metadata: authors: 0: author: Eden,Maya
count: World
docna: 0: docna: Is there enough redistribution ?
docty: Policy Research Working Paper
subtopic: Macro-Fiscal Policy,Taxation & Subsidies,Economic Adjustment and Lending,Labor Markets,Disability,Public Sector Economics,Services & Transfers to Poor,Economic Assistance,Access of Poor to Social Services,Public Finance Decentralization and
            Poverty Reduction
abstracts: cdata!: This paper asks whether there are
            welfare gains from additional redistribution. First, it
            derives a sufficient condition for the existence of welfare
            gains from a small increase in lump-sum transfers financed
            by a uniform increase in labor income taxes. A calibration
            suggests that, even under very conservative assumptions,
            most countries would benefit from such a scheme. Second, it
            asks whether, given existing tax revenues, there are gains
            from diverting public funds from government investment
            projects toward redistributive programs. The analysis
            suggests that the answer is highly sensitive to parameter
            values and the rate of return on government investment.
repnme: repnme: Is there enough redistribution ?
display_title: Is there enough redistribution
listing_relative_url: /research/2017/03/27264006/enough-redistribution
topicv3: Social Assistance m1327201 1466,Cash Transfers m1327437 1608,Taxation m1327615 1121,Labor Supply m1504055 2751,Elasticity of Labor Supply m1504056 2752
publishtoextweb_dt: 2017-03-13T00:00:00Z
no_of_pages: 35
author_keys: 0: author_key: 1172765
fullavailablein: node_id: 27264006
node_id: 27264006
model_extractions: WPS8003 Policy Research Working Paper 8003 Is There Enough Redistribution?

_Maya Eden_ Development Research Group Macroeconomics and Growth Team March 2017
Policy Research Working Paper 8003

# **Abstract**

This paper asks whether there are welfare gains from additional redistribution. First, it derives a sufficient condition for the existence of welfare gains from a small increase in lump-sum transfers financed by a uniform increase in labor income taxes. A calibration suggests that, even under very conservative assumptions, most countries would benefit from such a scheme. Second, it asks whether, given existing tax revenues, there are gains from diverting public funds from government investment projects toward redistributive programs. The analysis suggests that the answer is highly sensitive to parameter values and the rate of return on government investment.

This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at meden@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Is There Enough Redistribution?<sup>_∗_</sup>

**JEL Classification:** I38, H21

**Keywords:** Cash transfers; Optimal taxation; Targeting constraints

> _∗_ I am grateful to Eilyn Chong excellent research assistance and very helpful discussions. I thank Aart Kraay, Laura Ralston and Luis Serven for helpful comments. Contact address: Maya Eden, Development Research Group, The World Bank, 1818 H Street, NW, Washington D.C. 20433; Email: meden@worldbank.org.
# **1 Introduction**

The tradeoffs associated with redistributive policy are well-understood. On the one hand, if the marginal utility of consumption is higher at low income levels, then instituting transfers from rich to poor improves social welfare (from a utilitarian perspective). On the other hand, providing appropriate incentives for people to work puts limits on the efficient amounts of redistribution.

The characterization of the optimal redistributive policy has been shown to depend crucially on many aspects of the economic environment which may be difficult to measure, such as higher moments of the distribution of idiosyncratic productivity shocks (see, for example, Golosov et al. [2016]). Rather than characterizing the optimal policy, this paper studies the welfare implications of a simple perturbation to existing tax schemes. A calibration suggests that, for most countries, a small increase in transfers financed by a uniform increase in labor income taxes would improve social welfare.

Methodologically, the approach is similar to the pioneering work of Golosov et al. [2014], who develop a perturbation method which can be used to study a wide class of tax reforms. Here, I apply their method to the question of redistribution in a simple small open economy setting. The gains from redistribution can be calibrated based on the joint distribution of labor income and consumption, the elasticity of labor supply, the administrative cost of transfers, the labor income share, the share of taxes in GDP and the coefficient of relative risk aversion. While there is some controversy regarding the appropriate calibration of some of these parameters, I show that, in most countries, even conservative assumptions deliver positive welfare gains from additional redistribution.

The conclusion regarding the optimality of increasing tax revenues to finance additional government transfers is fairly robust. A related question is whether, given existing tax revenues, social welfare can be improved by shifting government spending from government investment programs to govern2
ment transfer programs. To address this question, the model is extended to include government capital that augments labor productivity. Here, the conclusions are more mixed: under a conservative parameterization, the welfare gains from additional government investment exceed the gains from additional redistribution. However, there are some realistic parameterizations for which the welfare gains from additional cash transfers are roughly equivalent to those of a government investment project with an annual rate of return of 20 percentage points. These estimates are broadly in the range of estimated rates of return to government investment, though these are notoriously imprecise (see Gramlich [1994] for a review). From a policy perspective, the findings suggest that while there is a strong case for increasing government spending through additional taxation, the optimal allocation of spending between transfers and government investment projects is sensitive to parameter values and the rate of return to government capital.

# **2 Model**

There is a continuum of price-taking households indexed _i ∈_ [0 _,_ 1]. The household supplies labor, _li,t_ , and accumulates savings, _ai,t_ .

In most models, inequality is generated by idiosyncratic shocks. It is therefore useful to assume that households face some uncertainty, captured by stochastic variables denoted _{si,t}i,t_ . The stochastic variables affect the disutility from labor. In this formulation, _si,t_ can be thought of as an individual productivity shock: a negative shock makes it more costly to supply the same amount of effective units of labor. It will be useful to denote the history of shocks by _s_<sup>_t_</sup> _i_<sup>= (</sup><sup>_si,_0</sup><sup>_, ..., si,t_).</sup> Household _i_ faces a proportional tax of _τt_ ( _li,t_ ) _≥_ 0 on its labor income and receives a government transfer of _bt_ ( _li,t_ ) _≥_ 0. Note that taxes and transfers may depend only on the household’s current labor income and not on 3
past labor income. This assumption is made mainly for simplification.<sup>1</sup> In addition, it will be useful to assume that at any time _t_ , the set of possible state variables _{ai,t, s_<sup>_t_</sup> _i_<sup>_}i∈_[0</sup><sup>_,_1]isfinite.</sup> While there is idiosyncratic uncertainty that generates inequality, I assume that there is no aggregate uncertainty. Thus, wages and aggregate quantities do not depend on the shocks _{si,t}i,t_ .

In principle, household decisions and state variables depend on the history of shocks, _s_<sup>_t_</sup> _i_<sup>.However,tosimplifynotation,Iwillomitthedependenceon</sup> _s_<sup>_t_</sup> _i_<sup>andwrite</sup><sup>_li,t_=</sup><sup>_li,t_(</sup><sup>_st_</sup> _i_<sup>),</sup><sup>_ai,t_=</sup><sup>_ai,t_(</sup><sup>_st_</sup> _i_<sup>_−_1</sup> ), etc.

Utility from consumption and leisure is separable. Households maximize expected utility subject to a budget constraint: s.t. _ai,_ 0 and: where _β ∈_ (0 _,_ 1) is the time discount factor. It is useful to denote the set of optimal labor supply decisions by _Li_ ( _ai,_ 0 _, {wt, rt, bt_ ( _·_ ) _, τt_ ( _·_ ) _}_<sup>_∞_</sup> _t_ =0<sup>).</sup>

**Production.** There are two production inputs: capital ( _Kt_ ) and labor 1 ( _Lt_ = �0<sup>_li,tdi_).Theproductionfunction,</sup><sup>_Ft_(</sup><sup>_Kt, Lt_),exhibitsconstantre-</sup> turns to scale and satisfies the standard Inada conditions. Factors are paid their marginal products. The wage rate, _wt_ , and the capital rental rate, _Rt_ , are given by:

> 1See Golosov et al. [2007] for a more general treatment.

4
**Capital markets.** I assume a small open economy in which the the global interest rate, _rt_ , pins down aggregate capital supply:

where _τt_<sup>_k_is the tax on capital income and</sup><sup>_δ∈_[0</sup><sup>_,_1] is the capital depreciation</sup> rate.

Note that, given the constant returns to scale assumption, the factor payments _Rt_ and _wt_ depend only on the capital-labor ratio, which is pinned down by equation 2. Consequently, changes in labor supply will be met by proportional changes in capital, and will thus have no effect on equilibrium wages.

**Social welfare.** Given a tax and transfer schedule, _{τt_ ( _·_ ) _, τt_<sup>_k, bt_(</sup><sup>_·_)</sup><sup>_}_,ase-</sup> quence of interest rates, _{rt}_<sup>_∞_</sup> _t_ =0<sup>,andasequenceofwages,</sup><sup>_{wt}∞_</sup> _t_ =0<sup>,social</sup> welfare is given by:

The government’s problem is to choose a tax and transfer schedule and a sequence of government debts, _{xt}_<sup>_∞_</sup> _t_ =1<sup>,to maximize social welfare,subject</sup> to _x_ 0, equations 1-2, the government budget constraint: the incentive compatibility constraints: 5
and the targeting constraint: The parameter _μ_ denotes the transfer multiplier: it costs the government 1 _/μ_ dollars to finance a transfer of 1 dollar. For example, if there are administrative costs, _μ <_ 1.

The targeting constraint (equation 6) states that it is impossible to perfectly target transfer recipients. The government must give the same transfer to all workers with _li,t ≤_ _<u>l</u>_ _~~t~~_<sup>,where</sup><sup>_<u>l</u>_</sup> _~~t~~_<sup>isexogenouslygiven.Itisusefultode-</sup> note by _It_ the set of workers for which _li,t ≤_ _<u>l</u>_ _~~t~~_<sup>.</sup> The following assumption will be crucial for the analysis that follows:

**Assumption 1** _Given existing taxes and transfers, the solution to the household’s optimization problem is unique. This unique solution will be denoted by a_<sup>_∗_</sup> _i,t_ +1<sup>_,l_</sup> _i,t_<sup>_∗andc∗_</sup> _i,t_<sup>_(withtheunderstandingthattheoptimalplanisafunction_</sup> _of the state variables)._

This assumption is in the spirit of Assumption 2 in Golosov et al. [2014]. However, it is not innocuous. It is possible to construct examples in which it will not hold at the optimal policy. In particular, depending on parameters, the optimal policy may implement an equilibrium in which some agents are indifferent with respect to two different choices of labor supply. Appendix A presents an example to illustrate the conditions under which this Assumption 1 is plausible.

In general, an optimal tax code may or may not satisfy the uniqueness assumption. However, this assumption will be crucial for the analysis that follows as it guarantees that labor supply decisions are continuous with respect to small changes in tax policy.

An additional assumption will be made: 1 **Assumption 2** _Let Tt_ = �0<sup>_τt_(</sup><sup>_l_</sup> _i,t_<sup>_∗_)</sup><sup>_wtl_</sup> _i,t_<sup>_∗di_+</sup><sup>_τ k_</sup> _t_<sup>_RtKtdenotethegovernment’s_</sup> _tax revenue. Then, given prevailing tax rates, a marginal change in the trans-_ 6
_fer size bt has no effect on current or future equilibrium tax revenues: ∂b∂Ttt′_<sup>= 0</sup> _for all t ≥ t_<sup>_′_</sup> _._ This assumption is reasonable, at least as an approximation, if transfer recipients generate a small share of aggregate labor income, especially if their labor income is taxed at a low (or zero) rate. Note that even if transfer recipients do not pay any taxes, their labor supply decisions may affect equilibrium tax revenues through capital taxation: lower labor supply would imply a lower capital stock which, if _τt_<sup>_k>_0,wouldgeneratelowertax</sup> revenues. However, this general equilibrium effect will be small if transfer recipients supply a small share of aggregate labor inputs (note that labor inputs are specified in effective units - thus, the assumption is that labor supplied by transfer recipients is relatively unproductive). Alternatively, if a large proportion of households are transfer recipients, then the assumption is a reasonable approximation if, for more productive households, transfers constitute a small share of income and thus an increase in transfers does not lead to a large change in the incentives to work.

Note that while Assumption 2 rules out scenarios in which transfers are distortionary, it does not preclude the possibility that there are distortions associated with the taxation used to finance those transfers. The distortionary costs of taxation operate through the labor supply elasticity: higher taxes reduce the incentives to work, resulting in lower equilibrium labor supply and, consequently, lower equilibrium capital supply.

It will be useful to introduce the following notation to describe different aggregations of the labor supply elasticity. Let _εL,t_ = _∂_<sup>_∂_</sup> ln(<sup>ln(</sup> _w_<sup>_Lt_</sup> _t_<sup><u>)</u></sup> )<sup>denotetheag-</sup> gregate elasticity of labor supply, and let _εi,t_ = _∂∂_ ln(ln( _lwi_<sup>_∗_</sup> _<u>,tt</u>_ )<sup>)denotetheindividual</sup> elasticity of labor supply. Note that both elasticities are equilibrium objects that may depend on the tax code and that, given Assumption 1, the elasticities are well-defined. The results will focus on _ε_ ̃ _L,t_ , which is defined as the maximum of the aggregate labor supply elasticity and a tax-weighted 7
average of individual labor supply elasticities: Note that the assumption of a finite set of state variables guarantees that the set of elasticities _{εi,t}i∈_ [0 _,_ 1] is finite, so the integral exists.

It will be convenient to introduce the notation _τi,t_ = _τt_ ( _li,t_<sup>_∗_)todenote</sup> equilibrium average tax rates. Given a sequence of capital tax rate _{τt_<sup>_k}∞_</sup> _t_ =0 and a sequence of deficits _{xt}_<sup>_∞_</sup> _t_ =0<sup>, it is said that the taxes</sup><sup>_{τi,t}i,t_are feasible if</sup> there exist a sequence of tax schedules _{τt_ ( _·_ ) _}_<sup>_∞_</sup> _t_ =0<sup>such that</sup><sup>_τi,t_=</sup><sup>_τt_(</sup><sup>_l_</sup> _i,t_<sup>_∗_(</sup><sup>_τt_(</sup><sup>_·_)).</sup> The analysis that follows focuses on the welfare gains from implementing a tax schedule that induces a uniform shift in average tax rates, _{τi,t_ + _η}_ for some _η >_ 0. Note that a uniform increase in the existing tax schedule of the ̃ form _τt_ ( _·_ ) = _τt_ ( _·_ ) + _η_ does not necessarily deliver the equilibrium tax rates _{τi,t_ + _η}_ , because a uniform shift may have differential effects on individual labor supply decisions at different tax brackets. Achieving an equilibrium in which the average tax rates are _{τi,t_ + _η}_ might necessitate a non-uniform adjustment of the tax code.

Before proceeding, it is worth motivating the focus on this particular perturbation of the tax code. In particular, one might find it more natural to discuss issues of redistribution by perturbing the tax schedule in a way that is more targeted at the very rich. One example is greater taxation of capital income, which is highly concentrated among wealthy households. In the small open economy setup considered here, capital taxation would be equivalent to labor taxation (in a closed economy setup, this equivalence would hold at the steady state). The reason for this is that the rate of return on capital is pinned down by global market conditions. Thus an increase in capital taxation would lead to a decline in the capital stock and, subsequently, a decline in wages. This would result in a decline in household income that is proportional to its labor income. Because the general equilibrium effects are 8
similar, I chose the more direct approach of analyzing an increase in labor taxes.

A second example of a perturbation that is more targeted at wealthy households is an increase in top tax rates (see Saez [2001]). While such a perturbation may be feasible and desirable, a uniform increase in tax rates is more likely to satisfy the incentive compatibility constraints. Intuitively, an increase in the top tax rate will have a smaller effect on the labor supply of highly productive households when lower incomes are taxed at a higher rate as well. In the context of this exercise, the focus on a uniform tax increase is more conservative. Countries that are shown to benefit from such a scheme would also benefit from a more progressive tax reform, assuming that such a reform is feasible.

The following proposition provides a sufficient condition for the desirability of greater redistribution of labor income: 1 **Proposition 1** _Let sT_ = (�0<sup>_τ_(</sup><sup>_li_)</sup><sup>_wlidi_+</sup><sup>_τ kRK_)</sup><sup>_/Ydenotetheshareoftax_</sup> _revenue in output, and let_ (1 _− α_ ) _denote the equilibrium labor income share. If the following condition holds:_ _then it is possible to improve welfare by an increase in transfers financed by a uniform increase in labor taxes: τi,t �→ τi,t_ + _η for some η >_ 0 _._ The proof is provided in the appendix. Intuitively, the optimality of increasing labor taxes depends on the elasticity of labor supply. The elasticity of labor supply governs the extent to which an increase in tax rates reduces labor inputs. A more elastic labor supply implies that an increase in taxes generates a smaller increase in tax revenues and, consequently, smaller increases in transfers and equilibrium welfare.

Additionally, the welfare gains from increasing transfers depend on the 9
Table 1: Benchmark calibration

|Parameter|Value or source|
|---|---|
|_u_<sup>_′_</sup>(_c_)|_c_<sup>_−γ_</sup>|
|_γ_|0.5|
| ̃_εL_|2|
|_μ_|0.6|
|Consumption distribution|Lakner and Milanovic [2016]|
|Labor income distribution|Consumption distribution for the bottom 90%|
|_sT_|Tax revenues to GDP, WDI|
|(1_−α_)|Labor income share, PWT|
|_I_|All households|

level of risk aversion, which is captured by the marginal utilities _u_<sup>_′_</sup> ( _ci,t_ ). To illustrate, note that if _μ_ = 1 and there is no consumption inequality, then the condition above is violated, and there are no gains from increasing redistribution. However, when there is consumption inequality, then �01<sup>_u′_(</sup><sup>_ci,t_)</sup><sup>_l_</sup> _L_<sup>_i,_</sup> _t_<sup>_tdi≤_</sup> <u>�</u> _It_ <u>1</u><sup>1</sup><sup>_di_</sup> � _It_<sup>_u′_(</sup><sup>_ci,t_)</sup><sup>_di_=</sup><sup>_E_[</sup><sup>_u′_(</sup><sup>_ci,t_)</sup><sup>_|i∈It_].Thus,for</sup><sup>_μ>_0suffi-</sup> ciently large, the left hand side will be positive, which is a necessary but not sufficient condition for the inequality to hold.

Note that the condition in Proposition 1 depends on a weighted average of marginal utilities, where weights are given by the individual shares of labor income ( _li/L_ ). An increase in transfers financed by a uniform increase in labor taxes is primarily financed by those whose labor income is higher. The amount of additional tax revenue collected from any given household is proportional to its labor income. Thus, the welfare cost of redistribution depends on the joint distribution of consumption and labor income.

# **2.1 Calibration**

This section assesses the condition in Proposition 1 for a sample of countries. Table 1 summarizes the calibration approach.

10
To calibrate the joint distribution of consumption and labor income, I use data from Lakner and Milanovic [2016]. The data set contains surveybased measures of _either_ household consumption or disposable income by decile. The calibration requires the joint distribution of labor income and 1 consumption for computing the term �0<sup>_u′_(</sup><sup>_ci,t_)</sup><sup>_li,t/Ltdi_.Iwillassumethat,</sup> for the bottom 9 deciles, consumption is given by disposable income and that income consists only of labor income ( _ci,t_ = (1 _− τt_ ( _li,t_ )) _wtli,t_ ). The top 10% is assumed to have no labor income and consumes only capital income. Formally, using _q_ ˆ _i_ to denote the estimated consumption or income of decile _i_ , I approximate: Note that ignoring the labor income of the top 10% vastly _understates_ the benefits from additional redistribution. The highest decile likely receives a relatively large share of aggregate labor income, and has the lowest marginal utility of consumption. Thus, instituting transfers from this group to the rest of the population is highly desirable from a welfare perspective. By ignoring the top 10%, the burden of financing the transfers falls only on the bottom 90%, which is less favorable. The assumption that the top decile receives no labor income is thus highly conservative in this context.<sup>2</sup> In addition, if taxes are progressive, then the approximation _li/L ≈ q_ ˆ _i/_<sup>�</sup> _j_<sup>_q_ˆ</sup><sup>_j_understatesthebenefitsfromadditionalredistribution.Underthe</sup> assumption that taxes are progressive, disposable income is more equally distributed than gross labor income. Note that the marginal benefits from

> 2At the same time, including the top 10% under the assumption that _ci,t_ = (1 _− τi,t_ ) _li,t_ may overstate the benefits from redistribution. The reason is that the top decile is likely to receive some capital income, which tends to be highly concentrated among the rich. Consequently, both the consumption level and the level of disposable income likely exceed labor income. The assumption _ci,t_ = (1 _− τi,t_ ) _li,t_ would therefore overstate the extent to which an increase in labor taxes would tax the rich. The most conservative approach is to ignore this group.

11
additional redistribution depend on gross labor income shares rather than on disposable labor income shares.

I assume that the marginal utility of consumption takes the form: Harsanyi [1955] establishes that, for the purpose of welfare analysis, it is appropriate to interpret _γ_ as the coefficient of relative risk aversion (rather than, for example, the intertemporal elasticity of substitution or a parameter that governs the degree of altruism). I choose _γ_ = 0 _._ 5, a highly conservative estimate of risk aversion. The macro literature typically uses _γ_ = 2, and much larger values are often used in the finance literature, which documents that large levels of risk aversion are necessary in order to account for the equity premium puzzle.

Estimated tax shares, _sT_ , are based on data from the World Development Indicators (WDI). Estimated labor shares, (1 _− α_ ), are taken from the Penn World Tables (PWT). I assume that _μ_ = 0 _._ 6, which is highly conservative given estimates of the administrative costs of cash transfer programs.

As a conservative benchmark, I assume that the targeting constraint does not allow any targeting of poor households, and that the government must transfer the same amount to every household. This is conservative because the welfare gains from cash transfer programs are higher when they target those who have higher marginal utilities of consumption.

Finally, there is substantial controversy regarding the labor supply elasticity. The micro literature typically finds an elasticity in the vicinity of 0.3, while the macro literature finds that a much larger elasticity of between 1 and 2 (see Keane [2011]). I take a conservative approach and assume that ̃ _εL_ = 2. Note that a high elasticity of labor supply implies that taxes are more distortionary.

12
**Results.** Despite its highly conservative assumptions, the calibration suggests that 103 out of the 111 countries in the sample satisfy the condition of Proposition 1, suggesting that there is too little redistribution in these countries. The exceptions are Botswana, Gabon, Mongolia, Norway, New Zealand, Senegal, Trinidad and Tobago and Turkey. Note that Proposition 1 provides a sufficient rather than a necessary condition; thus, additional redistribution may be desirable even in countries for which this condition is not satisfied, especially given the highly conservative calibration.

# **3 Extension: Government investment**

This section extends the model to allow for productivity-augmenting government investment. The purpose of this extension is to address the policy question of how government spending should be allocated between investment and cash transfer programs. Especially in developing countries, the ability to raise taxes may be limited but there may be more flexibility in how to allocate government spending and foreign aid.

To extend the model, assume that, in addition to private capital and labor inputs ( _K_ and _L_ ), output depends on the level of government capital, _G_ . The production function takes the form: where _h_ is a constant returns to scale production function. This specification guarantees that there are no rents and that, when capital and labor are paid their marginal products, the sum of factor payments is equal to output (provided that the services of government capital are free of charge).

The government budget constraint is modified as follows to include gov13
ernment investment: The government’s optimization problem is to maximize social welfare subject to the government budget constraint, the incentive compatibility constraints and the targeting constraint, taking taxes as given: s.t. _G_ 0, _x_ 0, equations 1-2, 5-6 and 12.

**Proposition 2** _Assume that (a) for all t, it is strictly optimal to set bt_ ( _li,t_ ) _>_ 0 _for i ∈ It; (b) the production function is Cobb-Douglas in K and L (h_ ( _K, L_ ) = _K_<sup>_α_</sup> _L_<sup>1</sup><sup>_−α_</sup> _), and (c) There is common elasticity of labor supply (εi,t_ = _εL). Then, at the optimum, the rate of return to government investment is:_

The proof is in the appendix. Note that, similar to the condition of Proposition 1, the rate of return on government investment implied by the welfare maximization problem depends on a weighted average of marginal consumption utilities. However, in contrast to the condition in Proposition 1, weights are proportional to disposable labor income and do not necessarily sum to 1. Intuitively, an increase in government capital increases labor productivity, which increases disposable labor income. The welfare gains from government investment are thus proportional to disposable rather than to gross labor income. Similar to equation 9, for the purpose of the calibration, 14
I assume that: To calibrate the optimal rates of return to government investment implied by proposition 2, it is necessary to take a stance on the values of _rt_ +1. I assume that _rt_ +1 is given by the Euler condition, which relates the interest rate to the rate of consumption growth.<sup>3</sup> The calibration based on the conservative parameterization described by Table 1 suggests that the optimal rate of return on government investment should be negative in most countries, and always below 2%. Given that most estimates suggest high rates of return to government investment, the conclusion is that, given current tax revenues, governments should divert resources from cash transfer programs to government investment projects.

However, this conclusion is a result of the highly conservative parameterization, and may not hold up in the presence of less conservative (and perhaps more realistic) parameters. Table 2 illustrates how the optimal rate of return to government capital varies with alternative calibration parameters.

Depending on the parameterization, the average optimal rate of return on government investment can be as high as 21%. This estimate is in the range of plausible estimates of rates of return to government investment projects, although those are notoriously imprecise. The calibrated optimal rates of return vary somewhat by country, and are reported in Appendix C.

> 3The Euler condition is _β_ (1 + _rt_ +1) = _u′_ ( _ct_ ) _/u′_ ( _ct_ +1). I assume that the rate of consumption growth is given by the rate of GDP per capita growth, which is calibrated based on data from the World Development Indicators.

15
Table 2: Average returns (percent): alternative calibrations

|_μ_| ̃_εL_|_I_|_γ_|Average optimal return|
|---|---|---|---|---|
|0.6|2|All|0.5|0|
|1|2|All|0.5|1|
|0.6|0.3|All|0.5|1|
|1|0.3|All|0.5|3|
|0.6|2|Bottom decile|0.5|1|
|1|2|Bottom decile|0.5|2|
|0.6|0.3|Bottom decile|0.5|3|
|1|0.3|Bottom decile|0.5|5|
|0.6|2|All|2|3|
|1|2|All|2|5|
|0.6|0.3|All|2|7|
|1|0.3|All|2|11|
|0.6|2|Bottom decile|2|8|
|1|2|Bottom decile|2|9|
|0.6|0.3|Bottom decile|2|18|
|1|0.3|Bottom decile|2|21|

16
# **4 Conclusion**

This paper establishes that, even under highly conservative assumptions, there is a strong case for further redistribution. In theory, the gains from redistribution depend on a variety of structural parameters, such as the social aversion to inequality; the ability to target transfers at the poor; and the administrative costs of transfer programs. The calibration establishes that even at very low levels of inequality aversion, an un-targeted transfer program in which transfer recipients receive only 60 cents on the dollar would improve social welfare. The implication is that under any realistic parameterization, there are gains from further redistributive policy.

The analysis relies crucially on two assumptions that guarantee the feasibility of increasing transfers and tax revenues. The first assumption states that, given prevailing tax policy, optimal individual labor supplies are uniquely pinned down. While this assumption seems plausible, it may be violated under an optimal policy. The second assumption states that an increase in transfers (holding taxes fixed) has a negligible effect on tax revenues. This framework therefore refocuses the debate on the optimal extent of redistribution. Given that further redistribution is desirable even under the most conservative parameterizations, opponents of further redistribution must either argue that, although desirable, increasing tax revenues is not feasible, or that increasing transfers would result in a large drop in tax revenues.

When total government spending is fixed, whether or not it is optimal to increase transfers to the poor depends on the welfare gains from other forms of government spending. I extend the model to include productivity-enhancing government investment, and establish that the optimality of diverting public funds from government investment projects to cash transfer programs is highly sensitive to parameter values.

To conclude, this paper refocuses the debate on the optimal scope of redistribution. First, it establishes that this debate is viable only in cases in which increasing tax revenues is not feasible. Otherwise, further redistribu17
tion is desirable under any realistic parameterization. Second, it establishes that, when increasing tax revenues is not feasible, the optimality of diverting funds from government investment projects to redistributive programs depends on the rate of return to government investment, the administrative cost of transfers, the elasticity of labor supply and the social aversion to inequality.

# **References**

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Emmanuel Saez. Using elasticities to derive optimal income tax rates. _The Review of Economic Studies_ , 68(1):205–229, 2001.

# **A An example to illustrate the conditions under which Assumption 1 holds**

**Example 1.** Consider the following simple example in which _ai,_ 0 = 0 for all _i_ and there are only two permanent states, _si,t ∈{_ 0 _,_ 1 _}_ (where _si,t_ = _si,_ 0 for all _t_ ). The utility function takes the form:

In this example, only agents with an _si,t_ = 1 shock are able to supply labor. Agents with _si,t_ = 0 are unable to supply any labor, because any positive amount of labor supply results in utility of _−∞_ . In addition, given that shocks are permanent and initial wealth is set to 0, they cannot borrow or consume out of savings. Thus, an optimal policy would allow them to choose _li,t_ = 0 and provide them with some government transfers to consume.

Assume that _β_ (1 + _rt_ ) = 1 for all _t_ . Under this assumption, a time invariant tax and transfer policy is optimal and the equilibrium can be solved in a static framework. Households with _si,_ 0 = 1 choose their permanent labor supply, _li,t_ = _l_ , subject to a tax rate _τ_ , and households with _si,t_ = 0 consume the government transfer. Assuming a production function with no capital inputs ( _F_ ( _K, L_ ) = _L_ ), the transfer is given by _b_ = _μτl_ .

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The _s_ = 1 household solves: _i,t_ The optimal labor supply decision in the region _l >_ 0 is: The optimal labor choice, _l_<sup>_∗_</sup> , is then given by: Of course, the government will not be able to finance any transfers unless _l_<sup>_∗_</sup> = _l_<sup>_o_</sup> _>_ 0. The size of the transfer is limited by the government budget constraint _b_ = _μτl_<sup>_∗_</sup> . Under the constraint that the government must induce productive agents to choose _l_<sup>_∗_</sup> = _l_<sup>_o_</sup> , the government’s optimization problem can be written as: s.t. _b_ = _μτl_<sup>_o_</sup> and the incentive compatibility constraint: Ignoring the incentive compatibility constraints yields an optimal tax rate of _τ_ = 0 _._ 5 (note that given log utility, it is optimal for the government to spend an equal share of aggregate labor income on each household). The tax rate _τ_ = 0 _._ 5 is consistent with the incentive compatibility constraint only for _μ_ sufficiently small. To see this, note that _b_ = _μτl_<sup>_o_</sup> and hence the incentive 20
compatibility constraint is satisfied only when: Or: Note that both sides of the inequality are negative. The inequality is satisfied for _φ >_ 0 sufficiently small and _μ <_ 1 sufficiently small. If we assume that the disutility from labor is low and that the administrative cost of transfers is large, then the incentive compatibility constraint will not be binding and the _si,t_ = 1 household will strictly prefer the choice _l_ = _l_<sup>_o_</sup> over the choice _l_ = 0.

However, it is easy to see that this will not be the case when _μ_ = 1. In this case, _τ_ = 0 _._ 5 is not a feasible policy choice because productive households will prefer not to work and receive the government transfer. The optimal tax rate, _τ_ , is then given by the incentive compatibility constraint: Thus, under the optimal policy, the solution to the productive household’s optimization problem is not unique: it is indifferent between _l_ = _l_<sup>_o_</sup> and _l_ = 0. To see this, note that if the household strictly prefers _l_ = _l_<sup>_o_</sup> , then the government can implement a small increase in the size of the transfer and in the tax rate. Given that _τ <_ 0 _._ 5, this alteration will improve social welfare.

To summarize, Assumption 1 holds in this example only when transfers are relatively costly ( _μ_ is small) and when the disutility of labor is not large ( _φ_ is small).

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# **B Proofs**

**Proof of Proposition 1.** Let _λt_ denote the Lagrange multiplier on the government’s budget constraint at time _t_ . Applying the envelope theorem and ignoring the targeting constraint, the government’s first order condition with respect to _bt_ ( _li_<sup>_t_)yields:</sup>

Thus, given Assumption 2 the targeting constraint implies that: Or: Consider the government’s optimization problem with respect to labor taxes, _τt_ ( _·_ ). It will be convenient to introduce the notation _τi,t_ = _τt_ ( _li,t_<sup>_∗_).</sup> This formulation assumes that the government can directly choose the tax rate faced by agent _i_ , rather than the tax schedule. Note that _τi,t_ must satisfy some incentive compatibility constraints, that induce agent _i_ to “reveal his type” - in other words, there must exist some tax schedule _τt_ ( _·_ ) such that, given that tax schedule, _τi,t_ = _τt_ ( _li,t_<sup>_∗_).</sup> The assumption that the solution to the household’s optimization problem is unique and that the set of state variables at any time _t_ is finite implies that, for a sufficiently small _η_ ̄ _>_ 0, it is possible to increase all labor tax rates ̄ by any _η ∈_ ( _−η,_ ̄ _η_ ) and distribute the tax receipts among agents in _It_ .

Let _W_ ( _η_ ) denote the welfare obtained from a uniform tax increase of _η_ : 22
s.t.

where _Tt_ is the equilibrium tax revenue given the tax schedule _{τi,t_ + _η}_ , and _b_<sup>_η_</sup> _t_<sup>issuchthatthattheincreaseintaxesandtransfersisbudgetneutral.</sup> The government can improve welfare by choosing _η >_ 0 if _W_<sup>_′_</sup> ( _η_ ) _>_ 0. Note that given the small open economy assumption, the capital labor ratio is pinned down by global market conditions, and hence both _Rt_ and _wt_ do not respond to changes in labor taxes. Thus, using the envelope theorem, this condition can be written as: Using equation 26, this condition holds if: 1 Note that taxes are given by _T_ = �0<sup>((</sup><sup>_τi_+</sup><sup>_η_)</sup><sup>_wili_+</sup><sup>_τkRtKt_)</sup><sup>_di_,andthus,</sup> using the observation that _wt_ and _Rt_ are pinned down by global market conditions and do not respond to changes in labor inputs or taxes: Rewriting: Note that, given that the capital labor ratio is pinned down at some _ζ_ , 23
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Or:

**Proof of Proposition 2.** Using _λt_ to denote the Lagrange multiplier on the government’s time _t_ budget constraint, an application of the envelope theorem implies that the first order condition with respect to _Gt_ +1 is:

where _wt_<sup>_′_</sup> +1<sup>(</sup><sup>_Gt_+1)isthederivativeofequilibriumwageswithrespecttogov-</sup> ernment capital. Note that this first order condition relies on the result that, for any _t_<sup>_′_</sup> _>_ 0, _wt_ + _t′_ does not depend on _Gt_ .

To simplify notation, it will be convenient to write _u_<sup>_′_</sup> <u>(</u> _<u>c</u>_ _~~t~~_<sup>) =</sup><sup>_E_[</sup><sup>_u′_(</sup><sup>_ci, t_)</sup><sup>_|i ∈_</sup> _It_ ]. Assuming that positive transfers are strictly optimal, the inequality in equation 26 holds with equality. Combining the two first order conditions yields:

To solve for _w_<sup>_′_</sup> ( _G_ ), note that, using the constant returns assumption and the assumption that factors are paid their marginal product: Using _L_<sup>_′_</sup> ( _G_ ) and _K_<sup>_′_</sup> ( _G_ ) to denote the derivatives of equilibrium labor and capital with respect to _G_ , taking a derivative of both sides of the above 25
equation with respect to _G_ yields: Note that _R_ is pinned down by the global equilibrium condition. As factors are paid their marginal products, _w_ =<sup>_<u>∂F</u>_</sup> _∂L_<sup>and</sup><sup>_R_=</sup> _∂K_<sup>_<u>∂F</u>_.Hence:</sup> Substituting into equation 43 yields: Rearranging yields: To calibrate the term<sup>_<u>∂T</u>_itisusefultonotethat:</sup> _∂G_<sup>,</sup> where _A_<sup> ̃</sup> = _Af_ ( _G_ ). Note that household decisions depend only on factor payments, which can be written as a function of _A_<sup> ̃</sup> rather than as a function of _A_ and _G_ . Thus, tax revenues depend only on _A_<sup> ̃</sup> .

Expanding the above yields: 26
Rearranging: Yielding: Or: The government’s optimization problem with respect to _xt_ +1 yields the standard Euler condition: 27
Thus, equation 55 can be rewritten as: To conclude the proof, it is left to show that, under the Cobb-Douglas assumption: To see this, note that the equalization of the marginal return to capital with the market rate implies that: Rearranging yields the following expression for the capital-labor ratio as a function of _A_ : Note that the derivative of the capital-labor ratio with respect to _A_ is: and hence: Wages are equal to the marginal product of labor: Taking logs: 28
The elasticity of the wage with respect to _A_ is therefore: As taxes are given by _T_ = � _τiwlidi_ + _τkRK_ , Note that the equilibrium rental rate, _R_ , does not depend on _A_ because the capital stock fully adjusts to meet the global rate of return. Rewriting yields: 29
# **C Optimal rates of return on government capital, by country**

The conservative calibration of the optimal rate of return to government investment uses the parameter values reported in table 1. The column labeled ”Not Conservative” reflects an alternative calibration in which _μ_ = 1; _γ_ = 2; _ε_ = 0 _._ 3; and transfers are targeted at the bottom decile of the distribution.

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Table 3: Optimal rate of return on government investment (%), by country

|Country|Conservative|Not conservative|
|---|---|---|
|Angola|0.5|23|
|Albania|0.2|21.1|
|Argentina|-0.4|14.8|
|Armenia<br>|0.7<br>|26.8<br>|
|Australia|-1|11.2|
|Austria<br>|-1.3|7.8<br>|
|Azerbaijan|0.3|34.3|
|Burundi|-0.2|7.9|
|Belgium|-1.3|7|
|Benin|-0.5|17.4|
|Burkina Faso|0|22.2|
|Bangladesh|0.6|36.6|
|Bulgaria|-0.8|16.4|
|Bolivia|1|37.2|
|Brazil|0.1|21.2|
|Barbados|-0.6|8.2|
|Bhutan|1.6|69.4|
|Botswana|-3.3|5|
|Central African Republic|-1.7|-4.7|
|Canada|-0.7|18.2|
|Switzerland|-0.4|30.3|
|Chile|-0.2|16.9|
|China|1|52.6|
|Cote d’Ivoire|-0.7|19.5|
|Cameroon|-0.8|20.5|
|Congo, Rep.|-0.4|37.2|
|Colombia|0.3|32.8|
|Comoros|-0.2|7.5|
|Cape Verde|-0.3|19.1|
|Costa Rica|0|34.8|
|Czech Republic|-0.6|15.2|
|Germany|-0.4|20.4|
|Djibouti|0.5|25.6|
|Dominican Republic|0.4|28.4|
|Ecuador|-1.6|14.5|
|Egypt, Arab Rep.|-1.4<br>|12.9|
|Estonia|0.5<br>31|59.8|
|Ethiopia|1.5|48.9|
|Fiji|-1.7|9.4|
Table 4: Optimal rate of return on government investment (%), by country (continued)

|Country|Conservative|Not conservative|
|---|---|---|
|France|-1|7.1|
|Gabon|-3|6.2|
|United Kingdom|-1.5|7.8|
|Ghana<br>|0|38|
|Guinea|-2|7.5|
|Guinea-Bissau|-0.1|9.4|
|Greece|-1.5|-0.2|
|Hong Kong SAR, China|0.2|26.7|
|Honduras|0.7|26.2|
|Croatia|-0.5|9.8|
|India|-1.2|43.9|
|Iran, Islamic Rep.|-0.4|20.7|
|Israel|-0.9|13.5|
|Italy|-1.7|2.6|
|Jordan|-0.4|7.7|
|Japan|-1.4|16.5|
|Kazakhstan|-0.1|26.4|
|Kenya|-0.6|23.6|
|Cambodia|0.5|36|
|Korea, Rep.|-0.7|27.6|
|St. Lucia|-1.4|5.8|
|Sri Lanka|0.7|41.8|
|Luxembourg|-1.3|7.4|
|Morocco|-1.1|15.7|
|Madagascar|-0.8|14.8|
|Mexico|-0.3|15.7|
|Mali|-0.3|43.2|
|Mongolia|-0.4|22.8|
|Mozambique|0|19.2|
|Mauritania|0.1|18.9|
|Malawi|-0.7|20.7|
|Malaysia|-0.6|29.1|
|Namibia|-0.3|16.9|
|Niger|-0.4|12.6|
|Nigeria|0.8<br>|65.3|
|Netherlands|-1<br>32|8.6|
|Norway|-1.3|4.3|
|Nepal|0.5|22.6|
|New Zealand|-1.6|6.1|
|Pakistan|-0.9|15.3|
Table 5: Optimal rate of return on government investment (%), by country (continued)

|Country|Conservative|Not conservative|
|---|---|---|
|Panama|0.3|48.6|
|Peru|-0.3|21.7|
|Philippines|-0.7|22.5|
|Poland|0.2|26.3|
|Portugal|-1.2|9.2|
|Paraguay|0.8|39.7|
|Russian Federation|-0.1|20|
|Rwanda|1.1|47.9|
|Sudan|0|24.7|
|Senegal|-1.2|6.9|
|Singapore|-0.6|23.4|
|Sierra Leone|0.1|22.8|
|El Salvador|-0.7|10.6|
|Slovak Republic|-0.2|28.6|
|Swaziland|-1.4|18.6|
|Syrian Arab Republic|-1|15.1|
|Chad|-0.1|14.7|
|Togo|0.1|18|
|Thailand|-1.7|16.5|
|Trinidad and Tobago|-2.1|6.8|
|Tunisia|-1.7|12.2|
|Turkey|-1.5|12|
|Taiwan|-1.5|13.8|
|Tanzania|-0.8|26.6|
|Uganda|0.3|31.2|
|Uruguay|-0.1|23.9|
|United States|-0.3|24.8|
|Uzbekistan|0.2|29.6|
|Venezuela, RB|-1|8.7|
|Yemen, Rep.|-1|0.9|
|South Africa|-1|9.8|
|Zambia|0.7|31.6|

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