Accounting for a Positive Correlation between Pension and Consumption Taxes *

Transcription

1 Accounting for a Positive Correlation between Pension and Consumption Taxes * Eungsik Kim Chul-In Lee Tepper School of Business Department of Economics Carnegie Mellon University Seoul National University (This version: March 20, 2017) PRELIMINARY DRAFT. PLEASE DO NOT CITE OR DISTRIBUTE WITHOUT PERMISSION OF THE AUTHORS. COMMENTS WELCOME. Abstract We attempt to account for a puzzling comovement between pension level and the consumption tax rate observed in the OECD data. First, using a standard overlapping generations model with lifetime uncertainty, we can find a set of optimal policy combinations of taxes and pension, but they cannot account for the data. Second, to resolve this puzzle, we consider welfare states where pension level is higher than the optimal level due to external and/or institutional reasons. In this setting, our analysis of optimal tax mix demonstrates that strengthening consumption taxation (relative to income taxation) can improve welfare, i.e., accounting for the proposed puzzle. Third, we also find that population aging further boosts the role of consumption taxation, reinforcing our main findings. Finally, our results lend support to recent pension reforms: when expanding welfare benefits, most countries tend to resort to consumption tax financing. JEL classification: E62, H21, H55 Keywords: Pension; Consumption Tax; Optimal Fiscal Policy; Overlapping Generations Model; Aging Population *We are grateful to conference participants at numerous places, including the Eastern Economic Association 43rd Annual Conference and the 2012 Korea Association of Public Finance Conference for comments and suggestions. address: eungsikk@andrew.cmu.edu address: leeci@snu.ac.kr 1

2 1 Introduction Public pensions usually take the largest share of fiscal expenditures of modern governments. Accordingly, understanding pensions permits an insight into the fiscal situation of a country. The recent worldwide trends of population aging and economic downturns reinforce this observation. In fact, many developed countries have expanded their welfare programs over time, especially public pensions. (See Figure 1) Currently, this generates the sustainability issue in the face of rising national debt and increasing the tax burden on young workers. 1 Figure 1: Public pension spending in OECD countries, Gonzalez-Eiras and Niepelt [9], D Amato and Galasso [7] and Song [21] show the size of social security has been expanded in a political economy model due to aging population, larger aggregate uncertainty, and more severe wealth inequality respectively. They all derive that the pension benefit level at politico-economic equilibrium is often larger than the optimal benefit level from the Ramsey problem. This implies that pension reform is required in some developed countries where the social security benefit is too generous. Although much of the recent discussion has dwelt on a proper reform of public pensions but pension reform has been unsuccessful in many countries around the world since an increasing elderly population in the era of population aging has a greater political voice than before, adding more rigidity to the institution of pension. 2 With limited room to cut social security spending due to pressures from an aging society, many developed countries have changed revenue measures, particularly raising the consumption tax, to advance fiscal consolidation. As a representative case, Japan has experienced a 1 The recent on-going sovereign debt crisis is partly related to excessive government expenditures for redistributive purposes. 2 The pension reform of France triggered by the Sarkozy administration in 2010 shows how difficult it is to amend the public pension system. The Sarkozy government was successful eventually in passing the law to postpone the retirement age from 60 to 62. However, there had been a huge nationwide turmoil to protest the reform. 2

3 tremendous expansion in gross public debt over the past two decades from 1995 to 2014: its gross public debt more than quadrupled and about 250 percent of GDP in The Japan s fiscal deficit issue has been largely contributed by its rapidly growing social security spending as a result of population aging. The country expands the public pension spending from 6.06 percent of GDP in 1995 to percent in To provide a stable source of revenue and reduce its high level of public debt, Japan raised the value added tax (VAT) by 3 % from 5% to 8% in An increase in the consumption tax offers not only more revenue to the society but also distributes the tax burden more fairly across generations since those entering retirements period pay a part of the pension benefit through consumption. The economic and financial crisis in 2009 leads to a substantial increase in the VAT rate in many developed countries to sustain the welfare programs and improve the financial consolidation. According to an OECD tax trends report in 2016 [18], 22 out of 33 OECD countries raised their VAT rate at least once between 2009 and These changes generated a surge of the unweighted OECD average consumption tax rate from 17.7% in 2009 to 19.2% in The anecdotal evidence above shows that countries tend to use the consumption tax as an fiscal consolidation strategy. This implies welfare states with more generous pension systems are likely to adopt a higher consumption tax rate due to the fiscal sustainablity issue. The crosssectional data from OECD countries in Figure 2 shows there is a positive correlation between the gross replacement rate of the public pension and the consumption tax rate. Lee [17] finds the similar trend between the replacement rate of the public unemployment insurance and the commodity tax rate with the OECD cross-country sample data. For the same OECD countries, we found out countries with a higher dependency ratio impose a higher consumption tax rate. (See Figure 3) This empirical fact is closely related to the previous empirical fact in Figure 2. There are relatively less number of young workers compared to retirees in a more rapid aging society which results in a shortage of tax revenues to operate the pension plan. Therefore, the government in the population aging might increase the consumption tax rate to fill the revenue gap. This paper attempts to account for the interesting co-movements of the consumption tax rate along with the pension benefit level and the dependency ratio. We also address what the welfare-improving tax policy is and why under the pension-side rigidities and the aging population. To answer these questions, we use the optimal taxation framework with the pension institution given out of model for two reasons. First, the governments, in the difficulty of the pension reforms, have modified the tax codes, especially the consumption tax relatively easy to administer compared to other taxes, to advance the financial consolidation and improve the social welfare. Second, it is very important to know what the optimal tax should be under the too generous but rigid pension systems and the aging population since it can provide many policy implications to developed countries which face the similar challenges to bring down public debt ratios and grapple with rising pension spending. We examine the role of consumption taxation in a standard two-period overlapping generations (OLG) model with fixed pension benefit (above the optimal level) and longevity shock to reflect the effect of aging population on pension and tax. The longevity shock means not all individuals can live up to the old-age period, parametrized by survival probability less than unity. 3

4 Figure 2: Value-added tax and gross replacement rate in OECD countries, (a) 2008 (b) 2010 (c) 2012 (d) 2014 Note. The gross replacement rate is defined as gross pension entitlement divided by gross pre-retirement earnings. We use data for male workers with average earnings from OECD countries in every two years between In our discussion of tax policies, we allow for income and consumption taxes. In the dual approach of optimal taxation, the government maximizes the ex-ante identical agents expected utility, using the tax mix where the labor and capital income taxes are considered along with the consumption tax. 3 Unlike the usual optimal taxation literature (e.g., Atkinson and Stiglitz [3]; Boadway et al. [5]) where the tax mix of direct and indirect taxes cannot be defined for the well-known indeterminacy issue, our framework permits a specific tax mix. Our model can be seen as a modified version of Atkinson and Sandmo s [2] OLG model, by adding the pension system and lifetime uncertainty. Using both analytical and quantitative analyses, we show the following. (i) With both pension and tax rates as choice variables of the government, it is impossible to account for the puzzle due to the usual indeterminacy. (ii) When one of the policy variables is given, the optimal policy mix is determinate and achieves the optimal stationary allocations defined by the well-known modified golden rule regardless of the survival probability parameter. (iii) We can account for the empirical observations when the pension is determined externally. Only, in this 3 Usually, tax mix refers to an optimal combination of income and consumption taxes. See Lee [17] for a review of the literature on tax mix. 4

5 Figure 3: Value-added tax and dependency rate in OECD countries, (a) 2008 (b) 2010 (c) 2012 (d) 2014 Note. The elderly dependency rate is defined as the ratio between the elderly population (aged 65 and over) and the working age (15-64 years) population. We use data from OECD countries in every two years between We exclude Japan because its dependency rate is extremely high to be considered as an outlier: about 40% in case, the consumption tax rate comoves with the pension level. This accounts for why countries with generous social welfare systems tend to adopt high consumption tax rates. (iv) Through a calibration exercise, we obtain that as population aging advances, strengthening consumption taxation helps social welfare, supporting the recent trend of rising consumption taxes in OECD countries in the era of population aging. (See Figure 4) These findings can account for why countries expanding their welfare system rely more on consumption taxes. The intuition behind main results is simple and clear. Given the institutionalized high pension level, we can get closer to the optimal pension level by strengthening consumption taxation so as to lower the real purchasing power of pension. However, consumption taxes cannot go up too high because they also lower the real value of labor income of working population, which is essentially the same with raising the labor income tax. To deal with this, labor income taxes need to be adjusted downward to some extent when consumption tax rates go up in response to an increase in pension benefit. Unlike the usual literature, the capital income tax rate is not necessarily zero in our setting because lifetime uncertainty in our model negatively affects 5

6 Figure 4: Value-added tax in OECD countries, Note. We use an unweighted average of VAT over all OECD countries in OECD Avg. savings incentives. 4 Meanwhile, regarding the relationship between population aging and pension level, the driving force is that as the elderly population becomes larger in the aging society, young generation s burden for subsidizing the current pensioners gets larger. Again, given the rigidity against reforming welfare programs, the government can only get closer to the optimal level by raising consumption tax rates. At the very least, we believe that this paper provides a new insight into the tax policy: when expenditure-side rigidities are present, a proper tax mix can improve welfare. This paper is organized as follows. Section 2 reviews the social security literature. Section 3 describes our general equilibrium model and presents its key features. Section 4 presents the indeterminacy issue arising from policy redundancy when the government has the ability to control all fiscal variables, and next, the stationary optimal tax mix under an externally given pension. In Section 5, we build a calibration model, and discuss the policy issues of interest: (a) how the consumption tax rate responds to pension benefit and (b) the population aging effects on the consumption tax and other fiscal variables. The final section summarizes the key results of this paper. 2 Literature Review There is an extensive literature about the pay-as-you-go pension system. One strand of the literature is to examine whether the unfunded pension plan is Pareto-efficient or not. (See Kotlikoff [12], Krueger and Kubler [13], others.) These normative papers examine both the positive intergenerational risk-sharing and redistribution and negative crowding out saving and 4 As we will see later, the nature of lifetime uncertainty here is a negative life span shock with the lifetime distribution truncated at the old-age period. This asymmetry in the distribution leads to a negative savings incentive. 6

7 labor supply effects of the pension and calculate the welfare gain of the social insurance system. The other strand of the social security literature focuses on why it exists instead of whether it should exist. This type of literature explains the public pension system arises as politicoeconomic equilibria in dynamic-voting models. Niepelt [9] shows the population aging enlarges the social security size and D Amato and Galasso [7] concludes that the intergenerational risk-sharing motive against the aggregate shock leads to a larger and persistent pension system. Both papers show the politically-driven pension level is higher than the optimal level selected by a benevolent government. In contrast to the previous literature, this paper makes a contribution by analyzing stationary optimal tax measures when the pension level driven by a political process is higher than the optimal point but there is downward rigidity on the pension system. There are some papers which study the relationship between the generosity of welfare programs and the VAT rates with the optimal taxation framework. Lee [17] accounts for the positive correlation between the public unemployment insurance level and the commodity tax rate observed in the data from OECD countries. The public pensions take the largest share of fiscal expenditures of modern governments and thus, understanding pensions permits an insight into the fiscal situation of a country and its decision on tax reforms. 5 This paper contributes to the literature by explaining the positive co-movement between the VAT rates and the public pensions with the optimal taxation framework. 3 Model 3.1 Environment To examine the optimal policy mix of taxes, we construct a standard overlapping generations model with a set of assumptions and conditions as follows. Our model can be seen as extending traditional OLG models (e.g., Atkinson and Sandmo [2]; Park [19]) with the new elements of lifetime uncertainty and exogenously given public pension. Time is discrete and indexed by t from 0 to infinity. In each period, agents are born and live two-periods labeled as young and old. Only young generations can work and old generations live relying on the savings from their youth and public pension without working. Each generation overlaps with the previous generations for one period. There is a population growth of which rate is denoted by n. The relative size of young workers to retirees is 1 + n. Agents are born with the same earning ability and preference for consumption and leisure. We introduce uncertainty in lifetime called longevity shock such while everyone lives the firstperiod, some individuals do not live through the second-period. The probability that an agent live until the second-period is defined as survival probability, which can also be interpreted as a degree of population aging in a society. Given this, our model becomes a two periods OLG model with lifetime uncertainty and pension. We assume the longevity shock is realized in the first period after labor supply decision to focus on the insurance function of the pay-as-you-go 5 The unweighted OECD public pension spending is five to ten times higher than the public unemployment spending during

8 pension system to smooth the first-period consumptions between states. The ex-ante identical individuals determine labor supply considering the lifetime uncertainty. A lower survival probability discourages agents to work since they lose an incentive to earn income when young for the future consumption. Agents who receive a good shock i.e. surviving will allocate their labor income between two periods whereas others who get a bad shock i.e. dying, will exhaust her labor income in the first period of their lives. This idiosyncratic lifetime uncertainty requires an insurance to smooth consumptions between two states to maximize the ex-ante expected utility of agents. Agents have access to the capital investment but it only transfers resource intertemporally. Thus, there is an incomplete market issue due to the longevity shock and lack of assets to transfer resources between states. The government can provide perfect insurance against the lifetime uncertainty with pay-as-you-go pension system since the shock is idiosyncratic, not systematic. The household behavior with the shock realization before saving is exactly identical with one under the assumption that agents receive the shock after saving but those not surviving can consume their saving for the first-period consumption. In both cases, agents decide their labor supply expecting the lifetime uncertainty and those who receive a bad shock consume all their saving when young. In the latter case, agents decide to save only considering the case of survival since they can reflect their saving into the utility function via the first-period consumption when dying. Thus, the individual decisions are equivalent in both cases. We assume agents can include their savings into their preferences although they die unlike the standard confiscatory bequest assumption in the OLG literature under which the government can take all the accidental bequests of those not surviving for managing public pensions. 6 In the real world, accidental bequests are vested first to the closest relatives of the dead by laws such as spouse, children, parents, siblings and cousins. Some financial assets require investors to designate beneficiaries in the preparation for sudden death like life insurance or retirement insurance. Many people make their will explicitly or at least share their plans about how to allocate their wealth in advance. There are many ways to use savings for old-age periods for personal benefits before dying. Hence, we assume agents get utility from their savings through the first-period consumption if dying in our two-period model. Exploiting the neo-classical growth model, firm s production function is constant returns to scale (CRS) and a single composite good and factor markets for labor and capital are perfectly competitive. Government s primary roles in this model are to collect taxes from both young and old generations to alter financing methods under the difficulty of pension reforms. This helps attain the economic close to one at the optimal pension level. there is no government consumption. Public pension acts as an insurance to help smooth consumption between states in the presence of lifetime uncertainty in an incomplete capital market where agents cannot perfectly cope with their lifetime uncertainty with private saving i.e. self-insurance. 6 We believe our assumption is more reasonable, but we check how the confiscatory policy changes the results. It turns out that the main conclusions of this paper remain the same. We will explain why in more detail in Appendix B.1. 8

9 3.2 Household Problem Ex-ante identical agents born in period t can live up to two periods at maximum, t and t + 1. The ex-ante identical individuals determine labor supply under the lifetime uncertainty. With a survival probability, a proxy for population aging, θ less than unity, agents live the second-period t + 1, i.e., lifetime uncertainty. 7 If individuals do not survive, they will consume their whole savings before dying. Therefore, the first-period consumption in this case, Ct n, is different from the first-period consumption conditional on survival, Ct s. An individual chooses a single composite good Ct s and Cn t, future consumption Z t+1, and labor supply l t in perfectly competitive goods and labor markets to maximize the expected value of lifetime utility (1) under constraints (2) and (3) : { } { } (1) max U (Ct s, Ct n, Z t+1, l t ) = θ U 1 (Ct s, l t ) + βu 2 (Z t+1 ) + (1 θ) U 1 (Ct n, l t ) }{{}}{{} subject to lifetiem utility of the case surviving two periods (2) (1 + τ c,t ) C s t + a t = (1 τ w,t ) w t l t for the case living only one period (3) (1 + τ c,t+1 ) Z t+1 = (1 + (1 τ r,t+1 ) r t+1 ) a t + m t+1 = (1 τ w,t)w t l t where Ct n and U (1+τ c,t ) 1 ( ) and U 1 ( ) are equivalent but with possibly different levels of consumptions conditional on states. Due to the assumption of the competitive economy, households take aggregate savings and labor supply, and then next period s return on capital as well as social security benefits as given. The expected utility is a sum of the lifetime utility when an agent lives until the second-period and the lifetime utility when an agent lives only the first-period. The utility function satisfies the regular conditions. In the first-period, agents supply labor l t and allocate their incomes into the first-period consumption Ct s and savings s t for the second-period consumption their saving is invested as the next period s capital stock. Individuals pay labor income taxes at the rate τ w,t and consumption taxes at the rate τ c,t in the first-period. All of these are summarized in the first-period budget constraint (2). If individuals survive, they face the second-period budget constraint (3), i.e., live on their savings and pension. They pay capital incomes taxes at the rate τ r,t+1 and consumption taxes at the rate τ c,t+1 in the second-period. However, individuals behaviors in the second-period are predetermined because they just consume their savings allocated in the first-period. If individuals do not survive, they are supposed to consume their whole savings before dying. Therefore, the first-period consumption, in this case, is a number of goods to be purchased by the net income, Ct n = (1 τ w,t)w t l t. The value of C (1+τ c,t ) t n is automatically determined by choosing l t to maximize the expected value of lifetime utility. 7 While θ represents lifetime uncertainty in general, a change in θ has a bit complicated meaning. When θ falls from unity to a number above 0.5, it involves greater randomness and a shorter lifespan. 9

10 Agents receive the longevity shock before allocating resources intertemporally. Thus, the lifetime uncertainty does not directly affect saving choice. Instead, it indirectly influences the amount of saving by varying labor supply decision. We check these results with the following Euler equations: (4) (1 τ w,t ) w t (1 + τ c,t ) ( ) ( θuc,t 1 + (1 θ) Uc,t 1 = θul,t 1 + (1 θ) U1 l,t ) (5) U 1 c,t (1 + τ c,t ) = β (1 + (1 τ r,t+1) r t+1 ) (1 + τ c,t+1 ) U 2 z,t+1 Equation (4) indicates the intra-temporal choice between current consumption and labor. Equation (5) shows the inter-temporal choices between current and future consumption. The survival probability, θ is absent in (5) i.e. no effect of longevity shock on saving choice. On the other hand, it direclty affects labor choice as seen in (4). 3.3 Firm Firms adopt a constant returns to scale (CRS) production technology. A firm chooses capital stock per capita k t and labor demand per capita lt d in perfectly competitive factor markets to maximize profit π t : [ ( ) ] (6) max π t = F k t, lt d w t lt d r t k t where w t is the wage rate in period t, r t is the interest rate in period t. Firm s optimization behavior creates two conditions related to wage rate w t and net interest rate r t : ) ) (7) w t = F l (k t, lt d, r t = F k (k t, lt d Marginal products of labor and capital are equal to wage and rental rates, respectively, at the optimum. 3.4 Factor Market Conditions Given the competitive factor markets, the labor market equilibrium in period t occurs when demand for labor by firms equal to supply of labor: (8) l d t = l t In the capital market, the next period s capital stock per capita is equal to private saving of the surviving old in period t, which derives a capital market condition: (9) (1 + n) k t+1 = θa t b t where b t is the per capita national debt issued in period t. Again, population growth and lifetime uncertainty modifies the conventional capital market equation only slightly. We can replace private saving a t in (9) with an individual s first period budget constraint (2) to derive a new capital market condition: (10) (1 + n) k t+1 = θ ((1 τ w,t ) w t l t (1 + τ c,t ) C s t ) b t 10

11 3.5 Resource Constraint The total resource of an economy in period t, the sum of output and the capital stock in that period, is consumed by the young generation born in period t and the old generation born in period t 1 who survives in period t and the rest turns into the capital stock for the next period, t + 1: (11) F (k t, l t ) + k t = θc s t + (1 θ) C n t + θ 1 + n Z t + (1 + n) k t+1 where we assume no capital depreciation for simplicity and it does not affect the entire subsequent results. Note that the current consumption comes from young and old generations, and among the young, θ proportion of them consumes Ct s, (1 θ) proportion of them consumes Cn t and the old θ generation is fewer than young generation as reflected by the factor 1+n for population growth and lifetime uncertainty. 3.6 Government Budget Constraint The total government revenues in time t come from newly issuing one-period risk-free bond and linear tax system and they are transferred in the form of public pension to the old generation born in period t 1 who survive in period t and the rest is used for the repayment of the existing bond. The government budget constraint in period t is: (12) b t + [τ w,t w t l t + τ c,t (θct s + (1 θ) Ct n )] }{{} from young generation + θ 1 + n (τ r,tr t a t 1 + τ c,t Z t ) = 1 + r t }{{} 1 + n b t 1 + θ 1 + n m t from old generation who survives Variables pertaining to old generation are discounted by the population growth rate n and life time uncertainty parameter θ applies to the surviving old population. 8 The existing bond, i.e., national debt in the previous period, is multiplied by 1+r t 1+n because of paying interest and population growth. 3.7 Equilibrium For all t, given factor prices w t and r t+1 and fiscal policy variables τ c,t, τ c,t+1, τ w,t, τ r,t+1, b t and m t+1, the ex-ante identical agents born in period t chooses C s t, Cn t, Z t and l t to maximize the expected value of her lifetime utility, and firms choose k t and l d t to maximize the profit. If the resulting feasible allocation C s t, Cn t, Z t, k t, l t and l d t and a set of prices satisfy the aforementioned capital and the labor market condition, and the resource constraint, and fiscal policies τ c,t, τ w,t, τ r,t, b t 1, b t and m t satisfy the government budget constraint, we obtain a competitive equilibrium in period t. The time subscript simply drops for the stationary equilibrium. 8 θ For instance, a factor 1+n represents the retirees are much fewer than the young workers because of population growth and lifetime uncertainty. 11

12 4 The Ramsey Problem To answer the questions in this paper, we discuss the optimal fiscal policy in a competitive economy. We construct the benevolent government s problems as follows. The government can choose fiscal policy instruments to maximize the ex-ante identical agents expected utility in four cases: (i) government can implement both pension and tax policies as choice variables without restrictions, (ii) government can implement fiscal policies without the restriction that either one of tax rates or pension level is externally given out of model, (iii) government can implement tax policy without restrictions in response to a variation of externally determined pension level and (iv) government can implement tax policy without restrictions in response to a variation of a degree of population aging with fixed pension benefit. We will examine what the optimal fiscal policy should be in each case. 4.1 The Indeterminacy Issue In this section, we discuss the optimal fiscal policy of the benevolent government which chooses taxes rates, pension level, and public bond size once at all to nudge households and firms to achieve the efficient allocation in the market. We assume there is no time-inconsistency problem pointed out by Kydland and Prescott [14] and thus, the benevolent government commits its announcement. We tweak the benevolent government problem a bit by replacing the household indirect utility function on its objective with the regular utility function. Thus, the benevolent government also selects economic allocations but the optimality conditions for both agents and firms should be placed as constraints in the new problem to respect the equilibrium outcomes. The programming of the benevolent government can be written as (13) max ( ) t 1 1+ρ t=0 θ ( U 1 (C s t, l t) + βu 2 (Z t+1 ) ) + (1 θ) U 1 (C n t, l t) +λ 1,t (household budget constraints) +λ 2,t (resource constraint) +λ 3,t (capital market clearing condition) +λ 4,t (intertemporal Euler equation) +λ 5,t (intratemporal Euler equation) where ρ 0 is the social discount rate which weights each generation as an exponential discounting, w t = F l (k t, l t ), r t = F k (k t, l t ) and lt d = l t and the government budget constraint is omitted by Walras s law. This new problem is equivalent to the original one because the two Euler equations are necessary and sufficient conditions by the strict concavity of the utility function and thus the regular utility function can be transfromed into the indirect utility function by plugging the allocation satisfying the household optimality conditions. In this two-period OLG model with lifetime uncertainty and pension, government cannot uniquely determine all five fiscal policy instruments three taxes, public debt and pension i.e. the usual indeterminacy problem happens. We let the bond satisfy the capital market clearing condition and thus, the government is restricted to choose only taxes and pension level. However, the indeterminacy issue still arises. The nature of indeterminacy lies in that uncountably many sets of fiscal policy instruments can achieve the same welfare. We summarize this point in Proposition 1. 12

13 Proposition 1. The indeterminacy issue. It is impossible to determine the optimal rates of taxes along with the optimal transfer and bond level at the steady-sate. proof. See Appendix A.1. The first-order conditions from the benevolent government problem are linearly dependent to each other. Hence, the number of unknowns are greater than the number of non-linear equations which results in indeterminacy issue almost surely. Intuition can be found from the following normalized lifetime budget constraint: three relative prices but four policy instruments. Hence, redundant fiscal policies can attain the same relative prices, i.e. the same allocations. (14) P t C s t + Z t+1 = ω tl t + m t+1 where P t = (1+(1 τ r,t+1)r t+1 )(1+τ c,t ), ω (1+τ c,t+1 ) t = (1+(1 τ r,t+1)r t+1 )(1 τ w,t )w t and m (1+τ c,t+1 ) t+1 = m t+1 (1+τ c,t+1 ). If pension level or one of the linear taxes is externally given, it is possible to find the finite sets of the optimal level of fiscal policy instruments. Corollary 1 outlines this. Corollary 1. The optimal policy mix. With one externally given policy variable, it is possible to determine the optimal rates of other policy tools. proof. There are the same number of relative prices and policy variables if either pension level or one of the linear taxes is externally given in (14). There is a more rigorous proof in Appendix A Optimal Pension Level Due to the indeterminacy issue in the previous section, we first set one of tax policies. With these three free policy instruments, we solve the benevolent government problem (13) and summarize the result in the following proposition. The optimal stationary capital and labor supply characterized by the modified golden rule can be achieved in the market. Proposition 2. The modified golden rule. Assuming one of tax policies is given, the optimal stationary outcomes characterized by the modified golden rule can be attained with the left policy variables. Under θ = 1, the optimal two endogenous taxes rates are zero as well if the exogenous tax rate is zero. proof. See Appendix A.3. It is worth noting that this proposition holds for any values of the exogenous policy and the survival probability and any specifications of the utility and production functions. Let r be the interest rate at the equilibrium steady-state without any policy interventions. If (1 + r) < (1 + n) (1 + ρ), the aggregate saving is higher than the optimal stationary level. Therefore, the benevolent government should achieve the modified golden rule level of capital investment into the production function by borrowing from the household private saving through bonds. As a result, the interest rate goes up to (1 + n) (1 + ρ) which improves the steady-state social welfare because households can earn a higher interest income. If the aggregate saving is lower than the optimal stationary level, the benevolent government should make a public saving. The government budget constraint at the optimal steady-state when we set τ c = 0 is (15) b + [τ w wl] + θ 1 + n [τ rra] = (1 + ρ) b + θ 1 + n m 13

14 Assuming θ = 1, the labor and capital income taxes rates are zero and thus, this equation degenerates to (16) m = (1 + n) ρb For the over-saving case, b is positive and thus m is negative. For the under-saving case, it is the opposite. When ρ = 0, the optimal stationary pension level is zero but the optimal bond level can be still positive or negative depending on the equilibrium aggregate saving level compared to the optimal one at the steady-state. Proposition 1 and 2 imply the government can reach the same optimal stationary allocation with tax policies under any pension levels because any three out of four fiscal policies can derive the same equilibrium. 4.3 Comovement between the Consumption Tax and Pension In this section, we let transfer be a parameter instead of one of the tax policies to examine the optimal consumption and other tax mixes when the externally given pension level increases. Let τ c,1, τ w,1 and τ r,1 be the optimal tax rates corresponding to an exogenous pension level m 1. We are interested in how the tax rates should change under a new pension level m 2 to attain the same stationary optimal outcome. When there is a downside-rigidity on the pension level, the benevolent government chooses the optimal consumption tax rate to respond positively to the pension level to maximize the social welfare at the steady-state. The following proposition summarizes this finding. Proposition 3. The optimal consumption tax. Let τ c,2, τ w,2 and τ r,2 be the optimal stationary tax rates corresponding to an exogenous pension level m 2. The optimal stationary tax policy attains the same allocation derived under τ c,1, τ w,1, τ r,1 and m 1 if satisfying the following conditions. When θ = 1, Proof. See Appendix A.4. m 2 m 1 = (1 + τ c,2) (1 + τ c,1 ), (1 + τ c,1) (1 + τ c,2 ) = (1 τ w,1) (1 τ w,2 ) and τ r,2 = τ r,1 m 2 m 1 = (1 + τ c,2 ), τ c,2 = τ w,2 and τ r,2 = 0 This proposition provides some important implications. First, the optimal tax mix indicates that the benevolent government should use the consumption tax financing to improve the economic well-being when there is a generous pension system. Conversely, the desirable tax policy is to weaken the consumption taxation if the pension level decreases. The consumption tax plays the central role of adjusting real pension value by modifying prices and thus the purchasing power of the transfer. Thus, the benevolent government can meet the same level of pension benefit measured in the single composite good by raising or dropping the consumption tax rate for too generous or too stingy pension systems respectively. In terms of tax burden, the consumption taxation distributes the economic cost of financing the public pension plan across generations. For example, an increase in the value-added tax makes retirees pay the transfer benefit by themselves through consumption in a higher pension 14

15 level. In the opposite case, a decline in the consumption tax rate benefits the old group by reducing their tax burden via good transactions. The optimal consumption tax direction in a response to the pension level accounts for the positive correlation between pension and consumption tax rate observed in the OECD data. It explains why the governments resort to the VAT under the expenditure-side rigidity and the pressure of increasing tax burden on young workers. Second, the optimal tax policy should involve the adjustment of the labor income tax rate along with the consumption tax. An increase in VAT reduces the real value of wages in fact. Hence, young workers lose an incentive to work, which requires compensation by cutting the labor income tax. Proposition 3 states the labor income tax should move in the reverse direction to the consumption tax to perfectly neutralize the young worker s moral hazard. When θ = 1, the wage tax should have the exactly opposite value compared to the commodity tax. This policy prescription attains the exactly same relative prices with the previous transfer level. Therefore, capital stock and labor supply remain the same and thus, the total output as well at the new optimum. Lastly, the zero capital income tax at the steady state when θ = 1 is consistent with Chamley s result [6]: for revenue-raising purposes, labor income taxes can be exploited but capital income taxes should not be used at the steady state for any configuration of parameter values. Raising revenue through capital income taxes lowers incentives to save, so it prevents capital accumulation and thus the total output. When θ < 1, it is worth noting that the optimal stationary capital income tax rate should be maintained even though the pension level changes. This result is coherent with the intuition that capital income taxes should not be utilized for the revenue-expanding purposes. The non-zero capital income tax is caused by the nature of the upper truncated lifetime uncertainty in our model. We will explain this in more detail at Section The Effects of Aging Population The aging population changes the socially efficient stationary allocation between the young and the old generations since it is not possible to keep the same amount of consumption for a relatively larger elderly group unless changing the labor supply of the young workers. Thus, the population aging described by an increase in the survival probability rate will affect the optimal tax rates which is the object of the analysis in this section. To isolate the effect of aging population on the redistribution of resources, we first consider an extreme case where labor supply is perfectly inelastic so that households are restricted to choose only consumption and saving. This simplified example helps derive a closed form solution of the optimal tax and transfer levels which decentralize the first-best allocation hedging the lifetime uncertainty. In this analysis, we first characterize the perfect risk sharing allocation from the social planner s problem. Then, we find the optimal transfer and income tax rates which attain the planner s allocation in the competitive market by fixing the consumption tax rate at zero due to the same indeterminacy issue discussed in Proposition 1. Finally, we examine the optimal consumption and other tax mixes when the externally given pension level varies as in Subsection 4.3. The content of Proposition is to define the first-best allocation perfectly hedging the lifetime uncertainty in the case of perfectly inelastic labor supply. 15

16 Proposition 4. The first-best allocation. When the labor supply is exogenously given, i.e l t = l for t, the optimal stationary allocations, C s, C n, Z and k, satisfy the following conditions: The consumptions bewteen states when young are identical: C s = C n = C The capital level is defined by the modified golden rule: (1 + F k ) = (1 + ρ) (1 + n) The consumption levels C and Z are defined by the resource constraint in (11) and the following intertemporal optimality condition: Appendix A.5 proves this finding. U 1 c = β (1 + F k ) U 2 z The first equation implies the first-best allocation of the social planner achieves the perfect consumption smoothing between states i.e. perfect risk sharing against the lifetime uncertainty. The price of giving one good to those surviving is 1/θ whereas it is 1/ (1 θ) for those not surviving in the planner s resource constraint. However, people who survive or not weighted by θ and (1 θ) in the planner s objective function and thus, the effective price of giving one good to either group is identical as one. Hence, the planner should maximize the social objective function by equating the consumptions in two states. Since the capital investment smooths consumption only over periods not states, there has been an incomplete market issue. The longevity shock is an idiosyncratic risk, not a systematic risk and thus, the benevolent government can achieve a Pareto-improvement by providing public insurance against the lifetime uncertainty in the asset market incompleteness with taxes and transfers in the form of pay-asyou-go pension system. We will discuss the optimal fiscal policy working as a public insurance below. It is worth noting that the optimal stationary capital is determined by the modified golden rule and the same regardless of the survival probability θ. The marginal cost of accumulating (1 + n) amount of the capital good is (1 + ρ) (1 + n) λ where λ is the marginal utility of one composite good in each period. Since we focus on the steady state, λ is the same across periods so we drop the time subscript. The social planner weights one generation at a higher rate by (1 + ρ) than its subsequent generation. The marginal benefit of the accumulation is (1 + F k ) λ because such accumulation increases one unit of capital good in the next period due to a population growth. Therefore, the optimal stationary capital is determined at the point where the marginal cost of capital accumulation is equal to its marginal benefit, (1 + ρ) (1 + n) = (1 + F k ), i.e. the modified golden rule. The social discount factor is independent of the survival probability rate by assumption. The production functions are also independent of the longevity shock since agents die after working. Since the marginal cost and benefit of capital accumulation are both not affected by the survival probability θ, the optimal stationary capital should be as well. The last optimality condition defines the intertempoal resource allocation along with the resource constraint and has some implications on the optimal capital income tax rates which we will discuss in the next. 16

17 The perfect inelastic labor supply assumption makes it possible to decentralize the firstbest allocation in the competitive economy with government intervention. Due to the same indeterminacy issue arised in Proposition 1, we let the benevolent government use income taxes and old-age transfer, not the consumption tax. In the following proposition, we summarize a closed form solution of the stationary optimal income tax rates and transfer which decentralize the perfect-risk sharing allocation at the steady-state. Proposition 5. Decentralization. When the labor supply is exogenously given, a closed form solution of the optimal stationary fiscal policies, τ w, τ r, m and b, is: τ w = 1 C F l l, τ r = 0, m = Z, and b = (1 + n) k Proof. See Appendix A.6. where C, Z and k are the optimal stationary allocations of the social planner in Proposition 4. The closed form solution gives out some important implications on the optimal fiscal policy. First, τ w is decreasing in C and thus, τ w will be affected by the survival probability via C. τ r = 0 so there are no intertemporal wedges. An individual saves considering only the case of surviving due to the assumption that she can exhaust her saving before dying. Thus, the idiosyncratic longevity shock does not distort the private saving. The old-age transfer m is always positive as long as θ > 0 because the Inada condition guarantees a positive consumption level when old in the social planner s problem if θ > 0. In the proof of Proposition 5, we observe the private saving is zero at all under the optimal income tax rates and pension level above. Hence, the benevolent government should always save instead of issuing bonds up to the capital level from the modified golden rule multiplied by the gross population growth rate (1 + n). Before examining the effect of the survival probability rate on the wage tax and transfer, we should note again that the optimal stationary capital level is identical regardless of the survival probability rate θ and thus, F l l as well. Therefore, the effects of the survival probability to the optimal intertemporal consumptions only matter for τ w and m. The resource constraint in (11) and the intertemporal optimality condition above characterize C and Z. From these two equations, it is straightforward to prove that C and Z are unique and functions of θ. By taking derivatives of the two equations with respect to θ, we obtain the following two equations: (17) and C θ + Z 1 + n + θ Z 1 + n θ = 0 (18) Ucc 1 C θ β (1 + F k) Uzz 2 Z θ = 0 By solving the two system of equations for Z θ Z 1+n U 1 cc Z θ = β (1 + F k ) Uzz 2 + θ 1+n U1 cc and C θ, < 0, and C θ = β (1 + F k) U2 zz Z Ucc 1 θ < 0 17

18 where Ucc 1 and Uzz 2 are negative due to the concavity assumption of the utility function. Thus, Z θ and C θ are both negative which means the stationary first-best consumption allocations are decreasing in the survival probability. The decreases in consumptions are seemingly a natural result since there are more people in each period due to the population aging but with the same resource because of the identical capital level regardless of θ. However, it is not natural whether the per-capital old consumption at the optimum will decrease or not under an aging population. We examine this as follows. Fixing the same resource share of the elderly group, the per-capita consumption level of the old will diminish as there are relatively more elderly people due to the aging population. The marginal utility of the old-consumption, Z, is higher than the marginal utility of the young-consumption, C. It is desirable to shift allocation from the young to the old but the new optimal old-consumption will be lower than its counterpart before the aging population. If not, the marginal utility of the young-consumption is rather higher than the marginal utility of the old-consumption. Hence, the population aging decreases consumptions for both young and old groups. This result provides the implications of the aging population on the wage tax rate and the old-age transfer as in Corollary 2. Corollary 2. The effect of population aging. As the population aging advances, the labor income tax increases whereas the old-age transfer decreases at the steady-state when the labor supply is perfectly inelastic. Proof. τ w is decreasing in C which is decreasing in θ. Therefore, τ w is increasing in θ. Z is decreasing in θ and m = Z. Hence, m is decreasing in θ as well. This result indicates that the wage tax rate should rise to transfer more from the young group to the old group in an aging population. However, per-capita old-age transfer should not be higher than its previous level before the aging population and essentially fall considering the utility loss of young workers. Another way of interpreting this result is that a lower old-age transfer but with a higher wage tax rate can also prevent the private saving which is necessary to attain the perfectly smoothed consumptions across states as discussed in Proposition 5. One of the main interests of this paper is to analyze how the aging population will affect the optimal consumption tax rate. Due to the well-known indeterminacy issue, we let transfer be a parameter instead of the consumption tax to study the optimal tax mixes when the externally given pension level rises. Assume that an economy starts with a survival probability θ 1 where the optimal labor income tax rate and transfer amount are denoted by τ w,1 and m 1. From Proposition 5, the benevolent government can achieve the first-best allocation without exploiting the consumption and capital income taxes, i.e. τ c,1 = τ r,2 = 0. As a comparative static analysis, the economy faces an aging population and thus the survival probability becomes θ 2 where θ 2 > θ 1, but the transfer remains at its previous level, m 1. Let τ w,2 and m 2 the optimal labor income tax rate and transfer amount corresponding to the survival probability, θ 2. From Corollary 2, m 1 > m 2. We are interested in how the tax rates should change under the new survival probability, θ 2, but with the externally fixed pension level m 1 to attain the stationary optimal outcome under τ w,2 and m 2. When there is a downside-rigidity on the pension level, the benevolent government can reach the same steady-state first-best allocation by moving the consumption tax rate positively to the aging population. The following proposition summarizes this finding. Proposition 6. The optimal consumption tax under population aging. Let τc, τw and τr be the optimal stationary tax rates under the survival probability, θ 2, and the externally fixed pension level m 1. 18