Income concentration and market demand

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1 This version: July, 00 Income concentration and market demand Corrado Benassi Alessandra Chirco + Marcella Scrimitore + Abstract We analyze the e ects of income concentration and income disersion on market demand and its elasticity. We show that, following an increase in income concentration towards the middle (measured by variations in mean reservingsread), the increase in demand faced by rms which serve at the margin middle income consumers, is associated with an increase in rice elasticity accordingly, the ositive e ects of the size of the market becoming wider are amli ed by a higher degree of cometition.our results hold for a large number of ossible income distributions. JEL Classi cation no: D3,D0. Keywords Personal income distribution, income disersion, rice elasticity of demand, lognormal distribution. Acknowledgements We warmly thank P.Figini, A. Leaci and D.Scolozzi for helful discussions. We also thank two anonymous referees for helful comments. We thank two anonymous referees for very useful criticsms and suggestions. C Benassi and A Chirco acknowledge nancial suort from the Italian Ministry of University and Research (National Research Project Cofin000). Corresonding author Alessandra Chirco, Diartimento di Scienze Economiche e Matematico-Statistiche, Università di Lecce, Ecotekne, via er Monteroni, 7300 Lecce, Italy. Tel.: ; alechirco@economia.unile.it. A revious version of this aer has been ublished in the following working aer series: Aunti, note e ricerche, Univesrità degli Studi di Bologna, sede di Rimini, n., 000. Any oinions exressed in thee aers included into the Quaderni del Diartimento di Scienze Economiche e Matematico-Statistiche are those of the authors. Citation and use of these aers should consider their rovisional character. Università di Bologna, Diartimento di Scienze Economiche + Università di Lecce, Facoltà di Economia

2 Introduction Among the many dimensions along which the agents heterogeneity can a ect market demand, income heterogeneity is surely a crucial one.the relationshi between the ersonal income distribution and market demand is usually studied, by ointing out how distributional changes yield their e ects via the shae of the consumers Engel curves which of course imlies emhasis on the size of the market for a given rice level (e.g., Lambert and Pfähler, 997). In this aer we take a di erent aroach, by emhasizing how income distribution a ects also the sensitivity of demand to rice changes. This focus on the rice elasticity of demand allows to draw some general relationshi between the behaviour of rms in noncometitive markets, and some key features of the distribution of income. The degree of market ower (as measured by the standard Lerner index) turns out to deend on such factors as the weight of the middle class and, more generally, the degree of income disersion. In articular, under fairly general conditions we show that, as incomes become more concentrated around the given mean, market demand and its elasticity both increase for a relevant intermediate range of rices. In this sense, the existence of a large middle class may suort a more cometitive market environment, conducive to lower monooly ro ts. A related imlication is that distributive changes can rovide an examle of a demand shift, able to deliver a negative comovement of demand and mark-u levels an issue widely debated both in the micro and the macro literature. Beyond the theoretical interest of studying the link between income disersion and market cometitiveness, our results may contribute to the debate on the imlications of the well known henomena of income olarization and shrinking middle class, which characterized several countries over the last two decades in some markets, income olarization may account for wider allocative ine ciencies. The aer is organized as follows. In Section we formalize distributive changes in terms of mean reserving sreads of the income distribution, and build a simle discrete-choice demand model. The co-movements of demand and rice elasticity are studied in Section 3, where we rely uon some roerties of the so-called income share elasticity function. Section resents a fully develoed examle, based on the lognormal distribution. Some nal comments are gathered in Section 5. For recent assessments of increasing income inequality and olarization, see, e.g., Atkinson (998), Gottschalk and Smeeding (000), and Pearson and Förster (000).

3 Personal income distribution and market demand We model income distribution as a continuous di erentiable unimodal density function f(y; µ), de ned over some ositive interval (y m ;y M ), 0 y m < y M. The arameter µ is a mean reserving sread. As is well known, in robability theory this is a measure of the degree of riskiness of a distribution. The reason why µ can be fruitfully alied to model income distributions, is that via changes in µ one can study the e ects of changes income disersion, as distinct from changes in aggregate (average) income loosely seaking, an increase in µ shifts income frequencies towards the tails of the density function, while a decrease in µ raises the frequency of central income values.using a mean reserving sread as de ned here, amounts to ranking equal-mean distributions by second-order stochastic dominance. In the literature on income distribution, it is well known that such ranking is equivalent to Lorenz dominance: µ is thus a roer inequality index satisfying the Pigou-Dalton s rincile of transfers (Atkinson, 970). Formally, letting h (y m ;y M ) denote the modal income, and letting subscrits denote artial derivatives, the following holds: f y (h; µ) =0 f y (y; µ) > 0 for y<h f y (y; µ) < 0 for y>h 9 = ; () Z y Z ym y m F µ (x; µ)dx 0; y m F µ (x; µ)dx =0 y < y M 9 >= >; () where F (y; µ) = R y y m f(x; µ)dx is the cumulative distribution function. We secialize our model by imosing some regularity conditions on F ( ;µ). First, we assume that the mean reserving sread is of the simle tye (Rothschild and Stiglitz, 97), i.e. the crossing of distributions imlied by () takes lace only once. Secondly, we assume that the shift of the frequencies towards the tails associated to an increase in µ is such that the old and new density functions intersect only twice. Accordingly, an increase in µ shifts unambiguously down the Lorenz curve. The link between inequality orderings and stochastic dominance has been recently sudied, e.g., by Formby et al. (999). 3

4 To see the imlications of these restrictions, consider the function F µ. Both assumtions are catured by this taking a shae like that exhibited in Fig. c: single crossing of the distributions (Fig. a) imlies that F µ crosses zero in the interior of (y m ;y M ) only once; double crossing of the densities (Fig. b) imlies that this function has only one maximum and one minimum over (y m ;y M ). It should be stressed that this behaviour is shared by many commonly used distributions subject to mean reserving shocks. 3 Figure about here This simle gure brings out a very general roerty of the e ects of changes in µ under our assumtions. Four intervals can be identi ed according as F µ and f µ have the same or the oosite sign. Indeed, for any y (y m ;y A ], F µ > 0 and f µ 0 (with equality only for y = y A ); for any y (y A ;y B ], F µ 0 (with equality only for y = y B )andf µ < 0; forany y (y B ;y C ], F µ < 0 and f µ 0 (with equality only for y = y C ); for any y>y C, F µ 0 (with equality only for y = y M )andf µ > 0 (clearly, this holds in the limit if y M = ). For ease of notation, we label intervals as A =(y m ;y A ], B =(y A ;y B ], C =(y B ;y C ], D =(y C ;y M ). Of course, the boundary values of these intervals in the interior of (y m ;y M ) are functions of µ, i.e. y i = y i (µ), i = A; B; C. Income distribution is immediately connected to market demand, whenever each consumer chooses discretely between buying or not buying one unit of the commodity, according as the quoted rice is lower or higher than his reservation rice the distribution of reservation rices across consumers can reasonably be thought of as mirroring somehow that of the consumers incomes. We assume throughout that the reservation rice coincides with income, so that market demand is Q(; µ) = F (; µ) (3) where oulation has been normalized to unity. This is clearly the sharest way to model the relationshi between income, reservation rices and demand. The weaker assumtion of strict roortionality of reservation rices to income comes out, e.g., in models for durabes, such as that suggested by Deaton and Muellbauer (980, ). In this case the distribution 3 To quote some examles, Beta, Gamma, Chi square, F, Lognormal, all follow this general attern (one maximum, one zero crossing and one minimum) if subject to aroriately de ned mean reserving sreads. Clearly, asymmetric distributions, while reserving the attern, trace out a more irregular function than that lotted in the gure.

5 of reservation rices is isomorhic to the income distribution, when the latter is lognormal (Cowell 995,.7-78). One should however notice that, indeendently of the seci c form of the income distribution, our general argument rests only on the idea that a mean reserving sread on incomes generates single and double crossings of cumulative and density distributions of the reservation rices which is always the case if the reservation rice is monotonically increasing in income. Our demand function is clearly continuous and continuously di erentiable. Moreover, Q = f(; µ) Q < 0 for < h(µ); Q > 0 for > h(µ); Q =0 for = h (µ) This de nition of market demand makes it clear that both its osition and itsfeaturesdeendonthearameterµ. Our focus is on studying how changes in income disersion a ect the otimal behaviour of noncometitive rms. As is well known, changes of this kind which have aarently taken lace in many countries may be attributed to long-run structural factors (such as the skill distribution of workers, the insitutional framework of wage negotiations, the relative weight of caitale vs labour income, etc.), as well as to changes in the shorter-run redistributive e ects of scal olicy. As to the latter, one might think of y as disosable income, and accordingly interret changes in µ as (equal yield) changes, e.g., in the degree of rogression of the income tax schedule as measured by the residual rogression index. 5 The crucial issue we are interested in is then how changes in µ translate into comovements of demand and its elasticity. This may have some relevance in its own, as ointing out a mechanism through which income distribution a ects the degree of cometition; however, it should also be recalled that in a non-cometitive general equilibrium setting the ro- or counter-cyclical In this sense the assumtion = y drastically simli es the exosition, with no substantial loss of generality. On the other hand, the binary-choice model of demand is widely alied in the literature (e.g., Anderson et al., 99); in the analysis of the relationshi between income distribution and demand it is very convenient, as continuous individual demand curves are standardly derived from omothetic references, which revent any discussion of distributional issues. 5 As is well known, residual rogression is (inversely) measured by the elasticity of osttax income to re-tax income; an increase in this index shifts unambiguously down the concentration curve (see, e.g., Lambert, 990, chs 7 and 9). This Lorenz dominance is equivalent, under a equal-yield constraint, to a mean reserving sread of the distribution of disosable income. 5

6 behaviour of demand elasticity is the key element to assess the role of demand in determining the equilibrium outut. 6 Given market demand (3), the (ositive) rice elasticity of demand is given by (; µ) = f(; µ) F (; µ) () By di erentiating with resect to µ one easily obtains µ fµ (; µ) µ(; µ) = f(; µ) + F µ(; µ) (; µ) (5) F (; µ) Simle insection of (5) reveals that changes in µ a ect di erently, deending on where lies in the four intervals A, B, C and D identi ed above. Ariori, the sign of µ is clearly unambiguous whenever f µ and F µ have the same sign, ambiguous otherwise: hence we can say that µ > 0 for A and µ < 0 for C; inintervalsb and D the sign of µ is otentially ambiguous. However, since µ changes sign going from A to C, it follows trivially by continuity that at least one oint b exists in the interior of B such that µ =0, and hence an interval B b ½ B exists where µ < 0 theleft boundary of B b being b. This allows us to establish the following Proosition For all distributions obeying () and (), and such that a change in µ generates single crossing of distributions and double crossing of densities, there exists a non-emty interval B b where the normalized demand function (3) and its elasticity () move in the same direction following a change in µ: Proof Follows trivially from the fact that, by the de nition of B, Q µ (; µ) = F µ (; µ) < 0 for B, while by the de nition of B b ½ B, µ(; µ) < 0 for B: b An immediate imlication of this roosition is that, if the income distribution is subject to a change in disersion as measured by µ, rmswhose equilibrium rice lies in the seci ed range face a ositive comovement in demand and rice elasticity. In articular, as incomes become less disersed, for all initial rices b B rms exerience an increase in both the level and the elasticty of demand. Of course, Proosition is a simle existence roof for b, which says nothing as to uniqueness in the set B and, a fortiori, over 6 For a general discussion of di erent ersectives on the comovements of market demand and its rice elasticity, see Benassi et al. (99, ch 5) and the references therein. 6

7 thewholerangeof the sign of µ in area D is clearly still ambigous. Sorting this out would enable us to determine the behaviour of over the whole range of. The roerties of the income distribution which deliver uniqueness of b are discussed in the next section. 3 Income share elasticity and the rice elasticity of demand Ideally, one would exect to in down a unique value b, such that µ > 0 for all <b and µ < 0 for all >b. Giventhat µ > 0 for A and µ < 0 for B, b µ crosses zero from above at the left boundary of B, b i.e.at b. In order to de ne the conditions for b to be unique, we rst notice that the derivative of µ with resect to is µ = (; µ) f(; µ) + µ + which, for µ =0collases to µ f µ (; µ) f (; µ) f(; µ) f µ(; µ) fµ (; µ) f(; µ) + F µ(; µ) F (; µ) (; µ) µj µ=0 = µ (; µ) (7) where µ (y; µ) is the derivative with resect to µ of the income share elasticity (Esteban, 986). The latter is de ned as (y; µ) =+ yf y (y; µ) f (y; µ) and measures the ercentage change of the income share accruing to individuals of income y, givenamarginalchangeiny. 7 Esteban shows that there is a one-to-one corresondence between f (y; ) and (y; ), so that any given distribution can be characterized in terms of. Therefore, given (7), at µ =0, sign µ (; µ) = sign [ µ (; µ)] (8) This is articularly convenient, as the function tyically exhibits some useful regularity roerties. R 7 y+h Formally, = lim h0 ¹ y xf (x; µ) dx, where¹ is the mean income (Esteban 986,.). 7 (6)

8 We are now in the osition to establish the following general roosition. Proosition If the distribution f (y;µ) and the corresonding income share elasticity (y; µ) are such that (a) µ (y; µ) is monotonically increasing in y and crosses zero, and (b) lim ym µ < 0, then there exists one value b such that µ(; µ) > 0 for <b and µ(; µ) < 0 for >b. Proof By Proosition there exists a b which is the lowest such that µ(; µ) crosses zero, obviously from above. Condition (a) together with (8) imly that µ (e; µ) =0at some unique e >b. Thisimliesthatb is the unique value of at which µ is zero. To see this, notice that by condition (b), if additional such oints existed, they should be even in number. Suose they are two (the roof alies trivially for any even number), and call them b and b, b < b. Obviously, µ will be ositive at b and negative at b. Two ossibilities arise: (i) b and b are both lower or higher than e; (ii) b < e <b.case(i) is ruled out by (8); case (ii)isruledoutby(8)together with condition (a). It should be noticed that conditions (a) and (b) of Proosition are veri ed for many widely used distributions, such as those quoted in f.note. One imlication of Proosition is that the interval b B identi ed by Proosition is unambiguously de ned as (b; y B ): Figure brings this out by showing a ossible behaviour of the sensitivity of the elasticity of market demand to µ; for di erent values of. Figure about here Why is it that, for rices lying between b and y B,anincreaseinincome concentration generates both an increase in demand and an increase in its elasticity (which for constant marginal costs would imly non cometitive rms setting a lower rice)? Prices in that interval are rices at which the higher income individuals getting oorer are still able to buy, while lower income individuals getting richer are eventually allowed to enter the market. The additional demand accruing at these rices is therefore due to the latter - those who descend into the middle class from the uer tail of the distribution were already buyers, and kee buying after the distributional change. However, this overall movement from the tails towards the central area of the distribution is such that for rices belonging to b B, there are more consumers actually buying, whose reservation rice is close to the set rice. This imlies, for examle, that non cometitive rms erceive a weaker incentive to exloit an intensive margin on higher income consumers, and a stronger 8

9 incentive to acquire new consumers at the margin by keeing lower rices. 8 Demand increases and becomes more elastic simly because there are indeed new consumers entering the market, but also more consumers whose decision to enter or exit the market is now very sensible to small variations in rices. Notice that these observations are consistent with the fact that a ositive comovementofdemandanddemandelasticityisobservedonlyin b B, i.e., it is eculiar of an intermediate ortion of the demand curve, as de ned by b B. Moreover, they aly to whatever unimodal distribution, once concentration towards central income values is considered, and this exlains the generality of our result. As a notable examle, in the next section we aly the results of Proositions and to the lognormal distribution. An examle: income disersion with lognormal distribution Assume that income is distributed lognormally. This is a articularly remarkable case, since as is well known the lognormal distribution is erhas the model most frequently used to describe actual income frequencies. 9 We standardize mean income equal to unity, so that the density and distribution functions take the form 0 f(y; µ) = y ¼ ln µ ex ³ln y+ ln µ lnµ F (y; µ) = R µ y f(x; µ)dx = + 0 lny+ln µ ln ( ) µ 8 This may o er a general exlanation for the emirical evidence discussed by Frankel and Gould (00), who nd a causal link running from income distribution in urban areas to retail rices: according to their estimates, greater inequality is indeed associated with an increase in retail rices aid by lower middle-class consumers. 9 It is well known that the lognormal distribution ts satisfactorily the actual income distribution for central income values, while it is unsatisfactory in the tails, i.e. for extreme income values (for an evaluation of the emirical erformance of various distributions, see e.g. Majumder and Chakravarty, 990). Since the henomenon we are interested in is eculiar of intermediate intervals, the lognormality assumtion seems worth investigating. We recall that, if reservation rices are roortional to incomes, they also are lognormally distributed. 0 Given a generic lognormal distribution f(y; µ) =(y ¼ ln µ) (ln y ³) ex ³ lnµ,the mean is ¹ = e ³ µ. Clearly, by imosing ¹ =one constrains the arameters µ and ³ accordingtotherestriction& = ln µ. Note, in articular, that income variance is ¾ = µ > 0. 9

10 R where (x) = x ¼ 0 e t dt is the so-called error function (Johnson et al. 99,.8). It can be checked that µ>isindeed a mean reserving sread, in that the conditions set out in () are satis ed. Also, single crossing of the cumulative distributions following a change in µ, aswellasdoublecrossing of the densities, are satis ed. In articular, for this lognormal distribution q we have y A (µ) =e ( lnµ+ln µ), yb (µ) = q µ and y C (µ) =e ( lnµ+ln µ).it is immediate to de ne the demand curve as Q(; µ) = µ the elasticity of which is (; µ) = ¼ ln µ ex ln+ln µ µ + ln ( ) µ ln + ln µ lnµ ln+ln µ ln ( ) µ The corresonding function µ(; µ) is derived in the Aendix, where it is shown to tend to minus in nity as. This function is in general analytically di cult to treat and indeed one advantage of Proosition is that it o ers a simle, general characterization of its qualitative behaviour in terms of the income share elasticity. Actually, the function takes the simle form given by (y; µ) = lny +lnµ ln µ It is easy to check that µ =lny=(µ ln µ), which is monotonically increasing in y and crosses zero at y = ¹ =. Therefore we can establish that a mean reserving shock generates a unique area of ositive comovement of demand and demand elasticity, the left boundary of which lies between y A (µ) =e (lnµ+ln µ) and, and the right boundary of which is q y B (µ) = µ. In order to assess the relevance of this henomenon, one can notice that a numerical aroximation erformed under the arbitrary value of µ =:5gives to this area a size such that about the 0% of the oulation has income (reservation rices) falling in it. The value is not wholly arbitrary, since µ =:5 yields a coe cient of variation = :5 very similar to the value of :37 recorded by Chamernowne and Cowell (998,.78) for the distribution of labour income in the UK. 0

11 5 Concluding remarks In this aer, through a simle discrete-choice model of consumers behaviour, we have derived some general results on how changes in demand elasticity may be associated with emirically relevant changes in income distribution. We have shown that the concentration of the households incomes in the range which one would roughly identify as middle class, results in a relevant segment of demand exanding and becoming more elastic. This may also contribute to exlaining why markets reviously atronized only by richer grous of consumers, tyically bene t from the middle class entering them, in terms of both market size and lower rices. According to our interretation, the latter e ect (tyical, e.g., of some durables) may be seen as a consequence (and not the cause) of the new consumers being indeed middle class. Moreover, the link from income distribution to the degree of cometitiveness may add a new ersective in evaluating the e ects of redistibutive olicies. Clearly, the relationshi between income distribution and market structure can be extended in several directions to quote some of them, the change in ro t margins may trigger entry and exit of rms; di erent distributions may alter the incentive to horizontal or vertical roduct di erentiation; and, similarly, if income distribution a ects rice elasticity, it may a ect the incentive towards rice discrimination. We believe that the framework we have develoed could fruitfully be enriched and alied to these research areas. References [] Anderson, S., De Palma, A., and Thisse, J.F. (99). Discrete Choice Theory of Product Di erentiation, MITPress,Cambridge (Mass). Atkinson, A.B. (970). On the Measurement of Inequality, Journal of Economic Theory,, -63. Atkinson, A.B. (998). The Distribution of Income in Industrialized Countries, in Income Inequality: Issues and Policy Otions: A Symosium, Federal Bank of Kansas City, Jackson Hole, Wyoming. Benassi, C., Chirco, A., and Colombo, C. (99). The New Keynesian Economics, Basil Blackwell, Oxford. Chamernowne, D.G. and Cowell, F.A.(998). Economic Inequality and Income Distribution, Cambridge University Press, Cambridge.

12 Cowell, F.A. (995). Measuring Inequality, LSE Handbooks in Economics, Prentice Hall - Harvester Wheatsheaf, London. Deaton, A. and Muellbauer, J. (980). Economics and Consumer Behavior, Cambridge University Press, Cambridge. Esteban, J. (986). Income Share Elasticity and the Size Distribution of Income, International Economic Review, 7, 39-. Formby, J.P., Smith, W.J., and Zheng, B.(999). The Coe cient of Variation, Stochastic Dominance and Inequality: A New Interretation, Economics Letters, 6, Förster, M. and Pearson, M. (000). Income Distribution in OECD Countries, OECD, Paris. Frankel, D.M. and Gould, E.D. (00). The Retail Price of Inequality, Journal of Urban Economics, 9, Gottshalk, P. and Smeeding, T.M.(000). Emirical Evidence on Income Inequality in Industralized Countries, in A.B.Atkinson and F.Bourguignon (eds), Handbook of Income Distribution, North Holland, Amsterdam. Johnson N.L., Kotz, S., and Balakrishnan, N. (99). Continuous Univariate Distributions, Wiley&Sons,NewYork,Vol.I. Lambert, P.J. (990). The Distribution and Redistribution of Income, Basil Blackwell, Oxford. Lambert, P.J. and Pfähler,W. (997). Market Demand and Income Distribution: A Theoretical Exloration, Bulletin of Economic Research, 9, Majumder, A. and Chakravarty, S.R. (990). Distribution of Personal Income: Develoment of a New Model and Its Alication to US Income Data, Journal of Alied Econometrics, 5, Rothschild, M. and Stiglitz, J.E. (97). Increasing Risk II: Its Economic Consequences, Journal of Economic Theory, 3, 66-8.

13 Aendix In this aendix we rove that, in the case of the lognormal distribution, lim y µ =. Writing out the whole exression, we get µ(y;µ) = 8 µ ex 8 ³ ln 5 ³ µ ¼ ln 5 µ ¼ µ¼ 3= µ + ex 8 ln ( ) µ ( ln y+ln µ) ln µ ln 3 µ lny+ln µ 8 >< >: ( ln ) µ lny+ln µ ( ln y+ln µ) ln µ (ln µ) ¼ µ ex 8 µ¼ 3= lny+ln µ ( ln ) µ This function can be written as ³ ln ³ µ ¼ ln y ln µ ¼(ln y) µ¼ 3= ³ ln 3 ³ µ ¼ ln 3 µ ¼ µ¼ 3= ( ln y+ln µ) ln µ ln ( ) µ lny+ln µ ln ( ) µ lny+ln µ lny+ln µ (ln µ) ¼ ln y ln ( ) µ 9 >= >; lny+ln µ ( ln ) µ + 8 e z 6 ¼( (z)) ln y 5 ¼( (z)) 3 ¼( (z))+ ¼e z ( lny) µ¼ 3 ( (z) ) lny+ln µ where z = and = ln µ, sothatln y = z. Substituting for the latter, we get 8 e z 6 ¼( (z))(8 z 3 z )+ 3 e z ( z) µ¼ 3 ( (z) ) which is in terms of z, and collases to e z 3 " ¼ ( (z)) z z # ¼e z z µ¼ 3 ( (z) ) = e z ¼ µ 3 ¼ 3 " ¼ ( ) z z # e z z ( (z)) (a.) We owe very useful suggestions about this roof to rofs A.Leaci and D.Scolozzi. 3

14 ¼ To ease notation, let k =,aconstant. Then(a.)canbewrittenas µ 3 ¼ 3 ³ z ¼ ( (z)) z e z 3 ¼ ( (z)) ke z 5 [ (z)] ½ h ¼( (z))z e = ke z z z i h ¼( (z))z e z i ¾ ¼ (a.) ( (z)) ( (z)) (z) Thus nding the lim y µ = amounts to studying the behaviour of (a.) as z tends to in nity. We now consider the curly brackets of (a.), to calculate h ¼( (z))z e z z i lim z ( (z)) and lim z As to the former, we can use Hôital s rule to obtain h ¼( (z))z e z i ( (z)) h ¼( (z))z e z z i lim z ( (z)) = lim z whereweusethefactthat,byhôital srule, ¼ ( (z)) z e z ¼ = ¼ (a.3) ( (z)) z lim = z+ e z ¼ As to the latter, notice that (a.) h ¼( (z))z e z i lim z ( (z)) =lim z ( z h ¼( (z))z e z z i ( (z)) ) =0 (a.5) whereweuse(a.3). Finally, the term outside the curly brackets tends to in nity. To see this, notice that ke z lim z (z) = k lim e z z z( (z)) z = using (a.). Putting togethere these limits, lim y µ =.

15 Fig. a. Single crossing of distributions Fig. b. Double-crossing of densities Fig. c. The function F µ (y; µ) Fig.. The function µ(; µ) 5