Politico-Economic
Equilibrium and
Economic Growth
Part 1,Background
? Data show that growth
performances are very different
cross countries,
? There are at least three
approaches to explain the
differences,
? Endowment Approach (e.g,
Acemoglu and Zilibotti,2001,
QJE)
? Multiple Equilibria Approach (e.g,
Azariadis and Drazen,1990,QJE)
Policy Approach (e.g,Barro,
Policy Approach
? Public policies may have huge
impacts on economic growth,
? For example,increasing capital
tax rate drops the interest rate
and then depresses physical
capital accumulation,Higher
wage tax rate has the same
effect on human capital
accumulation,
? The following question comes
naturally,why countries adopt
different public policies (or non-
optimal policy)?
Political Economy
? Public Policy is endogenously
generated from political
economy rather than an abstract
benevolent government,
? A number of theories tell us how
public policies are determined in
a political economy (a survey
see Persson and Tabellini’s
book,2000,MIT),
?The,Median Voter Theorem” is
the most popular one to pin
down the politico-economic
equilibrium,
Part 2,A Basic Model
? Environment,Consider a Three-
Period economy,Only one
generation lives in each period,
Every generation consists of a
skillful and a unskillful family (S
and U henceforth),S in the current
period gives birth to S in the next
period,so as U,
Skillful
Skillful
Skillful
Unskillful
Unskillful
Unskillful
Family
1
Generations
(or periods)
2
3
Preference
? S is altruistic towards her children
(also skillful) in the following
generations,so as U,Therefore,
S and U in the first two
generations would like to leave
bequest to the their own children
in the following generations,
Capital,Technology
and Policy
? Capital,The first generation is
born with certain level of capital,
Their bequests form the capital
held by the second generation,
whose bequest consequently form
the capital held by the last
generation,
? Technology,Assume small open
economy so that interest rate and
wage rate are both exogenously
given,
? Policy Set,Flat-rate capital
income tax is collected to balance
the transfer payment,which is
equally distributed between S and
The Third Generation’s Programming
? ?,.m a x 3 tscu i
? ? 333333 1 krrkwkc iiii ?? ?????
yields,? ?
33333,,?kkcc iii ?
The Second Generation’s Programming
? ? ? ?,.m a x 32 tscucu ii ??
? ? 2222232 1 krrkwkkc iiiii ?? ??????
yields,? ?
322222,,,??kkcc
iii ?
? ?33333,,?kkcc iii ?
Backward Solution,1
? ? ? ? ? ?,.m a x 3221 tscucucu iii ?? ??
? ? 1111121 1 krrkwkkc iiiii ?? ??????
The First Generation’s Programming
? ?322222,,,??kkcc iii ?
? ?33333,,?kkcc iii ?
yields,? ?
3211111,,,,???kkcc
iii ?
Backward Solution,2
? ?322222,,,??kkvv iii ?
? ?33333,,?kkvv iii ?
? ?3211111,,,,???kkvv iii ?
u
t
s
tt kkk ??
Backward Solution,3
? Note that in order to mimic the
competitive equilibrium,we assume
there is no strategic behavior between
S and U in each period,
? The model is closed by,
? Finally,we can get three generations’
indirect utility functions,
Once-and-for-all Voting
for Constant Tax Rate
? Voting Constitution,Voting is
once-and-for-all and only happens
in the first period,Moreover,
agents are restricted to vote for a
constant tax rate over three
periods of time,
? Median Voter,Assume U is the
majority,Under very general
conditions,the preference is
“single-peaked” and then the
Median Voter Theorem can be
applied here,Hence,a constant
capital income tax rate is
determined by the median voter,
i.e,the unskillful family in the first
Median Voter’s Choice
? Homogeneous Case,If the
wage rate of U is equal to S,the
preferred tax rate of U is equal to
the,first best” policy since S and
U are homogeneous,
? Heterogeneous Case,If the
wage rate of U is less than S,the
preferred tax rate of U is
generally larger than the,first
best” policy,Moreover,lower
wage rate of U induces higher
tax rate and then lower capital
returns in the following periods,
This depresses bequest motive
and hence capital accumulation,
? ? ? ????? ? ?,?,?,,m a xa r g 111? kkv
uu?
Implications on
Economic Growth
? Less capital accumulation
implies lower growth rate,Hence,
we can obtain,Inequality Does
Harm to Economic Growth,
? Up to now,inequality affects
growth through the distortion in
factor price,Additional
assumptions about technology
can enlarge the negative impacts
of inequality upon growth,
? For example,positive externality
of aggregate capital (Romer,
1986),strong complementarity
between skill and capital (KORV,
2000,Econometrica),etc,
Part 3,Different
Politico-Economic
Equilibria
? Once-and-for-all voting for
constant tax rate is assumed by
Alesina and Rodrick (1994,QJE),
This assumption substantially
simplifies analysis,However,in
reality we always observe
repeated voting and changing tax
rates,instead of once-and-for-all
voting and constant tax rate,
? Hence,more realistic assumptions
about the voting constitution and
more sophisticated concepts about
the political equilibrium have to be
made,
Once-and-for-all
Voting for Flexible Tax
Rates
? Relax the assumption for
constant tax rate over time
periods,Then the median voter’s
choice becomes
? ? ? ? ? ?321111?3 1 ?,?,?,,m a xa r g 3
1
???? ? kkv uutt
t ?
??
? Then the solution looks like the same
as the solution of traditional,Ramsey
Problem”,i.e.,the sequence of future
tax rates is determined at the
beginning period of time by the
government with commitment power,
? However,this equilibrium has intrinsic
problem since it is time-inconsistent
(Kydland and Prescott,1977,JPE),
Time Inconsistency
? Further relax the restriction of
once-and-for-all voting,we see the
preferred tax rate of U in the
second generation is contingent on
the distribution of capital at the
second period and hence deviates
from the,Ramsey Plan”,which is
just contingent on the initial capital
distribution,
? In the terminology of game theory,
the,Ramsey Plan” is an imperfect
“open loop” equilibrium,We need
to show a perfect,close loop”
equilibrium instead in the case of
repeated voting,which actually
looks very similar as the classical
“bequest game” studied by
Kohlberg (1976,JET),
Repeated Voting,
Backward Solution and Markov
Perfect Equilibrium
? ?32222?2,?,,m a xa r g 2 ??? ? kkv uu?
? Assume the tax rate is exogenously
given in the last period,U in the second
period chooses her favorite tax rate by
which yields,? ?
2222,kk u?? ?
? ?,.,,?,,m a xa r g 321111?1 1 tskkv ii ???? ??
? Then U in the first period chooses her
favorite tax rate by
? ?2222,kk u?? ?
? ?3211122,,?,,???kkkk iuu ?
which yields,? ?
1111,kk u?? ?
Digression,MPE
? Roughly speaking,an Equilibrium
is called,Markov” when it
depends only on current state
variables,Moreover,it is perfect
if it is indeed an equilibrium in
any period of time (more details
see Fudenberg and Tirole,1991,
MIT and 2001,JET),
? Note that MPE might not be the
only possible equilibrium in the
case of repeated voting,A much
larger strategy set is the strategy
contingent on historical events
(Goldman,1980,RES),One
example in macro is the trigger
strategy in Chari and Kehoe
(1990,JPE),
Existence and
Uniqueness,1
? Existence,If the utility function is
increasing,strictly concave and
additively separable,Leininger
(1986,RES) proves the
existence of MPE in a classical
“bequest game” by Generalized
Maximum Theorem (maximizing
upper semi-continuous function
yields upper hemi-continuous
correspondence),
? Uniqueness,If we restrict MPE
to be continuously differentiable
plans,uniqueness can be proved
in the,bequest game” (Kohlberg,
1976,JET) while existence of
MPE is not ensured in this case,
? Unfortunately,existence and
uniqueness of MPE have not yet
been strictly proved even in this
simplest politico-economic model,
? However,basic structures of the
bequest game and repeated voting
model are the same,both of them
focus on intergenerational conflict
(which is absent in the classical
Ramsey growth model with a
representative agent),So I guess
the method to prove the existence
and uniqueness would also be the
same,though there are two
dimensions of backward solution in
the politico-economic model,while
bequest game only considers one
dimension of backward solution,
Existence and
Uniqueness,2
Some Tricks
? There are some tricky assumptions
to avoid the analytical difficulties
when allowing repeated voting,
? Assume agents’ altruism only lasts
for one period and the voting in the
current period is for public policies in
the next period,then there will be no
intergenerational conflict since the
capital and tax rate in the next
generation are both decided by the
current generation,This approach is
taken by Persson and Tabellini
(1994,AER),
? We can also assume agents only
care about the level of bequest,not
the utility of following generations,
This approach is taken by Perotti
Part 4,Contributions
of Krusell et al,(1997)
? They provide an unified framework
to study the politico-economic
equilibrium in the classical Ramsey
growth model,which gives us a
platform to compare the equilibria
given by Alesina and Rodrick
(1994,QJE),Persson and Tabellini
(1994,AER) and Perotti (1993,
RES),as well as the,Ramsey
equilibrium” and Markov perfect
equilibrium that have not yet be
explored before,
? They also provide a numerical
method to solve the local dynamics
of these politico-equilibria
discussed above,
Nontrivial Extension
to the Infinite Horizon
? We expect to have stationary
equilibrium in the infinite horizon
? ?? ? ? ?tutttutt kkkk,,1 ?? ???
? ?? ? ? ? ? ? ? ??? ?? tsstttititttitit kkckkc ????,,,~,,1
? Particularly,we can recursively
solve the optimal bequest function as
? ? ? ?titittiti kkckkc,,,~ ??
? Once again we obtain MPE in the
infinite horizon,Note that here we
have to solve functional equations,
instead of applying backward
solution in the finite horizon model,
Numerical Solution,1
? Now the model looks very close
to a differential game,It is well
known that linear-quadratic
differential game can be easily
simulated since the MPE can be
restricted to (piecewise) linear
functions (not necessarily
continuous everywhere,see
Basar and Olsder,1982,AP),
? Then the following algorithm
seems not hard to figure,apply
“guess and iteration” procedure to
simulate linear bequest functions
and voting functions,which is
very similar to the,undetermined
parameter method” applied to
solve linear rational expectation
models in the 1970s,
? If the utility function is quadratic and
the production function is linear,
one loop,guess and iteration” is
sufficient to solve the MPE,as in
Cohen and Michel (1988,RES),
Moreover,in some special case,we
can even apply,undetermined
parameter method” to get analytical
solution (HRSZ,2003,AER),
? However,difficulty occurs when we
want to approximate utility and
production functions by linear-
quadratic formula,This is because
we do not know any feasible
equilibrium (usually the steady state)
around which Taylor approximation
can be applied,
Numerical Solution,2
? So a natural extension is to add a loop
into the algorithm,i.e.,first guess a steady
state and then check if the initial guess is
correct or not,
Numerical Solution,3
Guess the steady state
Guess the voting function
Solve the bequest function by standard
linear-quadratic control theory
Solve the new voting
function by differentiating
the indirectly utility function
Check if the voting
function get converged,
Check if the steady
state get converged
Results
? Tax rate is higher in repeated
voting than once-and-for-all
voting,while growth rate is high
in once-and-for-all voting than
repeated voting,
? The intuition here is
straightforward,when the voting
is sequential,higher bequest of S
leads to higher tax rates in the
future,So the main part of capital
accumulation (by S) is depressed
by repeated voting,
? Consequently,we can expect if S
is the majority in this economy,
growth rate will be higher in
repeated voting,This is
confirmed by Krusell et al.’s
Part 5,Possible Future
Research Directions
? Note that the identity of the median voter is
actually given by an implicit assumption
that the poor is always the poor,which
avoids an extremely difficulty problem,how
to identify the median voter in the case that
the identities of voters are endogenously
changing over time,
? The difficulty lies in the following aspects,
Firstly,it is hard (sometimes impossible) to
find the median voter when the
identification depends on multiple state
variables,Secondly,these state variables
are endogenously determined,Thirdly,
different expectations on identities of future
median voters affect the current choices
and then the evolution of state variables,
Finally,the identification of the median
voter,the expectation of the identities of
future median voters and the evolution of
Algorithm for Global
Solution
? Krusell et al.’s algorithm is only
suitable for finding local solutions,
Hence,they cannot well explore the
transitional dynamics in the politico-
economic equilibrium (especially
the evolutions of income
distribution and economic growth),
? A new algorithm can be easily
established for global solutions,
Instead of starting with a guess for
the steady state,we first guess a
very general function form (e.g,a
polynomial function that is able to
approximate any function) for the
voting function and then keep the
other,guess and iteration”
procedure the same as before,