Equilibrium Tax Rates under Ex-ante Heterogeneity and Income-dependent Voting

A.1 Steady-State Equilibrium

The distribution of households, μ(a, x, ψ), is time-invariant in the steady state, as are factor prices. We modify the algorithm suggested by Ríos-Rull (1999) in finding a time-invariant distribution μ. Computing the steady-state equilibrium amounts to finding the value functions, associated decision rules, and time-invariant measure of households. We simultaneously search for (i) the discount factor β that clears the capital market at the given annual real interest rate 4%; (ii) the disutility parameter B to match the average hours worked, 0.323; and (iii) the standard deviation of permanent productivity, σ _ψ , that matches the before-tax income Gini coefficient, 0.5. The current income tax rate τ ₀ is chosen to match the after-tax Gini coefficient in the data.^[15] The details are as follows:

1. Choose the grid points for asset holdings (a), ex-ante productivity (ψ), and an idiosyncratic productivity shock (x). The number of grids is denoted by N _a , N _ψ and N _x , respectively. We use N _a = 668, N _ψ = 5, and N _x = 15. This yields N _z = N _ψ × N _x = 75 for the number of grids for idiosyncratic productivity. The asset holdings, a, are in the range of [0, 90]. The grid points of assets are not equally spaced. We assign more grids in the lower asset range to better approximate the savings decisions of households near the borrowing constraint.

2. Pick initial values of β, B, and σ _ψ . Note that the variance of idiosyncratic productivity, σ _η , and persistence, ρ _x , are borrowed from Floden and Linde (2001). We construct five ability groups ψ _g (where the lower and upper bounds are set to ±2σ _ψ ): −1.42σ _ψ , −0.55σ _ψ , 0, 0.55σ _ψ and 1.42 σ _ψ , respectively. For the idiosyncratic productivity shock, we construct 5 (N _x by 1) vectors around each ψ _g . In a vector, elements (which denote individual productivity, ln z _j ) are equally spaced on the interval [ ψ g − 3 σ η / 1 − ρ x 2 , ψ g + 3 σ η / 1 − ρ x 2 ]. Then, we approximate the transition matrix of the idiosyncratic productivity using Tauchen’s (1986) algorithm.

3. Start with an initial amount of government transfers T. Given β, B, σ _ψ , and τ, we solve the individual value functions V at each grid of an individual state. In this step, we also obtain the optimal decision rules for asset holdings a′(a _i , x _j , ψ _g ) and labor supply h(a _i , x _j , ψ _g ). This involves the following procedure:

   1. Initialize the value functions V ₀(a _i , x _j , ψ _g ) for all i = 1, 2, …, N _a , j = 1, 2, …, N _x , and g = 1, 2, …, N _ψ .

   2. Update the value functions by evaluating the discretized versions:

      V 1 a i , x j , ψ g = max u 1 − τ 0 w h a i , x j , ψ g ψ x j + r a i + a i + T − a ′ , h a i , x j , ψ g + β ∑ j ′ = 1 N x V 0 a ′ , x j ′ , ψ g × π x x j ′ | x j , ψ g ,

      where π _x (x _j′|x _j ) is the transition probability of x, which is approximated using Tauchen’s algorithm.

   3. If V ₁ and V ₀ are close enough for all grid points, then we have found the value functions. Otherwise, set V ₀ = V ₁, and go back to step 3(b).

4. Using a′(a _i , x _j , ψ _g ) and π _x (x _j′, x _j ) obtained from step 3, we obtain the time-invariant measures μ*(a _i , x _j , ψ _g ) as follows:

   1. Initialize the measure μ ₀(a _i , x _j , ψ _g ).

   2. Update the measure by evaluating the (discretized version) law of motion for each ψ _g :

      μ 1 ( a i ′ , x j ′ , ψ g ) = ∑ i = 1 N a ∑ j = 1 N x 1 a i ′ = a ′ ( a i , x j , ψ g ) μ 0 ( a i , x j , ψ g ) π x ( x j ′ | x j ) .

   3. If μ ₁ and μ ₀ are close enough in all grid points, then we have found the time-invariant measure. Otherwise, replace μ ₀ with μ ₁ and go back to step 4(b).

5. Using decision rules and invariant measures, check the balance of the government budget. Total tax revenues are:

   Rev = ∫ a , x , ψ τ 0 ( w ψ x h + r a ) d μ ( a , x , ψ ) .

   If Rev is close enough to T, then we have obtained the amount of government transfers. Otherwise, choose a new T and go back to step 3.

6. We calculate the real interest rate, Gini coefficient, individual hours worked using μ* and decision rules. If the calculated real interest rate, average hours worked, and before-tax Gini coefficient are close to the assumed ones, we have found the steady state. Otherwise, we choose a new β, B, and σ _ψ , and go back to step 2.

7. Computing the steady-state equilibrium in other environments requires slight modifications of the benchmark algorithm.

   1. (Economy without ex-ante heterogeneity) The discount factor (β) and disutility parameter (B) are fixed at those values in the benchmark. By construction, the standard deviation of permanent productivity (σ _ψ ), is set to 0. We now search for the standard deviation of idiosyncratic productivity (σ _η ) that matches the before-tax income Gini coefficient. The other procedures are identical to those in the benchmark.

   2. (Economy with a larger ex-ante heterogeneity) The discount factor (β) and disutility parameter (B) are also fixed at those values in the benchmark. We set the ex-ante heterogeneity ratio ( σ ψ 2 / σ z 2 ) to 0.75. We search for the two parameters (the standard deviations of the permanent component ( σ ψ 2 ) and the stochastic component (σ _η )) that match the two moments (the before-tax Gini coefficient (0.5) and the assumed ratio of ex-ante heterogeneity (0.75)). The other procedures are identical to those in the benchmark.

A.2 Politico-economic Recursive Equilibrium

We use the one-time median voting equilibrium proposed by Corbae, D’Erasmo, and Kuruscu (2009). We compute all value functions along the transition from the current steady state to a new steady state. Computing the equilibrium paths during the transition amounts to finding the value functions, associated decision rules, and the distribution of households in each period. The details are as follows:

1. Compute the initial steady state under the current tax rate (τ ₀). Use the algorithm to find the initial steady-state equilibrium.

2. Construct the candidate tax rates (including the current tax rate) in increments of 1 percentage point ({τ _l , …, τ ₀ − 1%, τ ₀, τ ₀ + 1%, …, τ _h }), where τ _l and τ _h are lower and upper bound tax rates. Choose a new tax rate (τ) among candidates and compute the transition path as follows:

   1. Compute the final steady state under the new tax rate. Use the algorithm for the steady-state equilibrium above (step 3 to step 6).

   2. The economy starts from the initial steady state at time 1 and the economy is in the new steady state at period P. That is, the transition is completed after P − 1 periods. Choose a P big enough so that the transition path is unaltered by increasing P.

   3. Guess the capital-labor ratios during the transition, { K t / L t } t = 2 P − 1 and compute the associated prices { r t , w t } t = 2 P − 1 .

   4. Guess the path of government transfers { T t } t = 2 P − 1 during the transition. Note that the amounts of government transfers are different across the periods, since the decision rules and measures are different. Going backward, compute the value functions and policy functions for all transition periods by using V _P (⋅) from the final steady state. Using the initial steady-state distribution μ ₁ and the decision rules, find the measures of all periods { μ t } t = 2 P − 1 .

   5. Based on the decision rules and measures, compute the aggregate variables and total tax revenues. If the total tax revenue is close to the assumed transfers, we obtain the balanced-budget amount of transfers. Otherwise, choose a new path of government transfers and go back to 2(d).

   6. Compute the paths of aggregated capital and effective labor and compare them with the assumed paths. If they are close enough in each period, we found the equilibrium transition paths. Otherwise, update { K t / L t } t = 2 P − 1 and go back to 2(c).

   7. Choose a different tax rate and go back to 2(a) until we find all transition paths and corresponding value functions of all tax rates considered.

3. Find the most preferred tax rate (τ ⁱ ) for each individual i. Note that individuals differ in permanent productivity (ψ), idiosyncratic productivity shock (x) and asset holdings (a). If individual values under a certain tax reform (τ ⁱ ) are higher than the values under any other tax rates, i.e. V(a, x, ψ|τ ⁱ ; τ ₀) > V(a, x, ψ|τ; τ ₀) for all τ (≠τ ⁱ ), then this tax reform is most preferred by that individual.

4. For each tax rate, compute the fraction of individuals who prefer that tax rate the most. Then, we obtain the marginal distribution of most preferred tax rates. The cumulative distribution of households whose most preferred tax rates are the same or lower than each tax rate is generated by summing up the marginal distribution from the lowest tax rate (τ _l ). Similarly, the cumulative distribution of households whose most preferred tax rates are the same or higher is generated by summing up the marginal distribution from the highest tax rate (τ _h ). Both cumulative distributions in the politico-economic recursive equilibrium (τ ⁱ ) are above 50%. In other words, a majority prefer the tax rate τ ⁱ to any tax rate.

A.3 Voting Turnout Profile

As explained in Section 3, we assume that the turnout rate TR(⋅) is a log-linear function in labor productivity z = x ⋅ ψ:

where ln T R ̄ and ω represent the intercept and slope of ln TR(⋅), respectively. We also impose a restriction that the turnout rate should not exceed 100%. We search for two parameters (ω and ln T R ̄ ) that minimize the difference of the income quintile-turnout rates profile (5 moments) between the model and the data.

1. Compute the initial steady state under the current tax rate (τ ₀). Use the algorithm for the steady-state equilibrium.

2. Pick initial values of ω and ln T R ̄ . Compute the individual turnout rates and find the sum of squared errors of income quintile-turnout rates profile between the model and the data. Impose a penalty function when an individual turnout rate exceeds 100% so that these parameters cannot be selected. Pick other values of ω and ln T R ̄ and repeat the algorithm. Using a simplex method, we find values of ω and ln T R ̄ to minimize the errors.