DISTRIBUTIONAL GAME 1. Political Equilibrium

Taking the opponent’s economic policies as given, each political party P 2 fA; Bg chooses a combination of net taxation policy and public provision of health care for each group,fy_P^j; h^j_gPg^J_j=1; that maximizes its chances of winning elections subject to economic feasibility and non-negativity constraints. Parties take into account citizens’ expected voting decisions (stage 2) and individuals’

choices in competitive markets (stage 3). The policy choice problem of party Ais given by:

$^A = 1

The policy choice problem of political party B is symmetric. The First Order Conditions for both political parties P 2 fA; Bg are de…ned as:

[y^j] ^{j j}dV_P^j(y^j_P; h^j_gP)

Solving backwards, I characterize the Political Equilibrium of the game. The system of equations formed by the best responses for each political party and their feasibility constraints, simultaneously determine the Nash Equilibrium in the …rst stage of the game.Therefore, for both political parties, the equilibrium net taxation and in-kind transfers policies announced to groupj,(y^jN_P ; h^jN_gP), must satisfy:

8j 2 f1; :::; Jgand8P 2 fA; Bg. Those equilibrium conditions hold if and only if taxation policies announced by parties imply a positive level of net income for all groups,fy_P^jNg^J_j=1>0. .

Proof. Suppose that groupj is targeted no numeraire commodity,y_P^j = 0:By Envelope Theorem:

dV_P^j(y^j_P; h^j_gP)

dy^j =uc(0; h) =1 (67)

Thus equation (66) would not hold. Therefore, equilibrium net taxation policy must imply a positive available income for each group,y_P^jN >0;and then, in equilibrium, the multipliers associated to the non-negative constraints of net income are equal to zero, ^jN_yP = 0 8j 2 f1; :::; Jg and P 2 fA; Bg.

Furthermore, for each group j, politicians must decide whether targeting in-kind transfers.

In the case that party P 2 fA; Bg chooses not targeting in-kind transfers to group j, h^jN_gP = 0, politicians take into account that voters are expected to purchase health care in competitive markets with their available incomey_P^jN >0. For any positive net income targeted by partyP, the optimal behavior in competitive markets of an individual who belongs to groupj is characterized by:

uh(y^j_P phh^j_mP; h^j_mP) =phuc(y_P^j phh^j_mP; h^j_mP) (68) When individuals purchase health care through competitive markets and there is no public provision, by Envelope Theorem:

dV_P^jS(y^j_P; h^j_gP)

dy^j =uc(y_P^j phh^j_mP; h^j_mP) (69) dV_P^jS(y^j_P; h^j_gP)

dh^jg

=uh(y_P^j phh^j_mP; h^j_mP) (70) Given the expected behavior of voters in competitive markets, equilibrium condition (66) for a group j not targeted with in-kind transfers holds if and only if the targeted net taxation policy, y_P^jN, implies that the multiplier associated to the non-negative constraint of in-kind transfers must be zero, ^jN_hP = 0. Hence, the equilibrium condition for a groupj not targeted with in-kind tranfers that acquires health services in competitive markets, h^jN_mP, is given by:

uh(y_P^jN phh^jN_mP; h^jN_mP) =phuc(y_P^jN phh^jN_mP; h^jN_mP) (71) As an alternative, political partyP 2 fA; Bg could choose targeting in-kind transfers to group j, h^jN_gP > 0 and then ^jN_hP = 0. In that case, politicians take into account that voters in group j are expected not to make private purchases of health care in markets,h^jN_mP = 0, if and only if this condition holds:

uh(y_P^j; h^j_gP) phuc(y_P^j; h^j_gP) (72)

Otherwise, when the sigh of this condition is reversed, politicians expect that individuals make private purchases, h^jN_mP >0. The expected optimal behavior of individuals that suplement health services in competitive markets is given by:

uh(y_P^j phh^j_mP; h^j_gP +h^j_mP) =phuc(y_P^j phh^j_mP; h^j_gP+h^j_mP) (73) In the …rst alternative, when individuals do not purchase health care through competitive markets and there is public provision, by Envelope Theorem:

dV_P^{jN S}(y^j_P; h^j_gP)

dy^j =u_c(y^j_P; h^j_gP) (74)

dV_P^{jN S}(y^j_P; h^j_gP) dh^jg

=u_h(y^j_P; h^j_gP) (75) Thus, party P’s equilibrium condition (66) when group j is targeted with in-kind transfers, h^jN_gP > 0, and net income, y^jN_P > 0, such that individuals do not supplement health services, h^jN_mP = 0, is given by:

u_h(y_P^jN; h^jN_gP) =p_hu_c(y_P^jN; h^jN_gP) (76) Therefore, in equilibrium, condition (72) for groupj holds with equality.

Otherwise, when individuals purchase health care through competitive markets and there is public provision, by Envelope Theorem:

dV_P^jS(y^j_P; h^j_gP)

dy^j =uc(y^j_P p_hh^j_mP; h^j_gP+h^j_mP) (77) dV_P^jS(y^j_P; h^j_gP)

dh^jg

=u_h(y^j_P p_hh^j_mP; h^j_gP+h^j_mP) (78) Hence, party P’s equilibrium condition (66) when group j is targeted with in-kind transfers, h^jN_gP > 0, and net income, y_P^jN > 0, such that individuals do supplement health services with purchases in markets, h^jN_mP >0, is given by:

u_h(y^jN_P p_hh^jN_mP; h^jN_gP +h^jN_mP) =p_huc(y_P^jN p_hh^jN_mP; h^jN_gP +h^jN_mP) (79) Thus, the equilibrium net taxation and in-kind transfers policies for any group j, (y_P^jN; h^jN_gP) , must satisfy:

j jdV_P^j(y^jN_P ; h^jN_gP) dh^jg

=ph j jdV_P^j(y_P^jN; h^jN_gP)

dy^j (80)

8j 2 f1; :::; Jg and 8P 2 fA; Bg: In the pre-election stage, politicians announce policies such that the marginal bene…t of targeting one unit of in-kind transfers in terms of probability of winning elections is equal to the marginal opportunity cost. That cost is measured by the marginal decrease

in probability due to a reduction of targeted net income by p_h units. The presence of competitive markets allows the existence of multiple equilibrium policies for each groupj. In equilibrium, both political parties are indi¤erent to announce di¤erent combinations of net taxation policy and in-kind transfers for each social group j such that (80) holds. Therefore, the targeted consumption bundle of numeraire and health care to groupjimplicitly de…ned by (71), (76) and (79) is the same regardless of the choosen equilibrium policy. In Equilibrium:

uh(c^jN_P ; h^jN_P ) =phuc(c^jN_P ; h^jN_P ) (81) where c^jN_P = y^jN_P phh^jN_mP and h^jN_P = h^jN_gP +h^jN_mP 8j 2 f1; :::; Jg and P 2 fA; Bg; with y^jN_P >

0; h^jN_gP 0and h^jN_mP 0.

A.2. Distributive Politics

From the the First Order Conditions for both political parties P 2 fA; Bg; taking (61) for a pair of groupsk andk⁰ and arranging I get:

kdV_P^k(y_P^k; h^k_gP)

dy^k + ^k_yP = ^k0dV_P^k0(y_P^k0; h^k_gP⁰ )

dy^k0 + ^k_yP⁰ (82)

Given the equilibrium policies for each groupj discussed above, the relative treatment between groups in terms of numeraire are implicitly de…ned by:

kuc(c^kN; h^kN) = ^k0uc(c^k0N; h^k0N) 8k; k⁰ 2 f1; :::; Jg (83) wherec^kN =c^kN_P =y^kN_P phh^kN_mP and h^kN_P =h^kN_gP +h^kN_mP 8k; k⁰2 f1; :::; Jg and P 2 fA; Bg; with y_P^kN >0; h^kN_gP 0 andh^kN_mP 0:

Similarly taking the FOCs (62) for a pair of groupskandk⁰, the equilibrium patterns of health services across groups of voters are given by:

kuh(c^kN; h^kN) = ^k0uh(c^k0N; h^k0N) 8k; k⁰ 2 f1; :::; Jg (84) wherec^kN =c^kN_P =y^kN_P p_hh^kN_mP and h^kN_P =h^kN_gP +h^kN_mP 8k; k⁰2 f1; :::; Jg and P 2 fA; Bg; with y_P^kN >0; h^kN_gP 0 andh^kN_mP 0:

A.3. First Best Allocations: Allocative E¢ciency

The …rst-best problem consists of the maximization of the weighted average of individual util-itites with group-speci…c Pareto weights, ^j, subject to the economy feasibility constraint. The solution to this optimization problem yields the set of Pareto e¢cient allocations:

max

fc^j;h^jg^J_j=1 J

X

j=1

j ju(c^j; h^j) s.t

J

X

j=1 jc^j+

J

X

j=1 jqh^j

J

X

j=1

jw^j (85)

The FOCs for an interior optimum are given by:

[c^j] ^{j j}uc = ^j 8j2 f1; :::; Jg (86) [h^j] ^{j j}u_h = ^jq 8j2 f1; :::; Jg (87) The set of Pareto e¢cient allocation of resources, fc^j_{P O}; h^j_{P O}g^J_j=1, satis…es (86), (87) and the economy feasibility constraint such that:

uh(c^j_{P O}; h^j_{P O})

u_c(c^j_{P O}; h^j_{P O}) =q!M RS_h;c^j =M RT_h;c 8j (88) In a Pareto e¢cient allocation the rate at which individuals are willing to trade health services for numeraire commodity is equal across groups and equal to the rate at which the economy is able to transform numeraire into health care.

In the political equilibrium, the combination of choosen policies,fy^jN_P ; h^jN_gPg^J_j=1is such that (81) holds for all P 2 fA; Bg. Those equilibrium policies imply consumption bundles for all groups, fc^jN_P ; h^jN_P g^J_j=1, that satisfy the economy feasibility constraint given expected voting and competitive equilibrium behavior of citizens. In equilibrium:

uh(c^jN_P ; h^jN_P ) =phuc(c^jN_P ; h^jN_P )!M RS_h;c^jN =M RTh;c 8j (89) where c^jN_P = y^jN_P phh^jN_mP and h^jN_P = h^jN_gP +h^jN_mP 8j 2 f1; :::; Jg and P 2 fA; Bg; with y^jN_P >

0; h^jN_gP 0and h^jN_mP 0.

Therefore, the political process leads the economy to reach a Pareto E¢cient allocation.

B. EXTERNAL EFFECTS

Im Dokument The Political Economy of In-Kind Redistribution (Seite 37-41)