The Government Spending Multiplier in a Model with the Cost Channel

Transcription

1 The Government Spending Multiplier in a Model with the Cost Channel Salem Abo-Zaid 248 Holden Hall, Department of Economics, Texas Tech University, Lubbock, TX, Abstract This paper studies the government spending multiplier in the presence of the cost channel of the nominal interest rate. I find that the multiplier of normal times declines markedly when this channel is introduced. The drop in the multiplier is bigger for more flexible prices, bigger responses to inflation in the interest rate rule, more persistent government spending and lower intertemporal elasticity of substitution in the utility function. The rise in government spending leads to a rise in the nominal interest rate and, with the cost channel, to a rise in the marginal cost and inflation. In turn, this leads to a bigger rise in the nominal interest rate and the expected real interest rate than in the model that abstracts from the cost channel. Higher marginal costs also reduce the incentives to produce and, thus, weaken the response of output to government spending shocks. On the other hand, to the extent that firms borrowing costs rise at the zero-lower bound following government spending shocks, the cost channel makes the government spending multiplier larger than in a model that abstract from it. The cost channel strengthens the expected inflation channel, which leads to a bigger drop in the expected real interest rate and thus to a larger government spending multiplier. Therefore, by ignoring the cost channel, the spending multiplier is overestimated in normal times and underestimated in liquidity trap episodes. Keywords: Government Spending Multiplier; Cost Channel; Fiscal Policy; Monetary Policy; Zero-Lower Bound. JEL Classification: E52; E58; E62; E Introduction The main goal of this paper is to study the government spending multiplier in a model with the cost channel of the nominal interest rate. It is found that the cost channel matters a great deal for the size of the spending multiplier but the exact effects depend on the episode considered. In normal times, during which the nominal interest rate is not fixed, the cost channel reduces the multiplier considerably. Depending on the strength of this channel, the multiplier can drop from above one to zero or even a negative multiplier just because of the address: salem.abozaid@ttu.edu (Salem Abo-Zaid) August 14, 2015

2 mere introduction of the cost channel. This effect is stronger for less rigid prices, stronger response of the nominal interest rate to inflation, more persistent government spending and lower intertemporal elasticity of substitution in the utility function. However, when the nominal interest rate is fixed, as can happen because of the Zero-Lower Bound (ZLB) constraint, the cost channel makes the spending multiplier larger provided that the borrowing costs of firms rise in response to a government spending shock. For plausible calibration of the model, the liquidity-trap spending multiplier more than doubles in the presence of the cost channel comapred to its value in a model that does not account for this channel. This effect is stronger when prices are less rigid and the responses of the firms borrowing interest rates to government spending shocks are bigger. The intuition behind these results is as follows. When the nominal interest rate is free to adjust, the rise in government spending raises this interest rate and, as a result, the marginal costs of firms. In turn, this leads a a larger increse in inflation and inflation expectations than in a model that abstracts from this channel. However, since the nominal interest rate increases by more than one-for-one with inflation (and the expected infaltion rate), the rise in inflation triggers even a larger increase in the nominal interest rate. The result is a larger increase in the expected real interest rate, which implies a smaller rise in consumer spending. In turn, the multiplier will be smaller than in an otherwise model without the cost channel. In this respect, the cost channel weakens the expected inflation channel that is essential for a large government spending multiplier to materialize in this class of models. When the ZLB constraint is binding, the cost channel actually strengthens the expected inflation channel. A rise in the borrowing costs of firms raises the marginal cost, and thus inflation and expected inflation by more than in the absence of this channel. However, at the ZLB, the nominal interest rate that is relevant for consumer spending remains unchanged (and it is unaffected by the cost channel). Therefore, the decline in the expected real interest rate is larger than in the absence of this channel, which in turn implies a larger spending multiplier during this type episodes. Therefore, previous estimates of the multiplier that did not account for the cost channel underestimated the multiplier of stress times and overestimated the multiplier of normal times. This study extends an otherwise standard New Keynesian (NK) model by allowing for the cost channel to be operative. Following Christiano et al. (2005) and Ravenna and Walsh (2006), among others, I assume that the wage bill of firms is paid before production takes place, thus giving rise to borrowing by firms (which, in turn, leads to supply-side effects of the nominal interest rate on inflation as in the standard cost channel of monetary policy). In particular, the marginal cost of firms becomes a function of the nominal interest rate on borrowing. The basic setup assumes that one interest rate in the economy, but to address the change in firms borrowing interest rates at the ZLB, I assume that these rates differ from the policy interest rate. The deviation between these interest rates can possibly result from market power in the intermdiation sector, adjustment costs of loans and other costs associated with changes in the scale of operation of banks. See Elyasiani et al. (1995), Freixas and Rochet (1997) and Kopecky and VanHoose (2012), among others, for a discussion. 2

3 The paper adds to the recent literature on the government spending multiplier. Christiano et al. (2011) first show that the multiplier is modest (slightly above 1) when the nominal interest rate is governed by a standard Taylor rule. However, they show that the government-spending multiplier can be much larger than one if the nominal interest rate does not respond to an increase in government spending, as can happen when the nominal interest rate hits the zero lower bound. The rise in government spending raises output and inflation expectations. When the nominal interest rate is at the lower bound, that leads to a decline in the real interest rate and to a further rise in consumption, output and expected inflation, leading to a loop of rising inflation expectations and output. The multiplier, thus, can be large. Quantitatively, the benchmark calibration of their model with the ZLB delivers a multiplier of more than three times the size of the multiplier of normal times. One implication of this result is that the asymmetries between liquidity trap episodes and normal times can be very significant. Carlstrom et al. (2014) consider the effects of the duration of interest rate peg on the size of the spending multiplier within a New-Keynesian framework. They compare the results when the monetary-fiscal expansion lasts a certain number of periods ( deterministic duration ) with the results when the expansion is stochastic (with a mean that is equal to the fixed number of periods under the deterministic case). They find that the size of the stochastic multiplier is large and clearly larger than the deterministic multiplier. For example, when the interest rate is fixed for 4 periods (or the mean is 4), the stochastic multiplier is 66.9 and the deterministic multiplier is 2.3. Using a model similar to Carlstrom et al. (2014), but for normal times, Dupor and Li (2015) study the expected inflation channel and its implications for the government spending multiplier. They show that having a large multiplier requires a large response of expected inflation to government spending. However, according to their empirical evidence, that has not been the case in the U.S. over the periods and and, therefore, the expected inflation channel has no support in U.S. data or it has been too small to generate large fiscal multipliers. Furthermore, their analyses distinguish between passive monetary policy, which occurs when the central bank increases the nominal interest rate by less than one-for-one with the change in expected inflation and active monetary policy, when the opposite occurs. It is found that under a passive monetary policy the multiplier is above one and under the active policy, it is less than one. The size of the fiscal multiplier at the ZLB has recently been questioned by the empirical findings of Ramey and Zubairy (2015). The authors construct quarterly U.S. data for to test whether government spending multipliers differ according to the amount of slack in the economy and being near the ZLB. They find that the amount of slack in the economy does not affect the size of the multiplier. Further, the analysis does not point out to any statistically different multipliers between normal times and ZLB episodes, particulary when the full sample is used. Therefore, contrary to recent findings, government spending multipliers were not necessarily higher than average during the Great Recession. Auerbach and Gorodnichenko (2012), on the other hand, use U.S. for 1947:Q1-2008:Q4 and find large 3

4 differences in the size of spending multipliers between recessions and expansions. In particular, government spending is considerably more effective in recessions than in expansions, which aligns with the predictions of qunatitative models. Existing literature also considered other aspects that can be important for the size of the fiscal multiplier. Gali et al. (2007) extend a standard NK modelto account for rule-of-thumb consumers. They show that the exact agent s preferences matter for the size of the multiplier, including being larger than or less than unity. Monacelli and Perotti (2008) first document that variations in government purchases generate a rise in consumption, the real wage and the product wage, and a fall in the markup. They then build an otherwise standard business cycle model with price rigidity, in which preferences can be consistent with an arbitrarily small wealth effect on labor supply and show that the model can match the empirical evidence on the effects of government spending shocks well. This highlights again the role of preferences for the size of the multiplier. Zubairy (2014) studies the government spending multiplier in a stochastic general equilibrium model that features other distortionary taxes and finds that the impact government spending multiplier is In addition, private consumption responds positively to government spending shocks and the multipliers for labor and capital taxes on impact are 0.13 and 0.34, respectively. In light of the wide range of findings about the fiscal multiplier, Leeper et al. (2011) study the fiscal multipliers in five nested models: a simple real business cycle (RBC) model, the RBC model with real frictions, a standard new Keynesian model, the new Keynesian model with hand-to-mouth agents and an open economy version of the new Keynesian model. They use Bayesian prior predictive tools and conclude that model specifications and prior distributions can tightly limit the range of possible spending multipliers. Moreover, a passive monetary policy yields strong multipliers for output, consumption and investment. 1 Despite its empirical validity, the cost channel of the nominal interest rate has not been addressed in the context of the government spending multiplier. This paper aims for filling that gap in the literature. To this end, I first provide more updated evidence regarding the existence of a cost channel and then examine the fiscal multiplier in a computable model with this channel and compare it to a model that abstracts from it. I find that the cost channel became more significant since 1984, that the firms credit spread surged during the current ZLB episode and that firms s borrowing costs are responsive to government spending shocks, which supports the choice of a model that distinguishes between the policy interest rate and the borrowing rates of firms as a platform to study the implications of the cost channel for the size of the multiplier when the ZLB constraint is binding. The effects of the cost channel on the fiscal multiplier are robust to multiple specifications about agent s preferences, the degree of price rigidity, the responses to inflation in the interest-rate rule, the significance of the cost channel (in the empirically relevant range) and other parameters. In particular, the differences between the multipliers with and without 1 In frictionless real business cycle models this multiplier is typically less than one (see, e.g., Aiyagari et al. (1992), Aiyagari et al. (1993), Ramey and Shapiro (1998) and Burnside et al. (2004)). 4

5 the cost channel are bigger for less rigid prices. This happens because the coefficient of the nominal interest rate in the New-Keynesian Phillips Curve is bigger for lower price rigidity. When prices approach full stickiness, the cost channel does not affect the size of the fiscal multiplier as the marginal cost, through which the cost channel operates, becomes immaterial for inflation and the economy in general. The results of the paper are compared to those of Christiano et al. (2011), Carlstrom et al. (2014) and Dupor and Li (2015). I conduct the analyses for passive and active central banks as defined in the latter study. Interestingly, I find that the cost channel weakens the multiplier in the presence of active monetary policy but strengthens the multiplier of a passive monetary policy. The reason for these results are as follows. When the interest rate initially rises in response to government spending shocsk, the marginal cost rises, and so do inflation and the expected inflation rate. With a passive central bank, the nominal interest rate rises by less than one-for-one with the rise in the expected inflation rate, leading to a bigger drop in the expected real interest rate compared to the model with no cost channel. The result will be a higer spending multiplier. With an active monetary policy, the nominal interest rate rises by more than one-for-one with the rise in the expected inflation rate, which makes the rise in the expceted real interest rate with the cost channel lower than in an otherwise model without it, leading a lower multiplier. When the central bank is neutral, in the sense that the nominal interest rate rises by one-for-one with the expected inflation rate, the cost channel does not affect the size of the multiplier. The remainder of the paper is organized as follows. Section 2 outlines the model economy. Section 3 provides empirical evidence on the existence of the cost channel. Section 4 presents the government spending multiplier of normal times and the numerical results. Section 5 presents robustness analyses using alternative preferences and monetary policy rule. The spending multiplier of constant interest rate and evidence regarding the responsivness of firms s borrowing costs to government spending shocks are presented in 6. Finally, Section 7 concludes. 2. The Model Economy The economy is populated by a continuum of infinitely-lived households who derive utility from consumption and leisure. Households also hold cash and they are engaged in the intermediation sector by holding some of their assets in the form of interest-earning deposits. Intermediate-good firms operate in an environment of monopolistic competition. They hire labor as the only input to produce differentiated products, set prices in a staggered fashion and borrow from intermediaries to finance their wage bill as in a standard model with the working capital requirement. The products of these firms are sold to final-good firms (retailers) who package them into final goods using a one-to-one technology. The basic setup through which the cost channel is introduced follows Ravenna and Walsh (2006) and Christiano et al. (2005), and the reader may refer to these studies for more details. 5

6 2.1. Households The problem of the representative household is to choose consumption c t, labor supply n t, deposits D t and cash M t to maximize: E 0 t=0 β t ( [cγ t (1 n t ) 1 γ ] 1 σ 1 ) (1) 1 σ where β is the subjective discount factor, E 0 is the expectation operator, σ > 0 and γ (0, 1). The assumption of non-separable utility function follows Christiano et al. (2011), but I will also show the results with seperable preferences in what follows. At the beginning of the period, households make deposits at the financial intermediary and the remaining amount M t + W t n t D t, with W t being the nominal wage, is used for consumption. At the end of the period, households receive profits from the financial intermediary and intermediate-good firms as well as the principle and interest on their deposits at the intermediary. The corresponding constraints that households face and a full description of their problem is presented in Appendix A.1. The optimization by households leads to the following standard consumption Euler equation and the labor supply condition, respectively, expressed in log deviations from the steady state: [γ(1 σ) 1]ĉ t (1 γ)(1 σ) n 1 n n t = R t E t [ π t+1 [γ(1 σ) 1]ĉ t+1 (1 γ)(1 σ) n 1 n n t+1] (2) ĉ t + n 1 n n t = ŵ t (3) where x denotes the non-stochastic steady state and x t is the log deviation from this steady state for any variable x t, π t = P t /P t 1 is the gross inflation rate, w t is the real wage and R t is the gross nominal policy (or market) interest rate. As expected, these conditions hold regardless of the cost channel and they are similar to the corresponding conditions in Christiano et al. (2011). With logarithmic preferences (σ = 1), condition (2) restores the more familar form of the log-lineraized Euler condition The Production Sector As is standard in the literature, two types of firms operate in this sector: monopolistically competitive intermediate-good firms who produce differentiated products and perfectly competitive firms who transform intermediate goods into final goods using a constant return to scale technology Final-Good Firms Firms in this sector operate in a perfectly competitive environment. They purchase a continuum of intermediate goods from intermediate-good producers, indexed by j (0, 1), and assemble them into final goods using the following technology: 6

7 y t = ( 0 1 y ε 1 ε j,t with y j,t being the quantity of intermediate-good j that is purchased by a final-good firm and ε > 1 is the elasticity of substitution between two differentiated types of intermediate goods. Profit maximization by intermediate-good producers gives the following downward-sloping demand function for the product variety j: dj) ε ε 1 (4) y j,t = ( P ε j,t ) y t (5) P t where P t = ( 1 0 P j,t 1 ε dj) 1 1 ε minimization. is the Dixit-Stiglitz aggregate price level that results from cost Intermediate-Good Firms Firms in this sector are monopolistically competitive. Each firm j hires labor as the only input to produce a differentiated product using the following technology: y j,t = n j,t. (6) Firms in this sector borrow from intermediaries, at the beginning of each period at an interest rate of Rt L in order to pay their wage bills. This is in line with the working capital requirement as in Christiano et al. (2005), Tillmann (2008) and Ravenna and Walsh (2006), among others. Following these studies, I assume in this section that the intermediation sector is perfectly competitive and complete pass through from the policy interest rate to the loan rate. Therefore Rt L = R t. However, in what follows I also show the results when the loan rate is a markup over the market rate. Each period, a firm j has the probability 1 θ of changing its price as in Calvo (1983). With this characterization of the intermediate-good sector, the time t real profit function of firm j is given by: Π f j,t = y j,t R δ t w t n j,t (7) subject to (5)-(6). The parameter δ measures the strength of the cost channel. As is standard in the NK literature, optimization leads to the following forward-looking Philips Curve: π t = β E t π t+1 + κ mc t (8) and labor demand condition: with κ = (1 θ)(1 βθ) θ. Letting s t = wtnt y t mc t = ŵ t + δ R t. (9) be the labor share of output and using the log-linearized 7

8 version of the production function, condition (9) can be re-written as: mc t = ŝ t + δ R t, (10) which is the standard condition of the real marginal cost in models with the cost channel (see, for example, Ravenna and Walsh (2006) and Tillmann (2008)). The Philips Curve then reads: π t = β E t π t+1 + κ (ŝ t + δ R t ) (11) and, thus, the nominal interest rate directly affects the behavior of inflation Market Clearing and Monetary Policy In equilibrium, the resource constraint of the economy reads: y t = c t + g t (12) Denoting the steady-state values of government spending and output by g and y, respectively, log linearization of the resource constraint yields: ŷ t = (1 g)ĉ t + gĝ t (13) where g = g y is the steady-state government spending-output ratio. Government spending evolves according to the following AR(1) process: ĝ t = ρĝ t 1 + u t (14) with ρ being the AR(1) coefficient and u t N (0, σ 2 u). In addition, monetary policy is governed by the following interest-rate rule: In what follows, I will also consider other monetary policy rules. 3. The Cost Channel: Empirical Evidence R t = φ π π t + φ y ŷ t. (15) This section provides some evidence about the existence of the cost channel using pre-crisis U.S data (1960:Q1-2007:Q3). The sample ends in the third quarter of 2007 to avoid any biases in the results after the reach of the nominal interest rate the zero-lower bound. The estimation is then provided for the second half of the sample (1984:Q1-2007:Q3) to asses whether the cost channel has become weaker or stronger over time. I choose 1984 as the beginning of the second period because it is well known by now that many macroeconomic aggregates have experienced changes since that year. It also happens to be in the middle of the sample in hand. 8

9 In particular, I estimate condition (11). I use the 3 month treasury bill as a proxy for R t, the non-farm business labor share of output for s t and the rate of change in the GDP deflator for π t. Following Ravenna and Walsh (2006), the model is estimated using the Generalized Method of Moments with instrumental variables. All data have been detrended using the Hodrick and Prescott (HP) filter using a standard smoothing parameter of The results are reported in Table 1. Description 1960:Q1-2007:Q3 1984:Q1-2007:Q3 κ δ Price Duration κ Price Duration (0.015, 0.054) (0.005, 0.040) (0.991, 1.113) (1.818, 5.939) (4.567, 7.848) (5.148, ) (0.029, 0.070) (0.012, 0.049) (4.089, 5.869) (4.765, 8.431) Obs Table 1: Empirical results- estimation of condition (11). The list of instruments includes four lags of nonfarm business sector real unit labor cost, output gap, inflation rate (GDP deflator), commodity price index inflation, 10 year minus 3 month U.S. government bond spread, non-farm business sector hourly compensation inflation and 3-month T-bill rate. 95% confidence intervals in parentheses. The point estimates of κ for the full sample period are in line with previous findings and with the standard values of this parameter that are used in the literature on the government spending multiplier (e.g. Christiano et al. (2011), Carlstrom et al. (2014) and Dupor and Li (2015)). The cost channel is very strong and the value of δ is about unity, which is similar to the findings of Ravenna and Walsh (2006) and Chowdhury et al. (2006). Over the period 1984:Q1-2007:Q3, the cost channel became significantly stronger: δ exceeds 2 and the 95% confidence interval indicates that it can reach roughly 6. Given these findings, my initial estimates of the spending multiplier will consider values of δ between 1 and 6, with the focus being on the range 1-3 where the point estimates of this parameter lie. The botttom panel shows the estimation of the Phillips curve using the (non-farm business) unit labor cost (ULC), which is defined as total labor compensation divided by real output (wl/y). Given our linear technology, the ULC is essentially the real marginal cost. The results indicate a slightly higher κ than in the estimations with the labor share and the interest rate and they also show a reduction in the size of κ in the second sub period. The table also reports the corresponding price durations, which can be found by using the condition κ = (1 θ)(1 βθ) θ and the fact that the price duration is given by 1 1 θ. All cases indicate 9

10 that the price duration is above 4 quarter, mostly 6 quarters or more, and can reach as high as 12 quarters. These estimates are clearly higher than micro evidence about the duration of price contracts: Bils and Klenow (2004) find that the, even after excluding temporary price cuts (sales), half of prices last up to 5.5 months. Nakamura and Steinsson (2008) report a higher price duration: with the exclusion of product substitutions, the uncensored durations of regular prices of between 8-11 months. Allowing for price changes associated with product substitutions, the median duration until either the price changes or the product disappears is 7-9 months. Christiano et al. (2005) find a price duration of 2.5 quarters and Smets and Wouters (2007) suggest an average duration of prices of 3 quarters. All of these estimates lie within the 4 quarters range. Therefore, to be consistent with literature on government spending multipliers, my analyses will first use the values of κ as in Table 1. Then, I will also other values of κ that are consistent with the existing evidence about the average duration prices, particularly the micro evidence, which suggests average price durations between 2 and 4 quarters. 4. The Government Spending Multiplier The main analyses regarding the government spending multiplier are presented in this section. By using conditions (3), (8) and (9), the Philips Curve can then be written as: π t = β E t π t+1 + κ (ĉ t + n 1 n n t + δ R t ) (16) Following the literature, the model is solved using the method of undetermined coefficients. Under the assumption that φ π > 1, the solution yields: where: ŷ t = Aĝ t (17) A = g κ(φ π ρ) + (1 ρ)(1 βρ δκφ π )[γ(σ 1) + 1] (1 βρ δκφ π )[1 + (1 g)φ y ρ] + (1 g)(φ π ρ)κ [ 1 1 g + n 1 n + δφ y] The government spending multiplier is then given by: (18) or, alternatively, dy t dg t = A g. (19) dy t = g ĉ t. (20) dg t g ĝ t By setting δ = 0, we get the exact government spending multiplier as in Christiano et al. (2011). Clearly, the multiplier is affected by the presence of the cost channel, but the exact effect is not possible to tell without further information and it may depend on the size of 10

11 δ and other parameters. There is a clear case, however, when the role of the cost channel is easy to see: if prices approach full stickiness (θ 1), κ approaches zero, and the cost channel will not affect the size of the multiplier. This result is as expected since, according to condition (9), the interest rate does not affect the behavior of the marginal cost (hence, inflation), and the cost channel is immaterial. Indeed, as will be outlined below, the size of κ is very important for the size of the multiplier and for the effects of the cost channel on the multiplier. Condition (20) makes it clear that the size of the multiplier depends on the response of consumption to government spending disturbances. If consumption rises ( crowding in ), then the multiplier will be larger than one. If it falls ( crowding out ), the multiplier will less than one. And, if consumption does not respond to shocks to government spending, then the multiplier will be exactly Results Table 2 summarizes the parameter values that I use in the benchmark analyses. The parameters mostly follow Christiano et al. (2011), with the only exception being κ, which is set based on the empirical analyses that are presented above. These parameter values are widely accepted in the literature, but I will conduct comprehensive robustness analyses by changing the values of these and examine how the cost channel affects their effects on the spending multiplier. I will also compare my results to Christiano et al. (2011) using their value of κ = Parameter β σ ρ κ g φ π φ y n Value /3 Table 2: Values of the Parameters. Figure 1 shows the government spending multiplier for values of δ between 1 and 6 using the benchmark calibration of the model. Without the cost channel, the multiplier is slightly above one (as in Christiano et al. (2011)), but it decline rapidly as the significance of the cost channel rises. At roughly δ = 6, the multiplier becomes negative. This figure confirms the conjectures regarding the effects of the cost channel on the size of the multiplier and it suggests that not only this effect can be large, but it rises exponentially as the significance of this channel increases within the empirically plausible range. Within the range of δ = 1-3, the multiplier drops from 1.03 to 0.85, which is relatively muted. Notice, however, that this calculation is based on a value of κ = 0.835, which implies an average price duration of more than 6 quarters. As discussed above, this lies outside the range of average price durations that have been found in empirical work, particularly at the micro level. For this reason, I will experiment on the price rigidity parameter θ within the plausible range of average durations of price contracts and consider the implications of changing this parameter for the size of the government spending multiplier. 11

12 Multiplier Figure 1: The government spending multiplier for various values of δ. The key intiution behind the effects of the cost channel on the spending multiplier can be seen through the response of the expected real interest rate to government spending shocks. As shown in Appendix A.2, the expected real interest rate is give by: E t r t+1 = (φ π ρ)qĝ t (21) and, therefore, its response to a government spending shock depends on the response of inflation to this shock (captured by Q). In particular, since φ π > ρ, the expected real interest rate will rise by more than in the model without the cost channel as the response of inflation to spending shocks is bigger. Indeed, Figure B.1 shows that the response of the expected real interest rate to spending shocks is an increasing function of δ. With higher values of δ, the rise in the expected real interest rate is bigger and, therefore, the multiplier is lower. Furthermore, the shape of this graph is exactly the opposite of the multiplier (Figure 1), which further supports the intiution behind the results of this subsection. The rise in the actual real interest rate is also bigger when the cost channel is operative. The actual real interest will rise provided that φ π > 1, which is the standard assumption in this class of assumption. This results from the fact that the nominal interest rises by more than one-for-one the rise in the actual inflation rate. The cost channel makes Q larger and thus leads to a bigger actual real interest rate. The derivations are shown in Appendix A.2. When the nominal policy interest rate responds to output alongside inflation (φ y > 0), the same arguments hold. In this case, the expected real interest rate is given by: E t r t+1 = [(φ π ρ)q + φ y A] ĝ t (22) 12

13 which shows a larger increase in the expected real interest rate than in the model with response to inflation only. Therefore, the government spending multiplier in this case is lower than in the model with φ y = 0 even in the absence of the cost channel, which is in line with the findings of Christiano et al. (2011). Figure B.2 shows the numerical results with the cost channel, and they are very similar to the results of Figure 1. Since the effects of the cost channel are similar to the model with no response to output, I procced with the assumption that the nominal interest rate responds to inflation only Changing the Price Rigidity Parameter Figure 2a reports the results of varying δ for three values of the price rigidity parameter. 1 Defining the duration of prices as 1 θ, these values correspond to price durations of 2, 3 and 4 quarters, which are the most consistent with the empirical evidence. The impact of the cost channel in this case is clearly stronger than with higher degrees of price rigidity. With 4 quarters, the multiplier falls from 0.90 to 0.29 in this range of δ. With shorter price durations, the multiplier drops faster and it even becomes negative at values of δ between 2 and 3. Furthermore, the impact of the cost channel is magnified as the price duration drops and δ rises. This is well illustrated by the fact that the difference between the bottom and the middle lines is significantly bigger than the difference between the first and the second lines (even though the difference between the values of θ is smaller). (a) (b) 1 Multiplier =0.50 =0.67 = Multiplier =0 =1 =2 = Figure 2: The government spending multiplier, the cost channel and price rigidity. In Figure 2b, I show the results by varying the degree of price flexibility (1 θ) for the relevant values of δ. I use values of θ between 0.5 (a price duration of 2 quarters) and 1 13

14 (fully-rigid prices). The main insights can be summarized as follows. First, regardless of the cost channel, a decrease in price rigidity leads to a decline in the multiplier. For example, with no cost channel, the multiplier drops from 1.29 to In fact, even when prices have a high duration of 4 quarters (1 θ = 0.25), the multiplier is 0.90, which is significant drop from the multiplier of fully-rigid prices. Second, when prices approach full rigidity, the cost channel is immaterial, which is in line with out earlier discussions. Intiuitively, when κ approaches zero, the nominal interest rate does not affect the pricing behavior of firms. Third, the cost channel magnifies the drop in the multiplier as prices become more flexible. With a relatively small value of δ, the multiplier drops to nearly 0.50 at a price duration of 2 quarters and to 0.80 at a price duration of 4 quarters. With higher values of δ, the multiplier drops to either (roughly) zero or even a considerably negative value. With an intermediate value of δ, the multiplier is only 0.64 at a price duration of 4 quarters, which is half of its value with fully rigid prices Robustness In this subsection, I show that the effects of the cost channel on the size of the multiplier depends on other parameters as well. The results are summarized in Figure 3. Consider first the response of the nominal interest to inflation (φ π ). A bigger response of the interest rate to inflation leads to a higher rise in the nominal interest following a government spending shock, which lowers the size of the multiplier, with and without the cost channel. The cost channel makes this effect stronger, particularly as δ rises. The spending multiplier is lower for more persistent government expenditures, which results from a bigger negative wealth effect on consumption when government spending is more persistent as the present discounted value of taxes needed to finance these expenditures is bigger. This effect is stronger in the presence of the cost channel: the bigger negative wealth effect is coupled by a bigger substitution effect that results from a bigger rise in the interest rates, thus leading to a bigger drop in consumption and to a lower multiplier. The multiplier is increasing in both σ and γ. A higher γ increases the marginal utility of consumption and thus leads to a stronger response of consumption to government spending shocks. Given the non-separable preferences in consumption and labor, the the marginal utility of consumption is increasing in the amount of labor supplied. This effect is stronger for higher values of σ. The cost channel in this case weakens this effect: the desire to consume more is partially offset by the bigger rise in the interest rate in the presence of the cost channel. In fact, with δ = 4, the multiplier becomes virtually unaffected by the rise in γ, for δ = 5, the multiplier is decreasing in γ (but remains slightly positive), and with δ = 6, the multiplier drops markedly with the rise in γ and becomes negative even for γ = The same result holds for σ. Therefore, not only that the cost channel renders the multiplier lower, but it also flattens its increase in σ and γ, and even reverses the effects of these two parameters on the size of the multiplier. 14

15 Multiplier 0.8 Multiplier =0 =1 =2 = Multiplier Multiplier Figure 3: The government spending multiplier for various values the model s parameters. 5. Robustness Analyses: Model Specification In this section, I redo the analyses using a separable utility function as in Carlstrom et al. (2014) with the rest of the model specifics remaining unchanged. I then use this model to change the interest-rate rule. In particular, the second experiment considers an interest-rate rule where the central bank responds to the expected inflation rate and the strength of this response matters for the size of the multiplier. This modeling follows Dupor and Li (2015), and the analyses with the cost channel will allow good comparisons with these two recent studies. 15

16 5.1. Separable Utility Function The period utility function in this setup is given by: u(c t, n t ) = c1 σ t 1 σ χ n1+ν t 1 + ν which yields the following Euler and labor supply conditions: (23) R t E t π t+1 = σ(e t ĉ t+1 ĉ t ) (24) ν n t + σĉ t = ŵ t. (25) By using conditions (8), (9) and (25), the Phillips Curve can then be written as: π t = β E t π t+1 + κ (ν n t + σĉ t + δ R t ) (26) The solution for the spending multiplier, under the assumption φ π > 1, then yields: dy t dg t = κσ(φ π ρ) + σ(1 ρ)(1 βρ δκφ π ) κ [ν(1 g) + σ + δ(1 g)φ y ] (φ π ρ) + [σ(1 ρ) + (1 g)φ y ] (1 βρ δκφ π ) (27) Once more, the multiplier is affected by the cost channel, κ matters markedly, and the cost channel will not affect the size of the multiplier when κ = 0. Furthermore, the multiplier is clearly less than one, and that holds even when the cost channel is absent (δ = 0). With ρ y = 0 and ν = 0, which implies an infinite labor supply elasticity, the multiplier is exactly one, and the cost channel does not change this conclusion Multiplier Figure 4: The government spending multiplier for various values of δ. 16

17 I set β = 0.99, σ = 2, ν = 0.5, ρ = 0.8, κ = 0.034, φ π = 1.5, φ y = 0.0 and g = 0.2. The first four parameters are as in Carlstrom et al. (2014), κ follows the empirical findings and the rest follow Christiano et al. (2011). 2 As in the model with non-separable preferences, the spending multiplier is decreasing in the size of δ, and effects of the cost channel on the multiplier are exponentially rising in the size of this parameter. Overall, the shape of this graph is similar to that of Figure 1 and, therefore, the particular assumption about preferences is not behind the benchmark results of the paper. The analyses also reveal that the multiplier is very sensitive to changes in average price duration and that the cost channel becomes more significant for less rigid prices. For example, at δ = 3 and a price duration of 3 quarters, the multiplier is 0.27 as opposed to 0.87 in the model without the cost channel. For δ = 3 and a price duration of 2 quarters, the multiplier is compared to 0.85 in the model that abstracts from this channel An Alternative Interest Rate Rule: Active vs. Passive Monetary Policy The model is similar to the one from subsection, but with one modification. Dupor and Li (2015), the interest rate is governed by the following rule: Following R t = (1 + ψ)e t π t+1 (28) Monetary policy is active if the nominal interest rate rises by more than one-for-one following a rise in inflation expectations (which occurs when ψ > 0), passive if the interest rate rises by less than one-for-one (ψ < 0), and neutral if the interest rate rises by exactly one-for-one (ψ = 0). This specification directly addresses the expected inflation rate channel that has been frequently mentioned in the fiscal multiplier literature. The government spending multiplier is then given by: dy t dg t = κρψσ + σ(1 ρ)[1 βρ δκρ(1 + ψ)] κρψ[σ + ν(1 g)] + σ(1 ρ)[1 βρ δκρ(1 + ψ)] First notice that by setting δ = 0, we get the exact government spending multiplier as in Dupor and Li (2015). The spending multiplier has a value of 1 when the central bank is neutral (ψ = 0) and less that one for an active monetary policy (ψ > 0). The multiplier is larger than one if monetary policy is passive (ψ < 0) provided that the denominator is positive, which happens for most values of ψ that are between (-1) and zero. Clearly, ψ = 1 is the lower bound on this parameter, otherwise the nominal interest rate will decline following a rise in inflation expectations, which is not plausible. These observations hold whether or not the cost channel exists. The intuition behind these finding is as follows: an active central bank raises the interest rate by more than one-for-one in response to inflation, thus increasing the expected real interest rate and reducing consumer spending in response to a rise government spending. (29) 2 Carlstrom et al. (2014) analyze the multiplier at the ZLB and their multiplier does not include the last three parameters. For this reason, I use the values from the other study. 17

18 In particular (E t ĉ t+1 ĉ t ) declines. A passive central bank increases the expected interest rate by less than one-for-one, thus leading to a rise in consumer spending. And, a neutral central bank does not influence the expected real interest rate, thus leading to no change in consumer spending: the left-hand side of equation (24) does not change, and the right-hand side does not change either. The implications of the cost channel for the size of the multiplier can be seen in condition (29). With ψ < 0, both the numerator and denominator are larger than in the model without the cost channel. But since κρψσ < κρψ[σ + ν(1 g)], the numerator rises by more than the denominator when the cost channel is introduced, leading to a rise in the multiplier. With ψ < 0, the opposite happens, and the multiplier is lower. And, with ψ = 0, the cost channel is immaterial for the size of the multiplier. In this respect, this setup allows for a clearer illustration of the role of the cost channel in determining the size of the multiplier. (a) Passive (b) Active Multiplier Multiplier Figure 5: The government spending multiplier and the cost channel: active vs. passive monetary policy. The parameterization is based on Dupor and Li (2015) to allow for good comparisons with the latter study. I set β = 0.995, σ = 1, ν = 4, ρ = 0.75, κ = and g = 0.2, ψ = 0.5 for a passive central bank and ψ = 0.5 for an active central bank. The starting point of each graph replicates exactly the multipliers in Dupor and Li (2015): a multiplier of 2.25 with a passive central bank and a multiplier of 0.75 with an active central bank. As expected, for a passive central bank, the cost channel strengthens the impact of government spending shocks on consumption and increases the multiplier. The reason for this finding is the following: the cost channel increases the expected inflation rate beyond the corresponding increase in the expected inflation rate that would materialize when this channel is absent. But, since 18

19 the nominal interest rate rises by less one-for-one, the actual decline in the expected real interest rate is larger than when the cost channel is not operative, which in turn leads to a higher government spending multiplier. The opposite holds for the active central bank. The decline in the expected real interest rate with the cost channel is higher than otherwise, which makes the cost-channel multiplier lower. Furthermore, the impact of the cost channel on the size of the multiplier is more significant for passive monetary policy. This observation may reflect the fact that the government spending multiplier with the active bank is already (relatively) low and the cost channel has a relatively less ability to lower even further. The result about the active central bank is in line with the benchmark analyses (section 4). In the benchmark analyses, the central bank increases the nominal interest rate by more than one-for-one with the increase in the inflation rate. In particular, φ π = 1.5, which implies that the response to the expected inflation rate is φπ ρ = > 1. In both cases, the decline in the expected real interest with the cost channel is larger than otherwise, and the spending multiplier is lower accordingly. The gaps between the multiplier with the cost and without the cost channel is larger for higher values of ψ for the active central bank and for more negative values of ψ for the passive central bank. This is particularly true for the passive central bank; for example, with ψ = 0.7, the multiplier with no cost channel is roughly 6, and the multiplier with δ = 3 is above 11. Furthermore, the effects are stronger for more flexible prices. These results can be provided upon request. More generally, the analyses of this section support the findings of the benchmark model after changes in the model s specification and the interest-rate rule. The cost channel has implications for the strength of the expected inflation channel, and accounting for it provides a more accurate estimation of the strength of this channel and its implications for the government spending multiplier. 6. The Constant Interest-Rate Government Spending Multiplier So far, the anlayses assumed that the nominal interest rate is free to adjust and it is not at the zero lower-bound. In what follows, I relax this assumption by assuming that the nominal (policy) interest rate is fixed. In particular, it is assumed that the ZLB constraint on the nominal policy interest rate binds. As discussed in the introduction, it is well known by now that this class of models predicts a large spending multiplier when conventional monetary policy becomes powerless. Another assumption that is relaxed in this section is that the loan interest rate and the ploicy interest rate are exactly the same. If they were equal, then at the ZLB the cost channel would not affect the size of the multiplier, and the results of this analysis would converge to those of Christiano et al. (2011). Furthermore, if the loan rate equals the policy rate, then the loan rate is zero as well, which goes against the evidence that I will present in this section. 19

20 The first subsection presents the the fiscal multiplier in the model with a constant policy interest rate. The second subsection shows evidence regarding the size of the corporate credit spread and the effects of government spending multiplier on the this spread. The third subsection then presents the numerical findings with robusntess analyses. I close by a short description of the results with the separable utility function Analytical Analyses To account for possible deviations between the loan and the policy interest rates, I assume that there is a spread (η t ) between these two rates as follows: R L,t = R t + η t (30) which, at a constant policy rate, can be rewritten in log deviations from the steady state as: R L,t = η R + η η t (31) and, since the interest rate that is relevant for consumer is R t, condition (2) remains unaltered. On the other hand, the Phillips curve is now given by: π t = β E t π t+1 + κ (ĉ t + n 1 n n t + δ R L,t ) (32) The response of the credit spread to changes in government spending is governed by: η t = Zĝ t (33) Since we have no endogenous state variable and ĝ outsides the ZLB state, all endogenous control variables jump on impact, but they revert back to their steady state values when the shock expires (i.e. when the nominal interest rate is no longer at the ZLB). For this reason, following the literature, we can write E t π t+1 = pπ t and E t ĉ t+1 = pc t, where p is the probability that the ZLB remains in effect next period, and thus the expected duration of the ZLB is T = 1 1 p. The solution then yields: g(1 βp)(1 p)[γ(σ 1) + 1] gpκ + (1 g)pκλz A = (1 βp)(1 p) pκ(1 g) [ 1 1 g + n 1 n ] (34) and the government spending multiplier will be giveb by dyt dg t = A g, with λ = η δ, which at R+η the ZLB is given by λ = η 1+ηδ. With δ = 0, which implies λ = 0, or when the borrowing costs of firms do not respond to government spending shocks (Z = 0), the constant-interest rate government spending multiplier restores that of Christiano et al. (2011). The cost channel alters the size of the multiplier, but the exact effect depends on the response of the spread to government spending shock and the strength of the cost channel. Crucially, it also depends on the size of the denominator, which will be refered to as. 20