The Tradeoff Between Inequality and Growth

Size: px
Start display at page:

Download "The Tradeoff Between Inequality and Growth"

Transcription

1 ANNALS OF ECONOMICS AND FINANCE 4, The Trdeoff Between Inequlity nd Growth Jess Benhbib Deprtment of Economics, New York University 269 Mercer Street, 7th floor, New York, NY 10003, USA. E-mil: c 2003 Peking University Press Key Words: JEL Clssifiction Numbers: 1. INTRODUCTION Is there trde-off between inequlity nd economic growth? The theory nd the evidence re inconclusive. In the 1950es nd 60es prevlent view ws tht inequlity leds to higher svings becuse the rich sve proportiontely more thn the poor, nd tht this leds to increses the rte of investment nd growth see Kldor 1957, Kuznets More recently, it hs been rgued tht inequlity hurts growth becuse it leds to redistributive pressures, either through the medin voter who encts redistributive txes Tbellini nd Persson1993, or through generting socil conflict, exproprition, nd rent seeking behvior see Alesin nd Rodrik 1994, Alesin 1994, Benhbib nd Rustichini 1996, Benbou 1996, Perotti 1996, Acemoglu nd Robinson All such ctivities dilute the return on investment nd reduce the rte of growth 1. Another view is tht inequlity coupled with borrowing constrints nd finncil mrket imperfections prevents the tlented poor to undertke profitble investments in physicl nd humn cpitl, thereby limiting the full potentil for the growth of the economy Glor nd Zeir 1993, Bnerjee nd Newmn By contrst, there is lso recent literture which views inequlity 1 But see St. Pul nd Verdier1993, who rgue tht redistributive txes cn enhnce growth if they re chnneled to puiblic goods or eduction tht mkes the economy more productive /2002 Copyright c 2003 by Peking University Press All rights of reproduction in ny form reserved.

2 330 JESS BENHABIB s the result of growth spurts tht re ssocited with skill-bised technicl chnge Aghion On the empiricl side, the evidence on the trde-off between inequlity nd growth, despite lrge number of recent studies, remins inconclusive. In recent pper Deininger nd Squire 1996 document the frgility nd the non-robustness of the results obtined by using cross-country regressions see lso Levine nd Renelt 1992, Benhbib nd Spiegel 1994, Islm 1995 nd Esterly In ddition, nd to complicte mtters further, Forbes 2000 now finds significnt positive ssocition between inequlity nd growth in the short nd medium run. Redistributive pressures, insecure property rights nd socil conflict tht stem from significnt inequlity my well discourge investment nd hinder growth. On the other hnd, excessive interference with economic inequlity tht follows from differences in effort, productivity, enterprise nd inititive is lso likely to reduce growth nd investment. In this pper we present theoreticl politicl economy model of inequlity nd growth. We wnt to demonstrte tht while excessive inequlity cn disrupt the economy by inviting politicl interference through rent-seeking behvior nd pproprition, policies which support some modest inequlity to tke dvntge of productivity differences will led to the best growth rtes. Thus we show tht the reltion between inequlity nd growth my be mildly humpshped: growth my rise modestly t first, s we move wy from complete equlity, nd then drop gin s inequlity increses further. 2. THE SOCIAL PLANNER AND COOPERATIVE SOLUTIONS We strt by considering the policies of socil plnner, or government, in n economy with two clsses of gents. For simplicity, the utilities of the two sets of gents re logrithmic, they re incresing in consumption, nd they re qudrtic nd decresing in lbor. Output is produced with Cobb-Dougls production function, Ak α l 1 l 2 σ, using cpitl k, nd there re two types of lbor, l 1 nd l 2, with different productivity levels. We will not necessrily restrict the production function to constnt returns to scle in order to llow the possibility of endogenous growth with α = 1. Given the logrithmic utility of consumption nd the qudrtic disutility of work, the problem is well-defined, with the cse of constnt returns, α + + σ = 1, s specil cse. Let us first ssume tht the government ssigns weights nd 1, to the two gents nd cn choose their consumptions nd lbor supplies to mximize the discounted utilities of the gents. If we denote the vlue function by V k, the problem is given

3 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 331 by: V k = mx c 1,c 2,l 1,l 2 { ln c ln c 2 0.5B l l 2 2 +βv Ak α l 1 l 2 σ c 1 c 2 where we will set B = 1. 2 To solve the problem let y = Ak α l 1 l 2 σ. The first order conditions for consumption re: } c 1 1 = 1 c 2 1 = βv Ak α l 1 l 2 σ c 1 c 2 c i 1 = βα c i 1 y i = 1, 2 k where primes re next period vlues. Let c i = λ i y. Then, the solution for the consumption of the first gent must stisfy: λ 1 y 1 = βα λ 1 y 1 1 λ 1 λ 2 = βα y 1 λ 1 λ 2 y λ 1 = 1 βα, λ 2 = 1 1 βα We cn now solve for the optiml lbor supplies l i tht mximize the utility function 3 : l 1 = βv Ak α l 1 1 l 2 σ 1 = c 1 1 Ak α l 1 1 l 2 σ 2 l 1 = 1 βα 1 y 1 y l 1 1 l 1 2 = 1 βα 1 l 1 2 = 1 βα ; l 2 2 σ = 1 1 βα 2 In such cses where we require lbor supply to be inelstic we cn tke B = 0, nd l 1 = l 2 = 1. 3 Eqution 2 is simply the the stndrd lbor mrket condition which requires the mrginl disutility of lbor to equl the mrginl product of cpitl times the mrinl utility of consumption.

4 332 JESS BENHABIB We re now in position to obtin the vlue function for the government: V k = s ln k + I = ln [ 1 βα Ak α l 1 l 2 σ ] +1 ln [1 1 βα Ak α l 1 l 2 σ ] σ 1 1 βα = α ln k + α1 ln k + αβs ln k 1 βα + βs ln [βα Ak α l 1 l 2 σ ] + βi + ln 1 βα A l 1 l 2 σ + 1 ln 1 1 βα A l 1 l 2 σ +βs ln βα A l 1 l 2 σ βα 1 + σ + βi 1 Note tht the vlue function is concve in k. We cn now solve for s nd I by equting coefficients of ln k nd of the constnt terms s = 1 βα 1 α I = 1 β 1 [ ln + 1 ln 1 ] [ ln 1 βα A l + 1 β 1 1 l 2 σ ] +βs ln βα A l 1 l 2 σ βα 1 + σ 1 We note tht this lloction of consumptions nd lbor supplies cn be chieved in decentrlized setting by the socil plnner through simple redistribution of the initil cpitl stocks, so tht k 1 0 = k 0, k 2 0 = 1 k 0, provided we hve constnt returns to scle in production. For exmple, if lbor supply is inelstic B = 0, l 1 l 2 σ = 1 nd we set α = 1 to chieve blnced growth, the decentrlized setting described bove would implement the plnner s solution fter the initil cpitl ws llocted ccording to the specified weights to the two gents. 4 Then if ech gent consumed nd sved optimlly, expecting return on cpitl given by the mrginl product of cpitl, the first order conditions of ech gent with respect to c i tht is their Euler equtions, would be equivlent to the first order conditions of the socil plnner. In this cse ech gent s cpitl would grow t the rte βa 1. With incresing returns to scle, 4 For, σ 0, α 1, we pproch the stndrd endogenouis growth model. The Hessin of the utility function lnak t l 1 l 2 σ k t l l 2 2 will hve root pproching zero, ssocited with the blnced growth pth, with the other roots negtive, s the utilty of leisure is strictly concve.this cn be checked by computtion or deduced from the continuity of roots in prmeters.

5 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 333 s is well-known, the plnner s lloction in the context of decentrlized frmework would require system of txes nd subsidies. It my not be possible to design txes nd subsidies to identify or differentilly trget the segments of the popultion with different productivity levels however. Note tht if we bndon blnced growth by hving α < 1, nd focus on stedy stte levels of income nd consumption, we could reintroduce differentil lbor productivities nd still mintin constnt returns to scle. This structure would preserve the incentive issues ssocited with lbor supply in context where fully decentrlized implementtion of the socil plnner s solution is fesible by simple rellocting initil stocks. We cn now lso compute, for future use, the vlue function for the gent consuming shre λ 1 = 1 βα of the output: V 1 k = s ln k + I = ln 1 βα Ak α l 1 l 2 σ βs ln βα 1 Ak α l 1 l 2 σ + βi1 1 βα = α ln k + αβs ln k βα +ln 1 βα A l 1 l 2 σ + βs ln βα A l 1 l 2 σ + βi 1 s = α 1 βα 1 I1 = 1 β 1 ln 1 βα A l 1 l 2 σ βα 1 + βs ln βα A l 1 l 2 σ 3. THE NON-COOPERATIVE SOLUTION: MARKOV STRATEGIES We now consider sitution where ech gent cn choose to pproprite some consumption from the output, with the residul output, if ny, becoming the cpitl stock for the next period. This reflects sitution where property rights, nd in prticulr the returns to investment, re insecure, ill-defined or mnipulble. We envisge politicl system where pressure groups hve power to implement redistributive policies. Such policies cn rnge from outright exproprition to redistributive txtion, infltionry finnce tht sustins powerful groups, exchnge rte policies tht fvor certin constituencies over others, the provision of government employment trgeted to specific sections of the popultion, price controls, monopolistic mrketing bords creted to dvntge prticulr clsses, or other legisltive mesures tht cn ffect nd lter the brgining power of lbor nd/or

6 334 JESS BENHABIB cpitl. It is importnt to note tht while policies fvoring mny diverse groups cn coexist, they my no mens wsh out, or cncel ech other out. Ech one represents dilution of returns to investment tht tends to diminish nd discourge growth. These redistributive policies however my be the unvoidble result of the politicl process, nd it must be the job of good government nd purpose of effectively designed institutions to chieve the lest costly implementtion of the politicl consensus. Sustining productive politicl consensus will require specifiction of the equilibrium for the cse where the consensus breks down. We will model this s non-coopertive Mrkov-Nsh equilibrium where ech group pproprites consumption s best response to the competing group s pproprition. For simplicity we model the politicl power of the two groups s symmetric, without specifying limits to pproprition. The model could esily be modified to reflect unequl politicl power, nd upper different bounds on the ppropribility of output 5. The vlue function of the first gent is given by: } V s k = Mx c1 {ln c l1 s 2 + βv 1 λ 2 Ak α l1 s l2 s σ c 1 where the superscript s in li s, i = 1, 2, is used to indicte the lbor supplies chosen in the non-coopertive cse, nd which re different thn the lbor supplies l i, i = 1, 2, chosen in the coopertive cse. The first order conditions re given by: c 1 1 = βα c 1 1 y Let c i = λ i y. Then the bove simplifies to: k 1 λ 2 1 λ 1 λ 2 = βα 1 λ 2, A symmetric equilibrium then implies consumption shres, identicl for the two gents, of: λ = 1 βα 2 βα 5 To fully define the gme, outcomes must be explicitly specified for situtions where the gents ttempt to pproprite mounts tht sum to more thn the totl output vilble. We will overlook such considertions here, but for n explicit tretment of such issues see Benhbib nd Rustichini 1996.

7 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 335 We cn lso solve for the lbor supplies of the first nd second gents using their first order conditions for lbor supply: y l1 s = βv c 1 l s 1 = l s 1 1 βα 2 βα 1 λ 2 = c 1 1 y l1 s y yl s 1 = l s 1 1 λ 2 = 1 λ 2 y l s 1 2 βα 1 Solving the bove, the lbor supplies for the two gents re given by: l s 1 2 = 1 βα 1 = l 1 2 l s 2 2 = σ 1 βα 1 = 1 l 2 2 Given consumption shres nd lbor supplies we cn evlute the vlue functions ssocited with this non-coopertive equilibrium: V s k = s s ln k + I s = ln λak α l1 s l2 s σ βα 1 + βs s ln 1 2λ Ak α l1 s l2 s σ + βi s where nd I s = s s = α 1 αβ 1 = s ln λa l1 s l2 s σ + βs s ln βα A l1 s l2 s σ βα 1 1 β The cpitl ccumultion eqution for the socil plnner s solution is: k t+1 = αβ Ak α t σ σ σ 1 βα 1 βα while for the non-coopertive solution it is: αβ k t+1 = A k t α l 1 l 2 σ σ 2 αβ Note tht these ccumultion equtions led to stedy sttes if α < 1. They led to sustined blnced growth pths only if α = 1, ssuming tht prmeter configurtions under logrithmic preferences lso stisfy the sufficiency conditions.

8 336 JESS BENHABIB 4. SUSTAINABLE EQUILIBRIA Sustinble equilibri re consumption nd lbor pths tht yield higher utilities to the gents thn they could chieve unilterlly by breking the greement nd reverting to the non-coopertive equilibrium. One simple wy to model this is to ssume tht defection does not yield n initil period dvntge to the defector during which his opponent continues to consume nd supply lbor ccording to the greed upon coopertive pth. We my ssume tht the defection is instntly detected, so tht reversion to the non-coopertive equilibrium is immedite. By construction, symmetric lloction tht trets both gents identiclly will dominte the symmetric non-coopertive equilibrium, but given the dispersion in productivities, pertinent question is the degree of inequlity, prmetrized by, tht cn be sustined by the socil plnner, nd whether there is n optiml degree of inequlity tht ttins the highest level of growth. Before trying to nswer this question, let us note tht n lterntive specifiction of the model, discussed in the ppendix, would llow the defector one period dvntge before both gents revert to the non-coopertive equilibrium, nd mke defection more ttrctive. In such cse the degree of enforceble inequlity my in fct shrink. Under such circumstnces, the socil plnner s preferred solution, even under complete equlity, my not be enforceble nd other second-best solutions must be found Benhbib nd Rustichini However, the non-coopertive equilibrium described bove my not be the worst equilibrium to which the plyers cn defect: the thret or fer of reverting to n even worse equilibrium nrchy? my be ble to sustin more coopertion, even if it entils more inequlity. To evlute the degree of sustinble inequlity, we must compre the vlues of the coopertive nd the non-coopertive lloctions. Since s s = s, the comprison of the discounted sum of utilities hinge on I nd I s. We hve, since 0.5 l 1 = l s 1 nd l 2 = l s 2: I s = I 1 = ln +βs s ln 1 βα βα A 0.5 l l 2 σ A 0.5 l l 2 σ βα 1 β ln 1 βα A l 1 l 2 σ +βs ln βα A l 1 l 2 σ βα 1 1 β

9 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 337 Then the comprison of discounted utilities between the coopertive nd non-coopertive regimes for the first gent reduces to: ln 1 βα + βs ln βα βα 1 1 βα βα ln + βs s ln 2 βα 2 βα βs ln σ βα 1 or, ln βα βα 1 ln σ ln 2 βα Similrly for second gent, we obtin: ln βα 1 σ 1 1 βα 1 ln σ ln 2 βα Figure 1 illustrtes the growth rtes when α = 1, s well s the growth rtes s function of, for prmeter vlues βα = 0.98, = 0.15, σ = However, the sustinble growth rtes re those corresponding to vlues of for which both lines, representing the difference between the vlues of coopertion nd non-coopertion of ech gent, re non-negtive. Note tht this sustinble intervl in Figure 1 is skewed to the left becuse σ >, nd is the productivity of the gent ssigned weight. The gent with the higher productivity is less willing to cooperte if he is given low utility weight, becuse he cn do better in the non-coopertive sitution. However, note tht the growth rte is mximized if the productive gent is given low weight in fct zero weight, the lowest possible vlue for 1. The figure is provided only for [0.3, 0.7] for better visibility. This is becuse the optiml lbor ssignments of the gents re decresing 6 Of course these growth rtes would be trnsitory if α < 1. The level of the ttinble stedy stte when α < 1 would be positively relted to the vlue of the function ttined in the lower figures depicting the growth rte. Prlell results hold if we ssume tht the government ims t mximizing the stedy stte level of output insted of the growth rte: the government would then implement moderte level of inequlity t the boundries of the sustinble intervl. None of the qulittive fetures of the Figures chnge if we pick, ssuming constnt returns, βα = β 1 σ. This would imply, for + σ = 0.55, vlue of βα round 0.5. See Figure 3.

10 338 JESS BENHABIB in their utility weight. To mximize growth, it is best to ssign low utility weight to productive gents tht leves them with low initil stocks of cpitl so s to get more work out of them. If the only mechnism vilble to the socil plnner to elicit work, or more ppropritely, to elicit effective effort is decentrlized system of txes nd subsidies, then the welthier gents who receive higher utility nd higher consumption will lso supply less effort. This is n rtifct of the preference specifiction where leisure nd consumption re complements, so tht higher consumption is ccompnied by lower lbor supply. Of course pproprite non-wge subsides or the prestige ssocited with higher position my reverse this feture of the model. Alterntively, if ech gent were operting his or her own stock of cpitl in combintion with his or her lbor rther thn with one ggregte cpitl nd single production function, then there would be n dditionl tendency to llocte the cpitl towrds the group tht hd the more productive lbor. This would tend to prtilly offset the lbor supply effect of welth nd consumption, nd my led to n optiml lloction of cpitl, in the sense of mximizing the growth of output, tht would fvor the more productive groups. 10 Difference between the vlue of coopertion nd non coopertion s function of V Vs, V1 Vs Growth Rte s function of 1.24 Growth Rte FIG. 1. Figure 2 illustrtes the cse = 0.3, σ = In this cse the sustinble intervl is skewed to the right since the first gent is the more productive one, nd the growth is higher s becomes lower. These results

11 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 339 re not sensitive to the choices of α nd β. Under n lterntive specifiction where βα = 0.5 nd for = 0.3, σ = 0.15, very similr results obtin, s is cler from the Figure Difference in the vlue of coopertion nd non coopertion s function of V Vs, V1 Vs Growth rte s function of 1.24 Growth Rte FIG THE TRADE-OFF BETWEEN GROWTH AND INEQUALITY The min point of this nlysis is to study the optiml redistribution of welth tht mximizes the growth rte. Redistribution in terms of utility cn be implemented s relloction of the productive initil cpitl stock. As we see from the figures presented bove, when gents cnnot interfere with the politicl distribution mechnism, the growth mximizing lloction is to impoverish the most productive gents, thereby forcing them to supply more effort. However, such policy will not work becuse of politicl constrints: gents cn withhold effort, nd they cn resort to politicl rent-seeking ctivities to obtin higher consumption if the distribution of welth nd income is too unfvorble to them. We see tht outside the sustinble intervl for the prmeter, the economy will revert to noncoopertive equilibrium with low growth rte ssuming α = 1 given

12 340 JESS BENHABIB 1 Difference in the vlue of coopertion nd non coopertion s function of V Vs, V1 Vs Growth rte s function of 1.24 Growth Rte FIG. 3. by g s = αβ A 1 βα σ 1 βα 1 0.5σ. 2 αβ Note tht this growth rte is independent of. The vlue of tht mximizes stedy stte level of output or the growth rte of the economy must be chosen to mximize σ g = αβ Akt α σ σ 1 βα 1 βα nd must be t the boundry of the sustinble intervl in the figures bove. Whether the optiml is the upper or lower boundry depends on the reltive productivities of the two gents. If > σ, then we choose the lower bound for, nd if < σ, we choose the higher bound unless of course productive gents cn be offered non-wge compenstion outside the mrket tht gives them dditionl utility ssocited with their jobs, working conditions, nd the prestige of their positions. We should note tht our logrithmic specifiction for the preferences ssures tht gents sve constnt frction of their income. The totl svings rte remins constnt s we vry. In decentrlized economy,

13 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 341 the supply of lbor or effort then becomes the primry mechnism through which incentives ffect stedy stte output levels. If we plot growth s function of inequlity, we will observe growth rtes tht re low nd constnt for high degrees of inequlity, corresponding to non-coopertive outcomes. As inequlity decreses we my observe jump in the growth rte s we enter the rnge of sustinble coopertion. Therefter, growth rtes will decline s we move towrds perfect equlity, nd s we diminish the incentives for the more productive gents to supply effort. see Figure 4. We my conclude therefore tht the reltion between growth nd inequlity is hump-shped, nd tht growth mximizing socil plnner should im t moderte level of inequlity designed to elicit effort from the most productive gents in the economy Sustinble growth rtes Growth Rte FIG. 4. It my be possible for the government to improve the growth rte of the economy by enlrging the rnge of sustinble coopertive outcomes. This cn be chieve by mking devition to the non-coopertive equilibri more costly for the gents. If punitive costs for deviting from the coopertive outcome could be imposed, greter degree of inequlity could be sustined. In Figure 1 for exmple, incresing the costs of devition could rise the two curves representing the difference between the vlue of coopertion nd the vlue of devition, enlrging the rnge of for which the coopertive equilibrium is sustinble. In the cse depicted in Figure

14 342 JESS BENHABIB 1, this would permit the choice of higher vlue of, nd therefore of higher growth rte for the economy. Imposing such costs on devition in order to improve the growth rte however, my not lwys be politiclly fesible or desirble. APPENDIX: SUSTAINABLE EQUILIBRIA WITH A ONE PERIOD ADVANTAGE OF DEVIATION In this cse, the defecting gent the first gent in this cse considers his optiml consumption nd lbor supply ginst the cooperting gent the second gent consuming frction λ = 1 1 βα of the output nd supplying lbor l 2 = σ 1 1 βα, with the understnding tht both gents will revert to non-coopertive behvior from the next period on 1. The mximiztion problem for this gent is given s: mx c lnc d 1 0.5l1 d 2 + βα1 αβ 1 lnak α l1 d l 2 σ βα c d 1 ln 1 βα A 0.5 l + β1 β σ l 2 +βs s ln βα A 0.5 l σ l βα The first order conditions re: c d = βα 1 αβ Ak α l1 d 1 l2 σ βα c1 d nd they cn be reduced to: c d 1 = [1 1 1 βα 1 βα] Ak α l1 d l2 σ λ 1 dak α l1 d l2 σ It is lso strightforwrd to compute the optiml defection consumption of the second gent. It is given by: c d 2 = [1 1 βα 1 βα] Ak α l2 d l1 σ λ 2 dak α l2 d l1 σ Similrly, lbor supply of the defecting first gent is the solution to the first order conditions: l1 d = βα 1 αβ 1 Ak α l1 d 1 l s 2 σ βα c Ak α l d 1 1 l s 2 σ βα = c 1 Ak α l d 1 1 l2 σ βα 1 See lso Kitl nd Pohjol 1990, nd Benhbib nd Rustichini 1996.

15 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 343 They cn be simplified to: l d 1 2 = [1 1 1 βα 1 βα] βα = βα βα 1 βα = 1 βα 1 The optiml lbor supply of the defecting second gent ginst the first is, by symmetry, l s 2 2 = 1 βα 1 σ We cn lso compute the vlue function for the defection of the first gent: V D 1 = s D ln k + I1 D = ln [1 1 1 βα 1 βα] A l1 d l2 σ k α 0.5 l1 d 2 + βα 1 αβ 1 ln Ak α l1 d l2 σ {1 1 1 βα 1 βα} + β 1 β 1 ln 1 βα A l1 s l2 s σ +βs s ln A l1 s l2 s σ 0.5 βα 1 βα where s D = α 1 + βα 1 αβ 1 = α 1 αβ 1 = s s = s nd I D 1 = ln [1 1 1 βα 1 βα] A l1 d l2 σ l1 d A l d 1 l2 σ ln {1 1 1 βα [1 1 1 βα 1 βα]} + βα 1 αβ ln 1 βα A l s + β 1 β 1 1 l2 s σ +βs s ln A l1 s l2 s σ 0.5 βα 1 βα

16 344 JESS BENHABIB For the second gent, the vlue of defection is given by V1 D = s D k + I2 D, where σ 2 I2 D = ln [1 1 βα 1 βα] A l 1 l2 d 0.5 l2 d + βα 1 αβ 1 A l 1 l2 d σ ln {1 1 βα [1 1 βα 1 βα]} ln 1 βα A l s + β 1 β 1 1 l2 s σ +βs s ln A l1 s l2 s σ 0.5 βα 1 βα We cn, t this point, inquire whether coopertion cn be sustined for some specil cses. When β 0, we get: V D 1 V 1 = ln = 0.5 l d 1 l ln This quntity is positive if > e 1. This will be the cse for 0, 1. 2 Thus there cn be no coopertion s β 0. Furthermore one could show tht when = 0.5 nd = σ, then it is lwys better to cooperte s β 1. REFERENCES Acemoglu, Dron nd Robinson, Jmes A., 2002, Why did the west extend the frnchise? Democrcy, inequlity, nd growth in historicl perspective. Qurtely Journl of Economics 1154, Alesin, Alberto, 1994, Politicl models of mcroeconomic policy nd fiscl reforms. In: Stephn Hggrd nd Steven Webb, Voting Reform: Democrcy, Politicl Liberliztion, nd Economic Adjustment. Oxford University Press, New York, New York. Aghion, Philippe, 2000, Schumpeterin growth theory nd the dynmics of income inequlity. Econometric 703, Alesin, Alberto nd Rodrik, Dni, 1994, Distributive politics nd economic growth. Qurterly Journl of Economics 1092, Bnerjee, A. nd A. Newmn, 1993, Occuptionl choice nd the process of development. Journl of Politicl Economy 101, Benbou, Rolnd, 1996, Inequlity nd growth. In: Ben S. Bernnke nd Julio Rotemberg. NBER Mcroeconomics Annul Cmbridge, MA: MIT Press, Benhbib, Jess nd Spiegel, Mrk, 1994, The role of humn cpitl nd politicl instbility in economic development. Journl of Monetry Economics 34, Agin by symmetry, V2 D V 2 is negtive for the second gent if 1 > e 1.

17 THE TRADEOFF BETWEEN INEQUALITY AND GROWTH 345 Benhbib, Jess nd Rustichini, Aldo, 1996, Socil conflict nd growth. Journl of Economic Growth 1, Deininger, Klus nd Squire, Lyn, 1996, A new dt set mesuring inequlity. World Bnk Economic Review 103, Esterly, Willim, The middle clss consensus nd economic development. Journl of Economic Growth 6, Forbes, K., 2000, A ressessment of the reltionship between inequlity nd growth. Americn Economic Review 904, Glor, O. nd J. Zeir, 1993, Income distribution nd mcroeconomics. Review of Economic Studies 60, Islm, Nzrul, 1995, Growth empirics: A pnel dt pproch. Qurterly Journl of Economics 1104, Kitl, Veijo nd Pohjol, Mtti, 1990, Economic development nd greeble redistribution in cpitlism: Efficient gme equilibri in two-clss neoclssicl growth model. Interntionl Economic Review 31, No.2, Kldor, N., 1957, A model of economic growth. Economic Journl 67, Kuznets, S., 1955, Economic growth nd income inequlity. Review 451, Americn Economic Levine, Ross E. nd Renelt, Dvid, 1992, A sensitivity nlysis of cross country regressions. Americn Economic Review 824, Perotti, R., 1996, Growth, income distribution, nd democrcy. Journl of Economic Growth 1, Persson, Torsten nd Tbellini, Guido, 1994, Is inequlity hrmful for growth. Americn Economic Review 843, St. Pul, Gilles nd Verdier, Thierry, 1993, Eduction, democrcy, nd growth. Journl of Development Economics 422,

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10-438 UNVERSTY OF NOTTNGHAM Discussion

More information

2. Transaction Cost Economics

2. Transaction Cost Economics 3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function. Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS * Jorge Durán and Alexandra Rillaers **

PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS * Jorge Durán and Alexandra Rillaers ** PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS * Jorge Durán nd Alexndr Rillers ** WP-AD 2002-18 Corresponding uthor: Jorge Durán, Deprtmento de Fundmentos

More information

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule. Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Optimal Execution of Open-Market Stock Repurchase Programs

Optimal Execution of Open-Market Stock Repurchase Programs Optiml Eecution of Open-Mrket Stock Repurchse Progrms Jcob Oded This Drft: December 15, 005 Abstrct We provide theoreticl investigtion of the eecution of open-mrket stock repurchse progrms. Our model suggests

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Small Business Networking

Small Business Networking Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Public Capital Maintenance and Congestion: Long-Run Growth and Fiscal Policies

Public Capital Maintenance and Congestion: Long-Run Growth and Fiscal Policies Public Cpitl Mintennce nd Congestion: Long-Run Growth nd Fiscl Policies Evngelos Dioikitopoulos nd Srntis Klyvitis y Athens University of Economics nd Business Mrch 2008 z Forthcoming in Journl of Economic

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

This paper considers two independent firms that invest in resources such as capacity or inventory based on

This paper considers two independent firms that invest in resources such as capacity or inventory based on MANAGEMENT SCIENCE Vol. 5, No., December 006, pp. 93 99 issn 005-909 eissn 56-550 06 5 93 informs doi 0.87/mnsc.060.0574 006 INFORMS Strtegic Investments, Trding, nd Pricing Under Forecst Updting Jiri

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition and Monopoly Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

The Relative Advantages of Flexible versus Designated Manufacturing Technologies

The Relative Advantages of Flexible versus Designated Manufacturing Technologies The Reltive Advntges of Flexible versus Designted Mnufcturing Technologies George Normn Cummings Professor of Entrepreneurship nd Business Economics Tufts University Medford MA 055 USA e-mil: george.normn@tufts.edu

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade Lbor Productivity nd omrtive Advntge: The Ricrdin Model of Interntionl Trde Model of trde with simle (unrelistic) ssumtions. Among them: erfect cometition; one reresenttive consumer; no trnsction costs,

More information

CEO Pay and the Lake Wobegon Effect

CEO Pay and the Lake Wobegon Effect CEO Py nd the Lke Wobegon Effect Rchel M. Hyes nd Scott Schefer December 11, 2008 Forthcoming in Journl of Finncil Economics Abstrct The Lke Wobegon Effect, which is widely cited s potentil cuse for rising

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Lump-Sum Distributions at Job Change, p. 2

Lump-Sum Distributions at Job Change, p. 2 Jnury 2009 Vol. 30, No. 1 Lump-Sum Distributions t Job Chnge, p. 2 E X E C U T I V E S U M M A R Y Lump-Sum Distributions t Job Chnge GROWING NUMBER OF WORKERS FACED WITH ASSET DECISIONS AT JOB CHANGE:

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

JaERM Software-as-a-Solution Package

JaERM Software-as-a-Solution Package JERM Softwre-s--Solution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

The Development and Structure of Financial Systems

The Development and Structure of Financial Systems The Development nd Structure of Finncil Systems Shnkh Chkrborty y Tridip Ry z Revised: September 2006 Abstrct Firms rise externl nnce vi monitored bnk lons nd non-monitored borrowing in dynmic generl equilibrium

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Optimal Redistributive Taxation with both Labor Supply and Labor Demand Responses

Optimal Redistributive Taxation with both Labor Supply and Labor Demand Responses Optiml Redistributive Txtion with both Lbor Supply nd Lbor Demnd Responses Lurence JACQUET NHH Preliminry version Etienne LEHMANN y CREST Bruno VAN DER LINDEN z IRES - UCLouvin nd FNRS Mrch 28, 2011 Abstrct

More information

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct

More information

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans Eduction nd Optiml Dynmic Txtion: The Role of Income-Contingent Student Lons Sebstin Findeisen University of Mnnheim Dominik Schs CGS, University of Cologne First version: October 2, 20 This version: November

More information

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans University of Zurich Deprtment of Economics Center for Institutions, Policy nd Culture in the Development Process Working Pper Series Working Pper No. 421 Eduction nd Optiml Dynmic Txtion: The Role of

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Enterprise Risk Management Software Buyer s Guide

Enterprise Risk Management Software Buyer s Guide Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Why is the NSW prison population falling?

Why is the NSW prison population falling? NSW Bureu of Crime Sttistics nd Reserch Bureu Brief Issue pper no. 80 September 2012 Why is the NSW prison popultion flling? Jcqueline Fitzgerld & Simon Corben 1 Aim: After stedily incresing for more thn

More information

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358)

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358) C-crcs Cognitive - Counselling Reserch & Conference Services (eissn: 2301-2358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,

More information

PEARLS MONITORING SYSTEM

PEARLS MONITORING SYSTEM WORLD COUNCIL OF CREDIT UNIONS TOOLKIT SERIES Number 4 PEARLS MONITORING SYSTEM by: Dvid C. Richrdson World Council of Credit Unions October 2002 2001 The World Council of Credit Unions The World Council

More information

Portfolio approach to information technology security resource allocation decisions

Portfolio approach to information technology security resource allocation decisions Portfolio pproch to informtion technology security resource lloction decisions Shivrj Knungo Deprtment of Decision Sciences The George Wshington University Wshington DC 20052 knungo@gwu.edu Abstrct This

More information

College Admissions with Entrance Exams: Centralized versus Decentralized

College Admissions with Entrance Exams: Centralized versus Decentralized Is E. Hflir Rustmdjn Hkimov Dorothe Kübler Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Discussion Pper SP II 2014 208 October 2014 (WZB Berlin Socil Science Center

More information

Research Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model

Research Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model Mthemticl Problems in Engineering Volume 4, Article ID 678, pges http://dx.doi.org/.55/4/678 Reserch Article Competition with Online nd Offline Demnds considering Logistics Costs Bsed on the Hotelling

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Modeling POMDPs for Generating and Simulating Stock Investment Policies

Modeling POMDPs for Generating and Simulating Stock Investment Policies Modeling POMDPs for Generting nd Simulting Stock Investment Policies Augusto Cesr Espíndol Bff UNIRIO - Dep. Informátic Aplicd Av. Psteur, 458 - Térreo Rio de Jneiro - Brzil ugusto.bff@uniriotec.br Angelo

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

The Impact of Oligopolistic Competition in Networks

The Impact of Oligopolistic Competition in Networks OPERATIONS RESEARCH Vol. 57, No. 6, November December 2009, pp. 1421 1437 issn 0030-364X eissn 1526-5463 09 5706 1421 informs doi 10.1287/opre.1080.0653 2009 INFORMS The Impct of Oligopolistic Competition

More information

Unleashing the Power of Cloud

Unleashing the Power of Cloud Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,

More information

IIMK/WPS/121/FIN/2013/07. Abhilash S. Nair 1

IIMK/WPS/121/FIN/2013/07. Abhilash S. Nair 1 IIMK/WPS/2/FIN/203/07 EXISTENCE OF CAPITAL MARKET EQUILIBRIUM IN THE PRESENCE OF HERDING AND FEEDBACK TRADING Abhilsh S. Nir Assistnt Professor, Indin Institute of Mngement Kozhikode, IIMK Cmpus PO, Kozhikode

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Uncertain Lifetime and Intertemporal Choice: Risk Aversion as a Rationale for Time Discounting

Uncertain Lifetime and Intertemporal Choice: Risk Aversion as a Rationale for Time Discounting Uncertin Lifetime nd Intertemporl Choice: Risk Aversion s Rtionle for Time Discounting Antoine Bommier CNRS nd University of Toulouse (GREMAQ) June 2005 Abstrct This pper mkes explicit the links between

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

persons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining:

persons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining: COST- BENEFIT ANALYSIS OF NARCOTIC ADDICTION TREATMENT PROGRAMS with Specil Reference to Age Irving Leveson,l New York City Plnning Commission Introduction Efforts to del with consequences of poverty,

More information

Is Competition Among Charities Bad?

Is Competition Among Charities Bad? Is Competition Among Chrities Bd? Inkyung Ch nd Willim Neilson Tes A&M University, College Sttion, TX 7783 December Abstrct This pper studies tht the eect o incresed competition mong chrities or dontions,

More information

Health insurance exchanges What to expect in 2014

Health insurance exchanges What to expect in 2014 Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount

More information

On the Robustness of Most Probable Explanations

On the Robustness of Most Probable Explanations On the Robustness of Most Probble Explntions Hei Chn School of Electricl Engineering nd Computer Science Oregon Stte University Corvllis, OR 97330 chnhe@eecs.oregonstte.edu Adnn Drwiche Computer Science

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Trade liberalization and per capita income convergence: a difference-in-differences analysis

Trade liberalization and per capita income convergence: a difference-in-differences analysis Journl of Interntionl Economics 55 (2001) 203 228 www.elsevier.nl/ locte/ econbse Trde liberliztion nd per cpit income convergence: difference-in-differences nlysis Mtthew J. Slughter* Drtmouth College

More information

Decision Rule Extraction from Trained Neural Networks Using Rough Sets

Decision Rule Extraction from Trained Neural Networks Using Rough Sets Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown EE247 Lecture 4 Ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form exmple shown Then will ttend to high order ldder type filters incorporting zeros

More information

VoIP for the Small Business

VoIP for the Small Business Reducing your telecommunictions costs VoIP (Voice over Internet Protocol) offers low cost lterntive to expensive trditionl phone services nd is rpidly becoming the communictions system of choice for smll

More information

FDIC Study of Bank Overdraft Programs

FDIC Study of Bank Overdraft Programs FDIC Study of Bnk Overdrft Progrms Federl Deposit Insurnce Corportion November 2008 Executive Summry In 2006, the Federl Deposit Insurnce Corportion (FDIC) initited two-prt study to gther empiricl dt on

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

STATEMENT OF ALICE M. RIVLIN, DIRECTOR CONGRESSIONAL BUDGET OFFICE. Before the

STATEMENT OF ALICE M. RIVLIN, DIRECTOR CONGRESSIONAL BUDGET OFFICE. Before the STATEMENT OF ALICE M. RIVLIN, DIRECTOR CONGRESSIONAL BUDGET OFFICE Before the SUBCOMMITTEE ON ECONOMIC STABILIZATION COMMITTEE ON BANKING, HOUSING, AND URBAN AFFAIRS UNITED STATES SENATE MARCH 13, 1980

More information

Pay over time with low monthly payments. Types of Promotional Options that may be available: *, ** See Page 10 for details

Pay over time with low monthly payments. Types of Promotional Options that may be available: *, ** See Page 10 for details With CreCredit... Strt cre immeditely Py over time with low monthly pyments For yourself nd your fmily Types of Promotionl Options tht my be vilble: Not ll enrolled helthcre prctices offer ll specil finncing

More information

Credit Ratings, Collateral, and Loan Characteristics: Implications for Yield*

Credit Ratings, Collateral, and Loan Characteristics: Implications for Yield* Kose John New York University Anthony W. Lynch New York University nd Ntionl Bureu of Economic Reserch Mnju Puri Stnford University nd Ntionl Bureu of Economic Reserch Credit Rtings, Collterl, nd Lon Chrcteristics:

More information

Health insurance marketplace What to expect in 2014

Health insurance marketplace What to expect in 2014 Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum

More information

Industry and Country Effects in International Stock Returns

Industry and Country Effects in International Stock Returns Industry nd Country ffects in Interntionl Stock Returns Implictions for sset lloction. Steven L. Heston nd K. Geert Rouwenhorst STVN L. HSTON is ssistnt professor of finnce t the John M. Olin School of

More information

Does Trade Increase Employment? A Developing Country Perspective

Does Trade Increase Employment? A Developing Country Perspective Does Trde Increse Employment? A Developing Country Perspective By Sugt Mrjit City University of Hong Kong nd Center for Studies in Socil Sciences, Indi Hmid Beldi University of Texs t Sn Antonio 005 Address

More information

Participation and investment decisions in a retirement plan: the influence of colleagues choices

Participation and investment decisions in a retirement plan: the influence of colleagues choices Journl of Public Economics 85 (2002) 121 148 www.elsevier.com/ locte/ econbse Prticiption nd investment decisions in retirement pln: the influence of collegues choices Esther Duflo,b, *, Emmnuel Sez MIT,

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Strategic Sourcing in the Presence of Uncertain Supply and Retail Competition

Strategic Sourcing in the Presence of Uncertain Supply and Retail Competition Vol. 2, No. 10, October 2014, pp. 174 1760 ISSN 1059-147 EISSN 197-5956 14 210 174 DOI 10.1111/poms.1207 201 Production nd Opertions Mngement Society Strtegic Sourcing in the Presence of Uncertin Supply

More information