Heterogeneous Labor Skills, The Median Voter and Labor TaxesI Facundo Piguillem1,1,∗ EIEF

Anderson L. Schneider

Abstract This paper analyzes the most preferred sequences of labor taxes by the median voter in the standard neoclassical growth model. We consider an infinite horizon economy in which agents are heterogeneous with respect to both initial wealth and labor skills. We first provide a set of sufficient conditions for the existence of a Condorcet Winner. Then, we characterize the most preferred tax sequence by the median agent. First, we find that marginal labor taxes depend directly on the absolute value of the distance between the median and the mean value of the skills’ distribution. Second, and as opposed to the intuition stemming from standard representative agent economies, we find that optimal labor taxes are more volatile and that counter-cyclical taxation (e.g., increasing taxes in recession) might be optimal depending on the correlation between inequality and T F P . Finally, we calibrate the economy using data from six countries and in most of them, except for the US, pro-cyclical fiscal policy is optimal. Keywords: Median Voter, Business Cycle, Labor Taxes, Pro-cyclical Fiscal Policy, Tax shifting

I

The authors are grateful to Larry Jones and Narayana Kocherlakota for their support and encouragement. We would also like to thank V. V. Chari, Christopher Phelan, V´ıctor R´ıos Rull, Pierre Yared, seminar participants at Minneapolis Fed, Midwest Economic Theory Meetings, SED meetings, and workshop participants at the University of Minnesota for comments. Remaining mistakes are ours. ∗ Corresponding author Email address: [email protected] (Facundo Piguillem) 1 Contact Piguillem: Einaudi Institute for Economic and Finance (EIEF), Via Sallustiana 62, Roma, Italy. Email: [email protected]

Preprint submitted to Elsevier

December 5, 2012

1. Introduction One of the most commonly accepted prescriptions for fiscal policy over the business cycle is that labor taxes should exhibit little variance and their cyclical behavior should smooth deadweight burdens across time and states. However, such conclusions typically arise from representative agent economies, and thus, distributional aspects are absent. Any rational agent would prefer to receive the benefits of a particular fiscal policy and transfer the cost burden to others. This could be done in an intratemporal fashion by taxing groups with lower social weights more, or intertemporally by shifting the taxes to periods or states in which less favored groups would contribute relatively more. As Bassetto (1999) and Niepelt (2004) have shown, depending on the weights in the social welfare function, the intertemporal tax shifting and tax smoothing motives can be aligned or can act in opposite directions. Thus, any observed cyclical behavior could be optimal given some social weights. To provide a more precise answer on optimal cyclical behavior of labor taxes, we study a social welfare function that uniquely pins down the social weights by encompassing the key principle of any democratic institution: one man, one vote. We abstract from imperfections such as lack of commitment and market incompleteness. For this reason, we call the policy arising from this problem the optimal policy. If output is positively correlated with government surplus or negatively correlated with revenues (or taxes) we call a fiscal policy pro-cyclical. Thus, a counter-cyclical tax schedule, which is the focus of this paper, is called pro-cyclical fiscal policy. To be concrete, we study the Standard Neoclassical growth model with leisure choice, aggregate uncertainty, complete markets and heterogeneous agents. We assume that fiscal policies are chosen by majority rule. Agents vote once and for all at the beginning of time on sequences of capital and labor taxes. Building on Bassetto and Benhabib (2006) (henceforth B&B), we derive a median voter theorem and use the theorem to describe properties of equilibrium tax sequences. The theorem provides a precise statement of the role of redistribution considerations in determining policy. In Proposition 1, we provide a set of sufficient conditions for the existence of a Condorcet 2

Winner. We show that if heterogeneity is one-dimensional, either in skills or initial endowments, and agents have balanced growth preferences, then the median agent’s most preferred fiscal policy is the Condorcet Winner. If instead the heterogeneity is in both dimensions, Proposition 1 holds if we also assume that the distribution of initial holdings of capital is an affine function of labor skills. Next we characterize the Condorcet Winner. In Proposition 2 we show that marginal labor income taxes depend directly on the absolute value of the distance between the median and the mean value of the labor skills distribution. The results are extended to an environment in which skills evolve stochastically over time while the ranking among agents remains unchanged. In contrast to the representative agent’s environment, the correlation between labor taxes and employment (and output) is ambiguous over the business cycle. The intuition behind this result is similar to the tax shifting effect in Basseto (1999) and Niepelt (2004). More inequality reflects the fact that the median voter has become relatively less efficient than the average agent, and therefore, her relative contribution to revenues is smaller than in periods with less inequality. Thus, by increasing labor taxes in periods with high inequality and decreasing taxes in periods with low inequality, the median voter shifts the burden of taxation toward relatively more efficient agents. The median voter still wants to smooth distortions over states and time. If periods of high average productivity (and a larger tax base) are more likely to happen in periods with more inequality, the two effects are aligned, and therefore, labor taxes and aggregate productivity are positively correlated; therefore tax smoothing is optimal. Otherwise, the tax shifting and tax smoothing effects act in opposite directions and the sign of the correlation is ambiguous. Notice that a negative correlation between T F P and inequality is a necessary condition for a counter-cyclical tax schedule to be optimal. A sufficient condition is that the changes in inequality along the business cycle are large relative to the volatility of T F P . To shed light on the scale of our findings we solve the model economy numerically. We use constructed series for aggregate productivity (T F P ) and skills inequality for six countries: Argentina, Brazil, Chile, Canada, the UK and the US. The necessary condition

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for optimality of pro-cyclical fiscal policy is present in the sample: inequality is negatively correlated with T F P in all six countries, that is, inequality increases in recessions and decreases in booms. Further, the standard deviation of inequality is between 2 and 3.5 times larger in the developing countries than in the developed countries. This suggests the possibility of both more volatile labor income taxes and pro-cyclical (or less counter-cyclical) fiscal policy in developing countries. We find that for the poorer countries in our sample the optimal labor tax is twice as volatile as in the richer countries, while pro-cyclical labor taxes are optimal only for the US. The reason for this is that in Canada and the UK not only is the volatility of inequality low, but the volatility of T F P is also low. The incentives for tax smoothing and tax shifting are both low, but the latter dominates. The numerical simulations provide evidence that procyclical fiscal policy, at least in terms of revenue, should be the norm rather than exception. Our paper is related to the literatures on macro Median Voters and optimal policy with heterogeneous agents. As previously mentioned, we extend the B&B median voter result. B&B consider a more general class of Gorman aggregable preferences but do not consider leisure choice or uncertainty.2 Azzimonti et al. (2008) analyze majority voting over marginal taxes without aggregate uncertainty or capital accumulation, and are therefore able to characterize the best sequence of labor taxes for each type with taxes in the first two periods. Krusell & Rios-Rull (1999) study the steady state of a similar environment but with sequential voting, taxes on capital and labor income are constrained to be equal, and where only future taxes can be changed. They solve a Markov stationary equilibrium numerically and find, as in this paper, that the level of income taxation depends on the skewness of the income distribution.3 Three papers about the cyclical behavior of fiscal policy are closely related to ours: Bas2

Important contributions on median voter results related to fiscal policy include Meltzer & Richard (1981), the first paper in the macro literature, Alesina & Rodrik (1994), Persson & Tabellini (1994). 3 Azzimonti et al. (2006) provide an analytical characterization of time-consistent Markov-perfect equilibria in an environment similar to Krusell & Rios-Rull (1999), but individual heterogeneity is restricted to initial wealth. Cobae et al. (2009) in an environment with uninsurable idiosyncratic risk show that the increase in inequality in the US economy since 1983 can account for at least 2/3 of the observed taxes.

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setto (1999), Niepelt (2004) and Werning (2006). To the best of our knowledge Bassetto (1999) is the first paper to note that tax smoothing may not be optimal in environments with heterogeneous agents. However, the effect arises from intertemporal interest rate manipulation instead of a tax shifting motive. Niepelt (2004) shows how the relatively richer West Germans shifted the cost of German unification to future generations in order to allow formerly poor East Germans to become as rich as West Germans, and therefore, increase their contribution. Finally, Werning (2007) studies an environment similar to ours but with exogenous and arbitrary welfare weights. Our characterization can be seen as an special case of his. However, because we analyze the solution arising from a specific set of endogenous social weights we are able to provide a sharper characterization of labor taxes, which in turn provides a more precise mapping to the data. The paper proceeds as follows: Section 2 describes the environment. Section 3 characterizes the competitive equilibrium given a fiscal policy. In Section 4 we prove the existence of a Condorcet winner. Section 5 characterizes the Condorcet winner. Section 6 considers stochastic skills. Section 7 presents the numerical results and the last section concludes. 2. The Economy with Constant Skills There is a continuum of agents indexed by the labor skill parameter θ ∈ Θ ≡ [θ, θ] with θ > 0. Later we relax this assumption to allow for stochastic labor skills. The distribution R of θ is represented by the p.d.f. f (·) and the median type is denoted by θm ≤ Θ θf (θ)dθ = 1 by assumption. Uncertainty is driven by the publicly observable state st ∈ S, where S is finite. The state may affect the efficient production frontier. Let st = (s0 , ..., st ) be the history of shocks up to time t and Pr(st ) its marginal probability. We assume that Pr(s0 = s) = 1 for some s ∈ S. The output at time t is produced by competitive firms using capital and efficient labor. The resource constraint for each pair (t, st ) is

5

  t t−1 t C(s ) + K(s ) ≤ F L(s ), K(s ), s + (1 − δ)K(st−1 ) t

t

(1)

where the function F (·) is assumed to be homogeneous of degree one in both capital and labor for all st .4 Each agent has an endowment of one unit of time in each period and state. The agent of type θ uses l/θ units of time to produce l units of efficient labor that is rented to the ∞ firms. If agent type θ consumes the stream  {ct , 1 − lt /θ}t=0 ofconsumption and leisure, his P t t t discounted utility is given by ∞ t=0 β u c(s , θ), 1 − l(s , θ)/θ) , where:

u(c, le )=

 α 1−α 1−σ  [c le ]

if σ 6= 1

1−σ

(2)

 α log(c) + (1 − α) log(l ) if σ= 1 e In the initial period, agent type θ is endowed with k−1 (θ) > 0 units of capital. In each period the government levies an affine tax schedule on labor income given by τl (st )w(st )l(st ; θ) + T (st ), where w(st ) are the wage payments and the lump-sum tax T (st ) may be used for redistribution. Notice that the tax schedule is not individual specific. The government taxes capital returns net of depreciation at rate τk (st ) ∈ [0, τ ]. For the type of wealth distribution that we analyze, the lower bound will never bind. The upper bound on capital taxes is a technical condition required to guarantee that the best allocation for the median type exists. In order to reduce the arbitrariness of such an exogenous upper bound, we choose τ = 100%, so that the maximum levy corresponds to a loss of the full return net of depreciation. Profit maximization by the firms determines rental prices. Given a tax sequence, prices, and initial endowments, under complete markets agent type θ chooses his allocation to maximize utility subject to the budget constraint: 4

We could consider an exogenous stream of government expenditures, as long as government spending is not too high, without changing the main results. Since our focus is redistribution, restricting government consumption to zero prevents having to deal with valuations of positive marginal taxes net of the distortions in financing government expenditures.

6

X

  X   t t t t t t t t−1 p(s ) c(s ; θ) + k(s ; θ) ≤ p(s ) (1 − τl (s ))wt (s )l(s ; θ) + R(s )k(s ; θ) − T (3) t

t,st

where T ≡

t,st

P

t,st

p(st )T (st ) is the present value of the lump-sum taxes and R(st ) ≡ 1 + (1 −

τk (st ))(r(st ) − δ). Since markets are complete the government budget constraint can be written as: −T ≤

X

  t t t t t t−1 p(s ) τl (s )w(s )L(s ) + τk (s )(r(s ) − δ)K(s ) t

(4)

t,st

The usual definition of a competitive equilibrium follows: Definition 1. A competitive equilibrium given taxes {τl (st ), τk (st ), T (st )}∞ t=0 is a sequence of t t t ∞ prices {w(st ), p(st ), r(st )}∞ t=0 , individual allocations {c(s ; θ), l(s ; θ), k(s ; θ)}t=0 and implied

aggregate allocations {C(st ), L(st ), K(st )}∞ t=0 such that: 1. Given after-tax prices, {c(st ; θ), l(st ; θ), k(st ; θ)}t maximizes utility subject to (3); R R R 2. C(st ) = Θ c(st ; θ)f (θ)dθ, L(st ) = Θ l(st ; θ)f (θ)dθ and K(st ) = Θ k(st ; θ)f (θ)dθ; 3. Factor prices are equal to the marginal products for every st ; 4. The government budget constraint holds for every st ; and 5. The resource constraint holds for every st . 3. Characterization of the Competitive Equilibrium Here we characterize the economy given a fiscal policy. Let λ(θ) be the multiplier related to the budget constraint of type θ. The first order conditions with respect to individual consumption and labor yield:

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αβ t Pr(st |s0 ) [c(st ; θ)α le (st ; θ)1−α ] c(st ; θ)

1−σ

(1 − α)β t Pr(st |s0 ) [c(st ; θ)α le (st ; θ)1−α ] θ − l(st ; θ)

1−σ

= p(st )λ(θ)

(5)

= p(st )(1 − τl (st ))wt (st )λ(θ)

(6)

From the above we see that due to the complete markets assumption, over the business cycle the consumption-leisure ratio is independent of the type. Further, because of the homotheticity of preferences, the individual allocations are proportional to the aggregates. In particular, the individual allocations can be expressed as5 θ

t

c(s , θ) = R

θ Θ

−(1−α)(1−σ) σ

−(1−α)(1−σ) σ

ϕ1/σ (θ)

C(st ) = ω(θ)C(st )

(7)

ϕ1/σ (θ)f (θ)dθ

α(1−σ)−1

l(st , θ) θ σ ϕ1/σ (θ) ω(θ) 1− = R −(1−α)(1−σ) [1 − L(st )] = [1 − L(st )] θ θ σ θ ϕ1/σ (θ)f (θ)dθ Θ

(8)

where ϕ(θ) = 1/λ(θ) and capital letters represent aggregates. Thus, the equilibrium prices must satisfy: α−1 C(st ) p(s ) = αΦ [C(s ) (1 − L(s )) ] β t Pr(st |s0 ) (9) 1 − L(st )  α C(st ) t t t σ t α t 1−α −σ p(s )w(s )(1 − τl (s )) = (1 − α)Φ [C(s ) (1 − L(s )) ] β t Pr(st |s0 ) (10) 1 − L(st ) t

where Φ =

R Θ

θ

−(1−α)(1−σ) σ

σ

t α

t

1−α −σ



ϕ1/σ (θ)f (θ)dθ.

We normalize the initial price p0 so that Φ = 1.6 The other conditions for optimization in the problem faced by individual θ are: 5

To obtain these equations, we first integrate all first order conditions over θ to express the prices as functions of aggregate consumption and leisure. Then, replacing the prices in equation (5) with the aggregate allocations delivers the result. 6 In the proof of Lemma 1 we show that individual shares integrate to one for any normalization of initial prices.

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p(st ) =

X

R(st+1 )p(st+1 ), and

st+1

lim

t→∞

X

p(st )k(st ; θ) = 0

st

Using conditions (5)-(10) in each individual’s budget constraint yields7 :

ω(θ) =

f0 (θ, T, τk0 ) + θUL ] [uc0 W (1 − σ)V

(11)

where ≡

X

UL ≡

X

V

β t P r(st )u(C(st ), L(st ))

t

β t Pr(st )ul (C(st ), L(st ))

t,st

f0 (θ, T, τ0 ) ≡ R0 k−1 (θ) − T W Since for each type the expression for ω(·) depends on aggregate allocations and the tax schedule, the function can be rewritten as ω(Z; θ) ∈ R+ , where Z is a sequence consisting of aggregate allocations, initial tax on capital and the present value lump-sum transfer. The same is true for V and UL , which can be written as V (Z) and UL (Z), respectively. Let Z ∞ be the set of such sequences. From (11) we have that the share for type θ is equal to the present value of human and non-human wealth divided by the present value of the marginal utility of consumption for the average agent. More intuitively, using (11) and the fact that E(ω) = 1, the individual shares can be rewritten as:

ω(Z, θ) ≡ 1 +

ˆ T, τk0 )]] + (θ − 1)UL (z) f0 (θ, T, τk0 ) − E ˆ[W f0 (θ, uc0 [W θ (1 − σ)V (Z)

(12)

Therefore individuals who are wealthier than the average will have both consumption and leisure (measured in efficient units) higher than the respective aggregates.

7

See Appendix A for details.

9

Lemma 1. Z ≡ ({C(st ), L(st ), K(st )}t≥0 , T, τk0 ) is the aggregate allocation sequence (together with T and τk0 ≤ τ ) in an interior CE if and only if: 1. Z satisfies the resource constraint in (1) for all st ;   t+1 t+1 t+1 t t t 2. uc (C(s ), L(s )) ≥ βE [1 + (1 − τ )(Fk (s ) − δ)]uc (C(s ), L(s ))|s for all st ; 3. Evaluated at the aggregate allocation, the function ω : Z ∞ × Θ → R+ given by (12) is 1 t t such that ω(Z; θ) ∈ (0, 1−L(s t ) ) for all s , θ ∈ Θ, and L(s ) ⊂ Z.

Proof: Appendix A. Necessity comes from the reasoning above. Sufficiency is shown in Appendix A. The second condition comes from the upper bound on capital tax rates. The last condition ensures that consumption is non-negative and leisure is bounded by one. Notice that as opposed to Werning (2006) we do not need an implementability constraint for each agent. Given any aggregate allocation, since the government has access to lump sum taxation the budget constraint for the average agent can be made to hold with equality. Thus, we can drop the standard implementability constraint of Ramsey problems. In addition, because preferences are Balanced Growth, if the budget constraint is satisfied for one agent, it is satisfied for all of them. Thus, if we weren’t concerned about the non-negativity constraints or corner solutions, conditions 1 and 2 of Lemma 1 would be enough. However, for aggregate allocations that require large T , either positive or negative, it is likely that some individuals would choose corner solutions. Condition 3 ensures that almost all individual allocations are interior. Depending on the restrictions on the distribution of the initial endowment, we could relax the third condition to ω(Z; θ) ≥ 0. For example, this would be the case if initial endowments are non-decreasing in the skill level, making ω(·) strictly increasing in θ. Due to the homotheticity of preferences, Lemma 1 implies that two economies with different distributions of skills but the same mean and initial aggregate capital stock will have the same aggregate outcomes in equilibrium. Clearly, the distribution of ω in the economy will depend on the distribution of skills and the assumptions on initial endowments.

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4. Existence of Equilibrium: the Condorcet winner Let Ξ be the set of elements Z ≡ ({C(st ), L(st ), K(st )}st , T, τk0 ) that satisfy the conditions of Lemma 1. We begin by analyzing preference orderings over elements of Ξ. Given any Z ∈ Ξ, we can express the present discounted utility for each individual when aggregate allocations are given by Z. For agent type θ, denote this value by V (Z; θ). Then we have:

V (Z, θ) =

[ω(Z, θ)]1−σ V (Z) θ(1−α)(1−σ)

(13)

Given two competitive equilibrium allocations Z, Zˆ ∈ Υ, type θ prefers Z to Zˆ if and ˆ θ), or alternatively, log(V (Z; θ)/V (Z; ˆ θ)) ≥ 0. Equations (13) and only if V (Z; θ) ≥ V (Z; ˆ θ) as (11) can be used to compute the ratio V (Z; θ)/V (Z, V (Z; θ) = ˆ θ) V (Z;

[uc0 W0 (θ, T, τk0 ) + θUL (Z)] ˆ τˆk0 ) + θUL (Z)] ˆ [ˆ uc0 W0 (θ, T,

!1−σ

ˆ (Z) ΦV ˆ ΦV (Z)

!σ

Now we show under which conditions a Condorcet winner exists. The general strategy of the proof is similar to the one used to prove Proposition 2 in Benhabib and Przeworski (2006).

Proposition 1. (MVT) Assume balanced growth preferences in (2) and suppose the distribution of initial wealth is an affine function of the skill’s distribution, i.e., k0 (θ) = ν1 + ν2 θ. Consider any Z, Zˆ ∈ Ξ. If θm ∈ SZ,Zˆ , then either [θ, θm ] ⊆ SZ,Zˆ or [θm , θ] ⊆ SZ,Zˆ .

ˆ θm ) implies V (Z; θ) ≥ Proof. Take any Z, Zˆ ∈ Ξ; we shall show that V (Z; θm ) ≥ V (Z; ˆ θ) for at least 50% of the agents. A sufficient condition for this to happen is the V (Z; function

V (Z;θ) ˆ V (Z;θ)

being monotone.

ˆ τˆk0 ) can be written as W0 (θ, T, ˆ τˆk0 ) = a + Rνθ. Then consider the Notice that W0 (θ, T, following derivative:

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∂ log



V (Z;θ) ˆ V (Z;θ)

∂θ



"

ˆ uˆc0 UL + νRuc0 UˆL + Rν = (1 − σ) − ˆ τˆk0 ) + θUˆL] [uc0 W0 (θ, T, τk0 ) + θUL ] [ˆ uc0 W0 (T, " # ˆ uˆc0 ) a ˆuˆc0 (UL + Rνuc0 ) − auc0 (UˆL + Rν = (1 − σ) ˆ τˆk0 ) + θUˆL ] [uc0 W0 (θ, T, τk0 ) + θUL ][ˆ uc0 W0 (θ, T,

#

Therefore, the sign of the derivative does not depend on θ.

Next we highlight the key factors behind the proof of Proposition 1. First, as previously mentioned, given homothetic preferences, interior individual allocations of consumption and leisure are proportional to their respective aggregates. Therefore when comparing allocations ˆ the ratio of the proportionality factors ω(θ)/ˆ Z and Z, ω (θ) is key. Moreover, under the full insurance assumption the proportionality factors are constant over time and given by the value of the after-tax total wealth that individuals would have if they sold their entire endowment of labor to the firms. Since labor taxes are linear in labor income, the aftertax human wealth is linear in the productivity type. If there were no initial non-human wealth inequality, the function ω(θ)/ˆ ω (θ) would be monotone in the productivity type, and therefore, if the median type θm prefers Z to Zˆ then at least half of the remaining types will share her preferences over these two allocations. If the heterogeneity was only due to skills, the assumption in Proposition 1 would not be necessary.8 Which assumption can be relaxed? It is challenging to relax the assumption on the homotheticity of preferences. The main reason is that small perturbations on preferences would make the distribution of after-tax wealth in the economy matter significantly.9 Proposition 1 would also be true in an environment in which fiscal policy has no lumpsum component, but the government collects tax revenues in each period and redistributes 8

The linearity condition does not mean that the initial distribution of capital is itself linear. One kind of preferences that may be of interest for applied work is u(c) + v(1 − l), where both u(.) and v(.) are homothetic. Unfortunately, we are not able to prove Proposition 1 in that case. The proof breaks in the characterization of the market share ω. We cannot find a closed form solution for it as a function of the aggregates and θ. 9

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it trough a public good gt . In this case, the individual’s valuation of the public good should enter additively in the period utility function. 5. Characterization of the Condorcet Winner The characterization of the Condorcet winner comes from the maximization of the utility for type θm given that she has to pick a sequence of aggregate allocations, initial tax on capital and lump-sum transfers that can be supported as a competitive equilibrium. Therefore, to make sure that the conditions of Lemma 1 and Proposition 1 are satisfied, we assume the following: Assumption 1: The initial endowments are an affine and increasing function of skills among types: k−1 (θ) = γk + (K−1 − γk ) · θ with 0 ≤ γk ≤ K−1 . In addition, without a restriction on θ, Proposition 1 establishes the consensus result only for fiscal policies that support interior equilibria. Without any such restriction, some fiscal policies will generate aggregate allocations in which the decentralized competitive equilibrium exhibits a positive measure of agents supplying zero labor. Such aggregate allocations usually have the feature that the lump-sum component in the tax schedule is too large (a positive transfer), making it too costly for the lowest types to work. For more details see Piguillem & Schneider (2007). In order to avoid considering economies with non-interior allocations, we present a lemma used to impose a lower bound on the value of θ.

Lemma 2. Consider any Z satisfying conditions (1)-(2) of Lemma 1 and having both agf0 (θ, T, τ0 )/θ > 0. There exists θb < 1 gregate labor sequences bounded away from zero and W such that ω(Z; θ) ≤

1 1−L(st )

b and L(st ) ⊂ Z. for all st , θ ≥ θ,

Proof: Appendix A. Lemma 2 provides a minimum value for θ such that, even for the maximum feasible level of transfers −T > 0 (in a competitive equilibrium with aggregate labor bounded away from 13

zero), the lowest type will work a positive amount in any period and state of nature. It also imposes a restriction on the variance of skills distribution. Using Assumption 1, equation (11) states that individual labor supply is monotone in θ. f0 (θ, T, τ0 )/θ > 0 implies that the lowest skilled agent will have Notice that the condition W the lowest labor supply in all periods and states.10 b 1). Assumption 2: θ ∈ [θ, We partially characterize the solution to the following problem:

( P(M) :

max

{C,L,K,T,τ0 }

V (Z)

f0 (θm , T, τk0 ) − E[W f0 (θ, T, τk0 )]] + (θm − 1)UL [W +1 [(1 − σ)V (Z)]

)1−σ

    t t t t t−1 t  C(s ) + K(s ) + g(s ) ≤ F L(s ), K(s ), s + (1 − δ)K(st−1 ) (RC);      P  uc (st ) ≥ β st+1 Pr(st+1 |st )uc (st+1 )[1 + (1 − τ )(Fk (st+1 ) − δ)] (UB) s.t.   f0 (θ, T, τk0 ) − E[W f0 (θ, T, τk0 )]] + (θ − 1)UL + (1 − σ)V (Z) ≥ 0 (NN)  uc0 [W      τ ≤τ k0

The problem’s constraints ensure that the allocations can be implemented as a competitive equilibrium. The constraints (RC) and (U B) are standard. (RC) ensures that the allocation is feasible and (U B) discards allocations that require a capital tax higher than τ¯ to be implemented. Constraint (N N ) is non-standard and specific to our problem. It replaces the usual implementability constraint in Ramsey problems. As previously explained (see Lemma 1), we do not need an implementability constraint for each agent. The assumption on preferences and the availability of lump sum taxation ensures that the budget constraint for each agent is satisfied. However, the levels of lump sum taxation (or transfers) required to achieve this goal can be such that either consumption or leisure becomes negative for some 10

f0 (θ, T, τ0 )/θ < 0. We believe that in such cases the One may ask what would happen in cases where W labor supply will be strictly positive for the highest type. In this case we can show there exists θe > 1 such that the upper bound constraint in Lemma 1, Condition 3, will never bind. Furthermore, for the type of distribution that we analyze in the next section, the median voter will indeed prefer −T ≥ 0.

14

agents, rendering their feasibility sets empty. Assumption 2 and constraint (N N ) preclude this possibility. The objective function makes the trade-off between redistribution and efficiency faced by the median voter explicit. She cares not only about her share of the pie but also about the size of the pie. The indirect utility of the average agent, V (Z), captures the size of the pie. The agent’s share of the pie is represented by ω, which depends on the difference in after-tax total wealth between the mean and median type. Remark: If T ≤ 0 in the solution of P(M), then using E(ω(Z; θ)) = 1 constraint (NN) will not bind. The first thing to notice about P(M) is that if the distance between the mean and the median voter were zero, i.e. E(θ) = θm , and there were no government spending, taxes would be zero in all periods and states. The only way that the median voter can take advantage of nonzero marginal taxes is through the difference between the value of her wealth, human and non-human, and the mean wealth. When there is no difference she prefers marginal taxes to be zero. This would be true even if taxation were used to finance a public good. If government spending needs were sufficiently low, the chosen fiscal policy will use only lump-sum taxes to finance the stream of expenditures. Next, we turn the case where θm 6= E(θ). Lemma 3 establishes that constraint (NN) will never bind in the solution to P(M).

Lemma 3. If θm < 1 then T ≤ 0 in any solution to P(M). Proof: Appendix A. The intuition for Lemma 3 is simple. The main objective of the median voter is to achieve some degree of redistribution in her favor. This occurs only when the agent receives more resources than she pays. That is, because all the distortive taxes are linear and since θm < E(θ), she always pays less than the average agent. Thus, given that all agents receive 15

the same transfer, the only way for her to obtain some benefits from redistribution is to set the revenues from marginal taxation at a positive value (so that she pays less than the average agent) and the lump sum tax at a negative value (so that she receives the same as the average agent). When there is no government spending, the difference between payments and transfers is a net gain for the median agent. This in turn implies that at the optimal allocation ω(·) > θm . Next we state a lemma which will be used in Proposition 2. The result is an extension of the capital tax result in B&B. The notation st > st˜ denotes the histories that immediately follow st . Lemma 4. (The Bang-Bang Property) Assume σ ≤ (1 − θm )−1 . In the solution of the median voter’s problem, if there exists st˜ such that the implied tax τk (st˜) < τ then uc (C ∗ (st ), L∗ (st )) = β

X

˜

Pr(st+1 |st )[1 + Fk∗ (st+1 ) − δ]uc (C ∗ (st+1 ), L∗ (st+1 )) ∀ st > st

st+1

and therefore τk (st ) = 0 for all st > st˜. Proof: Appendix A. The proof of Lemma 4 depends on the return function in P(M) being increasing in the utility of the mean type. In general this might not be true (as B&B shows), but assumption σ ≤ (1 − θm )−1 ensures that it is indeed the case. When the condition fails the proof can be adapted to show that the constraint (UB) is always binding. The intuition behind Lemma 4 is straightforward. The term (1 − θm )−1 is decreasing in the potential gains from taxation (due to redistribution) for the median agent. When θm = 1 the potential gain is zero and when θm = −∞ it is infinity. Thus, the poorer the median agent, the more concerned she is about her share rather than the size of the pie. This effect interacts with her valuation of the allocations. The larger σ, the larger the value of getting closer to the mean with respect to future efficiency. 16

From the competitive equilibrium we know that FL (st )(1−τl (st )) = fore 1 − τl (st ) =

C(st )

(1−α) . E(θ)−L(st ) FL (st )α

C(st ) 1−α . E(θ)−L(st ) α

There-

Then we have the following:

Proposition 2. (Optimal Labor Tax) Suppose that σ ≤ (1 − θm )−1 and θm < 1. Then in P(M)’s solution there exists a history sbt such that, for all st > sbt the implied labor taxes are: 1. 0 < τl (st ) < 1. 2. τl (st ) depends on st only through L(st ): τl (st ) =

(1 − θm ) (θm − 1)[(1 − α)(1 − σ) − 1] + [1 − L(st )][(1 − σ) + σω(Z, θm )]

(14)

Proof: Appendix A. Since positive capital taxes reduce the payoff for the mean type, Lemma 4 implies that the first periods are sufficient to obtain full benefits from positive taxes on capital. After that, the median voter relies only on labor taxes to redistribute. Provided that skills inequality is not too large or, alternatively, σ ≤ (1 − θm )−1 , capital income taxes will eventually be zero. In contrast, labor income taxes are always positive and state dependent. Moreover, for all histories after which the upper bound constraint on capital taxes is not binding, labor taxes are given by (14). We also find a smoothing effect on labor taxes similar to Werning (2007). In particular, the elasticity of the labor tax with respect to aggregate labor is: ∂τl L [1 − σ(1 − ω)] = τl L > 0 ∂L τl 1 − θm where the inequality follows from σ ≤ (1 − ω)−1 The median voter increases taxes in “good times” and decreases them in “bad times”, that is, the labor tax schedule is pro-cyclical. This effect comes from the concavity of 17

the representative agent’s value function. For a given level of taxation, and therefore of redistribution, the median voter wants to smooth the sequence of consumption and leisure. The optimal way to dampen the movements in consumption and leisure sequences is to decrease (compensate) the distortions generated by labor taxation when other distortions affecting the leisure decision become larger. For instance, if the only source of uncertainty in the economy were aggregate productivity, the aggregate employment would be higher in booms and smaller in recessions. Thus, the median voter would increase taxes in booms and decrease them in recessions. The tax smoothing effect generates an unambiguously positive correlation between aggregate labor (and output) and taxation. This property of the fiscal policy it is not surprising, in fact, it is the standard finding in environments with a representative agent

11

. From this point of view, Proposition 2 can be seen as an extension

of the latter findings to an environment with heterogeneous agents. However, from equation (14) we can see that the key to the above argument is that the distribution of skills is constant over the business cycle. As it is apparent from equation (14), labor income taxes are potentially increasing in inequality. Thus, in environments where inequality changes over the business cycle, the validity of the above argument would depend crucially on the stochastic process followed by θ. In the next section we provide a simple extension that sheds light on this issue, while in Section 7 we show numerical simulations calibrated to six different countries. 6. Stochastic Labor Skills Suppose that agents’ initial types θ0i are distributed according to a skewed distribution on Θ = [θ, θ˙], but for each period after t = 0, skills of type i evolve stochastically, and may be correlated with the aggregate state. In particular, for each history st , skills are given by: 11

Chari et al. (1994) analyze the optimal (Ramsey) labor tax in a similar environment with the same preferences. They find that pro-cyclical labor taxes are optimal when risk aversion is high enough. More precisely, when σ < 6 they find that counter-cyclical labor taxes are optimal and when σ > 6 pro-cyclical labor taxation is optimal. For σ around 6, labor taxes are independent of the aggregate shocks in the economy.

18

θti (st ) = 1 − ρ(st )(1 − θ0i ). This specification allows for correlation between changes in the distribution of skills and aggregate productivity shocks. Notice that the process has been chosen such that the changes are a mean preserving spread and maintain the types’ ordering constant. It is straightforward to show a version of Proposition 2 under this extension. The result follows because the linearity restriction on initial types remains. By following the steps in the derivation of (14), it can be shown that the optimal marginal labor income tax is given by: τl (st ) =

ρ(st )(1 − θ0m ) ρ(st )(θ0m − 1)[(1 − α)(1 − σ) − 1] + [1 − L(st )][(1 − σ) + σω(Z, θ0m )]

(15)

By comparing equation (15) to (14) we see a new effect determining the business cycle properties of labor income taxes. This new effect is related to how the skills distribution changes over the business cycles. An increase in the distance between the mean and the median agent (a larger ρ) increases the gains from redistributive policies for the median voter, and therefore results in higher taxes. The intuition behind this result is similar to Basseto (1999) and Niepelt (2004). What the median agent cares about is the present value of transfers and the distortions generated to pay for them. Recall that in this economy, because of the Ricardian equivalence, the timing of lump sum transfers is irrelevant. As a result, the cyclical movement of taxes is only related to the intertemporal allocation of the burden of providing redistribution. More inequality reflects the fact that the median agent has become relatively less efficient than the average agent, and less inequality that she has become relatively more efficient. Thus, the median voter chooses to raise the level of taxation in periods in which she is relatively less efficient. In other words, the median voter shifts the burden of taxation to the periods in which she contributes less. We call this the intertemporal tax shifting effect. To understand how the tax shifting effect interacts with the smoothing effect, we compute the implied correlation between τl and L using a first order linear approximation around the 19

non-stochastic steady state: Cor(τl , L) =

 τ¯l  τ¯l [1 − σ + σω] std(L) [1 + τ¯l ((1 − α)(1 − σ) − 1)]Cor(ρ, L)std(ρ) + std(τl ) 1 − θ0m

(16)

where std(x) is the standard deviation of variable x and τ¯l is the steady state level of τl . The first thing to notice is that if std(ρ) = 0, the correlation is unambiguously positive, as in Section 5. However, when std(ρ) > 0 the sign depends on the sign of Cor(ρ, L) and the volatility of employment, std(L). If inequality and employment are positively correlated, the effects reinforce each other and labor taxes remain positively correlated with employment. However, if inequality rises in periods of low employment both effects act in opposite directions, making the sign of the correlation between employment and labor taxation ambiguous. The larger the relative volatility of inequality, the larger this effect. In other words, it could be optimal for a committed society to choose pro-cyclical fiscal policy, especially when recessions are accompanied by sizeable increases in inequality. 7. Quantitative Results In this section we present quantitative implications of the theory. We construct series for T F P and inequality for six countries: Argentina, Brazil, Chile, Canada, the United Kingdom and the US. The choice of countries is solely based on the availability of data for a sufficiently long period of time, to allow for meaningful computation of their statistical properties. Even with this reduced sample, we have to constrain the time period of our analysis to 1975-2009. Frequent (at least yearly) measures of inequality before 1975 are virtually non-existent. The T F P and inequality series provide estimates of the stochastic properties of A and θ. We calibrate the rest of the parameters, using the same value for each country, with standard values from the literature. 7.1. Data The median to mean ratio of skills, θ is the key variable determining the cyclical behavior of the equilibrium tax sequence. A suitable empirical counterpart would be the ratio of the 20

median gross hourly wages to its average. However, there are two drawbacks to this approach. First, measures of gross hourly earnings do not consider unemployed agents, even though they vote. There is ample evidence that the main factor driving inequality over the business cycle is precisely changes on unemployment.12 Second, data for a sufficiently long span of time and with consistent methodologies across countries is, to the best of our knowledge, unavailable. One measure of income inequality that is commonly studied and widely available is the Gini index. Nevertheless, the mapping from the Gini index to θ depends on the actual distribution of income in each particular country. The mapping could be one-to-one or there could be multiple mappings depending on additional parameters. The Pareto and the Lognormal distributions are widely used in studies of income inequality and provide a one-to-one mapping. Both distributions approximate of the upper tail almost perfectly. However, the Pareto distribution grossly fails to account for the shape of the lower tail. Because of this, we assume that the income distributions of each country are Log-normal. σx The Gini index for a variable x whose distribution is Log-normal is given by 2Φ( √ )−1 2

where σx is the standard deviation of x and Φ is the standard Normal distribution, and the 2 σx −1 Gini+1 median to mean ratio is e− 2 . Therefore, for each country we set θ = e−Φ ( 2 ) . A potential drawback of this strategy is that the changes in the Gini index could reflect changes in the distribution of income that do not necessarily affect the median to mean ratio. We have used gross earnings for Argentina, Chile, Canada and the US to compute θ. The main qualitative results shown in the next subsection remain the unchanged.13 12

See, for instance, Heathcote et al. (2010) for a comprehensive study of income inequality in the US. They find that inequality rises during recessions, and that the increase is mainly due to the increase in unemployment. 13 For instance, Guvenmen et al. (2012) show that in the last recession, the top 1% of the income distribution experienced the largest income drops. Since this 1% is not generally included in the usual surveys, CPS, PSID, etc., their findings are consistent with what other papers have found, e.g. Heathcote et al. (2010). Guvenmen et al. (2012) find that: 1) the variance of idiosyncratic shocks does not change much in recessions but instead the probability of large income drops increases, i.e. unemployment, 2) the higher the pre-recession income the lower the increase in the probability of an income drop. The fact that the top 1% experience more acute income drops during recessions tends to decrease income inequality. In this sense, if should weaken the negative correlation between inequality and TFP.

21

As mentioned before, due to the availability of data we restrict the sample to 19752009 and we compare the patterns for six countries, three developed (Canada, the United Kingdom and the US) and three developing (Argentina, Brazil and Chile). The statistical offices of the US (Bureau of Census, Table IE-1, 1975-2009), Canada (CANSIM, Table 2020705, 1976-2009) and the UK(National Statistics, 1977-2009) provide annual estimates of the Gini Index before taxes and transfers. For Argentina we use the estimated Gini index for gross per-capita incomes from the Encuesta Permanente de Hogares. For Brazil and Chile, we use the Standardized World Income Inequality Database.14 Although Brazil and Chile are two of the developing countries with the longest available data series, there are some missing values. In the case of Brazil we interpolate the four missing values using adjacent years. For Chile, we instead interpolate the missing values using median to mean ratio provided in the Encuesta de Ocupaci´on and Desocupaci´on del Gran Santiago. These interpolation strategies decrease the variance of the series. From this point of view they do not favor our results.

Figure 1: Gini coefficients in selected countries (1975-2009)

  68

68

63

63

58

58

53

53

US Canada UK

48

48

Argentina Brazil

43

43

Chile 38 1975

1980

1985

1990

1995

2000

2005

38 1975

2010

1980

1985

Year

1990

1995

2000

2005

2010

Year

   

In Figure 1 we show the original series for the Gini indexes. Two characteristics are

immediately apparent: the level of inequality and its volatility are larger in the poorer countries of the sample. Notice that the difference in levels seems to have diminished in 14

See http://myweb.uiowa.edu/fsolt/swiid/swiid.html for further details.

22

recent year. Brazil, Chile, Canada and the UK have Gini indexes of around 0.53 in 2009, while those Argentina and the US are around 0.46. Despite the trends apparent convergence, the inequality series show very different patterns during the period under consideration. Since we focus on cyclical behavior, we detrend the variables using the Hodrick-Prescott filter. As Marcet & Ravn (2004) point out, making cross-country comparisons using the same smoothing parameter in the HP filter for each country might lead to cycles in some countries that are either too smooth or too volatile. To deal with this potential problem we apply the Marcet & Ravn (2004) method to choose the smoothing parameters.15 We construct the Total Factor Productivity (TFP) series in each country in the standard way (T F P =

GDP ) K ν L1−ν

using GDP from the Penn World Tables 7.1 in constant prices of

2000. K is constructed using private investment, with the perpetual inventory method and assuming a depreciation rate of 5% per year. The data for investment is extracted from the Penn World Tables 7.1. L is total hours worked from the Total Economy Database. As with the inequality measure, we detrend T F P using the Marcet & Ravn (2004) method. Table 1 presents statistical properties of the constructed series. First, notice that for all countries in the sample, inequality and aggregate productivity are negatively correlated, that is, inequality increases in recessions and decreases in booms. Second, the volatility of inequality is higher in the less developed countries. In particular, the standard deviations for the Latin-American countries are between 2 and 3.4 times larger than in the developed countries. Using the intuition of equation (16), the necessary conditions for an optimal pro-cyclical fiscal policy are present in the data, and more so in the poorer countries of the sample. 15

The problem with using the same smoothing parameter for each country is that the signal to noise ratio of the cycles can be very different. For instance, if the cycles in Argentina are longer than the cycles in the US and we use the same smoothing parameter, the filter will shorten the cycles of Argentina, allocating part of the information to the trend. The Marcet & Ravn (2004) method adjusts the smoothing parameter in such a way that the noise to signal ratio remains constant across countries. This method requires a country as “numerarie”. We chose the US, and following Ravn & Ulhig (2002), we assign it a value of 7. The results are not sensitive to the alternative value of 100.

23

Table 1: Inequality measures. Smoothing Country

Gini1

θ01

Std(ρt )2

parameter

Cor(ρt , T F Pt )3

Argentina

0.46

0.69

0.046

12.6

-0.36

Brazil

0.58

0.51

0.037

7.3

-0.02

Chile

0.54

0.58

0.033

4.4

-0.13

Canada

0.49

0.65

0.022

10.9

-0.55

UK

0.50

0.63

0.018

9.7

-0.19

US

0.44

0.71

0.013

7.0

-0.10

1) Averages over the period 1975-2009. 2) Standard deviation of the cyclical component. The cyclical component is calculated as the difference between the log of the variable and its trend computed using the Holdrick-Prescott filter with the smoothing parameter shown in the fourth column. 3) Correlation between the cyclical component of ρ and the cyclical component of T F P .

7.2. Calibration We assume that uncertainty is described by a Markov chain with 9 states. Both T F P and labor skill shocks are assumed to take three possible values: A(s) ∈ {AL , AM AH } and ρ(s) ∈ {ρL , ρM , ρH }. To calibrate the values of A and ρ in each state and the Markov transition matrix for each country, we first estimate a VAR of order one in the pair composed of the cyclical component of ρ and T F P . Next we apply the multivariate version of Tauchen’s (1986) method to approximate the estimated coefficients for each country’s VAR with the nine states. The calibrated values of A, ρ and the transition matrix, together with the Matlab code that performs the calibration for each country are available upon request. The production function is Cobb-Douglas with capital share ν = 0.3, a standard value in the literature. The depreciation rate is set to δ = 0.05. For the baseline economy we use σ = 1 and we show alternative calculations with σ = 2. The parameter α is set to match L = 1/3 on average during the period considered for the US economy. Using this criterion, we find α = 0.45. It is worth mentioning that even though we keep this parameter constant across countries, the model generates only small differences in the proportion of leisure time

24

devoted to market work. Finally, the discount factor is set to 0.96, implying a annual interest rate of 4%. The parameters are summarized in Table 2. Table 2: Summary of the calibrated common parameters. Parameter (α, σ)

Description preference parameters

Value (0.45,1)

β

intertemporal discounting

0.96

ν

capital share

0.3

δ

depreciation

0.05

Usually, the cyclical properties of fiscal policy are defined according to the correlation between the fiscal surplus and some measure of GDP gap, where primary surplus is defined as: Tax revenue -Transfers. Therefore we define Surplust = Lt Wt τlt + Tt The Ricardian Equivalence holds in this economy, and therefore only the total amount of transfers T and not their distribution over time matters. We assume that transfers are equally distributed over time, that is Tt = (1 − β)T . Fiscal policy is counter-cyclical if the correlation between ∆Surplust /Yt and Yt is positive, (save in good times and borrow in bad times), and pro-cyclical if the correlation is negative.16 Since our main concern is labor taxes, initial wealth heterogeneity would add little content to the discussion at a large computational cost. Therefore, we do not consider heterogeneity in initial wealth. Still, since the problem is not recursive, we solve it using a two-step algorithm that explores the recursive property of the Lagrangian. For more details see Appendix B. 16

A previous version of this paper included results with endogenous government spending in the definition of surplus. We found that counter-cyclical government spending can make the government surplus countercyclical. However, there is ample evidence that government spending is in fact pro-cyclical. Since this fact is beyond the scope of this paper we have excluded those results. Providing a more precise answer on the overall behavior of the surplus would require modeling government spending in a more detailed and precise fashion as in Bachman et al. (2011).

25

7.3. Results: the optimality of pro-cyclical fiscal policy Next we highlight some statistical properties of the calibrated economies. In order to understand the implications of equation (16) we first present the implied correlations between labor taxes and aggregate employment in the calibrated economies (Table 3). We do this for two different specifications: one with constant labor skills and other with stochastic labor skills. The model with constant skills yields almost zero variation in labor taxes and remarkably similar across countries, in line with the findings of Chari et al. (1994). In addition, as shown in Section 5, the model with constant skills yields a correlation between labor and taxes equal to one for all countries in the sample. For the specification with stochastic labor skills, since the correlation between inequality and T F P is negative, the sign of the correlation between employment and labor taxes is a priori ambiguous. Table 3 shows that the tax shifting effect dominates in the calibrated model and the correlation between aggregate labor and labor income taxes becomes negative for all countries except the US. The positive correlation remains only in the US economy, where the volatility of inequality is small relative to the volatility of aggregate labor. Notice that the volatility of taxes increases as well. The standard deviations of the tax are still small but they become approximately twice as large in the richest and three times as large in the poorest countries of the sample. Finally, in the last column we present a sensitivity exercise using σ = 2. All other parameters are the same. The last column of Table 3 shows that the change in the sign of the correlations still happens.

26

Table 3: Labor income taxes and the business cycle. Constant Skills Country

Stochastic Skills

σ=2

Std(τ ) %

Std(L) %

Cor(τ, L)

Std(τ )%

Std(L) %

Cor(τ, L)

Cor(τ, L)

Argentina

0.3

0.9

1.00

1.1

1.1

-0.42

-0.52

Brazil

0.3

0.8

1.00

0.9

1.0

-0.35

-0.43

Chile

0.2

0.6

1.00

0.7

0.9

-0.40

-0.51

Canada

0.2

0.5

1.00

0.5

0.8

-0.56

-0.67

UK

0.2

0.5

1.00

0.4

0.6

-0.29

-0.41

US

0.3

0.9

1.00

0.4

0.9

+0.57

+0.48

Since the environment is frictionless, we call the outcome of the Median Voter’s problem an optimal policy because it arises when the social planner puts positive weight only on the median agent. As previously mentioned, there has recently been a debate about the procyclicality of fiscal policies in developing countries, see especially Alesina et al. (2008) and Ilzetzki & Vegh (2008). Such pro-cyclicality is considered a puzzle, as it is understood that fiscal policies should (if optimal) be counter-cyclical. The numerical simulations presented in Table 3 suggest that this is not necessarily true. The desire of the median voter to shift the burden of taxation to periods in which her contribution is smaller might reverse this intuition, so that the puzzle would be why fiscal policy is counter-cyclical. So far the analysis has provided intuition on how different exogenous shocks affect the optimal policy through equation (15). Next, we analyze their consequences for other commonly used indicators of fiscal policy cyclicality. Two commonly used indicators are the correlation between revenue and GDP and the correlation between the change in surplus and GDP.

27

Table 4: Revenues, Surplus and the business cycle. Constant θ (σ = 1) Cor(τ, Y )

    Cor ∆ Surp. ,Y Y

Argentina

+0.90

Brazil

σ=1

σ=2

Cor(τ, Y )

    Cor ∆ Surp. ,Y Y

Cor(τ, Y )

    Cor ∆ Surp. ,Y Y

+0.52

-0.26

+0.01

-0.32

-0.02

+0.84

+0.53

-0.10

-0.22

-0.14

-0.23

Chile

+0.79

+0.45

-0.28

-0.17

-0.32

-0.19

Canada

+0.76

+0.41

-0.60

-0.62

-0.65

-0.59

UK

+0.68

+0.44

-0.23

-0.35

-0.28

-0.36

US

+0.91

+0.55

+0.56

+0.39

+0.50

+0.37

Country

The first column of Table 4 shows the correlation between the labor tax and GDP. Since labor taxes are the only source of revenue in our model economy, looking at labor taxes or revenue is equivalent. Not surprisingly, since GDP and aggregate labor are highly correlated (see Table 5) we observe the same pattern as in Table 3. When skills are constant over the business cycle, fiscal revenue is counter-cyclical in all countries and becomes pro-cyclical when inequality changes over the business cycle. The second column of Table 4 presents the correlation between the change in the ratio revenue/GDP with the GDP gap. Notice that this indicator inherits the properties of the revenue in almost all countries, except for Argentina where the change in the surplus shows zero correlation with GDP. That is, the reduction in employment in recessions is not enough to compensate for the increase in taxes. Finally, notice that there are only minor differences in the implied correlations between the main aggregate variables across models with and without stochastic skills and across calibrations (see Table 5).

28

Table 5: Correlations between aggregate variables over the business cycle. Constant θ (σ = 1) Country

σ=1

σ=2

Cor(L, Y )

Cor(C, Y )

Cor(I, Y )

Cor(L, Y )

Cor(C, Y )

Cor(I, Y )

Cor(L, Y )

Cor(C, Y )

Cor(I, Y )

Argentina

+0.90

+0.48

+0.93

+0.89

+0.51

+0.93

+0.88

+0.58

+0.93

Brazil

+0.84

+0.34

+0.87

+0.80

+0.35

+0.88

+0.79

+0.47

+0.87

Chile

+0.79

+0.43

+0.84

+0.81

+0.43

+0.86

+0.81

+0.49

+0.87

Canada

+0.75

+0.39

+0.81

+0.85

+0.31

+0.87

+0.86

+0.35

+0.88

UK

+0.68

+0.37

+0.75

+0.72

+0.40

+0.76

+0.74

+0.41

+0.80

US

+0.91

+0.45

+0.94

+0.90

+0.46

+0.93

+0.91

+0.67

+0.94

To summarize, we find that pro-cyclical labor taxes are optimal only for the US, where tax smoothing incentives are stronger than tax shifting effects. For the rest of the countries in our sample the tax shifting effects appear to be a powerful force driving the cyclical properties of fiscal policy. The numerical simulations provide evidence that pro-cyclical fiscal policy, at least in terms of revenue, should be the norm rather than exception. Including government spending, if it were highly counter-cyclical, could potentially overturn our results. However, the empirical evidence shows that in most countries government spending is either weakly counter-cyclical or pro-cyclical (Alesina et al. (2008) and Ilzetzki & Vegh (2008)). 8. Conclusion We view the results regarding labor taxes as a neat characterization of an important component of fiscal policy. Most studies analyzing inefficiencies due to political constraints have followed the route of making strong ex ante assumptions about the institutional environment (e.g. specific game theoretic models of voting over a restricted set of tax instruments). This makes it hard to interpret the results since it is not clear whether the properties of the policies singled out as equilibria are chosen due to the institutional arrangements or are due to the assumed frictions. Allowing for the possibility of heterogenous agents may drastically 29

change the intuitions drawn from environments with a representative agent. It could also render the need for additional frictions to explain particular features of the data unnecessary. The fact that a pro-cyclical fiscal policy can be optimal with only heterogeneity and without any additional imperfections should make this point clear. When skills evolve stochastically over time, a temporary increase in inequality could imply either higher or lower labor taxes, depending on both the sign and magnitude of the correlation between inequality and aggregate productivity. We show that both possibilities are empirically plausible. On the one hand, if recessions generate a sufficiently large increase in labor income inequality, it could be optimal to shift the burden of taxation to those periods. On the other hand, if the increase in inequality is small it could be optimal to decrease taxes to smooth inefficiencies. Thus, this approach can be useful to explain potential empirical puzzles like the observed pro-cyclicality of fiscal policies in several countries. Finally, the findings presented might be useful to deal with other kinds of problems in dynamic economies. As is clear from Proposition 1, the existence of a Condorcet winner is a consequence of the Single Crossing Property. In this sense, we provide a set of sufficient conditions under which this property holds. For instance, the Single Crossing Property could be used in environments in which voting occurs sequentially, or in economies in which agents decide over objects other than income taxes.

30

[1] Alesina, A., F. Campante, and G. Tabellini (2008): “Why is Fiscal Policy Often Procyclical?,” Journal of the European Economic Association, 6, 1006–1036. [2] Alesina, A., and D. Rodrik (1994): “Distributive Politics and Economic Growth,” The Quarterly Journal of Economics, 109, 465–490. [3] Azzimonti, M., E. de Francisco, and P. Krusell (2006): “Median-voter Equilibria in the Neoclassical Growth Model under Aggregation,” Scandinavian Journal of Economics, 108(4), 587–606. [4] Azzimonti, M., E. De Francisco, and P. Krusell (2008): “Aggregation and Aggregation,” Journal of the European Economic Association, 6, 381–394. [5] Bachman, R., and J. Bai (2011): “Public Consumption over the Business Cycle,” NBER Working Paper No. 17230. [6] Bassetto, M. (1999): “Optimal Fiscal Policy with Heterogeneous Agents,” mimeo. [7] Bassetto, M., and J. Benhabib (2006): “Redistribution, Taxes, and the Median voter,” Review of Economic Dynamics, 9(2), 211–223. [8] Benhabib, J., and A. Przeworski (2006): “The Political Economy of Redistribution under Democracy,” Economic Theory. [9] Chari, V., L. Christiano, and P. Kehoe (1994): “Optimality fiscal policy in a business cycle model,” Journal of Political Economy, 102, 617–652. [10] Chari, V., and P. J. Kehoe (1999): “Optimal Fiscal and Monetary Policy,” in J.B. Taylor and M. Woodford (ed.), Handbook of Macroeconomics. [11] Corbae, D., P. D’Erasmo, and B. Kuruscu (2009): “Politico-economic consequences of rising wage inequality ,” Journal of Monetary Economics, 56, 43–61. [12] Gans, J., and M. Smart (1996): “Majority voting with single-crossing preferences,” Journal of Public Economics, 59, 219–237. 31

[13] Guvenen, F., S. Ozkan, and J. Song (2012): “The Nature of Countercyclical Income Risk,” NBER Working Paper No. 18035. [14] Heathcote, J., F. Perri, and G. Violante (2010): “Unequal We Stand: An Empirical Analysis of Economic Inequality in the United States: 1967-2006,” Review of Economic Dynamics, 13, 15–51. [15] Ilzetzki, E., and C. Vegh (2008): “Procyclical Fiscal Policy in Developing Countries: Truth or Fiction?,” NBER Working Paper, 14191. [16] Inman, R. (1987): “Markets, Government, and the ’New’ Political Economy,” Handbook of Public Economics. [17] Kramer, G. (1973): “On a Class of Equilibirum Conditions for Majority Rule,” Econometrica, 41(2). [18] Krusell, P., and J. Rios-Rull (1999): “On the Size of U.S. Government: Political Economy in the Neoclassical Growth Model,” The American Economic Review, 89, 1156– 1181. [19] Marcet, A., and M. O. Ravn (2004): “The HP-Filter in Cross-Country Comparisons,” CEPR Discussion Papers N 4244. [20] Meltzer, A., and S. Richard (1981): “A Rational Theory of the Size Government,” Journal of Political Economy, 89, 914–927. [21] Milgrom, P., and I. Segal (2002): “Envelope Theorems for Arbitrary Choice Sets,” Econometrica, 70(2). [22] Mirrlees, J. (1971): “An Exploration in the Theory of Optimum Income Taxation,” Review of Economic Studies, 38, 175–208. [23] Nieplet, D. (2004): “Tax smoothing versus tax shifting,” Review of Economic Dynamics, 7(1), 27–51. 32

[24] Persson, T., and G. Tabellini (1994): “Is Inequality Harmful for Growth?,” American Economic Review, 84, 600–621. [25] Piguillem, bution,

F.,

and

A.

L.

Schneider (2007):

optimal fiscal policy and corner solutions,”

“A note on redistri(www.econ.umn.edu/

∼

als/Research/T axescornersol.pdf ). [26] Ravn, M., and H. Uhlig (2002): “On adjusting the Hodrick-Prescott filter for the frequency of observations,” The Review of Economics and Statistics, 84(2), 371–375. [27] Tauchen, G. (1986): “Finite state markov-chain approximations to univariate and vector autoregressions,” Economics Letters, 20(2), 177–181. [28] Werning, I. (2006): “Optimal Fiscal Policy with Redistirbution,” Quarterly Journal of Economics. [29] Yared, P. (2010): “Politicians, Taxes, and Debt,” Review of Economic Studies, 77(2), 806–840.

33

Appendix A. Proofs Appendix A.1.

Proof of Lemma 1:

For the necessity part of the Lemma, it remains to show that the individual shares integrate to one regardless of the normalization of the initial price p0 . Recall, that ω(θ) is such that c(st ; θ) = ω(θ)C(st ) and θ − l(st ; θ) = ω(θ)[1 − L(st )] [C(st )α (1−L(st ))1−α ]1−σ . Replacing prices and individual allocations in the budget constraint for Let u(C(st ), L(st )) ≡ 1−σ each agent θ yields:

X

f0 (θ, T, τk0 ) β t Pr(st )ω(θ)α(1 − σ)u(C(st ), L(st )) = uc0 W

t

+

X

 −σ β t Pr(st )(1 − α) C(st )α (1 − L(st ))1−α

t

X



C(st ) (1 − L(st ))

α  
ω(θ) θ 1− 1 − L(st ) θ

f0 (θ, T, τk0 ) β t Pr(st )ω(θ)α(1 − σ)u(C(st ), L(st )) = uc0 W

t

+

X t

Let UL (Z) =

P

t

 1−σ X (1 − α) θ − ω(θ) β t Pr(st )(1 − α)(1 − σ)u(C(st ), L(st )) β t Pr(st ) C(st )α (1 − L(st ))1−α t 1 − L(s ) t

 1−σ (1−α) β t Pr(st ) C(st )α (1 − L(st ))1−α . Then, 1−L(st ) (1 − σ)ω(θ)

X

f0 (θ, T, τk0 ) + θUL (Z) β t P r(st )u(C(st ), L(st )) = uc0 W

t

or f0 (θ, T, τk0 ) + θUL (z) (1 − σ)ω(θ)V (Z) = uc0 W where V (Z) =

P

t

β t P r(st )u(C(st ), L(st )).

Thus, we obtain equation (11), ω(θ) =

f0 (θ, T, τk0 ) + θUL (Z)] [uc0 W (1 − σ)V (Z)

In what follows below, let Fl (st ) and Fk (st ) be the marginal product of labor and capital respectively. The intratemporal optimality condition for each individual can be expressed as:

(1 − L(st ))(1 − τl (st ))Fl (st ) =

1−α C(st ) α

Then:

C(st )

=

α[(C(st ) − Fl (st )L(st )) + τl (st )Fl (st )L(st ) + (1 − τl (st ))Fl (st )]

C(st )

=

α[(−K(st ) + (1 + Fk (st ) − δ)K(st−1 ) ± τk (st )[Fk (st ) − δ]K(st−1 ) + τl (st )Fl (st )L(st ) + (1 − τl (st ))Fl (st )]

Pr(st )β t (1 − σ)u(st )

=

Pr(st )β t uc (st )[−K(st ) + (1 + (1 − τk (st ))(Fk (st ) − δ))K(st−1 ) + Tˆt + (1 − τlt )Fl (st )]

34

where Tˆt ≡ τk (st )[Fk (st ) − δ]K(st−1 ) + τl (st )Fl (st )L(st ). Note that from the first to the second line we have used the fact that the production function is Cobb-Douglas, i.e, F (st ) = Fk (st )K(st ) + Fl (st )L(st ) plus the feasibility constraint. Since the final expression above is true in each period, using the intertemporal optimality conditions we have:

(1 − σ)

X

Pr(st )β t u(st ) = uc (s0 )(1 + (1 − τk0 )(Fk0 − δ))K0 − T +

t,st

or

X

Pr(st )β t ul (st )

t,st

h

i P uc (s0 )(1 + (1 − τk0 )(Fk0 − δ))K0 − T + t,st Pr(st )β t ul (st ) 1= P (1 − σ) t,st Pr(st )β t u(st )

But this is equivalent to

R θ

ω(θ)f (θ)dθ = 1.

Next we prove the sufficiency of conditions (1)-(3). First, use the function ω(·) and Z to construct the individual consumption and labor allocations. Set after-tax prices as:

p(st ) = β t Pr(st )uc (st );

p(s0 ) = uc (s0 )

p(st )w(st )(1 − τl (st )) = β t Pr(st )ul (st ) And r(st ) = Fk (st ); w(st ) = FL (st ); p(st ) =

X

p(st+1 )R(st+1 )

st+1

If 0 < ω(θ) ≤

θ 1−L(st )

for all st that the following necessary first order conditions are met for all st and l ∈ [0, 1]:

   Ul (c∗ (st ; θ), 1 − l∗ (st ; θ)/θ) + Uc (c∗ (st ; θ), 1 − l∗ (st ; θ)/θ) w(st )(1 − τl (st )) [l − l∗ (st ; θ)] ≤ 0

The transversality condition (Tvc) limt→∞

P

st

p(st )k(st ; θ) = 0 is satisfied because individual capital allocations are an affine

function of the aggregate capital stock. At the equilibrium prices, the Tvc is met, since the aggregate allocations are bounded in the product topology. Finally, using (11), we can recover the budget constraint for each θ. Condition (2) in the competitive equilibrium definition is satisfied by construction. As usual, the government budget constraint holds because of Walras’ law. Taxes on capital can be constructed in many ways, and taxes on labor are constructed using the definition of prices and w(st ) = FL (st ) 

Appendix A.2.

Proof of Lemma 2:

e be the set of allocations Z ≡ ({C(st ), L(st ), K(st )}t≥0 , T ≤ 0, τk0 ) with aggregate labor allocation bounded away Let Ξ from zero, the resources constraint satisfied for all periods, and the Euler equation satisfied with weak inequality. e let L(Z) ≡ inf{L(st )}t≥0 and define θ(Z) to be the solution to: For any Z ∈ Ξ,

inf

  θ ∈ [0, 1] θ s.t.  ω(Z; θ) <

e Claim: θ(Z) is bounded away from 1 for all Z ∈ Ξ.

35

1 1−L(Z)

Proof of the claim: Because of the linearity of ω(·) in types, it follows that ω(Z; θ = 1) = 1. {L(st )}t≥0 is bounded away from zero, and therefore

1 1−L(Z)

≥

1 1−

for some  > 0. The claim follows.

e Because of the claim, θb < 1. Then it is straightforward to check that θb has the property Define θb ≡ sup {θ(Z) : Z ∈ Ξ}. stated in the Lemma. In particular, if θ(Z) satisfies the second constraint in the inf problem above, then it satisfies that constraint for all L(st ) ⊆ Z) 

Appendix A.3.

Proof of Lemma 3:

Let Z ∗ be the allocation solving P (M ) and let Π∗ = {τk∗ (st ), τl∗ (st ), T ∗ } be the implied optimal fiscal policy. By contradiction, suppose that T ∗ > 0. Recall that the aggregate allocations coincide with the solution of the average agent, i.e., the agent with θ = 1. In addition, the optimal choices of the average agent do not depend (directly) on T , they depend only on the ˜ = {τ ∗ (st ), τ ∗ (st ), T ∗ − }, with  > 0, be an alternative fiscal policy. Assume that  is sufficiently marginal taxes. Thus, let Π k l ˜ be the competitive equilibrium allocation associated to Π. ˜ Then , clearly Z ˜ = Z ∗ , which in turn small such that T ∗ − ≥ 0. Let Z ˜ T ∗ , θm ) = ω(Z ∗ , T ∗ , θm ). ˜ = V (Z ∗ ). In addition, since ω is decreasing on T for all θ < 1: ω(Z, ˜ T ∗ − , θm ) > ω(Z, implies V (Z) Which contradicts Z ∗ being a solution to P (M ). Notice that if T ≤ 0 the proof would not go through. To decrease T could require increasing distortionary taxation in some periods or states. Lemma 3 states that at the solution the Median Voter receives some redistribution. In other words, the taxation policy increases her share in the economy. The following Lemma, which will be used later, proves this statement formally Lemma 5: If θm < 1 then in any solution of P(M) we have θm ≤ ω(Z, θm ) < 1. Proof. Clearly ω(Z, θm ) < 1 when θm < 1, so we only need to show that θm < ω(Z, θm ). Since T ≤ 0, it must be true that:

0

≤

uc0 [−(θm − 1)γk + (θm − 1)T ]

=

uc0 [(θm − 1)(K0 − γk ) − (θm − 1)K0 + (θm − 1)T ] h i ˜ 0 (θm , T, τk0 ) − E[W ˜ 0 (θ, T, τk0 )] − (θm − 1)uc0 E[W ˜ 0 (θ, T, τk0 )] uc0 W i h   ˜ 0 (θm , T, τk0 ) − E[W ˜ 0 (θ, T, τk0 )] − (θm − 1) uc0 E[W ˜ 0 (θ, T, τk0 )] + UL (Z) + (θm − 1)UL (Z) uc0 W h i ˜ 0 (θm , T, τk0 ) − E[W ˜ 0 (θ, T, τk0 )] − (θm − 1)(1 − σ)V (Z) + (θm − 1)UL (Z) uc0 W

= = =

˜ 0 (θ, T, τk0 )] + UL (Z) from the MKT constraint. The last inequality can be written as Where (1 − σ)V = uc0 E[W

h i ˜ 0 (θm , T, τk0 ) − E[W ˜ 0 (θ, T, τk0 )] + (θm − 1)UL (Z) ≥ (θm − 1)(1 − σ)V (Z) uc0 W

or, h i ˜ 0 (θm , T, τk0 ) − E[W ˜ 0 (θ, T, τk0 )] + (θm − 1)UL (Z) uc0 W (1 − σ)V (Z)

36

+ 1 = ω(Z, θm ) ≥ θm

Appendix A.4.

Proof of Lemma 4 and Proposition 2

In order to prove both Lemma 4 and Proposition 2 first we need some intermediate results. The objective function for the median voter problem is given by:

( m

V (Z, θ ) = V (Z)

where V (Z) =

P

t

f0 (θm , T, τk0 ) − E[W f0 (θ, T, τk0 )]] + (θm − 1)U L + (1 − σ)V (Z) uc0 [W [(1 − σ)V (Z)]

β t P r(st )u(st ) and UL =

P

t

)1−σ (A.1)

β t Pr(st )uL (st ).

The partial derivatives are given by:

VˆC (st ) = β t Pr(st )[ω(Z, θm )]1−σ

VˆL (st ) = β t Pr(st )[ω(Z, θm )]1−σ





(1 − σ) ω(Z, θm )



  (θm − 1)(1 − α) + 1 + σ uc (st ) 1 − L(st )

1−σ (θm − 1)[(1 − α)(1 − σ) − 1] + +σ [1 − L(st )]ω(Z, θm ) ω(Z, θm )



(A.2)

uL (st )

(A.3)

With some abuse of notation, let:

a(st ) = [ω(Z, θm )]1−σ

Lemma 6: If θm < 1 and 1 < σ ≤



(1 − σ) ω(Z, θm )

1 , 1−θ m



  (θm − 1)(1 − α) +1 +σ , t 1 − L(s )

b(θm ) =

−(θm − 1) (1 − L(st ))ω(Z, θm )

then a(st ) > 0 for all st in any solution of P(M) .

Proof. First, consider the case σ > 1. a(st ) is greater than zero as long as: (1 − σ) ω(Z, θm )



 (θm − 1)(1 − α) + 1 +σ >0 1 − L(st )

or σω(Z, θm ) (θm − 1)(1 − α) +1< t 1 − L(s ) σ−1 Since (θm − 1) < 0 the inequality above is true as long as σ ≤ σ≤

1 1−θ m

1 . 1−ω(Z,θ m )

From Lemma 5

1 1−θ m

<

1 , 1−ω(Z,θ m )

therefore

is a sufficient condition

Appendix A.4.1.

Proof of Lemma 4

The following slightly modifies the proof in Bassetto and Benhabib (2006). If the claim is not true, then {C ∗ (st ), K ∗ (st )}st >st˜ does not satisfy the first order conditions in the following problem:

max

{C(st ),K(st )} t ˜ s >st

s.t.

X

β t Pr(st )u(C(st ), 1 − L(st ))

st >st˜

     C(st ) + K(st ) + g(st ) ≤ F L(st ), K(st−1 ), st + (1 − δ)K(st−1 ) for all st > st˜   K ∗ (ste), K−1 , T ∗ , τ and {L∗ (st )} given k0 st >st˜

37

ˆ t ), K(s ˆ t )} t t˜ satisfying the constraints above that yields a higher Then, there must be an alternative allocation {C(s s >s value for the return function. Let st  st˜ with t > t˜ denote a history st that does not follow the history st˜. Since the utility ˆ t ), K(s ˆ t )} t t˜ and K−1 , T ∗ , for the median type is increasing in the value of the utility for the mean type, it follows that {C(s s >s τk0 , {L∗ (st )}t≥0 , {C ∗ (st ), K ∗ (st )}t
Appendix A.4.2. Case 1: 1 < σ ≤

Proof of Proposition 2

1 1−θ m

Because of Lemma 4, and since (NN) is not binding, the first order condition with respect to labor is (for t ≥ tˆ):

−

VLt (Z; θ) = FL (st ) Vct (Z; θ)

(A.4)

In the competitive equilibrium we know that

1 − τl (st ) = −

UL (st ) FL (st )Uc (st )

Combining the last two equations and using (A.2) and (A.3) generates

1 − τl (st ) =

Thus, if 1 < σ ≤

1 , 1−θ m

a(st ) a(st ) + b(θm )

by Lemma 6 we have a(st ) > 0, and therefore 0 < 1 − τl (st ) < 1 for all st .

Case 2: 0 < σ < 1. Suppose that the constraint (UB) is not binding for all t ≥ tˆ. Later we will check that constraints. We can write τl (st ) as:

τl (st ) =

−(θm − 1) (θm − 1)[(1 − α)(1 − σ) − 1] + [1 − L(st )][(1 − σ) + σω(Z, θm )]

(A.5)

Which implies that τl (st ) > 0 when 0 < σ < 1. In this case, τl (st ) < 1 follows from the intratemporal first order condition in the competitive equilibrium. Otherwise, the marginal productivity of labor should be negative. Next we claim that, if θm < 1 and 0 < σ < 1 then a(st ) > 0 for all st . First, notice that τl (st ) < 1 implies that a(st ) and a(st ) + b(θm ) must have the same sign. b(θm ) > 0 with τl (st ) > 0 implies the claim. Finally, since a(st ) > 0 for all t ≥ tˆ, constraint (UB) is not binding for large enough t

Appendix B.

Numerical Algorithm

We numerically approximate the solution to the following problem:

 max

ωc (θ m ),C,L,K

ω(θm ) θm 1−α

1−σ X

 β t Pr(st )

t,st

38

C(st )α (1 − L(st ))1−α 1−σ

1−σ

 P (θ m −1) t,st ρ(st )uL (st )   ω(θm ) ≤ 1 +  (1−σ)V (Z)     Resource constraint s.t.  non-negativity constraints       L(st ) ≤ 1

A straightforward extension of Lemma 5 in order to allow for stochastic labor skills shows that, in the solution to the problem above, ω(θm ) ∈ [θm , 1). Let λ be the multiplier related to the first constraint, which clearly binds in the solution. Let ξ(st ) be the multiplier on the resource constraint at history st . Then the Lagrangian is given by:

L=

X

Pr(st )β t

"

t,st

ω(θm ) θm 1−α

1−σ

#   X u(C(st ), L(st )) + λ (θm − 1) ρ(st )uL (st ) + (1 − σ)V (Z)(1 − ω(θm )) + t,st

X

    ξ(st ) F L(st ), K(st−1 ), st + (1 − δ)K(st−1 ) − C(st ) − K(st ) − g(st )

t,st

We then can rewrite L as:

L=

X

t

Pr(s )β

t

"

t,st

ω(θm ) θm 1−α

1−σ

t

t

t

t



m

t

u(C(s ), L(s )) + λ(1 − σ)u(C(s ), L(s )) 1 − ω(θ ) + ρ(s )(θ X

m

1−α − 1) 1 − L(st )

# +

    ξ(st ) F L(st ), K(st−1 ), st + (1 − δ)K(st−1 ) − C(st ) − K(st ) − g(st )

t,st

Taking λ and ω(θm ) as given, there exists a functional equation problem (FEP) with a modified return function that solves L above. Such return function is given by:

u b(C(st ), L(st ); λ, ω(θm )) ≡



ω(θm ) θm 1−α

1−σ

 u(C(st ), L(st )) + λ(1 − σ)u(C(st ), L(st )) 1 − ω(θm ) + ρ(st )(θm − 1)

1−α 1 − L(st )



Denote by V (K; λ, ω(θm )) the unique function solving the FEP. Using the product topology in the problem in question, we can apply Theorem 3 in Milgrom and Segal (2002). By setting

∂V (K;λ,ω) ∂ωc (θ m )

= 0 we obtain ω(θm ) = [θ(1−α)(1−σ) λ]−1/σ .

The numerical solution uses a two step algorithm. First, for a given λ, and therefore the ω(θm ) implied by first order condition, we solve the FEP using value function iteration for a grid of 300 points for the capital stock. In the second step, for each capital stock, we do a grid with 100 points for λ and find λ∗ (K) that attains related and

ω(θm )

∈

[θm , 1),

we can reduce the size of the grid for

λ0 s

∂V (K;λ,ω) ∂λ

= 0. Because λ and ω(θm ) are

in a great extent.

We check the numerical solution by evaluating the analytic first-order conditions from the original problem.

39