Download Growth or Stagnation? The Role of Public Education

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