Condition 3 is the most important restriction on the equilibrium. If this condition did not hold, then the lobby could change its contribution schedule and improve its welfare.
To establish this result, we can reason as follows. Suppose to obtain a contradiction that this condition does not hold for lobby n = 1 and, instead of q* , some q maximizes (12.12). Denote the difference in the values of (12.12) for n = 1 evaluated at q* and q by Δ > 0 (which is strictly positive by the hypothesis that (12.12) is violated). Then, consider the following contribution schedule for lobby in = 1:
where c1 (q) is a continuous positive function reaching its strict maximum at q =. Basically, this schedule is designed by lobby 1 to induce the politician to choose q instead of q* and, by design, it ensures greater utility for the politician at q than at q*. To see this, suppose that with this new schedule, the politician chooses q*; in this case, the payoff is:
On the other hand, if the politician chooses q, the payoff is:
This immediately shows that for any ε > 0, G () > G (q*). In fact, because c1 (q) is maximized at q = q, the politician strictly prefers the policy q =to any other feasible alternative, when faced with this contribution schedule for any ε > 0.
The change in the welfare of lobby 1 as a result of changing its strategy from1to 1 is:
Because Δ > 0, for small enough ε, the lobby gains from this change, showing that the original allocation could not have been an equilibrium.
The results in Proposition A2 appear far from the simple weighted utility maximization of Chapter 4. We next see that they in fact imply weighted utility maximization. Suppose that the contribution functions, thes, are differentiable. In practice, restricting to differentiable functions might be a simplifying assumption, although Bernheim and Winston (1986) showed that differentiable contribution functions have the desirable property of being robust to mistakes (or perturbations) and to coalition formation.
With differentiability, the politician’s maximization problem in (12.11) implies the following first-order condition for every policy choice, qs, within the vector q:
Similarly, from each lobby’s optimization, (12.12):
These two sets of first-order conditions basically state that both the politician and the lobbies are equating marginal cost to marginal benefits. For the politician, the benefits are increased contributions, whereas the costs are deviations from the social-welfare maximizing objective. For the lobby, the benefits are policies more in line with their interests, whereas the costs are increased contributions.
Combining these two first-order conditions, we obtain:
(12.13)
for all s = 1, 2, .. , S and n = 1, 2, .. , L. Intuitively, at the margin each lobby is willing to pay for a change in policy exactly as much as this policy brings them in terms of additional return. For this reason, the equilibrium of this type of lobbying game with differentiable contribution functions are sometimes referred to as “truthful,” in the sense that the contribution functions reflect the marginal benefits of policies to the lobbies.
The advantage of (12.13) is that it enables us to establish our main objective : the political equilibrium with lobbying can be characterized as a solution to maximizing the following function:
(12.14)
with respect to q.
In other words, the equilibrium maximizes a weighted social-welfare function, with individuals in unorganized groups getting a weight of a and those in organized group receiving a weight of 1 + a. Intuitively, 1/a measures how much money matters in politics and, the more money matters, the more weight groups that can lobby receive. As a → ∞, we converge to the utilitarian social-welfare function. Therefore, we can state the following proposition:
Proposition A.3: The lobbying game with full commitment on the side of the politicians and differentiable contribution functions leads to equilibrium policies that maximize the weighted utilitarian social-welfare function (12.14).
It is also useful to discuss the implications of the lobbying model for the two-class model. In a model with political divisions between the rich and the poor, it may be reasonable to think that, under certain circumstances, the rich are more organized and can form an effective lobby to influence policies. Specifically, we return to our baseline model and assume that the poor are unorganized but the rich are able to form an effective lobby. The results in this subsection imply that the lobbying equilibria are given by maximizing:
which has a first-order condition that can be written, again with complementary slackness, as:
(12.15)
As a → oo, we obtain the case of maximizing the utilitarian social-welfare function. As a → 0, equilibrium policy simply maximizes the utility of the rich agents, who become more influential in democratic politics because of their organized lobby. It is interesting, that in this case, irrespective of the value of a, we have that τ = 0, because even with the utilitarian social-welfare function, there should be no distortionary taxation, as discussed previously.
More interesting, it is possible to combine elements from the probabilistic voting model, where different groups have different amounts of political power, and the lobbying model. For example, we could have that equilibrium policy is given by:
where χns are political-power parameters coming from electoral politics.
Let us apply this model to our two-class model of redistribution and suppose that the rich are organized as a lobby and the poor are not. We can see immediately that there will be redistributive taxation (i.e., τ > 0) if the poor are sufficiently powerful in electoral politics (e.g., χp > χr) so as to offset the effects of the power of the rich that derive from their lobbying activities.
4. Partisan Politics and Political Capture
Another important approach to democratic politics incorporates the idea that political parties have broader objectives than simply winning power. First, political parties may also have ideologies, which would also have an effect on equilibrium policies. Second, an important question is whether certain groups can capture the political agenda (e.g., via lobbying as in the previous section) and how this could be influential in democratic politics. In this section, we introduce ideological parties (i.e., partisan politics) and show how they affect the implications of the Downsian political competition model; we also use this model to discuss issues of political capture. As long as there are no issues of probabilistic voting (i.e., ideological considerations on the side of voters), the predictions of the model of Downsian political competition apply as before, and there are strong forces toward convergence of policies to the preferences of the median voter. However, when there are either ideological considerations on the side of voters as well or problems of commitment on the side of parties, the ideological preferences of parties will also affect equilibrium policy. This provides another channel through which the reduced-form model of the distribution of political power in democracy can arise and another reason why certain groups may influence equilibrium policy more than their voting numbers suggest (i.e., because they are able to capture the agendas of political parties).
4.1 Electoral Competition with Partisan Parties
In the basic Downsian model of political competition, the objective functions of the parties were given by (4.1), which only valued the rent from coming to power. By ideological or partisan parties, we mean those that have preferences over policies as well as whether they come to power.
To formalize these notions, imagine a single dimension of policy, again denoted q from a convex and compact subset Q of R, and let there be two parties A and B. We now replace (4.1) with:
where WA (q) and WB (q) denote the “utility functions” of parties A and B, and R is a rent from being in office, which is assumed to be nonnegative. Parties now maximize their “expected utility,” taking into account the voting behavior of the citizens as summarized by the function P(qA, qB). This expected utility consists of their ideological preferences over policies that are implemented and the rent from coming to office.
To start, we consider the case where P(qA, qB) is given by (12.1): for example, because preferences are single-peaked and there are no ideological considerations on the side of the voters (we later come to probabilistic voting and thus to more smooth versions of (12.1)).
Suppose that the utility functions of the parties are smooth and strictly quasiconcave (i.e., single-peaked), with ideal policies qA and qB; that is,
In other words, ∂WA(qA)/∂qA = 0 and ∂WB(qB)/∂qB = 0.
A model of partisan politics along these lines was first formalized by Wittman (1983), who used it to argue that there may not be policy convergence when parties have ideological biases. We also use this model to discuss issues of capture of the political agenda by one of the groups.
Finally, we assume that both parties choose their policies (i.e., policy platforms) simultaneously. Therefore, the predictions of this model can be summarized by the corresponding Nash equilibrium, in which each party chooses the policy that maximizes its utility given the policy of the other party. Nash equilibrium policy platforms,, satisfy the following conditions:
and, simultaneously:
Intuitively, these conditions state that in a Nash equilibrium, takingas given,should maximize party A’s expected utility. At the same time, it must be true that takingas given,should maximize B’s expected utility.
The problem in characterizing this Nash equilibrium is that the function P(qA, qB), as shown by (12.1), is not differentiable. Nevertheless, it is possible to establish the following proposition, which was first proven by Calvert (1985) and shows that even with partisan politics, there is policy convergence; it is typically to the most preferred point of the median voter:
Proposition A.4 (Policy Convergence with Partisan Politics): Consider the partisan-politics model described above, with ideal points of the two parties qAand qB, and the ideal point of the median voter qM. Suppose also that the probability of party A winning the election is given by P(qA, qB), as in (12.1). Then:
• If R > 0, or if qA ≥ qM ≥ qB, or if qB≥ qM≥ qA, the unique equilibrium involves convergence of both parties to the median (i.e., qA = qB = qM), and each party wins the election with probability ½ .
• If, on the other hand, R = 0 and qAand qBare both to the left or to the right of qM, there is no convergence to the median. In particular, when VM(qA) > VM(qB), the equilibrium policy is qAand when VM(qA) < Vm(qB), the equilibrium policy is qB.
Therefore, the basic result is that although there can be exceptions when there are no rents from coming to office and both parties have the same type of ideological bias, there are strong forces toward policy convergence. As the following discussion illustrates, the source of these powerful forces is (12.1), which implies that the policy that comes closer to the median voter’s preferences will win relative to another policy.
Proposition A.4 is relatively straightforward to prove and here we simply provide an outline and the basic intuition. Start with the first case in which the preferences of the median voter are intermediate with respect to the ideal points of the two parties. Consider first the situation in which qA = qM≠ qB. Then, we have that P(qA, qB) = 1, and party A is winning for sure. The utility of party B is given by WB(qM). Now imagine a deviation by party B to qB = qM. We have that P(qA, qB) = ½ , so the utility of party B changes to R/2 + WB(qM) > WB(qM); hence, the deviation is profitable, and qA = qM ≠ qB cannot be an equilibrium. (In the case in which R = 0, the argument is different, and now party A can change its policy to something slightly away from q M toward its ideal point qA, still win the election, and implement a policy closer to its preferences.)
Similarly, consider a situation in which qA≠ qM≠ qB and suppose without loss of any generality that qA > qM > qB and VM(qA) > VM(qB), so that we again have P(qA, qB) = 1. It is clear that we must have qA≥ qM; otherwise, party A could find a policy q‘A such that VMq’A > and q’A≥ q M preferable to any qA E(qM, qB). But then party B is obtaining utility WB(qA) and by changing its policy to qB = qM, it obtains utility R + WB(qM) if qA > qM and R/2 + WB(qM) if qA = qM. By the fact that qA ≥ qM, both of these are greater than its initial utility, WB(qA); hence, no policy announcements with qA≠ qM≠ qBcan be an equilibrium. Therefore, the equilibrium must have qA = qB= qM- that is, convergence to the median. Intuitively, the median voter’s ideal point is preferable to each party relative to the other party’s ideal point and, moreover, increases their likelihood of coming to power. Therefore, no policy other than the median voter’s ideal point can ever be implemented in equilibrium.
Next, we consider the case in which q B> q A> qM (other configurations give analogous results). Now, suppose that we have qA = qA. What should party B do? Clearly, any policy qB > qA loses the election. On the other hand, qB= qAwins the election with probability ½ and is preferable. But, in fact, party B can do better. It can set qB= qA — ε, which is closer to the median voter’s preferences and, by the fact that voters’ preferences are single-peaked, this is preferable to q Aand therefore wins the election for party B. Although this policy is worse for party B than qA (because qB > qA), for ε small enough, the difference is minuscule, whereas the gain in terms of the rent from coming to power is first-order. This argument only breaks down when R = 0 and, in this case, the best that party B can offer is qB= qA (or any other policy qB> qA for that matter because it does not care about coming to power; in either case, qA is the equilibrium policy).
Therefore, the policy convergence to the median is a rather strong force that demonstrates that the assumption about objectives of parties in the Downsian model is not as restrictive as it may first appear. However, there can be exceptions, especially when rents from coming to power are nonexistent.
4.2 Electoral Competition with Partisan Parties and Probabilistic Voting
Nevertheless, these results depend crucially on the form of the P(qA, qB) function, which created strong returns to being closer to the most preferred point of the median voter. In the previous discussion we learned that in the presence of ideological considerations on the side of the voters, P(qA, qB) can become a continuous function. If that is the case, then policy convergence breaks down. To understand this, suppose that P(qA, qB) is a continuous and differentiable function and suppose that it reaches its maximum for each party at qM (i.e., being closer to the median voter’s preferences is still beneficial in terms of the probability of being elected - the fact that we make this point which maximizes winning probabilities the median voter’s ideal point is simply a normalization without any consequences). In that case, the Nash equilibrium of the policy competition game between the two parties is a pair of policies, such that the following first-order conditions hold:
The first term on both lines is the gain in terms of the utility of winning times the change in the probability of winning in response to a policy change. The second term is the product of the current probability of winning times the gain in terms of improvements in the party’s utility because of the policy change. When these two marginal effects are equal to each other, each party is playing its best response. When both parties are playing their best responses, we have a Nash equilibrium.
Although (12.17) characterizes the Nash equilibrium implicitly for any function P(qA, qB), it is not informative unless we put more structure on this function. To do this, let us follow the analysis of probabilistic voting above and assume that parties maximize their vote shares given by (12.6), πA =λnFn(Vn(qA) — Vn(qB)). In that case, the equilibrium condition for party A in (12.17) can be written as:
with a similar condition for party B.
The interest of the partisan-politics model is that under some circumstances, it also leads to the reduced-form model of the distribution of political power in democracy used in Chapter 4, potentially giving more power to the rich than the MVT. To highlight this possibility in the simplest way, we next assume that both parties have preferences aligned with those of one of the social groups (for example in our two-class model, the rich). We denote this gr
oup that has captured the platforms of both parties by “1.” Then, we have that:
In that case, the equilibrium is again symmetric, and, using (12.18), we obtain the equilibrium policy, in this case denoted q*, satisfying:
implying
where the second line uses the fact that in equilibrium, each party comes to power with probability ½; thus,= ½ .
This analysis then implies that the equilibrium policy is the solution to maximizing the weighted utilitarian social-welfare maximization:
(12.19)
where
In other words, the group whose preferences are represented by the party platforms has a greater weight in politics. The model also highlights that this effect is more likely to be pronounced when parties do not value coming to power (i.e., R is small), whereas when coming to office matters to the parties, the results are similar to the baseline probabilistic voting model.
Economic Origins of Dictatorship and Democracy Page 52