Fiscal Policy in an Incomplete Markets Economy

Transcription

1 Fiscal Policy in an Incomplete Markets Economy Francisco Gomes LBS and CEPR Alexander Michaelides LSE, CEPR and FMG Valery Polkovnichenko UT at Dallas December 2007 Preliminary, please do not quote or distribute. We are grateful for comments made by Marco Bassetto, Jeff Campbell, David Marshall, Christina De Nardi, Martin Schneider, Harald Uhlig and seminar participants at the Chicago FED, LSE and the 2007 SED meetings. Michaelides gratefully acknowledges financial support from the ESRC under grant RES during this project. Polkovnichenko acknowledges support from a McKnight Business and Economics research grant at the University of Minnesota and hospitality of the Financial Markets Group at LSE. All remaining errors are our own. Address: London Business School, Regent s Park, London NW1 4SA, United Kingdom. Phone: (44) fgomes@london.edu Address: Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, United Kingdom. A.Michaelides@lse.ac.uk. Address: Department of Finance and Managerial Economics, University of Texas at Dallas, School of Management SM31, P.O.Box , Richardson, TX polkovn@utdallas.edu.

2 Fiscal Policy in an Incomplete Markets Economy Abstract We study the quantitative implications of fiscal policy decisions in an heterogeneous agent model with incomplete markets, and where equity and government debt are not perfect substitutes. This set-up allows us to study the impact on macroeconomic activity, cross-sectional wealth distribution, asset prices and the risk premium, in a unified framework. For a given level of government expenditures, a 10% increase in government debt to GDP decreases the capital stock between 1.0% and 5.8%, depending on how the new debt is financed. As a result, output falls by between 0.4% to 1.7%, while the cost of government debt increases by between 18 to 83 basis points, inducing households to hold the extra bonds. This elasticity is consistent with the existing empirical evidence. Given the crowding out of investment the return on capital also rises. In the new equilibrium, the ratio of government debt to capital is now higher thus the equity return increases by less than the riskless rate, implying a lower equity premium. A 2.5% increase in the capital income tax rate, to finance additional government expenditures, leads to a 0.91% reduction in the aggregate capital stock, and a 0.33% reduction in output. The lower capital stock is associated with a higher return on capital and a lower equity premium. We find that tax rate changes have a small (although not negligible) impact on the equity premium, and a more significant impact on the riskless rate. Our results identify the portfolio re-allocation behavior of households (asset substitution channel), as extremely important for determining the impact of fiscal policy decisions on capital accumulation, and aggregate economic activity in general. On the other hand, the crowding-out effect of taxes through the tightening of liquidity constraints is very small since the households potentially affected by these constraints own a very small fraction of the capital stock. JEL Classification: E21, G11. Key Words: Fiscal Policy, Household Heterogeneity, Incomplete Risk Sharing, Life-Cycle Models, Limited Stock Market Participation.

3 1 1 Introduction What are the long-run effects of changes in taxation and government debt on investment, output, wealth inequality and asset prices? We study fiscal policy decisions in a general equilibrium model with incomplete markets, heterogeneous agents and where government debt and capital are imperfect substitutes. Markets are incomplete because of both aggregate uncertainty and idiosyncratic productivity shocks. The idiosyncratic shocks are not perfectly diversifiable due to the presence of borrowing constraints and, in the extended version of the model, finite horizons. In the simpler model agents are heterogeneous along one single dimension: the idiosyncratic productivity shocks. In the more complete model we also have heterogeneity in age, preferences and investment opportunities. Finally, while government bonds are (one-period) riskless, equity is a claim on the productive capital stock and thus earns a random return. Therefore, households also make portfolio decisions and, in equilibrium, the two assets earn a different rate of return. Our results will show that this imperfect asset substitution is an extremely important feature of the analysis. Models where the return on capital and the interest rate on government bonds are identical, will either significantly underestimate the former, or overestimate the latter, or both. Typically since this return is calibrated to match the return on capital, those models strongly exaggerate the cost of government debt. In addition, our results identify the portfolio re-allocation behavior of households (i.e. asset substitution), as an important channel for determining the impact of fiscal policy decisions on capital accumulation, and aggregate economic activity in general. In addition, this set-up will allow us to study the differential impact of fiscal policy decisions on both rates of return, and on the equity premium. Therefore, our model presents a unified framework for studying the quantitative impact of fiscal policy on macroeconomic activity, cross-sectional wealth distribution and asset prices. As a result, our assessment explicitly takes into account the important links between these different elements, and how they might interact in reaction to policy decisions. Before discussing our results, it is important to state that the analysis in this paper is not normative. The goal of this paper is to provide a quantitative assessment of impact of these different policies along a wide range of important dimensions. These results can then be used to inform policy makers. We start by identifying the important economic mechanisms in the context of an infinite-horizon model, where all agents are ex-ante identical but receive different idiosyncratic shocks and face borrowing constraints. Next, we present an overlapping generations model where we carefully attempt to capture the cross-sectional dispersion in wealth and consumption which will also help us to match aggregate moments better. More precisely, in the overlapping generations model there

4 2 is less risk sharing, and as a result the equilibrium risk premium is higher. In addition, to the extent that this economy delivers different wealth accumulation results, and thus implies a different calibration of preference parameters, that can also have a significant impact on the quantitative predictions of the model. Finally, our model will also capture another important empirical fact: a significant fraction of households do not participate in the stock market, either directly or through pension funds. Moreover, non-participation is much more pervasive among poor households. 1 Therefore we will include two types of agents in our economy, stock market participants and non-participants. While, for tractability reasons, we assume this separation exogenously (as in Guvenen (2003) and Basak and Cuoco (1998)), we carefully replicate the large differences in wealth heterogeneity between these two groups. 2 In our model, the differences in wealth accumulation arise from preference heterogeneity: differences in elasticities of intertemporal substitution and discount rates. We consider three different fiscal policy decision variables: lump-sum taxes, distortionary (capital income) taxes, and public debt. 3 To be clear, our analysis focuses on long-run (steady state) effects resulting from permanent policy changes, and thus should be kept separate from discussions about the timing of taxes and Ricardian Equivalence. We find that, for a given level of government expenditures, an increase in the government debt relative to GDP by 10 percentage points causes a permanent reduction in the capital stock of between 1.0% and 5.8%, depending on how the new debt is financed. As a result, output (GDP) falls by between 0.4% to 1.7%, while the interest rate on government bonds increases by between 18 to 83 basis points, inducing households to hold the extra government debt. This elasticity is consistent with the empirical results in Engen and Hubbard (2004) where it is found that an increase in the government debt to GDP ratio by one percentage point raises the return on government debt by 3 basis points, and with those in Laubach (2005) who finds a somewhat larger impact of 5 to 6 basis points for 1 percent increase in debt to GDP ratio. 4 Given the crowding out of the capital 1 For example, in the 2001 SCF the overall participation rate is 45% and it is 88.84% among households with wealth above the median, and only 15.21% for those with wealth below the median. 2 Gomes and Michaelides (2006) show that is important to match the differences in wealth accumulation between these two groups, to avoid counterfactual implications. Moreover, they show that, for the observed wealth accumulation of nonstockholders, a small fixed cost is enough to keep them out of the stock market. Therefore, the same result should hold in our model, if we were to introduce such cost. 3 We will refer to lump-sum taxes and taxes on labor income interchangeably since in the model, for tractability reasons, we do not include a household labor supply decision and, consistent with the data, household-level labor income is uncorrelated with stock returns (Heaton and Lucas (1997)). 4 The effect of government debt on interest rates remains ambiguous empirically, as Engen and Hubbard (2004) note in their review of this (large) empirical literature. Nevertheless, these recent results are in line with the magnitudes generated in our model.

5 3 stock, the return on capital rises. Since, in the new equilibrium, the ratio of government debt to capital is now higher the equity return increases by less than the riskless rate and thus the equity premium falls. We obtain reductions in the equity premium between by 33 to 42 basis points across several experiments. These numbers are likely to be even larger in an economy that actually matches the observed equity premium. Next we consider tax rates increases to finance additional government expenditures. A 2.5% increase in the capital income tax rate leads to a 0.91% reduction in the aggregate capital stock, and a 0.33% reduction in output. As households save less the equilibrium rates of return must increase to clear financial markets. Since the demand for capital is downward slopping while the supply of government bonds is fixed, the return on capital increases by less than the riskless rate (7 and 13 basis points respectively), and thus the equilibrium equity premium falls. The magnitude of this effects is economically very small: a 6 basis point decrease from a 2.5% increase in the capital income tax. Thus we can conclude that tax rate changes have a small impact on the risk premium, although they have a more important effect on the riskless rate (higher increase, and relatively to a much smaller base). Despite the previous conclusion, we show that is very important to study the effects of fiscal policy decisions in a model with a positive equity premium. More precisely, we show that, when we account for the fact that capital and government bonds are not perfect substitutes, the quantitative impact of fiscal policy decisions is significantly altered, relative to an otherwise identical model. When the two assets earn different rates of return there is an additional important channel in the model: the portfolio re-allocation behavior of households. To illustrate this effect it is easier to consider the case of lump-sum taxes. Lump-sum taxes correspond to negative riskless bond holdings, with the tax payments behaving like fixed coupon payments. In a model where bonds and equity are not perfect substitutes, when lump-sum taxes increase households must compensate for this by decreasing equity holdings. In equilibrium this results in a lower level of the capital stock. We show that this effect is quantitatively very large. A 10% increase in the ratio of government debt to GDP decreases the capital to GDP ratio by 1.0% if the interest payments are financed by higher lump-sum taxes. 5 To put this number in perspective, the crowding-out effect would be 5.8% if distortionary capital income taxes were used instead. Naturally, even if the two assets are perfect substitutes, lump-sum taxes still affect capital accumulation because of the presence of liquidity constraints. However, we solve such an economy 5 Elmendorf and Kimball (2003) analyze (in a two period, partial equilibrium model) a different effect from redistributing labor income taxes across time, namely that under certain conditions revenue-neutral deferral of taxes and higher taxation reduce labor income risk and lead to higher investment in the risky asset.

6 4 and show that this effect is economically negligible (less than 0.01% crowding out). Intuitively, the households that are most affected by these constraints own a very small fraction of the capital stock. Therefore, the previous numbers are driven by the portfolio re-allocation channel, and not by the direct liquidity constraints channel. It is important to clarify that this result does not negate the importance of borrowing constraints in the model. In fact, without them the equity premium would be much smaller. Therefore, although their impact alone is negligible, the importance of the asset substitution channel is strongly affected by their existence (or any other mechanism that helps to deliver the equity premium). Our model is part of a growing literature studying fiscal policy decisions in production economy. Baxter and King (1993) consider an infinite-horizon representative-agent model without aggregate uncertainty. Aiyagari (1995) and Conesa, Kitao, and Krueger (2007) study economies with heterogeneous agents, idiosyncratic shocks and borrowing constraints, but again without aggregate uncertainty. Therefore, these models do not capture the asset substitution channel discussed in our paper as, in these economies, government bonds earn the same rate of return as the capital stock. Ludvigson (1996) consider an economy with aggregate uncertainty but with a single representative agent, while Chari, Christiano and Kehoe (1994) characterize optimal fiscal policy in a model with heterogeneous agents and aggregate uncertainty. 6 However, in their set-up, idiosyncratic risk is perfectly diversifiable, allowing them to determine the optimal allocations by solving the corresponding Ramsey problem. All of these papers, however, incorporate a labor-leisure decision which is absent in our analysis, but on the other hand, they do not consider limited stock market participation. The closest paper to our is probably Heathcote (2005), who also considers an incomplete markets production economy with heterogeneous agents, aggregate uncertainty, and no labor supply decision. 7 As in our model, incomplete markets arise because of idiosyncratic productivity shocks and liquidity constraints. However, in his set-up, aggregate uncertainty is exclusively driven by tax rate shocks and therefore capital and government bonds are perfect substitutes. There is also a important difference regarding the scope of the analysis. Heathcote (2005) studies the time-series reaction to temporary tax rate changes, while we compare steady-state responses to permanent changes. Our economy generates a structure similar to the recently-used saver-spender models where, by assumption, two groups of agents have different savings behaviors. In those models, the savers are life cycle rational optimizers who behave according to the Permanent Income Hypothesis, while the 6 Shin (2005) considers a similar set-up in an economy without capital. 7 As a result, he also uses the methodology developed by Krusell and Smith (1998) and den Haan (1997) for solving the model.

7 5 spenders are exogenously assumed to consume their current income (or pension) every period. This representation has motivated applications of these models to different policy evaluation studies. For example, Abel (2001) and Diamond and Geanakoplos (2003) in the context of social security reform, Mankiw (2000) in a fiscal policy model, and Gali et. al (2004) on the evaluation of monetary policy. Naturally, the analysis in this paper does not allow us to consider all interesting aspects of fiscal policy decisions. For example, we abstract from optimal tax smoothing as studied by, among others, Angeletos (2002), Aiyagari, Marcet, Sargent, and Seppala (2002), Lucas and Stokey (1983), or Barro (1979). We also ignore the potentially important role of private information, as consider by, for example, Golosov and Tsyvinski (2007) and Golosov, Kocherlakota and Tsyvinski (2003). Finally, since ours is not a normative analysis, our results are also unrelated to the discussion on the optimal level of capital income taxation. 8 Chamley (1986) and Judd (1985) argue that, in the context of a Ramsey problem, the optimal tax rate on capital income should be zero. 9 Aiyagari (1995) and Conesa, Kitao, and Krueger (2007) shows that this result is no longer valid when we have incomplete markets, as in our model. Golosov, Kocherlakota and Tsyvinski (2003), Klein, Quadrini, and Rios-Rull (2005) and Chen and Lee (2007) argue that private information, limited commitment or limited enforcement can also justify a positive capital income tax rate. Our paper is not part of this debate, we simply acknowledge that capital income taxes do exist, and as such we study their impact on economy. The paper is structured as follows. In section 2 we discuss the model with infinitely lived agents and consider cases with and without aggregate uncertainty. Section 3 outlines the OLG model, its calibration and discusses the baseline results. Section 4 studies the impact of fiscal policy decisions for a given level of government expenditures, while Section 5 compares the impact of using the different fiscal policy instruments to finance changes in government expenditures. Section 6 provides the concluding remarks. Technical details of the computational procedure are provided in the appendix. 2 Infinite-Horizon Models To build intuition, we first consider a fairly standard growth model with infinitely lived, heterogeneous households. The only aspect of heterogeneity across households arises from different histories of uninsurable idiosyncratic earnings shocks. As we will see shortly, this model has some difficulties in simultaneously matching important facts about macroeconomic variables and asset returns. 8 In addition, such a discussion in a model with an exogenous labor supply, such as ours, would be meaningless. 9 Atkeson, Chari and Kehoe (1999) and Lucas (1990) argue the same point.

8 6 However, we emphasize that achieving such an ambitious goal is not the point at this stage of our analysis. Instead, this model simply serves as a starting point for understanding the interaction of household portfolio and savings decisions with fiscal variables, when capital and government bonds are not perfect substitutes. The model with infinitely lived households allows us to start building the intuition behind our main results in a transparent way. Later in the paper we consider a more complicated overlapping generations model with limited participation, which better accounts for asset prices, macroeconomic variables and cross-sectional distributions of wealth and consumption. That model will constitute our baseline model in the sense that it will deliver our specific quantitative predictions. However, all of the economic intuition and insights already arise in the models that we consider in this section. The common elements of the two models considered in this section are as follows. Households are infinitely-lived. They receive wage income, subject to uninsurable idiosyncratic shocks, and cannot borrow against this income. They can invest in two alternative assets: a claim to the risky capital stock (equity) and a riskless (for one period) government bond. Firms are perfectly competitive, and combine capital and labor, using a constant returns to scale technology, to produce a nondurable consumption good. The government taxes wages and capital income to finance government expenditures and the interest payments on public debt. We will consider two versions of this infinite horizon model: one where the capital stock and government debt are perfect substitutes due to the absence of aggregate uncertainty (Aiyagari (1994)), and another where capital is riskier than government debt, due to the presence of aggregate uncertainty (Krusell and Smith, 1997). The latter model nests the former and for brevity we only describe the model with aggregate uncertainty, noting the relevant differences where appropriate. 2.1 Production technology Production function The technology in the economy is characterized by a standard Cobb-Douglas production function, with total output at time t given by Y t = Z t Kt α L 1 α t (1) where K is the total capital stock in the economy, L is the total labor supply and Z is a stochastic productivity which follows the process Z t = G t U t G t = (1+g) t

9 7 Secular growth in the economy is determined by the constant g (>0), while the productivity shocks U t are stochastic. In the model without aggregate uncertainty we set U t =1. Firms make decisions after observing aggregate shocks. Therefore, they solve a sequence of static maximization problems with no uncertainty, and factor prices (wages, W t, and return on capital, R K t ) are given by the firm s first-order conditions W t =(1 α)z t (K t /L t ) α (2) and R K t = αz t (L t /K t ) 1 α δ t (3) where δ t is the depreciation rate. The depreciation rate is constant in the model without aggregate uncertainty and is stochastic in the extended model Stochastic depreciation Standard frictionless production economies cannot generate sufficient return volatility, since agents can adjust their investment plans to smooth consumption over time (see Jermann (1998) or Boldrin, Christiano and Fisher (2001)). This usually motivates adjustment costs for capital, which create fluctuations in the price of capital and increase return volatility. 10 Since we have incomplete markets, different stockholders have different stochastic discount factors. They will therefore disagree on the solution to the optimal intertemporal decision problem of the firm (see Grossman and Hart (1979)). This is not a concern here because there is no intertemporal dimension to the firm s problem, but introducing adjustment costs would change that. 11 Recent papers with production economies and incomplete markets have captured the adjustment cost effect by assuming a stochastic depreciation rate for capital (Storesletten et al. (2007), Krueger and Kubler (2006), and Gottardi and Kubler (2004)). We follow the same route and assume that the depreciation rate is given by δ t = δ + s η t (4) where η t is an i.i.d. standard normal and s is a scalar. Therefore, δ t is a general measure of economic depreciation, combining physical depreciation, adjustment costs, capital utilization and 10 Adjustment costs are also very important for a realistic characterization of aggregate investment flows. See, for example, Abel and Eberly (1994) or Eberly (1997). 11 Guvenen (2005) introduces adjustment costs in a model with restricted stock market participation, but in his model there is perfect risk sharing among stockholders. Therefore, there is a unique stochastic discount factor for pricing capital.

10 8 investment-specific productivity shocks. 12 with the productivity shock U t. In the baseline case we assume that η t is uncorrelated 2.2 Government debt The government s budget constraint is B t+1 =(1+R B t )B t + G c t T t (5) where G c is government consumption, B is public debt, R B is the interest rate on government bonds, and T denotes the tax revenues. Tax proceeds arise from proportional taxation on capital (tax rate τ K ), proportional taxation on labor (tax rate τ L ). In this type of models there is the potential for government debt to become non-stationary. The main problem lies with the simple observation that the evolution of B t+1 depends on B t but through a multiplication by a time-varying coefficient that is on average greater than one because the riskless rate has a positive mean. As a result, if taxes and government consumption are stationary, then government debt becomes nonstationary. Moreover, it is not obvious what normalization may be used to make B t stationary. One solution is offered by Heathcote (2005) who makes taxes (and household decisions) depend on government debt: high government debt relative to its long run average implies higher taxation. This requires the addition of one extra state variable in the model, and imposes a restriction on the path of tax rates in response to other shocks in the economy. 13 Therefore we instead assume that the government debt is constant over time with government consumption adjusting endogenously to satisfy (5) period-by-period. 2.3 Households and financial markets Preferences Households have Epstein-Zin preferences (Epstein and Zin (1989)) defined over a single non-durable consumption good. Let C t and X t denote consumption and wealth (cash-on-hand) in period t, respectively. Household preferences are defined by { V t = (1 β)c 1 1/ψ t + β ( E t (V 1 ρ t+1 ) ) } 1 1/ψ 1 1 1/ψ 1 ρ (6) 12 Hercowitz (1986) and Greenwood, Hercowitz and Huffman (1988) use the same approach to model fluctuations in capital utilization, while Greenwood, Hercowitz and Krusell (1997) use it to model investment-specific technological shocks as a reduced form for vintage capital models. 13 While still feasible in the setting without aggregate productivity or depreciation shocks, the computational burden of the additional state variable required by this method is a serious obstacle when we consider a model with aggregate shocks in this section or an overlapping generations economy later in the paper.

11 9 where ρ is the coefficient of relative risk aversion, ψ is the elasticity of intertemporal substitution, and β is the discount factor Labor endowment Let i index the households. All households supply labor inelastically and are subject to idiosyncratic productivity shocks such that, the process for individual labor income (Ht i)isgivenby Ht i = W tl i t (7) where L it is the household s labor endowment (labor supply scaled by productivity), and W t is the aggregate wage per unit of productivity. The household s labor productivity is subject to idiosyncratic uninsurable transitory shocks (ε i ), such that L i t = ε i t and where ln ε i is independent and identically distributed with mean 0.5 σ 2 ε and variance σ2 ε Wealth accumulation There are two financial assets: a one-period riskless asset (government bond), and a risky investment opportunity (capital stock). The riskless asset return is Rt B = 1 1, where P B denotes the Pt 1 B government bond price. The return on the risky asset is denoted by Rt K. In the model without aggregate uncertainty the return to capital is constant and equal to the return on risk-free bond. Total liquid wealth (cash-on-hand) can be consumed or invested in the two assets. At each time (t), agents enter the period with wealth invested in the bond market, Bt i, and (potentially) in stocks, St i, and receive Li t W t as labor income. Cash-on-hand at time t is given by X i t = Ki t (1 + (1 τ K)R K t )+Bi t (1 + (1 τ K)R B t )+Li t (1 τ L)W t (8) Households cannot borrow against their future labor income, and cannot short any asset. More precisely, X it 1 1 α Gt B i t 0 K i t 0 In the presence of deterministic growth we need to normalize the trending variables in this economy. We choose the following normalization kt+1 i = Ki t+1 1, b i t+1 = B it+1 1, c i t = C it 1, and 1 α 1 α 1 α Gt Gt Gt ( x i t = 1 α. Then, defining ω t = and w t = Wt, the individual budget constraint ) 1 G t G t α Gt 1

12 10 C i t + Ki t+1 + Bi t+1 = Xi t = Ki t (1 + RK t (1 τ K)) + B i t (1 + RB t (1 τ K)) + L i W t (1 τ L ) (9) becomes, after dividing through by the normalizing factor, c it + k it+1 + b it+1 = x it =(1+R K t (1 τ K)) k it ω t +(1+R B t (1 τ K)) b it ω t + L i w t (1 τ L ) 1 ω t (10) Labor taxes are non-distortionary in our model because there is no household-level labor-leisure decision. As a result we will preferentially refer to them as lump-sum taxes, which is what they effectively are. Our main comparison in this paper is therefore between the effects of lump-sum taxation and the effects of distortionary (capital income) taxation. Naturally, it would also be interesting to include distortionary labor income taxes in the model, however this would require the inclusion of a labor supply decision, and the model would no longer be tractable (especially in the presence of aggregate uncertainty). Given the empirical evidence that the labor supply elasticity of prime-age males is very low, this is a useful benchmark for more complicated future models that might include those endogenous decisions. 2.4 Equilibrium The equilibrium consists of endogenously determined prices (bond prices, wages, and equity returns), a set of value functions and policy functions, ({V,b,k}), and rational expectations about the evolution of the endogenously determined variables, such that: 1. Firms maximize profits by equating marginal products of capital and labor to their respective marginal costs: Equations (2) and (3). 2. Individuals choose their consumption and asset allocation by maximizing equation (6). 3. Markets clear and aggregate quantities result from individual decisions. Specifically: k t = ktdi, i b t = b i tdi. (11) i The aggregation equation for labor supply is redundant since there is no labor-leisure choice. Once these two equations are satisfied, Walras law implies that total expenditure (government consumption, investment, and household consumption) must equal total output: c G t + k t+1 (1 δ t)k t + c i t ω di = U tk α (1 + g) t L1 α t. (12) t ω t i i

13 11 4. The government budget [equation (5)] is balanced every period to sustain a given ratio of government debt to GDP. Specifically b t+1 = c G t + 1 ω t { (1 + R B t )b t k t R K t τ K b t R B t τ K w t τ L } (13) 6. Market prices expectations are verified in equilibrium. Analytical solutions to this problem do not exist and we therefore use a numerical solution method (details are given in Appendix 1) The dynamic programming problem In the presence of aggregate uncertainty the model is similar to Krusell and Smith (1997). Households are price takers and maximize utility given their expectations about future asset returns and aggregate wages. Under rational expectations, the latter are given by equations (2) and (3): future returns and wages are determined by future capital and labor, and by the realizations of aggregate shocks. Labor supply is exogenous, as are the distributions of the aggregate shocks. The capital stock, however, is endogenous. Forming rational expectations of future returns and wages is, therefore, essentially equivalent to forecasting the future capital stock. Capital accumulation is determined by the cross-sectional asset wealth distribution. We would therefore need to include this as a state-variable in the household s optimization problem. Krusell and Smith (1998) and den Haan (1997) suggest that, for this class of incomplete-markets economies, it is possible to approximate this infinite-dimensional state variable with a small set of moments. As discussed in the appendix, our model can accurately approximate the information contained in this distribution using its lagged mean (last-period s aggregate capital stock, k t 1 )andthe state-contingent realizations of the two aggregate shocks (productivity shock, U t, and stochastic depreciation, η t ): k t =Γ K (k t 1,U t,η t ) (14) Since government bonds are only riskless over one period, households must forecast future bond prices (Pt B B ). The forecasting rule for Pt is P B t =Γ P (P B t 1,k t 1,U t,η t ) (15) This introduces four additional state variables in the individual s maximization problem (P B t 1, k t 1, U t,andη t ).

14 12 The individual optimization problem now becomes: subject to the constraints, and with the laws of motion, V t (x i t ; k t,u t,η t,p B t )= Max {k i t+1,bi t+1 } { (1 β)(c i t )1 1/ψ (16) +β ( [ E t (ωt+1 ) 1 ρ V 1 ρ t+1 (xi t+1 ; k t+1,u t+1,η t+1,pt+1 B 1 1/ψ 1 ρ )]) k i t+1 0 b i t+1 0 c i t + b i t+1 + k i t+1 = x i t } 1 1 1/ψ x i t+1 = 1 [ ] k i ω t+1(1 + (1 τ K )Rt+1)+b K i t+1(1 + (1 τ K )Rt+1)+L B i t(1 τ L )w t+1 t+1 Rt+1 K = R(k t+1,u t+1 ) w t+1 = W (k t+1,u t+1 ) k t+1 = Γ K (k t,u t+1,η t+1 ) P B t+1 = Γ P (k t,u t+1,η t+1,p B t ) 2.5 Calibration Decisions are made at an annual frequency. Capital s share of output (α) is set to 36%, and the average annual depreciation rate (δ) is 10%. The tax rates were chosen to match the different aggregate revenue shares: respectively the share of capital income taxation and labor income taxation to GDP, in our main model (the OLG model that we will discuss later). In the context of our model calibrating the revenue shares is more important than calibrating the tax rates. This is particularly the case for labor income taxes since, as previously discussed, we don t have a labor supply decision in the model and thus they are effectively lump-sum taxes. In our baseline economy this implies a mean capital tax rate of 12.5% and a mean labor income tax rate is 15%. The capital income tax rate is significantly lower than in the data. 14 This is driven by one of the assumptions in the OLG model, 100% bequest tax, which we plan to change in the next version of paper. This will allow us to bring τ K closer to the data without changing the ratio of capital income revenues to GDP. 14 For detailed calculations of the capital income tax rate in the US see Trabandt and Uhlig (2006), Carey and Rabesona (2002) or Mendoza, Razin and Tesar (1994).

15 13 The volatility of idiosyncratic shocks (σ ε ) is set equal to 20 percent percent per year. 15 The aggregate supply of bonds is approximately 35% of GDP. Our calibration is based on the average value of U.S. Treasury securities held by the U.S. public which is 35% of GDP, according to numbers from the Congressional Budget Office (from 1962 to 2003). In the model with aggregate uncertainty the parameter s (the stochastic depreciation volatility) determines the return of equity volatility and is set at 14%, while the aggregate productivity shock follows a two-state Markov Chain with a standard deviation of of 1% and the transition probability of changing the state set to 0.4 to match the duration of the business cycle. 2.6 Model Without Aggregate Uncertainty The model in this section is very close to one studied in Aiyagari (1994 and 1995) Benchmark results In the absence of aggregate uncertainty (no depreciation shocks and no productivity shocks), the return from holding government bonds or stocks is the same (R K = R B ). The normalized individual optimization problem is then: { V (x i t; R K )=Max (1 β)(c i t) 1 1/ψ + β ( [ E t (ωt+1 ) 1 ρ V 1 ρ (x i t+1; R K ) {kt+1 i } ]) 1 1/ψ 1 ρ } 1 1 1/ψ subject to the constraints and laws of motions given above. Individual savings (capital and bond holdings) have to add up to the total capital stock and total government debt in the economy, since debt and capital are perfect substitutes. The solution to this problem, for the special case where ρ =1/ψ, is well understood since the seminal contributions by Aiyagari (1994 and 1995). 16 We solve a version of this model with the parameters values that we use/calibrate in later sections. Since, at this stage, our interest is to understand the qualitative mechanism behind the effects of fiscal policy decisions, we just state these parameter values and discuss their calibration later. More precisely, the set of parameters is {ρ = 5,ψ = 0.4,β = 0.99,g = 0.01}, which yields the results reported in table 1 (column Model 15 This approximately reflects the combined one-year volatility of permanent (8%) and transitory (10%) shocks estimated by Carroll (1992). Of course, this will underestimate labor income risk since, in this version of the model, all shocks are transitory. Later on, in the OLG model, we introduce separate permanent and transitory shocks. 16 The effect of government debt and transfers on welfare has been analyzed by Aiyagari and McGrattan (1998) and Floden (2001) is a similar model with CRRA preferences, while Domeij and Heathcote (2004) investigate the effect of capital gains taxation on welfare, both in steady state, and in the transition to a new steady state, and argue that distributional effects of capital gains taxation can distort the welfare conclusions relative to a representative agent model. (17)

16 14 I ). The model matches aggregate quantities relatively well, given that, as we just explained, it was not fully calibrated to do so. Since there is no aggregate uncertainty, all assets earn the same rate of return, which therefore represents both the return on capital and the interest rate on government bonds. As a result this economy will either significantly underestimate the former, or overestimate the latter, or both. In this case we have an equilibrium real rate of return of return of 5.66% which, most notably, strongly exaggerates the cost of government debt. Having established a benchmark case, we now proceed to our comparative statics exercise. More precisely, we now consider the responses to changes in the three key fiscal variables in the model: the tax rates on labor and capital, and the level of government debt. In this section we only consider changes to these variables in isolation, with government expenditures acting as a residual to satisfy the government budget constraint. Later on, in our OLG economy, we will consider alternative scenarios, in addition to these Changes in tax rates Since in our model labor income taxes are effectively lump-sum taxes it is easier to study them first. From the government s budget constraint, for a fixed level of government debt, lower taxes lead to a lower level of government expenditure. In a complete markets representative agent model, changing lump sum taxes does not affect the capital stock since in equilibrium the marginal product of capital must equal the discount rate. As a result, the lower taxation translates one for one into lower government consumption, and one for one into higher private consumption, consistent with the Permanent Income Hypothesis (PIH) logic. Therefore, aggregate output and aggregate returns also remain unchanged. In Table 3 (columns 2 and 3) we show that this is not the case in our economy. When we set the lump-sum tax rate equal to zero, both the capital stock and output increase (by 0.11% and 0.04% respectively), and the return on capital decreases (by 1 basis point). The economic magnitudes are very small, especially considering the magnitude of the change in taxes: from 15% to 0%. 17 However, they will be significantly larger later on when we consider more realistic economies. Therefore it is important to understand what drives this effect. In this version of the model, decreasing τ L to zero has real effects in this economy simply because of the liquidity constraints. In the presence of the liquidity constraint and uninsurable idiosyncratic risk, consumption does not rise one for one with disposable income, since the agent saves some of the permanent increase in wages in anticipation of potentially binding liquidity constraints in the future (as in Deaton (1991)). Therefore, we can conclude that liquidity constraints will increase the 17 In fact, that is exactly the reason why we considered such an extreme change in tax rates. In subsequent models we will consider smaller changes, as these will have much larger economic effects.

17 15 distortionary impact of taxes (even when there is none in their absence), but the effect is negligible unless we have additional frictions or additional sources of market incompleteness in the economy. Intuitively, the households that are most affected by these constraints own a very small fraction of the capital stock. Capital taxation naturally has distortionary effects even in an otherwise frictionless model, since it affects the relative price of savings (future consumption). As shown in Table 3 (columns 4 and 5), in our economy, a 2.5% decrease in the capital tax rate increases capital accumulation by 1.6%, while the return on capital falls by 16 basis points. In a complete markets economy, R K would have decreased by 15 basis points to keep the equality (1 + (1 τ K )R K t )= 1 β so, again, the difference due to liquidity constraints is economically very small Changes in the government debt In our third experiment we now decrease the level of government debt. For given tax rates, a lower steady-state level of government debt implies a higher steady-state level of government expenditures. From equation (5), in the new steady-state, G t = Rt B B t τ K Rt B B t = Rt B (1 τ K) B t where τ K Rt B B t captures the loss in tax revenues. For expositional purposes, it is easier to consider a simpler economy at first. In a frictionless model without capital, the increase in G would simply lead to a one-for-one reduction in private consumption: C t = G t = Rt B (1 τ K) B t (18) since Y t = C t +G t and Y t remains unchanged. Moreover, since there is only one asset in the economy, total savings must fall exactly by the same amount as government debt ( B t = i Bi t di). At the household level, wealth decreases by the value of the reduction in government bonds, leading to both lower individual consumption and lower savings, consistent with the new equilibrium.

18 16 In this simple economy the budget constraint (equation (9)) reduces to 18 C i t + Bi t+1 = Xi t = Bi t (1 + RB t (1 τ K)) = X i t = Bi t (1 + RB t (1 τ K)) Since the P.I.H. holds the marginal propensity to consume does not depend on wealth, and we have C i t = mpc i X i t = mpc i B it (1 + R B t (1 τ K )) Aggregating across all agents we have Ct i di = mpc i Bt i (1 + RB t (1 τ K))di i and combining this with equation (18) we obtain Rt B (1 τ K) B t =(1+Rt B (1 τ K)) i i mpc i Bt i di (19) Rt B(1 τ K) (1 + Rt B (1 τ K )) = mpc i di i where the last equivalence follows from the independence between mpc and wealth. Since the RHS does not depend on B then the equilibrium rate of return is unchanged. Now let us consider a production economy. In equilibrium, household savings are used to supply the capital demanded by firms, and to buy the government bonds supplied by the government (X C = K + B). If we have no frictions or aggregate uncertainty, then the previous results still holds. At the initial rate of return we have C t = G t and (Xt i Ci t )= i Bt+1 i i so that investment remains constant, the return on capital/bonds remains constant, and all of the above applies. In other words, there is still no crowding-out in the economy. However, this result breaks down if we introduce frictions (as in this section), or aggregate uncertainty (as in the next section). In the presence of borrowing constraints and labor income 18 Note that X it di = B it (1 + Rt B (1 τ K))di i i = B it di + B it Rt B (1 τ K)di i = B + C i confirming that total wealth still equals total savings plus total consumption, at the previous rate of return,

19 17 uncertainty, the marginal propensity to consume is a decreasing function of wealth, due to the precautionary savings motive. When wealth falls, household consumption now falls by more than in the previous case, since the liquidity constraints become more binding. Therefore, for a given equilibrium rate of return, household savings decrease by less than in the previous case. Repeating the previous analysis we see the rate of return can no longer remain constant in the new equilibrium. Assume again that investment and the equilibrium rate of return both remain constant, so that equation (18) would still hold. Since the marginal propensity to consume is no longer constant equation (19) now becomes Rt B (1 τ K) B t = [mpc i Xt i ]= [mpc i (1 + Rt B (1 τ K)) Bt i di + mpc i Xt i di] i i where mpc i now denotes the initial marginal propensity to consume (i.e. with the initial level of wealth). Since mpc i is a decreasing function of X it we now have Rt B (1 τ K) B t > (1 + Rt B (1 τ K)) mpc i Bt i di i Rt B(1 τ K) B i t < mpc (1 + Rt B i di mpc i di (1 τ K )) i B t i In the above, the direction of the inequality has changed because we consider B t < The above inequality implies that the rate of return must decrease in the new equilibrium, since the LHS is increasing in Rt B.20 Intuitively, since households savings no longer falls one for one with government bonds, asset markets cannot clear at the previous equilibrium rate of return. A lower rate of return induces firms to demand more capital and thus match the excess household savings. Therefore, investment and output also increase, and there is crowding-out in this economy (in this case crowding-in, since we have B <0). In the equilibrium we must have Y t = C t + I t + G t where, relative to the previous economy, we now have I t > 0and Y t > 0. Since, in steady-state, I t = K t < Y t then we must have C t < G t 19 The last term might not hold with equality since individual portfolio decisions are indeterminate in an economy without aggregate shocks, and therefore the cross-sectional distributions of Bt i and Ki t are not well-defined (only Bt i + Kt i is well-defined). This equation will hold with equality if we assume that the changes in government bond holdings are equally spread across households, i.e. are independent from their initial wealth levels, and therefore independent from their marginal propensity to consume. 20 The RHS is just the integral over the initial values of the marginal propensity to consume, so it is a constant.

20 18 Aggregate consumption does not fall as much, because of the crowding-in effect and the corresponding increase in output. This is exactly what we find in Table 3 (columns 7 and 8), although once again the effect is economically negligible. Even though we consider an extreme case where we set B = 0, the return on capital only falls by 1 basis point, and the capital stock only increases by 0.11%. The consumption share of GDP decreases by 1.9% which is slightly less (in absolute terms) than the 12.66% increase in government expenditures Summary The main lessons from these comparative statics are threefold. First, in the presence of borrowing constraints lump sum taxes have an effect on capital formation but, in the absence of other frictions or aggregate uncertainty, the quantitative magnitude of this effect is very small. Second, in this economy, the rates of return are not significantly affected from changing tax rates or the available level of government debt. Third, in such a world, government debt barely affects aggregate capital accumulation: the crowding-out effect is economically very small. 2.7 Model With Aggregate Uncertainty / Imperfect Substitutability We now introduce aggregate uncertainty in the previous model through aggregate productivity and depreciation shocks. As a result, the returns on government bonds and capital are no longer the same. At the micro level this implies that households now have a portfolio decision, in addition to their saving decision, and they need to form expectations about the evolution of the aggregate capital stock. At the macro level we can now try to match the rate of return on capital, without imposing a counterfactually high rate of return on government bonds. The calibration (e.g. values for the preference parameters) is identical to the one in the previous model Benchmark results Table 1 (in column Model II ) reports the equilibrium macroeconomic quantities obtained in this economy. Since we are keeping the same parameter values, instead of re-calibrating the models, they deliver different equilibrium outcomes. In particular, output is higher now and therefore it is best to compare the other variables in terms of their GDP shares. Consumption is now lower, while investment, capital and government expenditures all constitute a larger fraction of aggregate output. Finally consumption growth is smoother than GDP growth, and the orders of magnitude match well with the long run data from Campbell (1999).

21 19 Table 2 reports the main asset pricing moments implied by the model, along with their empirical U.S. counterparts again taken from Campbell (1999). The model generates a small equity premium of 2.44%, while simultaneously generating a moderately low risk free rate (2.68%) with standard deviations close to those observed in the data. These statistics simply reflect the difficulty in simultaneously matching the equity premium within a relatively standard model. While it would be possible to increase risk aversion to obtain a higher equity premium, this would imply a more volatile consumption process and a higher riskless rate. While the equity premium in this model is less than half of its empirical counterpart, we will show that this already has important quantitative implications for our results Changing the lump-sum tax rate Table 4 (columns 2 and 3) compares the baseline economy (with τ L = 15%), with an otherwise identical economy where the lump-sum tax rate has been decreased by 2.5%. As before, a lower lump-sum tax rate results in lower government expenditures, which translate directly into higher household consumption. In the model without aggregate uncertainty the presence of borrowing constraints would induce a small increase in savings/investment (0.11%) and very modest change in the equilibrium rate of return (1 basis point). This was even in the extreme case of τ L =0, thus a decrease in the tax rate of 15 percentage points! In contrast, when we introduce aggregate uncertainty, a much smaller tax rate change change (2.5%) generates an almost 4 times larger response: investment and capital both increase by 0.42%, and the return on capital decreases by 4 basis points. Therefore, the results are more than an order of magnitude higher than in the one-asset model. Why do the results change so dramatically in the presence of aggregate uncertainty? This is because there is a second channel operating here, in addition to the borrowing constraint channel. In this model capital and bonds are no longer perfect substitutes and therefore households also make a portfolio allocation decision. Lump-sum taxes are essentially equivalent to a negative position in riskless bonds: a non-contingent future payment. Therefore, a lower lump-sum tax rate reduces the value of this implicit negative bond position, and consequently households respond to this by shifting their portfolio allocations more towards stocks. Therefore, investment and capital increase by more than in the economy without aggregate uncertainty. Comparing results in Tables 3 and 4 from corresponding columns with lower τ L we we see that this effect is very large. 21 Since we consider an economy with aggregate uncertainty, we can also study the impact of fiscal 21 It is important to remember that in the first case we had τ L = 15%, while in the second case we have τ L = 2.5%.