# Lecture 3 The dynamics of human capital

```modifeid November 14, 2002
Lecture 3 The democratic dynamics of human capital
A. PUNE dynamics
We have observed that, if there are constant returns to scale (b+c=1) then , under
laissez-faire the distribution of human capital remains constant over time, except for a
multiplicative growth factor. On the other hand, if at every period educational finance is
determined by a political equilibrium (PUNE), then, regardless which party wins the
election, the coefficient of variation of the distribution of human capital is strictly
monotone decreasing. Our task, now, is to study whether that coefficient of variation
tends to some positive number, or to zero. In the latter case, we say that wages approach
equality in the long-run.
It is convenient to normalize the distributions of human capital that occur over
time to have constant mean. Thus, if Ft is the probability distribution of human capital at
date t, t=0,1,…., and its mean is t define the normalized distributions
t
t 
Fˆ (h)  F ( t h) ,

0
(3.1)
which all have mean 0. This transformation does not affect the coefficients of variation
of the distributions, so we will now study the dynamics of the coefficients of variation of
t
the sequence {Fˆ } . Denote the coefficient of variation of Fˆ t by t.
Our first observation is:
Proposition 3.1 Let b  c  1. Then
(a) the distribution function Fˆ t 1cuts the distribution function Fˆ t once from below.
That is,
t 1
t
t 1
t
(h )(0  h  h  Fˆ (h)  Fˆ (h) and h  h Fˆ (h)  Fˆ (h)).
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(b) Fˆ t 1second-order stochastic dominates Fˆ t
t
(c) The sequence { } is monotone decreasing, and hence converges.
Proof. See appendix.
We have already observed part (c). Part (b) follows from part (a). Part (a) is
proved by again exploiting the fact that both educational investment functions give a
positive investment to families at the lowest level of human capital. See the Appendix
for the proof.
Part (a) of the proposition is illustrated in figure 3.1. What is not clear is whether
these CDFs converge to the one with all its mass at the mean, indicated by the heavy line
in the figure, or if the convergence stops before that.
Consider the manifold of quasi-PUNEs, illustrated again in Figure 3.2. Fix a
pivot type h* at which the probability of victory is positive for both parties. At any
PUNE with h* as the pivot, those who vote Left are predicted to be the interval [0,h*);
1
1 F(h*)  2
hence the probability of Left victory is
, where  is the error term.

2
2
Because the educational investment function is weakly monotone increasing, whichever
party wins, it follows that the mapping of parent’s human capital to child’s human capital
is strictly increasing, and so descendents occupy exactly the same rank in their
distribution of human capital as their ancestors.
It follows that
t
0
Fˆ (St (h*))  F (h*) .
Let St (h*) be the tth descendent of h*.
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Let us therefore fix a sequence of quasi-PUNEs in which, at every date, the pivot
type is the descendent of our given h* from date 0. Then the probabilities of Left victory
will be constant in the sequence.
Our dynamic analysis will investigate two such sequences. One sequence is
denoted A(h*) , where the pivot is St(h*)at date t and the quasi-PUNE lies on the lower
boundary of the manifold. The other sequence, denoted B(h*), again has the pivot as the
tth descendent of h*, but it lies on the upper boundary of the manifold. Recall from
Lecture 2 that the A sequence is one where politics are ideological, in the sense that the
militants are as powerful as they can be in the intra-party bargaining game, and the B
sequence is one in which politics are opportunist.
Our main theorem is:
Theorem 3.1 . Let b+c=1. For any h*&gt;0, the limit CV of the distribution of human
capital for the sequence A(h*) is zero, and the limit CV of distribution of human capital
for the sequence B(h*) is positive.
Thus our claim is that ideological politics produce equality of wages in the longrun, while opportunist politics do not.
The proof is a bit involved, and so I will not attempt to present it here; it is
relegated to the Appendix. Let me, however, attempt to motivate the result.
In the sequence B(h*), both parties play the ideal policy of the pivot voter, which
I graph in Figure 3.3. Because investment is constant for h&gt;h*, it follows that, over
time, the ratio of the human capitals of any two dynasties h1&gt;h2 that are greater than h*
bt
h 
approaches one. This is because their ratio at date t is simply  1  , which approaches
h2 
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one, because b&lt;1.
4
The question is, what happens to the ratios of human capital in
dynasties that are smaller than h*?
Suppose that the educational investment function illustrated in figure 3.3 passed
through the origin, instead of above it. Then, on the interval [0,h*), the investment
function would be exactly the laissez-faire investment function, and hence the
distribution of human capital in that interval would stay the same over time. Thus,
intuitively, the issue is: How rapidly does the vertical intercept of the investment function
approach the origin? (It does approach the origin.) The proof shows that the intercept
approaches the origin ‘fast,’ and so the coefficient of variation of the distribution stays
bounded away from zero.
Indeed, it converges to a positive number.
Of course the distributions of human capital need not converge to a distribution:
there could be constant growth. But the mean-normalized distribution functions do
indeed weakly converge to a distribution function.
Now for the first claim of the theorem, where we study ideological politics. I
have noted that in both the A and B sequences, the probabilities of victory are constant
over time, and they are positive for both parties. It follows that each party wins the
election an infinite number of times in these sequences.
Consider the sequence of total resource bundles that is realized in the stochastic
process that unfolds in A(h*), which we denote
L
R
R
L
X 0 ,X1 ,X 2 ,X 3 ,...
(3.2)
where the sub-script denotes the date and the super-script the victorious party. We have
noted that, whichever party wins, the coefficient of variation strictly decreases at each
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date. Consider a sequence where we replace the Right policies with the laissez-faire
policy; thus:
L
LF
LF
L
X 0 ,X1 ,X 2 ,X 3 ,...
(3.3)
Recall that we are studying the normalized distributions, and if the laissez-faire policy is
enacated at date t then the normalized distribution of human capital is identical at dates t
and t+1. Therefore, the limit coefficient of variation of the sequence (3.3) is at least as
large as the limit CV of the sequence (3.2). But we may now drop the laissez-faire
policies from (3.3), since they leave the distribution of human capital unchanged, and
conclude that the limit CV of (3.3) is the same as the limit CV of the sequence consisting
of all Left policies.
We therefore need only show that, if Left wins every election, the CV approaches
zero. That will prove the result.
Let us recall what the Left policy looks like, from figure 2.2. At each date, there
t
will be a critical type hL : below this type, investments in education are the same for all,
and above h*, investments in education are the same for all. The proof proceeds by
t
*t
showing that the gap between hL and h becomes an arbitrarily small fraction of the
population. Thus, for large t, investment in virtually all children becomes the same, and
this produces convergence to equality of wages. The fraction of the population where
investment is strictly increasing becomes vanishingly small, and the amounts invested in
all types approach the same value.
I do not think the fact that ideological politics engenders equality of human
capital in the long-run is intuitively obvious. It is perhaps not surprising that if Left
ideological politics dominate, we get equality: but equality will occur even if Right wins
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an infinite number of elections -- even if Right wins the election 99% of the time! The
reason is, as I have explained, that even when Right wins, the coefficient of variation of
human capital decreases, and when Left wins, it decreases ‘a lot’. And so, even if Left
only wins 1% of the time, the CV approaches zero.
Let me present the results of a simulation. I begin with F0 being the lognormal
distribution with mean 40 and median 30, an approximation to the US distribution of
income, in units of thousands of dollars, in 1990. I take b=c==0.5. I examine the
sequence of quasi-PUNEs defined, at each date, by taking h* to be the median wage, and
which is located half-way between the upper and lower boundaries of the  manifold.
Each party wins with probability one-half at each date. Our theorem does not tell us what
will happen to the CV in this sequence, because it does not lie on one of the boundaries
of .
In Table 3.1, I present the sequence of CV’s for nine iterations of six generations
each. The reason these iterations differ is that there is a different realization of the
random variable that determines which party wins the election at each date. We notice
that CV’s appear to converge rapidly to zero. This is a conjecture. I was not able to
extend these simulations beyond six generations, because of limits of precision in the
computation.
Figure 3.4 shows what the CDF of the normalized distribution of human capital
looks like after five consecutive Left victories.
It is getting very close to an equal
distribution, with all the mass at one point.
I do not know what happens to the coefficient of variation if we take PUNEs
which lie in the interior of the quasi-PUNE manifold. My conjecture is that if we are
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near the lower boundary, we converge to equality, and if we are near the upper boundary
we do not.
I also do not know exactly where the true PUNEs lie in the manifold. I know the
set of PUNEs is non-empty. It is even conceivable that there are no PUNEs on the upper
boundary of . It is therefore conceivable that in all true PUNEs, the CV converges to
equality. This would be an important result, but it remains an open question.
Theorem 3.1 assumes the case of constant returns to scale. We are not proposing
that returns to scale are indeed constant. If, in reality, they are decreasing, then we
conjecture that convergence to equality will occur much more rapidly with ideological
politics than with opportunist politics or with laissez-faire: that is what our theorem
suggests.
Recall that we studied the constant-returns case because of its sharp attribute
that, under laissez-faire, the distribution of human capital remains unchanged.
It is
easier to study whether or not a sequence of coefficients of variation converges to zero
than to compare the speeds of convergence of different processes.
The final contribution to this section will be a short report on what happens if
returns to scale are increasing -- that is, if b+c&gt;1. I report on a simulation where
b=c=.75, =.5, and the initial distribution is again the lognormal with mean 40 and
median 30. I examine, again, the sequence of quasi-PUNEs which is defined by the point
midway between the top and bottom boundaries of the manifold, at each date, where the
pivot is always the median of the distribution. For two six generation simulations, see
Table 3.2. The coefficient of variation falls, but it is not monotone decreasing: Right
victories sometimes increase inequality. The interesting comparison is with laissez-faire:
at date 3, the CV of human capital is already 3 x 1011. So democracy appears to have a
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radical equalizing effect, in comparison to laissez-faire, when returns to scale are
increasing.
B. Hotelling-Downs dynamics
I now proceed to a comparison of this model of ruthless political competition with
the model of Hotelling-Downs applied to our problem. As you know, the HotellingDowns model only (roughly speaking) has an equilibrium when policies are
unidimensional. So let us suppose that the policy space is the space of affine total
resource functions, that is
X(h)  ah  (1 a), 0  a 1.
Here, 1 a is a constant marginal tax rate, and (1 a) is a lumpsum transfer to all
citizens. As always, consumption and educational investment are divided in proportions
1:c.
The constraint on a assures us that X is non-decreasing and has derivative no
larger than one.
Denote the median of the distribution of human capital by m. The indirect utility
function of types on policies a now becomes
v(a;h)  Log(ah  (1 a)),
which is single-peaked in a. Consequently, since both parties are unremittingly
opportunist, the unique political equilibrium has them both proposing the ideal policy of
the median type, which is
1, if m  
.
a  
0, if m  
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That is, if the median is less than the mean, X(h)=, while if the median is greater than
the mean, then X(h)=h, the laissez-faire policy.
Let us suppose we begin with a distribution in which the median wage is less than
the mean wage. Then the policy X(h)= wins. This generates a distribution of human
t
t
capital at date 1. As long as m   , there will be constant investment in all children,
and if this continues forever, then we converge to equality of wages.
On the other hand, if there is some date at which the median is greater than the
mean, then the laissez-faire distribution is implemented, and the distribution of human
capital remains unchanged. Therefore the median is greater than the mean at the next
date, and so on forever. Thus, if the median is ever greater than the mean, then the
coefficient of variation is constant and positive forever after.
Our problem thus becomes: When is it the case that the median is less than the
mean forever? We have a nice result:
Theorem 3.2 . Let b+c=1. Let F be the date 0 distribution of h, with median m and
mean . Under Hotelling-Downs politics, on the unidimensional policy space, the CV of
the distribution of human capital converges to zero if and only if
log m   log h dF (h).
The proof uses some well-known inequalities. In fact, the arguments are very
similar to those that show the CES production function approaches the Cobb-Douglas
production function as the elasticity of substitution approaches one.
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The condition &quot;log m   log h dF (h)&quot; is stronger than the condition &quot;m  &quot; . (Just
note that log m   log h dF (h)  m  exp(  log h dF(h))  m   h dF (h)  , where the
last implication follows from Jensen’s inequality [for convex functions].) But the
converse direction is generally false. So the critical inequality for the theorem is one that
can be interpreted as strong positive skewness of the distribution F (because ‘m&lt;’ is
commonly called positive skewness).
It so happens that the lognormal distribution with mean 40 and median 30
satisfies precisely the equation
log m   log h dF (h).
So, with that distribution, wages will converge to equality under Hotelling-Downs.
However, if the median were any larger, the limit CV would be positive.
There are two notable differences between Hotelling-Downs politics and what I
will modestly call democratic politics, &agrave; la PUNE. The first is that convergence to
 equality never occurs with democratic opportunist politics, but it sometimes occurs with
the unidimensional opportunist politics of Hotelling-Downs.
The second is that,
whether or not convergence to equality occurs with Hotelling-Downs depends on the
initial distribution; but with democratic politics, if political competition is sufficiently
ideological, there is convergence to equality independent of the initial distribution.
Let me comment on the first difference. Why do we get convergence to equality
with opportunist politics, in some cases, but never with democratic politics? The reason
is that, on the unidimensional policy space, doing well by the median voter means doing
well by the poor. That is, the median voter’s ideal policy, in the case m&lt;, is the best
policy for the poor as well: radical redistribution to the mean. However, with democratic
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opportunist politics, as we saw in figure 3.1, doing well by the pivotal voter does not
mean doing well by the poor. With an infinite dimensional policy space, the pivotal voter
does best for himself by depriving the poor.
Thus, I claim, the Hotelling-Downs model gives us a false picture of the dynamics
of democracy.
Although the PUNE model is not so easy to handle, the analysis shows
that the Hotelling-Downs model is a poor approximation to reality -- if we think of real
politics as being, more or less, no-holds-barred, as our infinite-dimensional policy space
captures. Until a simpler model than PUNE comes along, I think that serious political
analysis requires that we use it.
We can now summarize what we have learned about convergence to equality ,
according to the three possibilities of decreasing, constant, or increasing returns to scale,
and Downsian versus democratic politics:
(Table 3.3 here)
C. Stochastic talent
Until now, we have assumed that the wage formation process is deterministic.
We now relax this assumption, and allow for a random talent element in children. Thus
we assume that the wage of child is given by
h h r ,
b c
where  is a positive random variable with mean one. Let us assume that  is i.i.d for all
families.
If the parent knows the realization of  for her child before she engages in politics,
then nothing about political equilibrium changes from our earlier analysis: the  talent
Graz-Schumpeter Lec 3
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term just appears as an additive constant in parental utility. Equilibrium policies are the
same as before. However, the distribution of human capital will not be the same as in
the earlier model.
Suppose we are in the sequence of equilibria B(h*). Then, in the earlier model,
the human capital of distant descendents of the first ancestor, Eve, depends on the Eve’s
rank in the initial distribution, whereas in the sequence of equilibria A(h*), the long-run
descendent of Eve has human capital that is independent of Eve’s rank. We could say
that in the A distribution, equality of opportunity is achieved in the limit, while in the B
distribution it is not. That property is inherited by the model with random talent.
Distributions of human capital will not tend to equality of wages, because of stochastic
talent: but in the ideological political equilibrium, equality of opportunity is approached,
in the sense that the distant descendant’s human capital becomes independent of his
Eve’s human capital, while equality of opportunity is not approached, if politics are
opportunist.
Of course, we could interpret the random variable  as defining effort, as well as
talent, in which case the equal-opportunity language is more appropriate. For according
to equality of opportunity, it is appropriate for a person’s income to depend on her effort,
but not on her ancestor’s social class.
D. Endogenous Growth
In the proof of theorem 3.1, exogenous growth was permitted, in the sense that we
could assume that the educational technology at date t is defined by
h  h r ,
t
b c
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where t is a time superscript, and the sequence {t} is exogenously given. Exogenous
technical change means that the value of t is independent of political decisions.
Because we have not included capital in the wage function, we must assume that the
influence of capital on marginal productivity, and hence on the wage, is captured in t.
Thus, investment must be uninfluenced by political decisions. This is unrealistic, but
perhaps no more unrealistic than the assumption that labor is inelastically supplied in our
model: investment is also inelastically supplied.
Because we are concentrating on the role of education, however, it is
inappropriate to ignore the effects of education on the technology, and so we should
recognize the possibility of growth that is endogenous in our model, in the sense of the
production function’s being influenced by decisions on educational policy. Thus, we
now consider a modified educational technology given by
h  h r r ,
t
b c
d
(3.4)
where r is the average educational investment in generation t’s children. We may think
of r as influencing the quality of the technology, either in the sense that a higher level of
education produces better R &amp; D, or because a more educated work force is capable of
using a more sophisticated technology, which is therefore built, or both.
I noted earlier, that in our first model, the social and private returns to education
were identical. This is no longer the case with (3.4). Thus, society may have an interest
in investing more in your child than you wish to invest, because of the external effects
your child, among many others, will have on aggregate productivity.
Thus, we can no longer expect that parties will propose to disaggregate the total
resource bundle into consumption and investment just as each family would like. One
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consequence is that we will no longer be able to simplify the analysis by looking at the
reduced policy space T*: we shall have to work on the policy space T, which preserves
the distinction between the consumption and investment policies.
Let us first note that, under a laissez-faire policy, as before, the distribution of
human capital is unchanged from one generation to the next, except for a multiplicative
constant. For if parents decide privately on educational investment, then each
appropriately assumes aggregate investment is fixed and unchangeable by her action, and
so an h parent invests
c
h in her child’s education, and so
1  c
c
 c  c d
h h 
 h r ,
1  c 
b
whence the ratio of human capitals of the sons of h1 and h2 is:
 h  h b c
1
1

 ,
  
  

h
2 
h2 
just as in the first model. So under laissez-faire the CV of distribution of human
converges to zero, stays constant, or explodes, as returns are decreasing, constant, or
increasing, respectively. It is therefore again appropriate for us to study the constantreturns case in this model, because of the clean benchmark it provides.
Suppose c were close to zero, but d were significantly positive. Then there are
small private returns to education, but significant social returns. Political parties, which
represent large coalitions, will be interested in making educational investments.
Individual parents would also want society to invest in education, but they would want
almost all of their family’s total resource bundle devoted to consumption. Indeed, in this
case, we might conjecture that society would invest approximately equally in all children.
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In this case, if b 1 then wages would tend to equality, because the ratio of the human
capitals in any two dynasties would approach one. Such policies, we conjecture, would
be Pareto efficient, in the policy space, and hence both parties would advocate policies of
this form.
So we can surmise that, in the case c&gt;0 and d&gt;0, there will be more of a tendency
to equality of wages than in the case where returns to investment are only private. We
will therefore concentrate on the case of equilibria in sequence B(h*), where politics are
dominated by opportunism. Will we achieve equality of wages in the long-run, even
with opportunistic politics, if d&gt;0 ? If so, then our previous work strongly implies that
any sequence of equilibria with ‘invariant pivot’ will bring about equality of wages.
As I remarked, this problem is more difficult than the first one we studied,
because we no longer can work on the reduced policy space where parties are choosing
only the total-resource-bundle function. We must return to policy space with elements
(,r) . Our trade-off will be to work on subspace of T, namely the set of policies
Tˆ  {(,r) T | 0  1, 0  r1} .
Thus, we now require the consumption and investment functions each to have bounded
derivatives in the interval [0,1], which is not a constraint on T. Of course, all of the
policies in PUNEs in the earlier problem were in fact in Tˆ as well, so this might not
appear to be a serious restriction. Nevertheless, it does simplify the optimizations that we
will have to perform.
Let F be the probability measure of human capital at a given date. It is
convenient to define the function
Graz-Schumpeter Lec 3
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h
Q(h)   xdF(x)  h(1 F(h)) .
0
Note that Q()   and Q is increasing.
We have two theorems that characterize what the PUNEs in the sequence B(h*)
look like, depending on the size of   Q(h*) .
Theorem 3.3 Suppose that:
(A1) (c  d)h*    Q(h*) , and
(A2)
d 1 F(h*)
.

c
F(h*)
Then the investment in all types is a constant given by:
r* (h)  r  (c  d)y,
where y 
  h *Q(h*)
. Consumption is given by:
1  (c  d)
h  h * y, if h  h *
.
 * (h)  
y, if h  h *

If we begin , in the sequence B(h*), with an economy in which (A2) and (A1)
hold, then (A2) holds forever. If (A1) held forever, we would surely converge to
equality of wages, because the same amount is being invested in all children.
Unfortunately, the r.h.s. of (A1) approaches zero over time, and the l.h.s. does not. So
eventually we leave the regime of Theorem 3.1.
We therefore require an analysis of what occurs in the sequence B(h*) when (A1)
no longer holds. We have:
Graz-Schumpeter Lec 3
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Theorem 3.4 Suppose that:
(B1) (c  d)h*    Q(h*) , and
(B2)
d r0  Q(h1 ) 1 F (h*)

,
c
r0  h1 F (h*)  F(h1 )
where (r0,h1) is the solution of the system of equations:
(a) r0    Q(h*),
(b)
1
c
d


(1 F (h1 )).
h *h1 r0  h1 r0  Q(h1 )
Then the PUNE at B(h*) has both parties proposing the policy (r*, *) given by:
r0  h, for 0  h  h1
r * (h)  
 r0  h1 , for h  h1
and

0, for h  h1

 * (h)  h  h1 , for h1  h  h *
 h * h , for h  h *.

1
The policy is illustrated in Figure 3.5.
t
*t
We see that investment is constant for h&gt;h1, and since at every date, h1  h , it
follows that the ratio of wages of any two dynasties with initial human capital larger than
t
h* tends to one. The question is, what happens to the value h1 ? We are able to show:
Theorem 3.5 Let b+c=1. Suppose that for all t=0,1,2,… the time-dated versions of
conditions (B1) and (B2) of theorem 3.4 hold. Then :
t
t
(1) lim F (h1 )  0,
(2) the CV of human capital approaches zero in the dynamic process,
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t
(2) lim h1 
18
(c  d)
*t
lim h and
1 (c  d)
(3) condition (B2) approaches condition (A2) [of Theorem 3.3].
Because of conclusion (1), it follows that, for large t, the same is invested in
virtually all children, and hence wages tend to equality. To be precise, given any two
original Eves with non-zero human capital, there exists a date T, such that, for all t &gt;T,
the same amount is invested in the education of the descendents in these dynasties, and
hence the ratio of their human capitals tends to one. Thus conclusion (2) follows
immediately from (1).
We may therefore conclude that if the ratio
d
is sufficiently large, then wages
c
converge to equality, even with opportunist politics. ‘Sufficiently large’ means, in the
case where (A1) holds, that (A2) is true, and in the case where (B1) holds, that (B2) is
true.
Because of (3), it follows that condition (A2) is essentially what is needed
concerning the ratio d/c.
I next provide a simulation. I begin with the lognormal distribution with mean 40
and median 30, and choose h* to be the median. It is convenient to re-normalize, and set
the median equal to unity, at all dates: this does not affect the coefficients of variation. I
choose   0.75, b  c  0.5, d  0.6. The critical ratio
d
 1.2. At date 0, we are in the
c
regime of Theorem 3.2: both (B1) and (B2) hold. Table 3.1 presents the results of a three
generation simulation:
Graz-Schumpeter Lec 3
19
Table 3.3
The last column presents the ratio at the r.h.s. of condition (B2) at each date; we require
that this ratio be less than or equal to 1.2. We see that the convergence claimed in
conclusion (1) of Theorem 3.3 appears to be occurring rapidly: by date 2, the same
amount is being invested in virtually all children. From condition (2) of that theorem,
t
we know that h1 is converging to approximately 0.452.
Another glimpse of the speed of convergence is provided by looking at the CDFs
of human capital in dates 0 and 2 of the above simulation. They are provided in Figure
3.6. The medians have been normalized to one in these graphs.
(figure 3.6 here)
We see the convergence to equality is rapid. And recall, this is in the PUNE
where politics are most opportunistic, which is when the convergence to equality is least
rapid, in the manifold of quasi-PUNEs.
It is interesting to observe what the ratio
easy to verify that
r(h)
looks like in these solutions. It is
(h)
r(h)
 c for all h. This means that every type would like to
(h)
redistribute the total resource bundle assigned to its family towards consumption and
away from educational investment. This desire will be strongest among the poorest
types. We thus see, in this model, an important role for the publicness of education:
Graz-Schumpeter Lec 3
20
assuming the usual free-rider psychology, it would not be possible to realize these
solutions with private financing of education. Political parties here overcome the freerider problem because they represent large coalitions of citizens. Parties in PUNEs
always propose constrained Pareto efficient solutions.
Indeed, we note that , in this opportunist solution, after-tax income is zero for a
poor section of the population. This is to be interpreted as consumption’s being driven
down to subsistence level.
One might hastily conclude that it is empirically incorrect that every type – even
the rich-- would like to shift the resources dedicated to its family away from education
towards consumption. But it is not clear that this is indeed incorrect. The wealthy send
their children to universities, but pay only a fraction of university tuitions. In the United
States, if a wealthy child attends a public university, then a very large fraction of her
tuition is paid for by taxpayers, and, indeed, by non-wealthy taxpayers. If the child
attends a private university, a fraction of the tuition, sometimes substantial, is paid for by
endowment funds. At many private universities in the United States today, students are
admitted independently of their parents’ ability to pay; when such a student is admitted,
the university finances the short-fall out of endowment income. University attendance
would doubtless fall by a great deal if all expenses had to be privately financed, which is
the same as saying that families would not invest in their children what society invests in
them. This conjecture is consistent with our analysis. And why are relatively poor
families willing to pay taxes to support public universities? Because of the positive
externality the education thus provided bestows on their children, who may not attend the
Graz-Schumpeter Lec 3
21
university, but whose wages will benefit from the education of more privileged. This
too, then, is consistent with our model.
Thus, to repeat this important point, it is perfectly consistent to say that, given the
total resources devoted to her family (after-tax income plus educational investment), each
parent would rather redistribute towards consumption and away from investment, and
that the members of political parties are content, on average, with what their parties
recommend to invest publicly in education.
What happens if the ratio d/c is not as large as the conditions in theorems 3.3 and
3.4 require? We know that if d=0, we definitely do not get convergence to wage
equality—that was our first model in the equilibria B(h*). I have not analyzed the
PUNEs in the intermediate situation, when d is positive but not large relative to c.
Doubtless, when d /c becomes sufficiently small, convergence to equality no longer
occurs with opportunist politics.
E. Conclusion
Let me sum up. We began by analyzing the model where returns to education are
completely private. We showed that, if politics are ideological, then in a sequence of
PUNEs where the probability of each party’s victory remains constant over time,
convergence to equality of human capital occurs, where by that convergence we mean
that the ratio of the levels of human capital in any two dynasties approaches unity. If
politics is very opportunistic, then convergence to equality of human capital never occurs.
These statements are independent of the initial distribution of human capital. In contrast,
we showed that in a unidimensional model, with opportunist politics, convergence to
Graz-Schumpeter Lec 3
22
equality of wages does occur, if and only if the initial distribution of human capital is
strongly positively skewed. That result was given to argue for the value of working on a
large policy space. Large policy spaces exist in reality, and it makes a difference to
model politics thusly.
We then considered an educational technology with endogenous growth, one in
which the general level of education has a positive effect on all wages. One
interpretation is that the sophistication of machines and technique is positively related to
the level of education, and hence, so is labor productivity and hence wages. We showed
that if the ratio d/c is sufficiently large, then, over time, even in the PUNEs with
opportunist politics, we tend to a state in which the same amount is invested in all
children, which induces equality of wages.
All these statements are true for the constant-returns case, when b+c=1. If
decreasing returns are the reality, then we presume that our theorems transform into
statements about the relative speed of convergence to equality of wages. And if there is a
stochastic talent or effort element in children, then we argued that statements about
convergence to equality transform into statements about convergence to a state in which
the tth descendant’s human capital is independent of the human capital of his distant
ancestors.
Graz-Schumpeter Lec 3
23
Ft(h)
t=1
t=0
t=2
0
Figure 3.1
The normalized CDF of human capital at various dates
h
Graz-Schumpeter Lec 3
24
total resource
X(h*)
boundary
where
opportunists
dominate
B(h*)
ˆ

boundary
where
militants
dominate
A(h*)
h*
Figure 3.2 The manifold of quasi-PUNEs
h*
Graz-Schumpeter Lec 3
25
Figure 3.3 The policy played by both parties on the upper boundary of manifold 
r(h)
c
1  c
h
h*
Graz-Schumpeter Lec 3
Table 3.1 Coefficients of variation in six-generation simulations when b=c= 0.5=
26
Graz-Schumpeter Lec 3
27
gen
0
1
2
3
4
5
6
mean
40.
31.4903
25.5745
20.6967
16.9311
13.854
11.3373
median
30.
29.4005
25.8911
21.9691
18.2861
15.1532
12.5166
cvar
0.777778
0.238905
0.133496
0.0552047
0.045739
0.040081
0.0363719
winner
Right
Right
Left
Right
Right
Right
none
gen
0
1
2
3
4
5
6
mean
40.
31.4903
24.4413
19.9764
16.026
13.1141
10.8325
median
30.
29.4005
25.8911
21.7406
18.1286
14.9711
12.3349
cvar
0.777778
0.238905
0.111755
0.0876068
0.0647746
0.0598669
0.0400345
winner
Right
Left
Right
Left
Right
Left
none
gen
0
1
2
3
4
5
6
mean
40.
31.4903
24.4413
19.401
15.6823
12.8274
10.4978
median
30.
29.4005
25.8911
21.7406
18.0099
14.8266
12.1695
cvar
0.777778
0.238905
0.111755
0.0756528
0.0605814
0.0492044
0.0478734
winner
Right
Left
Left
Left
Left
Right
none
gen
0
1
2
3
4
5
6
mean
40.
31.4903
25.5745
20.0601
16.4099
13.3159
10.9721
median
30.
29.4005
25.8911
21.9691
18.2861
15.1532
12.4772
cvar
0.777778
0.238905
0.133496
0.0825377
0.0712898
0.0535628
0.0378218
winner
Right
Right
Left
Right
Left
Left
none
gen
0
1
2
3
4
5
6
mean
40.
30.1078
23.3832
18.6632
15.2701
12.5011
10.3844
median
30.
29.4005
25.3509
21.1867
17.5126
14.4383
11.8592
cvar
0.777778
0.209562
0.104315
0.0740434
0.0672418
0.051507
0.0346534
winner
Left
Left
Left
Right
Left
Left
none
gen
0
1
2
3
4
5
6
mean
40.
30.1078
23.3832
19.1164
15.4042
12.6056
10.4662
median
30.
29.4005
25.3509
21.1867
17.6068
14.5167
11.9462
cvar
0.777778
0.209562
0.104315
0.0841093
0.0640802
0.0599651
0.038533
winner
Left
Left
Right
Left
Right
Left
none
gen
0
1
2
3
4
5
6
mean
40.
30.1078
24.5092
20.0248
16.3769
13.2326
10.8889
median
30.
29.4005
25.3509
21.4133
17.9313
14.9242
12.3143
cvar
0.777778
0.209562
0.127354
0.0967003
0.0802714
0.0565481
0.0382226
winner
Left
Right
Right
Right
Left
Left
none
gen
0
1
2
3
4
5
6
mean
40.
31.4903
24.4413
19.401
15.6823
12.8334
10.5026
median
30.
29.4005
25.8911
21.7406
18.0099
14.8266
12.1876
cvar
0.777778
0.238905
0.111755
0.0756528
0.0605814
0.0574073
0.0552878
winner
Right
Left
Left
Left
Right
Right
none
gen
0
1
2
3
4
5
6
mean
40.
31.4903
24.4413
19.401
15.8728
12.9463
10.5949
median
30.
29.4005
25.8911
21.7406
18.0099
14.8723
12.2253
cvar
0.777778
0.238905
0.111755
0.0756528
0.0675324
0.0520871
0.0500408
winner
Right
Left
Left
Right
Left
Right
none
Graz-Schumpeter Lec 3
28
Table 3.2 Coefficients of variation when b=c= 0.75, =0.5
gen
0
1
2
3
4
5
gen
0
1
2
3
4
5
mean
40.
46.3957
62.1499
92.1395
178.626
497.265
mean
40.
46.3957
62.1499
92.1395
168.301
439.997
median
30.
40.6303
57.4749
94.2662
186.659
525.59
median
30.
40.6303
57.4749
94.2662
186.659
508.532
cvar
0.777778
0.431824
0.397763
0.293562
0.389374
0.50656
cvar
0.777778
0.431824
0.397763
0.293562
0.245359
0.358216
winner
Left
Right
Left
Right
Right
none
winner
Left
Right
Left
Left
Right
none
Graz-Schumpeter Lec 3
b+c &lt; 1
Laissez-faire (LF)
CV 0
29
Democracy
CV  0,
Downsian politics
CV 0
faster than LF
b+c =1
b+c &gt;1
constant CV
CV explodes
Theorem 1
CV decreases to 0.4
(simulation)
Theorem 2
if b&lt;1:
strong skew  CV  0;
otherwise, CV explodes
Table 3.3: Dynamic behavior of coefficient of variation of human capital by regime type


Graz-Schumpeter Lec 3
30
Figure 3.5 The solution of Theorem 3.3

r
1
1
h
h1
Graz-Schumpeter Lec 3
Figure 3.6 CDFs in a dynamic simulation with endogenous growth
31
Graz-Schumpeter Lec 3
32
This is generation 5 afterfiveLeftvictories
CDF
1
0.8
0.6
0.4
0.2
10
20
30
40
50
Figure 3.4 The CDF of human capital after five Left victories
```