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 (. Acemoglu and Zilibotti, 2001, QJE)
Multiple Equilibria Approach (. Azariadis and Drazen, 1990, QJE)
Policy Approach (. Barro, 1990, JPE)
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 U.
The Third Generation’s Programming
yields:
The Second Generation’s Programming
yields:
Backward Solution: 1
The First Generation’s Programming
yields:
Backward Solution: 2
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, . the unskillful family in the first period.
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.
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
Then the solution looks like the same as the solution of traditional “Ramsey Problem”, ., 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
Assume the tax rate is exogenously given in the last period. U in the second period chooses her favorite tax rate by
which yields:
Then U in the first period chooses her favorite tax rate by
which yields:
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 (1993, RES).
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
Particularly, we can recursively solve the optimal bequest function as
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, ., 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 simulation.
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 state variables are all interacted.
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 (. 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.
