Tax Buyouts. Marco Del Negro Federal Reserve Bank of New York

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1 Tax Buyouts Marco Del Negro Federal Reserve Bank of New York Fabrizio Perri University of Minnesota, Federal Reserve Bank of Minneapolis CEPR and NBER Fabiano Schivardi Universita di Cagliari, Einaudi Institute for Economics and Finance and CEPR October 2009 PRELIMINARY Abstract The paper studies a fiscal policy instrument that can reduce fiscal distortions without affecting revenues, in a politically viable way. The instrument is a private contract, offered by the government to each individual citizen, whereby the citizen can choose to pay a sum of money upfront in exchange for a given reduction in her tax rate for a pre-specified period of time. The key issue is the pricing of the tax-rate reduction. We consider a calibrated private-information economy and show that, under simple and non-distortionary pricing mechanisms, the contract is signed by a significant fraction of the population, which leads to sizeable increases in labor supply, consumption, TFP and welfare. jel codes: e62,h21 key words: Taxes, Private Information, Distortions The paper previously circulated under the title of On the Privatization of Public Debt. Views expressed here are our own and do not necessarily reflect those of the Federal Reserve Banks of Minneapolis and New York, or the Federal Reserve System. We thank seminar participants at the Bank of Italy, NYU, the Bank of Canada, Penn State University, the University of Rochester, The Federal Reserve Bank of Philadelphia, Universita di Salerno, the SED, Midwest-Macro and ESSIM meetings for very helpful comments

2 1 Introduction The paper proposes a instrument for fiscal policy, a private contract between the government and private citizens whereby the agent agrees to pay the government a lump-sum in exchange for a given reduction in her tax rate in the coming fiscal year. We refer to this contract as a tax buyout: the contract makes it possible for the agent to effectively buy out a portion of her distortionary taxes via a lump-sum payment. The motivation for the paper arises from the debate on the detrimental role of high (distortionary) taxation on economic growth and labor supply (see for example Prescott 2004). This debate is particularly relevant in many countries, including the U.S., where the current and perspective levels of fiscal deficits (and often debt) are high enough that the government cannot contemplate a tax reduction without considering its implication in terms of fiscal sustainability. Some authors (Mankiw and Weizierl, 2004) have argued, using essentially dynamic Laffer-curve arguments, that in the United States (and, one would suspect, a fortiori in Europe) the reduction in tax rates would in large part finance itself. This is a risky bet, however, as shown by the experience of the U.S. in the eighties. In absence of a growth effect that is strong enough to re-equilibrate tax revenues, the reduction in taxes must come at the cost of reduced government spending, an option often politically hard to follow. This paper proposes a contract between the government and the households that reduces the distortionary taxation while keeping the government proceeds unchanged. The fact that the contract is revenue neutral implies that those who do not participate in the contract are not affected by it, abstracting from general equilibrium effects. Participation to the contract is entirely voluntary: those individuals that have most to gain from a reduction in the marginal tax rate self-select into the program. In some dimensions, our contract is similar to some of the mechanisms described in the optimal capital taxation literature (Judd 1985, Chamley 1986). From the government s perspective it involves an exchange between lump-sum taxation today (achieved by means of unanticipated capital taxes) and distortionary taxation in the future. Differently from these mechanisms, from the agents perspective the exchange is not the result of expropriation on the part of the government, but of free choice. In a world of perfect information the contract simply amounts to an exchange of tax revenues collected in a distortionary fashion for revenues collected lump-sum. In such a world, the amount of foregone revenues is straightforward to compute, since the 1

3 government knows each agent s ability. This amount would give the price of the contract. By construction, the contract is revenue neutral for each agent, and therefore for the economy as a whole. In such a world which arguably is not the one we live in the government could collect all revenues lump-sum in the first place. Therefore we focus on a world of asymmetric information where the price of the contract cannot be made contingent on agents abilities since the government has only partial information (if any) about these abilities. Using a simple model economy, we show that even if the government has no information whatsoever on individual abilities, the introduction of the contract is Pareto-improving. The intuition for this result is that the inefficiencies generated by distortionary taxation, once removed, create a surplus that the government can share with the agents. Therefore the government can price the contract high enough to make positive revenues, yet low enough to be attractive to the agents with high ability and high income prospects. If the first part of the paper (section 2) explores the workings of the contract from a theoretical standpoint within a tractable framework, the second part (section 4) obtains quantitative results using a richer model, which includes a number of features that are going to be relevant for the design and pricing of this contract. In this paper we focus exclusively on taxation of labor income in order to narrow the set of issues we have to deal with. Also, we want to understand how much mileage we can get from the introduction of the contract on labor income only. The idea of the contract applies to capital taxation as well, however. 2 A simple model with heterogeneous productivity In this section we consider a small open economy populated by a continuum of agents, each with different productivity and initial level of wealth. The model is in the same spirit of the one considered by Heathcote, Storesletten and Violante (2009). Agents productivity is a function of two components: i) a fixed component A, which is constant over time and distributed according to the CDF F A (A), and ii) a transitory one ɛ, which is i.i.d. across time and agents according to the CDF F ɛ (ɛ) and independent from A. We assume that the fixed component A is uninsurable while the transitory one ɛ can 2

4 be perfectly insured. 1 Agents are endowed with and initial amount of wealth w. The government has an outstanding debt b 0 and a constant flow of government spending g, financed via labor taxes. We consider the simple case where the tax rate is constant and equal to τ. Private agents and the government have access to the world capital market where they can borrow and lend at the risk free rate r. We assume that this exogenous gross risk-free rate equals the inverse of the discount rate: β(1 + r) = 1. Our modification to this otherwise standard set-up is that in each period the government offers each agent a contract, whereby the agent agrees to pay an amount of resources upfront in exchange for a reduction in her tax rate. If the agent does not buy the contract, she faces the pre-contract tax rate τ: status quo is always an option. Under perfect information the contract is designed in such a way that the value of resources extracted from each agent remains the same as in the pre-contract regine, regardless of the amount of tax reduction the agent chooses to purchase. That is, the contract is revenue neutral from the government s perspective, agent by agent. We show that under these conditions agents will want to pay all their taxes upfront in lump-sum form. The logic of this result is straightforward: The contract gives agents the opportunity to turn distortionary taxation into lump-sum taxation, and revenue neutrality implies that all the benefits from the reduction in distortions accrue to the agent. The economy with perfect information is a useful starting point for our analysis, as it highlights the basic insights of the contract, that will hold also in a more realistic setting. However, under perfect information, a benevolent government could more simply use lump-sum taxation where the lump-sum taxes are a function of the fixed component of ability A in the first place. We then turn to study an economy with asymmetric information, where the government does not have perfect knowledge of the agents abilities and therefore cannot implement (ability dependent) lump-sum taxation. In the most extreme case, we assume that the government has no information at all on abilities, so that the contract cannot be made contingent on them. The question is whether the introduction of the contract can be Pareto-improving in spite of this constraint. We show that this is the case. The intuition for this result is that the inefficiencies generated by distortionary taxation, once removed, create a surplus that the government can share with the agents. Therefore the government can price the contract high enough to make positive revenues, 1 It can be shown that this economy behaves very similarly to an economy in which agents can borrow and lend at a fixed rate r and are hit by permanent and transitory shocks 3

5 yet low enough to be attractive to the high ability agents. The next section describes the economy and characterizes its stationary equilibrium. Section 2.2 describes the introduction of the contract under perfect information, and section 2.3 considers the asymmetric information case. 2.1 The economy Agents preferences and technology are given by: β t u(c t, l t ), (1) t=0 and y t = Aɛ t l t, (2) respectively, where c t and l t are consumption and labor in period t, y t is labor income, β > 0 is the time discount factor and u is a standard utility function. Given the assumption that the temporary shock ɛ t is perfectly insurable, we consider a typical component-planner problem who chooses consumption and labor for all agents that have the same fixed level of ability A and wealth w. To simplify notation, define w = r w as the flow value of wealth. In addition, the assumption β(1 + r) = 1 implies that the component-planner problem for this (A, w)-cohort is stationary and can be simplified as follows: subject to: max {c(a,ɛ,w,τ),l(a,ɛ,w,τ)} u(c(a, ɛ, w, τ), l(a, ɛ, w, τ))df ɛ (ɛ) (3) (w + (1 τ)aɛl(a, ɛ, w, τ) c(a, ɛ, w, τ) ) df ɛ (ɛ) = 0 (4) where c(a, ɛ, w, τ) > 0, l(a, ɛ, w, τ) [0, 1], and the initial wealth level is given. The government sets constant labor taxes τ to finance constant exogenous current expenditures g and service its outstanding debt b. government budget constraint is: ( rb + g = τ A ) ɛl(a, ɛ, w, τ) df ɛ (ɛ) In the stationary equilibrium the df A (A). (5) Stationary equilibria for this economy, in which we impose β(1 + r) = 1, are characterized by policy functions c(a, ɛ, w, τ) and l(a, ɛ, w, τ), and a Lagrange multiplier 4

6 λ(a, w, τ) associated with (4) satisfying: u c (c(a, ɛ, w, τ), l(a, ɛ, w, τ)) = λ(a, w, τ) (6) Aɛ(1 τ)u c (c(a, ɛ, w, τ), l(a, ɛ, w, τ)) = u l (c(a, ɛ, w, τ), l(a, ɛ, w, τ)) (7) as well as the budget constraint (4), and a tax rate τ satisfying The economy with the contract: perfect information At the beginning of each period each agent is offered a menu of contracts {δ, d} from which she can buy a reduction δ in its tax rate for the current fiscal year in exchange for the payment of a premium d. Under perfect information the fixed effect A is known to both the government and the agent, while the temporary shocks ɛ are by definition not known to either. The government chooses d such that the contract is revenue neutral for each agent in expected value. The government is making sure that in expected terms the amount of resources extracted from each agent does not change with the introduction of the contract. Therefore d is a function of both A and δ (as well as w, non stated explicitly to simplify notation) that satisfies: 2 d(a, δ) = ( τaɛl(a, ɛ, w, τ) (τ δ)aɛl(a, ɛ, w d, τ δ) ) df ɛ (ɛ) (8) where l(a, ɛ, w d, τ δ) is the policy function for labor under the new regime. Note that the government is exploiting its knowledge of the agents policy function in designing the contract. In expression (8) we have used the same policy function for labor supply described in the previous section, with the only difference that the arguments w and τ are replaced by w d and τ δ, respectively. Indeed, one can see that for any given δ these policy functions satisfy the first order conditions as well as the new budget constraint: ((w d) + (1 τ + δ)aɛl(a, ɛ, w d, τ δ) c(a, ɛ, w d, τ δ) ) df ɛ (ɛ) = 0. Moreover, since the contract is such that the government s intertemporal budget constraint is unchanged for each agent, it is also unchanged in the aggregate: ( τaɛl(a, ɛ, w, τ) dfɛ (ɛ) ) df A (A) = ( (τ δ)aɛl(a, ɛ, w d, τ δ) df ɛ (ɛ) + d(a, δ) ) df A (A) 2 For simplicity, we abstract from differences in wealth, so that we can easy notation. All arguments go through when explicitly conditioning on differences in wealth. (9) (10) 5

7 To simplify the notation, in the remainder of the paper we will use a short-hand notation l A (δ, d) = ɛ l(a, ɛ, w d; τ δ) df ɛ (ɛ) and c A (δ, d) = c(a, ɛ, w d; τ δ) df ɛ (ɛ), and λ A (δ, d) = λ(a, w d, τ δ) to denote aggregate (ability adjusted) labor supply, consumption, and Lagrange multiplier, respectively, for a (A, w)-cohort. Thus, l A (0, 0) is labor supply without the contract. Also, from now on we will loosely refer to the planner for the (A, w)-cohort as the agent. Next we turn to the question of the optimal choice of δ, the amount of tax reduction: Proposition 1 If the government prices the contract according to (8) agents will choose to buy the maximum possible tax reduction. Moreover, labor supply will increase for each agent. The proof uses the fact that the contract can be represented by the combination of a compensated price change and a rightward shift of the budget constrain (defined in terms of consumption and leisure). As a consequence, the old (without contract) optimal allocation remains feasible after the purchase of the contract. Hence, purchasing the contract cannot but raise the agent s utility. Moreover,at the new prices the lower tax wedge implies that agents will choose to work more, so that her utility is strictly greater than before the contract. 2.3 The economy with the contract: asymmetric information This section studies the economy with asymmetric information. We start by assuming that the government has no information whatsoever on A, and hence needs to offer everyone the same price for the contract d(δ). We want to show that the contract can be Pareto improving even under asymmetric information. Namely, we want to show that there exists a function d(δ) such that: (i) a positive mass of agents takes the contract, and (ii) the government budget is still balanced. The second condition implies that those agents who do not take the contract are made no worse off by its introduction. We will prove (i) and (ii) for a linear pricing function of the form d(δ) = dδ. Of course, more general pricing functions will allow the government to improve on the outcome obtained with the linear pricing function. We will explore this aspect numerically in the next section. 6

8 First, we discuss two results on the optimal choice of δ for the case in which the function d(δ) is only restricted to be either weakly concave or convex over the range of δ. Lemma 1 Under a linear pricing function d(δ) = dδ, an agent of ability A will either not buy into the contract or buy the maximum amount allowed. Proof: see the Appendix In what follows, we make the further assumption of separable utility. Indeed, almost all of the results go through without this assumption, but in one case Lemma 2 Assume that the utility function is separable in consumption and leisure. Call δ A the optimal choice of δ for given pricing function d(δ) = dδ. Let us have two agents with abilities A 1 > A 2. Then δ A1 δ A2. According to Lemma 1, linear pricing functions are completely characterized by the upper level of tax reduction that can be bought and by the per-unit price of such reduction, { δ, d}. Lemma 2 states that if agent with ability A 2 buys the (maximum allowed) tax reduction, so will A 2 For each level of ability we define two (per-δ-unit) prices, d A and d A. The former is the lowest price at which the government is not losing resources from offering the contract to agent A. The latter is the price for which agent A is indifferent between taking and not taking the contract. Lemma 3 Under a linear pricing function d(δ) = dδ, for each level of ability A there exist: i) A per-unit price d A such that agent A is willing to enter the contract and the government is neither losing nor gaining resources from the agent. ii) A per-unit price d A for which agent A is indifferent between taking and not taking the contract. If agent A decides to enter the contract, the government is gaining resources from the agent. The lemma is proven in the appendix. A consequence of Lemma 3 is: 7

9 Corollary 1 For d (d A, d A ) the government is gaining resources from offering the contract to agent A, and agent A is willing to take the contract. We have just shown that for each level of ability there is a price that is high enough to make government s revenues positive, yet low enough to be attractive to the agents. The removal of the inefficiency due to distortionary taxation creates a surplus that the government and the agent can share. Building on this result, we now show that the government can attract into the contract the upper tail of the ability distribution, and still make positive revenues. Once we have shown this, it follows that the contract is Pareto-improving: The government could either rebate these excess revenues to all agents, or lower the price even further to attract more agents into the contract. Here we pursue the latter route, and show that if the price is low enough government s excess revenues are driven to zero. Define A and A as the lower and upper bounds of the support of the ability distribution, where we assume that 0 < A < A <. Define the marginal ability A(d) as the level of ability for which d A = d. This function is implicitly defined by the equation u(c A ( δ, d δ), l A ( δ, d δ)) = u(c A (0, 0), l A (0, 0)) Such level is well defined for any d d A. The standard regularity conditions on the utility functions imply that the mapping A(d) is continuous. As a consequence of Lemma 2 agents with ability less than A(d) will not take the contract, while agents with ability greater than A(d) will. Therefore government s revenues from the introduction of the contract can be written as: R(d) = A A(d) Continuity of A(d) implies that: [ (τ δ)al ] A ( δ, d) τal A (0, 0) df A (A) + d δ(1 F A (A(d))) (11) Lemma 4 R(d) is continuous for d < d A. From Lemma 3 we have that the government can attract to the contract the upper tail of the ability distribution, and still extract excess revenues from all of these high ability agents if d > d A : 8

10 Corollary 2 For d [d A, d A ) all agents with ability A [A(d), A] will take the contract, where A(d) is strictly less than A. Moreover, R(d) > 0 in this interval. Our final assumption is that, when the contract is priced at zero, the government is loosing money R(0) < 0. This is simply assuming that we are not on the declining portion of the Laffer curve, where a tax decrease leads to an increase in revenue, in which case the Government should simply lower taxes. The above corollary, the continuity of R(d), and above assumption imply the key theoretical result of this section: Proposition 2 There exist a pricing function d(δ) = d δ, with d (0, d A ), such that: (i) a positive mass of agents with A [A(d ), A] enter the contract; (ii) the government s budget is balanced ( R(d ) = 0 ). Since those who enter the contract are better off, and those who do not enter the contract are no worse off given that the government still balances the budget, we have shown that the contract is Pareto improving. Figure 1 below gives an illustration of the proposition, as it plots the government losses as a function of d, the price of the contract. 3 Starting from the right the figure shows that as the price d is large losses are close to 0, as only few very reach agents elect to purchase the buyout. As the price falls more households participate and initially the revenues from higher participation offset the losses from lower prices, so that government revenues increase (losses fall). As the price is lowered further though the price effect dominates the increased participation, and losses increase. When the price reaches d > d losses are 0, so the government does not lose from introducing the contract and a positive fraction of the population takes it. Yet, an unpleasant feature of linear pricing, the case considered in this section, is that all agents who enter the contract will purchase the maximum amount, and therefore pay the same price, regardless of the level of ability. But the gains from the contract increase with ability. In fact, it is the case that high ability agents are not only extracting all the surplus from the reduction of their own distortions, but also appropriating some surplus from those lower ability agents that enter the contract. In the next section we study alternatives to linear pricing. 3 The figure is drawn for a log-normal distribution of abilities 9

11 Figure 1: Government losses and the price of the buyout 2.5 x d* Price of the contract, d 10

12 3 Non linear pricing In the previous section we showed that by introducing a tax-buyout contract with linear pricing the government can achieve allocations that are Pareto improving and at the same time balance the budget. Yet we also showed that linear pricing implies essentially offering only one possible tax-buyouts scheme one contract to agents: All those who buy the contract buy the maximum amount and consequently pay the same price. But there is no reason why the government should limit itself to linear pricing. In fact, we know from the theory of static Pareto efficient taxation (Mirrlees 1971, Stiglitz 1982) that the optimal tax schedule entails different marginal tax rates for individuals with different abilities so to achieve self-selection. A single contract cannot achieve that in general. Starting from proportional taxes and a continuum of abilities the case considered in the previous section the post-contract equilibrium entails two different marginal tax rates for the entire population, which is obviously far from the continuum suggested by the optimal taxation literature. It is therefore interesting to find out how much the government would leave on the table, in terms of efficiency, by offering only one contract. In the remainder of this section we will consider a discretized version of this continuous set of contracts {δ, d(δ)} consisting of a sequence of N contracts {δ i, d(δ i )} with i = 1,..., N and N potentially large. We call this discretization many contracts, whence the title of the section. We now describe an algorithm for computing the set of N contracts that delivers self-selection in the economy described above, where by self-selection we mean that the pricing scheme is such that any two agents A i and A j will choose different contracts, {δ i, d(δ i )} and {δ j, d(δ j )}, respectively. This self-selection implies that, consistently with the optimal taxation literature, agents with different abilities will face different marginal tax rates. We will later discuss why the tax-buyouts scheme cannot in general achieve the informationally constrained Pareto efficient frontier. The algorithm works as follows. Imagine a discretization {A i } N i=o of the interval [A, Ā] with A 0 = A and A N = Ā. Fix the maximum amount δ that the government is allowed to offer. From the optimal taxation literature we know that in general the highest ability individual A N should have the lowest marginal tax rate, hence it is natural that the maximum tax reduction should be targeted toward agent N, hence δ N = δ. Fix the price d(δ N ) associated with the maximum tax reduction, and assume that d(δ N ) is such that agent N is willing to buy the contract (from the previous section we know that if 11

13 agent N faces a pre-contract marginal tax rate of τ, for δ N τ she will be willing to buy the contract for any d(δ N ) δa N l AN (0) where l AN (0) is the pre-contract labor supply). The problem of the government is to choose the remaining contracts {δ i, d(δ i )} N 1 i=0 so that households {A i } N i=o will self-select into purchasing only one contract. We tackle the problem in a pairwise fashion: Given the contract tailored for agent i+1, {δ i+1, d(δ i+1 )}, we choose {δ i, d(δ i )} so that: (a) agent i + 1 is indifferent between contract i + 1 and i, and (b) agent i prefers contract i to i + 1. We construct {δ i, d(δ i )} as follows: We know that since A i+1 > A i in order to have separation we need that δ i+1 > δ i (in this discussion we maintain the assumption that pre-contract marginal tax rates are the same across agents). We therefore start with a candidate δ i = δ i+1 ε δ, with ε δ a very small number. If d(δ i ) = d(δ i+1 ), agent i will prefer contract i + 1. Hence we lower d(δ i ) by multiples of ε d another very small number until agent i prefers contract i over contract i + 1. For d(δ i ) < d(δ i+1 ) agent i + 1 may also prefer contract i over contract i + 1, violating (a). We therefore further lower δ i to δ i = δ i+1 2ε δ, and, if needed, lower d(δ i ) to make it appealing to agent i. We repeat the two steps until the gap between δ i+1 δ i makes agent i + 1 indifferent between the two contract (condition a) while the difference d(δ i+1 ) d(δ i+1 ) makes condition (b) hold. Once we have chosen the contract for agent i, we repeat the same steps for agent i 1, and so on. By construction the self-selection works pairwise: agent i prefers her contract to i + 1 or i 1. It turns out that it works for all (i, k): agent i prefers her contract to any other contract k (we check this condition numerically). The intuition for this result is that indifference curves for the agents in the consumption/income space are steeper for the less able agents (see Stiglitz 1982). Therefore if agent i + 1 is indifferent between her contract and the i contract, while agent i is indifferent between her contract and the i 1 contract, then agent i + 1 will strictly prefer her contract to i 1, and viceversa. In fact, in our numerical example we find that the disutility from taking contract k for agent i increases with the distance A k A i. As we go down the ability ladder, at some point an additional constraint comes into consideration: agent i must prefer contract i to not taking the contract at all (this constraint also limits the government s choice of d(δ N ), the price charged to the highest ability agent). The informationally-constrained social planner does not face such a constraint. Therefore the allocations achievable with the tax-buyouts scheme are generally less efficient in a Pareto sense than those achieved by the social planner. 12

14 Once computed the {δ i, d(δ i )} N i=0 schedule that achieves self-selection, we can use it to analyze its properties numerically. Figure 2 compares the many contracts to the one contract ( δ =.13) cases, where by one contract we refer to one of the linear pricing scheme discussed in the previous section. The value of δ in the linear case and the {δ i, d(δ i )} N i=0 schedule in the non-linear case are chosen so that the fraction of agents entering the tax buyout scheme is roughly the same in the two cases. In all panels the horizontal axis displays the the pre-contract income (Al A (0)), which is a monotone function of ability. Panel (a) displays the chosen δ as a function of income. The lines stop at the ability level (expressed in terms of pre-contrcat income) below which agents prefer not to purchase the contract. As shown in the previous section, in the linear case all agents buying the contract choose the maximum level of δ. In the non-linear case δ i increases in the pre-contract income. In the example considered here the increase is approximately linear. Panel (b) displays the associated price of the contract d(δ) as a function of income. In the linear case all agents buying the contract pay the same price. In the non-linear case, the price d(δ) is a convex function of δ, which is what we would expect from the household s maximization problem with respect to δ. Panel (c) shows the cumulated government losses from the offering contract, starting from the highest ability individuals (hence, we accumulate losses going from the right to the left). By construction, in both cases the total government losses from the contract are approximately zero (i.e, both lines reach zero for the indifferent agent). The shape of the cumulated government losses is very similar for the linear and the non-linear pricing schemes in this specification. In both cases the government loses revenues on high ability individuals: In the linear pricing scheme δ is lower than in the non-linear pricing scheme, but so is the price. In both cases, consistently with the theory, the government makes positive revenues on the agents that are close to being indifferent. Panel (d) shows utility relative to pre-contract. The figure shows that the non-linear scheme is a Pareto improvement over the linear one: for all agents utility increases. For the close-to-indifferent agents, not surprisingly, utility is very close under the two schemes, but as the level of ability increases, the non-linear scheme delivers progressively higher utility. As figure (c) makes clear, the many contracts environment allows the government to deliver higher utility to high ability individuals without having much greater losses. Of course, the lower distortions for high ability individuals under many contracts result in higher labor supply and output than in the one contract case. 13

15 Figure 2: One versus Many Contracts I 0.21 (a) 12 (b) Delta Price of the Contract Pre Contract Income Pre Contract Income 3 x 10 3 (c) 0.14 (d) Cumulated Government Losses from the Contract Utility Relative to Pre Contract Pre Contract Income Pre Contract Income 14

16 Figure 3 compares the same one contract scheme ( δ =.13) to a many contracts environment where the top ability individual is offered a contract with a tax reduction of at most.13 (δ N =.13). As ability decreases, δ i also decreases (again, in an approximately linear fashion). Unlike in the previous example, this set of contracts {δ i, d(δ i )} N i=0 makes high ability individuals worse off than under the one contract scheme. As is clear from panel (b), the highest ability individual is now paying for the same tax reduction a higher price than under the linear scheme. As a consequence, if for high levels of ability the government is losing money under the one contract scheme, the government is losing less money under many contracts. Panel (c) shows that indeed the solid line (cumulative losses under many contracts ) lies below the dotted line (cumulative losses under one contract ) for all ability levels to the right of the peak of the dotted line that is, for all ability levels at which the government is losing money under one contract. The extra resources make it affordable to both charge a lower price to individuals that before were close to being indifferent, and extend the contract to less able people. As shown by panel (d), both sets of people are better off under the many contracts scheme. Although the interval of abilities that could not afford the buyout under the linear scheme, but can under many contracts, appears small, there is enough population mass in that interval that aggregate output is larger under many contracts, even though higher ability individuals are subject to more distortions. 4 Quantitative analysis In this section we present a richer model economy in order to evaluate more precisely the effects of making a tax buyout contract available to US households. Our setting follows closely recent works that use calibrated life-cycle heterogenous agents economies in order to analyze policy issues (See for example Conesa, Kitao and Krueger, 2008 or Heathcote, Storesletten and Violante, 2008). We consider a discrete time small open economy inhabited by overlapping generations of finitely lived households. 4 Production is carried out by a representative firm which uses a constant return to scale technology to produce a single good, used for consumption and investment. Finally a government levies taxes, provides transfers and public spending and might offer tax buyouts. We now describe in more detail households, firms and government, define equilibrium, describe 4 The small open economy assumption is done for computational convenience. In the sensitivity analsysis section we explore how our results depend on it. 15

17 Figure 3: One versus Many Contracts II (a) (b) Delta Price of the Contract Pre Contract Income Pre Contract Income (c) (d) 3 x Cumulated Government Losses from the Contract Utility Relative to Pre Contract Pre Contract Income Pre Contract Income 16

18 the calibration procedure and finally perform the experiment of introducing a tax buyout. 4.1 Households Each period a new generation of mass 1/(N +1) of unitary households is born. Preference of an household at birth are represented by ( N ) E 0 β t u(c h, l h ) + β N+1 V N+1 h=1 where c h and l h are consumption and labor effort of the household at age h, u is a standard utility function, β > 0 is the time discount factor (possibly household specific) and V N+1 represents the value of retirement (to be specified later). 5 Each household is born with a wealth endowment a 0 and is endowed with 1 unit of time in each period of its life. For the first N years of life households can work in the market and each unit of time spent working yields e h efficiency units of work, where e h is given by e h = e Az hε h p h where A is a individual specific fixed effect, z h is a deterministic age effect, ε h is a idiosyncratic transitory shock and p h is a idiosyncratic persistent shock. All these 4 states are exogenous, in the sense that they do not depend on any household decision. For simplicity we will denote with the letter S = (A, z t, ε t, p t ) these exogenous states and by S the set of all possible values taken by S Retirement During retirement households are not allowed to work. The only role played by retirement in our setup is to affect the wealth accumulation motive during working life. We capture this effect in this simplified fashion V N+1 = V N+1 (a, S N ) u(φa + T R (a, S), 0) (1 β) + βφ (12) where φa represents the annuity value of wealth and T R (a, S) represents the annuity value of all social security payments for an agent entering retirement with wealth a and 5 To keep notation light in this section we omit t subscripts (indicating different calendar dates) and i subscript (indicating different households) 17

19 state S The key parameter here is 0 < φ 1 which captures in a reduced form way the length of retirement. A small φ implies a small annuity value (the effect of φ in the numerator of 12) but a large total utility (the effect of φ in the denominator in 12), thus capturing the idea of a long retirement period, which induces agents to accumulate more wealth. Similarly a large φ captures the idea of short retirement period. 6 In our exercise φ is treated as a parameter and is set to match age-wealth accumulation profiles from the SCF data Working age Households have access to uncontingent borrowing and lending (at an exogenous rate of r) subject to a borrowing constraint, so in each period they decide how much to borrow, save and consume, how much to work and how much tax reduction to buy (tax buyout). Notice that we are allowing agents to purchase the buyout every period, and that the buyout affects only tax rates in that period. The tax buyout contract in its most general form is represented by a set of possible tax rate reductions and by a pricing function d(δ h, O h ) which maps quantity of tax reduction purchased δ h and household observables O h into prices. Observables include variables that can be contracted upon, such as age and past level of income; it is convenient to represent observables as function of the households states, i.e. as O h (a h, S h ). Notice that this formulation allows the pricing of the contract to be non linear, dependent on household characteristics. The problem of an agent of age h with wealth a and state S can be written in a recursive fashion as V h (a h, S h ) = max u(c h, l h ) + βev h+1 (a h+1, S h+1 ) δ h,a h+1,l h s.t. h = 1, N c h + d(δ h, O h ) + a h+1 + T (we h l h, ra h, δ h ) = (1 + r)a h + we h l h δ, a ā 0 where w is the wage rate per efficiency unit and ā represents the minimum level of wealth agents are allowed to hold. The function T (we h l h, ra h, δ) represents the total taxes paid by an agent with labor income we h l h, capital income ra h and tax buyout purchase δ. Let the function T (we h l h, ra h ) be the current tax code (without the buyout). Then the function T (we h l h, ra h, δ) = max( T (we h l h, ra h ) wlδ, 0) This implies, in very 6 In the limit as φ = 1 the model reduces to the special case of retirement which lasts exactly one period 18

20 practical terms, that when an agent who has purchased δ units of the buyout files taxes, she simply computes the tax bill under the standard tax code and then subtract from the tax bill a fraction δ of labor earnings. The solution to this problem can be represented by age dependent decision rules for labor effort l h (a h, S h ), consumption c h (a h, S h ), next period assets a h+1 (a h, S h ) and tax buyouts δ h (a h, S h ). 4.2 Firm Output is produced by a representative, competitive and profit maximizing firm which hires labor L at price w and capital K at price (1 + r) and uses a standard constant returns to scale technology so that its profits are given by K α L 1 α + (1 κ)k (1 + r)k wl where 0 < α < 1 is the share of capital in the technology and κ is the rate of depreciation of capital used in production. 4.3 Government In each period the government has to finance a constant flow of spending g, transfers to retired household and interest on debt rb and its receipts are given by taxes, sales of tax buyouts and issue of new debt b. Let F h (a, S) be a distribution of households of age h over assets a and exogenous states S, then the budget constraint can be written as rb + g + T R (a, S)dF N+1 (a, S) (13) N h=1 a,s S,a ( T (we(s)l h (a, S), ra, δ h (a, S)) ) + d(δ h (a, S), O h (a, S)) df i (a, S) + b 4.4 Equilibrium A steady state equilibrium is the following set of objects: Decision rules of households: l h (a h, S h ),a h+1 (a h, S h ) δ h (a h, S h ), and of firms for K(r, w) and L(r, w) Prices w, r 19

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