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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 15 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 16