Review, 92 (June): 613-624.

 [5] Lahart, Justin and Evans, Kelly. 2008. “Trapped in the Middle”, Wall Street Journal, April 19

 [6] Lowenstein, Roger, 2008. “Triple-A Failure”, New York Times, April 27

 [7] Luttmer, E., 2005. “Neighbors as Negatives: Relative Earnings and Well-Being”, Quarterly Journal

    of Economics, 20(3), 963-1002

 [8] O’Brien, Jeffrey. 2007. “What’s your house really worth?” Fortune, Feb 15

 [9] Steinhauer, Jennifer., 2005. “When the Joneses Wear Jeans”, New York Times, May 29

[10] Surowiecki, James., 2007. “Paulson’s plan” The New Yorker, December 7




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