Business Cycle Implications of Capacity Constraints

under Demand Shocks

Florian Kuhn∗ Chacko George†‡

October 15, 2015

Abstract

When capacity constraints limit the production of heterogeneous firms, demand

shocks can endogenously generate a number of important business cycle regularities: recessions are deeper than booms, economic volatility is countercyclical, the

aggregate Solow residual is procyclical and the fiscal multiplier is countercyclical.

The model’s main mechanism is that the share of firms at their production limit is

strongly procyclical. A baseline calibration of a basic New Keynesian DSGE model

with capacity constraints delivers more than 25% of the empirically observed asymmetry in output, 18% of the additional cross-sectional dispersion in recessions and

around 25% of the additional aggregate volatility, and more than 50% of the fluctuations in the Solow residual. The model implies fluctuations in the fiscal multiplier

of around 0.12 between expansions and recessions.

JEL codes: E13, E22, E23, E32

Keywords: Capacity constraints, asymmetric business cycles, economic volatility, Solow residual, fiscal multiplier

∗Department of Economics, Binghamton University, PO Box 6000, Binghamton, NY 13902-6000, email:

floriankuhn@utexas.edu

†FDIC, Center for Financial Research, 550 17th St NW, Washington, DC 20429, email:

cgeorge@fdic.gov

‡We thank Matthias Kehrig, Olivier Coibion, Andy Glover and Saroj Bhattarai for helpful suggestions

and comments. Florian Kuhn is grateful to his advisors Matthias Kehrig and Olivier Coibion for their

guidance. Naturally any errors are our own. Opinions expressed in this paper are those of the authors

and not necessarily those of the FDIC.
1 Introduction

This paper studies how to reconcile within a simple framework four disparate business cycle

facts: the asymmetry of business cycle fluctuations, the countercyclicality of aggregate and

cross-sectional volatility, the acyclicality of utilization-adjusted total factor productivity, and

counter-cyclical fiscal multipliers. Together, these empirical findings characterize recessions as

times when output is especially low, volatility is high, and fiscal policy is particularly effective.

While previous work has considered mechanisms that can account for each fact in isolation,

these potential explanations are generally at odds with other facts. For example, one can appeal

to asymmetric business cycle shocks to explain the asymmetry in business cycles, but this would

not, by itself, account for the observed countercyclicality in the dispersion of cross-sectional firm

productivity. Rather than trying to combine all of the mechanisms that could potentially account

for each fact individually into an unwieldy model, we instead show that a single mechanism —

occasionally binding capacity constraints— can endogenously generate each of these business

cycle facts when introduced into an otherwise standard business cycle model.

In the model, firms choose their capital capacity before the realization of idiosyncratic and

aggregate demand shocks. After learning about these, they may vary their utilization of capital

in a way that is increasingly costly as the utilization rate increases. When the economy experiences positive shocks to the demand for firms’ products, they increase their capital utilization

and output. With capital predetermined, this endogenous choice of utilization gives rise to procyclical measured total factor productivity even when business cycles are driven by shocks other

than TFP. At the same time utilization-adjusted factor productivity may remain acyclical, as

documented by Basu et al. (2006).

The combination of predetermined capital and convex utilization costs yields an upper bound

on any individual firm’s production. Large, positive aggregate shocks, then, increase the number

of firms at their capacity constraint. This adds extra concavity to aggregate production as a

function of demand and helps explain the three remaining business cycle facts. First, booms

are “smaller” than downturns, in the sense that average deviations of output from trend are

smaller in absolute value when the economy is far above trend than far below trend. In the

calibrated model, capacity constraints generate around one quarter of the observed asymmetry

of U.S. business cycles.

Second, capacity constraints provide a channel through which fiscal multipliers can be countercyclical. Higher government spending that increases demand for firms products will have

larger effects when the economy is in a downturn than in an expansion. During downturns,

few firms are capacity constrained and they can therefore readily expand production. During a

boom, on the other hand, firms are already producing at their capacity constraint which reduces

the expansionary effects of fiscal policy. Quantitatively, while the extent of countercyclicality of

fiscal multipliers remains a point of contention empirically, the model here suggests a difference

of about 0.12 between the multiplier in recession and expansion, respectively.

Third, the upper limit to production reduces cross-sectional and aggregate volatility when

many firms have high capacity utilization. Idiosyncratic demand shocks generate a non-trivial

distribution in the measured productivity of firms. The share of firms at their capacity constraint

affects the variance of this distribution: since all constrained firms look very similar in terms of

their productivity, a higher share of constrained firms implies a lower variance in the distribution

of productivity. Recessions, during which few firms are capacity constrained, are then periods of

high cross-sectional productivity dispersion. Occasionally binding capacity constraints therefore

1
provide a previously unexplored channel through which cross-sectional productivity dispersion

can endogenously move in a countercyclical manner even in the absence of second-moment shocks.

Similarly, when the economy-wide utilization is already high, additional demand shocks do not

move aggregate output much. Firms at their constraint are in the flat part of their production

function and hence many of them do no respond to changes in demand. This explains why

aggregate volatility, as measured by the conditional growth rate of aggregate output, is higher

in recessions.

Understanding the properties of recessions matters in the assessment of their welfare costs.

For example, while symmetric fluctuations reduce welfare, this loss is more severe if fluctuations exhibit asymmetry and the cost of a downturn is concentrated in a short period of time.

Increased volatility in recessions can similarly reduce the welfare of risk-averse agents, and, as

recent literature has shown, can have adverse economic effects of its own. The question of how

economic fluctuations originate and are transmitted also has important implications for fiscal

policy because the efficacy of government spending in general depends heavily on the cause of

downturns. For example, the government multiplier is generally acyclical in standard models,

whereas in models of uncertainty shocks, government spending can actually be less effective in

recessions than in normal times.

The main contribution of this paper is to show that capacity constraints can explain several important features of the behavior of output under few additional assumptions. Second,

capacity constraints suggest a novel explanation as to why productivity dispersion among firms

is countercyclical. Third, while the traditional Keynesian literature has long emphasized idle

capacities as one likely source of high fiscal multipliers when aggregate demand is low, there

has been relatively little work on integrating this mechanism into modern DSGE models. This

paper provides such a model. Fourth, we document how much this model, in addition to being

qualitatively consistent, can contribute quantitatively to the explanation of the four business

cycle facts. Finally, we add some empirical evidence to previous work on output asymmetry and

find that large recessions on average deviate 30% more from trend output than large booms.

A number of papers study the effects of variable capacity utilization in general equilibrium

frameworks. Work by Fagnart et al. (1999), Gilchrist and Williams (2000), Alvarez-Lois (2006)  ́

and Hansen and Prescott (2005) investigates capacity constraints with heterogeneous firms. The

main difference to the present paper is that they consider shocks to aggregate TFP under puttyclay technology or irreversibilities, whereas we focus on fluctuations in aggregate demand under

standard Cobb-Douglas production in which capacity constraints arise endogenously rather than

as an assumption on production technology. The closest models are Fagnart et al. (1999) and

Alvarez-Lois (2006), who explicitly model the pricing decision of monopolistically competitive  ́

firms. Fagnart et al. (1999) focus on the amplification of TFP shocks under putty-clay technology and flexible prices, whereas Alvarez-Lois (2006) looks at the response of firm mark-ups  ́

when prices are set one period in advance as well as the internal propagation of the putty-clay

mechanism. Gilchrist and Williams (2000) emphasize the asymmetric effects on output following

large TFP shocks and the hump-shaped response that is generated through the effects of vintage capital. Hansen and Prescott (2005) generate asymmetries by including a choice along the

extensive margin of operating or idling plants.

A strand of papers considers variable capacity utilization in a representative-agent framework

(Greenwood et al. (1988), Cooley et al. (1995), Bils and Cho (1994), Christiano et al. (2005)).

In contrast, the environment with heterogeneous firms allows us to consider occasionally binding

capacity constraints, as well as price setting and demand shocks in the monopolistic competition

2
framework. This firm heterogeneity in turn is driving several of the results in our model, as we

show in section 5.

A recent paper that also looks at the interplay of cross-sectional and aggregate asymmetries is

Ilut et al. (2014), albeit under a different mechanism. They show that under ambiguity aversion

(or more generally any concave reaction of employment growth to expected profitability), news

shocks can tightly link countercyclical volatility at the micro and macro level. Their explanation

involving firms’ decision making offers a complementary alternative to the approach in this paper

focusing on firms’ production technology.

The paper is structured as follows: In the next section 2 we review the stylized facts established by recent literature. In section 3 we illustrate in a stylized example how capacity

constraints can generate these facts qualitatively. We embed this mechanism in a full DSGE

model in section 4, and discuss quantitative results in section 5. Section 6 concludes.

2 Four business cycle regularities

In the following we review the evidence for the four business cycle facts (asymmetry in output,

countercyclical profitability dispersion, strong dependence of the Solow residual’s cyclicality on

factor utilization, a countercyclical fiscal multiplier) that previous literature has found. Since

business cycles can be “asymmetric” in many ways, we discuss the specific type of asymmetry

we are interested in and then provide additional evidence from US output series.

Large deviations in output from trend are likely negative The question of whether

business cycles are asymmetric is fairly old. However, as noted by McKay and Reis (2008), it is

also too broad to answer — there are many different ways in which business cycle asymmetry

could theoretically manifest itself. As they emphasize, one should therefore be specific in exactly

which way one wants to assess asymmetries. Previous literature can be loosely grouped into four

ways to research this question: By looking for asymmetry in 1) output growth 2) output levels

3) employment growth 4) employment levels. It is worth recalling that asymmetry in levels and

growth rates need not be associated. As discussed for example in Sichel (1993), a time series

exhibits asymmetry in levels if, say, troughs are far below trend but peaks are relatively flat.

Asymmetry in growth rates would be characterized by, say, sudden drops and slow recoveries.

Correspondingly, these two types of asymmetry have been dubbed “deepness” and “steepness”,

respectively, in the literature.

Our reading of the literature is that there is no strong evidence for asymmetry in output

growth rates which most papers have focused on (e.g. DeLong and Summers (1986), Bai and Ng

(2005), McKay and Reis (2008)). As documented by Sichel (1993) and Kn ̈uppel (2014), there is

evidence for skewness in output levels. Employment tends to behave more skewed than output

over the cycle: Prior work has found asymmetry in both employment growth and in employment

levels (e.g. Ilut et al. (2014), McKay and Reis (2008)).

The focus of this paper is on the claim that large deviations of output from trend are more

likely to be negative than positive. This means we are interested in the behavior of output levels,

for which there is some evidence of asymmetry (Sichel (1993)).

In Table 1 we report a number of additional observations about the relative magnitude of

“strong” booms and recessions. Specifically, we use a detrended output series to construct three

measures of differences in large output deviations. For the first measure, we pick an integer

3
N and compare the N/2 largest (i.e. positive) deviations with the N/2 smallest (i.e. negative)

deviations by comparing their means. Here, if business cycles are asymmetric in levels, we would

expect the mean deviation in strong recessions to be larger than the mean deviation in strong

expansions. Second, in the next column we count how many of the N periods with the largest

absolute deviations from trend were positive versus negative. If output is asymmetric as defined

above, we would expect the number of periods with negative output deviations to be larger.

As a third measure we report the overall skewness of the series (using all periods), defined as

the sample estimate of E



(x − μ)

3/σ3



. This coefficient of skewness is a less direct measure of

only large output deviations, but all else equal we would expect the coefficient of skewness to be

negative.

We construct these measures for a range of specifications in which we vary the time-series

representing “output”, the length of the series, the trend filter, as well as the number N of extreme

periods considered. The baseline specification uses HP-filtered postwar data. HP filtering often

constitutes the weakest case in terms of differences between expansions and recessions since at

the edges of the sample this detrending method tends to attribute parts of the cyclical movement

into the trend. For almost all specifications in Table 1 we see that large deviations from trend

are more likely to be negative.1 On average across all specifications, recessions appear around

30% deeper than booms are high.

In section 5 we calibrate our model to an HP-1600-filtered quarterly US GDP series, corresponding to the quarterly baseline specification in Table 1. The model will yield trend deviations

of 3.24% in an expansion and −3.45% in a recession and thus covers a little more than a quarter

of the observed asymmetry under the baseline specification.

Cross sectional and aggregate volatility are countercyclical The second fact is

connected to a range of findings that associate recessions with increased microeconomic and

macroeconomic volatility. On the microeconomic level, recent literature has found strong evidence for countercyclicality of cross-sectional dispersion among firms in several measures. Eisfeldt

and Rampini (2006) show that capital productivity is more dispersed in recessions. Bloom (2009)

and Bloom et al. (2012) include empirical evidence associating times of low aggregate production to higher dispersion in sales growth, innovations to plant profitability, and sectoral output.

Directly related to levels of firm productivity, Kehrig (2015) finds that the distribution of plant

revenue productivity becomes wider in recessions; Bachmann and Bayer (2013) reach a similar

result for innovations to the Solow residual in a dataset of German firms.

Broadly, there have been two, not mutually exclusive, approaches to explain the negative correlation of profitability risk with output. One fruitful strand of literature starting with Bloom

(2009) investigates the effect of exogenous increases in aggregate, cross-sectional, or policy uncertainty on economic conditions. A different set of papers has considered the reverse direction of

causality, studying under which conditions a bad aggregate state can cause firm-level dispersion

to increase endogenously; examples include Bachmann and Sims (2012), Decker et al. (2015),

and Kuhn (2014).

On the macroeconomic level, the fact that many aggregates exhibit increased volatility in

recessions has been documented for many real and financial indicators of economic activity (see

1

In fact the only specification in which negative output deviations are not larger than positive deviations

is for annual GDP when we start the series in 1929 and use an HP filter which, at the beginning of the

sample, picks up the Great Depression as part of the trend.

4
Table 1: Strong recessions larger than strong expansions

Specification Mean pos vs neg # pos vs neg Skewness

Quarterly GDP

Baseline 2.73% vs −3.43% 16 vs 24 −0.46

N = 20 3.12% vs −4.33% 6 vs 14 −0.46

N = 80 2.28% vs −2.87% 40 vs 40 −0.46

Until 2007 2.71% vs −3.36% 18 vs 22 −0.46

Linear filter 7.99% vs −12.70% 6 vs 34 −0.81

Rotemberg filter 4.19% vs −5.68% 6 vs 34 −0.33

Rotemberg filter, N = 80 3.74% vs −5.14% 29 vs 51 −0.33

Annual GDP

Baseline 3.20% vs −4.40% 3 vs 7 −0.35

N = 6 3.37% vs −4.83% 0 vs 6 −0.35

N = 20 2.99% vs −3.55% 13 vs 7 −0.35

Until 2007 3.20% vs −4.41% 4 vs 6 −0.35

From 1929 16.69% vs −11.61% 6 vs 4 +1.00

Linear filter 7.29% vs −12.51% 2 vs 8 −0.88

Linear filter from 1929 20.50% vs −31.08% 3 vs 7 −0.91

Rotemberg filter 6.23% vs −13.50% 1 vs 9 −0.87

Rotemberg filter from 1929 16.15% vs −36.95% 1 vs 9 −1.22

Monthly industrial production

Baseline 4.52% vs −5.90% 50 vs 70 −0.65

N = 40 5.48% vs −7.57% 7 vs 33 −0.65

N = 240 3.71% vs −4.45% 124 vs 116 −0.65

Until 2007 4.39% vs −5.58% 56 vs 64 −0.65

From 1919 11.35% vs −13.59% 54 vs 66 −0.55

Linear filter 17.03% vs −22.69% 33 vs 87 −0.52

Rotemberg filter 7.47% vs −11.23% 46 vs 74 −0.62

Notes: “Mean pos vs neg”: Mean of the N/2 largest periods vs mean of the N/2 smallest

periods. “# pos vs neg”: Out of the N periods with largest absolute value, how many

were positive and how many were negative. “Skewness”: Coefficient of skewness defined as

E



(x − μ)

3/σ3



.

For all three series in the baseline, N corresponds to a little less than 1/6 of observations,

series were HP filtered and starting date is January 1949. “Quarterly GDP”: N = 40,

end date 2014:4, HP(1600)-filtered. “Annual GDP”: N = 10, end date 2013, HP(100)-

filtered. “Monthly industrial production”: N = 120, end date 2014/02, HP(10, 000)-filtered.

Alternative specifications differ from respective baseline only along listed dimensions.

5
Figure 1: GDP and TFP measures from Basu et al. (2006)

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

−6

−4

−2

0

2

4

6

Year

Growth Rate (Percent)

TFP (simple)

TFP (purified)

Real GDP

Notes: Annual series for growth rates of GDP (blue solid line), simple TFP as measured by the Solow

residual (red dash-dotted line), and purified TFP as constructed by Basu et al. (2006) (green dashed

line). Data from Basu et al. (2006). Correlation between output growth and simple TFP growth is 0.74,

correlation between output growth and purified TFP growth is 0.02.

for example Bloom (2014)’s survey article). In the context of our model, where we focus on the

variance of output growth, two recent results are Bloom et al. (2012) who find that recessions are

associated with a 23% higher standard deviation of output compared to the long-run average, and

Bachmann and Bayer (2013) who find a difference of around 35% between booms and recessions.

Looking at the periods with the largest trend-deviations we find a difference of around 40%

between expansions and recessions in the US data, as documented in table 4 in section 5.2.3.

Factor utilization makes TFP look more procyclical The simple Solow residual is

strongly procyclical, but much less so if corrected for factor utilization. For this stylized fact

we draw on Basu et al. (2006) who discuss ways to improve the measurement of aggregate

productivity. In particular, they construct a measure for aggregate technology that accounts

for potentially confounding influences of returns to scale, imperfect competition, aggregation

across sectors and (especially relevant here), utilization rates of factor inputs. Their uncorrected

productivity measure, the Solow residual, is strongly procyclical: Correlation between output

growth and simple TFP is 0.74. The corrected measure does not exhibit this strong association

with aggregate production, as the correlation of purified TFP with (contemporaneous) output

growth is 0.02. Figure 1 visualizes Basu et al. (2006)’s results.

Since the mechanism considered in this paper hinges strongly on the effect of adjustment

in factor input utilization, we recalculate the above correlation coefficients using data provided

by John Fernald2(see Fernald (2012)) which corrects only for intensity of capital and labor

utilization. This allows us to check if utilization is indeed responsible for the difference in

2Data available at www.frbsf.org/economic-research/economists/jfernald/quarterly tfp.xls

6
cyclicality between the simple and the purified productivity measure (or if instead the difference

stems mainly from the other ‘purifying’ steps taken by Basu et al. (2006)). Additionally, this

dataset spans 15 more years at the end of the sample and is at a quarterly frequency. Again,

simple TFP is strongly procyclical, with a correlation of 0.83, whereas utilization-corrected TFP

has a coefficient of −0.03.

Our takeaway from this finding is that not correcting for factor input utilization strongly

increases the relationship between measured aggregate productivity and output. While we do

not want to weigh in on the question of which type of shocks drive business cycles, we focus

on demand shocks in order to take the extreme stance of constant physical productivity. This

allows us to assess how much cyclicality in measured TFP can be generated even when the

model’s correlation of output with physical TFP is zero.

As suggested by Wen (2004) and Basu et al. (2006), demand shocks under variable capacity

utilization are a possible explanation of this fact. Alternatively, Bai et al. (2012) provide an

example of a search model in which demand shocks can show up as productivity shocks when

search effort is a variable margin.

The government spending multiplier is countercyclical The cause of asymmetries in

the business cycle in our model is directly relevant for the effectiveness of policy. Our contribution

about capacity constraints and business cycle asymmetries thus complements the literature on

cyclical fiscal multipliers. Empirically estimating the level and cyclicality of the government

multiplier is difficult because of severe endogeneity issues. Nevertheless, recent empirical work

on government multipliers has found significant cyclicality in fiscal multipliers, although the

exact size of fluctuations is not identified very precisely. On one end of the spectrum, Auerbach

and Gorodnichenko (2012b) estimate the fiscal multiplier in a regime-switching model and find

large swings over the cycle ranging from around 0 during a typical boom to around 1.5 during a

typical recession, albeit with large confidence intervals. Other papers identifying the multiplier

in structural VARs are Mittnik and Semmler (2012) and Bachmann and Sims (2012) who also

find significant cyclicality. Auerbach and Gorodnichenko (2012a), Ilzetzki et al. (2013) and

Corsetti et al. (2012) all find evidence for state-dependence of the fiscal multiplier in crosscountry comparisons. Nakamura and Steinsson (2014) use regional variation in the US to identify

a positive relationship between the local spending multiplier and the unemployment rate. Ramey

and Zubairy (2014) find that the estimated magnitude of multiplier fluctuations over the cycle

is sensitive to the exact specification of the employed empirical model.

Not too much is known about the particular transmission channel through which aggregate

conditions affect the multiplier. As Sims and Wolff (2015) point out, several papers model the

difference between government spending when interest rates are at the zero lower bound and

spending during normal times. Historically however, episodes at the zero lower bound have

been relatively rare; and the empirical estimates go beyond these times indicating that the fiscal

multiplier also fluctuates with the business cycle when interest rates are positive. Sims and Wolff

(2015) explicitly consider multiplier fluctuations over the business cycle in a medium-scale RBC

model. Their mechanism is based on households’ higher willingness to supply additional labor

in recessions. The model by Michaillat (2014) generates a labor multiplier, in which a search

friction causes overall employment to respond stronger to government hiring in recessions than

in booms.

Here, we focus on the effect of underutilized capacity which complements mechanisms in

7
these papers. Our calibrated model implies average fluctuations of the fiscal multiplier of around

0.12, with the fiscal multiplier increasing with the size of recessions.

3 A Simple Example

In order to illustrate the aggregate effects of capacity constraints in a framework of heterogeneous

firms in this section we outline a stylized example. Firms choose their capacity before their

random demand is realized. A given capacity is associated with an upper bound to production,

so that if a firm’s demand is greater than this bound, that firm will be constrained and produce

just at capacity.

Formally, there is a continuum of ex-ante identical firms indexed by i ∈ [0, 1]. Each firm

can rent capital (or “capacity”) ki at a real rental price of R at the beginning of the period. A

firm’s production yiis a function of utilized capital  ̃ki, which for simplicity is specified as linear.

Capital utilization is free here, however it is subject to the constraint that utilized capital is less

than capacity, that is, yi =  ̃ki s.t.  ̃ki ≤ ki. Finally, a firm faces random demand bi which is

distributed according to a cumulative distribution function F(b). The price for each firm’s good

is constant and normalized to 1.

A firm’s sales after realization of bi will then be yi = min {bi, ki}. The firm uses this fact

when deciding on the amount of capacity to rent in order to maximize expected profits. The

problem can be written as

max

k

−Rk +

Z k

0

b df(b) + [1 − F(k)] k.

The resulting choice for ki (if interior) requires 1 − F(ki) = R, such that for any firm there is a

chance of 1 − R that the capacity constraint binds. Denote the cutoff value for bi at which the

firm just produces at capacity as  ̄bi = ki.

Since all firms face the same problem, they choose the same capacity ki = k and therefore

face the same cutoff  ̄b = k. The demand shocks bi then induce a distribution over yi with a mass

1 − F(

 ̄b) concentrated at point  ̄b.

We can introduce aggregate fluctuations into the example by shifting the mean of the distribution F(b), which allows us to show how aggregate shocks qualitatively generate the four

stylized facts outlined in the previous section. For concreteness, consider the case that demand

b is distributed uniform(0, 1). This implies that the optimally chosen capacity is ki = k = 1 − R.

Output fluctuations Aggregate output under a uniform distribution over b between 0 and 1

is Y =

R 1−R

0

b db + R(1 − R) = 1

2

(1 − R2). Now there is an unexpected shift in the mean by ,

so that b is drawn uniformly from the interval (, 1 + ). Aggregate production then becomes

Y =

1

2

(1−R2)+(1−R)−

1

2



2

. By inspection, we can see that output fluctuations are asymmetric.

The presence of the second-order term implies that for small values of , positive and negative

output changes are about the same size, while for large values of , positive output changes are

smaller than negative ones.

Fiscal multiplier: An unexpected small increase in demand can be captured by a marginal

increase in . If this increase in demand represented a change in government policy, the resulting

increase in output would measure the (marginal) fiscal multiplier. With the second derivative

d

2Y /d

2 = − an additional small increase in aggregate demand affects output less, the higher

aggregate demand already is.

8
The government multiplier and the asymmetry in output are therefore closely related. They

are not quite measuring the same thing however. The difference between a large boom and

recession is given by the average effect of an increase in demand (that is, the difference in output

between aggregate states), while the multiplier is determined by the marginal effect (that is, the

effect of a small demand shock on output at different aggregate states).

Figure 2 displays the mapping from demand shocks biinto output yi for an interest rate of

0.3 such that the implied capacity constraint is at 0.7. The three sets of points represent the

case without aggregate shock ( = 0) as well as aggregate shocks of  = ±0.1.

Figure 2: Distribution of yiin numerical illustration

0 0.35 0.7

0

0.35

0.7

1

Firm output yi Firm demand b

i

No shock, ε = 0

Negative shock, ε = −0.1

Positive shock, ε = 0.1

Notes: The figure plots simulated output levels yi (X-axis) for a sample of 100 firms depending on their

respective realized demand bi (Y-axis). Blue •: no aggregate shock, firm output uniformly distributed

between 0 and 0.7, and a mass point at 0.7. Green +: For a positive demand shock  = 0.1, additional

firms get pushed into their capacity constraint. Output expands less than proportionally, dispersion in

output (and profitability) decreases, aggregate capacity utilization and Solow residual increase. Red ∗:

The opposite is true for a negative demand shock  = −0.1. The left tail of the distribution becomes

wider and the mass of firms at capacity decreases.

Cross-sectional and aggregate volatility: The example also illustrates that aggregate fluctu9
ations affect differences between firms. An individual firm’s profitability can be measured as

yi/ki = yi/(1 − R). Since the factor input cost R is the same for all firms this means that the

relative cross-sectional variance in profitability at any point is equal to the relative variance in

output. While the analytic expression for Var(yi) as a function of  is somewhat involved, the

intuition is straightforward: the greater the mass of firms at the capacity constraint, the smaller

the variance in profitability of the overall distribution. In the extreme case of a very large negative shock (corresponding to  < −0.3 in the example), no firm is capacity constraint and thus

dispersion is greatest.

In this stylized example one needs an additional assumption in order to see that the growth

rate of output as a measure of aggregate volatility varies with the state of the economy . In

particular, we need to prevent firms from being fully flexible in adjusting their choice of k between

periods. Imagine therefore that firms have a fixed capacity level for two periods. If in the first

period  is positive, then a further shock in the second period will tend to have relatively small

effects, because the relatively large mass of firms at their constraint will not change production –

the intuition here is analogous to why fiscal multipliers are smaller in a boom. In the full model

of section 4 there will be a price adjustment friction that takes the role of giving persistence to

aggregate demand shocks.

Measured aggregate productivity: Simple measured aggregate TFP is Y /K =

1

2

(1 + R) +  −

1

2



2/(1 − R), hence it increases with  due to more intensive use of installed capacity. Measured

aggregate TFP is hence endogenously procyclical while TFP corrected for utilization is trivially

given by Y /K ̃ = 1 by definition of the production function.

This example illustrates the basic mechanics with which capacity constraints can qualitatively

generate deep recessions along with meek booms, countercyclical fiscal multipliers and a more

dispersed productivity distribution in recessions. All of these features arise from a simple shock

structure that is perfectly symmetric over time and across firms.

4 Model

We now embed capacity constraints in a New-Keynesian model of aggregate demand shocks to

look at the effects in general equilibrium. While the intuition from the previous section about

their qualitative implications fully carries through, only a general-equilibrium model will be able

to inform us about the size of asymmetries generated by capacity constraints quantitatively.

The main difference relative to the example in the previous section is that the capacity

constraint arises endogenously due to convex capital utilization costs. This type of utilization cost

can be justified by empirically relevant features such as overtime pay or increased depreciation.

By introducing capacity constraints in this way, a firm’s maximal production is given by its

willingness to supply goods rather than an assumed technological constraint. To this end, firms

not only choose their capacity, but also their goods price at the beginning of the period before

any shocks are realized.

The full model also includes the following standard features: Labor constitutes a second

flexible factor of production in addition to utilized capital, and an individual firm’s demand now

comes from a final goods aggregator. Finally, there is a central bank setting nominal interest

rates.

There are several reasons why we model firms as setting their price in advance. First, it

keeps the model tractable since all firms face the same environment at the time of their decision

10
and hence choose the same price. Second, it will allow us to endogenize capacity constraints

as the quantity firms are willing to supply at the set prices. Third, in this context it provides

a convenient way of introducing price rigidities which allow preference shocks to affect output

through changes in relative prices, as is usual in New Keynesian models.3

In order for firm supply to constitute an upper bound to production we will specify that, when

supply and demand do not coincide at the set price, quantity traded is given by the minimum

of supply and demand, and hence determined by the ‘short’ market side. This rule differs in

particular from an alternative in which the price setter is required to satisfy the other market

side’s demand or supply at the given price. Fagnart et al. (1999) use a similar setup and discuss

the implications for planned and traded quantities in more detail.

4.1 Timing

The timing within a period is as follows:

1. Households enter a period t with an amount of aggregate capital Kt. At the beginning of

the period, before any shocks are realized, a capacity rental market opens where households

supply Kt and firms rent their capacity for this period, kit. Simultaneously, firms choose

their price pit. (Later in equilibrium, because all firms are the same at the beginning of

the period, kit = Kt and pit = pt.)

2. All idiosyncratic and aggregate shocks are realized.

3. The remaining markets open: Firms make their decisions about labor demand and capacity

utilization; households decide on their labor supply and desired savings in capital and

bonds. Households also receive firm profits and pay taxes. The monetary authority sets

the nominal interest rate as a function of inflation. The period ends.

4.2 Final goods aggregator

The final good Y is assembled from a continuum of varieties indexed by i ∈ [0, 1] according

to a standard CES function with parameter σ measuring the elasticity of substitution between

intermediate goods

Y =

Z

b

1

σ

i

y

σ−1

σ

i

di

 σ

σ−1

.

The weights {bi} are realizations of iid random variables with mean 1.

The perfectly competitive final goods aggregator takes intermediate goods prices as given.

It has a nominal budget of I ≥

R

piyi di, where piis an intermediate variety’s nominal price.

The aggregator also takes into account the capacity constraint that limits the supply of some

varieties. Denoting this upper limit4 by  ̄y, it therefore has to consider a continuum of inequality

3Kuhn (2014) shows that in general it is important to model firms’ pricing behavior explicitly when

considering cross-sectional profitability measures: Differences in pricing can prevent firms’ profitability

from tracking their physical productivity, as highlighted by Foster et al. (2008).

4

In equilibrium the upper bound  ̄y is equal to the intermediates’ maximum supply dictated by costly

capacity utilization y

s and will indeed be the same for all firms. One could solve the aggregator’s problem

more generally using a variety-specific  ̄yi at the cost of more notation, but considering a  ̄y constant across

varieties is enough here.

11
constraints yi ≤ y ̄ ∀i. The problem can then be expressed as

max

{yi},λ,{μi}

Z

b

1

σ

i

y

σ−1

σ

i

di

 σ

σ−1

+ λ



I −

Z

piyi di



+

Z

μi ( ̄y − yi) di.

After taking first-order conditions (see appendix A), one has

y

d

i = bi

IU P

σ−1

U

p

σ

i

with IU ≡

R

yi<y ̄

piyi di the budget spent on unconstrained varieties and P

1−σ

U ≡

R

yi<y ̄

p

1−σ

i

di a

price index over unconstrained varieties.5

4.3 Firms

We solve the firm’s problem backwards: We first determine a firm’s optimal utilization and labor

input given its realization of bi and chosen capacity and price, and then the optimal k and p

choices that maximize expected profits.

Technology The intermediate goods firms’ production function is y =  ̃k

αl1−α, where l is

the hired labor input.6 There is a quadratic real cost of utilizing capital which depends on the

utilization rate  ̃k/k and total capacity k given by

cu 

 ̃k

k

, k!=

χ

2

 

 ̃k

k

!2

k.

This formulation ensures that the utilization costs scale linearly with k and hence the optimal

utilization rate is going to be independent of firm size. There is a Rotemberg-type quadratic real

cost of adjusting the nominal price p depending on the relative change p/p−1 through

C



p

p−1



=

ξ

2



p

p−1

− 1

2

.

We employ this cost because it is the simplest possible way of introducing persistent nominal

rigidities — its tractability in the context of this model stems from the fact that all firms choose

the same price in equilibrium. Additionally, the price adjustment cost adds an intertemporal

dimension to the firm’s problem and thus generates some internal propagation of shocks (if ξ = 0

the firm’s problem is reduced to an infinite sequence of one-shot problems).

5As noted by Fagnart et al. (1999) the demand function for the constrained varieties is undefined, and

y

d denotes demand for the unconstrained varieties.

6

In this section the firm index i is suppressed to save notation. It will reappear in the section on

aggregation below.

12
Cost function The cost function describes the cheapest way for a firm to produce a fixed

output level y given the marginal cost of the input factors which are in turn determined by the

level of capacity k and the real wage w. It is given by

C (y) = min

k,l  ̃

wl +

χ

2

 

 ̃k

k

!2

k

s.t.  ̃k

α

l

1−α ≥ y.

The first-order conditions give optimal input factor quantities as

 ̃k =



α

1 − α

w

χ

k

 1−α

α+2(1−α)

y

1

α+2(1−α) (1)

l =



1 − α

α

χ

w

k

−1

 α

α+2(1−α)

y

ψ

α+2(1−α) (2)

and so the cost function as

C (y) = α + 2 (1 − α)

2α

"

χ

α



α

1 − α

w

2(1−α)

y

2

k

−α

# 1

α+2(1−α)

.

Supply function and cutoff b Because the firm incurs convex utilization costs, there

will be some cutoff quantity of output more than which the firm will find it unprofitable to

produce. Since output increases with the level of demand shock, the cutoff output quantity will

be associated with a cutoff level of demand shock; here we solve for both output and demand

cutoffs, labeled y

s and b, respectively.

The firm considers the level of output y

s

that maximizes profits given its price and cost

function, but ignoring its level of demand. In other words, the firm thinks about how much it

would produce if demand for its variety was infinite. With P denoting the nominal price of the

final good, it considers its maximal operating profits

max

y

p

P

y − C (y)

which is solved by

y

s =



α

χ

 1 − α

w

2(1−α)

α  p

P

 α+2(1−α)

α

k. (3)

The convexity of the capital utilization cost function ensures that supply given w, p/P and k is

finite.

As mentioned above, there are no contractual arrangements that would require firms to

produce more than they desire, so that actual quantity traded is given by

y = min ny

d

, ys

o

. (4)

This defines a cutoff value  ̄b for the idiosyncratic demand shock at which y

s = yd as

 ̄b

IU P

σ−1

U

p

σ

≡



α

χ

 1 − α

w

2(1−α)

α  p

P

 α+2(1−α)

α

k.

13
Any firm with b >  ̄b will be constrained due to costly utilization, while firms with b <  ̄b just

satisfy demand. An algebraically useful implication is that y

d

can be written as

y

d = (b/ ̄b)ys

. (5)

Operating profits, expected profits, and value function Depending on realized

demand b, operating profits as a function of p and k are given by

π(p, k, b) = ( p

P

y

d

(p, b) − C



y

d

(p, b); k



if b ≤  ̄b

p

P

y

s

(p, k) − C (y

s

(p, k)) = p

P

y

s

(p, k)

α

2

if b >  ̄b

At the beginning of the period the firm can compute expected profits by integrating over b:

E [π(p, k, b)] = Z  ̄b

0

p

P

y

d

(p, b) − C



y

d

(p, b); k



df(b) + Z ∞

 ̄b

p

P

y

s

(p, k)

α

2

df(b).

It can now choose its price and capacity in order to maximize expected operating profits

minus the rental cost of capacity and the (expected discounted sum of future) costs of price

adjustment. In fact, only the price adjustment cost makes the firm problem truly dynamic. The

problem is summarized in the firm’s value function

V (p−1) = max

p,k

E [π (p, k)] − [R − (1 − δ)] k −

ξ

2



p

p−1

− 1

2

+ βE [V (p)] . (6)

4.4 Households

There is a price-taking representative household. She maximizes lifetime utility given by

E

"X∞

t=0

β

t



log Ct − φt

L

1+ε

t

1 + ε

#

where Ctis consumption and Ltis hours worked in period t. There is a random weight φt shifting

the relative preference of consumption and leisure and which will serve as an aggregate demand

shock. This formulation of preferences is consistent with existence of a balanced growth path.

Because consumption and leisure are separable, the household’s labor supply function is not

concave. This allows us to isolate the effects of variable capacity utilization from the mechanism

that drives the results of Sims and Wolff (2015).

Separability between consumption and leisure precludes the concavity in the household’s

labor supply function that drives the results in Sims and Wolff (2015) which helps us isolate the

effects of variable capacity utilization on the firm side.

Besides working, the household also earns income from renting capital Kt to firms as well as

from holding one-period bonds issued by the central bank. Her real bond demand in t is denoted

with St, and central bank pays a nominal interest rate of Rt on these bonds. The household also

collects all profits from firms πet ≡

R

πit − [R − (1 − δ)] kit −

ξ

2



pit

pi,t−1

− 1

2

di and finances any

government spending with a lump-sum transfer of Gt. Combining all these payments in units of

final goods yields her (real) flow budget constraint

Ct + St + Kt+1 =

Rt−1

Πt

St−1 + Rt−1Kt + wtLt + πt − Gt.

14
The variable Πt ≡ Pt/Pt−1 denotes inflation.

Her optimality conditions are the labor supply equation

wt = φtL

ε

tCt

, (7)

the Euler equation

1

Ct

= βRtE



1

Ct+1Πt+1 

, (8)

as well as a no-arbitrage condition between nominal assets and capital

RtE



1

Ct+1Πt+1 

= E



Rt

Ct+1 

. (9)

4.5 Central bank and government

The central bank sets nominal interest rates in accordance with a simple Taylor rule such that

inflation fluctuates around its long-run mean of zero:

log (Rt) = log (1/β) + CBrf log (Πt). (10)

The parameter CBrf determines how strongly the central bank reacts to inflation.

A government undertaking fiscal policy constitutes the second part of the public sector. It

can buy goods Gt from the final goods firm which it then consumes. It runs a balanced budget

by collecting lump-sum taxes Gt from the household. Since the government’s only purpose is to

allow us to assess the size of the fiscal multiplier, we fix Gt = 0 for all t.

4.6 Aggregation and equilibrium

Firms use their first-order necessary conditions from maximization of their value (6) to determine

optimal price and capacity (pit, kit) at the beginning of the period. Since, before realization of

period t shocks, all firms share the same state variables, they choose identical prices and capacities

such that pit = pt and kit = kt ∀i. Additionally, firms’ decisions about utilization and labor in

(1) - (2) and quantity traded in (4) are monomial in min bi/

 ̄b, 1

	

. This makes integration over

i straightforward and gives aggregate capital utilization costs and labor demand as

CU =

α

2

p

P

y

s

 Z  ̄b

0



b

 ̄b

 2

2−α

df (b) + 1 − F



 ̄b

!

(11)

L

d =

1 − α

w

p

P

y

s

 Z  ̄b

0



b

 ̄b

 2

2−α

df (b) + 1 − F



 ̄b

!

(12)

and final goods supply using the aggregator’s production function as

Y =  ̄b

1

σ−1 y

s

("Z  ̄b

0

b

 ̄b

df (b) + Z ∞

 ̄b



b

 ̄b

 1

σ

df (b)

#) σ

σ−1

. (13)

In equilibrium, the final goods price Pt as well as the producer price pt are not determined in

levels. These prices, however, only matter relative to each other or their respective values from

15
the previous period. We therefore define the real price of intermediate goods as rpt = pt/Pt

,

inflation as Πt = Pt/Pt−1, and producer price inflation as Πppi

t = pt/pt−1. These relative prices

in turn are related according to

Π

ppi

t = Πt

rpt

rpt−1

(14)

as can easily be derived from their definition.

Equilibrium then is defined in the usual way using agents’ optimality conditions and clearing of aggregate markets. Notably, the clearing of aggregate markets is unaffected by the

fact that predetermined prices prevent intermediate goods markets from clearing. Specifically,

we define as equilibrium a sequence of prices nRt, Rt, wt, rpt

, Πt, Π

ppi

t

o∞

t=0

, and of quantities



Yt, Ct, CUt, Ld

t

, ys

t

, kt, Kt, Lt

	∞

t=0 and cutoffs 

 ̄bt

	∞

t=0 that satisfy the firms’ two optimality conditions derived from (6), their supply (3), aggregate factor demands and final goods supply

(11)-(13), the household’s optimality conditions (7)-(9), the Taylor rule (10), the definition of

producer price inflation (14), as well as market clearing for labor and capital, an aggregate resource constraint, and the aggregator’s zero-profit condition. Note that for this definition we

have already imposed y

s =  ̄y.

Appendix B collects these equilibrium conditions.

5 Calibration and results

5.1 Calibration

In the following we simulate the model and show that the qualitative results from the example

hold up in general equilibrium. Table 2 summarizes the calibration of model parameters in

two groups: The first group contains parameters that have direct empirical interpretations or

standard values in the literature, whereas the second group consists of parameters that are

specific to the model.

The first group of parameters is set to conventional values found in the literature. Capital’s

share of income α is set to 1/3, and capital depreciation is δ = 2.6% implying an annual rate of

10%. Based on estimates of the average mark-up between around 10% and 30% , the macroeconomic literature uses values for the elasticity of substitution between goods σ between 4 as

for example in Bloom et al. (2012) and 10 as for example in Sims and Wolff (2015). We hence

choose an interior value of 6. Households have a discount factor of β = 0.99 such that the annual

steady-state interest rate is around 4%. The parameter ε set to 1/2 targets a Frisch elasticity of

labor supply of 2 which is also a standard value in macroeconomic models. The aggregate shock

follows an AR(1) process in logs such that log (φt) = ρφ log φt−1 + uφ where uφ is a mean-zero

normal random variable with variance σ

2

φ. We set the persistence parameter ρφ = 0.9. The

standard deviation of innovations σφ = 0.004 is chosen to match the empirical standard deviation of quarterly postwar US GDP of 1.8% when detrended with an HP(1600) filter. The price

adjustment cost parameter ξ is set to 75, corresponding to the estimate in Ireland (2001). The

coefficient measuring how the central bank reacts to inflation is set to 1.75 as in Sims and Wolff

(2015).

The second group of parameters describes the utilization cost function and the variance of

idiosyncratic shocks. We assume the distribution of the iid idiosyncratic shock bi to be lognormal and set the parameter σb governing its variance to match the variance of innovations

16
Table 2: Baseline Calibration

Parameter Value Meaning Calibration

Standard parameters

α

1

3

yi =  ̃k

α

i

l

1−α

i Capital share

β 0.99 Hh discount factor Standard (quarterly)

δ 0.026 Capital depreciation Standard (quarterly)

ε

1

2

Inv. Frisch elas. labor Standard

σ 6 E. of S. intermediates Literature: σ ∈ [4, 10]

ρφ 0.9 Shock persistence Standard (quarterly)

σφ 0.004 Shock variance sd(Yt) = 1.8%

ξ 75 Scale price adj. cost Ireland (2001)

CBrf 1.75 Taylor rule Sims and Wolff (2015)

Model-specific parameters

χ 1 Scale utiliz. cost See text

σb 0.67 sd idiosync. shocks sd(∆TFPi) = 0.185

to firm profitability in the data. In particular, both Syverson (2011) and Ilut et al. (2014) find

a standard deviation of innovations to the log of firm TFP of around 0.185. We match this

to average growth rates in the firms’ measured TFP in the model. We are unaware of direct

empirical estimates for the parameter χ; here we set the parameter to 1. We are not very

concerned with this issue for two reasons. First, varying the parameter over the admissible range

for determinacy implied by the Blanchard-Kahn conditions changes quantitative results only

minimally. Second, the fact that the utilization cost parameter is not well identified inside the

model can also be observed in other models, see for example Christiano et al. (2005). We show

sensitivity of the results to this parameter in section 5.2.6.

A central feature of the model is that the fluctuating share of capacity constrained firms

generates extra concavity in aggregate production. This causes effect sizes to increase with the

magnitude of aggregate fluctuations. For example, if aggregate shocks are small, the response

of output to a positive shock is similar to the response to a negative shock. Relative differences

between booms and recessions increase as the aggregate shock becomes larger. Model results are

therefore somewhat sensitive to the variance σ

2

φ of innovations to φ. In the baseline calibration

we take a conservative stance by detrending the empirical GDP series with an HP-1600 filter,

which implies a relatively moderate standard deviation of 1.8% for its cyclical component. If,

on the other hand, the underlying growth trend of the empirical series were better described by

a linear trend, then the time-series standard deviation of the cyclical component is 4.7%, which

significantly amplifies output asymmetry and the fiscal multiplier in our results. We consider

this alternative calibration in section 5.2.6.

17
5.2 Results

5.2.1 Impulse response functions

We simulate business cycles by a shock to the household’s preference weight φ governing her

relative taste for consumption and leisure. While we acknowledge many other possible shocks

that can cause aggregate fluctuations, as discussed above we focus on this preference shock as

a simple way to generate demand-side effects through distorted relative prices, which in turn

allows us to assess how much movement in the measured Solow residual is generated even by

a non-technology shock. The model is solved with a second-order approximation around the

non-stochastic steady state using the software package Dynare (see Adjemian et al. (2011)). An

approximation of at least second order is necessary here since we want to account for the nonlinearities generating differences between positive and negative shocks. Under linearization these

differences would be lost.

Figure 3 displays simulated impulse response functions following a 1-standard-deviation increase in the leisure preference of households φ. For approximations of order higher than 1 the

effect size of a shock will in general depend on the state of the economy at the time of impact.

The standard way of computing impulse responses in such a case is through simulation, which

approximates an ‘average’ effect of the shock across many simulated states.7

Most notable is the strong reaction to the shock on impact in period 1. With prices set one

period in advance the usual “New Keynesian” effect of demand shocks via relative prices is fully

concentrated in period 1. What remains of the shock in periods 2 and later is primarily driven

by the supply side effect of reduced household willingness to work and reduced capital stock from

period 1, as well as the fact that firms’ price adjustment costs prevent a full alignment of relative

prices in period 2.

As expected, capacity utilization drops along with aggregate output. The share of firm below

their capacity constraint F



 ̄b



decreases as well. This is not only due to the reduction in demand

for intermediates, but also due to the increase in firms’ willingness to supply their respective

variety: With nominal intermediate goods prices fixed at p, the decrease in the aggregate price

level P leads to a temporarily high relative price.

5.2.2 Output asymmetry

We now turn to an assessment of the implications for the stylized facts in general equilibrium.

Quantitatively, the model explains around 1/4 of the observed asymmetry in output, and explains

fluctuations in the fiscal multiplier of around 0.12.

For the difference between large positive and negative deviations in output, following the

approach from the empirical section, we choose an integer N of around 1/6 of the observations

(N = 1666 out of 10, 000 simulated periods) and compare the mean of the N/2 periods with

7More precisely, one chooses an appropriate ‘burn-in’ period and a large number I of simulations

indexed by i. For each simulation one simulates the model forward such that the model economy is at

some random point Si,0 of its ergodic state set. Next, one draws a sequence of aggregate shocks {Zi,t}

T

t=1

of length T equal to the desired time horizon of the impulse response, and simulates the model forward

twice starting from Si,0: Once, using only the shocks {Zi,t}, and once using the same shocks where for Zi,1

an additional 1-sd shock the exogenous state variable has been added. The simulated impulse response

is then just the difference between the two simulations, averaged over all I repetitions. For more details

see, for example, Adjemian et al. (2011).

18
Figure 3: Impulse Response Functions

1 5 10

−0.02

−0.01

0

Output

1 5 10

−0.2

0

0.2

Investment

1 5 10

−0.02

−0.01

0

Labor

1 5 10

−3

−2

−1

0

x 10−3 Consumption

1 5 10

−15

−10

−5

0

5

x 10−3 Utilized capital

1 5 10

0

0.5

1

1.5

x 10−4 Relative price p/P

1 5 10

−0.01

0

0.01

0.02

0.03

Firm supply ys

1 5 10

−4

−3

−2

−1

0

x 10−3 Share constr. firms

Notes: Simulated impulse response functions for a positive 1-sd shock to the leisure preference φt in

period 1. Y-axes show log-deviations from the non-stochastic steady state. A description of the simulation

procedure is given in footnote 7.

19
Table 3: Asymmetry of Output Levels and Other Variables

Model Data

Variable Mean pos vs neg Skewness Mean pos vs neg Skewness

Levels

Output Y 3.24% vs −3.45% −0.11 2.73% vs −3.43% −0.46

Labor L 3.35% vs −3.54% −0.12 3.11% vs −3.99% −0.40

Investment I 18.97% vs −23.12% −0.43 13.08% vs −17.12% −0.53

Consumption C 1.46% vs −1.47% −0.02 2.23% vs −2.24% −0.00

Growth rates

Output ∆Y 3.16% vs −3.15% 0.01 2.30% vs −2.25% 0.19

Labor ∆L 3.45% vs −3.43% 0.01 1.63% vs −2.03% −0.55

Investment ∆I 23.32% vs−23.04% 0.02 10.00% vs −11.30% −0.21

Consumption ∆C 0.36% vs −0.37% −0.06 1.94% vs −1.85% 0.38

Notes: Measures of asymmetry as defined in Table 1 (baseline specification). “Levels” measured in

log-deviations from simulation mean (model) or from HP-1600 trend (data), respectively. “Growth

rates” measured as log-differences. Data for Output, Investment and Consumption from BEA NIPA

tables (Real gross domestic product, personal consumption expenditures, gross private domestic

investment”, respectively). Data for labor from BLS statistics as hours of all persons in the nonfarm

business sector. All data are quarterly.

highest output to the N/2 periods where output is lowest. As shown in Table 3, the average

large recession in that sense is −3.45% below trend, whereas the average large expansion is 3.24%

above trend. Output is also negatively skewed with a coefficient of −0.11.

Comparing this to the empirical equivalents in Table 1, the differences between positive and

negative output deviations in the model cover around a quarter of those in the data. In the

model, recessions are 0.21 percentage points (or a bit more than 6%) deeper than expansions.

As the model was calibrated to match the standard deviation of HP(1600)-filtered, the closest

comparable measure is the first row of Table 1 showing a relative difference of 23%, or 0.7

percentage points.

Regarding the other aggregate time series also listed in Table 3, the model generates asymmetry in levels of investment as well as levels of hours worked, but not for the level of consumption

nor the growth rate of output — all these patterns are consistent with empirical findings discussed in section 2 and replicated in the Table. The simulation does not exhibit asymmetry in

growth rates of employment, even though there is some empirical evidence for this (e.g McKay

and Reis (2008)). The reason behind this is that in the model with its frictionless labor markets

the employment and output series move together very closely.

5.2.3 Cross-sectional and aggregate volatility

To assess the relation between profitability dispersion and output, we consider the cross-sectional

standard deviation of log(profitabilityi). Profitability is measured as firm i’s priced Solow residual

piSRi = piyi/(k

α

i

l

1−α

i

) which has the interpretation of “revenue in dollars per input factor

basket”. As discussed above, this measure uses rented capacity as a measure of capital input

— of course firms’ true physical productivity yi/(

 ̃k

α

i

l

1−α) is constant by construction. Since a

20
firm’s profitability is only a function of its price and demand shock, and all firms choose the

same price, profitability dispersion in any given period only depends on the variance of realized

demand between firms up to capacity min bi,

 ̄b

	

with

Var (log(piSRi)) = 

α

2 − α

2

Var log min bi,

 ̄b

	

(see appendix C).

We then consider the fluctuation of this measure over the business cycle in the simulations

to assess the question, how much wider does the firm distribution of profitability become in

recessions? Kehrig (2015) finds that for the six recessions in his data ranging from 1972 to 2009,

profitability was 2.84% more dispersed in recessions compared to the long-run. If similarly we

look at the 1/6 of simulation periods in which output is lowest we find that profitability dispersion

increases by 0.51% in recessions, implying that the model captures 18% of the cross-sectional

volatility.

Profitability dispersion in the model is only a function of the cutoff level  ̄b, such that its

correlation with output will mirror the correlation of  ̄b with output. In the simulations the

correlation corr(sd(log(SRi))t, Yt) = −0.93 is correspondingly strong, and higher than the −0.4

to −0.5 that have been measured in Kehrig (2015) and Bloom et al. (2012). This high correlation

in the model results from the close comovement between aggregate output and the level of

constrained firms we saw in the impulse response functions.

Turning to aggregate volatility, we construct a measure of aggregate volatility from the

simulated output series. For this, we look at the variance in the growth rates of output in

recessions and expansions, respectively. Specifically, we compute sd(log(Yt+1/Yt)) conditional

on Yt being in its lowest or highest quintile. We expect the variance of output growth to be large

in recessions: When a large number of firms are far away from their capacity constraint, output

effects of a shock of a given size are stronger. In the model here, the fact that firms can adjust

their capacity levels and prices quickly in response to an aggregate shock dampens this effect: It

allows firms to lower their capacity after the realization of a bad shock, which in turn increases

the number of firms at their constraint. If it took firms longer to react, say with a ‘time to

build’ of two periods instead of one, we would expect to see a significantly stronger movements

in aggregate conditional volatilities.

Table 4 lists the volatility of several model time-series in the first two columns. Going from

boom to recession, the standard deviation of output growth increases from 1.49% to 1.66%.

The model’s investment and labor series exhibit countercyclical conditional volatilities as well,

whereas consumption volatility stays constant over the cycle. In the model, aggregate risk

as measured by the volatility of output increases by 10.8%. We also construct the empirical

analogues of the volatility measures using US data, which are shown in columns 3 and 4 of

Table 4. As in the baseline empirical specification of Table 1, we consider as recessions the 20

quarters since 1949 in which detrended output was lowest. In the data, output volatility in a

recession is 39.5% higher in recessions than in booms and thus fluctuates a bit stronger than

in the model. Additionally, in the US series both investment and consumption exhibit cyclical

volatilities, whereas in our model households are generally able to smooth consumption very well

as they do not face any frictions.

Using different empirical strategies, Bloom et al. (2012) find that recessions are associated

with a 23% higher standard deviation of output (compared to normal times), and Bachmann

21
Table 4: Aggregate risk: Conditional Volatilities of Aggregate Variables

Model Data

Variable Expansion Recession Expansion Recession

Output Y 1.49% 1.66% 0.95% 1.41%

Labor L 1.59% 1.77% 0.68% 1.08%

Investment I 9.66% 11.97% 5.21% 5.91%

Consumption C 0.19% 0.20% 1.08% 0.68%

Notes: Standard deviation of growth rate in expansion/recession in the model. For time series X,

conditional volatility in recession is computed as the standard deviation of growth rates following

a recessionary quarter; i.e. we compute sd(log Xt+1 − log Xt|Xt in recession). Analogous for expansions. Recessions and expansions as defined in Table 1 (baseline specification) and section 5.2.2;

in particular output is among the lowest/highest 20 periods (data) and lowest/highest 833 periods

(simulated model series).

and Bayer (2013) obtain a difference of around 35% between booms and recessions, in line with

the empirical values found here. Based on these estimates, the model covers between a third to

a quarter of observed fluctuations in aggregate output volatility.

5.2.4 Aggregate Solow residual

We construct the aggregate Solow residual in a similar way as its firm-level equivalent. We compute the uncorrected Solow residual as SRsimple,t = Yt/



Kα

t L

1−α

t



using aggregate capital in the

denominator, and the corresponding version corrected for utilization as SRcorr,t = Yt/



K ̃ α

t L

1−α

t



where K ̃

t =

R

i

 ̃kitdi is defined as the aggregate utilized capital. Figure 4 displays the log deviations from the mean for output as well as both Solow residual for a subset of the simulated

periods. As in Basu et al. (2006) and Fernald (2012), the correlation between the simple TFP

measure with output is strong with a value of 0.76. On the other hand, utilization-corrected

productivity barely moves over the cycle.8 The standard deviation of simple TFP growth in

the simulated series is 0.52%. This value is a little smaller than the corresponding measure in

John Fernald’s quarterly dataset where the uncorrected Solow residual grows with a standard

deviation of 0.87%.

5.2.5 Fiscal multiplier

Finally, we consider the cyclicality of the contemporaneous fiscal multiplier dYt/dGt. In constructing it we follow Sims and Wolff (2015) by averaging the state variables over those periods

in which production is in its lowest quintile. We compare output in this “average bad state” to

output in the same state, but with an additional small positive shock to government spending.

More formally, if S is the aggregate state, and S + ∆G the aggregate state after small fiscal

spending shock ∆G, the government multiplier is computed as Y

S+∆G − YS



/∆G. The value

8Strictly, even utilization-corrected TFP fluctuates over time because of changes in the composition of

input factors and their allocation between firms. Since corrected TFP has a very small variance (it has a

standard deviation of 0.00018), however, even a tiny amount of noise —like measurement error— renders

it acyclical.

22
Figure 4: Output and TFP measures

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

T

Deviations in %

Output

simple TFP

util.−corrected TFP

Notes: Output (solid blue line) is Yt, simple TFP (red dashed line) is measured as Yt/(Kα

t L

1−α

t

), corrected

TFP (green dash-dotted line) is measured as Yt/(K ̃ α

t L

1−α

t

). Y-axis displays log-differences from nonstochastic steady state. X-axis displays a window of 100 periods out of the 10, 000 simulation periods.

23
of the multiplier when output is in its top quintile is computed the same way. We obtain values

of 1.07 for the multiplier in a recession, and 0.95 for a multiplier in a boom. These results are

close to what Sims and Wolff (2015) find in their DSGE model using a different mechanism.

5.2.6 Role of heterogeneity, discussion and sensitivity

Variance of idiosyncratic shocks The variance of idiosyncratic demand shocks, parameterized by σb, directly influences how many firms are capacity constrained. It is instructive

to consider how model results depend on this parameter. Figure 5 shows this for several outcomes.

The graph in upper left displays the share of constrained firms in steady state. Unsurprisingly,

the wider the distribution of idiosyncratic demand shocks, the more firms face a level of demand

exceeding their capacity. The next two graphs show output deviations from steady state for

booms and recessions (top right), and the relative size of these deviations to each other (bottom

left), respectively. Notably, output asymmetry is non-monotonic in σb. Why is this? What matters is the average change in the share of constrained firms over the cycle, and not its absolute

level. Those differences in F



 ̄bt



between expansion and recessions are largest for an interior

value of σb. At a low value of 0.3 there are practically no constrained firms in equilibrium, and

recessions are around 3.5%, or 0.13 percentage points, larger than expansions. (Even when there

is no heterogeneity between firms there is some concavity in production through the convex capacity utilization cost.) Increasing the standard deviation σb to around 0.75 makes recessions

more than 6% larger than expansions. For high values of σb, output asymmetry is reduced again

because, despite a larger share of constrained firms in steady-state, the change in this share over

the cycle is smaller.

A similar pattern can be observed for the fiscal multiplier in the bottom right graph of

Figure 5. When virtually no firms are capacity constrained, the timing of government spending

does not matter for its effect on output — all firms can increase their production in response

to government demand. The cyclicality of the multiplier is strongest when the fluctuations in

F



 ̄bt



over the cycle are large. In this case comparatively many firms have idle capacities in a

recession and can respond to an increase in government demand.

Summarizing, the firm heterogeneity causing capacity constraints to bind matters in this

model because it generates cyclicality in the fiscal multiplier and cross-sectional profitability

dispersion, and amplifies the deepness of recessions.

Sensitivity to utilization cost In the baseline calibration we chose the scale parameter

of the utilization cost function to be 1 in lack of more direct empirical estimates. Table 5 shows

the specification is not very sensitive with respect to parameter values. The table displays results

for recessionary and expansionary output deviations and multipliers for alternative values for χ.

Both statistics increase only minimally in the parameter. There is an upper limit for its domain

near χ = 1.4 implied by determinacy of the model (otherwise the Blanchard-Kahn conditions

are violated).

Effect size Is it possible for the same mechanism to deliver stronger effects? One can

think of several factors potentially affecting the results.

First, the model is only solved locally, i.e. any effects of aggregate fluctuations are captured

by evaluation of the first and second derivative of the equilibrium conditions at the steady state.

24
Figure 5: Varying Idiosyncratic Shock Variance σb

0.2 0.4 0.6 0.8 1

0.95

0.96

0.97

0.98

0.99

1

F(bbar) − share of firms with yd < ys

0.2 0.4 0.6 0.8 1

0.02

0.025

0.03

0.035

0.04

Trend deviations large

recession & expansion

Recession

Expansion

0.2 0.4 0.6 0.8 1

0.02

0.03

0.04

0.05

0.06

0.07

Percentage Difference between

large recession & expansion

0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

dY/dG for

recession & expansion

Notes: X-axes display value for σb. On Y-axes: Top left – Fraction of unconstrained firms F(

 ̄b) in

the non-stochastic steady state. Top right – Absolute log deviations of recessions and expansions from

non-stochastic steady state. Bottom left – Log difference between absolute deviations in recession and

expansion (i.e. log difference of the curves in top right). Bottom right – Government multiplier in recession

and expansion.

25
Table 5: Sensitivity with Respect to Cost Function

Parameter value Output asymmetry Multipliers

χ = 0.01 3.29% vs −3.40% 0.66 vs 0.75

χ = 0.1 3.26% vs −3.41% 0.73 vs 0.84

χ = 0.5 3.25% vs −3.44% 0.85 vs 0.97

Baseline 3.24% vs −3.45% 0.95 vs 1.07

χ = 1.4 3.24% vs −3.44% 1.01 vs 1.13

Notes: Alternative values of utilization cost parameter χ. Baseline: χ = 1. All remaining parameters are held constant, except for the aggregate shock variance σφ which needs to be adjusted to

match its targeted moment. The value of χ = 1.4 is close to the largest admissible value to not

violate determinacy of the model.

Any higher-order concavity in the relation between shock size and output is lost when moving

away from the steady-state and could only be recovered through a global solution method.

Second, the model has little internal propagation due to the one-period-ahead choices of

prices and capacity. Firms are thus very quick to adjust to aggregate shocks, such that it is hard

for individual shocks to “add up” over time. In fact it is predominantly the innovation to the

aggregate state variable φt that matters for chance of binding capacity constraints. Since the

model is solved up to a second-order approximation, effect sizes increase linearly in the size of the

aggregate shock. As an illustration, if one detrends quarterly GDP since 1949 with a linear filter

(instead of the HP(1600) filter used in calibration) this implies a considerably higher standard

deviation of the detrended series of 4.7% instead of 1.8%. When this standard deviation is used

in the model, the corresponding output deviations in recession and expansion strengthen to -

9.57% and 8.12%, respectively, and the recessionary and expansionary fiscal multipliers become

1.17 and 0.85, respectively. Similar effects can be expected from increasing the “time to build”

(and price-set) from one period to a longer horizon.

6 Conclusion

This paper includes capacity constraints in a DSGE framework under demand shocks and shows

that the model replicates diverse stylized facts of US output: Recessions are deep; they are

times of high volatility both in the aggregate and the cross-section; and they are times when

fiscal policy is particularly effective. Since firms choose their utilization after capacity has been

installed, the mechanism also reproduces an endogenously procylical Solow residual.

A calibrated New Keynesian model yields differences in output between booms and recessions

of around 0.21 percentage points, such that the model explains more than a quarter of the 0.7

percentage-point difference we find empirically. While the empirical literature has not settled on

the size of fluctuations in the government spending multiplier over the cycle, in our basic model

we find a multiplier of on average 0.95 in booms and 1.07 in recessions. The multiplier increases

with the severity of recessions.

The model contains a minimal set of ingredients for the mechanism of capacity constraints

to qualitatively deliver the stylized facts. An interesting expansion of this approach would be

to gain more realism in the description of firm behavior. A stronger intertemporal dimension

26
could be added to the problem by including more heterogeneity in price-setting and investment

behavior. This could yield new testable implications for the mechanism and make the crosssectional aspects of the model more accurate.

27
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30
A Aggregator’s demand function

The aggregator’s problem is to

max

{yi}

1

i=0

Z

b

1

σ

i

y

σ−1

σ

i

di σ

σ−1

+ λ



I −

Z

piyidi+

Z

μi[ ̄y − yi] di

such that the first-order necessary conditions with respect to yi are given by

Z

b

1

σ

i

y

σ−1

σ

i

di 1

σ−1



bi

yi

 1

σ

= λpi + μiyi ∀i.

For any given variety i either we have to consider two cases. If the aggregator is unconstrained

in this variety, i.e. μi = 0, then

λ =

Z

b

1

σ

i

y

σ−1

σ

i

di 1

σ−1

(bj/yj )

1

σ ,

whereas the aggregator is limited to purchasing  ̄y of variety i if

μi =

Z

b

1

σ

i

y

σ−1

σ

i

di 1

σ−1



bi

yi

 1

σ

− λpi > 0

.

For any two varieties i, j with μi = μj = 0 then the relationship

yi

yj

=

bi

bj



pj

pi

σ

holds. Integrating over all i one then has

I =

Z 1

0

piyidi =

Z

i∈U

p

1−σ

i

bidi

yj

bj

p

σ

j +

Z

i∈C

piydi  ̄

= P

1−σ

U

yj

p

σ

j

bj

+

Z

i∈C

piy,  ̄

where U ≡ {i : μi = 0} and C ≡ {i : μi > 0} are index sets over unconstrained and constrained

varieties, respectively, and PU ≡

R

i∈U

p

1−σ

i

bidi 1

1−σ

is a price index over unconstrained varieties.

Demand for an unconstrained variety j is then given by

yj = bj



I −

R

i∈C

piydi  ̄



P

σ−1

U

p

σ

j

= bj

IU P

σ−1

U

p

σ

j

,

where IU ≡ I −

R

i∈C

piydi  ̄ are the aggregator’s expenses over unconstrained varieties.

31
B Equilibrium conditions

First-order conditions for pit and kit (i-subscripts suppressed in the following):

ES

"

ξ



Π

ppi

t − 1



Π

ppi

t + y

s

t

rpt(σ − 1) Z  ̄bt

0

b

 ̄bt

df (b)

#

= ES

"

y

s

t

rpt

(

1 − F



 ̄bt



+ σ

Z  ̄bt

0



b

 ̄bt

 ψ

α+ψ(1−α)

df (b)

)

+ ξ



Π

ppi

t+1 − 1



Π

ppi

t+1#

Rt − (1 − δ) = ES

"

α (ψ − 1)

ψ

rpt

y

s

t

kt

(



1 − F



 ̄bt

 +

Z  ̄bt

0



b

 ̄bt

 ψ

α+ψ(1−α)

df (b)

)#

Firm supply y

s

:

y

s

t =



α

χ

 1

ψ−1



1 − α

wt

 ψ(1−α)

α(ψ−1)

rp

α+ψ(1−α)

α(ψ−1)

t

kt

Aggregate supply and factor demands from firms:

Yt =  ̄b

1

σ−1

t

y

s

t

("Z  ̄bt

0

b

 ̄bt

df (b) + Z ∞

 ̄bt



b

 ̄bt

 1

σ

df (b)

#) σ

σ−1

L

d

t =

1 − α

wt

rpty

s

t

 Z  ̄bt

0



b

 ̄bt

 ψ

α+ψ(1−α)

df (b) + 1 − F



 ̄bt

!

CUt =

α

ψ

rpty

s

t

 Z  ̄bt

0



b

 ̄bt

 ψ

α+ψ(1−α)

df (b) + 1 − F



 ̄bt

!

Household optimality conditions (Euler equation, no-arbitrage, labor supply):

1

Ct

= βRtE



1

Ct+1Πt+1 

RtE



1

Ct+1Πt+1 

= E



Rt

Ct+1 

wt = φtL

ε

tC

τ

t

Definition of producer price inflation:

Π

ppi

t = Πt

rpt

rpt−1

Market clearing conditions:

kt = Kt

L

d

t = Lt

32
Taylor rule:

log (Rt) = log (1/β) + CBrf log (Πt)

Aggregate resource constraint:

Yt = Ct + CUt +

ξ

2



Π

ppi

t − 1

2

+ [Kt+1 − (1 − δ) Kt] + Gt

Aggregator’s zero-profit condition It = PtYt:

Yt = rpty

s

t

 R  ̄bt

0

bdf(b)

 ̄bt

+



1 − F



 ̄bt

!

C Variance of firm profitability

A firm’s profitability is given as

piSRi =

piyi

k

α

i

l

1−α

i

= pi

 

 ̃ki

k

!α

=

p

2

i

P

 

min bi,

 ̄b

	

 ̄b

! α

2−α 

α

2

1

χ

α 

1 − α

w

1−α

.

Since all firms set the same price pi = p, it follows for the variance of log profitability

Var (log(piSRi)) = Var 

α

2 − α

log min bi,

 ̄b

	 + log "

1

 ̄b

 α

2−α p

2

P



α

2

1

χ

α 

1 − α

w

1−α

#!

=



α

2 − α

2

Var log min bi,

 ̄b

	 .

33