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New Dynamic Public Finance: A User's Guide

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1. 376 Diamond The actual collection of wealth tax revenue is irrelevant and we could have considered a constraint on Q consistent with there being no sav ings After some manipulation we can sign the tax on capital income sign R Q sign s s 45 Thus there is a tax on savings since there would be an increase in savings if a type A decided to imitate a type B See Diamond and Mirrlees 1982 for a special case To explore tax smoothing despite an age structure of workers that prevents its optimality for a single cohort one could examine OLG models with an assumption that taxes are period specific and cannot be age specific or how age specific taxes change over time 5 Ramsey vs Mirrlees In contrasting Ramsey and Mirrlees approaches GTW draws three dis tinctions The first is that the Ramsey approach has a representative agent while the Mirrlees approach has a heterogeneous population Since income distribution matters this aspect of the Ramsey approach implies that Ramsey models can generate insight into influences rel evant for tax policy but should not be viewed as generating answers to what taxes should be But then I think that is true generally of models As Alfred Marshall put it 1948 page 366 it is necessary for man with his limited powers to go step by step breaking up a complex question studying one bit at a time and at last combining his partial solutions into a more or less complete solution
2. Ry lt 0 35 ule A y A QAY Bulc A y A 0 A ulc B y B A Bulc B y B 6 A This problem has the FOCs for consumption levels 7 y u c A A O A Az 36 7 it c B y B 8 B yrr c B y B 0 A Az 37 m W Bulc A y A 0 A Ar R 38 ZaPu c B y B 0 B yBu c B y B 0 A Az R7 39 While there is no tax on savings for the high type for the low type we have Zguelci B 1 B 0 B wule B y B 0A _ Agu C2 B 2 B 0 B yu ca B y2 B 82 A BR 40 Thus the sign of the wedge u c B i B 0 B _ u c2 B y2 B 6 B I depends on that of U 2 B 2 B O2 B _ u c2 B 2 B 02 A 41 uc c B y B 0 B u c1 B y B 0 A BR Thus there can be a negative wedge for a suitable impact of additional labor in both periods on the intertemporal consumption MRS The Comment 375 GTW example with nonseparable utility and second period uncertainty has both of these elements in it providing both positive and negative pushes on the wedge I have followed GTW in examining individual marginal incentives at the point of the optimal allocation assuming full government con trol full observability of consumption and earnings A similar insight comes from considering the same model except that while the govern ment can observe savings it can not observe who is saving implying lin
3. because our main goal is to pro vide the reader with an overview of three implications of the dynamic Mirrlees literature that differ from those of Ramsey s Our workhorse model is a two period economy that allows for aggregate uncertainty regarding government purchases or rates of returns on savings as well as idiosyncratic uncertainty regarding workers productivity The mode is flexible enough to illustrate some key results in the litera ture Moreover its tractability allows us to explore some new issues We aim to comprehensively explore the structure of distortions and its dependence on parameters within our dynamic Mirrleesian economy Papers by Albanesi and Sleet 2006 Golosov and Tsyvinski 2006a and Kocherlakota 2005 include some similar exercises but our sim ple model allows us to undertake a more comprehensive exploration Although some of our analysis is based on numerical simulations our focus is qualitative We do not seek definitive quantitative answers from our numerical exercises rather our goal is to illustrate qualitative features and provide a feel for their quantitative importance The presence of private information regarding skills and the stochas tic evolution of skills introduces distortions in the marginal decisions of agents We focus attention on two such wedges The first wedge is a consumption labor wedge or simply a labor wedge that measures the difference between the marginal rate of substitutio
4. 2 changes sig nificantly As a increases the shock to skill becomes smaller and the level of wedges at the top falls To see this effect compare the upper dotted line for a 0 05 with the bottom dashed line for a 0 95 To summarize the discussion above we conclude that the size of the second period shock only has significant effects on labor wedges for the agents who experience that shock and only in that period Intuitively the skill distribution for agents not affected by the shocks matters only New Dynamic Public Finance A User s Guide 343 indirectly and therefore the labor wedge for those agents is affected only to a small degree We now proceed to characterize the effects of the size of shocks on the capital wedge The intertemporal wedge becomes smaller and flat ter when amp increases compare for example the lower curve associ ated with 0 95 to the highest curve associated with 0 05 The reason is that consumption becomes less volatile in the second period when the skill drop is smaller The inverse Euler equation then implies a smaller distortion The intuition for this result is simple If there were no skill shocks in the second period 1 then as we discussed above the capital wedge is equal to zero The higher is the wedge in the sec ond period the further away from the case of constant skills we are therefore the distortion increases Also note that low a large shocks in the second p
5. 217 230 Werning I 2005 Tax Smoothing with Redistribution Federal Reserve Bank of Min neapolis Staff Report 365 Comment Kenneth L Judd Hoover Institution and NBER Professors Golosov Tsyvinski and Werning have given us an excel lent overview of recent work applying the Mirrlees 1971 approach for income taxation to questions in the theory of taxation of dynamic and stochastic environments I am delighted to see this renewed interest in optimal taxation problems The work discussed in this paper shows us that there is great value in this effort and also how mutch is left to be done My comments will focus on three issues First I will comment on the relationship between this work and the earlier literature Second I want to discuss a possibly heretical interpretation of Mirrlees work Third I will discuss the problems facing future work This literature has worked to emphasize the difference between the dynamic Mirrlees literature and the Ramsey literature In particular these papers often interpret any difference between a marginal rate of substitution and the corresponding marginal rate of transformation as a tax However the dynamic Mirrlees approach is not strictly compa rable to the Ramsey approach In the Ramsey approach as executed in for example Atkinson and Stiglitz 1976 Diamond and Mirrlees 1971 and Judd 1985 1999 assume that a full set of private markets exist and that prices are determined comp
6. Marek 2005 Efficient Allocations in Dynamic Private Information Economies with Persistent Shocks A First Order Approach Mimeo University of California Santa Barbara Kingston Geoffrey 1991 Should Marginal Tax Rates be Equalized Through Time Quarterly Journal of Economics 106 3 911 24 Klein Paul Per Krusell and Jose Victor Rios Rull 2005 Time Consistent Public Expen ditures Mimeo University of Pennsylvania Kocherlakota Narayana R 2005 Zero Expected Wealth Taxes A Mirrlees Approach to Dynamic Optimal Taxation Econometrica 73 5 1587 1621 Kocherlakota Narayana 2006 Advances in Dynamic Optimal Taxation In Advances in Economics and Econometrics Theory and Applications Ninth World Congress Volume I 269 299 New York NY Cambridge Univ Press Lucas Robert E Jr and Nancy L Stokey 1983 Optimal Fiscal and Monetary Policy in an Economy without Capital Journal of Monetary Economics 12 55 93 Mirrlees James A 1971 An Exploration in the Theory of Optimum Income Taxation Review of Economic Studies 38 2 175 208 Phelan Christopher 2005 Opportunity and Social Mobility Federal Reserve Bank of Minneapolis Research Department Staff Report 323 Phelan Christopher and Ennio Stacchetti 2001 Sequential Equilibria in a Ramsey Tax Model Econometrica 69 6 1491 1518 Prescott Edward C and Robert Townsend 1984 Pareto Optima and Competitive E
7. and private information Many of the more recent results build on the insights first developed in those papers The New Dynamic Public Finance literature extends previous models by focusing on the stochastic evolution of skills and aggregate shocks Thus relative to the repre sentative agent Ramsey approach commonly pursued by macroecon omists it places greater emphasis on individual heterogeneity and uncertainty whereas relative to traditional work in public finance it places uncertainty at the aggregate and individual level at the fore front of the analysis Werning 2002 and Golosov Kocherlakota and Tsyvinski 2003 incorporated Mirrleesian framework into the standard neoclassical growth model Werning 2002 derived the conditions for the optimal ity of smoothing labor income taxes over time and across states Build ing on the work of Diamond and Mirrlees 1978 and Rogerson 1985 Golosov et al 2003 showed that it is optimal to distort savings in a general class of economies where skills of agents evolve stochastically over time Kocherlakota 2005 extended this result to an economy with 322 Golosov Tsyvinski and Weming aggregate shocks We discuss these results in section 4 Werning 2002 Shimer and Werning 2005 and Abraham and Pavoni 2003 study optimal taxation when capital is not observable and its rate of return is not taxed da Costa and Werning 2002 Golosov and Tsyvinski 2006b and da Costa 2005 consi
8. for how the choice of the welfare function affects optimal taxes in static framework 13 That is we use that E 1 x gt 1 E x when Var x gt 0 where x in our case is the marginal utility u c i j 14 For example if an agent considers changing her labor then in general she also con siders changing her savings Golosov and Tsyvinski 2006a Kocherlakota 2005 and Albanesi and Sleet 2006 showed that such double deviations would give an agent a higher utility than the utility from the socially optimal allocations and therefore the optimal tax system must be enriched with additional elements in order to implement the optimal allocations 15 A formalization of such a game and an equilibrium concept sustainable equilibrium is due to Chari and Kehoe 1990 They formulate a general equilibrium infinite horizon model in which private agents are competitive and the government maximizes the wel fare of the agents Benhabib and Rustichini 1997 Klein Krusell and Rios Rull 2005 Phelan and Stacchetti 2001 and Fernandez Villaverde and Tsyvinski 2004 solve for equilibria in an infinitely lived agent version of the Ramsey model of capital taxation 360 Golosov Tsyvinski and Werning 16 See also Sleet and Yeltekin 2005 who prove similar result when agents shocks fol low an i i d process and the government is benevolent 17 The assumption of uniformity of distribution of skills is not innocuous Saez 2001
9. lt 1 risk aversion is lower than in our baseline and consump tion and work effort are substitutes in the sense that u lt 0 that is an increase in labor decreases the marginal utility of consumption When c gt 1 the reverse is true risk aversion is higher and consumption and labor are complements in that u gt 0 For both reasons the latter case is considered to be the empirically relevant one We first consider lt 1 cases Figure 5 8 shows the schedules for 1 0 9 0 7 0 65 The baseline with o 1 is plotted as a dotted line Lower o correspond to the lower lines on the graph We notice that a lower o pushes the whole schedule of labor distor tions down Intuitively with lower risk aversion it is not optimal to redistribute or insure as much as before The economy moves along the equality efficiency tradeoff towards efficiency The results for capital taxation are more interesting First a lower gis associated with a uniformly lower schedule of capital distortions Second lower o introduces a non monotonicity in the schedule of capi tal distortions so that agents with intermediate skills have lower capi tal distortion than those with higher or lower skills Finally for all the cases considered with o lt 1 we always find an intermediate region where the intertemporal wedge is negative To understand this result it is useful to think of the case without uncertainty in the second period For this case Atkinson and St
10. probability events which would not be worthwhile Just as incomplete markets are a reality so too incomplete use of incentives is a reality I see no reason to believe that assuming such a reality is necessarily worse than deriving it when trying to model something as complex as tax policy 6 Concluding Remarks It is good to have macroeconomists looking at the same issues as public finance economists In the spirit of encouraging further comple mentary analysis let me say that there is a great deal of current inter est in annuities and taxation This might appeal to macroeconomists as well After all as Benjamin Franklin wrote in a letter to M Leroy 1789 Our Constitution is in actual operation everything appears to promise that it will last but in this world nothing is certain but death and taxes Acknowledgments I am indebted to Ivan Werning for very clarifying comments and to the National Science Foundation for financial support under grant SES 0239 080 Endnote 1 Nonseparability over time in the utility of consumption is also plausible Mirrlees and I 1986 explored an extreme Leontieff case of intertemporal nonseparability and 2000 a standard of living model Comment 379 References Atkinson A B and J E Stiglitz 1976 The Design of Tax Structure Direct versus Indi rect Taxation Journal of Public Economics 6 55 75 Corlett W J and D C Hague 1953 Complementarity and the Ex
11. ule A Bule A oly 4 0 4 Boly A A 2 ule B Pule B vly B 6 A Boly B A From the FOCs for consumption levels there is no tax on savings WIBI pp _ HTA ii u le2 B we A Now consider the FOCs for earnings z W viy A 0 4 0 4 Az 11 m W Bo ly A O A 81A An R 12 nv ly B 0 B 0 B wo ly B 814 8 4 Az 13 1 Bv ly B 0 B 0 B yBo ly B 0 A 0 A Am R 14 370 Diamond Taking a ratio of FOCs there is no intertemporal earnings wedge for the high type consistent with no marginal taxation of the highest type on all margins AE A O AN A _ BR 15 vina A 0 A 15 Turning to type B let us define A as the wedge V1 B AO AB gp 7 YLy2 B 6 B 76 B 9 F 16 If A is negative then the first period marginal disutility of earning is larger than the discounted second period marginal disutility From the ratio of FOCs the sign of A depends on the difference in intertemporal MRS for type B and for type A if imitating type B A 202 Lys B 6 B 1 0 B wv tfy B 8 A A g0 LY2 B 8 B 8y B wo Ly2 B 6 A 82 A _ UTyi B 6 B 6 B v Ly2 B B 8 B with 17 v Ly2 B 8 A 8 B v Ty2 B B A B and y gt 0 the sign of A is the same as that of v Ty2 B 8 A 9 B _ v Tn B 6 A B faa v y2 B _ B 6 A o Lyy B 61B A
12. 4 5 The Government s Role As Insurance Provider In the previous discussion we assumed that a government is the sole provider of insurance However in many circumstances markets can provide insurance against shocks that agents experience The presence of competitive insurance markets may significantly change optimal policy prescriptions regarding the desirability and extent of taxation and social insurance policies We assumed that individual asset trades and therefore agents con sumption are publicly observable In that case following Prescott and Townsend 1984 Golosov and Tsyvinski 2006b show that allocations provided by competitive markets are constrained efficient and the first welfare theorem holds The competitive nature of insurance markets even in the presence of private information can provide optimal insur ance as long as consumption and output are publicly observable Note that individual insurance contracts between agents and firms would feature the same wedges as the social planning problem we studied providing another motivation for focusing on wedges rather than taxes that implement them In this paper we do not model explicitly reasons why private insur ance markets may provide the inefficient level of insurance Arnott and Stiglitz 1986 1990 Greenwald and Stiglitz 1986 and Golosov and Tsyvinski 2006b explore why markets may fail in the presence of asymmetric information 5 Numerical Exercises We no
13. Bule Boly 4 4 subject to G c y Ea Rc i Ry i 0 Pule A oly A 0A gt Bulc B Poly B A 21 This problem has the FOCs for consumption levels uw c A 22 x W Bu c A Am R 23 2 y Bu tc B A R 24 372 Diamond Adding the last two equations and taking a ratio to the first equation we have TG cigs os m4 WU co A 2g W u c2 B In contrast without a wedge the individual would see a gain from savings if w c TawleyA rpa A a Thus we have implicit marginal taxation of savings provided u c A lt u c B as follows from the need to have c A gt c B to induce type A not to imitate type B The underlying argument does not need the additive structure of preferences provided that preferences are such that keeping c A enough larger than c B to just induce the higher labor supply implies a lower marginal utility of consumption at the higher consumption level That is consider the condition ulc y A ule y 6 and c gt c implies 27 aulc y 0 _ duc y 6 ac ac Then the argument above goes through if the binding incentive compatibility constraint is that the high skill worker not imitate the low skill worker then the optimum has a positive wedge on intertem poral consumption This parallels the result that Mirrlees and I have found in the special case that labor is a zero one variable and the low skill pe
14. If v z 2 then the sign of A is the same as that of y2 B 0 A 0 B _ yi B A 6 B 19 y2 B B A y B B A or simplifying that of 2 y 62 o 20 6 A aA Comment 371 Thus with power function disutility of labor and the same age earnings profile for both types we have tax smoothing as in Werning 2005 But tax smoothing requires the same age earnings profile for everyone If higher earners have steeper age earnings profiles x2 20 A B then A is negative and there is heavier marginal taxation of second period earnings and a wedge in the intertemporal earnings tradeoff Without a power function there may not be tax smoothing even with the same age earnings profile 4 Taxing Savings with Uncertainty GTW explore the case for taxing savings in models with uncertainty about future productivity I will present a simple model of that and then contrast the route to taxing savings in this model to one with fewer government controls With the same notation as above consider a one type model with uncertainty about second period skill but not first period skill This is a simpler version of GTW analysis Let z now stand for the probability of having skilliin the second period We continue to assume that the only binding incentive compatibility constraint is that type A not want to imitate type B which now refers only to the second period Maximize ule vly 8 Ex
15. In this section we argue that the dynamic Mirrlees literature and Ramsey literature are both prone to time consistency problems However the nature of time inconsistency is very different in those two approaches An example that clarifies the notion of time inconsistency in Ramsey models is taxation of capital The Chamley Judd Judd 1985 Chamley 1986 result states that capital should be taxed at zero in the long run One of the main assumptions underlying this result is that a govern ment can commit to a sequence of capital taxes However a benevolent government would choose to deviate from the prescribed sequence of taxes The reason is that once capital is accumulated it is sunk and taxing capital is no longer distortionary A benevolent government would choose high capital taxes once capital is accumulated The rea soning above motivates the analysis of time consistent policy as a game between a policy maker government and a continuum of economic agents consumers New Dynamic Public Finance A User s Guide 335 To highlight problems that arise when we depart from the benchmark of a benevolent planner with full commitment it is useful to start with Roberts 1984 example economy where similar to Mirrlees 1971 risk averse individuals are subject to unobserved shocks affecting the marginal disutility of labor supply But unlike the benchmark Mirrlees model the economy is repeated T times with individuals having per f
16. commitment power The more information is revealed by agents about their types the stronger is the incentive of the government to deviate from the originally promised tax sequences This motivated several authors to study optimal taxation in environments where the govern ment cannot commit Optimal taxation without commitment is techni cally a much more challenging problem since the simplest versions of the Revelation Principle do not hold in such an environment One of the early contributors was Roberts 1984 who studies an economy where individuals have constant skills which are private information Bisin and Rampini 2006 study a two period version of this problem Sleet 324 Golosov Tsyvinski and Werning and Yeltekin 2005 and Acemoglu Golosov and Tsyvinski 2006 show conditions under which even the simplest versions of the Revelation Principle can be applied along the equilibrium path We discuss these issues in section 4 3 A Two Period Mirrleesian Economy In this section we introduce a two period Mirrleesian economy with uncertainty 3 1 Preferences There is a continuum of workers that are alive in both periods and max imize their expected utility E u c o n Kule vn where c represents consumption and n is a measure of work effort With two periods the most relevant interpretation of our model is that the first period represents relatively young workers say those aged 20 45 while the second period rep
17. constraints that are considered for the next round and possibly dropping some of those that were not binding and repeat the procedure Comment Peter Diamond Massachusetts Institute of Technology and NBER 1 Introduction This interesting and stimulating paper referred to as GTW discusses four issues when capital should not be taxed when labor taxes should be constant over time and states of nature the sources of concern about limited government commitment and the methodology of modeling for tax analysis And it contains calculated examples I will touch on three of these issues leaving out the complex issue of how policy feasi bility and desirability are influenced by the nature of the political pro cess in democratic states In the macro tradition the analysis focuses on settings with stochastic shocks To bring a public economics perspec tive I will consider the first two issues in deterministic models with heterogeneous populations Then I will consider a stochastic model to add to the intuition about taxing savings For clarity of presentation I work with models with only two types of workers and assume that the binding incentive compatibility constraint is that type A not imitate type B I do not consider sufficient conditions for this pattern of con straints to be correct 2 Taxing Savings Atkinson Stiglitz 1976 showed that in the presence of optimal non linear earnings taxes it was not optimal to also use disto
18. extremely promising in the 70s and early 80s but received relatively less applied 358 Golosov Tsyvinski and Werning interest later One common explanation for this is that the approach made quantitative and applied work difficult and demanding We hope that this time around the recent surge in interest combined with the more advanced quantitative techniques and computing power avail able today may soon create enough progress to make solving realistic quantitative models feasible Recent quantitative work is promising in this regard e g Golosov and Tsyvinski 2006a Farhi and Werning 2006a but more is needed Another direction for future research is to relax the assumption of mechanisms operated by benevolent social planners A relevant ques tion in this context is whether the normative insights of the dynamic Mirrlees literature apply to the positive real world situations where politicians care about reelection self enrichment or their own individ ual biases and cannot commit to sequences of future policies A related question is under what conditions markets can be better than optimal mechanisms The potential misuse of resources and information by the government may make mechanisms less desirable relative to markets Certain allocations resulting from anonymous market transactions can not be achieved via centralized mechanisms Nevertheless centralized mechanisms may be preferable to anonymous markets because of the addit
19. is low for agents with low skills in the first period yet is quite high for agents with high skills The rea son is that it turns out that lower skilled workers are quite well insured Their consumption is not very volatile in the second period It follows 340 Golosov Tsyvinski and Werning 0 10 20 30 40 50 Figure 5 1 Consumption Allocation Middle Dotted Line Shows First Period Consumption Outer Solid Lines Are Second Period Consumption 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 a no a n Figure 5 2 Effective Labor Allocation Dashed Line Is for First Period Solid Lines Are for Second Period Top Is High Shock Bottom Low Shock New Dynamic Public Finance A User s Guide 341 distortion t 1 distortion high t 2 0 5 0 5 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 10 20 30 40 50 10 20 30 40 50 distortion tow t 2 distortion capital 0 05 0 5 0 04 0 4 0 03 0 3 0 02 0 01 0 2 10 20 30 40 50 10 20 30 40 50 Figure 5 3 Benchmark Implicit Marginal Tax Rates that the intertemporal distortion required is smaller Note that figure 5 1 shows that consumption uncertainty in the second period increases with the first period shock 5 3 Effects of the Size of Second Period Shocks We now consider the effects of an increase in the size of the adverse second period shock affecting agents This is an important exercise as it allows us to identify forces that distinguish the dynamic Mirrlees taxa tion in whic
20. is that at the opti mum taxes on capital average out to zero and raise no revenue That is the conditional average over j for 7 i j given by equation 11 is zero when the Inverse Euler equation 8 holds At first glance a zero average tax rate may appear to be at odds with the positive intertempo ral wedge 1 i defined by equation 7 found in Proposition 3 but it is not Savings are discouraged by this implementation The key point is that the tax is not deterministic but random As a result although the average net return on savings is unaffected by the tax the net return R s 1 7 j s is made risky Indeed since net returns are negatively related to consumption see equation 11 there is a risk premium com ponent in the language of financial economics to the expected return This tax implementation makes saving strictly less attractive just as the positive intertemporal wedge 1 suggests In some applications the number of shocks that agents face is small and with a certain structure that allows for simple decentralizations Golosov and Tsyvinski 2006a study a model of disability insurance where the only uncertainty agents face is whether and when they receive a permanent shock that makes them unable to work In this scenario the optimal allocation can be implemented by paying disabil ity benefits to agents who have assets below a specified threshold i e asset testing the benefits 4 4 Time Inconsistency
21. section 6 we examine another source for departures from the perfect tax smoothing bench mark 4 3 Tax Implementations In this section we describe the general idea behind decentralization or implementation of optimal allocations with tax instruments The general goal is to move away from the direct mechanism justified by the rev elation principle to study constrained efficient allocations and find tax systems so that the resulting competitive equilibrium yields these allo cations In general the required taxes are complex nonlinear functions of all past observable actions such as capital and labor supply as well as aggregate shocks It is tempting to interpret the wedges defined in 5 7 as actual taxes on capital and labor in the first and second periods Unfortunately the relationship between wedges and taxes is typically less straightforward Intuitively each wedge controls only one aspect of worker s behavior labor in the first or second period or saving taking all other choices fixed at the optimal level For example assuming that an agent supplies the socially optimal amount of labor a savings tax defined by 7 would ensure that that agent also makes a socially optimal amount of savings However agents choose labor and savings jointly In the context of our economy taxes in the first period T y can depend only on the observable labor supply of agents in that periods and taxes in the second period T y Y k s c
22. the first period a plausible condition is sufficient for a positive intertemporal consumption wedge The insight paralleling the argument through the inverse Euler condition is that with this condition less future work and lower future consumption will result in a higher marginal utility of consumption and a greater incentive to save unless the condition is not satisfied and the impact of hours worked on the mar ginal utility of consumption overcomes the higher level of consumption Easing the incentive compatibility constraint then comes from making the return to saving smaller Additivity makes this argument easy to make but the underlying argument has much greater generality GTW explore a class of nonseparable period utility functions in their numerical results They work with the utility function u c y 6 ce 1 o And they have many first period productivity 374 Diamond levels not just one This utility function satisfies the condition above that at equal utilities marginal utility of consumption is higher at the consumption labor pair that has higher consumption and labor Their finding of a negative wedge at some skill levels comes from a direct impact of nonseparability on the desired wedge as can be seen in the optimization in a model with first period variation and no conditional uncertainty about second period productivities Maximize Em ula 0 1 Bul V O subject to G Er c D Rc y
23. work effort function o n Kn y with x gt 0 and y 2 1 The first order conditions are then w M w i DN Aa 9 nO E OCO en lroa agl roo where A and s are first and second period multipliers on the resource constraints and where we define n 1 wiir viii zo A t x i n E wa yi R for notational convenience Combining and cancelling terms then leads to 2 y i 8 ai J rO x u c i s W t which proves that perfect tax smoothing is optimal in this case We sum marize this result in the next proposition derived by Werning 2007 for a more general dynamic framework nal K w c 0 n i t s 1 Proposition 4 Suppose the disutility of work effort is isoelastic v n kn y Then when idiosyncratic uncertainty for skills is concentrated in the first period so that 0 j i 0 i then it is optimal to perfectly smooth mar ginal taxes on labor t t s T Intuitively tax smoothing results from the fact that the tradeoff between insurance and incentives remains constant between periods and across states As shown by Werning 2007 if the distribution of skills 332 Golosov Tsyvinski and Werning varies across periods or aggregate states then optimal marginal taxes should also vary with these shifts in the distribution Intuitively the tradeoff between insurance and incentives then shifts and taxes should adjust accordingly In the numerical work in
24. 0 0 u c DO The consumption labor wedge distortion at t 2 for type i j in state s is t i 1 5 Zij 9 06 i se l 0 eG j SNC a The intertemporal wedge for type 7 is 1 i 1 ue 7 New Dynamic Public Finance A User s Guide 327 Note that in the absence of government interventions all the wedges are equal to zero 4 Theoretical Results and Discussion In this section we review some aspects of the solution to the planning problem that can be derived theoretically In the next sections we illus trate these features in our numerical explorations 4 1 Capital Wedges We now characterize the intertemporal distortion or implicit tax on capital We first work with an important benchmark in which there are no skill shocks in the second period That is all idiosyncratic uncer tainty is resolved in the first period For this case we recover Atkinson and Stiglitz s 1976 classical uniform taxation result implying no inter temporal consumption distortion Capital should not be taxed Then with shocks in the second period we obtain an Inverse Euler Equation which implies a positive intertemporal wedge Diamond and Mirrlees 1978 Golosov Kocherlakota and Tsyvinski 2003 4 1 1 Benchmark Constant Types and a Zero Capital Wedge In this section we consider a benchmark case in which the skills of agents are fixed over time and there is no aggregate uncertainty Specifically assume that N i 1 Vi a
25. 007 Inequality and Social Discounting Journal of Political Economy forthcoming Farhi Emmanuel Narayana Kocherlakota and Ivan Werning 2005 Estate Taxes and Estate Subsidies Work in Progress MIT Fernandez Villaverde Jesus and Aleh Tsyvinski 2004 Optimal Fiscal Policy in a Busi ness Cycle Model without Commitment Mimeo Harvard University Golosov Mikhail and Aleh Tsyvinski 2006a Designing Optimal Disability Insurance A Case for Asset Testing Journal of Political Economy 114 2 257 279 Golosov Mikhail and Aleh Tsyvinski 2006b Optimal Taxation with Endogenous Insur ance Markets Quarterly Journal of Economics forthcoming Golosov Mikhail Narayana Kocherlakota and Aleh Tsyvinski 2003 Optimal Indirect and Capital Taxation Review of Economic Studies 70 3 569 587 362 Golosov Tsyvinski and Werning Greenwald Bruce C and Joseph E Stiglitz 1986 Externalities in Economies with Imper fect Information and Incomplete Markets Quarterly Journal of Economics 101 229 264 Judd Kenneth L 1985 Redistributive Taxation in a Perfect Foresight Model Journal of Public Economics 28 59 83 Judd Kenneth L 1989 Optimal Taxation in Dynamic Stochastic Economies Theory and Evidence Mimeo Stanford University Judd Kenneth L 1999 Optimal Taxation and Spending in General Competitive Growth Models Journal of Public Economics 71 1 1 26 Kapi ka
26. 5 studies capital income taxation and owner ship in this context 4 See also Diamond Helms and Mirrlees 1980 for an early quantitative study of mod els in which taxes are not linear 5 A few papers have departed from the representative agent setting For example the analysis of optimal capital taxation in Judd 1985 allowed some forms of heterogeneity 6 One exception is Werning 2005a who studies tax smoothing and capital taxation in a model with heterogeneous agents subject to aggregate shocks Another one is Kocherla kota 2005 who extends the inverse Euler equation to the case of aggregate uncertainty and includes a numerical illustration of the optimum with two skill types 7 See also Kingston 1991 and Zhu 1992 for perfect tax smoothing results within a representative agent Ramsey economy with proportional taxation 8 See also Brito et al 1991 9 See Kocherlakota 2006 for another review of the literature 10 It is straightforward to extend the model by allowing the third period to explicitly distinguish retired individuals from older workers Indeed if we assume no labor deci sion in the third period nothing is lost by ignoring it and lumping consumption into the second period as we implicitly do here 11 The Revelation Principle guarantees that the best allocations can always be achieved by a mechanism where workers make reports about their types to the planner 12 See Diamond 1998 and Tuomala 1990
27. Barro 1979 and aggregate states of nature New Dynamic Public Finance A User s Guide 321 Lucas and Stokey 1983 As shown by Werning 2007 this notion does not depend on the representative agent assumption as it extends to economies with heterogenous agents subject to linear or nonlinear taxa tion Thus in our setup perfect tax smoothing obtains as long as all idio syncratic uncertainty regarding skills is resolved in the first period In our numerical exercises we also consider the case where idiosyn cratic uncertainty persists into the second period We find that labor wedges then vary across aggregate shocks Thus perfect tax smooth ing where the wedges for each skill type are perfectly invariant to aggregate states does not hold Tax rates vary because individual skill shocks and aggregate shocks are linked through the incentive con straints Interestingly aggregate shocks do not increase or decrease tax rates uniformly In particular we find that a positive aggregate shock from a higher return on savings or a lower government expenditure lowers the spread between labor wedges across skill types in the sec ond period 2 An Overview of the Literature The dynamic Mirrleesian literature builds on the seminal work by Mirrlees 1971 Diamond and Mirrlees 1978 Atkinson and Stiglitz 1976 and Stiglitz 1987 These authors laid down the foundation for analyzing optimal non linear taxation with heterogeneous agents
28. This PDF is a selection from a published volume from the National Bureau of Economic Research Volume Title NBER Macroeconomics Annual 2006 Volume 21 Volume Author Editor Daron Acemoglu Kenneth Rogoff and Michael Woodford editors Volume Publisher The MIT Press Volume ISBN 0 262 01239 1 978 0 262 01239 3 Volume URL http www nber org books acem06 1 Conference Dates April 7 8 2006 Publication Date May 2007 Chapter Title New Dynamic Public Finance A User s Guide Chapter Author Mikhail Golosov Aleh Tsyvinski Ivan Werning Chapter URL http www nber org chapters c11181 Chapter pages in book 317 388 5 New Dynamic Public Finance A User s Guide Mikhail Golosov MIT and NBER Aleh Tsyvinski Harvard University and NBER Ivan Werning MIT and NBER 1 Introduction New Dynamic Public Finance is a recent literature that extends the static Mirrlees 1971 framework to dynamic settings The approach addresses a broader set of issues in optimal policy than its static counterpart while not relying on exogenously specified tax instruments as in the represen tative agent Ramsey approach often used in macroeconomics In this paper we show that this alternative approach can be used to revisit three issues that have been extensively explored within repre sentative agent Ramsey setups We show that this alternative approach delivers insights and results that contrast with those from the Ramsey approach First it is o
29. accord ing to those reports Workers make skill reports i and j to the planner in the first and second period respectively Given each skill type i a reporting strategy is a choice of a first period report i and a plan for the second period report j j s as a function of the true skill realiza 326 Golosov Tsyvinski and Werning tion j and the aggregate shock Since skills are private information the allocations must be such that no worker has an incentive to misreport his type Thus the allocations must satisfy the following incentive con straint u c i of Mi nO gyja c i j s Bela male f e s 4 0 i 9 i j y i 2u c gi yvof WED BY lci i popeo ekia GIDS sj 0 i j for all alternative feasible reporting strategies i and j j In our applications we will concentrate on maximizing a utilitarian social welfare function The constrained efficient planning problem maxi mizes expected discounted utility EO BO co aes j s 0 sii Joep i t subject to the resource constraints in 1 and 2 and the incentive con straints in 4 Let c y k denote the solution to this problem To understand the implications of these allocations for the optimal tax policy it is important to focus on three key relationships or wedges between marginal rates of substitution and technological rates of trans formation The consumption labor wedge distortion in 1 for type is v y i
30. alysis as an explanation why we have income taxation instead of say lump sum taxes We do not need information economics to understand why taxes need to be different for people with different abilities to pay The key accomplish ment in Mirrlees 1971 is that he did not restrict the functional form of the tax policy He made the exogenous assumption that taxes depended Comment 383 only on income but avoided any further simplification such as linearity The asymmetric information story is useful as a way to motivate the search for the optimal nonlinear tax schedule and is a story that may apply in some tax problems and mechanism design problems but we should not take it literally in these tax models I conjecture that the commodity tax literature can be similarly moti vated That literature typified by Diamond and Mirrlees assumes that different goods are taxed at different rates but that for each good all individuals pay the same constant marginal tax rate If the government can observe only transactions not final consumption and cannot keep track of each individual s participation in each transaction then any nonlinearity in the tax system would be a source of arbitrage profits Therefore it is likely that the only feasible tax system would have con stant tax rates In fact most countries have a hybrid system where they do not attempt to measure each individual transaction except in the case of the labor and capital markets where t
31. an depend on labor supply in both first and second period as well as agents wealth In competi tive equilibrium agent solves max os i y BS wet j syo s ta A ale jl nol subject to c i K lt y T UD c i j 8 lt y i j 8 RASK TY yi j 8 KC 8 New Dynamic Public Finance A User s Guide 333 We say that a tax system implements the socially optimal allocation aO yO ci j 8 y i 7 if this allocation solves the agent s prob lem given T y0 and T y 0 y i j 8 k 2 8 Generally an optimal allocation may be implementable by various tax systems so T y i and T y y j k 8 may not be uniquely determined In contrast all tax systems introduce the same wedges in agents savings or consumption leisure decisions For this reason in the numerical part of the paper we focus on the distortions defined in section 3 and omit the details of any particular implementation In this section however we briefly review some of the literature on the details of implementation Formally the simplest way to implement allocations is a direct mecha nism which assigns arbitrarily high punishments if individual s con sumption and labor decisions in any period differ from those in the set of the allocations c i c i j j that solve the planning program Although straightforward such an implementation is highly unrealistic and severely limits agents choi
32. as important to recognize incompleteness when analyzing taxes In the context of taxes he thought it was unrealistic to think that policies could be contingent on a full set of types However he thought it was important to know what the optimal policy would be if policies could be contingent on a full set of types as a first step to thinking about what to do with fewer powers
33. bor wedge high t 2 17n 10 20 30 40 50 10 20 30 40 50 Labor wedge low t 2 Capital wedge 7 eater BS fo hte Sy 0 6 i ie 0 15 0 1 0 05 10 20 30 40 50 Figure 5 6 Varying Risk Aversion The effects of higher risk aversion on the intertemporal wedge are the outcome of two opposing forces 1 a direct effect for a given consump tion allocation a higher risk aversion increases the wedge the capi tal wedge results from the Inverse Euler equation by applying Jensen s inequality which is more powerful for higher 2 an indirect effect with higher curvature in the utility function u c it is optimal to insure more lowering the variability of consumption across skill realizations which reduces the capital wedge For the cases we considered the direct effect turned out to be stronger and the capital wedge increases with risk aversion 5 6 Effects of Changing Elasticity of Labor Supply We further investigate the properties of the optimum by consider ing three modifications of the disutility of labor Figure 5 7 shows the results Our benchmark case as before is v P plotted in bold in the figure We also display two more inelastic cases v P and v l l plotted with dashed lines 346 Golosov Tsyvinski and Werning Labor wedge t 1 Labor wedge high t 2 10 20 30 40 50 10 20 30 40 50 Labor wedge low t 2 10 20 30 40 50 Figure 5 7 Changing Elasticity of Labor Regardi
34. ces A significant body of work attempts to find less heavy handed alternatives One would like implementations to come close to using tax instruments currently employed in the United States and other advanced countries Here we review some examples Albanesi and Sleet 2006 consider an infinitely repeated model where agents face i i d skill shocks over time and there are no aggregate shocks They show that the optimal allocation can be implemented by taxes that depend in each period only on agent s labor supply and capi tal stock or wealth in that period The tax function T y k is typically non linear in both of its arguments Although simple their implemen tation relies critically on the assumption that idiosyncratic shocks are i i d and cannot be easily extended to other shocks processes Kocherlakota 2005 considers a different implementation that works for a wide range of shock processes for skills His implementation sepa rates capital from labor taxation Taxes on labor in each period t depend on the whole history of labor supplies by agents up until period and in general can be complicated non linear functions Taxes on capital are linear and also history dependent Specifically the tax rate on capital that is required is given by written for simplicity for the case with no aggregate uncertainty AGLO a TORTES BRau e i j 334 Golosov Tsyvinski and Werning Incidentally an implication of this implementation
35. cess Burden of Taxa tion Review of Economic Studies 21 1 21 30 Diamond P 2003 Taxation Incomplete Markets and Social Security Cambridge MA MIT Press Diamond P and J Mirrlees 1978 A Model of Social Insurance with Variable Retire ment Journal of Public Economics 10 295 336 Diamond P and J Mirrlees 1986 Payroll Tax Financed Social Insurance with Variable Retirement Scandinavian Journal of Economics 88 1 25 50 Diamond P and J Mirrlees 2000 Adjusting One s Standard of Living Two Period Models In P J Hammond and G D Myles eds Incentives Organization and Public Eco nomics Papers in Honour of Sir James Mirrlees Oxford Oxford University Press Diamond P and J Mirrlees 1982 Social Insurance with Variable Retirement and Pri vate Saving MIT Working Paper no 296 1995 unpublished Forthcoming in Journal of Public Economics Kaplow L 2006 On the Undesirability of Commodity Taxation Even When IncomeTax ation Is Not Optimal Journal of Public Economics 90 1235 Laroque G 2005 Indirect Taxation is Superfluous under Separability and Taste Homo geneity A Simple Proof Economics Letters 87 141 144 Marshall A 1948 Principles of Economics eighth edition New York The Macmillan Com pany Saez E 2002 The Desirability of Commodity Taxation under Non Linear Income Taxa tion and Heterogeneous Tastes Journal of Public Economics 83
36. d period are functions of s The probability of state s is denoted by H s The resource constraints are LaO K SRK G 1 S i y i aG 1 2 SR s K G s forallses 2 Lj where K is capital saved between periods t 1 and t 2 and K is the endowed level of capital An important special case is one without aggregate shocks In that case we can collapse both resource constraints into a single present value condition by solving out for K Ffeo no Lilet j yG lr p nosR G 5G 3 2 2 i i 3 4 Planning Problem Our goal is to characterize the optimal tax policy without imposing any ad hoc restrictions on the tax instruments available to a government The only constraints on taxes arise endogenously because of the infor mational frictions It is convenient to carry out our analysis in two steps First we describe how to find the allocations that maximize social wel fare function subject to the informational constraints Then we discuss how to find taxes that in competitive equilibrium lead to socially effi cient allocations Since we do not impose any restrictions on taxes a priori the tax instruments available to the government may be quite rich The next section describes features that such a system must have To find the allocations that maximize social welfare it is useful to think about a fictitious social planner who collects reports from the workers about their skills and allocates consumption and labor
37. d route is to take insights into the nature of optimal taxation from Mirrleesian models and incorporate them in a simplified fashion in Ramsey style models augmented with heterogeneity and idiosyn cratic uncertainty regarding skills The work by Conesa and Krueger 2005 and Smyth 2005 may be interpreted as a step in this direction These papers compute the optimal tax schedule in a model where the tax function is arbitrarily restricted but flexibly parameterized to allow for wide range of shapes including progressive taxation Work along these lines using state of the art computational models could explore other tax features such as certain differential treatments of capital and labor income or some forms of history dependence Another quantitative direction for research is to consider the implica tions of the new approach for classic macroeconomic questions such as the conduct of fiscal policy over the business cycle We only perfuncto rily touched on this topic but there is much more to be done to consider many of the issues that macroeconomists studied in the Ramsey tradi tions Ideally one could derive a rich set of quantitative predictions similar in spirit to the quantitative Ramsey analysis in Chari Chris tiano and Kehoe 1994 The main reason we stress the potential value of quantitative work is as follows In our view the approach to optimal taxation pioneered by Mirrlees 1971 and Atkinson and Stiglitz 1976 was seen as
38. dels that can capture some empirically relevant fea tures of skill dynamics such as those studied in for example Stores letten Telmer and Yaron 2004 The main difficulty is that it is currently not tractable to solve multiple period models with such a rich structure New Dynamic Public Finance A User s Guide 357 for skill shocks Most current studies impose simplifying assumptions that provide illustrative insights but remain unsuitable for quantitative purposes One recent route around this problem is provided by Farhi and Werning 2006a who study tax reforms in a dynamic Mirrleesian setting to evaluate the gains from distorting savings and provide a simple method which remains tractable even with rich skill processes There is also some early progress in analyzing dynamic Mirrlees models with persistent shocks using a first order approach in Kapicka 2005 A quantitative analysis could also be used to address and evalu ate the importance of a common challenge against the New Dynamic Public Finance literature that it delivers tax systems that are too complicated For example one could compare the level of welfare obtained with the fully optimal scheme to that which is attained when some elements of the tax system are simplified For example it may be interesting to compute the welfare losses from a tax code close to the one in the United States and other countries or other systems with limited history dependence A relate
39. der economies where individual borrowing and lending are not observable so that non linear distortions of savings are not feasible but the government may still uniformly influence the rate of return by taxing the observable capital stock Unlike the taxation of savings less work has been done in studying optimal labor wedges in the presence of stochastic skills shocks Batta glini and Coate 2005 show that if the utility of consumption is linear labor taxes of all agents asymptotically converge to zero Risk neutral ity however is crucial to this result Section 5 of this paper explores dynamic behavior of labor wedges for risk averse agents in our two period economy Due to space constraints we limit our analysis in the main body of the paper only to capital and labor taxation At this point we briefly mention recent work on other aspects of tax policy Farhi and Werning 2007 analyze estate taxation ina dynastic model with dynamic private information They show that estate taxes should be progressive Richer parents should face a higher marginal tax rate on bequests This result is a consequence of the optimality of mean reversion in consumption across generations which tempers the intergenerational transmission of welfare Rich parents must face lower net rates of return on their transfers so that they revert downward towards the mean while poor parents require the opposite to revert upwards Albanesi 2006 con siders optimal taxation
40. e complex With respect to this issue he felt that there was room for a middle ground In their view it was essential to bring heterogeneity and skill shocks into the models In such models it turned out to be convenient analytically to start by studying the case where the government is only restricted by the informational friction and not in addition by restrictions on the set of tax instruments He suggested that restrictions on tax instruments should be considered but only after the basic models were well under stood He also noted that in some cases the tax systems that emerged from their approach were reasonably simple citing recent work on disability insurance by Mikhail Golosov and Aleh Tsyvinski as well as work by himself and Robert Shimer Golosov said that they were sympathetic to Kenneth Judd s comment that it was important to think about the interaction of private market arrangements and government policies He said that this was the reason why they had deliberately used the term wedges rather than taxes in the paper However he emphasized that there are many circumstances 386 Discussion where even if markets are perfectly functioning they would fail to yield efficient outcomes due to externalities Greg Mankiw asked Peter Diamond what the evidence was for the statement he had made that high type people are more patient Dia mond responded that the assumptions on preferences that are made in these models imply that hig
41. e full insurance The agent s consumption would be equalized across realizations of shocks Labor of agents would be increasing with their type It is obvious that when skills are unobserv able the unconstrained optimal allocation is not incentive compatible as an agent with a higher skill would always prefer to claim to be of a lower type to receive the same consumption but work less The optimal allocation with unobservable types balances two objectives of the social planner Providing insurance and respecting incentive compatibility constraints New Dynamic Public Finance A User s Guide 339 The optimal allocation for the benchmark case with unobservable types is shown in figure 5 1 and figure 5 2 There is no bunching in either period Agents of different skills are allocated different consump tion and labor bundles First note that there is a significant deviation from the case of perfect insurance agents consumption increases with type and consumption in the second period for an agent who claims to have a high shock is higher than that of an agent with the low shock The intuition for this pattern of consumption is as follows It is optimal for an agent with a higher skill to provide a higher amount of effective labor One way to make provision of higher effective labor incentive compatible for an agent is to allocate a larger amount of consumption to him Another way to reward an agent for higher effort is to increase his continuati
42. e move beyond the simple models In particular the optimal tax problem becomes multidimen sional in some cases forcing the authors to consider far more incentive 384 Judd constraints than is necessary in the usual one dimensional models This is because the single crossing property that is heavily exploited in the one dimensional literature has no analogue for even two dimensional problems Therefore if there are N types of taxpayers we need to exam ine N incentive constraints instead of N Judd and Su 2006 have examined this problem in more complex cases and find cases far more challenging than the ones in this paper and argue that the multidimensional optimal tax problem is generally far more difficult They show that the solution to an optimal taxation problem will generally not satisfy the linear independence constraint qualification a fact that greatly increases the difficulty of solving these problems numerically Fortunately the last decade has seen many advances in the field of mathematical programming with equilibrium constraints which can be applied to these problems Again I congratulate the authors for their users guide to an approach that can potentially provide major insights into the design of rational public policy and encourage other young researchers to follow their lead References Atkinson A B and J E Stiglitz 1976 The Design of Tax Structure Direct vs Indirect Taxation Journal of Public Ec
43. ear taxation of savings This decrease in observability lowers social welfare since the incentive compatibility constraint becomes more restrictive when the potential imitator can simply modify savings A parallel result is then that the optimum includes taxation of savings not subsidization That is one can see the same underlying mecha nism that savings adjustment makes the incentive compatibility con straint harder to meet and so one should discourage savings in this slightly different setting First consider the individual savings problems 1 if planning to produce the output level of type A when type A and 2 if planning to produce the type B output even if type A Note that what was previ ously consumption is now the net of tax wage Denote the net of tax return on savings by Q Define the indirect utility of consumption functions V c A B Q Max u c s 2 Bulc A Qs 42 1 Pulc B Qs V c c B Q Max ufc s Bulc B Qs 43 Note that the optimal savings levels s depend on the same vari ables as the indirect utility functions V With these preferences and much more generally since c A gt c B we have s gt s The social welfare maximization now becomes Maximize _V fe A B Q oly 8 Er poly B0 subject to G Ea c Rc i y R y i s 1 QR 44 V c A B QI 2 Boly A 8A 2 V a c B QI 2 Boly B A
44. ectly persistent types Under full commitment a benevolent planner would choose the same allocation at every date which coincides with the optimal solution of the static model However a benevolent gov ernment without full commitment cannot refrain from exploiting the information that it has collected at previous dates to achieve better risk sharing ex post This turns the optimal taxation problem into a dynamic game between the government and the citizens Roberts showed that as discounting disappears and T gt the unique sequential equilib rium of this game involves the highly inefficient outcome in which all types declare to be the worst type at all dates supply the lowest level of labor and receive the lowest level of consumption This example shows the potential inefficiencies that can arise once we depart from the case of full commitment even with benevolent governments The nature of time inconsistency in dynamic Mirrlees problems is therefore very dif ferent from that in a Ramsey model In the dynamic Mirrlees model the inability of a social planner not to exploit information it learns about agents types is a central issues in designing optimal policy without commitment A recent paper by Bisin and Rampini 2006 considers the problem of mechanism design without commitment in a two period setting They show how the presence of anonymous markets acts as an additional constraint on the government ameliorating the commitment proble
45. ed efficient allocation satisfies an Inverse Euler Equation 1 1 1 hamli 8 w c BR brai D If there is no uncertainty in second period consumption given the first period shock the condition becomes E S A u c BR w which is the standard Euler equation that must hold for a consumer who optimizes savings without distortions Whenever consumption remains stochastic the standard Euler equa tion must be distorted This result follows directly by applying Jensen s inequality to the reciprocal function 1 x in equation 8 gt u c BR u C 9 Proposition 3 Suppose that for some i there exists j such that 0 lt atjli lt 1 and that c i is not independent of j Then the constrained efficient alloca tion satisfies w c lt BR Zui GID 4 i gt 0 The intuition for this intertemporal wedge is that implicit savings affect the incentives to work Specifically consider an agent who is con templating a deviation Such an agent prefers to implicitly save more than the agent who is planning to tell the truth An intertemporal wedge worsens the return to such deviation The Inverse Euler Equation can be extended to the case of aggregate uncertainty Kocherlakota 2005 At the optimum 330 Golosov Tsyvinski and Werning 1 1 wei N E pz rezanaca jo If there is no uncertainty regarding skills in the second period this expression reduces to w c BYR 3 u c i slus so that the int
46. em of maximizing utility subject only to the economy s resource constraints This extreme case emphasizes the more general point that a key determinant of distortions is the desire to redistribute or insure workers with respect to their skills As a result the level of taxation is affected by the distribution of skills and risk aversion among other things 320 Golosov Tsyvinski and Werning 1 3 Numerical Results We now summarize the main findings from our numerical simulations We begin with the case without aggregate uncertainty We found that the main determinants for the size of the labor wedge are agents skills the probability with which skill shocks occurs risk aversion and the elasticity of labor supply Specifically we found that the labor wedges in the first period or for those in the second period not suffering the adverse shock are largely unaffected by the size or probability of the adverse shock these parameters affect these agents only indirectly through the ex ante incentive compatibility constraints Higher risk aversion leads to higher labor wedges because it creates a higher desire to redistribute or insure agents As for the elasticity of labor supply we find two opposing effects on the labor wedge A lower elasticity leads to smaller welfare losses from redistribution but also leads to less pre tax income inequality for a given distribution of skills making redistribution less desirable Turning to the capita
47. eriod significantly steepens the capital wedge profile We conclude that the shape and size of the capital wedge responds significantly to the size of the shocks that an agent may experience in the future 5 4 Effects of the Probability of Second Period Shocks and Uncertainty We now consider the effects of changing the probability of the adverse second period shock This exercise is of interest because it allows us to investigate the effects of uncertainty about future skill realizations on the size and shape of wedges In figure 5 5 we show in bold the benchmark case where 2 21 0 5 dashed line correspond to 2 21 0 7 and 0 9 while the dotted lines correspond to 2 2 0 3 and 0 1 respectively We first notice that the effects of the change in the probability of the adverse shock on labor wedge are similar to the case of increase in size of the adverse shock That is as the probability z 2 of a drop in skills rises the informational friction increases and so does the labor wedge For the intertemporal wedge there is an additional effect of chang ing the probability of the adverse skill shock The wedge is the highest when uncertainty about skills is the highest At the symmetric baseline case with 7 21 0 5 Intuitively the reason is that the uncertainty about next period s skill is maximized at 7 2 0 5 It is uncertainty about future skills rather than the level of next period s skill shock that matters f
48. ertemporal marginal rate of substitution is undistorted However if the agent faces idiosyncratic uncertainty about his skills and consumption in the second period Jensen s inequality implies that there is a positive wedge on savings u c 0 lt uly IDR s u c i j 8 4 2 Tax Smoothing One of the main results from the representative agent Ramsey frame work is that tax rates on labor income should be smoothed across time Barro 1979 and states Lucas and Stokey 1983 This result extends to cases with heterogenous agents subject to linear or nonlinear taxation Werning 2007 that is where all the unobserv able idiosyncratic uncertainty about skills is resolved in the first period To see this take j 1 i Oi We can then write the allocation entirely in terms of the first period skill shock and the second period aggregate shock The incentive constraints then only require truthful revelation of the first period s skill type i 2 By ue i spre BED his gt 10 YG Yo i 8 wei o li Jeze 6 Jes for all i i Let y i i represent the Lagrangian multiplier associated with each of these inequalities The Lagrangian for the planning problem that incorporates these constraints can be written as u c t of uk New Dynamic Public Finance A User s Guide 331 X lasy i ue oro 4e pf ueso 4E2 vii i of a pof gi ea uet sn 22 aon To derive the next result we adopt an iso elastic utility of
49. etitively Even Mirrlees 1971 assumes that workers are paid their marginal product implicitly assuming that there is no market power in labor markets In these analyses taxes are then chosen to distort market outcome so as to accomplish a realloca tion of resources desired by the government In the dynamic Mirrlees approach outlined in this paper there are no private markets for insur ance and government policy is used for both conventional purposes of raising revenue for government expenditures and redistribution as well as to replace or at least offer a substitute for the missing private markets 382 Judd This point is acknowledged in this paper The authors are often care ful to refer to distortions as wedges staying away from the question of what they signify Section 4 5 correctly argues that in many cases pri vate markets will attain a constrained Pareto allocation and that these private outcomes will have many of the same wedges often called taxes in the dynamic Mirrlees literature If the government does not enjoy an advantage in either transaction costs or information then no govern ment policy can attain a Pareto superior allocation This does not mean that the dynamic Mirrlees approach as executed so far has no value The point here is that we should do as Mirrlees did assume that private markets work and then find the policy that best achieves the goal taking into account the presence of a private market I suspect tha
50. for insurance and sig nificantly increases the size of both labor wedges However the effect on capital wedges could be ambiguous as the uncertainty about future skills also matters 3 Capital wedges are affected by the degree of uncertainty over future skills 4 A lower elasticity of labor decreases the capital wedge but could have ambiguous effects on labor wedge for a given skill distribution 5 If utility is nonseparable between consumption and labor the capi tal wedge may become negative The sign of the wedge in that case depends on whether labor is complementary or substitutable with con sumption and on whether an agent expects to experience a higher or a lower shock to skills in the future 6 Aggregate Uncertainty In this section we explore the effects of aggregate uncertainty In section 4 2 we showed that if agents types are constant it is optimal to perfectly smooth labor taxes i e the labor wedges are constant across states and periods The literature on new dynamic public finance virtually has not explored implications of aggregate uncertainty 6 1 Baseline Parameterization We use unless otherwise noted the same benchmark specifications as in the case with no aggregate uncertainty Additional parameters that we have to specify are as follows We assume that there are two aggre gate states s 2 The probability of the aggregate states are symmetric 1 2 1 2 We take the number of skills in the fir
51. ggregate shocks We also argued that this approach not only provides a workable alternative to Ramsey models but that it also comes with several sig nificant advantages over its predecessor First while Ramsey models have provided several insights into optimal policy their well under stood limitation regarding the ad hoc nature of tax instruments may make interpreting their prescriptions problematic In contrast the main premise of the Mirrleesian approach is to model heterogeneity or uncer tainty creating a desire for insurance or redistribution and an infor mational friction that prevents the first best allocation and determines the set of feasible tax instruments endogenously In particular although a simple non discriminatory lump sum tax component is never ruled out the optimum features distortions because these improve redistri bution and insurance Second we also argued that this approach has novel implications for the type of dynamic policy issues that macro economists have been interested in capital taxation smoothing of labor income taxes and the nature of the time consistency problem In addi tion some new issues may arise directly from the focus on richer tax instruments such as the progressivity of taxation In what follows we outline what we think are largely unresolved questions that we hope are explored in future research One remaining challenge is the quantitative exploration of the theory using calibrated mo
52. h skilled people have higher earnings and that people who discount the future less heavily have higher sav ings rates Given this he said the statement follows from the empiri cal correlation between savings rates and earnings Mankiw responded that this correlation may be due to consumption smoothing Diamond thought that it was unreasonable to think that consumption smoothing explained the entire correlation James Poterba remarked that the paper had potential implications for the design of the tax period He observed that many people had argued in favor of a lifetime income tax He noted that such a tax seemed to dilute the information on what happens period by period Poterba asked if the paper was pushing in the opposite direction by advocating that the government should exploit high frequency information Werning responded that some of their results were supportive of tax smooth ing but that temporary shocks to individuals generally did move the optimal tax system away from a completely smooth tax He conjectured that it might be possible that a lifetime income tax accompanied with side programs like unemployment insurance to deal with temporary shocks might be close to what the theory suggests is optimal Kenneth Rogoff remarked that the discussants had emphasized the importance of knowing how robust the results of the paper were along several dimensions He noted that another important dimension to generalize the model was the internationa
53. h skills stochastically change over time from a dynamic case in which types of agents do not change over time We consider a range of shocks From a very large shock 0 05 that makes an agent almost disabled in the second period to a small drop 0 95 that barely changes the agent s skill In figure 5 4 the bold line corresponds to the benchmark case of 0 5 the dashed lines correspond to 0 6 0 8 0 9 and 0 95 while the dotted lines correspond to 0 3 0 1 and 0 05 respectively 342 Golosov Tsyvinski and Werning Labor wedge t 05 0 4 03 0 2 0 1 10 20 3 40 50 Labor wedge low t 2 Capital wedge Figure 5 4 Varying amp We now describe the effects of an increase in the size of the skill shocks on the labor wedges First notice that the size of the second period shocks practically does not affect the first period wedge sched ule 7 0 and the shape and the level are preserved Even when agents experience a high shock to their skills e g 0 05 the schedule of labor wedges in the first period is essentially identical to the case when an agent experiences a very small shock 0 95 Similarly we don t see large changes in the marginal labor wedge schedule 7 1 in the second period for the high realization of the shocks i e if skills remain the same as in the previous period Interestingly the marginal tax on labor in the second period after a downward drop 7
54. he monitoring costs are moderate This reinterpretation is important because it frees us from unneces sary constraints on the models we look at There is currently a kind of orthodoxy that tries to draw a sharp line between models with exog enous and endogenous institutions arguing that the latter is obviously better However a closer examination of the problem such as in this tax case reveals shades of gray It is not clear which is better An analysis that exogenously specifies a set of policy instruments corresponding to the ones we see used or using false assumptions about informational costs in order to derive an endogenous set of instruments Tax problems like the ones examined in this paper quickly become extremely com plex Demanding analyses with fully endogenous sets of instruments will severely limit the range of problems we can examine The models discussed in this paper are obviously limited in many ways In particular there are too few periods in the dynamic dimension and there is usually no capital accumulation There is great potential in this literature but only if we address the mathematical difficulties We must give up focusing on simple problems that can be solved ana lytically or characterized in simple ways and exploit computational tools if we are to attain quantitatively substantive results This won t be easy For example the numerical approach used in this paper is indica tive of the challenges that we face when w
55. iglitz 1976 show that when preferences are separable savings should not be taxed but that in general whenever preferences are non separable some distortion is optimal Depending on the details of the allocation and on the sign of u this distortion may be positive or negative We now turn to the case with o gt 1 and consider o 1 2 3 see figure 5 9 The baseline with o 1 is plotted as the dotted line Away from the baseline higher correspond to lower lines on the graph 348 Golosov Tsyvinski and Werning distortion t 1 distortion high t 2 0 5 0 5 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 10 20 30 40 50 10 20 30 40 50 distortion low t 2 distortion capital 0 05 0 5 A 0 4 S 0 0 3 0 2 0 1 0 05 10 20 30 40 50 Figure 5 8 Nonseparable Utility with oS 1 We notice that higher o pushes the whole schedule of labor distor tions up The intuition is again that higher risk aversion leads to more insurance and redistribution requiring higher distortions A higher is associated with a uniformly higher schedule of capital distortions and these are always positive Second higher omay create a non monotonicity in the schedule of capital distortions with the high est distortions occurring for intermediate types It is not only the value of the o that determines the sign of the wedge We found that for the case where the skill shocks in the second period have an upward trend so that 1 5 and g 1 i e an agent may ex
56. inal taxes rates are unaffected This result is relevant for thinking about balanced growth in an extension of the model to an infinite horizon It is also convenient in that it allows us to normalize without any loss of generality the second period shock for our numerical explorations We now discuss how we choose parameters for the benchmark exam ple We use the following baseline parameters We first consider the case with no aggregate uncertainty Assume that there is no discounting and that the rate of return on savings is equal to the discount factor R B 1 We choose the skill distribution as follows In the first period skills are distributed uniformly Individual skills in the first period 8 i are equally spaced in the interval 6 The probability of the realiza tion of each skill is equal to z i 1 N for all i We choose baseline parameters to be 0 1 1 and N 50 Here a relatively large number of skills allows us to closely approximate a continuous distri bution of skills In the second period an agent can receive a skill shock For computational tractability we assume that there are only two pos sible shocks to an agent s skill in the second period N i 2 for all i Skill shocks take the form of a proportional increase 6 i 1 9 i or 338 Golosov Tsyvinski and Werning proportional decrease i 2 9 i For the baseline case we set 1 and 1 2 This means that an agent i
57. ion and higher labor Relative differences in con sumption would become larger and increase the desire for redistribution given our constant relative risk aversion specification of preferences In the figure 5 11 we plot the intertemporal wedges for the case with government expenditures thin line and for the case of no government expenditures bold line As in the case of labor wedges we see that the size of the wedge is higher in the case of government expenditures Figure 5 10 Labor Distortion 352 Golosov Tsyvinski and Weming Intertemporal Wedge 0 09 0 08 0 07 0 06 0 05 0 04 0 03 0 02 o2 o 10 15 20 25 3 Figure 5 11 Intertemporal Distortion We could have considered a case of transitory changes in govern ment expenditures i e keep government expenditure deterministic but make it higher or lower in the second period versus the first This case is very similar to the one above given our simple linear savings technology as it is the present value of government expenditures that matters rather than the distribution of them across time 6 4 Effects of Aggregate Shocks to Government Expenditures We now consider the effects of stochastic shocks to government expen ditures In this specification we have G 0 2 G 1 0 3 G 2 0 2 and H 1 0 7 u 2 0 3 In figure 5 12 we plot labor wedges The solid line is 7 the dotted line is T Av 1 1 i e high type in state 1 the dashed line i i
58. ional insurance they provide to risk averse agents Acemoglu Golosov and Tsyvinski 2006 approach these questions with a model that combines private information regarding individual skill types with the incentive problems associated with self interested rulers Finally we close by emphasizing that the New Dynamic Public Finance approach can be used to analyze a large variety of new topics rarely explored within Ramsey settings For instance one recent line of research focuses on intergenerational issues Phelan 2005 and Farhi and Werning 2007 consider how intergenerational incentives should be structured while Farhi and Werning 2006b and Farhi Kocherla kota and Werning 2005 derive implications for optimal estate taxa tion This is just one example of how this approach promises more than just new answers to old questions but also leads to new insights for a large set of unexplored questions Acknowledgments For comments and suggestions we thank Daron Acemoglu V V Chari Peter Diamond Kenneth Judd James Mirrlees and Michael Woodford New Dynamic Public Finance A User s Guide 359 Endnotes 1 However see Diamond and Mirrlees 1978 1986 1995 for early work with dynamic economies with private information 2 Judd 1999 extends the analysis to cover cases where no steady state may exist 3 Aiyagari et al 2002 and Werning 2005 study tax smoothing of labor income taxes when markets are incomplete Farhi 200
59. l dimension Rogoff felt that this was especially important in the context of a world in which both financial and human capital were increasingly mobile Daron Acemoglu remarked that the Mirrlees approach to optimal tax ation was notso much in the business of writing exact models that could make precise predictions but rather concerned with understanding general principles He felt that the real power of the Mirrlees approach was that it was making an explicit effort to understand what the con straints on taxes are He noted that even though the Ramsey approach often yielded nice insights the question about why lump sum taxes were ruled out always remained He noted that in the dynamic setting lump sum taxes sneak in through the back door in that the optimal tax mimics a lump sum tax Golosov agreed with Acemoglu s assessment Discussion 387 Peter Diamond said that while Werning had stressed the role of shocks and Kenneth Judd had talked about insurance markets his own comments stressed the role of predictable differences between people He emphasized that there were many predictable differences between people and that in these cases what insurance markets cannot do comes to the fore He noted that the conclusions of optimal tax theory were likely to change once it was taken into account that the adjustments made by workers in response to shocks are in practice not always smooth in the number of hours worked Diamond also remarked that it w
60. l wedge we find that two key determinants for its size are the size of the adverse future shock and its probability A higher elasticity of labor may decrease the savings wedge if it decreases the desire to redistribute More significantly we derive some novel pre dictions for capital wedges when preferences over consumption and labor are nonseparable The theoretical results in dynamic Mirrleesian models have been derived by assuming additively separable utility between consumption and labor In particular the derivation of the Inverse Euler optimality condition which ensures a positive capital wedge relies on this separability assumption Little is known about the solution of the optimal problem when preferences are not separable Here we partially fill this gap with our numerical explorations The main finding of the model with a nonseparable utility function is that the capital wedge may be negative We show that the sign of the wedge depends on whether consumption and labor are complements or sub stitutes in the utility function as well as on whether skills are expected to trend up or down We now describe the cases with aggregate uncertainty Most of our numerical findings are novel here since aggregate shocks have remained almost unexplored within the Mirrleesian approach When it comes to aggregate shocks an important insight from repre sentative agent Ramsey models is that tax rates on labor income should be smoothed across time
61. ll shocks becomes less valuable The economy then behaves closer to the benchmark where there are no new skill shocks where perfect tax smoothing obtains We now turn to figure 5 13 which shows the intertemporal distor tion In that figure the upper dashed line is 4 0 7 the solid line is 4 0 5 and the lower dotted line is 4 0 3 We see that intertemporal wedge becomes higher with higher 4 6 5 Effects of Rate of Return Shocks In this section we consider the effects of shocks to returns We consider a case in which R 1 1 and R 2 4 In figure 5 14 we plot labor distortions We plot labor wedges as follows The solid line is yC the 354 Golosov Tsyvinski and Werning Intertemporal Wedge Figure 5 13 Intertemporal Distortion Labor Wedge Figure 5 14 Rate of Return Shocks New Dynamic Public Finance A User s Guide 355 dotted line is A a 1 1 e wedge for the high shock type in state 1 the dashed line is AS 2 1 i e wedge for the low type in state 1 the dotted line with thick dots is AG 1 2 i e wedge for the high type in state 2 the dashed line with thick dots is T Ae 2 2 i e wedge for the low type in state 2 As in the case of government expenditure shocks here we also observe that the spread between wedges on low and high type in a bad state are higher We now turn to the analysis of the behavior of the capital wedge under aggregate uncertainty Figure 5 15 plots the inte
62. m Acemoglu Golosov and Tsyvinski 2006 depart from Roberts 1984 framework and consider instead of a finite horizon economy an infi nite horizon economy This enables them to use punishment strategies against the government to construct a sustainable mechanism defined as an equilibrium tax transfer program that is both incentive compatible for the citizens and for the government i e it satisfies a sustainabil ity constraint for the government The best sustainable mechanism implies that if the government deviates from the implicit agreement citizens switch to supplying zero labor implicitly punishing the gov ernment The infinite horizon setup enables them to prove that a ver sion of the revelation principle truthful revelation along the equilibrium path applies and is a useful tool of analysis for this class of dynamic 336 Golosov Tsyvinski and Werning incentive problems with self interested mechanism designers and with out commitment The fact that the truthful revelation principle applies only along the equilibrium path is important since it is actions off the equilibrium path that place restrictions on what type of mechanisms are allowed these are encapsulated in the sustainability constraints This enables them to construct sustainable mechanisms with the rev elation principle along the equilibrium path to analyze more general environments and to characterize the limiting behavior of distortions and taxes
63. n and trans formation between consumption and labor The second wedge is the intertemporal or capital wedge defined as the difference between the expected marginal rate of substitution of consumption between peri ods and the return on savings In this paper our focus is distinctively on these wedges which are sometimes termed implicit marginal tax rates rather than on explicit tax systems that implement them How ever we do devote a section to discussing the latter 1 2 Ramsey and Mirrlees Approaches The representative agent Ramsey model has been extensively used by macroeconomists to study optimal policy problems in dynamic set New Dynamic Public Finance A User s Guide 319 tings Examples of particular interest to macroeconomists include the smoothing of taxes and debt management over the business cycle the taxation of capital in the long run monetary policy and a variety of time inconsistency problems This approach studies the problem of choosing taxes within a given set of available tax instruments Usually to avoid the first best it is assumed that taxation must be proportional Lump sum taxation in particular is prohibited A benevolent government then sets taxes to finance its expenditures and maximize the representative agent s util ity If instead lump sum taxes were allowed then the unconstrained first best optimum would be achieved One criticism of the Ramsey approach is that the main goal of the go
64. n the second period can only receive an adverse shock a We also assume that there is uncertainty about realization of skills and set 7 117 2 217 1 2 The agent learns his skill in the second period only at time t 2 We chose the above parameterization of skills to allow a stark characterization of the main forces determining the optimum We choose the utility function to be power utility The utility of con sumption is u c c 1 o As our baseline we take o 1 so that u c log c The utility of labor is given by u 1 as our benchmark we set y 2 We use the following conventions in the figures below 1 The horizontal axis displays the first period skill type i 1 2 50 2 The wedges distortions in the optimal solutions are labeled as fol lows a Distortion t 1 is the consumption labor wedge in period 1 A b Distortion high t 2 is the consumption labor wedge in period 2 for an agent with a high skill shock AG 1 c Distortion low t 2 is the consumption labor wedge in period 2 for an agent with a low skill shock AZ 2 d Distortion capital is the intertemporal capital wedge q i 5 2 Characterizing the Benchmark Case In this section we describe the numerical characterization of the opti mal allocation Suppose first that there were no informational friction and agents skills were observable Then the solution to the optimal program would featur
65. nd that 6 i i j i In this case the constrained efficient problem simplifies to max Z ue 0o HO swear we ko subject to the incentive compatibility constraint that Vi 1 N and i e 1N naroa eluea soh ufe i l si alue pal alt and subject to the feasibility constraint 328 Golosov Tsyvinski and Werning Flao norEe0 v0 h ERK G1 7G i 2 2 We can now prove a variant of a classic Atkinson and Stiglitz 1976 uniform commodity taxation theorem which states that the marginal rate of substitution should be equated across goods and equated to the marginal rate of transformation To see this note that only the value of total utility from consumption u c Bu c enters the objective and incentive constraints It follows that for any total utility coming from consumption u c i Bu c a it must be that resources c i 1 R c are minimized since the resource constraint cannot be slack The next proposition then follows immediately Proposition 1 Assume that the types of agents are constant A constrained efficient allocation satisfies u c BR w c i Vi Note that if B R then c c Indeed in this case the optimal allocation is simply a repetition of the optimal one in a static version of the model 4 1 2 Inverse Euler Equation and Positive Capital Taxation Wenow return to the general case with stochastic types and derive a necessary condition for optimality The Inver
66. ng the effect on labor distortions intuitively there are two opposing forces On the one hand as labor becomes more inelastic wedges introduce smaller inefficiencies Thus redistribution or insur ance is cheaper On the other hand since our exercises hold constant the skill distribution when labor supply is more inelastic the distribu tion of earned income is more equal Hence redistribution or insurance are less valuable Thus combining both effects there is less uncertainty or inequality in consumption but marginal wedges may go either up or down In our simulations it seems that the first effect dominated and the labor wedges increased when the elasticity of labor was reduced The distortion on capital unambiguously goes down since consump tion becomes less variable 5 7 Exploring Nonseparable Utility We now consider a modification to the case of non separable utility between consumption and labor When the utility is nonseparable the New Dynamic Public Finance A User s Guide 347 analytical Inverse Euler results that ensured a positive intertemporal wedge may no longer hold Indeed the effects of nonseparable utility on the intertemporal wedge are largely unexplored 5 7 1 Building ona Baseline Case We start with the specification of the utility function that can be directly comparable with our baseline specification ce je u c leg Here the baseline case with separable utility is equivalent to o 1 When
67. ngs is part of the optimum The GTW exploration of the taxation of savings focuses on uncer tainty about future earnings as a source of the desirability of taxation of savings It is true that people are uncertain about future earnings It is also true that people differ in discount rates The case for not taxing savings does not survive either issue with plausible character izations lt PBR 8 Comment 369 3 Earnings Tax Smoothing With uncertainty about future earnings different workers will realize different age earnings profiles and this uncertainty can require varying implicit taxes on earnings over time over worker ages In contrast GTW show tax smoothing when everyone has the same age earnings profile and the disutility of labor is a power function A failure of tax smoothing also comes without uncertainty if we allow different age earnings profiles for different workers In this example there is no wedge on the intertemporal consumption decision However there are different consumption earnings wedges in the two periods and so a wedge on the intertemporal earnings decision With the same notation as above consider a two types model with two periods of earnings and the only binding incentive compatibility constraint that type A not want to imitate type B with that imitation done for the entire life Maximize u e Bule D oly Poly 8 0 subject to G Er c Rc i y R y i 0 9
68. of entrepreneurs In her setup an entrepreneur exerts unobservable effort that affects the rate of return of the project She shows that the optimal intertemporal wedge for the entrepreneurs can be either positive or negative da Costa and Werning 2005 study a monetary model with heterogeneous agents with privately observed skills where they prove the optimality of the Friedman rule that the optimal inflationary tax is zero The analysis of optimal taxation in response to aggregate shocks has traditionally been studied in the macro oriented Ramsey literature Wer ning 2002 2007 reevaluated the results on tax smoothing in a model with private information regarding heterogeneous skills In his setup all idiosyncratic uncertainty after the initial period is due to unobserv able shock In Section 6 for the two period economy introduced in this paper we explore the extent of tax smoothing in response to aggregate New Dynamic Public Finance A User s Guide 323 shocks when unobservable idiosyncratic shocks are also present in the second period Some papers for example Albanesi and Sleet 2006 Kocherlakota 2005 and Golosov and Tsyvinski 2006a consider implementing optimal allocations by the government using tax policy Those analyses assume that no private markets exist to insure idiosyncratic risks and agents are able to smooth consumption over time by saving at a mar ket interest rate Prescott and Townsend 1984 show that the fi
69. of the whole riddle The more the issue is thus narrowed the more exactly can it be handled but also the less closely does it correspond to real life Each exact and firm handling of a narrow issue however helps towards treating broader issues in which that narrow issue is contained more exactly than would otherwise have been possible With each step exact discussions can be made less abstract realistic discussions can be made less inexact than was possible atan earlier stage I view a realistic discussion as best drawing intuitively on mul tiple models of different aspects of a question This is very different from taking literally the answer generated by a single model even one viewed as the best available single model This is especially true when the best available model is visibly highly limited in key dimensions as is the case when a representative agent model is being analyzed for normative tax analysis Comment 377 Asecond distinction they draw is between linear taxes and nonlinear taxes Since some taxes are linear in practice it seems worthwhile to analyze how to set linear taxes as well Since it is often the case that linear taxes operate in the presence of nonlinear ones it is important to learn about that interaction But not all linear taxes are in a setting where there are nonlinear taxes making a separate analysis also worth while In Massachusetts it is not constitutional to have progressive tax ati
70. omplete Markets Mimeo MIT Werning ivan 2007 Optimal Fiscal Policy with Redistribution Quarterly Journal of Eco nomics forthcoming Zhu Xiaodong 1992 Optimal Fiscal Policy in a Stochastic Growth Model Journal of Economic Theory 58 250 89 Appendix Numerical Approach In this appendix we describe the details of the numerical computations that we performed in this paper The major conceptual difficulty with computing this class of models is that there are a large number of incentive constraints and there iS no result analogous to static models that guarantee that only local incentive compatibility constraints can bind to reduce them Our computational Strategy in this regard is as follows 1 We start with Solving several examples in which we impose all of the IC con straints This Step gives us a conjecture on what kind of constraints may bind 2 We then impose constraints that include deviations that bind in step 1 In fact we include a larger set that also includes constraints in the neighborhood of reporting Strategies to the ones that bind 3 Finally once the optimum is computed we check that no other constraints bind This approach is very much like the active set approach in constrained opti mization one begins with a set of constraints that are likely to be the binding ones one then Solves the smaller problems checking all constraints and add ing the constraints that are violated in the set of
71. on value i e allocate a higher amount of expected future consumption for such an agent We now turn our attention to the wedges in the constrained efficient allocation In the unconstrained optimum with observable types all wedges are equal to zero We plot optimal wedges for the benchmark case in figure 5 3 We see that the wedges are positive indicating a significant depar ture from the case of perfect insurance We notice that the consumption labor wedge is equal to zero for the highest skill type in the first period and for the high realization of the skill shock in the second period 1 8 7 8 1 0 This result confirms a familiar no distortion at the top result due to Mirrlees 1971 which states that in a static con text the consumption labor decision of an agent with the highest skill is undistorted in the optimal allocation The result that we obtain here is somewhat novel as we consider an economy with stochastically evolv ing skills for which the no distortion at the top result have not yet been proven analytically We also see that the labor wedges at the bottom t 8 7 8 1 ACE 2 are strictly positive A common result in the literature is that with a continuum of types the tax rate at the bottom is zero if bunching types is not optimal In our case there is no bunching but this result does not literally apply because we work with a discrete distribution of types We see that the intertemporal wedge
72. on of a single kind of income apart from an exempt amount Some would love to see the same restriction in the U S constitution More generally political economy considerations may call for restrictions in the taxes considered I wonder if the very minor distinctions in income taxation by age of the worker in current U S law are not a reflection of the difficulty in setting so many tax parameters as would be needed with different income taxes for each age of a worker or pairs of ages for a working couple Or maybe this is just the lag of practice behind theory as we saw in the roughly two decade lag in the United States in collecting tolls only one way on some bridges and tunnels The third distinction drawn by GTW is between a given restricted set of tax tools referred to as an ad hoc restriction and deriving the set of tax tools from an underlying technology asymmetric informa tion in the Mirrlees case I think this distinction is overdrawn First if we assume that for some transactions asymmetric information extends to the parties engaged in transactions then taxation of a transaction might vary with the size of the transaction but cannot vary with the presence of other transactions Then nonlinear taxation based on total earnings is not feasible Assuming that without this constraint there would be higher taxation of larger transactions and that such taxation can be prevented by repeated transactions then we are left with linear taxa
73. onomics 6 55 75 Diamond Peter and James A Mirrlees 1971 Optimal Taxation and Public Production American Economic Review 61 8 27 175 208 Judd Kenneth L 1985 Redistributive Taxation in a Perfect Foresight Model Journal of Public Economics 28 59 83 Judd Kenneth L 1999 Optimal Taxation and Spending in General Competitive Growth Models Journal of Public Economics 71 1 1 26 Judd Kenneth L and Che Lin Su 2006 Optimal Income Taxation with Multidimen sional Taxpayer Types Hoover Institution Working Paper Mirrlees James A 1971 An Exploration in the Theory of Optimum Income Taxation Review of Economic Studies 38 2 175 208 Discussion Ivan Werning began by saying that he agreed with most of what the discussants had said He noted that part of the discussants comments had focused on bringing new issues to the table He and his coauthors felt that this was exactly what was nice about their approach to the tax problem namely that it could address issues that could not have been addressed before using the traditional Ramsey approach Werning observed that their approach provided scope for making normative assessments on the effects of policies related to unemployment com plementing the positive analysis from the previous day s discussion on unemployment in Europe Werning agreed that the optimal tax systems that emerge from the class of models they studied were in some cases quit
74. or the size of the capital wedge 344 Golosov Tsyvinski and Werning Labor wedge t 1 Labor wedge high t 2 0 5 a 0 4 0 3 0 2 0 1 0 10 20 30 40 50 10 20 30 40 50 Labor wedge low t 2 Capital wedge 10 20 30 40 50 Figure 5 5 Varying the Probability of Skill Drop 7 2 5 5 Effects of Changing Risk Aversion We proceed to explore effects of risk aversion on optimal wedges and allo cations This exercise is important as risk aversion determines the need for redistribution or insurance for an agent Specifically we change the risk aversion parameter in the utility function The results are shown in figure 5 6 Our benchmark case of logarithmic utility o 1 is shown in bold With dotted lines we plot lower risk aversions o 0 8 0 5 0 3 and 0 1 and with dashed lines we plot higher risk aversions o 1 5 and 3 The immediate observation is that a higher degree of risk aversion leads to uniformly higher distortions The intuition is again rather simple We know that if o 0 so that utility is linear in consumption and an agent is risk neutral private information about the skill would not affect the optimal allocation and the unconstrained allocation in which all wedges are equal to zero can be obtained The higher is risk aversion the higher is the desire of the social planner to redistribute and insure agents Therefore all distortions rise New Dynamic Public Finance A User s Guide 345 Labor wedge t 1 La
75. perience a positive skill shock the results are reversed In particular for o lt 1 we found that capital wedges were always positive whereas for o gt 1 they were negative over some region of skills Intuitively the trend in skills matters because it affects the trend in labor We obtained similar results with the alternative specification of util ity also common in macroeconomic models New Dynamic Public Finance A User s Guide 349 distortion t 1 distortion high t 2 0 6 0 6 0 4 0 45 0 2 l 0 2 0 0 10 20 30 40 50 10 20 30 40 50 distortion low t 2 distortion capital 10 20 30 40 50 Figure 5 9 Nonseparable Utility with o2 1 Garan D u c l a This utility function was used by Chari Christiano and Kehoe 1994 in their quantitative study of optimal monetary and fiscal policy 5 8 Summarizing the Case with No Aggregate Uncertainty The exercises above give us a comprehensive overview of how the opti mal wedges depend on the parameters of the model We now sum marize what seems to be most important for the size and the shape of these wedges 1 Labor wedges on the agent affected by an adverse shock increase with the size or the probability of that shock However labor wedges in other periods and labor wedges for agents unaffected by the adverse 350 Golosov Tsyvinski and Werning shock are influenced only indirectly by this variable and the effects are small 2 Higher risk aversion increases the demand
76. provides a calibrated example of distribution of skills Diamond 1998 also uses Pareto distribution of skilis Here we abstract from the effects of varying the skill distribution 18 Two notable exceptions are Kocherlakota 2005 and Werning 2005a 19 We thank Ken Judd for pointing this to us References Abraham Arpad and Nicola Pavoni 2003 Efficient Allocations with Moral Hazard and Hidden Borrowing and Lending Mimeo University College London Acemoglu Daron Mikhail Golosov and Aleh Tsyvinski 2006 Markets versus Govern ments Political Economy of Mechanisms Mimeo MIT Aiyagari S Rao Albert Marcet Thomas J Sargent and Juha Sepp l 2002 Optimal Taxation without State Contingent Debt Journal of Political Economy 110 6 1220 1254 Albanesi Stefania 2006 Optimal Taxation of Entrepreneurial Capital under Private Information Mimeo Columbia University Albanesi Stefania and Christoper Sleet 2006 Dynamic Optimal Taxation with Private Information Review of Economic Studies 73 1 30 Arnott Richard and Joseph Stiglitz 1990 The Welfare Economics of Moral Hazard In H Louberge ed Information and Insurance Essays in Memory of Karl H Borch 91 121 Norwell MA Kluwer Arnott Richard and Joseph E Stiglitz 1986 Moral Hazard and Optimal Commodity Taxation Journal of Public Economics 29 1 24 Atkinson Anthony B and Joseph E Stiglitz 1976 The De
77. ptimal to introduce a positive distortion in savings that implicitly discourages savings Diamond and Mirrlees 1978 Rog erson 1985 Golosov Kocherlakota and Tsyvinski 2003 This contrasts with the Chamley Judd Judd 1985 Chamley 1986 result obtained in Ramsey settings that capital should go untaxed in the long run Second when workers skills evolve stochastically due to shocks that are not publicly observable their labor income tax rates are affected by aggregate shocks Perfect tax smoothing as in Ramsey models Barro 1979 Lucas and Stokey 1983 Judd 1989 Kingston 1991 Zhu 1992 Chari Christiano and Kehoe 1994 may not be optimal with uncertain and evolving skills In contrast it is optimal to smooth labor distortions when skills are heterogenous but constant or affected by shocks that are publicly observable Werning 2007 Finally the nature of the time consistency problem is very different from that arising within Ramsey setups The problem is essentially about learning and using acquired information rather than taxing sunk capital A benevolent government is tempted to exploit information collected in the past Indeed capital is not directly at the root of the problem in that even if the government 318 Golosov Tsyvinski and Werning controlled all capital accumulation in the economy or in an economy without capital a time consistency problem arises 1 1 User s Guide We call this paper a user s guide
78. py ufc Bute i aly 8 subject to G Pm Rc i ys S0 1 ufe A Bulc A oly A 6 A ufc B B u c B viy B 8 A with notation c i consumption in period j of household i y i earnings in period j of household i i skillin period j of household i number of workers of type i discount factor of household i 1 plus the return to capital government expenditures A y LaGrange multipliers OQ DMA 368 Diamond This problem has the FOCs for consumption levels 4 W wie lA An 2 x y Byu le A Az R 3 m y u c B An 4 78 WBa u c B Ax R 5 Taking the ratio of FOCs for A there is no tax on savings on the high type w c A lt a ra T PAR 6 ulea This is the familiar no marginal taxation condition at the very top of the earnings distribution Now let us turn to type B Taking the ratio of FOCs we have Zg y Ww c B Ba PgR 7 xs z vB te The plausible case is that high earners have a lower discount of future consumption lt B resulting with z y gt 0 in u c B u co B That is type B would save if that were possible at zero taxation of savings so there is implicit marginal taxation of savings If and only if B B does this imply no taxation of savings for type B Saez does his analysis with linear taxation of savings and concludes that since higher earners have higher savings rates taxing savi
79. qui libria with Adverse Selection and Moral Hazard Econometrica 52 21 45 Roberts Kevin 1984 Theoretical Limits to Redistribution Review of Economic Studies 51 177 195 Rogerson William P 1985 Repeated Moral Hazard Econometrica 53 1 69 76 Saez Emmanuel 2001 Using Elasticities to Derive Optimal Income Tax Rates Review of Economic Studies 68 1 205 229 Shimer Robert and Ivan Werning 2005 Liquidity and Insurance for the Unemployed Mimeo MIT Sleet Christopher and Sevin Yeltekin 2005 Credible Social Insurance Mimeo Carnegie Mellon University Smyth S 2005 A Balancing Act Optimal Nonlinear Taxation of Labor and Capital in Overlapping Generation Economies Mimeo Harvard University New Dynamic Public Finance A User s Guide 363 Stiglitz Joseph 1987 Pareto Efficient and Optimal Taxation and the New New Wel fare Economics In A Auerbach and M Feldstein eds Handbook on Public Economics 991 1042 North Holland Elsevier Science Publishers Storesletten Kjetil Chris I Telmer and Amir Yaron 2004 Cyclical Dynamics in Idiosyn cratic Labor Market Risk Journal of Political Economy 112 3 695 717 Tuomala Matti 1990 Optimal Income Taxation and Redistribution Oxford University Press Clarendon Press Werning Ivan 2002 Optimal Dynamic Taxation PhD Dissertation University of Chicago Werning Ivan 2005 Tax Smoothing with Inc
80. resents relatively older workers and retired individuals say those older than 45 3 2 Skills Following Mirrlees 1971 workers are at any time heterogenous with respect to their skills and these skills are privately observed by workers The output y produced by a worker with skill and work effort n is given by the product effective labor y n The distribution of skills is independent across workers For computational reasons we work with a finite number of skill types in both periods Let the skill realizations for the first period be 8 i fori 1 2 N and denote by z i their ex ante probability dis tribution equivalent to the ex post distribution in the population In the second period the skill becomes 6 i j for j 1 2 N where j i is the conditional probability distribution for skill type j in the second period given skill type i in the first period 3 3 Technology We assume production is linear in efficiency units of labor supplied by workers In addition there is a linear savings technology New Dynamic Public Finance A User s Guide 325 We consider two types of shocks in the second period 1 a shock to the rate of return and 2 a shock to government expenditures in the second period To capture both shocks we introduce a state of the world s e S where S is some finite set which is realized at the beginning of period t 2 The rate of return and government expenditure in the sec on
81. rson does not work Diamond and Mirrlees 1978 1986 2000 The insight paralleling the argument through the inverse Euler con dition is that when less future work with lower future consumption results in a higher marginal utility of consumption and so a greater incentive to save making savings less available eases the incen tive compatibility constraint Additivity makes this argument easy to make but the underlying condition is plausible and has much greater generality To see this argument I go through the same steps as above The opti mization becomes Maximize ule y 8 La Pulc i y 1 8 3 Comment 373 subject to G c y Ea Rc i Ry i lt 0 28 Pulec A y A 6 A 2 Bulc B y B A This problem has the FOCs for consumption levels ule y 4 29 z y Bule A y A 6 A Ar R 30 npu c B y B 0 B ypu c B y B 6 A An R 31 Adding the last two equations and taking a ratio to the first equation we have clei y1 A Vic ie2 A y2 70A ague lez B 2 B 7 62B vuele 128 RA A 32 In contrast without a wedge the individual would see a gain from savings if ulei y 0 lt BR 33 Mau C2 A 2 A O2 A ngu c2 B yo B A2 B Thus the sign of the wedge depends on the sign of wu c A y A A uc B y B 8 A 34 which is signed by the condition above Thus in a setting where every one is the same in
82. rst wel fare theorem holds in economies with unrestricted private markets and the efficient wedges can be implemented privately without any gov ernment intervention When markets are very efficient distortionary taxes are redundant However if some of the financial transactions are not observable the competitive equilibrium is no longer constrained efficient Applying this insight Golosov and Tsyvinski 2006b and Albanesi 2006 explore the implications of unobservability in financial markets on optimal tax interventions We discuss some of these issues in section 4 In step with theoretical advances several authors have carried out quantitative analyses of the size of the distortion and welfare gains from improving tax policy For example Albanesi and Sleet 2006 study the size of the capital and labor wedges in a dynamic economy However they are able to conduct their analyses only for the illustrative case of iid shocks to skills Moving to the other side of the spectrum with permanent disability shocks Golosov and Tsyvinski 2006a show that the welfare gains from improving disability insurance system might be large Recent work by Farhi and Werning 2006a develops a general method for computing the welfare gains from partial reforms starting from any initial incentive compatible allocations with flexible skill pro cesses that introduce optimal savings distortions All the papers discussed above assume that the government has full
83. rtemporal distor tion 1 for various values of the shock to the rate of return R 1 solid line the benchmark case of no uncertainty and R 2 1 2 2 3 and 4 dotted lines We see that distortions decrease with the rate of return shock R Intu itively a higher R leads to more resources and with more resources the planner can distribute them in a way that reduces the relative spread in consumption making the desire for redistribution lower given our CRRA preferences and thus lowering the need to distort We also explored the effects of upwards shocks for R 2 1 1 2 2 3 and 4 on labor distortions Qualitatively they are similar to the ones in figure 5 14 Intertemporal Wedge 0 05 0 045 0 04 Figure 5 15 Intertemporal Distortion Varying R 356 Golosov Tsyvinski and Werning 6 6 Summary We can now summarize the main implications of our analysis There are two main points to take away from this section 1 aggregate shocks lead to labor wedges differing across shocks and 2 a positive aggre gate shock either a higher return on savings or lower realization of government expenditures leads to lower capital wedges and toa lower spread between labor wedges 7 Concluding Remarks In this paper we reviewed some main results from the recent New Dynamic Public Finance literature We also provided some novel explo rations in the determinants of capital and labor wedges and how these wedges respond to a
84. rting linear consumption taxes provided that all consumer preferences are sepa rable between goods and labor and all consumers have the same sub utility function of consumption Laroque 2005 and Kaplow 2006 have extended this result showing that with the same preference assumptions in the presence of any income tax function that gives rise to an equilibrium if there are distorting consumer taxes then a move 366 Diamond to nondistorting consumer taxes can be done along with a permutation of the income tax that leaves every consumer with the same utility and the same labor supply while the government collects more revenue If labor supply is smooth with uniform transfers to all consumers no jumps in labor supply then this revenue gain can be used to make a Pareto improvement GTW explore this issue by solving a social welfare optimization with quantities as control variables and incentive compatibility constraints as well as a resource constraint or constraints if there is uncertainty about aggregate resources Then they compare the MRS between first and second period consumptions at the optimal allocation to the MRT The comparison allows calculation of the wedge between them reflecting implicit marginal taxation of savings They consider two other wedges between consumption and earnings in each of the two periods reflecting the implicit marginal taxation of earnings They compare these two labor wedges to find condition
85. s Mimeo Getulio Vargas Foun dation da Costa Carlos and Ivan Werning 2002 Commodity Taxation and Social Insurance Mimeo MIT da Costa Carlos and lv n Werning 2005 On the Optimality of the Friedman Rule with Heterogeneous Agents and Nonlinear Income Taxation MIT Working Paper Cam bridge MA MIT Diamond Peter A L J Helms and James A Mirrlees 1980 Optimal Taxation in a Sto chastic Economy Journal of Public Economics 14 1 29 Diamond Peter A 1998 Optimal Income Taxation An Example with a U Shaped Pat tern of Optimal Marginal Tax Rates American Economic Review 88 1 83 95 Diamond Peter A and James A Mirrlees 1978 A Model of Social Insurance with Vari able Retirement Journal of Public Economics 10 3 295 336 Diamond Peter A and James A Mirrlees 1986 Payroll Tax Financed Social Insurance with Variable Retirement Scandinavian Journal of Econontics 88 1 25 50 Diamond Peter A and James A Mirrlees 1995 Social Insurance with Variable Retire ment and Private Saving Mimeo MIT Farhi Emmanuel 2005 Capital Taxation and Ownership when Markets Are Incom plete Mimeo MIT Farhi Emmanuel and Iv n Werning 2006a Capital Taxation Quantitative Implications of the Inverse Euler Equation Mimeo MIT Farhi Emmanuel and Ivan Werning 2006b Progressive Estate Taxation Mimeo MIT Farhi Emmanuel and lv n Werning 2
86. s 7 AG 2 1 i e low type in state 1 the dotted line with thick dots is Tal i 2 i e high type in state 2 the dashed line with thick dots is 1 2 2 Ge low type in state 2 The most important observation is that there is a difference in taxes across realizations of government expenditure This contradicts one interpretation of perfect tax smoothing which would lead one to expect wedges to remain constant across these shocks This finding is new to both the literature on dynamic Mirrlees taxation and to the Ramsey taxation literature For example Ramsey models call for smoothing labor tax distortions across states of the economy As reviewed in sub New Dynamic Public Finance A User s Guide 353 Figure 5 12 Shocks to Government Expenditure section 4 2 without unobservable idiosyncratic shocks tax smoothing also obtains in a Mirrleesian model Interestingly the distortions do not move in the same direction for the low and high types This is in contrast to the comparative static exercise in figure 5 10 where lower government expenditure leads to lower taxes overall Here instead the spread between the distortions on the low and high types becomes smaller when government expen ditures are low Our intuition is that when government expenditure is low resources are more abundant As a consequence the contribu tion to output from labor the source of inequality becomes relatively smaller Thus insuring the new ski
87. s where earnings are marginally taxed the same in both periods These labor wedges are also examined with aggregate uncertainty about the resource constraint in order to compare wedges across states of nature The comparison of labor wedges across periods is really a fourth wedge between earn ings in the two periods That is in this four good model there are two separate own rates of interest in earnings and in spending The Atkinson Stiglitz condition for non use of distorting consump tion taxes has naturally received a great deal of attention particularly with the interpretation of present and future consumption goods and so the taxation of savings That is under these assumptions using the vocabulary of GTW there is no wedge between MRS and MRT for consumptions in different periods With no wedge for intertemporal consumption unless the implicit marginal taxation of earnings is con stant over time there is a nonzero wedge between earnings in different periods Below I will offer a simple example of an optimal model with no wedge in intertemporal consumption but a wedge in intertemporal earnings that is non constant marginal taxation of earnings Despite the great interest in the Atkinson Stiglitz result there remain arguments in favor of taxing savings with nonlinear earnings taxes One obvious argument would be that preferences do not exhibit the sepa rability between consumption and labor used in the theorem Then the Corlett Hag
88. se Euler Equation This optimality condition implies a positive marginal intertemporal wedge We consider variations around any incentive compatible allocation The argument is similar to the one we used to derive Atkinson and Stiglitz s 1976 result In particular it shares the property that for any realization of i in the first period we shall minimize the resource cost of delivering the remaining utility from consumption Fix any first period realization 7 We then increase second period util ity u c i j in a parallel way across second period realizations j That is define u i j A u c i j A for some small A To compensate we decrease utility in the first period by BA That is define u i A u c i BA for small A The crucial point is that such variations do not affect the objective function and incentive constraints in the planning problem Only the resource constraint is affected Hence for the original allocation to be optimal it must be that A 0 minimizes the resources expended New Dynamic Public Finance A User s Guide 329 aia DEG j AGI i le 0 A Eu leali j AGI 2 j for all i The first order condition for this problem evaluated at A 0 then yields the Inverse Euler equation summarized in the next proposi tion due originally to Diamond and Mirrlees 1978 and extended to an arbitrary process for skill shocks by Golosov Kocherlakota and Tsyvinski 2003 Proposition 2 A constrain
89. sign of Tax Structure Direct vs Indirect Taxation Journal of Public Economics 6 55 75 Barro Robert 1979 On the Determination of the Public Debt Journal of Political Econ omy 87 5 940 971 Battaglini Marco and Stephen Coate 2005 Pareto Efficient Taxation with Stochastic Abilities Mimeo Cornell and Princeton Benhabib Jess and Aldo Rustichini 1997 Optimal Taxes without Commitment Journal of Economic Theory 77 231 259 Bisin Alberto and Adriano Rampini 2006 Markets as Beneficial Constraints on the Government Journal of Public Econontics 90 4 5 601 629 Brito Dagobert L Jonathan H Hamilton Steven M Slutsky and Joseph E Stiglitz 1991 Dynamic Optimal Income Taxation with Government Commitment Journal of Public Economics 44 1 15 35 Chamley Christophe 1986 Optimal Taxation of Capital in General Equilibrium Econo metrica 54 607 622 New Dynamic Public Finance A User s Guide 361 Chari V V and Patrick Kehoe 1990 Sustainable Plans Journal of Political Economy 94 783 802 Chari V V Lawrence J Christiano and Patrick Kehoe 1994 Optimal Fiscal Policy in a Business Cycle Model Journal of Political Economy 102 4 617 652 Conesa Juan Carlos and Dirk Krueger 2005 On the Optimal Progressivity of the Income Tax Code Journal of Monetary Economics 53 7 1425 1450 da Costa Carlos 2005 Yet Another Reason to Tax Good
90. st period to be N 30 As before skills are equispaced and uniformly distributed We set R 1 6 2 Effects of Government Expenditure Fluctuations We now turn to analyzing the effects of government expenditures There is a sense in which return and government expenditure shocks are similar in that they both change the amount of resources in the sec New Dynamic Public Finance A User s Guide 351 ond period that is for a given amount of savings K they are identi cal Comparative statics in both exercises however are different in that they may induce different effects on savings In the exercises that follow we assume that there are no return shocks and R 1 R 2 1 6 3 Effects of Permanent Differences in G We first consider a comparative static exercise of an increase in gov ernment expenditures Suppose we increase G G 1 G 2 0 2 i e there is no aggregate uncertainty Figure 5 10 shows labor wedges for this case We plot in bold the benchmark case of no government expenditures G G 1 G 2 0 and using thin lines the case of G G 1 G 2 0 2 solid lines correspond to the first period distortion dashed lines to the second period distortion of the low types and dot ted lines to the second period distortion of the high types We see that higher G leads to higher labor wedges Intuitively if the wedge schedule were not changed then higher expenditure would lead to lower average consumpt
91. t this is a much more difficult problem explaining why this path has not been taken but the insights in the work summarized in this paper will help us tackle the more complex problem This paper makes the common assertion that Mirrlees endogenized the tax instruments by basing his analysis on an informational friction more specifically Mirrlees assumed that the government could observe income but could not observe either hours or wages This is argued to be superior to starting with an exogenously restricted set of tax instruments I disagree with this characterization of Mirrlees 1971 In fact wages and hours are not only observable but are often used by the government Many workers punch a time clock recording when a worker begins his work and when he finishes and his income is the product of the measured hours and a wage rate known to both worker and employer If wages and hours could not be observed then we could have neither minimum wage laws nor laws regarding overtime pay Of course wages and hours would be difficult to measure for many individuals and impossible for some occupations such as profes sors However ignoring the wage and hours information that could be obtained cheaply is particularly odd in any analysis such as Mir rlees 1971 where the objective is to shift money to the poor since they are the ones more likely to have jobs with easily observed wages and hours For these reasons I do not view Mirrlees an
92. tion derived not assumed Second there is the issue of admin istrative costs which are assumed to be zero for observables in the Mirrlees model We can recast asymmetric information as assuming that the administrative cost is infinite for what are otherwise labeled non observables This can be a helpful recasting We could track the identity for each purchase of gasoline the way we do each payment of earnings But that would be expensive but becoming less so par ticularly if we do not allow purchases for cash If expensive enough gasoline purchase should be subject to linear taxes as they are Hav ing a more basic model deriving what tax structure might otherwise be assumed is not necessarily a virtue if the basic model has critical incompleteness 378 Diamond In GTW there are two periods with a stochastic change in worker skill between the two periods This allows taxes to be set differently in each period But if skills evolve more rapidly than taxes are set because of administrative costs perhaps then the modeling needs to recognize an explosion of types depending on all the stochastic realizations of opportunities that might occur within a year Plausibly we are in the same basic position as with the assumption of a complete set of markets no one can list all the states that might occur So we can not envision trading on all of them even apart from the cost in today s resources of preparing in this way for distant and or low
93. ue 1953 style analysis in a 3 good model current work current consumption and future consumption can examine whether a move towards taxing savings or towards subsidizing savings raises Comment 367 welfare But we do not know much about the relevant cross elasticities although the commonly used assumptions of atemporal and intertem poral separability strike me as implausible Another argument for taxing savings one that is based closely on empirical observations is due to Saez 2002 He argues that there is a positive correlation between labor skill level wage rate and the savings rate In a two period certainty setting with additive preferences this is consistent with those with higher earnings abilities having less discount of future consumption In terms of the conditions of the Atkinson Sti glitz theorem Saez preserves separability but drops the assumption that the subutility function of consumption is the same for everyone I begin my formal analysis echoing Diamond 2003 by showing this result in a two types model with labor only in the first period illustrat ing the Atkinson Stiglitz result at the same time Consider the following social welfare function optimization Assume full nonlinear taxation and two types of households with the only binding incentive compatibility constraint being type A considering imitating type B I do not analyze sufficient conditions for this to be the only binding constraint Maximize
94. vernment is to mimic lump sum taxes with an imperfect set of instruments However very little is usu ally said about why tax instruments are restricted or why they take a particular form Thus as has been previously recognized the represen tative agent Ramsey model does not provide a theoretical foundation for distortionary taxation Distortions are simply assumed and their overall level is largely determined exogenously by the need to finance some given level of government spending The Mirrlees approach to optimal taxation is built on a different foun dation Rather than starting with an exogenously restricted set of tax instruments Mirrlees s 1971 starting point is an informational friction that endogenizes the feasible tax instruments The crucial ingredient is to model workers as heterogenous with respect to their skills or pro ductivity Importantly workers skills and work effort are not directly observed by the government This private information creates a trade off between insurance or redistribution and incentives Even when tax instruments are not constrained distortions arise from the solution to the planning problem Since tax instruments are not restricted without heterogeneity the first best would be attainable That is if everyone shared the same skill level then a simple lump sum tax that is an income tax with no slope could be optimally imposed The planning problem is then equivalent to the first best probl
95. w turn to numerical exercises with baseline parameters and per form several comparative static experiments The exercises we conduct strike a balance between flexibility and tractability The two period New Dynamic Public Finance A User s Guide 337 setting is flexible enough to illustrate the key theoretical results and explore a few new ones At the same time it is simple enough that a complete solution of the optimal allocation is possible In contrast most work on Mirrleesian models has focused on either partial theoretical characterizations of the optimum e g showing that the intertempo ral wedge is positive Golosov Kocherlakota and Tsyvinski 2003 or on numerical characterizations for a particular skills processes e g i i d skills in Albanesi and Sleet 2006 or absorbing disability shocks in Golosov and Tsyvinski 2006a In a recent paper Farhi and Werning 2006a take a different approach by studying partial tax reforms that fully capture the savings distortions implied by the Inverse Euler equa tion The problem remains tractable even with empirically relevant skill processes 5 1 Parameterization When selecting parameters it is important to keep the following neutral ity result in mind With logarithmic utility if productivity and govern ment expenditures are scaled up within a period then 1 the allocation for consumption is scaled by the same factor 2 the allocation of labor is unaffected and 3 marg

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