Persistent Government Spending and Fiscal Multipliers: the Investment-Channel Martial Dupaigne TSE Université Paul Valéry Montpellier, GREMAQ and IDEI Patrick Fève TSE University of Toulouse I-Capitole, GREMAQ and IDEI December 4, 204 Preliminary Version Abstract This paper investigates the size of government spending multipliers in various smallscale DSGE setups with endogenous labor supply and capital accumulation. Contrary to models without capital, the response of investment to government spending shocks strongly affects short-run multipliers on output and consumption. We analytically characterize the short-run investment multiplier, which in equilibrium can be either positive or negative, and show that the investment multiplier increases with the persistence of the exogenous government spending process. The threshold level of persistence is given by the dynamic adjustment of consumption at equilibrium. We also connect the response of investment to the output and consumption multipliers. Keywords: Government Spending Multipliers, DSGE models, Capital Accumulation, Labor Supply, Market Imperfections. JEL classification: E32, E62. Introduction In the current painful fiscal consolidation programs in the Euro Area and following the stimulus packages facing the Great Recession for instance, the ARRA in the US, there Address: TSE, GREMAQ Université de Toulouse I Capitole, Manufacture des Tabacs, bât. F, 2 allée de Brienne, 3000 Toulouse, France. email: patrick.feve@tse-eu.fr. We would like to thank Treb Allen, Levon Barseghayan, Ryan Chahrour, Kerem Cosar, Sebastian Di Tella, Franck Portier, Jean-Guillaume Sahuc and Edouard Schaal for valuable remarks and suggestions. This paper has benefited from helpful discussions during presentations at various seminars and conferences. The traditional disclaimer applies.
has been a renewed deep academic and policy interest in studying the effects of government activity. Understanding the responses of the economy to government spending and transfers appears as the most important priority for both the future of the macroeconomic research agenda and the policy making see e.g., Poterba, 200, Reis, 200, or Rogoff, 200. A key quantity that has attracted considerable attention is the government spending multiplier, i.e. the response of GDP consecutive to a unit increase in government spending see Ramey 20b for a recent survey. However, no single figure is behind this concept, and a large uncertainty is surrounding its measurement. The value of the multiplier depends on many factors such as the econometric approach, the identification strategy, the underlying model, the nature and duration of the fiscal change, or the state of the economy see among others, Cogan et al., 200, Uhlig, 200, Christiano, et al., 20, Ramey, 20a, Auerbach and Gorodnichenko, 202, Coenen et al., 202, Fève et al. 203, or Erceg and Lindé, 204, leaving the decision maker in trouble. In this paper, we analytically study issues related to size of government spending multipliers output, consumption and investment in a Dynamic Stochastic General Equilibrium DSGE context. We consider various versions of a tractable business cycle model with physical capital accumulation, endogenous labor supply and stochastic government spending. This canonical model is sufficiently simple given its functional forms on utility and production functions to get analytical and insightful results. However, this model shares the key ingredients that can be found in the modern applied DSGE literature : the utility is separable between consumption and leisure, 2 a constant return-to-scale technology combines labor and capital inputs, 3 and the stochastic process of non productive government spending is exogenous and persistent. We stress an important channel of government spending multipliers through the interplays of capital accumulation and the persistence of the government spending shock. We show that when government spending is more persistent than the equilibrium adjustement speed of consumption, saving increases to sustain future consumption plans. So, private investment increases and then the output multiplier is magnified. Our analysis implies that there exists a threshold value for the persistence parameter such that the response of private investment is zero. 4 This very particular situation allows Abstracting from real and nominal frictions, our model displays the same core modeling assumptions than Smets and Wouters 2003, 2007 medium-scale DSGE models for the Euro Area and US. Moreover, these core assumptions are present in most of current DSGE models see Coenen et al., 202. 2 Consumption and leisure are deliberately maintained as normal goods. 3 We do not consider capital utilization, but it is easy to show that none of our results are altered by varying capital utilization. Results are available from the authors upon request. 4 As usual in the relevant literature, we specify an Auto-Regressive process of order one for government spending and then examine the effect of the autoregressive parameter on the short-run responses of real quantities to the government spending shock. More elaborated processes can be considered to account for instance for the typical shape of the ARRA see Uhlig, 200. Government spending can also include noisy news shocks to account for the anticipated part in the conduct of fiscal policy see Fève and Pientrunti, 2
to retrieve the value of the multiplier when physical capital is held constant. This popular analytical version of the multiplier has been extensively used in the literature see e.g. Hall, 2009, Woodford, 20, Christiano et al. 20, Fève et al. 203. In frictionless setups, constant capital multipliers only result from the intra-temporal allocations the marginal rate of substitution between consumption and leisure, the marginal productivity of labor and the aggregate resources constraint and thus ignores expectations about the timing of government policy. 5 In addition, the response of private consumption is negative and thus the output multiplier is less than one. This has reinforced the view that in frictionless real business cycle models this multiplier is typically less than one see Christiano et al, 20. Depending on the persistence of the government spending shock, we show that this common wisdom is not true when physical capital can adjust over time. If the persistence is low, we obtain that the short-run multiplier 6 is smaller that the one we would obtain with constant capital. Conversely, when the persistence is high for example, the autoregressive parameter close to one the short-run multiplier largely exceeds the one with constant capital, and it can take value above unity for some model s calibrations. 7 We also connect our results with the non-stochastic steady-state multiplier. This other concept of the multiplier allows for a total adjustment of physical capital. We show that this multiplier can be obtained as the limit case of the long run response of aggregate variables after a permanent shock to government spending. investment channel is totally taking into account. In this case, all of our previous results are reinforced, as the To complement our results, we also focus our analysis on two key parameters of DSGE modeling: the intertemporal elasticity of substitution in consumption and the Frisch elasticity of labor supply. The intertemporal elasticity of substitution in consumption only modifies the size of the constant capital multiplier, but does not alter expectations about the effect of the government spending. For example, if consumption displays a large degree of intertemporal complementarities, then the crowding out effect on private consumption is mitigated and the output multiplier with constant capital takes larger values exceeding unity. This effect is just reinforced when government policy displays a persistent time pattern. The elasticity of labor supply plays in two directions. First, when this elasticity is lower, the constant capital multiplier is smaller because the labor supply is less responsive after the negative income effect. Second, a smaller elasticity of labor supply reduces the adjustment speed of consumption for a given level of physical capital. This implies that 204. 5 This is not true a sticky price version in which expectations matters. See the discussion in Christiano et al. 20. 6 The short-run multiplier is obtained from the impact response of real quantities to an unexpected shock on government spending. 7 We do not concentrate our analysis on the size of the multiplier, for example focussing under which conditions the model yields an output multiplier greater than one. Our analysis essentially focusses on the main mechanisms that modify the size of the multiplier. 3
the threshold value of the autoregressive parameter on government spending must be higher to insure a positive response of saving. Due to the combination of these two factors, the output multiplier can be thus very small when labor supply is less elastic. Finally, we consider two types of market imperfections. These two model versions nests our original model by adding an additional single parameter, so it is simple to inspect the mechanisms at work. First, we consider external endogenous discounting, assuming that an increase in aggregate consumption leads more impatient agents see Schmitt-Grohé and Uribe, 2003, in a small open economy setup. Endogenous external discounting reinforces the investment channel and magnifies our previous results. As government spending crowds out private consumption, households become more patient and thus save more. In this economy, the threshold value on the persistence parameter is smaller, making the government spending policy more effective. It is worth noting that external endogenous discounting does not modify the constant capital multiplier. 8 Second, we consider imperfect financial markets under the form of hand-to-mouth consumers see Galí et al., 2007. The fraction of these households affects the multiplier in two ways. First, when this fraction increases, the output and consumption mechanically increases. Second, the share of these households modifies the discount factor of the aggregate economy, as we now combine static agents non-savers with forward looking agents savers. This creates a second amplification effect of government spending, making the presence of hand-to-mouth households a very relevant propagation mechanism. Our results extend the existing literature in the following directions. Aiyagari et al. 992 provides insightful intuitions why more persistent government shocks can lead to higher multiplier, but we make more progress in at least three directions. First, we determine analytically under which conditions private investment can increase after a positive shock to government spending our threshold value both depends on preferences and technology. Second, we show how other aggregate variables output, consumption are affected and we analytically decompose the short-run multipliers into a static component the constant capital multiplier and a term related to expectations about future government spending policy. Third, we extend your results to economies with externality and financial market imperfections. Baxter and King 993 and Leeper et al. 20 reports very useful quantitative findings about multipliers in calibrated DSGE models, but without any explicit characterizations of the main driving mechanisms. In Baxter and King 993 for example, investment can either increase or decrease depending on the persistence of the government shock. In this paper, we determine analytically the threshold value and characterize this value with respect to the deep model s parameters. Leeper et al. 20 show that the persistence of the government spending shock is essential for obtaining a large output mul- 8 This parameters does not alter intra-temporal allocations and only affects the dynamic adjustment of consumption. 4
tiplier. Our results show under which conditions a larger multiplier can be obtained. In addition, Leeper et al. 20 also find that the fraction of hand-to-mouth consumers matters a lot for multipliers. Again, we are able to disentangle the two key mechanisms at work intra-temporal and inter-temporal when considering that a fraction of households has no access to financial markets. The paper is organized as follows. In a first section, we consider a prototypical model with complete depreciation to highlight the key mechanisms at work and determine the constant capital multiplier. A second Section extends the previous concept to dynamic economies and show how the persistence of the government spending policy impacts the multiplier. In third section, we extend the model in two directions. We first allows for nonunit intertemporal elasticity of substitution in consumption and details how the multipliers are modified. Second, we study the effect of labor supply elasticity and show that this elasticity both altered intra and inter-temporal allocations after a shock on government spending. In a fourth section, we consider two types of market imperfections and inspect how they modify multipliers. A last section concludes. A Prototypical Model with Government Spending We first consider a business cycle model with physical capital accumulation, endogenous labor supply and exogenous non productive government spending, but with complete depreciation of capital and linear utility in leisure. Despite its abusive simplicity, this model contains the key ingredients that we want to push forward.. Setup The inter-temporal expected utility function of the representative household is given by E t i0 β i {{log c t+i + η n t+i } + Vg t+i } where η > 0, β 0, denotes the discount factor and E t is the expectation operator conditional on the information set available as of time t. Time endowment is normalized to unity, c t denotes period-t real consumption and n t represents the household s labor supply. We follow Hansen 985 and Rogerson 988 and assume that utility in leisure displays indivisibility, so that in our simple setup with perfect financial insurance, the utility is linear in leisure. 9 The function V is increasing and concave in g t. Government spending 9 This assumption simplifies a lot the computation of the solution because the real wage and the real interest rate depend only on real consumption. In terms of the size of the multiplier, this specification boosts the response of the economy to a government spending shock, through a negative wealth effect on labor supply. We investigate in section 3 the role of the elasticity of labor supply. 5
delivers utility in an additively separable fashion and does not affect optimal choices on consumption and leisure. Without any normative perspective, this additive term in utility allows government spending to be useful. The representative firm uses capital k t and labor n t to produce the homogeneous final good y t. The technology is represented by the following constant returns to scale Cobb Douglas production function 0 y t Ak θ t n θ t, 2 where A > 0 is a scale parameter and θ 0,. We consider the full depreciation case at this stage and will relax this assumption later on. The capital stock evolves according to k t+ x t 3 Finally, the final good can be either consumed, invested or devoted to unproductive government spending y t c t + x t + g t, 4 where g t denotes exogenous government spending. For a given level of output, an increase in government spending reduces the resources available for consumption and investment. Agents may respond to this change by changing their consumption, investment, and labor supply decisions. The dynamic equilibrium of this economy is summarized by the following equations k t+ y t c t g t 5 y t Ak θ t n θ t 6 ηc t θ y t 7 n [ t ] yt+ βθe t 8 c t Equations 5 and 6 define the law of motion of physical capital after substitution of the aggregate resource constraint 4 into 3 and the production function equation 2. Equation 7 equates at equilibrium, the marginal rate of substitution between consumption and leisure with the marginal product of labor. Equation 8 represents the Euler equation on consumption. Before analyzing fiscal multipliers in our dynamic stochastic economy, we present a k t+ restricted economy which provides an useful benchmark. 0 We can consider a more flexible specification of this production function, allowing for a non unitary elasticity of substitution between inputs. All our main results are left unaffected. Results are available from the authors upon request. c t+ 6
.2 Constant Capital Government Spending Multipliers We first consider a variant of our economy where the supply of physical capital is held fixed. Absent capital accumulation, allocations can be set statically, period by period, for successive values of government spending g t. Fiscal multipliers are evaluated from the repeated static version of our model and depend on steady state ratios. To make sure the different economies we study have similar scales, we set the fixed levels of investment and capital, as well as the share of government spending in output, equal to their steady-state values in the variable-capital economy. Definition Constant capital government spending multipliers refer to changes in aggregate variables consecutive to a unit increase in government spending expenditures g in an economy defined by intra temporal allocations only. The constant-capital output multiplier is denoted y. The constant-capital investment multiplier x is null by assumption. The ressource constraint y c + x + }{{ }}{{ } 0 ties the constant-capital consumption multiplier c to the constant-capital output multiplier, c y. An increase in government spending plays a negative income effect because households have access to less final goods, ceteris paribus. This static economy shows how much less they choose to consume and how much more they choose to work and produce. In that fixed capital economy, deep parameters move the constant capital multipliers on output and consumption in the same direction, although their signs are opposite. Proposition When capital is held constant in the economy described in section., the output multiplier equals y > 0 9 + θ βθ ḡ/ȳ θ where ḡ/ȳ denotes the steady state share of government spending. The consumption multiplier equals c < 0. 0 + θ βθ ḡ/ȳ θ 7
Proof: See Appendix A. Proposition shows that an increase in government spending always reduces private c consumption, < 0, assuming a strictly positive consumption-to-output ratio. This implies that the output multiplier y is smaller than one as in Hall, 999 and Woodford, 20, investment being held constant and possibly zero in fixed-capital economies. The output multiplier does tend to one when the capital share θ because in that limit case, output is linear in labor and capital is irrelevant. Note that the constant-capital output multiplier remains below one despite an infinitely y elastic labor supply, as implied by the linear disutility of labor. The value of in Proposition needs therefore to be interpreted as an upper bound, as will be shown in Section 3. Constant capital multipliers serve as a useful first step for two reasons. First, many positive models used to study fiscal stabilization policy do not model capital accumulation. Models with nominal rigidities display forward-looking inflation dynamics; capital accumulation adds a backward-looking dimension which imposes numerical solutions. Hence, fiscal multipliers in this model without capital accumulation share some features with existing multipliers in the literature. Second, these multipliers will show up as special cases of the economies with dynamic features we now study. 2 Dynamic Government Spending Multipliers We now analyze our simple model with capital accumulation and stochastic government spending. Households face a dynamic problem on top of the static consumption-leisure tradeoff seen in Section??. Increases in government spending still have a negative income effect which may lead households to consume less and work more. But households can now transfer resources from one period to the other through the physical asset and display richer saving or borrowing behaviors. To solve the intertemporal rational expectations equilibrium of this model, we need to specify how government spending evolves over time. The log of governùent spending g t in deviation from its deterministic steady state value ḡ follows a simple stochastic process log g t ρ log g t + ρ log ḡ + ε t The previous equation equation simply rewrites ĝ t ρĝ t + ε t The consumption-to-output ratio is equal to βθ ḡ/ȳ. A positive ratio at non-stochastic steady-state thus implies that ḡ/ȳ < βθ. 8
where ĝ t log g t log ḡ g t ḡ/ḡ, ρ and ε is a white noise shock to government spending with zero mean and variance equal to σ 2 ε. The log-linear approximation of this economy around its the non-stochastic steady state is simple enough to fully characterize the time series properties of aggregate variables. Proposition 2 In the economy with capital accumulation described in section., equilibrium consumption ĉ t in relative deviations from steady-state follows a first-order autoregressive process while equilibrium investment x t follows an autoregressive process of order two. Denoting L the lag operator, the stochastic process of consumption and investment write θs g βθ 2 θl ĉ t θ + θs c βθρ [ θ + ρl + θρl 2 ] x t s g ρ θ βθρ ε t 2 ε t. 3 The stochastic process of output is a linear combination of two autoregressive processes on order one and one autoregressive process of order two: ŷ t s c ĉ t + s x x t + s g ĝ t, where s c βθ ḡ/ȳ, s x βθ and s g ḡ/ȳ are the consumption to output ratio, the investment to output ratio and government spending to output ratio, respectively. Proof: See Appendix B. This analytic characterization delivers the impulse response function of each aggregate variable following a government spending shock ε 0 > 0. The main results of this paper, laid out in the next section propositions 3 and 4, characterize the investment channel in impact fiscal multipliers. 2. Impact Government Spending Multipliers The impact response of investment to a government spending shock has an ambiguous sign. Most of the existing literature on fiscal multiplier including Hall, 2009, Christiano at al. 20 or Woodford, 20 mentions crowding-out type effects on investment, in which the rise in real interest rate consecutive to an increase in government spending reduces investment. Implicitly, this effect describes the response of savings represented in blue in Figure, i.e. movements along the capital demand schedule. But policies which stimulate employment also raise the marginal product of capital, and therefore shift the demand for capital services in red, making the shift in equilibrium 9
Figure : The ambiguous response of capital to a government spending shock next period next period real interest rate real interest rate K s K s K d next period capital input K d next period capital input capital ambiguous. The capital demand effect is not specific to our setup. It is present as soon as factors are substitutable and employment increases. 2 The actual sign of the investment response is determined by a cutoff rule on the persistence of government spending, as illustrated in the following proposition: Proposition 3 In the economy with capital accumulation described in section., the impact response of investment to an increase in government spending, x 0, can have both signs. There exists a threshold value ρ < of the persistence of government spending such that when ρ ρ, investment does not react to a change in government spending; the impact investment multiplier x 0 is strictly positive for any ρ > ρ ; the impact investment multiplier x 0 is strictly negative for any ρ < ρ. Proof: Short run investment multipliers write x h sx x t+h s g ε t for h 0,, 2,.... The impact investment multiplier is obtained using the stochastic process described in the previous x 0 proposition: ρ. Since 0 < β < and 0 < θ <, this multiplier is an increasing function of ρ and has the sign of its numerator, ρ θ. Hence, ρ θ. ρ θ βθ The increase in government spending, which acts as a non-permanent drain on resources, has two opposite effects on investment. On the one hand, households want to smooth their consumption and eat part of the existing capital a crowding-out like effect. On the other hand, it stimulates employment and the marginal productivity of capital, increasing the demand for capital services. What matters for capital accumulation and investment is in fact the expectations of next period labor input. The more persistent the shock, the larger is that expectation. Capital accumulation is therefore desirable when government spending and employment are highly persistent, while households facing very temporary fiscal shocks 2 This effect would not be present in models with Leontieff technologies and under-utilization of capital. 0
exhibit negative savings. When the persistence parameter of government spending is equal to the threshold, ρ θ, the crowding-out and crowding-in effects exactly cancel out. In that case, capital accumulation will never be affected and fiscal multiplier are identical to those of the constant capital economy, as reported in Proposition. It is important to understand that the persistence of government spending shocks does not drive investment through a present value mechanism, but through dynamics and expectations. An arbitrarily large realization of an white noise government spending shock will drive investment down a lot; on the contrary, an arbitrarily small realization of a random walk government spending shock will increase investment by a tiny amount, even when the present value government spending increases more in the first scenario than in the second one. As in Proposition 3, a cutoff rule will remain valid in all the extensions we consider latter. While in other versions of the model, the threshold value will be a possibly complicated function of the underlying parameters, it is particularly simple in this economy with complete depreciation and linear disutility of labor: ρ is equal to θ, the capital share. For a given value of the persistence parameter ρ, the impact investment multiplier is a decreasing function of the elasticity of next period output with respect to today s investment, θ. This means that, when returns to capital are low, agents need to invest a lot today to relax future ressource constraints. We can also perform comparative statics with respect to the discount factor β, and see that impact investment and output multipliers both increase when agents value the future more, strengthening the discounted utility benefits of current investment. The next proposition will show why the response of investment is crucial to understand fiscal multipliers. Proposition 4 The impact government spending multiplier on output in the economy with capital accumulation y 0 combines the static output multiplier in the economy y and the impact investment multiplier x 0 : y 0 y + x 0. The same decomposition holds for the impact consumption multiplier c 0 c + x 0. Proof: From Proposition 2 and appendix B, the impact output multiplier, y 0 ŷ t s g ε t, equals y 0 βθ 2 y βθ θ + θ θ βθ ḡ/ȳ βθρ y + βθ ρ ρ θ. The impact βθ ρ c consumption multiplier, 0 sc ĉ t s g ε t equals c 0 θ θ βθ ḡ/ȳ βθ 2 c + θ θ βθ ḡ/ȳ βθρ. βθ θ βθ ρ
According to that decomposition, the response of investment may amplify or dampen the multiplier on output and consumption, with respect to the constant capital case. Such a decomposition is already present in Aiyagari, Christiano & Eichenbaum 992, and holds as long as investment and government spending are composed of the same good other wise, the partial derivative with respect to x t and g t would differ. While Proposition 4 characterizes the relative values of impact multipliers with and without capital, it does have an implication for the absolute value of the output multiplier in the general model. Remember that the constant-capital output multiplier is smaller than one. Therefore, a positive impact investment multiplier is a necessary condition for the impact output multiplier to exceed one A large part of the literature on fiscal multipliers focuses on constant-capital effects. Taking into account the investment channel offers an alternative potential amplification mechanism. The paper by Leeper et al. 20 consider a large-scale DSGE model a multi-sector open economy model with real and nominal frictions, non-optimizing agents and rich monetary and fiscal policy rules and explores which parameters matter for the fiscal multipliers. Their quantitative analysis pinpoints the persistence of government spending as the parameter with the highest predictive power, as our results show. They also find that investment decreases unless government spending are highly persistent, which is a consequence of our cutoff rule. 2.2 Long-run Government Spending Multipliers We turn to the long-run response of the economy. For that reason, we consider permanent shocks to government spending, i.e. when ρ. The long-run response of real quantities strikingly exemplifies how investment shapes fiscal multiplier, any adjustment in physical capital being completed in the long-run. It is important to notice that the asymptotic results which follow are robust to any form of rigidity that would disappear in steady state, including nominal contracts, habit persistence, adjustment costs, and so on. Consumption dynamics is the easiest to study. Proposition 2 has established that consumption follow a first-order autoregressive progress. Consumption therefore converges monotonically over time and returns from below to its initial steady state value. 3 When ρ, Proposition 2 implies that the growth rate of investment follows an autoregressive process of order one: θl L x t s g ε t. After a θ permanent 3 The invariance of steady-state consumption is a direct consequence of the infinite elasticity of labor supply assumed so far. While a permanent increase in government spending stimulates labor supply and equilibrium employment, it does not affect the steady-state real rate interest rate which is pinned down by the psychological discount factor β, nor the steady-state real wage rate which in turns depends on the capital share θ. With Hansen-Rogerson preferences, consumption is therefore not affected and the steady-state output multiplier increases with β and θ, which jointly determine the steady-state capitalistic intensity k ȳ. 2 βθ
government spending shock, investment raises gradually and eventually converges to its new steady state value. Finally, the response of output combines the jump in government spending and the monotonic increases in consumption and investment. Proposition 5 computes the asymptotic responses of output, consumption and investment. Proposition 5 In the economy with capital accumulation described in section., the asymptotic multipliers associated to permanent changes in government spending are given by c 0 x y βθ βθ > 0 βθ > The asymptotic, constant-capital and impact multipliers on output, y as follows: y y 0 > lim > y ρ., y The asymptotic, constant-capital and impact multipliers on investment, x x 0, rank as follows: Proof: y x sx s g lim h x t+h ε t + x βθ lim ρ y 0 θ θ+θs c x x 0 > lim > x ρ. and y 0, rank, x and sx sg s g βθ y, which implies βθ βθ c + x +. Regarding the ranking of output multipliers, Proposition 4 implies βθ 2 βθ investment multipliers, Proposition 3 gives x 0 while Proposition computes y θ θ+θs c <. For ρ θ is null by definition. x and ρ βθ When shocks to government spending are permanent, the long-run fiscal multiplier taking into account capital adjustments exceeds the fiscal multiplier of a similar economy where capital is held fixed. The economic intuition for this general result is straightforward: the permanent increase in government spending raises employment, and the marginal productivity of capital. This shift in the real rate of return stimulates capital accumulation. On the supply side, this additional capital amplifies the increase in output, while on the demand side steady-state investment is higher. Notice that this investment channel is present as soon as employment increases in response to the government spending shock. The steady-state output multiplier can in fact be very large when agents are very patient and/or returns to 3
capital very high. The asymptotic multiplier also exceeds the impact multiplier, which is itself larger than the constant-capital multiplier when government spending are very persistent, as implied by the cutoff rule. The impact multiplier does incorporate a response of investment, but does not account for any adjustment of the capital stock yet. In a sense, the fixed-capital multiplier and steady-state multiplier are two polar benchmark where either consumption or investment react to the change in government spending, but not simultaneously as they do in the general case. Remark that both benchmarks can be expressed as functions of great ratios whose counterparts are observable: y + θ + θ θ θ 4 y βθ x/ȳ 5 with θ the share of capital income in value added. 3 Extensions under Incomplete Depreciation In this section, we extend our analysis to an incomplete depreciation setup, where the capital stock evolves according to the law of motion k t+ δ k t + x t, where 0 < δ < is a constant depreciation rate. Using this law of motion on capital, we will also consider more general specifications of the utility function. We will show that two results established in the previous section are left unaffected. First, short-run fiscal multipliers combine a constant-capital effect and the response of investment. Second, the sign of the investment channel obeys a cutoff rule. From now on, we will generically denote P the vector of deep parameters the set of which will be environment specific, 4 with the exception of the persistence of government spending ρ. Using this notation, the cutoff rule and impact output multiplier write x 0 ρ C P U P ρ y 0 U P C P K P U P ρ 6 7 with C P, K P and U P three reduced form coefficients, functions of the parameter vector P. The constant-capital output multiplier, denoted K P, is defined as in the previous section see the definition. U P is the unstable root of the dynamical system and is always larger than one. The third reduced from coefficient, C P, which shows up as a 4 Including incomplete depreciation, this vector is given by P {β, δ, ḡ/ȳ, θ}. 4
threshold on ρ in the numerator, drives the expected dynamics of consumption when capital is held fixed. It appears in a log-linear version of the Euler equation embedding optimal labor choices, of the form E t ĉ t+ C P ĉ t X P k t+. In the environment studied in the previous section with complete depreciation for instance, P contains the value of β, θ and ḡ/ȳ. The reduced form coefficients respectively equal K P + θ x/ȳ ḡ/ȳ, C P θ < and U P >. In what follows, P also βθ θ contains the depreciation rate δ and parameters specific to each economy studied. Throughout the section, we will emphasize if a specific modification of the environment affects the constant-capital multiplier K P, the impact response of investment x 0 which can occur through a shift in the cutoff C P or through a change in the denominator due to U P, or both. 3. Non-unit Intertemporal Elasticity of Substitution in Consumption The desire to smooth consumption through saving or borrowing is one of the factors which shape investment in our economy. To investigate the role of consumption smoothing, we allow for a more general specification of utility with respect to consumption. The instantaneous utility rewrites uc t, n t c σ t σ + η n t + v g t 8 where σ 0, [ ], + 0 is the inverse of the intertemporal elasticity of substitution in consumption.this economy reduces to the benchmark studied in Sections and 2 when σ and δ. We note P σ the vector of relevant deep parameters excluding the persistence of government spending, i.e. P σ {β, δ, ḡ/ȳ, θ, σ}. Utility function 8 modifies two optimality conditions: the static consumption-leisure choice 7 and the Euler equation 8. These equations rewrite η θ y t n t c σ c σ t βe t [ δ + θ y t+ k t+ t 9 ] c σ t+ 20 Next proposition shows how the parameter non-unit intertemporal elasticity of substitution in consumption affects the fiscal multipliers. Proposition 6 With the utility function 8 and incomplete depreciation,. The constant capital government spending multipliers on output and consumption equal 0 < y + θ θσ sc + θσ[ θ βδθ β δ ȳ] K P σ < ḡ c K P σ < 0 5
The constant capital output multiplier is positive but smaller than unity while the constant capital consumption multiplier is negative. Both multipliers increase with the curvature of the utility function σ i.e. the multipliers are decreasing functions of the intertemporal elasticity of substitution. 2. The impact multipliers on investment, output and consumption are given by x 0 ρ CPσ UP σ ρ y 0 K P σ UPσ CPσ UP σ ρ > 0 c 0 [K P σ ] UPσ CPσ UP σ ρ < 0 θ + θ [ β δ] β with C P σ < and U P + θ θ [ β δ] σ βcp σ >. Notice that the threshold value of persistence C P σ is invariant to the intertemporal elasticity of substitution. 3. The long run government spending multipliers following a permanent shock equal y, s x c 0, x s x s x where s x βδθ β δ is the steady state share of investment. They are invariant to the intertemporal elasticity of substitution. The steady-state output multiplier always exceeds the constant-capital one. Proof: See Appendix C. This proposition shows that the impact output multiplier y 0 is a decreasing function of the intertemporal elasticity of substitution /σ. However, the output multiplier is only affected through the constant capital multiplier K P σ, while the investment multiplier is invariant to the consumption smoothing parameter σ which affects neither C P σ nor U P σ. The parameter σ pins down how the raise in aggregate savings is statically broken down between a decrease in consumption and an increase in output leisure reduction. In the constant capital economy, agents with large σ want their consumption profile to be extremely smooth and reduce their consumption less when they face higher government spending and future taxes in fact, c 0 0 when σ +. They produce the required resources through a stronger increase in labor supply, which yields larger constant-capital output multipliers. The results obtained with the utility specification 8 also hold in other setting with richer intertemporal consumption decisions. For instance, the minimal consumption model uc t, n t logc t c m + η n t + v g t 6
where c m 0 is a minimal level of consumption, can be interpreted as a proxy for habits in consumption decision without adding a new state variable that can complicate the derivation of the policy function. In the log-linear approximation of the model, we have σ c m / c, where c m / c is the steady-state share of minimal consumption. When interpreted as a proxy for habit, setting σ 4 is equivalent to an habit parameter equal to c m / c 0.75, a value commonly obtained when it comes to estimated versions of mediumscaled DSGE models see e.g. Smets and Wouters, 2007. In this case, combined with very persistent government spending, the model yields multiplier that exceed unity. Notice finally that the cutoff persistence of government spending, C P σ, decreases with the depreciation rate δ. A large depreciation rate means that capital can adjust quickly. The persistence of government spending above which capital accumulation becomes optimal is therefore lower. 3.2 Finite Elasticity of Labor Supply We have just investigated various levels of the intertemporal elasticity of substitution when utility is linear in leisure. We now consider finite elasticities of labor supply when utility is log in consumption. 5 The instantaneous utility rewrites uc t, n t log c t η + ϕ n+ϕ t + v g t 2 with ϕ 0 the inverse of the Frischean elasticity of labor supply and η > 0 a scale parameter. This representation of preferences allows to investigate the role and consequences of labor supply elasticities, because the parameter ϕ can takes values between zero infinite elasticity, as in Hansen, 985, and Rogerson, 988 and section and infinity inelastic labor supply. In the latter case, the wealth effect of government spending disappears. We denote P ϕ the vector {β, δ, ḡ/ȳ, ϕ}. Additive separability between consumption and leisure implies that the Euler equation is unchanged with respect to Section. after accounting for incomplete depreciation. The only optimality condition affected is the static consumption-leisure choice 7, which rewrites ηn ϕ t θ y t n t c t. 22 Solving the model gets more complicated because the dynamics of consumption is no longer autonomous. This is due to the fact that the real wage and the real interest rate do not depend only on consumption at equilibrium. The responses of aggregate variables to an unexpected shock on government spending, reported in Appendix D, point out the interplay 5 We can also combine non-unit intertemporal elasticity of substitution in consumption with finite elasticity of labor supply. To simplify the exposition, we prefer to consider each mechanism in isolation. Results are available from the authors upon request. 7
of government spending persistence and capital adjustment in short and long run output multipliers. Proposition 7 With a finite elasticity of labor supply as in 2 and incomplete depreciation,. The constant capital government spending multipliers on output and consumption equal 0 < y + θ+ϕ θ sc + θ+ϕ θ [ βδθ β δ ȳ] K P ϕ < ḡ c K P ϕ < 0 The constant capital output multiplier is positive but smaller than unity while the constant capital consumption multiplier is negative. Both multipliers decrease with the labor supply parameter ϕ i.e. the multipliers are increasing functions of the labor supply elasticity. 2. The impact multipliers on investment, output and consumption are given by with C P ϕ +βθ θ θ+ϕ ȳ k x 0 ρ CPϕ UP ϕ ρ y 0 K P ϕ UPϕ CPϕ UP ϕ ρ > 0 c 0 [K P ϕ ] UPϕ CPϕ UP ϕ ρ < 0 + θ θ+ϕ [ β δ] < and U P ϕ >. The complete expression of the unstable root of the system is given in Appendix D. Notice that the threshold value of persistence C P ϕ decreases with the elasticity of labor supply /ϕ. 3. The long run government spending multipliers following permanent shocks equal y s x + ϕs c, c ϕs c, s x + ϕs c x s x s x + ϕs c where s x βδθ is the steady state share of investment, s β δ c is the steady state share of consumption s c s x s g and s g ḡ/ȳ. The steady state output multiplier y is a decreasing function of ϕ. The steady state output multiplier always exceeds the constant-capital one and is greater than one if ϕ < sx s c. Proof: See Appendix D. The elasticity of labor supply shifts the constant capital multiplier, as did the intertemporal elasticity of substitution, because hours worked respond less when ϕ is large in the limit case ϕ +, labor supply is inelastic and K P ϕ 0. This labor supply parameter also affects the threshold persistence of government spending for which investment is 8
not affected by the government spending shock: C P ϕ increases with the labor supply elasticity parameter ϕ. 6 For a given persistence of government spending, higher labor supply elasticity stimulates employment, hence the marginal product of capital which itself boosts investment. Symmetrically, the less elastic is employment, the more persistence it takes for investment to increase after a government spending shock as in apparent in Figure 2. Note however that the long run multiplier exceeds unity as long as ϕ < sx s c automatically satisfied when the utility is linear in labor supply. a condition Figure 2: Threshold persistence value and convex disutility of labor 0 x 0 > 0 ρ 0 ρ ϕ ρ + 4 Market Imperfections In the environment we have studied so far, equilibrium allocations are optimal and there exists a tight link between the stable and unstable roots. We study in this section two variants of the benchmark model which embed market imperfections and break that link. 4. Endogenous External Discounting We consider a formulation of endogenous discounting where households do not internalize the fact that their discount factor depends on their own levels of consumption. 7 We assume that the discount factor depends on the average level of consumption per capita, c t, which individual households take as given: 8 β t+ β c t β t 6 The elasticity of labor supply also affects the unstable root of the system, U P ϕ. The expression of the unstable root is relatively simple in the case of linear utility in leisure, but not when the elasticity of labor supply is finite. As previously mentioned, when utility is linear in labor, both the equilibrium wage rate and the equilibrium rate of interest do not depend on the capital stock, making the dynamics of consumption autonomous. This technical reason explains that the stable root of the system is in that case equal to the threshold value C P ϕ. Since the product of the stable and unstable root always equal /β in frictionless setups, the unstable root is easy to compute. 7 See Schmitt-Grohé and Uribe 2003 for an application in a small open economy setup. 8 This specification has the attractive feature to simplify a lot the computation of the equilibrium as discounting is not internalized. Moreover, as shown in Schmitt-Grohé and Uribe 2003, the dynamic and quantitative implications of such a specification is very similar to an endogenous internal discounting, whereas much more simple for the analytical computation of the general equilibrium. 9
with β 0. As usual, we assume β c t / c t < 0, i.e. agents are more impatient when aggregate consumption increases. The foundations of this specification relies both on jealousy or catching up with the Joneses effect, as the individual household is more impatient and wants to consume more today when the aggregate or reference social group does. Here we denote ω the elasticity of the discount factor with respect to consumption. We also assume β c β, which implies that the long-run multiplier is not affected by this modification in discounting. This economy nests our benchmark economy when ω 0 and we label P ω {β, δ, ḡ/ȳ, ω} the relevant vector of structural parameters. Notice, that we consider again linear utility in leisure ϕ 0. The only condition modified is the Euler equation on consumption β c t E t δ + θ y t+. c t k t+ c t+ In equilibrium, individual and average per capita variables are identical, c t c t, and this equation rewrites βc t E t δ + θ y t+ c t k t+ c t+ As before, the short run multipliers are obtained by solving the log-linear approximations about the non-stochastic steady state of the FOCs and equilibrium conditions. Although endogenous discounting does not alter the multiplier with constant capital, this mechanism increases the short-run response of the economy to a government spending shock for a given persistence level. Proposition 8 With endogenous external discounting and incomplete depreciation,. The constant capital government spending multipliers on output and consumption are identical to the benchmark model, i.e. 0 < y + θ c K P ω < 0 θ sc + θ[ θ βδθ β δ ḡ ȳ] K P ω < The constant capital output multiplier is positive but smaller than unity while the constant capital consumption multiplier is negative. 2. The impact multipliers on investment, output and consumption are given by x 0 ρ CPω UP ω ρ y 0 K P ω UPω CPω UP ω ρ > 0 c 0 [K P ω ] UPω CPω UP ω ρ < 0 ω The threshold value of persistence C P ω < is a decreasing function + θ [ β δ] θ of ω, the elasticity of the discount factor with respect to aggregate consumption. The θ + θ unstable root U P ω [ β δ] > is invariant to ω. β 20
3. The long run government spending multipliers following permanent shocks are identical to the benchmark model, i.e. y, s x c 0, x s x s x where s x βδθ β δ is the steady state share of investment. The steady-state output multiplier always exceeds the constant-capital one. Proof: See Appendix E. The multiplier with constant capital is the same as in the benchmark model, K P + θ θ sc + θ[ θ βδθ β δ ḡ, because the discount factor does not modify the intra- ȳ] temporal allocation. The impact output multiplier is affected by endogenous discounting through the second term, the impact response of investment. Endogenous discounting modifies C P ω, the persistence of government spending for which investment is not affected by the government spending shock. Notice that we simply obtain C P ω ωc P. This change in dynamics is easy to understand. After a positive shocks on public spending, individual households reduce their consumption due to the negative wealth effect effect. In equilibrium, all households take the same decision, so average per capita consumption c t is reduced. This makes agents more patient since the discount factor is a decreasing function of the aggregate consumption. Households have an additional incentive to save and are more willing to increase their capital stock, which amplifies the investment channel. Proposition 8 shows that the threshold with endogenous discounting equals ω the threshold with constant discounting. In other words, for a whole range of persistence ρ, investment would increase when discounting is endogenous while it would decrease in our benchmark environment. The unstable root U P ω, contrarily to the stable root, is left unchanged. Therefore, the more elastic the discount factor to aggregate consumption, the larger is the impact response of investment. 4.2 Hand-to-Mouth Consumers The last environment we consider introduce a fraction of hand-to-mouth consumers. These non-savers, which have the same preferences as savers but no access to stores of value, consume each period their entire disposable income, which corresponds to the labor income net of taxation, plus some government transfers see Galí et al., 2007. For presentation clarity, we only consider the labor income into their budget constraint: c ns t w t n t 2