Political Economy of Immigration and Fiscal Sustainability^†

1. Policy Implementation of the Winning Candidate

Once one of the three candidates is elected by plurality voting, the third and final step of the political process is decision making by the winning candidate. The winning candidate implements his preferred policies on taxation and immigration. As there is no mechanism to resolve commitment problems, all voters recognize ex ante that the winning candidate will pursue his optimum after being elected, and all other pledges are considered as 'cheap talk' under any voting equilibrium. Let the optimal policy of the decisive candidate at time t be denoted by a triplet for the winning candidate’s party d ∈ {s, u, o}. Then, solves the following maximization problem:

where Φ_t+1 is the perception of today’s winning candidate of the policy variables in the next period. Obviously there can be multiple different rules of perception Φ_t+1 that satisfy sub-game perfection; thus, the complexity of this optimization problem depends on the multiplicity of the expectation of the policy rule of the winning candidate in the next period. If there are multiple reasonable perceptions on future policies, this period’s winning candidate must then consider multiple possibilities stemming from his decision today.

In order to refine this type of complexity and reduce the number of perceived future policy rules, this paper relies on the Markov-perfect property.10 That is, the perception function Φ_t+1 is assumed to be a function of the state variable for the period t + 1 only without depending on anything else, such as signals to eliminate any types of “sunspot” equilibria. With the introduction of this Markov-perfect property, the winning candidate expects that the next period’s winning candidate will make a decision according to a rule identical to those of individuals living today. Formally, the winning candidate’s problem is expressed as follows

if the winning candidate belongs to the cohort d ∈ {s, u}. Intuitively, this means that today’s winning candidate decides on policy variables with the belief that the winning candidate of the next period will decide on policy variables in exactly the same manner used by this period’s candidates, and, in any equilibrium, this belief is precisely realized.

If the winning candidate is an old retiree, his decision rule is simple given that his utility consists only of the consumption of the lump-sum payment this period. Specifically, the decision when winning the election is given by the solution of the following problem:

where b_t is represented as a function of only the policy variables and the states by solving (4)~(12).

2. Voting Process

One step backward from the policy implementation procedure in section Ⅲ-A-1, now assume that one candidate is selected from each of the three political parties. The winning candidate will be determined by single plurality voting, meaning that the result of voting mainly depends on the sizes of the political parties. However, when analyzing the voting process, the possibility of strategic voting must be considered, as there are three parties. That is, if a voter believes that the candidate from his cohort has no possibility of winning the election, he may have an incentive to vote for the next-best candidate in order to prevent the worst candidate from winning the election.

Also, in this model all of the voters in the same cohort have identical preferences, implying that voters in the same political party vote identically. This setup means we must concentrate on the behavior of a representative voter instead of analyzing the complicated voting equilibrium within a cohort.

Formally, let denote the voting decision of a representative voter in cohort i ∈ {s, u, o}, where stands for the probability that the voter in party i votes for the candidate from party j, and Δ² is a 2-simplex. Then, voting profile represents voting equilibrium if it solves the following problem: 11

for all i ∈ {s, u, o}, where P^j(e_t) is the probability that the candidate from party j wins the election given voting profile e_t. That is, under a state of voting equilibrium, each voter chooses for whom to vote so that the expected value of himself is maximized given the other voter’s decision and the perceived future policy of the winning candidate for the next period.

In order to reduce the possibility of multiple equilibria for a given set of state variables, I only consider the equilibria in which none of the voting decisions is a weakly dominated strategy. In addition, I assume the following tie-breaking rules for the election:

These tie-breaking assumptions, together with the assumption that equilibria with weakly dominated strategies are excluded, guarantee that there exists a unique purestrategy voting equilibrium for each state vector of (μ_t−1, s_t). In all of these equilibria, the members of the largest (not necessarily majority) cohort always vote for the candidate from their own party regardless of the formation of a coalition; thus, a political coalition is potentially formed only by the two smallest parties.12

1. Preferences of Candidates on Policy Variables

After substituting the equilibrium allocation with closed-form solutions, the utility of an individual in each cohort can be represented by the following indirect utility functions with respect to policy variables and states:

where b_t and take the form in (22)~(24) for i ∈ {s, u}. Note that the continuation value is zero for young workers as β = 0 is assumed. The ideal policies of each political party, therefore, can be analyzed with these indirect utility functions. The following proposition displays the old retiree’s preferred policies.

Proposition 1. If an old retiree wins the election, the implemented policy, as a result of his optimum, is characterized as follows:15

Proof. See Appendix B.1.

The implication of this proposition is clear: as old retirees do not have any labor endowment, they are pure beneficiaries of the welfare state. Therefore, if the candidate from the old retirees wins the election, he will choose policy variables such that the tax revenue of the government per capita is maximized. Figure 5 displays the relationship between the tax rate and the lifetime value of the old retirees. The tax rate is decided at 1 / (1 + v), which is the Laffer rate of the economy. Regarding the skill composition of immigrants (σ_t), the old retiree chooses σ_t = 1 if s_t is small enough and σ_t = 0 if s_t large enough, , as production and thus tax revenue is maximized when the proportion of skilled workers in the labor force equals α, the exponent in the Cobb-Douglas production function. To gain a better understanding of this implication, we assume that μ → ∞. Then, the decision of σ_t converges to α, which means that the old retirees want the skill composition of the economy’s workers to be consistent with the Cobb-Douglas exponent and the wage rates of skilled and unskilled labor groups to be equalized. Figure 5 shows that the value of the old retirees is maximized when σ_t is optimal, as in Proposition 1. Finally, the immigration volume is decided at μ_t = μ. This result relies on the assumption of constant-returns-to-scale technology. In other words, as long as the skill composition satisfies the optimality condition, the tax revenue is monotone increasing in the volume of immigration because the production efficiency is robust with regard to the scale of factor inputs.

FIGURE 5.

TAX RATES AND THE INDIRECT UTILITY OF THE OLD RETIREES

The next two propositions show the optimality of the young workers with each skill level:

Proposition 2. If a skilled young worker wins the election, the implemented policy, as a result of his optimum, is characterized as follows:

where

Proof. See Appendix B.2.

Proposition 3. If an unskilled young worker wins the election, the implemented policy, as a result of his optimum, is characterized as follows:

where

Proof. The proof is omitted as it is symmetric to the proof of Proposition 2. Proof is provided simply by substituting s_t with 1 − s_t, σ_t with 1 − σ_t, and α with 1 − α in proposition 2.

The main implication of Propositions 2 and 3 is that young workers consider not only the lump-sum transfer but also their labor income. For example, when s_t is sufficiently small such that there are only a small number of skilled young workers in the economy, the wage rate of the skilled labor is so high that skilled young workers become net contributors to the welfare state, meaning that they prefer a zero tax rate. The upper panel in Figure 6 shows the utility of the skilled young workers for different tax rates and σ_t, the skill composition of the immigrants, when their optimal tax rate equals zero. Given a zero tax rate, the transfers also equal zero; thus, the utility of the skilled young workers depends only on their labor income. Because the wage rate for skilled labor is increasing in the input of unskilled labor due to skill complementarity, skilled workers prefer as many unskilled immigrants as possible.

The lower panel in Figure 6 displays the opposite case, i.e., where s_t is large enough. In this case, skilled workers now become net beneficiaries of the welfare state because there are already excessively many skilled workers such that the equilibrium wage for skilled labor is too small, meaning that skilled workers prefer a strictly positive tax rate . Nevertheless, their ideal tax rate is still smaller than the Laffer rate because they are in any case taxpayers, while the old retirees are not. In this case, how the skilled young workers select σ_t depends on two factors. First, their wage rate increases as more unskilled labor enters the economy, as in the previous case. Second, the direction of the changes of the lump-sum transfer depends on the pre-immigration skill composition of the economy. Specifically, the transfer increases in σ_t if s_t is relatively small and decreases if s_t is large. Regardless of the size of s_t, however, it can be shown that the wage effect dominates the transfer effect. Thus, skilled workers choose to approve of unskilled immigrants only.

FIGURE 6.

TAX RATES AND THE INDIRECT UTILITY OF THE SKILLED YOUNG WORKERS

The optimum level of unskilled young workers is symmetric to that of skilled workers, as shown in Figure 7. They set a zero tax rate when s_t is so large that the unskilled workers are net contributors to the welfare state, and they prefer a positive tax rate when s_t is small and they are net beneficiaries of the welfare state. In either case, unskilled workers prefer the maximum number of skilled immigrants.

FIGURE 7.

TAX RATES AND THE INDIRECT UTILITY OF THE UNSKILLED YOUNG WORKERS

In summary, old retirees want to maximize the government’s tax revenue by levying a high tax rate (at the Laffer rate) and balancing the skill composition of workers for efficient production. On the other hand, young workers prefer a lower (even zero) tax rate and want all immigrants to have opposite skills to maximize their disposable labor income.

2. Political Equilibria

Essentially, the result of the election depends on the relative sizes of the three political parties, as described in Section Ⅲ-B, while there is no guarantee that the largest party always wins the election given that the two smallest parties potentially form a coalition when the largest party’s preferred policy is believed to be the worst for either of the smaller groups. Therefore, not only the relative size of the electorate but also the preference of each party with regard to the preferred policies of one another will determine the result of the election.

The pattern of political equilibria depends on the relative values of the parameters. For instance, a simple case is when (the size of the native old population exceeds that of the native young population), in which the candidate of the old retirees’ party always wins the election by a majority. While equilibria can be represented in a closed form in this non-forward-looking case, I will describe only one form of equilibrium with interesting implications rather than considering every possible case of parameter values and states. Specifically, the analysis concentrates on the case in which

In this case, the patterns of political equilibrium can be described by the range of s_t, as follows:

Case I.

In this case, the population of native unskilled young workers constitutes the majority, meaning that the winning candidate’s policy decision maximizes the lifetime value for the unskilled young. Because s_t is so small in this case that the unskilled young workers become net beneficiaries of the welfare state, the choice of the winner’s policy variables is , where is defined as in Proposition 3. Whether a political coalition arises is immaterial as it cannot take the majority.

Case II.

In this case, the party of the unskilled young workers is still the largest but does not secure a majority. However, the smallest parties (the skilled young workers and the old retirees) do not have any incentive to form a coalition because they perceive that the policies of the other party are the worst. For the old retirees, the preferred policy of the skilled young includes a zero tax rate, which is the worst for the old retirees as it leads to a zero welfare state. For the skilled young workers, the policies of σ_t for both the unskilled young workers and the old retirees are equally the worst because both prefer σ_t = 1, while the old retirees even prefer a higher tax rate. Therefore, for both the old retirees and the skilled young workers, it is the next-best choice to let the unskilled workers win the election, thereby forgoing the possibility of a political coalition. As a result, the unskilled young workers' party wins the election and the winner implements according to Proposition 3.

Case III.

In this case, the population of old retirees is the largest, the unskilled workers are the second, and the skilled workers form the smallest group. Therefore, it is important as to whether the skilled and unskilled workers have an incentive to form a coalition with each other. It is clear that the skilled workers have an incentive to vote strategically for the unskilled candidate. The preferred tax rate of the unskilled workers is τ^u ∈ (0, 1 / (1 + v)), which is advantageous for the skilled workers as it levies fewer taxes than the old retirees would levy. Both the old retirees and the unskilled workers prefer the maximum number of skilled immigrants. Therefore, the skilled workers are willing to vote for the political party of the unskilled workers. Because the population of unskilled workers exceeds that of skilled workers, the coalition is formed such that the unskilled candidate represents the overall coalition. Therefore, the unskilled workers win the election.16 The policy implementation becomes (τ_t, μ_t, σ_t) = (τ^u, μ, 1) according to Proposition 3.

Case IV.

In this case, the population size is still the order of the old, the unskilled, and the skilled. The only difference from the previous case is that now the unskilled workers are net contributors and thus prefer zero taxes, as also wanted by the skilled workers. Therefore, the coalition is formed so that the unskilled party wins the election, and the policy of (τ_t, μ_t, σ_t) = (0, μ, 1) is implemented according to Proposition 3.

Case V.

In this case, the old retirees are the largest group, the skilled are the second-largest, and the unskilled are the smallest in size. Also, because both skilled and unskilled workers are net contributors to the welfare state, both groups prefer a zero tax rate; thus, the old retirees equally dislike both young candidates and want neither to win. The skilled workers obviously prefer the unskilled workers’ policy profile to that of the old retirees. Therefore, the important problem is whether the unskilled prefer the skilled workers’ ideal policy to the old retirees’. If the unskilled workers prefer skilled workers, the skilled candidate would then represent the coalition. Otherwise, the unskilled workers would have no reason to vote for the skilled workers, meaning that the skilled workers strategically vote for the unskilled workers' candidate, and the unskilled candidate will represent the coalition.

For the unskilled workers, the preferred policies of the other two parties have the following trade-offs: Though the skilled workers’ preferred policy, denoted by (τ_t, μ_t, σ_t) = (0, μ, 0), is preferable because it levies zero taxes, it will allow many unskilled immigrants in and make the competition in the unskilled labor market tighter. Though the old retirees’ preferred policy, denoted by (τ_t, μ_t, σ_t) = (1 / (1 + v), μ, 0), is beneficial because it will allow in a maximum number of skilled immigrants, it levies high taxes on labor income. Therefore, the problem is whether the unskilled workers can bear the high tax rate (at the Laffer rate) preferred by the old retirees in order to allow as many skilled immigrants to enter as possible. Formally, this trade-off is characterized by the threshold tax rate, denoted by 𝜏̃(μ_t−1, s_t) and determined implicitly by solving for the following equation:

V^u(μ_t−1, s_t ; τ_t = 𝜏̃, μ_t = μ_t, σ_t = 1) = V^u(μ_t−1, s_t ; τ_t = 0, μ_t = μ_t, σ_t = 0), where the left-hand side is the utility of the unskilled workers when the tax rate is at 𝜏̃ and the maximum number of skilled immigrants are approved, and the right-hand side is the utility of the unskilled workers when the skilled young workers’ candidate wins the election.17 In other words, 𝜏̃ is the maximum tax rate that the unskilled young workers can bear in return for approving the maximum number of skilled immigrants. If 𝜏̃ < 1 / (1 + v), then the tax rate to be levied by the old retirees is excessively burdensome to the unskilled, meaning that the unskilled will strategically vote for the skilled workers’ party. Otherwise, if 𝜏̃ ≥ 1 / (1 + v), the unskilled workers then prefer the old retirees’ policy at least as much as the skilled workers’; thus, a coalition is formed such that the unskilled candidate represents the overall coalition and wins the election.

As a result, the skilled young workers' candidate wins the election with the support of the unskilled party if 𝜏̃ < 1 / (1 + v), which implements (τ_t, μ_t, σ_t) = (0, μ, 0) according to Proposition 2. If 𝜏̃ ≥ 1 / (1 + v), then the unskilled workers win the election with the support of the skilled workers and the policy implementation is (τ_t, μ_t, σ_t) = (0, μ, 1) via Proposition 3.

Case VI.

In this case, the order of electorate size is as follows: skilled, the old retirees, and then the unskilled. As in the previous case, the old retirees do not have an incentive for strategic voting because all of the young workers prefer a zero tax rate. Therefore, the problem is whether the unskilled are willing to vote strategically for the old candidate or not, and this decision depends on the level of the unskilled workers’ threshold tax rate 𝜏̃. If 𝜏̃ ≤ 1 / (1 + v), he unskilled do not participate in the coalition with the old retirees and the skilled party wins; thus , the policy implementation will be (τ_t, μ_t, σ_t) = (0, μ, 0) according to Proposition 2. If 𝜏̃ > 1 / (1 + v), then the old retirees win the election with the support of the unskilled workers, and the resulting policy implementation is given by (τ_t, μ_t, σ_t) = (1 / 1 + v), μ, 1) according to Proposition 1.

Case VII. α − (1 − α)μ < s_t ≤ α + αμ

In this case, the order of the electorate size is identical to that in the previous case (the skilled, the old retirees, and then the unskilled). The difference is that now s_t is so large that the old retirees prefer a “balanced” skill composition of immigrants at instead of allowing skilled immigrants only. This makes the unskilled workers less favorable to the old retirees, thereby decreasing the threshold tax rate for strategic voting. Formally, the new threshold tax, denoted by , is determined implicitly by the following equation:

If , the unskilled do not participate in the coalition with the old retirees and the skilled party wins; thus, the policy implementation will be (τ_t, μ_t, σ_t) = (0, μ, 0) according to Proposition 2. , then the old retirees win the election with the support of the unskilled workers; thus, is implemented via Proposition 1.

Case VIII.

In this case, the electorate size is identical to that in the previous case (the skilled, the old retirees, and then the unskilled). However, now s_t is so high that the old retirees want to implement σ_t = 0. Therefore, neither the old retirees nor the unskilled workers have an incentive to support each other. As a result, no coalition is formed and the skilled candidate wins the election as the largest party. The policy of (τ_t, μ_t, σ_t) = (0, μ, 0) is implemented according to Proposition 2.

Case IX.

In this case, the skilled young workers’ party constitutes the majority of the electorate. The skilled young candidate wins the election and implements (τ_t, μ_t, σ_t) = (0, μ, 0) via Proposition 2.

The nine cases above describe the patterns of political equilibria depending on the skill composition and the age composition of the economy. In short, a highly unbalanced skill composition makes the larger skill group more likely win the election, and the old retirees are likely to win when the young-to-old ratio is small enough and the skill composition of the young voters is relatively balanced. The forward-looking case will be shown to exhibit a similar pattern of equilibria in the next section.