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DISCUSSION PAPERS IN ECONOMICS Working Paper No. 98-22 Growth or Stagnation? The Role of Public Education Kenneth R. Beauchemin Department of Economics, University of Colorado at Boulder Boulder, Colorado October 1998 Center for Economic Analysis Department of Economics University of Colorado at Boulder Boulder, Colorado 80309 © 1998 Kenneth R. Beauchemin Kenneth R. Beauchemin 1 First Draft: June 1997 Revised: October 1998 1Department of Economics, University of Colorado at Boulder, Boulder, CO 80309. Internet: [email protected] gratefully acknowledgethe excellent research assistanceof Juan Blyde. I am also thankful to JoAnne Feeney, Jack Robles and participants of the 1998 meetings of the Society for Economic Dynamics for helpful discussions and comments. This researchwas partially supported by a Grant-in-Aid from the Council on Researchand Creative Work at the University of Colorado. Abstract This paper presents a political-economic cumulation. theory of growth and human capital ac- Age heterogeneity is put forth as the primary source of disagreement between individuals over various levels public education expenditures. An overlap- ping generations model with with two-period lived agents is constructed to capture the heterogeneity. With a growing population, the equilibrium quantity of public education reflects the preferences of youth and is therefore foward looking. As such, policy preferences are a function of intertemporal elasticities, utility discounting and population growth. Despite foward looking behavior, it is shown that sufficiently rapid population gro~h can trigger stagnation (zero growth) in the form of a corner solution to the public policy problem. The model therefore complements existing models that associate slow per capita output growth with high population growth. Key words: EconomicGrowth; Public Education; Political Economics JEL classification: 040; E62 1 Introd uction As is the case with economic growth rates, government expenditures on education vary substantially throughout the world. In the data set of Barro and Lee (1994), annual per capita real GDP growth ranges from -1.5% to 6.4% averaged over 196085; the ratio of government spending on eduction to GDP ranges from 1.2% to 8.3% averaged over the same period. The positive association between various measures of schooling and per capita growth is well documented in the empirical growth literature.! Given that the majority of primary and secondary education is provided publicly throughout the world, understanding the political-economic process governing public investment in human capital is an important ingredient in a theoretical account of differences in per capita income growth across countries. This paper presents a political-economic model that jointly detertnines public education expenditures and per capita output growth. The model is intentionally spare in detail to isolate the role of public education in generating economicgrowth and vice versa. It is characterized by an infinite sequence of overlapping generations comprised of two-period lived agents where individuals within generations are identical. Given that age is the only source of heterogeneity, it is also the only source of political rivalry regarding education policy. Although there are no explicit altruistic linkages betweengenerations,the young can be conceptualizedas a combinationof parent and child; they are not only the recipients of publicly provided education, they are also producers of output and voting taxpayers. Thus, the structure conveniently embodies a limited form of altruism while retaining the analytical simplicity associated with two-period lived agents. The competing policy preferences of the young and the old are aggregated us1See, for example, Barro (1991), Benhabib and Speigel (1994), and Barro and Sala-i-Martin (1995). Bils and Klenow (1996) question the strength of these correlations citing the presenceof reversecausality. In this paper, schooling both causes,and is caused by, growth. ing majority rule voting. Positive population growth subsequently implies that the young agent is the decisive voter with the implication that the quantity and quality of public education is in the hands of the immediate beneficiaries. It also means that education policy is forward looking and determined much in the same way as other investment decisions. Consequently, the amount of public investment in human capital undertaken by a society is largely a function of the ease with which agents willingly substitute consumption between time periods along with their rate of time preferencein utility discounting. Unlike ordinary investment decisions,however,the young must forecastthe policy choiceof the ensuinggeneration. In political-economic equilibrium, their forecastsare correct. Despitethe fact that population growth gives public education its strongest support in terms of age demographics, it is shown that sufficiently high population growth paradoxically leads to drastically limited public education expenditures with economic stagnation the end result. It is common for growth models to exhibit the feature that high population growth is coincident with low economic growth. In the standard neoclassical growth model, for example, population growth dilutes the capital stock, depressing the steady state capital stock per labor unit resulting in lowertransitional growth rates. In the Becker, Murphy and Tamura (1991) model with endogenousfertility choice,two stable steady states are generated due to increasingreturns to human capital accumulation. One of these steady states -often thought of as "development trap" -is associated with low levels of human capital and high fertility with the other yielding the opposite attributes. The present work introduces a separate, but complementary, explana- tion for observed negative correlations between measures of population growth and economicgrowth acrosscountries. Generally speaking, the approachto economic growth here mirrors that of Lucas (1988), Jones and Manuelli (1990), and Rebelo (1991) in that differencesin growth rates are due to differencesin rates of human capital accumulation. As in the models 2 of Glomm and Ravikumar (1992) and Saint-Paul and Verdier (1993), human capital is the only factor of production and is accumulated within a political-economic process that determines expenditures on public education. The political competition in these papers, however, originates in intm-generational income and wealth heterogeneity. To retain analytical tractability, they assume that agents consume only once during their lifetimes. In contrast, the focus here is on the intergenerational redistribution of resources arising from public education. Consequently, the public policy problem involves an inter-temporal tradeoff that only exists when agents are given the ability to consume in each period of life. The next section presentsthe model of human capital accumulation and growth. Section 3 introduces the public policy choice mechanism and defines the politicaleconomic equilibrium. The main results of the paper are presented in section 4. Section 5 contains some concluding remarks. 2 Economic Growth In this section, a simple one-sector overlapping generations growth model with twoperiod lived agents is constructed. The economy is dated in discrete time by the index t = 1,2, ...with Nt individuals born at the beginning of period t. The population is assumedto grow at a constant rate with the grossgrowth rate given by n =-!YL>1 Nt-l for all t. All agents within a generation are identical. Unlike the standard overlapping generations model of Diamond (1965), the accumulable factor here is instead embodied in the work force as human capital. As commo~y assumed,the rate of return on human capital investment does not dirninish as human capital is accumulated. In contrast to narrowly defined physical capital, human capital eases the accumulation of further knowledge and is therefore not subject to diminishing returns. Accordingly, an agent of generation t is assumed 3 h:-1 ~ to produce an amount of output equal to his or her human capital in each period of life: t Yt+i i h~+i 0,1 (where the notational conventionusessubscripts to denotecalendar time and superscripts to indicate the generation). Aggregate output is obtained by summing the output of the young and the output of the old: ~ Nth~ + Nt-lh~-l t = 1,2, Each agent is assumed to inherit the human capital attained by the previous generation. This intergenerational linkage is essentially the same as the ones used by Glornrn and Ravikumar (1992), Persson and Tabellini (1994) and Saint-Paul and Verdier (1993) and accords with Lucas' (1988) view that human capital is a "social activity, involving groups of people in a way that has no counterpart with the accumulation of physical capital." Additionally, Glomm and Ravikumar note that the assumption is supported empirically by a number of studies that find positive correlations between parental educational attainment and that of their children, inherited stock can subsequentlybe augmentedthrough publicly provided education. It is assumed that gross public investment in human capital is a linear function of total expenditures per pupil (i.e. per young agent) In the second period of life, therefore, an agent's human capital is given by + Agt h~+l where 9t is the period-t public education expenditures per pupil and A> the productivity of public investments. (2) a measures Education expenditures are financed with revenueraised by a proportional output tax applied uniformly to the young and the old so that 9t TYt T(h:+ ~ht-l) Nt 4 (3) where T is the output tax rate. The young in this model can be viewed a combination of parent and child in that they attend school and produce output. An alternative approach assumes 3-period lived agents with the middle-aged agents serving as parents that care about the quality of the young generation's education. In this setup, the young generation -the children attending school -do not produce or vote. By the time they reach old age, agents no longer have children in school and lose interest in supporting public education. The two structures yield identical results. By accepting the "parent/child" fiction, however, the young internalize the benefits and costs of their own education obviating the need for a separate assumption regarding parental altruism. Next, let r be the function mapping the policy choice variable T and the exogenous population growth rate n into the gross growth rate of per-capita output. Substituting equations (2) and (3) into this definition of growth yields the expression n l+n )-1 T which shows per capita gro\vth to be decreasing in n and increasing in T. Note that ~ gives the proportion of young agents (or pupils) in the total population. Since this proportion is increasing in n, economic growth decreaseswith increasesin the relative size of the school-agedpopulation. The reason for this is simple. Given a tax. rate T, an increase in population growth implies that the number of agents benefiting by public education (the young) increases relative to the contributors (the total population) diluting the quality of public education as measured by expenditures per pupil. This dampening influence of a more rapidly growing population on per-capita output growth is similar to the mechanismat work in a neoclassicalgrowth model where additional investment is required simply to maintain the capital stock per capita as it is diluted by population growth. With tax rates endogenizedin the sections that 5 follow, it is shown that equilibrium public education expenditures generally depend on population growth, resulting in a richer relationship betweeneconomicgrowth and population growth. 3 Political Economy The model presented thus far has been entirely descriptive in that the productive resourceof human capital is accumulatedas the consequenceof a fixed public saving rule. Determining public education expenditures as a proportion of aggregate output (T) in a political-economic equilibrium first requires an assessment of policy prefer- ences of the two competing agents (young and old). Subsequently, a description of how those preferences are aggregated into a policy outcome is required. The policy choice of the old agent is trivial. The output tax applied uniformly to the population directs resources away from the old to enhance the productivity of the young, which entails no benefit for the old. Since the old lack any altruistic motives, they clearly favor an output tax of zero. In contrast, the young internalize the future productivity benefit of formal schooling and face a nontrivial intertemporal tradeoff betweenpresent and future consumption. In order to model policy outcomesin actual economies,it is necessaryto aggregate the competing policy preferences. Perhaps the most straightforward and sensible aggregation mechanism is majority rule yoting. In an economy with a growing population, i.e. n > 1, majority rule voting leads to policy outcomes that are weighted in favor of the younger members of society. In terms of the current model, the younger generation perpetually chooses the educational investment proportion of output. An alternative would be to weight the preferences of each currently alive generation but allow for more weight on the preferences of the young. In light of the above discussion, the choice does not matter. Since the old always prefer T = 0, shifting the 6 political franchise towards the old systematically lowers the equilibrium tax. rate but leaves the qualitative implications of the theory unaltered. Given that the model is not taken to be an accurate quantitative representation of actual economies, voting is assumedto be strictly majority rule so that the equilibrium policy outcome is simply the one preferred by the young. The policy problem of the young agent is not only framed by his or her desire for a more productive future, but also by a future with low (ideally zero) taxation. However, the ideal lifetime tax profile of any generation t is not feasible since the tax rate applying in old age will be chosen by the ensuing generation of yoWlg agents, i.e. members of generation t + 1. Given an amount of inherited human capital h~ > 0, the preferred policy of the young agent in period t maximizes discounted lifetime utility: ut u (c:) + {3u (C:+l) where /3 E (0,1 is the subjective discountfactor. The assumptionson the momentary utility function are standard: u is twice continuouslydifferentiable with u' > 0, u" < o. It is also assumedthat the lifetime utility function is homothetic. Consumption in each period of life is equal to after-tax output: c~ = (1 -Tt) t Ct+l h~ , 'Tt+l) h:+1 . (1 FUrthermore, it is easily shown that an agent's growth in human capital from youth to old age follows the growth rate of per capita output so that h:+1 = r (Tt; n) h: The homotheticity property of U greatly simplifies the young agent's policy problem by making it independent of the level of human capital. Therefore, in deciding on the optimal tax rate, the young agent is not required to forecast the policy response 7 :argmax u' of the ensuing generation to the change in the future level of human capital caused by a current change in the tax rate (and hence the current level of public education). This means that tax policy cannot be used by the young to strategically manipulate the policy preferences of the ensuing generation in order to help achieve their own future policy objectives.2 Dropping the subscriptsand superscripts and expressingone period aheadvariables with primes, the young agent's preferred tax rate as a function of the future tax rate 7' is given by U (1- 'I/J(7') T: + {3u((1- 7') r (7; n)) T (4) The first order condition of the young agent's problem is given by (1 T)~,B(l 7') r' (7; n) u' ((1- 7') r (7; n)) (5) and T~O (6) Letting r denote the tax rate at which (5) holds with equality, the solution to the agent's problem is given by max{O,r}. The solution for every T' E [0,1] gives the single-valued policy function 1/Jmapping from the unit interval into itself. Whether the policy function is continuous in [0,1] depends on the exact specification of u. It is possible that 1/J(7') -* 1 as 7' -* 1 while 1/J(1) = O. That is, a young agent's best policy response to a forecast of 7' = 1, or equivalently, a forecast of zero consumption in old-age, is a tax rate of zero in youth to maximize consumption in youth. For this reason, 'I/Jmay have a discontinuity at T' = 1 but it is always continuous in [0,1 The following result regarding the slope of the policy function proves useful in further characterizing the political-economic equilibrium:3 2SeeBeauchemin (1998) for an example of an overlapping generationseconomywhere this property does not hold and fiscal policy is used to manipulate the policy preferencesof future generations. 3Proofs of all propositions are presented in the appendix. 8 Proposition 1 Consider the anticipated tax rate r E [0, 1) and the equilibrium policy U , II(( ~~~:~:,~~)T'):I ' , I-T r I-T r < U function 1/;. If 1/;(7') = 0, then 1/; (7') = O. Otherwise, -u' I-T' r) :> 1 implies 1/;' (r) ~ 0 In constructing a political-economic equilibrium, it must be recognized that young agents across generationsface identical policy problems. Also, in order to avoid indeterminacy, it is necessaryto close the model with the requirement that agents' forecastsof future tax rates are realized. Together, intergenerationalsymmetry and perfect foresight imply that an equilibrium tax rate, r*, satisfiesr* ing the equilibrium per capita growth rate A (~) 7 = 7' yield- r*. In closing this section, the political-economic equilibrium is formally stated as follows: Definition 1 A symmetric political-economic equilibrium is given by the func- tion'l/; : [0,1] -t [0,1] and a tax rate 7* such that (i) 'I/; (r) satisfies (4) , and (ii) 7* is a fi1;ed point 0/'1/;, i.e. 7* = 'I/;(7*). 4 Characterizing the Equilibrium In what follows, the degree of curvature in the period utility function is shown to be a fundamental determinant of public education expenditures and growth. It useful, therefore, to restrict u as follows: u(c) where -~ 1-0L- 1-0" , for Inc, for a = 1 a > 0, a =1= 1 = 0"determines the curvature and the willingness of agents to substitute consumption across time periods; the "intertemporal elasticity of substitution" is approximated by 1/0". With relatively large values of 0", agents want consumption in different periods to look very similar. Consequently, they dislike growth -positive 9 or I. negative -in their lifetime consumption profile. Conversely, with smaller o-'s, agents are more tolerant of an upward (or downward) sloping lifetime consumption profile. Given the anticipated tax rate 7', an interior solution to the young agent's policy problem is now represented by 'l/J(7') 1 () (7') 1 + A (~) (}(7') (7) T') 5r:;;:! () (T') It is easily shown that the policy function 'l/Jin (7) is single-valued and continuous in interval [0, 1) It is also monotonic. This result follows directly from proposition 1 and is stated below: Proposition 2 Consider (i) a < 1, then 'l/J'(7) a a E (0,00) and the corresponding :::; 0, (ii) a > 1, then 'l/J'(7) policy function 'l/J. If :;.:::0, and (iii) a > 1, then = 0, for all 7' E [0, 1 With exception of log preferences (0does not have a closed-form solution. 1), the equilibrium condition r' 'I/J (7"*) Given the properties of 'If; already stated, however, existence and uniqueness of the political-economic equilibrium is readily established. Proposition 3 A political-economic equilibrium with isoelastic period utility exists and is unique. 4.1 Intertemporal Substitution Figure la illustrates the three qualitatively distinct types of political-economic equilibria by plotting the policy function 'I/Junder the parameter settings 0" E {.5, 1, 2}.4 4The ancillary parameter settings are fJ = .277and A = 3.2. Although not intended as an accurate quantitative portrayal, these parametersare chosensensibly. The first is chosento approximate 10 7') The three separate cases generate quantitative differences in a young agent's optimal policy response to a change in the after-tax rate of return to public investment represented by the terD1 (1 r' (7) in the first-order condition (5) exogenous increase in the anticipated tax rate 7'. Consideran With this increase comes a si- multaneous reduction in the rate of return to public investment in human capital and (human) wealth A reduction in the rate of return encourages additional current consumption (lower preferred T) at the expense of future consumption through intertemporal substitution, and reductions in both current and future consumption (higher preferred r) as the consequence of diminished wealth. The policy function slopes downward in the first case (0" = 0.5) becausethe substitution effect prevails. In the third case (0" = 2), the policy function slopes upward because the wealth effect dominates. Only with log utility (0" = 1), do the two effects cancel resulting in a flat policy function. Equilibrium, characterized by the fulfillment of expectations, is depicted by the crossing of the policy function with the 45-degree line. Note that the equilibrium tax rate, and hence the economic growth rate, echoes the intensity of the young agent's desire to smooth consumption. This gives rise to an inverse relationship between a and T' With agents that view consumption as strong (weak) complements~ the economy exhibits less (more) growth. 4.2 Population growth and economic growth As previouslydiscussed,high population growth in this model can retard per capita output in a neoclassicalmanner by diluting a given quantity of public investment and thereby reducing the quality of education. Endogenouspolicy choiceintroduces another channel through, which population growth influences per capita economic an annual discount factor of .95 when a generation is 25 years in length. The latter is calibrated so that equilibrium education expenditures as a fraction of GDP are similar to those observed across countries. 11 growth. By lowering education quality, population growth also reduces the rate of return to public investment, which has been shown to alter the young agent's policy preferences.Thus, population growth combines with the defining characteristics of intertemporal preferences in the political process to playa leading role in determining equilibrium growth. Assuming that the economy begins from a position of strictly positive per capita growth, the following proposition establishes the set of a, defined by a maximal value, that is both necessary and sufficient for an exogenous increase in the population growth rate to decrease equilibrium growth per capita. Proposition 4 Consider an economy with a gross population growth rate, n > 1, and a strictly positive equilibrium tax rate, T* > 0, so that per capita output growth is also positive: (~) T* > o. An exogenousincreasein the population growth rate decreasesper capita output growth if and only if a< where a 1 + QT. QT. (~) T* association is relatively large for economies with small growth rates -in other words, for economies with rapid initial population growth (n large) and low productivity of public investments (A small). These features, along with others, are commonly used to describethe economyof a less developedcountry. In contrast, this set is smaller for a developed economy -that is, one with the reverse properties. In this sense, a developed (or industrialized) economy is less likely to experience a decrease in per capita growth as the result of higher population growth.5 5For quite different reasons,the positive predicted associationbetween population growth and economic growth in this case,is shared with some Schumpeterian researchand development based 12 For example, real per capita GDP in the United States increased by 68.4% during the 25-year period from 1960-85 -rougWy the span of a generation. Interpreted literally, proposition 4 states that a small increase in U.S. population growth during this period would have decreasedper-capita output if, and only if, 0" < 1.684/.684 = 2.46. In contrast, Argentina's real per capita GDP increase of 18% translates into the condition 0" < 1.180/.180 = 6.55, which is much more likely to be met that than the corresponding one for the United States. Growth or Stagnation? The questionstill remains: caneconomicgrowth disappearaltogether? A numerical example depicted in figure 1 b confirms that it can. The example is constructed using a higher population growth rate compared to the previous example (displayed in figure la). In all three economies, equilibrium public education expenditures are zero: 1/1(r') = O. With the policy function 'lp (r') monotonically nonincreasing in the first two cases (0"~ 1), 1/1is simply the zero function: 1/1(r/) = O. In the third case (0" > 1), 1/1(r/) > 0 for sufficiently large r/: the policy function is zero at the origin ensuring stagnation. More generally, there is a threshold population growth rate -or equivalently, a threshold proportion of young agents -past which the political process ensures that economic growth is zero. Proposition 5 Per-capita output growth is positive if and only if ~ To fix ideas, consider the case where A < A{3. 1; in other words the governmenttrans- forms goods into human capital unit for unit The nonzero growth condition then states that the relative size of the young cohort cannot exceedthe discount factor. For example, {3 = .277 corresponds to an annual discount factor of .95 when a model models. See Aghion and Howitt (1998). It should be emphasized,however,that this is only true for a subset of economieswithin the general class considered here. 13 period (or generation) is 25 years in length. In this case, stagnation sets in if the school-age population accounts for more that 27.7% of the total population. The more efficient a government is at converting output into human capital (captured by higher values of A), the more room an economy has for population growth before the onset of stagnation.6 A couple of comments regarding this result are in order. First, stagnation cannot occur without the political process in this model. This means that high population growth alone cannot causestagnation; it must result from a corner solution in the policy problem of the young. Second, a strict interpretation of the model is that zero growth corresponds to zero funding of public education; a result that is counterfac- tual. Stagnant economiesare observedeven in the presenceof positive government expenditures on education. The strength of the result is partly due to the model's simplicity; it could be softened somewhat by assuming a constitutionally determined minimum tax rate in place of the zero tax rate set by majority rule voting. Stagnation, in such an environment, is still represented by a corner solution, but one that is marked by "low" growth and little public investment in education. Furthermore, by allowing human capital to depreciate, the corner solution with minimal levels of educationcould also exhibit zero growth. What is the contribution of public educationin accounting for the great differences in growth rates across counties? The discussionabove strongly suggeststhat such heterogeneity must originate in similar variation in population growth. This necessary condition is satisfied. In addition to wide variation in economic growth rates and national resource commitments to public education, the data set of Barro and Lee also revealstremendous cross-countryvariation in population growth rates. Average 6Although the sparsestructure of the model and the low resolution offered by two-period lifetimes is inadequate to take the quantitative implications seriously,the qualitative properties carryover to a richer setting. 14 annual population growth rates during the 1960-85 period take on values between 1.1% and 19.3% and are fairly uniform in distribution. It is also consistent with the observednegative, but weak, co~elation of population growth with per capita output growth.7 In proposition 5, economic stagnation is unambiguously the consequence of high rates of population growth Proposition 4, on the other hand, shows that increases in population growth can actually increase equilibrium per capita output growth provided that population growth is initially low and that the productivity of public investment in public education is high. Roughly speaking, the role of public education strengthens previously defined theoretical roles for rapid population growth to coincide with miniscule rates of economic growth while preserving the possibility that the two growth rates bear a more moderate overall statistical association. 5 Concl tiding Remarks This paper addressed itself to understanding the joint determination of public education expenditures and per capita income growth. The model is successful in that it is consistent with a standard set of empirical regularities documented by modern growth theory. By construction, the theory is consistent with the fact that growth is generally increasing in the fraction of aggregate output dedicated to public education. The main implication of the theory is an outgrowth of the political process and is consistent with another growth fact: a stagnant or slowly growing economy is often coincident with a rapidly growing population. 7The correlation of average annual population growth and per capita real GDP growth during the 1960-85period is -.215. Although the model features an exact correspondencebetweenthe rate of population growth (or fertility) and the proportion of the population that is school-aged,one does not exist in practice due to the complicating factors of mortality and immigration. It should be noted that in the same data set, the correlation between the relative size of the school-aged population and per capita growth is slightly stronger at -.357. 15 The existenceof large cross country differencesin population growth suggestsa role for this theory in explaining the wide dispersion in per capita income growth. Its ability in this regard rests largely on the usefulness of majority rule (or weighted) voting as a descriptor of collective choice throughout the world. Since examplesof nondemocratic societiesabound, there is reasonfor suspicion. However, there is no reason to suspect, a priori, that nondemocratic countries tend to favor the interests of the old with respect to the single issue of public education. A formal investigation of these largely institutional relationships is beyond the scope of the current paper. Because stagnation is coincident with high population growth in this model, it bears similarities to models with endogenous fertility that display multiple steady states one which is associated with a "development trap" of high population growth and low economic growth (as in Becker, Murphy, and Tamura (1991)). Unlike the results of Becker, et. al., a continuum of long-run growth rates are possible in the present model. The two explanations do, however, complementeach other. Although the present model does not require a direct appeal to "history" or "luck" in determining an economy's equilibrium growth state -it is instead a function of population growth mediated by a political process -it makes no attempt to explain population growth. A simultaneous determination of demographics, public education policy and growth is the subject of future research. 16 .u" +u' Appendix Proof of Proposition 1 There are two cases. First, consider the case where the first order condition (5) holds with inequality. In this case, 1/J(7') = 0 in a neighborhood of 7' so that 1/Jis constant. That is, 1/J'(7') = o. Next, considerthe first order condition (5) holding with equality evaluated at the preferred tax rate 'l/J(7-') u' (1 -'I/J (r')) -{3 (1 7') r' (1/J(7')) u' ((1 -7') r (1/J(7'))) Differentiating this expressionwith respectto 7' and noting that r" (1 7/J(7')) 7/J'(7') -,B {-r' 7') r' ('ljJ(7')) Ull ((1 +(1 (7/J(7')) u' ((1- r') r (1/J(r'))) [(1 o. 0 yields 7') r (7/J(7'))) r') r' ('1/1 (r')) '1/1'(r') -r ('1/1 ( r'))] 0 which can be rearrangedas ((1 -T') r ('ljJ(T')))] Note that the expressionin curly brackets on the left hand side of the equation above is negative by the strict concavity of u. Thus, since r' > 0, sign (1/;' (r')) sign ((1 -T') r (1/J(T')) u" ((1 -7') Alternatively, 1/J'(7') < > ~ 0 (1 -7') r (1/;(7')) u" ((1 17 r (1/J(T'))) + u' ((1 -T') r (1/J(T')))) or, equivalently, u" ((1 -r') r (1/J(1"')))(1 -r') r (1/J(1"')) < u' ((1- 7') r ('1/1 (7'))) =1 > for any 7' such that first order condition (5) holds with equality. 0 Proof of Proposition 2 The proof follows directly from proposition 1. 0 Proof of Proposition 3 A symmetric political-economic equilibrium exists provided that 'If;crossesthe 45degreeline depicted in figure 1. For uniqueness, it suffices to show that 'If;(r*) -r* = 'If;(7) -7 cannot occur whenever r* # 7, i.e. multiple crossings are not allowed. There are two cases to consider: 0" ~ 1 and 0" > 1. Case (i). Suppose that a :$ 1. In this case, 7/Jis continuous and monotonically nonincreasing in [0,1). For 7/J (0) > 0, there exist a r* > 0 such that 7/J(r*) = r* with 7/J (7) -7 > 0 for 7 < r* and 7/J(7) -7 < 0 for 7 > r* so that a single crossing of the 45-degfee line is guaranteed. For 7/J(0) = 0, r* = 0 with 1p(7) -7 Case (ii). < 0 for 7 > O. Suppose that a > 1. In this case, 7/Jis continuous and monotonically nondecreasing in [0,1). With only these properties, it is possible that 7/J(z) > z for all z E [0,1). To ensure existence and uniqueness, it suffices to show that 7/Jis convex and that 7/J'(r') -+ 00 as r' -+ 1. Beginning with the latter, observe that 1/J'(7') = -(1 + a) 0' (1+-~or > 0, where () = (a{3)-;} (1 -T')~ and a ==A [~] the economy. Differentiating () gives Of = I-a a 18 is the (net) per capita growth rate of with u > 1. Observing that () -t 'l/J' (r') -t 00 as T' -t 1. Convexity 0 and ()' -t 00 as T' -t is easily demonstrated 1, it is also true that by twice differentiating the expression (7) yielding '" (1+a(})(1+a) 'l/J (7") = [2a((}I)2_(1+a(})(}'I] -> 0 (1 + a(})4 provided that 8" < O. Twice differentiating 8 and applying the condition a > 1 gives 0" (7') = - u-1 1 (a{3)-q (1 -7')- l;i:£: ~ < 0 U2 as desired. Thus 1/;" (r) > O. 0 Proof of Proposition 4 Considerthe interior equilibrium tax rate defined by the first-order condition (5): u' (1 -r*) where a = A (~). := Q{3(1 -r*) u' ((1 -r*) (1 + Qr*)) Since the left-hand side is both increasing in T and independent of population growth, the optimal tax rate decreases with an exogenous increase in n if and only if the right hand side decreasesgiven T*. Differentiating the right-hand side with respect to n gives the expression {3(1 -T*) [u' (. ) + au" (.) (1- 7*)7*]~ which is negative if and only if u' ( .) + au" (. ) (1 -T*) T* > O. 19 dn or a< 1 + aT* aT* 0 Proof of Proposition 5 Recall that the net per capita growth rate is A (~) 1/J( r*) implying zero growth if, and only if, r* = 1/J(r*) = o. Equivalently, growth is positive if, and only if, 'I/J(0) > 0, or 1 -(} (0) 'I/J(0) = 1 + A (~) (}(0) > 0 , It follows that 'I/J(0) > 0 is equivalent to the 1- A l+n n f:J] -~ n > 0 ~ -:;-:- < AfJ. 0 20 [11] Lucas, R. E., 1988, "On the Mechanics of Economic Development," Journal of Mon- etary Economics,22, 3-42. [12] Persson T. and, Tabellini, G., 1994, 'is Inequality Harmful for Growth? Theory and Evidence," American Economic Review, 48, 600-621. [13] Rebelo, S., 1991, "Long Run Policy Analysis and Long Run Growth," Journal of Political Economy, 99, 500-521. [14] Saint-Paul, G. and Verdier, T., 1993, "Education, Democracy and Growth," Journal of Development Economics, 42, 399-407, 22