Sectoral Comovement After a Monetary Shock: A DSGE Approach
Teun Brinkman
Master Thesis - Reseach Master Economics and Econometrics March 30, 2010
Abstract
Empirical evidence shows that after a contractionary monetary shock, durable output declines much more than nondurable output. Furthermore, some durable categories (e.g.
housing) decline more than others (e.g. business structures). In this paper, I analyze explanations for these observed movements in two nested DSGE models with a nondurable sector and either one or two durable sectors. Assuming that durable prices are ‡exible, Barsky et al. (2007) showed in a standard sticky-price model that an adverse monetary shock leaves output unchanged and leads durable output to increase. When this model is extended with a second durable sector with sticky prices, a second problem arises: extreme negative comovement between durables. I look at the role of credit market frictions and sticky nominal wages in the two models. Both frictions solve the …rst problem since they cause a decline in output after a contractionary monetary shock. The high income elasticity of durable goods leads durable output to decline more than nondurable output. In a model with two durables, only when wages are very sticky we see all sectors positively comove. Sticky wages make output prices in all sectors sluggish, thus dampening the e¤ect of monetary policy on relative prices. In contrast, credit frictions are unable to account for a positive correlation between durables, but can help explain why speci…c sectors contract more than others.
Keywords: Sticky prices, durables, comovement, credit, sticky wages JEL: E31, E32, E52
This thesis is written during an internship at De Nederlandsche Bank. I gratefully acknowledge the comments by Pierre Lafourcade, Vincent Sterk and Joris de Wind from De Nederlandsche Bank and Ben Heijdra and Jan Jacobs from the University of Groningen.
1 Introduction
Monetary policy has di¤erent e¤ects on di¤erent sectors. Empirical research shows that after a contractionary monetary shock, output of durable goods such as cars, refrigerators and houses declines much more than nondurable output.1 Furthermore, and what is less discussed is that some durable categories (e.g. residential investment) decline more after a monetary shock than others (e.g. investment in structures).2 In this paper, I analyze credit market frictions and sticky nominal wages as explanations for these observed movements. I do this in two nested dynamic stochastic general equilibrium (DSGE) models with a nondurable sector and either one or two durable sectors.
The motivation for this study is that the standard approach to analyze the e¤ects of monetary policy, the New-Keynesian sticky-price model (see e.g. Walsh, 2003 and Galí, 2008), is unable to explain observed sectoral comovement. Barsky, Kimball and House (2003, 2007) extend this model with a durable sector and assume that durable prices are more ‡exible than nondurable prices.3 Completely in contrast with empirical evidence, they show that a contractionary mon- etary shock leads durable output to increase, while nondurable output decreases. In the case of completely ‡exible durable prices, total output remains largely unchanged. Carlstrom and Fuerst (2006) refer to this contrast as the ‘comovement puzzle’.4
A second comovement problem arises when the model by Barsky et al. (2007) with ‡exible durable prices is extended with a second durable goods sector with sticky prices.5 In this paper, I show that a monetary shock in this model causes extreme negative comovement between these two durable sectors. Speci…cally, after an unexpected rise in the interest rate, the ‡exibly priced durable output expands and the sticky priced durable output declines.
These two comovement problems show that standard monetary business cycle model based on sticky nondurable prices fails to explain sectoral comovement after a monetary shock.6 The explanation for these theoretical outcomes can be found in the demand for durable goods. Ser- vice ‡ows of long-lived durables are spread out over time. As Barsky et al. (2007) show, the value of a durable good is therefore largely independent to temporary disturbances such as a
1See e.g. Barsky, Kimball and House (2003), Erceg and Levin (2006) and Monacelli (2009).
2In a short empirical section I show results that suggest this.
3We do not know whether durable prices are more ‘‡exible’ than nondurable prices. What we do see is that the price of durables relative to other prices declines after a monetary shock, especially for residential investment.
See e.g. Barsky et al. (2003) and Bils and Klenow (2004). In a short empirical section I also show this.
4The problem is not new. Murphy, Schleifer and Vishny (1989) refer to positive co-movement as one of the stylized facts that real business-cycle models should try to explain.
5Barsky et al. (2007) show that it is irrelevant for their results whether durables enter the utility function or the production function. The reason is that all assets are owned by the same representative household.
6A comment on sticky-price models. In these models output goes down due to a decline in labor supply. This makes it hard to explain why unemployment goes up after a negative monetary shock. Walsh (2003, p.217) is also critical about the role of price rigidities in the business cycle: ‘(...) adjusting production is also costly and it is di¢ cult to see why shutting down an assembly line is less costly than reprinting price catalogs.’ Price adjustment might however be more costly for nondurable goods than for durable goods.
monetary shock. Consequently, households and …rms are almost indi¤erent towards the timing of new durable purchases. As a result, the price of durable goods plays a very important role in equilibrium. Most important, to a …rst-order approximation, the sole determinant of labor supply is the wage per unit durable (as opposed to per unit nondurable).7 In the model by Barsky et al. (2007), ‡exible durable prices and ‡exible wages imply that the wage per durable does not react to a monetary shock. Labor supply therefore remains constant, implying that a monetary shock has no e¤ect on output. In a model with two durables, a small change in their relative price leads durables to react strongly in opposite directions.
In this paper, I show that the reason for the comovement puzzle is that the sticky-price model by Barsky et al. (2007) with ‡exible durable prices does not contain frictions leading output to change after a monetary shock. Speci…cally, I examine two dimensions that are capable of producing such an output e¤ect: credit market frictions and sticky nominal wages. Because of the sensitivity of durable demand to wage income, these frictions a¤ect durables much more than nondurables relative to the benchmark.
Credit frictions have been analyzed in this context as collateral constraints by Carlstrom and Fuerst (2006), Monacelli (2009) and Sterk (2009). The credit market frictions I propose are di¤erent and can be seen as a liquidity constraint. With this credit friction, which is used among others in Christiano and Eichenbaum (1992), …rms and households need to borrow money at the beginning of the period to pay for respectively wages and new durables. A rise in the nominal interest rate increases the cost of labor and the cost of new durables. The wage per durable therefore declines, in turn leading employment and total output to contract. Following the drop in income, the sensitivity of durable demand to wage income implies that for large enough credit frictions, durable output falls more than nondurable output. These credit frictions do not explain why di¤erent durable sectors move in the same direction after a shock, as I explain in this paper.
As I also show, in contrast with wage stickiness, for some parameter calibrations, credit is able to explain why speci…c durable sectors contract after a monetary shock.
Sticky wages (as in Erceg, Henderson and Levin 2000) have been analyzed by Carlstrom and Fuerst (2006, 2009). Sluggish input prices cause an increase in real production costs after an adverse monetary shock. This leads output prices to decline less, thus making the nonmonetary good (leisure) more attractive and work (and output) less so. Again, the decline in wage income essentially results in a reduction in durable demand and thus in positive correlation between durable and nondurable output. For the model with two durable goods, sticky wages dampen the e¤ect of monetary policy on the relative price between durables. For an average wage adjustment of over 7 quarters, sticky wages help explain the second comovement problem.
The general account for the results in this paper is that, although the output price of durables might be perfectly ‡exible, the role of prices of important inputs such as credit or labor reduces the total ‡exibility of the price of durables.
7The nominal wage divided by the price of durables.
Types of both frictions have been analyzed by Carlstrom and Fuerst (2006, 2009) but they distract from the fundamental features by adding frictions such as investment adjustment costs and habit persistence. In this paper, I solely focus on credit frictions and sticky wages. Another novel contribution of this paper is the examination of a model with two durable goods. This paper is the …rst to deal with the problem of negative comovement between durables, and also the …rst to suggest solutions in the form of credit frictions and sticky wages. Other new features are the following. First, the treatment of this type of credit friction in the context of sectoral comovement. Second, in a short empirical section, this paper is the …rst to look at the pattern of output and relative prices of six di¤erent consumption and investment categories around Romer dates of unexpected monetary contractions (Romer and Romer 1989).
The research in this paper relates to recent literature on monetary business cycle models such as the seminal work of Christiano, Eichenbaum and Evans (2005). They assume however that both nondurable and durable goods, modelled as productive capital, have the same price.8 The credit market frictions analyzed here also relate to the literature on the ‘…nancial accelerator’
of monetary policy as in Bernanke, Gertler and Gilchrist (1999) and Iacoviello (2005) who emphasize the role of imperfect …nancial markets in monetary transmission.
Only the comovement problem between durable and nondurable output is analyzed before, mainly by Barsky et al. (2003, 2007). Barsky et al. (2003), Carlstrom and Fuerst (2006, 2009) and Monacelli (2009) proposed solutions. Barsky et al. (2003) suggested labor market immobility, with dissatisfying results.9 With high labor market immobility, durable output does not expand after a monetary contraction. However, it does not contract either, due to its still lower relative price. Carlstrom and Fuerst (2006) calculate that an unrealistic low value for the intratemporal substitution elasticity between durables and nondurables of 0:029 is needed for durable output to contract, a value very close to Leontief preferences. Carlstrom and Fuerst (2006, 2009) introduce wage stickiness in the model by Barsky et al. (2007) with the same results as shown in this paper. Carlstrom and Fuerst (2006) also analyze a type of credit constraint where durable consumption is restricted by wage income with satisfying results.
Monacelli (2009) considers collateral constraints in a model with heterogenous agents, as in Kiyotaki and Moore (1997) and Iacoviello (2005), as a solution to the comovement problem. The relatively impatient agents are constrained lenders and a monetary shock causes a tightening of their collateral requirement. As Sterk (2009) shows, …nancial market equilibrium also causes patient agents to reduce savings which they invest in the other main asset which are durable goods. This e¤ect causes durable output again to increase and makes output eventually go up after a contractionary monetary shock under the extreme case of perfectly ‡exible durable prices.
I proceed as follows. In the next section I present some stylized facts about sectoral comove- ment after a monetary shock. In the third section I describe the DSGE model which I use to explain these stylized facts. The fourth section prepares the model for the simulation analysis
8Carlstrom and Fuerst (2006) have refered to this assumption as ‘quite heroic’.
9They also suggested wage stickiness and collateral constraints without deriving any results.
by describing the loglinearized version of the nonlinear model and explaining the parameter cal- ibrations. Fifth, I simulate the DSGE models and show how credit and sticky wages enter the model by Barsky et al. (2007). I also show how sensitive these results are for di¤erent parameter calibrations. In the last section I conclude and give suggestions for future research.
2 Stylized facts
In this section, I present the stylized facts that are the central subject of analysis in the DSGE- model I present in the next section. How do di¤erent sectors comove after a monetary shock?
And what happens with the relative prices of these output categories? Empirical research has been done with respect to durable output as one category (Erceg and Levin, 2006) and in particular housing construction (Barsky et al., 2003 and Leamer, 2008). However, no research has been focusing on all main output and price categories in the US national accounts.10
In this section, I look at the pattern of output and relative prices of di¤erent sectors around the Romer dates of unexpected monetary contractions. I do this speci…cally for six consumption and investment categories: (1) durable consumption, (2) nondurable consumption, (3) services, (4) investment in structures, (5) investment in software and equipment and (6) residential invest- ment. The most nondurable categories are nondurable consumption and services. Relative prices are computed by dividing the price indices by the GDP price index. The series are quarterly data from 1947Q1 to 2008Q4 taken from the National Product and Income Accounts (NIPA) of the United States Bureau of Economic Analysis (BEA).
I calculate the average pattern of these di¤erent output and price series around unexpected monetary contractions between 1948Q3 and 2004Q4.11 These monetary events or shocks are identi…ed as Romer dates (Romer and Romer, 1989) supplemented by the Boschen-Mills index (Boschen and Mills, 1995, updated by Weise, 2008). This approach is used among others by Gertler and Gilchrist (1994) and Barsky et al. (2003). Both indices are narrative indicators of Federal Reserve policy. The Romer dates I use are 1955Q3; 1968Q4; 1974Q2; 1978Q3; 1979Q4 and 1988Q4.12 In Bernanke and Mihov (1998) it can be seen that the Romer dates are close to the point where the Boschen-Mill index declines to 2 which implies "aggresive anti-in‡ation"
policy. Using this index I add 1994Q4 and 2000Q2 as contractionary monetary events.
The standard empirical approach to look at the e¤ects of unexpected monetary shocks is the Vector Autoregression (VAR) model. Since this model can only take up a limited number of dependent variables I choose the alternative approach.13 There are three important disadvan-
1 0All data used in this paper is for the United States.
1 1This selection is made since I will look at 6 quarters before the shock until 16 quarters after the shock.
1 2Two additional notes. First, the original Romer dates are months. Second, the Romer date October 1947 is discarded since it ocurred before 1948Q3. For this reason I also did not look beyond 2004Q4 for unexpected monetary contractions.
1 3A ‡exible approach that does use a VAR model is used by e.g. Gertler and Gilchrist (1994). In this case a …xed baseline VAR (with e.g. real GDP, prices and the Federal Funds rate) is extended each time with an
tages of the approach employed here (see also Barsky et al. 2003). First, it is unclear whether the Romer dates are really exogenous events. Shapiro (1994) has shown that Romer dates are not really exogenous since they always occur when in‡ation is high and unemployment is low.14 Bernanke and Mihov (1998) show in a VAR-model that Romer dates are often too late to cap- ture the true moment of monetary tightening. In order to deal with this issue, I also look at the pattern of the di¤erent series before a monetary event. A second disadvantage is that the e¤ects of the shocks are not corrected for the magnitude of the shock, where in a VAR the e¤ects are always shown relative to a standardized shock to the exogenous part of the interest rate. With the approach I use, there is no obvious way to scale the shocks and the results.
A third disadvantage, which Barsky et al. (2003) note, is that some series do not recover after a monetary shock, showing large deviations from the linear trend. I argue that this might be due to structural breaks in the growth rate of some variables over the past 65 years. I partially circumvent this problem comparing the logarithm of the series to its dynamic trend which is identi…ed by the Hodrick-Prescott (HP) …lter. The resulting cyclical component of the series is mean reverting.
I proceed as follows. I take the log of the real output and relative price series and use the HP-…lter to extract the cyclical component.15 For the eight monetary event dates t, I take the cumulative quarterly growth of the cyclical component of variable x, which is xt+j xt. I take the following window around the event dates j = 6; :; 0; ::; 16. I summarize the eight patterns by taking the average and standard deviation of all series over the eight monetary events and compare them to the long-term trend.16
Figures 1 and 2 show the results. Concerning the validity of the results, we observe that especially for prices the standard deviation bounds are relatively wide. Furthermore, some output categories (especially durable goods and residential investment) are already declining before the monetary shocks, suggesting some endogeneity in the Romer dates. Despite the aforementioned di¢ culties, this appraoch is able to identify some results about sectoral comovement around an unexpected monetary shock.
First of all, the results con…rm that durable output declines more after a shock than non- durable output. Services and nondurable consumption decline later and much less than the other four categories after a Romer date. Second, within the durables category we see that the response is di¤erent for di¤erent durables. Residential investment seems to decline more and structures decline less than the other durables.17 Third, looking at the relative prices in …gure 2 we see that
additional dependent variable. This only partially solves the problem since each time an additional dependent variable is added the VAR needs to be completely re-speci…ed.
1 4Recent research by Bordo and Haubrich (2009) shows that all except one US recession (1945) coincided with tight monetary policy. This suggests that it is di¢ cult to claim that the Romer dates are exogenous.
1 5Since I use quarterly data I use a smoothing parameter equal to 1; 600.
1 6The long-term trend of the cyclical component of the HP-…ltered series is 0 for all j.
1 7Another interesting observation is that that structures and residential investment seem to have an opposite pattern around the monetary shock, hinting at some substitution between these two sectors across the business cycle.
Figure 1: The average deviation of output categories from trend around Romer dates (Note: The trend is the HP-…ltered trend with = 1; 600. Solid and dotted lines are respectively the average log deviation and the average log deviation plus/minus the standard deviation. The six original Romer dates are supplemented by two dates identi…ed by the Boschen-Mills index (see text).)
Figure 2: The average deviation of relative price categories from trend around Romer dates (Note: The relative price is the de‡ator divided by the GDP de‡ator. The trend is the HP-…ltered trend with = 1; 600. Solid and dotted lines are respectively the average log deviation and the average log deviation plus/minus the standard deviation. The six original Romer dates are supplemented by two dates identi…ed by the Boschen-Mills index (see text).)
of all these categories the relative price of residential investment declines most obvious. This con…rms the assumption that the relative price of durables versus nondurables declines after a monetary shock, which is an important force in the model by Barsky et al. (2007).
In the rest of this paper I try to …nd explanations for these stylized facts of sectoral comove- ment by means of a DSGE model. Why do durables decline more than nondurables after a monetary shock? And why do some durables decline more than other durables?
3 Model
In this section, I describe the complete DSGE model that will be used for the simulation analysis in section …ve. The complete model contains one nondurable sector and either one or two durable sectors. This model can be calibrated such that the second durable sector does not play a role.
In the next subsection, I present the baseline model without credit frictions and sticky wages.
This model is very similar to the model used in Barsky et al. (2007).18 In this subsection, I also present some analytical results. First, I describe why durable goods play such an important role in the equilibrium. Second, I give an analytical explanation for both comovement problems.
In the second and third subsection, the baseline model is extended with respectively credit market frictions and sticky nominal wages. Again, analytical results are presented suggesting whether these frictions solve the comovement problems. These results are not de…nitive, and to get a better understanding of the functionings of this model, the model will be calibrated and simulated in the next few sections.
Concerning notation, I follow some conventions. Real variables are denoted by small case letters, nominal variables are denoted by upper case letters. This also applies to variables denoted by Greek letters. Real variables are always the nominal value divided by the aggregate price level Pt at time t. This aggregate price level is de…ned as follows:
Pt P
jPj;tyj;t yt
(1)
Where Pj;t is the price of …nal goods in sector j and yt is aggregate output or real GDP de…ned as the sum of the di¤erent sectoral outputs yj;t:19
1 8The two main di¤erences between this model and Barsky et al. (2007) are the following. The …rst is that in Barsky et al. (2007) …rms can use capital from a …xed capital stock, so there are diminishing returns to labor input. In my model I abstract from this feature to keep the model simple. The result is that in my model output is more sensitive to changes in employment. The second main di¤erence is that I use an interest-rate rule for monetary policy where Barsky et al. (2007) use a money growth rule. By using an interest rate rule I can abstract from money. Carlstrom and Fuerst (2006) have shown that an interest rate rule does not change the outcome of the model by Barsky et al. (2007). Another disadvantage of money rules is shown in Galí (2008): interest rates declines after a monetary contraction.
1 9This de…nition is not the same as the chain-weighted de…nition of real output employed in the U.S. national accounts. However, if all sectors grow at the same rate, as is the case in this paper, the two de…nitions converge.
yt
X
j
yj;t (2)
The steady state of variable xt will be written as x, so without time-subscript t. A period t represents a quarter of a year throughout this paper. All agents in this model are in…nitely- lived. An …nal important note is that in the model by Barsky et al. (2007), prices are not very sensitive to output changes. For this reason, in the analytical results I present in this section, I treat prices as exogenous for quantities.20
3.1 Baseline
In this subsection I describe the baseline model without credit frictions and sticky wages. First, the households in this model economy are described. I pay attention to the important role of durable goods demand in the model. Second, I describe …nal good …rms, intermediate good
…rms and labor unions. In the baseline model labor unions have no market power and play no important role. The behavior of monopolistically competing labor unions is introduced in the last subsection where sticky wages are introduced. The model is closed by means of a monetary authority who sets the interest rate and market clearing on the …nal goods markets.
Households and …nal good producers are homogenous so I look at them from the perspective of respectively the representative household and representative …nal goods producer.
3.1.1 Households
Each time t the representative household buys three types of goods in the …nal goods markets:
nondurable goods ct, durables with ‡exible prices df;t and durables with sticky prices ds;t. Households supply homogenous labor nt to the labor unions at the wage Wth, where labor unions sell di¤erentiated labor l at a wage rate Wt(l) to intermediate good …rms.21 In this baseline model labor unions have no market power and wages are ‡exible so in equilibrium Wth Wt= Wt(l).22
The objective of the representative household is expected lifetime utility:
E0
X1 t=0
t log ct+ !flog df;t+ !slog ds;t
n1+t 1 +
!
(3)
In this utility function < 1 is the time discount factor. The parameter > 0 can be interpreted as the inverse intratemporal elasticity of labor supply. The parameters !f; !s; > 0
2 0Prices are not really exogenous for quantities in this model, since we deal with a general equilibrium model.
We will see below that a drop in output will a¤ect monetary policy, which in turn a¤ects price adjustment. Still, this is a reasonable approximation for the …rst period after the shock.
2 1By assuming this additional wage rate WthI am able to seperate the wage-setting decision from the household problem. The new agents responsible for wage-setting are the labor unions. Below I show the wage rate Wthcan be seen as the (nominal) marginal disutility of working. When households would set the wage rate, sticky wages would be modelled by households staggeredly adjusting their wage rate to this disutility of working.
2 2Buying di¤erent types of labor, intermediate good …rms pay the wage Wt as I show below.
are set such that the steady state values of durable output and hours worked are correctly calibrated.23
The household budget constraint is:
Rt 1Bt 1+ Wthnt+ ft + lt= Bt+ Pc;tct+ Ps;txs;t+ Pf;txf;t (4) Where Bt is the only …nancial asset held at the monetary authority which gets returned at the gross nominal interest rate Rt.24 Further, ft and ltare respectively total nominal …rm and labor union pro…ts which are lump-sum transfers received at the end the period.
The durable stock accumulation equation or stock-‡ow relation for durable goods j = f; s can be written as:
dj;t= xj;t+ (1 j) dj;t 1 (5)
Where xj;t is new durable investment of durable type j = f; s. The parameter j is the durable-speci…c depreciation rate.
Households maximize total lifetime utility in (3) with respect to fct; xs;t; xf;t; ds;t; df;t; Bt; ntg1t=0
subject to (4) with Lagrange multiplier Ptt and the equations in (5) with Lagrange multipliers
tqj;t for j = f; s. The …rst-order conditions for respectively ct; Btand nt are:25
1
ct = pc;t t (6)
t = Et Rt
t+1 t+1 (7)
nt = wth t (8)
Where t Pt
Pt 1 is the aggregate in‡ation rate. We see that the real marginal disutility of working nt
t is equal to the household wage wth. The Lagrange multiplier tcan be interpreted as the marginal utility of real wealth. The …rst-order conditions for xj;tand dj;tare for j = f; s:
pj;t = qj;t (9)
tqj;t = Et
!j
dj;t
+ (1 j) t+1qj;t+1 (10)
The Lagrange multiplier qj;t can be interpreted as the real asset price of durable good j, which can also be seen as the secondary market price. This variable is comparable to Tobin’s marginal q. We see below that credit frictions drive a wedge between the secondary and primary market price of durable goods by changing (9).
2 3When !s is set equal to zero after the optimum is determined, the model contains only one durable sector.
2 4I assume in this paper that the monetary authority directly interacts with households and …rms. This is the standard way to model …nancial markets (see Walsh 2003 and Galí 2008).
2 5Where I de…ne the real variables pc;t Pc;t
Pt and wht WPth
t .
Durable goods In this subsection I will show that durable goods play a central role in both comovement problems.
Solving (10) forward, the shadow value j;tof durable j = f; s can be written as:
j;t
X1 k=0
( (1 j))kEt
!j
dj;t+k
= tqj;t (11)
This shows that the current value of durable good j depends on the value of future durable stocks. For low values of j a temporary shock has therefore small e¤ects on this shadow value and it is therefore nearly constant in the stochastic equilibrium, so we can write j;t j. Also, households are indi¤erent to the timing of the purchase of new durables and small changes in relative prices cause enormous adjustment in new durable output.26
What is the e¤ect of the nearly constant shadow value on the other optimality conditions for households? First, by inserting (11) into (8) we get the labor supply equation:
nt
j
wth
qj;t (12)
We see here that labor supply almost only changes with respect to the household wage rate per durable good j. Also, nondurable prices have little e¤ect on labor supply. When the wage per durable does not change after a monetary shock, labor supply remains unchanged. I show below that when both wages and durable prices are ‡exible the wage per durable is a constant.
A second implication of this condition, which is not stressed when Barsky et al. (2007) describe the comovement problem, is that when the wage per durable does change, there is a large e¤ect on durable expenditure. The reason is that the shadow value of durables j;t is not completely constant but marginally adjusts to restore the optimum. As a result, durable expenditure need to adjust a lot.
Second, following (6) we see that nondurable output is very sensitive to the relative price between durables and nondurables:
1
jct pc;t
qj;t (13)
This shows that nondurable output is very sensitive to the relative price between durables and nondurables. Again, this condition also shows that durable output needs to change by much when this relative price changes. In section two we have seen that this relative price between durables and nondurables is likely to decline after a monetary shock. This empirical …nding suggests that the only reason why durable output can decline must be an income e¤ect, which has to be found in a change in the wage per durable in (8).
Third, combining (10) for j = f; s shows the comovement problem between durables. When a relative price change between di¤erent durables is given, we see that the shadow values of both durable goods must change to restore optimality:
2 6As already mentioned, I treat prices as exogenous with respect to quantities in this section.
qs;t
qf;t = s;t
f;t
(14) A change in the shadow value (11) implies a large change in the current durable stocks and even larger changes in durable expenditure. This condition shows that the intratemporal substitution elasticity between durable goods is very high. This optimality condition explains the extreme negative comovement between the two di¤erent durable goods categories.
Finally, we can insert condition (11) into the Euler condition for intertemporal optimality (7). In this case we get:
Et
qj;t+1
qj;t t+1 EtfRtg (15)
So the expected change in the nominal secondary market durable price of durables is propor- tional to the nominal interest rate. In other words, the real interest rate in durable goods is a constant. This problem is also noted in Barsky et al. (2003, 2007) and Sterk (2009). It implies that already one period after the monetary shock the durable price starts rising again. Since this research focuses mainly on the e¤ects of a monetary shock at impact I do not deal with this undesirable feature of the model.27
3.1.2 Firms
Goods production takes place in three sectors: a nondurable sector, a durable sector with ‡exible prices and a durable sector with sticky prices. Each sector has a seperate supply channel of …nal and intermediate good producers.
Final good producers The representative …nal good producer in sector j buys di¤erentiated goods i 2 [0; 1] from intermediate good …rms i 2 [0; 1] in sector j and sells homogenous goods to the households. Final good producers face perfect competition on output and input markets.
The representative …nal good producer in sector j minimizes total production cost with respect to all inputs fyt(i)g1i=0:
min
fyt(i)g1i=0
Z 1 0
Pj;t(i) yj;t(i) di
Where Pj;t(i) and yj;t(i) are respectively price and output of intermediate good …rm i in sector j. They minimize cost subject to a production technology which is a Dixit-Stiglitz aggre- gator:
2 7In practical terms the modelling choice is between a realistic pattern for durable in‡ation or for the nominal interest rate. In this model I choose for the nominal interest rate. A declining nominal interest rate is not able to explain the role of credit frictions in equilibrium. In the models by Barsky et al. (2007) and Sterk (2009) the e¤ect is the other way around: the nominal interest rate follows durable price in‡ation. The explanation for this can be found in the parameters of monetary policy rule: strong stabilization policy will lead to an immediate drop in the nominal interest rate.
yj;t= Z 1
0
yj;t(i)
"p 1
"p di
"p
"p 1
The parameter "p> 1 is a technology parameter which will coincide with the intratemporal elasticity for intermediate goods demand. This parameter is assumed to be equal across sectors.
Final good …rms have no market power so we have the zero-pro…t condition:
Pj;tyj;t= Z 1
0
Pj;t(i) yj;t(i) di:
In appendix A I show that cost minimization and this zero-pro…t condition lead to a set of demand equations for intermediate good i in sector j, and a …nal goods price index Pj;t:
yj;t(i) = Pj;t(i) Pj;t
"p
yj;t (16)
Pj;t = Z 1
0
Pj;t(i)1 "pdi
1 1 "p
(17)
Intermediate good producers In all three sectors j heterogeneous intermediate good pro- ducers are indexed as i 2 [0; 1]. They face monopolistic competition on the output market and perfect competition on the input market. Intermediate good producer i in sector j buys dif- ferentiated labor types l 2 [0; 1] from labor unions and sells intermediate good i to …nal good producers. Labor unions buy homogenous labor from households at wage Wth and sell it as di¤erentiated labor for wage Wt. In this baseline model I assume no market power for labor unions and I assume that wages are perfectly ‡exible, so in equilibrium Wt Wth.
Intermediate good …rm i in sector j minimizes total production cost subject to labor inputs fnj;t(i; l)g1l=0:
min
fnj;t(i;l)g1l=0
Z 1 0
Wt(l) nj;t(i; l) dl (18)
Where nj;t(i; l) is labor type l used by …rm i in sector j. Furthermore, Wt(l) is the wage rate for labor type l, which is based on the assumption that labor type l is mobile across …rms and sectors so there is only one wage rate for each labor type.28
The minimization problem is subject to a constant returns to scale production technology and a Dixit-Stiglitz labor input aggregator:
yj;t(i) = nj;t(i) (19)
nj;t(i) = Z 1
0
nj;t(i; l)"w"w1dl
"w
"w 1
2 8Which implies that Wj;t(i; l) = Wt(l).
The technology parameter "w > 1 coincides with the intratemporal elasticity of labor de- mand.29 In appendix B I show that minimization in two stages leads to a set of labor demand equations for labor type l and a wage index comparable to the price index in (17):
nt(l) = Wt(l) Wt
"w
nt (20)
Wt = Z 1
0
Wt(l)1 "wdl
1 1 "w
(21) Also shown in appendix B, we get a relation between the economy-wide real wage rate and real marginal costs 'j;t(i) for …rm i in sector j:
wt= 'j;t(i) (22)
This condition shows that marginal costs are equal across …rms and sectors, so 'j;t(i) = 't. Finally, given that all intermediate good …rms use the same technology combined with labor market clearing, we have one economy-wide production function (see also appendix B):
yt= nt (23)
At the output side, intermediate good …rms maximize pro…ts by staggered price adjustment leading to sticky prices. Firm i in sector j adjust its price Pj;t(i) every period with probability 1 j as in Calvo (1983). The adjustment price is ePj;t, which is identical across …rms in sector j. If a …rm does not reset its price with probability j, it sticks to the previous periods price Pj;t 1(i).30 At the end of the period intermediate good …rm pro…ts are distributed to the owners, the households.
Firm i in sector j maximizes pro…ts with respect to ePj;t subject to …nal good …rm demand in (16) and given the probability that this reset price will be lasting the coming periods. Real pro…ts are:
Et
X1 k=0
( j )k t+k
Pt+k Pj;t+kjt(i) yj;t+kjt(i) Pt+k't+kyj;t+kjt(i)
Where xj;t+kjt(i) is the value of variable x at t + k given that the last reset period was t, so Pj;t+kjt(i) = ePj;t. In appendix C I show that the optimal price will be:
Pej;t= "p
"p 1 EtP1
k=0( j )kPj;t+k"p Pt+k
t+kyj;t+kPt+k't+k EtP1
k=0( j )kPj;t+k"p Pt+k
t+kyj;t+k (24)
2 9In the baseline model I assume that labor unions have no market power. This sets "w! 1 so that "w"w1 =
"w
"w 1 = 1.
3 0Based on the parameter j, prices last for (1 j) 1 quarters on average in sector j. This is computed as follows:
(1 j) X1 k=1
k kj = (1 j) 1
Flexible durable prices and sectoral comovement What happens with ‡exible durable prices in this model? When durable prices are ‡exible, f = 0, and we can write (24) as:
Pef;t= "p
"p 1Pt't
This shows that all intermediate good …rms in sector f charge the same price. Following the price index in (17) the aggregate price in sector f is equal to ePf;t. When we combine these results with the labor demand in (22) we can write:
wt pf;t
="p 1
"p
This shows that with ‡exible durable prices the wage rate per durable is a constant. Com- bining this result with the labor supply in (12) and recalling ‡exible wages and no labor union market power wht = wt we have that:
nt j"p 1
"p
(25) This shows that with ‡exible wages and ‡exible durable prices labor supply is nearly con- stant in the stochastic equilibrium. Following the aggregate production function in (23) the model implies monetary neutrality: a monetary shock does not not a¤ect total output. Given the sensitivity of durable and nondurable output to their relative price in (13), durable and nondurable output negatively comove.
3.1.3 Monetary policy
The monetary authority sets the nominal interest rate according to a nonlinear Taylor rule to target in‡ation and output:
Rt
R = t yt
y
y
exp ("t) (26)
The policy parameters and y are the Taylor coe¢ cients for the response to deviations in aggregate in‡ation t and real output yt from their steady state or target values and y.31 The monetary policy shock "t is the only source of uncertainty in this model and is identically and independently distributed with expectation 0 and variance 2".32
3 1To simplify I do not allow for di¤erent monetary policy responses to sector-speci…c output and prices. Alter- natively, one could say that in this case the sector-speci…c weights are equal to the shares of the sectors in total output. This can be seen below in the loglinearized equations for real output and the aggregate price level.
3 2Standard in the literature (see Carlstrom and Fuerst 2006, Monacelli 2009), either an interest-rate smoothing process or a AR(1)-term in "t is applied. Since sticky nondurable prices account for the internal propagation in this model and a realistic nominal interest rate pattern I choose to keep the model simple and abstract from interest-rate smoothing or an AR(1) error term.
3.1.4 Market clearing on the …nal good markets
Market clearing on the …nal good markets implies equality between consumption and production:
yc;t = ct ys;t = xs;t
yf;t = xf;t
3.2 Credit market frictions
In the previous subsection I have shown why a monetary shock in the baseline model leads to negative comovement between durables and nondurables and between di¤erent durables. In this subsection I show how the model changes when credit market frictions for households and
…rms are introduced. In the model with credit frictions I assume that credit is necessary for the consumption of new durable goods and wage payments since agents have not enough funds at the beginning of the period. This gives the credit friction a role as a liquidity constraint. A similar friction is also used in Christiano and Eichenbaum (1992), Keen (2004) and Christiano et al. (2005).
Households need to borrow an amount Lj;t for durable j = f; s which cannot be less than a fraction j of the actual expenditure on this durable good:
Lj;t jPj;txj;t (27)
This constraint is only binding when the cost of borrowing is higher than the cost of reducing savings. If borrowing and reducing savings would be equally expensive, households would be indi¤erent between these two options. Borrowing would not be costly in that case since excess money borrowed can be stored in a savings account at a rate equal to the borrowing rate.
In this paper I choose to model this in the same way as e.g. Christiano and Eichenbaum (1992) by assuming that loans are repaid within-the-period, where savings are repaid over-the- period. With repayment within-the-period, taking out excess loans would be costly since interest is only paid for savings over over-the-period.33
When the constraint in (27) is binding, the total expenditure for durable good j = f; s that enters the household budget constraint is equal to:
jRtPj;txj;t+ 1 j Pj;txj;t
This changes the …rst-order condition (9) as follows:
3 3Another possibility to model this is to assume a di¤erent savings and borrowing rate. In this case some additional structure to the model is needed to drive a wedge between these two rates in the steady state. A possibility is a …nancial intermediary who incurs costs for lending money and collecting deposits. These costs then drive a wedge between the two interest rates. I did not choose this option to keep the model simple.
jRt+ 1 j pj;t= qj;t (28) So credit for durable goods drives a wedge between the primary and secondary market price.
The result is that demand for durable j depends on the nominal interest rate. With ‡exible durable prices and wages we can compute (25) in the same way but now the nominal interest rate enters the equation:
nt = j
jRt+ 1 j
"p 1
"p
Since labor supply is sensitive to the price of durable goods, the credit frictions implies that the nominal interest rate a¤ects labor supply. When j> 0 labor supply depends negatively on the nominal interest rate and a positive unexpected shock to the nominal interest rate causes output to decline. In the simulation analysis I show that a high enough value of the parameter
j can explain why durable output declines more than nondurable output after a contractionary monetary shock.
For …rms the setup and the e¤ects are quite similar. The amount Lw;tthey borrow is at least a fraction w of their total wage expenditureR1
0 Wt(l) nj;t(i; l) dl:
Lw;t w
Z 1 0
Wt(l) nj;t(i; l) dl
Again these loans are repaid within-the-period so the constraint is binding in equilibrium.
Total production cost is then:
wRt
Z 1 0
Wt(l) nj;t(i; l) dl + (1 w) Z 1
0
Wt(l) nj;t(i; l) dl
This results in the following marginal costs equation. Marginal cost depends here partially on the nominal interest rate:
( wRt+ (1 w)) Wt= Pt't (29)
Again we can compute labor supply (25) in the same way and again the nominal interest rate enters the equation:
nt = j
( wRt+ (1 w))
These results show that when there is a positive role for credit in equilibrium, ‡exible prices and wages are not a su¢ cient condition for monetary neutrality. As I show below in the simula- tion analysis the size of these credit frictions can also cause durable output to decline more than nondurable output.
Credit frictions also a¤ect comovement between di¤erent durable goods. With credit frictions in both durables (14) becomes:
