Fiscal Stimulus Payments and Economic Activity in a Model of Liquidity-Constrained Households Preliminary and incomplete.

Transcription

1 Fiscal Stimulus Payments and Economic Activity in a Model of Liquidity-Constrained Households Preliminary and incomplete. Ralph Lütticke & Lien Pham-Dao February 15, 214 Abstract Fiscal stimulus payments have become a commonly used policy to counteract recessions and alleviate their welfare consequences. We examine the eects of such payments on individual consumption and the macroeconomy in a model with incomplete markets, liquid und illiquid assets, and a nominal rigidity. The positive consumption response of wealthy but liquidity-constrained households is strongly amplied with endogenous asset prices, for helicopter drops of money push up the price of illiquid physical capital. At the same time the response falls in money holdings as ination reduces their value. Overall, the former eect dominates because households primarily save in the high-return illiquid asset, generating a high marginal propensity to consume out of extra cash. Yet, output hardly increases because ination depresses prots of those rms that are not able to adjust their prices - partly counteracting the eect of the stimulus. As a result, the multiplier on scal stimulus payments remains well below the average marginal propensity to consume out of extra cash. Keywords: Fiscal Stimulus Payments, Liquidity, Nominal Rigidities, Incomplete Markets. JEL-Codes: E32, H31, E12, E22 We would like to thank Christian Bayer and Thomas Hintermaier for helpful comments. Bonn Graduate School of Economics, Department of Economics, Universität Bonn. Address: Adenauerallee 24-42, Bonn, Germany. rcluetticke@uni-bonn.de.

2 1 Introduction Fiscal stimulus payments have become a commonly used policy to counteract recessions and alleviate their welfare consequences. During the last two downturns in the U.S. of 21 and 28-9, for example, U.S. households received one-o payments from $5 to $1 of which a large share is spent on nondurables in the quarter that they were received. 1 Kaplan and Violante (211) show that such a high marginal propensity to consume (MPC) out of extra cash can be explained by wealthy but liquidity-constrained households. These so-called wealthy hand-to-mouth households own sizable amounts of illiquid wealth, but consume all of their disposable income each period. Kaplan and Violante document that almost 1/3 of U.S. households can be classied as such according to the Survey of Consumer Finances (24). This behavior emerges in an incomplete markets model with two assets one liquid and the other illiquid, where the latter pays a higher return but is costly to trade. The relative return dierence in combination with transaction costs substantially increases the fraction of households that are constrained in their consumption in comparison to a plausibly calibrated one-asset model. More importantly, wealthy hand-to-mouth households also have higher MPCs out of extra cash than their poor counterparts with zero net worth, for the former are wealthier and thus target a higher consumption path. We show that this intensive margin is strongly amplied with endogenous asset prices, where we interpret illiquid assets as productive capital and liquid assets as money, the return of which is pinned down by a monetary authority. In the period of the one-o stimulus payments, ination rises to bring the money market in equilibrium, because households spend part of this extra cash on consumption and investment. Ination then remains high until the stock of real money is back at its long-run value. This increases the relative return dierence between money and capital for some time, leading to a prolonged boom in investment that pushes up the price of the latter. This additional wealth eect not only reinforces the consumption response of wealthy hand-to-mouth households, but also makes all owners of capital richer increasing their consumption path as well. The opposite result obtains for households rich in money. Higher ination reduces not only the real value of the payment by the government, but also the value of all money holdings. Accordingly, the consumption response falls in money holdings. Taking everything into consideration, we nd an average consumption response of 45% out of a $5 stimulus payment to all households, which is roughly twice as large as the results that Kaplan and Violante (211) obtain in a partial equilibrium model with 1 See Johnson et al. (26) and Parker et al. (211) for estimates of both episodes. 1

3 xed returns. This average response hides contrasting welfare eects that result from the dierential impact of the stimulus on the price of money and capital. In the short run, however, all households prot from such a policy as wages rise as well and thereby more than compensate households for higher ination in our calibration. The prolonged boom in investment reduces capital dividends so that households rich in capital start to suer welfare losses after 5 years. Average welfare thus falls over time but it remains positive. Fiscal stimulus payments do not only move prices, but also aect output in our economy because of a nominal rigidity. We assess the eects of such a demand stimulus in a scenario where the central bank provides helicopter drops of money in the rst period, and from then on follows Friedman's k%-rule. 2 The relatively high MPCs in comparison to a one-asset framework do not, however, translate into an output multiplier of similar size. Our model generates multipliers of about 2%, which is comparable in size to the study by Oh and Reis (212) who report the scal transfer multiplier for transfers across agents in a model with one asset only. The limited eectiveness of scal stimulus payments in such a New Keynesian framework follows from the price rigidity that makes the economy demand-driven in the rst place. The increase in ination reduces prots of those rms that cannot adjust their prices, negatively aecting consumption. Our results represent the upper-bound of this eect as we assume, for a start, that all prots are consumed each period to simplify the numerical solution. However, even if one were to assume the opposite extreme that lower prots did not aect consumption at all, the multiplier on scal stimulus payments would still be below one. Neglecting the response of prots, our model suggests that output would increase by almost 5% of the stimulus payment on impact. The remainder of the paper is organized as follows. Section 2 develops our model, and Section 3 discusses the solution method. Section 4 explains the calibration of the model. Section 5 presents the numerical results. Section 6 concludes. 2 Model Following the insight by Kaplan and Violante (211), we consider an incomplete markets model with a liquid and an illiquid asset. In contrast to their seminal paper, asset returns are now endogenous, for we interpret the illiquid asset as productive capital with a return equal to its marginal product and the liquid asset as money, the return of which is pinned down by a monetary authority setting the ination rate. In addition, our model 2 Helicopter drops of money have actually been proposed in the UK, among others, by Willem Buiter, based on insights derived from his previous work, e.g. Buiter (27). 2

4 also features a nominal rigidity that makes the economy demand-driven in the short-run. In particular, we follow the New Keynesian literature in assuming that monopolistically competitive rms face a Calvo (1983) price setting friction. 3 For tractability reasons, we assume two seperate types of agents: (worker-)households and entrepreneurs. Households supply capital and labor, and are subject to idiosyncratic shocks to their human capital, i.e. their labor productivity. Households can self-insure in money and in less liquid physical capital. Illiquidity is understood in the spirit of Kaplan and Violante's (211) model of wealthy hand-to-mouth consumers, where trading the illiquid asset is subject to a friction. We model this friction as infrequent participation in the capital market. Every period a fraction of households is randomly selected to trade physical capital. All other households may only adjust their money holdings. 4 While money is subject to an ination tax and pays no dividend, capital can be rented out to the intermediate-good-producing sector on a perfectly competitive rental market. This sector combines labor services, capital, and nal goods into intermediate goods in roundabout production, see Basu (1995), and sells them to the entrepreneurs. Entrepreneurs capture all pure rents in the economy. For simplicity, we assume entrepreneurs to be risk neutral. They obtain rents from the adjustment of the aggregate capital stock due to convex capital adjustment costs and, more importantly, from dierentiating the intermediate good. Facing monopolistic competition, they set prices above marginal costs for these dierentiated goods. Price setting, however, is subject to a pricing friction à la Calvo (1983) so that entrepreneurs may only adjust their prices with some positive probability each period. The dierentiated goods are nally bundled again to the composite nal good used for consumption, investment, and as production input by perfectly competitive nal-good producers. The model is closed by a monetary authority that provides money in positive net supply and adjusts money growth according to Friedman's k%-rule. All seignorage is wasted. 3 This model setup builds on previous joint-work with Christian Bayer and Volker Tjaden, where we assess the aggregate consequences of shocks to household income uncertainty in a similar framework. See Bayer et al. (214). 4 We choose to exclude trading as a choice, and hence we use a simplied framework relative to Kaplan and Violante (211) for numerical tractability. Random participation keeps the households' value function concave, thus makes rst-order conditions sucient, and therefore allows us to use a variant of the endogenous grid method as algorithm for our numerical calculations. See Bayer et al. (214) for proofs. 3

5 2.1 Households There is a continuum of ex-ante identical households of measure one indexed by i. Households are innitely lived, have time-separable preferences with time-discount factor β, and derive felicity from consumption c it. They maximize the discounted sum of felicity: V = E max {c it } The felicity function takes CRRA form with risk aversion ξ: β t u (c it ). (1) t= u(c it ) = 1 1 ξ c1 ξ it, ξ >, where c it is household i's demand of the bundled consumption good. It is obtained from bundling varieties j of dierentiated consumption goods according to a Dixit-Stiglitz aggregator, ( c it = η 1 η cijt ) η η 1 dj. Each of these dierentiated goods is oered at price p jt so that the demand for each of the varieties is given by c ijt = ( pjt P t ) η c it, ( ) 1 where P t = p 1 η 1 η jt dj is the average price level. Households derive income from supplying labor and from renting out capital. household is endowed in each period with h it eciency units of labor, which evolve according to an AR(1)-process. log h it = ρ h log h it 1 + ɛ it, ɛ it N (µ, σ h ) (2) We assume households to inelastically provide all of their hours of labor. Fiscal stimulus payments are small in comparison to total wealth and hence the wealth eect on labor supply should be negligible. Thus total labor input supplied is given by h it di. Following the literature on idiosyncratic income risk, we assume that asset markets are incomplete. Households can only trade in nominal money, m it, that does not bear any interest and in capital, k it, to smooth their consumption. Holdings of both assets 4 A

6 have to be non-negative. 5 Moreover, trading capital is subject to a friction. This trading friction allows every period only a randomly selected fraction of households, ν, to participate in the asset market for capital. Only these households can freely rebalance their portfolio. All other households obtain dividends, but may adjust their money holdings only. For those households participating in the asset market, the budget constraint reads: c it + m it+1 + q t k it+1 = m it π t + (q t + r t )k it + w t h it + τ t π t, m it+1, k it+1, (3) where m it are real money holdings, k it are capital holdings, h it is the stochastic endowment with eciency units of labor, q t is the price of capital, w t is the wage rate, r t is the rental rate or dividend, and π t = Pt P t 1 holdings of household i at the end of period t by m it+1 := m it+1 P t lump-sum payment by the government. is the ination rate. We denote real money. Finally, τ t is the nominal For those households that cannot trade in the market for capital the budget constraint simplies to: c it + m it+1 = m it π t + r t k it + w t h it + τ t π t, m it. (4) Note that we assume that the depreciation of capital is replaced through maintenance such that the dividend, r t, is the net return on capital. Since a household's saving decision will be some non-linear function of that household's wealth and productivity; the price level, P t, and accordingly aggregate real money, M t+1 = M t+1 P t, will be functions of the joint distribution Θ t of (m t, k t, h t ). This makes Θ t a state variable of the households' planning problem. This distribution evolves as result of the economy's reaction to the scal stimulus shock, which we model as an unexpected one-o payment. With this setup, the dynamic planning problem of a household is then characterized by two Bellman equations V a in case the household can adjust its capital holdings and 5 The non-negativity requirement on money holdings reects the natural borrowing limit. Any other borrowing limit might lead in case of a suciently large deation to a violation of the requirement that households need to be able to repay their debt. 5

7 V n otherwise: V a (m, k, h; Θ) =max k,m a u[c(m, m a, k, k, h)] + β [ νev a (m a, k, h, Θ ) + (1 ν)ev n (m a, k, h, Θ ) ] V n (m, k, h; Θ) =max m n u[c(m, m n, k, k, h)] + β [ νev a (m n, k, h, Θ ) + (1 ν)ev n (m n, k, h, Θ ) ] (5) In line with this notation, we dene the optimal consumption policies for the adjustment and non-adjustment cases as c a and c n, the money holding policies as m a and m n, and the capital investment policy as k. See Appendix A for the rst order conditions. 2.2 Intermediate Goods Producers We follow Basu (1995) in assuming that the intermediate-good-producing sector operates a gross production function with constant returns to scale instead of a value added production function. Production of the intermediate good requires pre-products X t, time-constant aggregate labor N, and capital K t. Pre-products are acquired on the market for the bundled nal-consumption good.total gross output of the intermediategood sector is: Y G t = X γ t N α(1 γ) K (1 α)(1 γ) t. Let MC t be the relative price at which the intermediate good is sold to entrepreneurs. The intermediate-good producer maximizes prots, MC t Y G t X t = MC t X γ t N α(1 γ) K (1 α)(1 γ) t X t, choosing the amount of pre-products. The optimal amount of pre-products is then given by: X t = γmc t Y G t = (γmc t ) 1 1 γ Using this term, we can express net output or GDP in this setting, as: N α K 1 α t. (6) ] Y t = Yt G Xt = [(γmc t ) γ 1 γ (γmc t ) 1 1 γ N α Kt 1 α. (7) The real wage and the user costs of capital are given by the marginal products of 6

8 labor and capital: 2.3 Entrepreneurs w t = α(1 γ)γ γ 1 1 γ 1 γ MCt r t + δ = (1 α)(1 γ)γ γ 1 1 γ 1 γ MCt ( Kt / N ) 1 α Entrepreneurs dierentiate the intermediate good and set prices. (8) ( N/Kt ) α. (9) They are risk neutral and have the same discount factor as households. We assume that only the central bank can issue money so that entrepreneurs do neither participate in the money nor capital market. This assumption gives us tractability in the sense that it separates the entrepreneurs' price setting problem from the households' saving problem. It enables us to determine the price setting of entrepreneurs without having to take into account households' intertemporal decision making. Under these assumptions, the consumption of entrepreneur j equals her current prots, Π jt. By setting prices of nal goods, entrepreneurs maximize expected discounted future prots: E β t Π jt. (1) t= Entrepreneurs buy the intermediate good at a price equalling the nominal marginal costs, MC t P t, where MC t are the real marginal costs at which the intermediate good is traded due to perfect competition, and then dierentiate them without the need of additional input factors. The goods that entrepreneurs produce come in varieties uniformly distributed on the unit interval and each indexed by j [, 1]. Entrepreneurs are monopolistic competitors, and hence charge a markup over their marginal costs. They are, however, subject to a Calvo (1983) price setting friction, and can only update their prices with probability θ. They maximize the expected value of future discounted prots by setting today's price, p jt, taking into account the price setting friction: max {p jt } (θβ) s EΠ jt,t+s = s= (θβ) s EY jt,t+s (p jt MC t+s P t+s ) (11) s= s.t. : Y jt,t+s = ( pjt P t+s ) η Y t+s, where Π jt,t+s are the prots and Y jt,t+s is the production level in t + s of a rm j that set prices in t. 7

9 We obtain the following rst-order condition with respect to p jt : (θβ) s p jt EY jt,t+s P t 1 s= where µ is the static optimal markup. η η 1 }{{} µ P t+s MC t+s =, (12) P t 1 Recall that entrepreneurs are risk neutral, and that they do not interact with households in any intertemporal trades. Moreover, aggregate shocks to the economy are small and homoscedastic. Therefore, we can solve the entrepreneurs' planning problem locally by log-linearizing around the zero ination steady state, without having to know the solution of the households' problem. This yields after some tedious algebra, see e.g. Galí (28), the New Keynesian Phillips curve: where log π t = βe t (log π t+1 ) + κ(log MC t + µ), (13) κ = (1 θ)(1 βθ). θ Besides dierentiating goods and obtaining a rent from the markup they charge, we assume that entrepreneurs also obtain and consume rents from adjusting the aggregate capital stock. Since the dividend yield is below their time-preference rate, in equilibrium entrepreneurs never hold capital. The cost of adjusting the stock of capital is ( ) 2 Kt+1 Kt + K t+1. Hence, entrepreneurs will adjust the stock of capital until the φ 2 K t following rst-order condition holds: 6 q t = 1 + φ K t+1 K t. (14) 6 Note that we assume capital adjustment costs only on new capital (or on the active destruction of old capital) but not on the replacement of depreciation. Depreciated capital is assumed to be replaced at the cost of one-to-one in consumption goods, and replacement is forced before the capital stock is adjusted at a cost. This dierential treatment of depreciation and net investment simplies the equilibrium conditions substantially, for the user cost of capital and hence the dividend paid to households does not depend on the next periods stock of capital, and the decisions of non-adjusters are not inuenced by the price of capital q t. 8

10 2.4 Government We assume that money supply is given by Friedman's k% rule, i.e. M t+1 M t = (θ 1 /π t ). (15) Here M t+1 are real balances at the end of period t (with the timing aligned to our notation for the households' budget constraint). The coecient θ 1 1 determines steady state ination. The government uses seigniorage revenues S t for wasteful consumption G t of nal goods. In order to nance the total one-o nominal payment τ t to households, the scal authority prints additional money. 2.5 Goods, Money, Asset, and Labor Market Clearing The labor market clears at the competitive wage given in (8); so does the market for capital services if (9) holds. The money market clears, whenever the following equation holds: (θ 1 /π) M t = [νm a(m, k, h; q, π) + (1 ν)m n(m, k, h; q, π)] Θ t (m, k, h)dmdkdh, (16) with last end-of-period real money holdings given by M t := m t Θ t (m t, h t )dm t dh t. In the period of the scal stimulus payments, the money market clearing condition needs to take into account payments, τ t, on both the giving and the receiving end: (θ 1 /π) (M t +τ t ) = [νm a(m, k, h, τ; q, π) + (1 ν)m n(m, k, h, τ; q, π)] Θ t (m, k, h)dmdkdh. Last, the market for capital has to clear: (17) q t = 1 + φ K t+1 K t = 1 + νφ K t+1 K t (18) K t K t Kt+1 := k (m, k, h; q t, π t )Θ t (m, k, h)dmdkdh K t+1 = K t + ν(k t+1 K t ), where the rst equation stems from competition in the production of capital goods, the 9

11 second equation denes the aggregate supply of funds from households trading capital, and the third equation denes the law of motion of aggregate capital. The goods market then clears due to Walras' law, whenever both, money and capital markets, clear. 2.6 Recursive Equilibrium A recursive equilibrium in our model is a set of policy functions {c a, c n, m a, m n, k }, value functions {V a, V n }, pricing functions {r, w, π, q}, aggregate capital and labor supply functions {N, K}, distributions Θ t over individual asset holdings and productivity, and a perceived law of motion Γ, such that 1. Given V, Γ, prices, and distributions, the policy functions {c a, c n, m a, m n, k } solve the households' planning problem, and given the policy functions {c a, c n, m a, m n, k }, prices and distributions, the value functions {V a, V n } are a solution to the Bellman equations (5). 2. The labor, the nal-goods, the money, the capital, and the intermediate-good markets clear, i.e. (8), (13), (17), and (18) hold. 3. The actual law of motion and the perceived law of motion Γ coincide, i.e. Θ = Γ(Θ, s ). 3 Numerical Implementation We compute the transitional dynamics after an unexpected one-o scal stimulus shock with the help of Krusell and Smith (1998)-rules. We consider an economy that is in steady state before period t = with τ t =. In t =, all households receive an unexpected scal transfer, τ, that shocks the steady state. There are no more shocks from t = 1 onwards. From then on, households anticipate how prices evolve on the path back to the long-run equilibrium of the economy. These prices are, of course, a function of all states including the joint distribution Θ t (m, k, h). Hence, we assume that households predict future prices on the basis of a restricted set of moments as in Krusell and Smith (1997, 1998). Specically, we make the assumption that households condition their expectations only on last period's aggregate real money holdings, M t, and the aggregate stock of capital, K t. The reasoning behind this choice goes as follows: (17) determines ination, which in turn depends on the current money stock and, in case of a stimulus shock, on the additional money printed to nance scal policy. Once ination is xed, the Phillips 1

12 curve (13) determines markups and hence wages and dividends. These will pin down asset prices by making the marginal investor indierent between money and physical capital. If asset-demand functions, m a,n and k, are suciently close to linear in human capital, h, and in non-human wealth, m, k, at the mass of Θ t, we can expect approximate aggregation to hold. For our exercise, the two aggregate states M t and K t are sucient to describe the evolution of the aggregate economy. Households use the following log-linear forecasting rules for current ination, π t, and the price of the illiquid asset, q t : log π t = β 1 π + β 2 π log M t + β 3 π log K t (19) log q t = β 1 q + β 2 q log M t + β 3 q log K t. (2) The law of motion for real money holdings, M t, then follows from the monetary policy rule and is given by: log M t+1 = log M t + (log θ 1 log π t ). The law of motion for K t results from (18). To nd the deterministic law of motion in response to a zero-probability scal stimulus payment shock, we need to solve for the market clearing prices each period. Concretely, this means the posited rules, (19) and (2), are used to solve for the households' policy functions. Having solved for the policy functions conditional on the forecasting rules, we then simulate the model for t =,..., T periods, keeping track of the actual distribution Θ t. The simulation starts in steady state and the transfer shock hits in t =. We then calculate in each period t the optimal policies for market clearing ination rates and asset prices assuming that households resort to the policy functions derived under rule (19) and (2) from period t + 1 onwards. Having determined the market clearing prices, we obtain next period's distribution Θ t+1. We next re-estimate the parameters of (19) and (2) from the simulated data and update the parameters accordingly. Subsequently, we recalculate policy functions and iterate until convergence in the forecasting rules. The posited rules (19) and (2) approximate the aggregate behavior of the economy in response to a transfer shock well. The minimal within sample R 2 is %. See Appendix C for details. 3.1 Solving the household planning problem In solving for the households' policy functions we apply an endogenous gridpoint method as originally developed in Carroll (26) and extended by Hintermaier and Koeniger 11

13 (21), iterating over the rst-order conditions. We approximate the idiosyncratic productivity process by a discrete Markov chain with 11 states, using the method proposed by Tauchen (1986). 7 Details on the algorithm can be found in Bayer et al. (214). 4 Calibration We calibrate the model to the asset distribution observed in the U.S. economy. particular, we target the share of wealthy hand-to-mouth households corresponding to the estimates by Kaplan and Violante (211). Those households may hold sizable amounts of the illiquid but nothing of the liquid asset. Moreover, we also target mean capital holdings to get reasonable returns in general equilibrium. For the calibration of the supply side of the model, we take standard parameter values that are widely used in the New Keynesian literature. Table 1 summarizes our calibration. In detail, we choose the parameter values as follows. 4.1 Households For the felicity function, u = 1 1 ξ c1 ξ, we set the coecient of relative risk aversion ξ = 4 as in Kaplan and Violante (211). The time-discount factor, β, and the asset market participation frequency, ν, are jointly calibrated to match the ratios of liquid and illiquid assets to output. We equate illiquid assets to all capital goods relative to nominal GDP, and thus obtain as an average over the period 198 to 212 an annual capital-to-output ratio of 29%. In our baseline calibration, this implies a annual real return for illiquid assets of 5%. We equate liquid assets to base money and not to inside claims, because the net value of inside claims does not change with ination. Specically, we take the average of the St. Louis Ajdusted Monetary Base from the Federal Reserve Bank of St. Louis for the years 198 to 212. For details on the steady state asset distribution, see Appendix B. The calibrated participation frequency ν = 5% is slightly larger than the value that Kaplan and Violante (211) obtain in their state-dependent participation framework for workers. 4.2 Intermediate, Final, and Capital Goods Producers We parameterize the production function of the intermediate good producer according to the U.S. National Income and Product Accounts (NIPA). In the U.S. economy, inter- 7 We solve the household policies for 5 points on the grid for money and 8 points on the grid for capital using equi-distants grids on log scale. For aggregate money and capital holdings we use a grid of 5 points each. In 12

14 Table 1: Calibrated Parameters Parameter Value Description Target Households β.98 Discount factor K/Y = 29% (annual) ν 5% Participation frequency M/Y = 15% (annual) ξ 4 Coecient of rel. risk av. Standard value Intermediate Goods γ.45 Share of pre-products NIPA: Intermediate consumption α 65% Share of labor Income share of labor of 62% δ 1.35% Depreciation rate NIPA: Fixed assets & durables Final Goods κ.4 Price stickiness Avg. price duration of 6 quarters µ.5 Markup 5% markup (standard value) Capital Goods φ 3 Capital adjustment costs Relative investment volatility Monetary Policy θ Money growth 2% p.a. Income Process ρ h.987 Persistence of productivity σ.6 STD of innovations 1/3 HTM-consumers 13

15 mediate consumption, i.e. the total amount of pre-products used in production, makes up roughly 45% of gross output. Hence, we set α =.45. The labor and capital share including prots (about 2/3 and 1/3) align with standard calibrations. To calibrate the parameters of the entrepreneurs' problem, we use standard values for markup and price stickiness that are widely employed in the New Keynesian literature. The Phillips curve parameter κ implies an average price duration of 6 quarters, assuming exible capital at the rm level. The steady state marginal costs, exp( µ) =.95, imply a markup of 5%. For simplicity, we set the entrepreneurs' discount factor equal to the households' discount factor. We calibrate the adjustment cost of capital, φ = 3, to match an investment to output volatility of Central Bank We assume the central bank to follow Friedman's k% rule. We set the average growth rate of money, θ 1, such that our model produces an average annual ination rate of 2% in line with the usual ination targets of central banks and roughly equal to average ination in the U.S. between 198 and Income process We calibrate households' income process to the fraction of wealthy hand-to-mouth households following the estimates by Kaplan and Violante (211), i.e. such households may hold substantial illiquid wealth but no liquid assets. Depending on the measure of liquid and illiquid assets, Kaplan and Violante obtain a share of 2 3% of wealthy hand-to-mouth households and 3 4% of constrained households in total. Our calibrated income process generates a fraction of 1/3 of constrained households, of which 2/3 are wealthy hand-to-mouth. We set the annual standard deviation of persistent shocks to idiosyncratic productivity to.12. The annual autocorrelation is.95 a standard value in the literature. 8 This value for φ is comparably high. However, if we assumed that capital adjustment costs only referred to investment relative to the capital stock of the households adjusting capital, then the adjustment cost parameter would be φν 15, and hence well in the range of the estimates that e.g. Ireland (23) obtains. 14

16 5 Quantitative Results In the following, we consider a policy experiment that consists of an unexpected, one-o payment of $5 to each household in the economy paid out in t =. This is about the amount of the tax rebate authorized by the U.S. Congress in 21. We assume that the policy is nanced by the central bank printing the required money, but otherwise following Friedman's k% rule. We rst show how a household's consumption response to such a payment depends on her asset position and moreover how changes in equilibrium prices amplify this response in our model. Second, we assess the aggregate eects of such a policy. Finally, we show that it has contrasting welfare eects for households with dierent portfolio compositions, because prices of the liquid and illiquid asset move in opposite directions after the one-o stimulus. 5.1 Individual Consumption Response Figure 1.1 plots the marginal propensity to consume out of extra cash as a function of a household's asset position keeping prices xed. Throughout most of the asset-space households consume only about 5-2% out of the $5 payment by the government. This includes the diagonal along which most households would be clustered if we were to consider net worth only. With two assets, however, a large fraction of wealthy households prefers to hold illiquid, rather than liquid assets. This makes them constrained in their consumption in every period in which they are not able to tip into their capital account. In addition, these wealthy hand-to-mouth households have a higher MPC than their poor counterparts without any assets. The MPC increases in capital because richer households have a higher target consumption. For households very rich in capital but without any liquid assets it actually reaches 85%, whereas households with no assets at all only consume around 1% out of the extra cash. The MPCs in Figure 1.2 take into account the eects of price changes that happen in the very period of the stimulus payment and rational expectations about the future path of prices. The transformation of the graph is striking. The consumption response of wealthy hand-to-mouth households increases markedly, reaching more than 1% for high holdings of capital. Overall, the response is larger throughout the asset-space because of higher wages that more than compensate households for higher ination in our calibration. Figure 2 plots the impulse responses of ination and the price of capital to the stimulus shock. On impact, the latter increases by almost.5 percentage points instantaneously increasing the value of all capital holdings. The spike in ination, in 15

17 Figure 1: Marginal Propensity to Consume out of Extra Cash 1.1 With Fixed-Prices 1.2 With Equilibrium-Prices 2 2 MPC in % 1 MPC in % Capital Holdings Money Holdings Capital Holdings Money Holdings Notes: The graphs refer to the MPCs out of $5 extra cash in the period in which the payment is received. We plot the MPCs as a function of the asset position human capital has been integrated out. Capital and Money holdings are measured in terms of average quarterly income. On the bottom of each graph, the density of the distribution over idiosyncratic asset holdings, Θ(m, k), is displayed. contrast, reduces the value of money holdings by about.45 percentage points. Theoretically, households who predominantly hold money (and a lot of it) may actually be worse o because of the policy. Taking everything into consideration, the average MPC out of the stimulus policy considered here is higher in general equilibrium than with xed prices 45% versus 3%. This can easily be seen by looking at the distribution over capital and money holdings displayed in the bottom of Figure 1. Households hold more than twenty times more wealth in capital than in money. Money because of its liquidity has mainly value for short-run consumption smoothing, but capital is the asset of choice for saving purposes because of the relative return dierence. 5.2 Aggregate Eects The equilibrium price eects, which lead to an amplication of the households' consumption response, can be understood by looking at Figure 3 and 4. Figure 3 shows the 16

18 Figure 2: Impulse Responses of Prices.5 Ination π t.6 Price of Capital q t Percentage Points Percentage Points Quarter Quarter Notes: Impulse responses to an unexpected one-o payment of $5 to all households by the government. This stimulus is nanced by the central bank printing the required money. impulse responses of aggregate capital and real money balances, and Figure 4 does so for net output and its components. In period t =, the goverment pays out $5 to each household in the economy, thereby increasing nominal money supply by.83%. Households then consume, invest, or hoard this extra cash. All else equal, money demand thus falls short of supply. This disequilibrium is resolved by ination, which rises until real money supply is suciently reduced. In equilibrium, real money demand then increases by only.8% on impact, see Figure 3. Figure 4 shows how households spend the $5 payment by the government. They consume almost 45% out of it see Section 5.1 for a detailed analysis of the heterogeneous consumption responses and invest roughly 3.5% of it in capital. The rest is hoarded in money. The investment response is, of course, limited by assumption as only a fraction of households, ν = 5%, is allowed to trade capital each period. This fraction, however, matches the fraction of households that optimally adjusts in the model of Kaplan and Violante (211) for a stimulus of equal size. In their model, households may always adjust their illiquid asset holdings, but adjusting them is subject to a transaction fee. The prolonged period of ination moreover increases the relative relative return difference between money and capital for some time, leading to an equally long boom in investment. Consequently, capital peaks only after 5 quarters. These persistent dynamics extend the impact of the scal stimulus on output beyond the very period in which the payments occur. The positive eect on output remains until the capital stock has 17

19 Figure 3: Impulse Responses of Capital and Money.25 Capital K t.8 Real Money M t Percentage.15.1 Percentage Quarter Quarter Notes: Impulse responses to an unexpected one-o payment of $5 to all households by the government. This stimulus is nanced by the central bank printing the required money. returned to its steady value. What is surprising given the boom in investment and households' sizable consumption out of extra cash is the small output multiplier of the scal stimulus scheme. Net output increases by only 2% of the payments on impact, see Figure 4. 9 Key for understanding this limited impact on output is the response of total consumption. Total consumption consists of consumption by households and by entrepreneurs in our model. The latter consume all of their prots by assumption. Average prots, however, fall with higher ination, because a fraction of rms cannot adjust their prices to compensate for higher marginal costs, MC t P t. All in all, total consumption slightly falls by about 2% on impact. The consumption response turns positive after 3 quarters, as the change in ination becomes smaller each period and prots recover. Our results represent the upper bound of this countervailing eect on consumption that emanates from falling prots. For tractability reasons, we assume entrepreneurs to be seperate agents, who consume all of their prots each period. Typically, households are assumed to own the rms, and hence prots are an additional income source that accrues to households. In such an environment, households would still be negatively aected by falling prots, but they would try to smooth consumption. All else equal, consumption would hence not fall one-to-one with prots. Even if one were to assume the opposite extreme that lower prots did not aect 9 Oh and Reis (212) report scal multipliers between 2% to 6% for transfers from rich to poor households in a standard incomplete markets model with one-asset only. 18

20 Figure 4: Impulse Responses of Output and its Components Percent of Transfer Net Output Y t Percent of Transfer Consumption C t Total Consumption Household Consumption Entrepreneur Consumption Quarter Quarter 4 Investment I t.5 Seignorage S t Percent of Transfer Percent of Transfer Quarter Quarter Notes: Seignorage: S t = M t Mt 1 π t ; Consumption by entrepreneurs: Yt G (1 MC t ) Impulse responses to an unexpected one-o payment of $5 to all households by the government. This stimulus is nanced by the central bank printing the required money. 19

21 households' consumption at all, the output multiplier would still be below one. Neglecting prots, our model suggests that output would increase by 48% of the scal stimulus on impact. 5.3 Welfare Eects The opposite movement of asset prices after the scal stimulus shock strongly aects consumption, as discussed in Section 5.1, and this has direct consequences for welfare. To quantify and understand the eect of scal stimulus payments on welfare, we trace agents over the S periods after the scal stimulus shock, and track their period-felicity u it +t to calculate for each agent with individual state (m, k, h) in period T the discounted expected felicity stream over the next S periods as: [ S ] v S (m, k, h) = E β t u T +t (m T, k T, h T ) = (m, k, h), t= where u T +t is the felicity stream in period T + t under the household's optimal saving policy. For large S, v S approximates the actual household's value function. We then determine an equivalent consumption tax that households would be willing to face over the next S quarters to eliminate the scal stimulus shock in time T as: ( v shock S CE = v no shock S ) 1/ξ + 1. (21) Figure 5 displays the consumption equivalents, CE, that households would demand or be willing to pay for not to receive stimulus payments by the governments in t = at dierent points in time. Who wins and who looses from such a policy dramatically changes over time. After the rst year capital owners mainly prot from the stimulus shock because of higher capital prices in particular wealthy hand-to-mouth households for the extra cash also alleviated their consumption constraint. This picture changes as the prolonged boom in investment depresses capital dividends, which is an important source of income for households rich in capital. They suer welfare losses from year 5 onwards. A higher capital stock again increases the wage rate, which is the only income for households with zero assets. Hence, poor households gain the most from the stimulus payment in the long run as they prot from higher wages for an extended period. On average, households gain around.2 CE after 25 years. 2

22 Figure 5: Welfare over Time After 1 Year After 5 Years CE CE Capital Holdings 1.5 Money Holdings Capital Holdings 1.5 Money Holdings 1.5 After 1 Years After 25 Years CE CE Capital Holdings 1.5 Money Holdings Capital Holdings 1.5 Money Holdings 1.5 Notes: Welfare costs in terms of consumption equivalents (CE) as dened in (21). The graphs refer to the conditional expectations of CE with respect to money and capital. Human capital has been integrated out. On the bottom of each graph, the Θ(m, k)-distribution is displayed. 21

23 6 Conclusion The Great Recession and the policy response to it have led to a renewed interest in the study of scal policy. This paper contributes to this literature in assessing the eects of scal stimulus payments on households and the macroeconomy. Following the insight by Kaplan and Violante (211), we consider an incomplete markets model with a liquid and an illiquid asset, where the latter pays a higher return but trading it is subject to a friction. In such a setup, wealthy hand-to-mouth households emerge. They hold sizable amounts of the illiquid asset, but choose to not hold any liquid assets constraining such housholds in their consumption in every period in which they do not have access to their illiquid funds. We show that endogenous asset returns strongly amplify the marginal propensity to consume out of extra cash in a model with money and illiquid physical capital. Fiscal stimulus payments lead to ination, which increases the relative return dierence between both assets. An investment boom follows and pushes up the price of capital, increasing the consumption response of all capital owners including the wealthy hand-to-mouth households as they become richer. Money holdings, in contrast, decline in value, but the former eect dominates as households hold more than twenty times more capital than money. Taking these price eects into account increases the average marginal propensity to consume out of extra cash by 1/3. We then calibrate the model to the U.S. economy, where 2% of U.S. households can be classied as wealthy hand-to-mouth according to Kaplan and Violante (211), to examine the eects of a $5 scal stimulus payment as authorized by the U.S. congress in 21. Households consume on average 45% of this payment, but output is hardly aected as prots fall because of higher ination. We understate the eect on output, because consumption falls one-to-one with prots in our model for tractability reasons. If prots did not aect consumption at all, the multiplier on scal stimulus payments could be as large as.5 at the other extreme. References Basu, S. (1995). Intermediate goods and business cycles: Implications for productivity and welfare. American Economic Review, 85(3): Bayer, C., Lütticke, R., Pham-Dao, L., and Tjaden, V. (214). Precautionary savings, illiquid assets, and the aggregate consequences of shocks to household income risk. 22

24 Buiter, W. H. (27). Seigniorage. Technical report, National Bureau of Economic Research. Calvo, G. A. (1983). Staggered prices in a utility-maximizing framework. Journal of monetary Economics, 12(3): Carroll, C. (26). The method of endogenous gridpoints for solving dynamic stochastic optimization problems. Economics Letters, 91(3): Galí, J. (28). Monetary Policy, ination, and the Business Cycle: An introduction to the new Keynesian Framework. Princeton University Press. Hintermaier, T. and Koeniger, W. (21). The method of endogenous gridpoints with occasionally binding constraints among endogenous variables. Journal of Economic Dynamics and Control, 34(1): Ireland, P. N. (23). Endogenous money or sticky prices? Journal of Monetary Economics, 5(8): Johnson, D., Parker, J., and Nicholas Souleles, N. (26). Consumption and tax cuts: Evidence from the randomized income tax rebates of 21. American Economic Review, 96: Kaplan, G. and Violante, G. L. (211). A model of the consumption response to scal stimulus payments. NBER WP Krusell, P. and Smith, A. A. (1997). Income And Wealth Heterogeneity, Portfolio Choice, And Equilibrium Asset Returns. Macroeconomic Dynamics, 1(2): Krusell, P. and Smith, A. A. (1998). Income and Wealth Heterogeneity in the Macroeconomy. Journal of Political Economy, 16(5): Oh, H. and Reis, R. (212). Targeted transfers and the scal response to the great recession. Journal of Monetary Economics, 59:S5S64. Parker, J. A., Souleles, N. S., Johnson, D. S., and McClelland, R. (211). Consumer spending and the economic stimulus payments of 28. Technical report, National Bureau of Economic Research. Storesletten, K., Telmer, C. I., and Yaron, A. (24). Cyclical Dynamics in Idiosyncratic Labor Market Risk. Journal of Political Economy, 112(3):

25 Tauchen, G. (1986). Finite state markov-chain approximations to univariate and vector autoregressions. Economics Letters, 2(2):

26 A First Order Conditions A.1 Euler Equations Denote the optimal policies for consumption, for money holdings and capital as c i, m i, k, i {a, n} respectively. The rst order conditions for an inner solution in the (no-)adjustment case read k : u(c a) q m a : u(c a) m n : u(c n) [ =βe ν V a(m a, k ; z ) k [ =βe ν V a(m a, k ; z ) =βe + (1 ν) V n(m a, k ; z ] ) k ] + (1 ν) V n(m a, k ; z ) m m [ ν V a(m n, k; z ) + (1 ν) V n(m n, k; z ] ) m m Note the subtle dierence between (23) and (24), which lies in the dierent capital stocks k vs. k in the right-hand side expressions. Dierentiating the value functions with respect to k and m, we obtain V a (m, k; z) k V a (m, k; z) m V n (m, k; z) m V n (m, k; z) = r(z) u[c n(m, k; z)] k (22) (23) (24) = u[c a(m, k; z)] (q + r) (25) = u[c a(m, k; z)] π 1 (26) = u[c n(m, k; z)] π 1 (27) + βe [ ν V a[m n(m, k; z), k; z ] + (1 ν) V n [m n(m, k; z), k; z ] k k = r(z) u[c n(m, k; z)] + β(1 ν)e V n{[m n(m, k; z), k; z], k; z } k ] (28) + βνe u{c a[m n(m, k; z), k; z], k; z } (q(z ) + r(z )) Such that the marginal value of capital in non-adjustment is dened recursively. Now we can plug in the second set of equations into the rst set of equations and 25

27 obtain the following Euler equations (in slightly shortened notation) u[c [ a(m, k; z)] q(z) =βe ν u[c a(m a, k ; z )] [q(z ) + r(z )] + (1 ν) V n (m a, k ; z ] ) k u[c a(m, k; z)] u[c n(m, k, ; z)] [ =βeπ (z ) 1 ν u[c a(m a, k ; z )] + (1 ν) u[c n(m a, k ; z ] )] [ =βeπ (z ) 1 ν u[c a(m n, k; z )] + (1 ν) u[c n(m n, k; z ] )] (29) (3) (31) B Asset Distribution Table 2 summarizes the wealth distribution implied by our model. As with any incomplete markets model that does not resort to heterogeneity in preferences or extremely skewed processes for idiosyncratic productivity, we fail to match the skewness in wealth documented for the U.S. Whereas the fraction of wealth held by the richest quintile is about 8% in the U.S., the top quintile in our model only holds 47% of total wealth. The same discrepancy holds for the Gini coecient, where our model falls short by 4%.46 versus.79. Table 2: Asset Distribution Quintiles 1st 2nd 3rd 4th 5th Gini-Coe. Fraction of Total Wealth held in Money held in Capital These shortcomings are not of great importance for our transmission mechanism. Our calibration matches that 1/3 of households do not hold any liquid assets, as estimated by Kaplan and Violante (211). This share is important for the average marginal propensity to consume out of extra cash as their seminal paper explains. Moreoever, we also match aggregate capital holdings, which is important to get reasonable returns in general equilibrium. 26

28 C Equilibrium Forecasting Rules Tables 3 displays the equilibrium laws of motion for the Krusell-Smith equilibrium. The R 2 is above for both laws of motion and hence they approximate the aggregate behavior of the economy well. Table 3: Laws of Motion for Asset Prices β 1 x β 2 x β 3 x R 2 Ination π t Capital price q t