Superstar Firms and Aggregate Fluctuations
Qazi Haque
The University of Adelaide & CAMA
Oscar Pavlovy
University of Tasmania & CAMAMark Weder
Aarhus University & CAMA

November 20, 2023

Abstract
The rise of market power in the last decades is primarily driven by the largest(superstar) …rms. The current paper examines the impact on aggregate ‡uctuations of these multi-product superstars through their interaction with smaller
competitors. Market share reallocation and product scope adjustments generateheterogeneous markup dynamics across …rms and time. Moreover, the prevalenceof superstar …rms increases the parameter space for macroeconomic indeterminacy.
The endogenous ampli…cation mechanism of product creation improves the …t of theestimated general equilibrium model and implies animal spirits play a non-trivial
role in driving U.S. business cycles.

1

Introduction

Firms are not identical. Many markets are polarized and populated by a few relatively
big …rms mixed in with a greater number of smaller …rms that extort less market power.
Empirically such dispersion is well documented.1 In the last thirty years, this polarizationof markets has become more accentuated. De Loecker et al. (2020) report a steady andsigni…cant increase of market power but this increase of the average markup in the U.S.
was foremost driven by the …rms in the top percentiles.
What are the e¤ects of increasing product market concentration on the workings of themacroeconomy? And what is the role of the competition among big and smaller …rms?
This paper proposes a dynamic economy that emphasizes the role of …rm heterogeneity
Preliminary draft. We thank seminar participants at Aarhus Universitet, The University of Adelaide, ASSET 2023, Cardi¤ University, Le Mans Université and University of Sydney for their excellent
comments. Weder acknowledges generous support from the Australian Research Council under the grant
DP200101963. Keywords: Superstar …rms, animal spirits, multi-product …rms, business cycles, Bayesianestimation. JEL Classi…cation: E32.
y
Tasmanian School of Business and Economics, University of Tasmania, Sandy Bay TAS 7005, Australia. oscar.pavlov@utas.edu.au.
1
See for example Bartelsman et al. (2013), De Loecker et al. (2020), Baqaee and Farhi (2020), Kehrig
and Vincent (2021) and Edmond et al. (2023).

1

and, in particular, it examines the in‡uence of superstar …rms on aggregate ‡uctuations.
What makes superstar …rms special? In our arti…cial economy, large …rms have access todi¤erent technologies and markets that results in more market power and larger market
shares. Neary suggests the following characteristics of superstars:“So far, the advantage of superstar …rms has not been speci…ed exactly.
One interesting and important case is where the superstar technology involves the ability to produce a large number of products. In that case, thesmall number of superstar …rms are multi-product …rms, while the remaining
insiders which constitute the competitive fringe are single-product …rms. This
con…guration is consistent with the empirical evidence [...].” [Neary, 2010, p
15]
We follow Neary’s suggestion and propose a model in which …rm heterogeneity manifests itself in a grouping of …rms being able to produce multiple products and to gain
larger market power relative to mono-product …rms. This characterization parallels empirical work as in Broda et al. (2010) and particular Bernard et al. (2010) who reportthat a considerable fraction of U.S. manufacturing …rms produces in multiple …ve-digit
SIC industries and these …rms account for well over 80 percent of total sales. The theoretical framework then allows to shed light on the e¤ects of the superstar …rm environment
involving changes in market concentration and also of the dispersion of market power. Of
particular importance to us is explaining in a uni…ed way the observed market power by a
restricted group of …rms, their interactions with conventional …rms and the impact of thatenvironment on the macroeconomic dynamics – either via the propagation of impulses
or multiple equilibria. We show that a rising market share of superstar …rms allows for
greater divergence in markups between large and small …rms. Our theory predicts thatsuperstars charge a higher price, they set larger markups and grab a larger market share.
Our paper, thus, provides a quantitative theory of superstar …rms’divergence from therest and it can explain various empirical …ndings such as in De Loecker et al. (2020). We
also show how this divergence increases the parametric space for macroeconomic indeterminacy. This indeterminacy implies that pro…t-seeking businessmen’animal spirits canlead to self-ful…lling macroeconomic outcomes. This indeterminacy mechanism is novel as
it comes about from the superstars’endogenous and time-varying product creation evenwhen we keep constant the number of …rms. The estimated version of the model suggeststhat the endogenous ampli…cation mechanism of product creation within superstar …rms
is empirically important and that a non-trivial portion of U.S. aggregate ‡uctuations
is driven by realized animal spirits, i.e. non-fundamental swings between euphoria andpessimism.
This article is in …ve parts. It begins by presenting the baseline model from which
we have stripped o¤ various bells and whistles that we insert into the full model when2

estimating it. This approach allows us to highlight the main mechanisms that drive
our results. Section 3 analyses the local dynamics by presenting the parametric zones
for indeterminacy and impulse responses of macroeconomic variables to animal spirits(i.e. to a animal spirits or expectational error shock). The fourth Section presents theBayesian estimation of the fuller model. We end the article by listing our conclusions.

2

Model

The economy is populated by two groups of …rms. One group consists of smaller monoproduct …rms. We will coin them ordinary …rms. The other grouping are superstars: they
produce multiple products, and consequently, have more market power. Both groups of…rms produce di¤erentiated goods and adjust their markups according to ‡uctuations intheir market shares. The …rms’goods are bought by perfectly competitive …rms that weld
the varieties together into the …nal good that is used for consumption purposes or added
to the capital stock. People rent out labor and capital services. Firms and households
are price takers on factor markets.

2.1

Final goods

Similar to Shimomura and Thisse (2012), …nal output is a combination of products produced by M ordinary …rms and N superstar …rms.2 M and N are constant for now but
this will be relaxed later. With this assumption on …rm numbers, we can pinpoint to therole of time-varying product scopes as opposed to …rm dynamics of entry and exit. Final
output Y is then
Y =

M
X

x(i)

1

+

i=1

in which

N
X
j=1

Y (j)

1

!

1

(1)

> 1 stands for the elasticity of substitution and x(i) is the amount produced bymono-product …rm i: Since superstar …rms are multi-product …rms, Y (j) is a composite
good! 1
Z S(j)
1(2)
Y (j) =
x(j; s) ds
0in which S(j) stands for the product scope and x(j; s) denotes variety s of superstar
j. The CES aggregators imply a love of variety e¤ect of 1=(

1): This variety e¤ectin (2) provides the bene…t of product creation for the superstar …rm.3 The …nal pro…t2

We suppress the time index in these static equations for notational ease.
More broadly, the love of variety can be interpreted as a stand-in for other e¢ ciency gains of productcreation within multi-product …rms. Pavlov (2021) discusses an alternative way of modelling e¢ ciency
gains of product creation without the love of variety.
33

maximization problem yields two demand functions
x(i) =p(i)P

x(j; s) =p(j; s)PY;Y

and the aggregate price index
M
X

P =

p(i)1

+

j=1

with
P (j) =NX

P (j)1j=1

Z St (j)p(j; s)1

ds

0

2.2

!11

!11

:

Intermediate good …rms

Varieties supplied by superstar …rms are produced using labor h(j; s) and capital services
U k(j; s). U stands for the utilization rate set by the owners of physical capital. Superstar…rm j maximizes pro…ts(j) =Z S(j)

p(j; s)x(j; s)wh(j; s)rU k(j; s)ds

0

subject to the production technology
Z S(j)

x(j; s)ds =Z S(j)[U k(j; s) h(j; s)1

s ]ds

0<

f

< 1:

0

0

Here, w is the wage and r the rental rate for labor and capital services. The variety-level…xed cost s is paid each period and it restricts the amount of varieties the …rm produces.
The …rm-level …xed cost f provides economies of scope and helps to pin down steadystate pro…ts. Ordinary …rm i only produces a single variety and its production technology
isx(i) = U k(i) h(i)1in which the …xed costoo is calibrated so that regular …rms have zero pro…ts at the steadystate. The …rst-order conditions arew = (1r=) U k(j; s) h(j; s)U

1k(j; s)1= (1h(j; s)1

=

U

1

r w1

) U k(i) h(i)

(3)

1

(4)

k(i)

1

h(i)1

where
(1

)
4

(5)

are the marginal costs that are the same for both …rm types. In the spirit of Yang and
Heijdra (1993), …rms take into account the e¤ect of their prices on the aggregate priceindex P .4 Due to each variety having the same production technology, superstar j charges
the same price for all of its varieties i.e. p(j; s) = p(j; k) = p(j). The markups are then

(j)

p(j)

=

1

P (j)
P

1

1

and
(i)

p(i)
P

1

p(i)

=
1

1

P (j)
P

1

p(i)
P

1

:

1

1

From the above demand functions we …nd that
P (j)
P

1

p(i)
P

1

and

=

P (j)Y (j)
PY

(j)

=

p(i)x(i)PY(i):

The markups are thus positively related to the …rms’market shares and for the superstar…rm this market share is increasing in the number of varieties S(j).
Superstar j maximizes pro…ts
(j) =

p(j)
p(j)

P Y (j)[S(j) s +f]with respect S(j) and takes into account the e¤ect of its product scope on its own prices,
@ (j)
= 0,
prices of other …rms, and the aggregate price index. The …rst-order condition, @S(j)implies
s =

PY

p(j)
p(j)

2

@ (j)
+ Y (j)
@S(j)

p(j)
p(j)

@P
:
@S(j)

(6)

@ (j)
@P
It can be shown that @S(j)> 0 and @S(j)< 0: The term on the left-hand side represents

the direct cost of expanding the product scope. The …rst term on the right-hand side
represents the gain to market share due to the love of variety in the CES aggregator(2). The second term indicates that pro…ts are reduced due to the higher product scope
reducing the aggregate price index. Again, as demonstrated in Pavlov and Weder (2017),
given the Dixit and Stiglitz (1977) demand system and …xed costs structure, a form ofvariety e¤ect is required for multi-product …rms to exist.
4

Our economy can also be interpreted as a representative sector where …rms take into account thee¤ect of their prices on the sectoral price index. We abstract from explicitly modelling sectors to keep
the presentation tidy.

5

2.3

Symmetric equilibrium

In the symmetric equilibrium each superstar …rm produces the same number of varieties
S(j) = S, charges the same price p(j) = ps ; and has the same market share (j) =s:Similarly, for the ordinary …rm p(i) = po and (i) = o hold. The markups arrange to
(1
(1

s =

s)
s)

1

>

(1
(1

o =

o)
o)

1

>

(7)

1

and
1
z = Sps> o = po1

with the …nal good set as the numeraire P = 1: Superstar …rms have larger market sharesand markups than regular …rms due to producing multiple products. Since both ordinaryand superstar …rms hire labor and capital services from the same factor markets and bothhave constant returns to scale (abstracting from …xed costs) production functions, from
(3) and (4) we obtain
w
1
Ko U
1
Ks U
=
=
r
Ho
Hs
in which Hs = N Shs and Ho = M ho : Therefore, all …rms choose an identical capital-labor
intensities

Ks
Ko
=
:
Ho
Hs

From the marginal costs (5) we get
ps =

s

(1

)

1

r w1

> po

thus, superstars charge a higher price than their ordinary counterparts. Moreover, summing production and demand functions of ordinary …rms
MXx(i) =

i=0

M
X
i=0

p(i)
P

Y =

po
U Ko Ho1
M o

Y =

j=1

0

j=1

0

p(j; s)P

U k(i) h(i)1oi=0then applying symmetry yields

Similarly, for superstar …rms
N Z S(j)N Z S(j)X
X
x(j; s)ds =M
X

Y ds =M o :Z S(j)
NX
j=1

which yields

U k(j; s) h(j; s)1

s

0

ps
(U Ks Hs1
N S s N f ):N s
Lastly, the …rst-order condition (6) rearranges for the product scopeY
S = f ( s ; o ; N; M; )s pswhere the function f > 0. We delegate details on the derivation of the product scope
to Appendix A.1. Finally, the capital and labor markets must be in equilibrium, that isY =K = Ko + Ks and H = Ho + Hs :
6

ds

f

!

2.4

PeoplePeople are personi…ed by a representative agent who chooses sequences of consumption
Ct and hours worked Ht to maximize lifetime utility
1
X

t

ln Ct

t=0

in whichis the discount rate,

Ht1+
1+

> 0;

> 0;

0

denotes the disutility of working andis the inverseof the Frisch labor supply elasticity. The agent owns all …rms and receives the pro…ts
that they potentially generate. The budget is thus constrained by
wt Ht + rt Ut Kt +tt

It + Ct

in which It is investment that adds to the capital stock
Kt+1 = (1t )Kt + It

and the depreciation rate varies according to
t =

1

Ut

> 1:

The …rst-order conditions from the agent’s maximization problem combine for the labor
supplyHt Ct = wt
the Euler equation
1
1=(rt+1 Ut+1 + 1
Ct
Ct+1

t)

and the optimal rate of capital utilization
rt = Ut

1:

The steady state versions of these equations then pin down

3

= (1=

1 + )= .

Dynamics

Let us now analyze the local dynamic properties of the model. The equilibrium conditions
are log-linearized and the dynamical system is arranged to
"
#
" #
b t+1
bt
K
K
=
J
bt+1
bt :
C
C

Hatted variables denote percent deviations from their steady-state values and J is the 2 2
Jacobian matrix of partial derivatives. Note that Ct is a non-predetermined variable andthat Kt is predetermined. Indeterminacy, and the potential presence of animal spirits,
7requires both roots of J to be inside the unit circle. For easier comparison to previous
studies, the standard parameters are calibrated at a quarterly frequency as= 0:3,= 0:99, = 0:025 and = 0. We directly calibrate the steady state markups and theelasticity of substitution : From (7), these parameters then determine the market shares
1

s

s = 1

1

s

1

o

> o=1
o

1

:

Since market shares sum to unity, M o + N s = 1; we can then calibrate the market shareof superstar …rms, N s , to pin down the number of …rms at the steady state
M=

1

N s
o

1

N=
s

o
1+ M
N s

:

It is then straightforward to show that for each calibration ofo , the lower bound onis1): As approaches this lower bound, the number of regular …rms approaches
in…nity and their markups become constant at =(1) as in the monopolistic competio =( otion version of the model. This is also where the love of variety 1=(
CES aggregator (2) is at its maximum. For the upper bound,both o =( orespectively.

o1)(1 + M) andN ss =( s1) implied by themust be smaller thanN s1)(1 + M) to guarantee Mo> 1 and N > 1;

Figure 1 plots the feasible parameter space for indeterminacy and determinacy. We
can clearly see that a rise in the market share of superstars increases both zones and allows
for greater di¤erences between the markups of superstar and regular …rms. Animal spirits
arise more easily in situations where substantial markup di¤erences, as documented bymany studies mentioned previously, exist between small and large …rms. When markups
approach unity, the implied variety e¤ect 1= (1) becomes too small and indeterminacy

cannot arise. That is, low markups imply a small gain to product creation and a weak
endogenous ampli…cation mechanism.
The way indeterminacy arises is best explained via the equilibrium wage-hours locus.
Product creation within superstar …rms makes this locus upwardly sloping and indeterminacy arises when it becomes steeper than the labor supply curve. The reason is thepresence of love of variety in the CES aggregator (2). The composite good from eachsuperstar can be created more e¢ ciently the greater the product scope. Endogenous
variations in superstars’ product scopes thus generates an endogenous e¢ ciency wedgethat expands production possibilities. If people feel optimistic about the future path ofincome and consumption, the labor supply curve shifts up along the upwardly sloping
wage-hours locus, thereby raising employment and output, validating the initial optimistic
expectation.

8

Figure 1: Indeterminacy (blue) and determinacy (orange) zones.

Figure 2 illustrates the dynamics of the model in response to an animal spirits shockthat raises output one percent above its steady state. We set the two markups at 1.4
and 1.8 as well as the market share of superstars at 60 percent which, as we argue below,
matches broadly the reported values in De Loecker et al. (2020) and Bernard et al.
(2010). Conditional on an animal spirits shock, product creation and the markups of large…rms are procyclical whereas smaller …rms’ markups are countercyclical. This pattern
is not unlike the unconditional correlations of markups and output that are reported
by Burstein et al. (2023). Unlike, Minniti and Turino (2013) and Pavlov and Weder(2017), where markups are countercyclical, the number of …rms here is constant. Productscope adjustments together with …rm heterogeneity thus provide a novel indeterminacy
mechanism for market share reallocation and markup dynamics without entry and exit.
The model’s multiplicity carries over to environments with endogenous entry and exit asin the following section.

4

Estimation

So far, we have shown that superstar …rms can lead to macroeconomic instability which
opens the possibility of animal spirits driving business cycles. The current section estimates both the indeterminate and the determinate model to assess the importance of the
product creation mechanism within superstars on explaining aggregate ‡uctuations. In
doing so, we examine the importance of animal spirits versus other fundamental shocks
and see whether they help to replicate the basic business cycle facts by comparing themodel’s second moments to the U.S. quarterly time series counterparts. Appendix A.4
9

1

1

0.5

0.5

0

0

0.04
0.03
0.02
0.01
0

-0.5

-0.5
5

10

15

20

5

10

15

20

6

0.15

0.01

4

0.1

0

2

0.05

-0.01

0

0

-0.02

-2

-0.05
5

10

15

20

10

15

20

0.3

0.04

0.2

0.02

0.1

0

0

10

15

20

5

10

15

20

5

10

15

20

-0.03
5

0.06

5

1

0.5

-0.02

0

-0.1
5

10

15

20

-0.5
5

10

15

20

Figure 2: Impulse responses to an ouput sunspot shock, percent deviations from thesteady state, s = 1:8; o = 1:4; = 4:5:

sets out the exact data sources. The models used here are extended by exogenous growth,fundamental aggregate supply and demand shocks, external habits in consumption, separable love of variety, and endogenous entry and exit of ordinary …rms.

4.1

The extended model

Since entry and exit of superstar …rms is not likely to be signi…cant at business cycle
frequencies, we have continued with the assumption that N is constant, but Mt now
adjusts via free entry of ordinary …rms that forces their pro…ts to zero. That is, eachperiod …rm i’s pro…ts aret (i) =

pt (i)
pt (i)t

Pt Yt t (i)t o = 0which in symmetric equilibrium boils down topo;t = ( o;t1)o;t Ytoto determine the number of ordinary …rms. Similar to product scope adjustments, entry and exit enlarges the indeterminacy region due to the e¢ ciency gains in productaggregation from the love of variety in the CES aggregator (1).5
The love of variety governs the gain to product creation for superstar …rms and is thecentral ampli…cation mechanism in our model, leading to equilibrium indeterminacy as
5

The entry decision is static to keep the model tractable. Appendix A.2 shows that indeterminacy
remains when we introduce dynamic entry as in Bilbiie et al. (2012).

10explained in Section 3. Similar to Pavlov and Weder (2017), here we separate the varietye¤ect from the elasticity of substitution

and we do it for two main reasons. First,

isolating the variety e¤ect allows us to estimate the direct bene…t of product creation
separately from the parameter that primarily determines the markup elasticities and thesteady state number of …rms. Second, it gives us an option to estimate a determinate
version of the model where the gain to product creation is too small for animal spirits toplay a role. Speci…cally, the CES aggregators are now
(

Yt =

1) 1

Mt

Mt
X

1

xt (i)

+N

(

1) 1

and
St (j)(

1) 1

Z St (j)xt (j; s)10in which

Yt (j)

1

j=1

i=1

Yt (j) =NX> 0 debotes the love of variety. Setting

CES aggregators from Section 2. As

!

!

1

1

ds

= 1=(

1) would bring back theis a key parameter that transports the economy

between the determinacy and indeterminacy zones, in the following estimation we will
let data decide its magnitude.
We include a mix of fundamental supply and demand disturbances to the model. The…rst such fundamental shock takes the form of labor augmenting technological progress
At and it a¤ects all …rms equally. Aggregate output is now
Yt =

ps;t
[(Ut Ks;t ) (At Hs;t )1
N s;ts N StfN] =po;t[(Ut Ko;t ) (At Ho;t )1
Mt o;tMt o ]:

Technological progress is non-stationary and follows the process
ln At = ln At 1 + ln at
ln at = (1A ) ln a +AA ln at 1 + "twhere 0A < 1 governs the persistence of the shock, ln a is the average growth rateAand "t is an i.i.d. disturbance with variance 2A . Next, shifts of marginal e¢ ciency ofinvestment zt a¤ect the transformation of investment to physical capital as in Greenwood
et al. (1988)
Kt+1 = (1t )Kt + zt It :The technological shifter follow the exogenous process
ln zt =zz ln zt 1 + "t :As laid out by Justiniano et al. (2011), this shock can be a stand-in for disturbances of…nancial markets on investment behavior. Intuitively, a positive shock to zt represents
a boom in …nancial markets that reduces borrowing costs for …rms, leading to a rise ininvestment.
11

The …rst fundamental demand disturbance is a taste shockt to the agent’s period

utility that increases the marginal utility of consumption as in Christiano (1988). Lifetime
utility becomes
E0

1
X

t

t ln(Ct

Ht1+
1+

bCt 1 )

t=0

where E0 denotes the expectations operator and parameter 0

b < 1 determines thedegree of external habits in consumption. The taste shock follows the processt =t 1 + "t :This shock a¤ects the economy’s labor wedge, i.e. the gap between the marginal rateof consumption-leisure substitution and the marginal product of labor and hence can beinterpreted as a stand-in for other shocks that a¤ect this wedge. The second demand
shock is to government expenditures, Gt , …nanced by lump sum taxes. Consequently, theeconomy’s resource constraint becomes
Yt = Ct + It + Gt :Lastly, government spending Gt follows a stochastic trend
Agt = (Agt 1 ) ag (At 1 )1ag

where ag governs the smoothness of the trend relative to the trend in output. Then,
detrended government spending is gt Gt =Agt and follows
ln gt =

gg ln gt 1 + "t :As in Pavlov and Weder (2017), the non-fundamental animal spirits shock is modelled
as an expectation error to output that is unrelated to any fundamental changes in theeconomy6 . Under indeterminacy, the economy’s response to fundamentals is not uniquelydetermined, and we model the behavior of output as
Ybt = Et 1 Ybt +where the parameters

A,

z,

A
A "t +zz "t +"t +gsg "t + "tandg determine the e¤ect of technology, investment,
preference and government shocks on output. The term "st is i.i.d., independent of fundamentals and comes with variance 2s . It can be thought of pro…t-seeking businessmen

exercising their animal spirits.
6

Farmer et al. (2015) show that estimation results are robust to the choice of expectation error.

12

4.2

Bayesian estimation

The model is estimated via Bayesian methods with the procedure largely following Farmer
et al. (2015) and Pavlov and Weder (2017). U.S. data includes quarterly real per capita
growth rates of output, consumption, investment, government spending and the logarithmof per capita hours worked from 1990:I-2019:IV as observables. We use credit spread data
to identify investment shocks z^t as in Justiniano et al. (2011). Concretely, we adopt thecredit spread between BAA corporate bonds and the market yield on 30 year Treasury
securities to identify disturbances to the marginal e¢ ciency of investment
spreadt = {bzt

{ < 0:

We focus on the period post 1990 to coincide with the rise of superstar …rms as reported
by De Loecker et al. (2020) and abstract from the COVID-19 pandemic as our small
scale model is not designed to deal with its complexities.
We follow Bilbiie et al. (2012) and de‡ate Yt ; Ct ; It ; and Gt in the model by a dataconsistent price index to obtain variables that are more comparable to the data, which
does not take into account the welfare improvements of product variety at quarterly
frequency. For example, data-consistent output is
Ytd

Pt Yt
pt

P t Yt
Pt Yt
Mt t;o +
N t;s
po;t
ps;t

which removes the welfare gains coming from entry and product scope adjustments. The
measurement equation is thus
22
3
ln Yt ln Yt 1
66 ln Ct ln Ct 1 7 66
7 66 ln It ln It 1 7 66
7 66 ln Gt ln Gt 1 7 = 6 cd
6
7 6 G t4 ln Ht ln H 5 64
spreadt

3
2
3 2 m:e: 3
Ycd t Ycd t 1 + b
at
ln a
"t
7
c
c
d
d
6
6
7
7
7
C t C t 1 +b
at
7 6 ln a 7 6 0 7
7 6 ln a 7 6 0 7
Ibd t Ibd t 1 + b
at
7+6
7+6
7
7 6
7 6
7
cd t 1 + abg t abg t 1 + b
G
at 7 6 ln a 7 6 0 7
7 4 0 5 4 0 5
bt
5
H
0
0
{b
zt

where agt = Agt =At = (agt 1 ) ag at 1 , "m:e:is a measurement error restricted to account fort
not more than ten percent of output growth and ln H is the logarithm of the average
hours worked over the sample period. We calibrate a subset of the model parameters
to avoid identi…cation issues. We set the quarterly growth rate of labor augmenting
technological progress to 0:34 percent to be consistent with the growth rate of per capita
real GDP over the sample period. The share of government expenditures in output G=Y
is 0:19 and the values of standard parameters remain the same as in previous sections.
In line with Barkai (2020), we calibrate the share of …xed costs in output so that thesteady state pro…ts are ten percent. We calibrates,o,, and N s as follows. A large

portion of …rms are multi-product producers. Bernard et al. (2010) report that close
13

Table 1: Prior and posterior distributions
Name
b
A
z

g
ag
s
A
z

g
m:e:
A
z

g

{

Range
R+
[0,1)
[0,1)
[0,1)
[0,1)
[0,1)
[0,1)
R+
R+
R+
R+
R+
[0; 0:18][-3,3][-3,3][-3,3][-3,3][-20,0]Prior
Density
Normal
Beta
Beta
Beta
Beta
Beta
Beta
Inverse Gamma
Inverse Gamma
Inverse Gamma
Inverse Gamma
Inverse Gamma
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform

Mean
0.075
0.5
0.5
0.5
0.5
0.5
0.5
0.1
0.1
0.1
0.1
0.1
0.09
0
0
0
0
-10

Std. Dev.
0.05
0.1
0.2
0.2
0.2
0.2
0.2
Inf
Inf
Inf
Inf
Inf
0.05
1.73
1.73
1.73
1.73
5.77

Mean
0.19
0.52
0.00
0.79
0.95
0.99
0.86
0.26
0.70
0.07
0.47
0.81
0.18
-0.53
1.60
0.67
0.09
-4.71

Posterior
90% Interval[0.16,0.22][0.43,0.60][0.00,0.01][0.73,0.86][0.93,0.97][0.98,0.99][0.65,0.99][0.23,0.29][0.63,0.78][0.04,0.10][0.42,0.52][0.72,0.89][0.18,0.18][-0.63,-0.42][0.72,2.57][0.54,0.79][-0.01,0.19][-6.73,-2.75]

This table presents the prior and posterior distributions for model parameters and shocks. Inf
implies two degrees of freedom for the inverse gamma distribution. Standard deviations are inpercent terms. Log-data density (modi…ed harmonic mean): -780.60.

to half of U.S. manufacturing …rms produce in multiple …ve-digit SIC industries. These…rms account for well over 80 percent of total sales. We accordingly calibrate the market
share of superstars to 60 percent, i.e. N s = 0:6. A conservative interpretation of thecomposition of markups reported in De Loecker et al. (2020) suggests a markup of large…rms to be around s = 1:8 as this is the rough average for the (revenue weighted) top
75 to top 90 percentiles of …rms. Smaller …rms, say the top 50 percentile, have seen asteady markup at aroundo = 1:3. Lastly, we agnostically pick a number for= 5 thatfalls in the middle of its admissible values. We take these numbers as our benchmark
calibration. We can demonstrate robustness.
The remaining parameters are estimated. These include the love of variety, , external habits, b; the coe¢ cient mapping the credit spread to investment shocks, {, andparameters that govern the stochastic processes:

A,

z,

,

g,

ag ,

s,

A,

z,

,

g,

, g , and m:e: . Table 1 presents the initial prior and posterior distributions.
We employ a normal distribution, truncated at zero, for the variety e¤ect . Since thisA,z,parameter is central to our ampli…cation mechanism which generates indeterminacy, we
set the mean to 0:075 to give a prior probability of determinacy of about 50 percent.
A wide uniform distribution is employed for the expectation error parameters A , z ,, g and the credit spread coe¢ cient {. The shock processes follow the standard
inverse gamma distribution. The Metropolis-Hastings algorithm is employed to obtain

14

Table 2: Business cycle dynamicsxln(Yt =Yt 1 )ln(Ct =Ct 1 )ln(It =It 1 )ln(Gt =Gt 1 )ln(Ht =H)
spreadt

x

0.58
0.47
1.66
0.77
6.16
0.60

Data
(x; ln(Yt =Yt 1 ))
1
0.67
0.79
0.25
0.20
-0.58

ACF
0.29
0.38
0.62
0.24
0.99
0.85

x

0.79
0.63
2.97
0.83
4.80
0.56

Model
(x; ln(Yt =Yt 1 ))
1
0.49
0.82
0.13
0.12
-0.25

ACF
0.49
0.48
0.62
0.04
0.99
0.79

Business cycle statistics for the arti…cial economy are calculated at the posterior mean. x
denotes the standard deviation of variable x, (x; ln(Yt =Yt 1 )) is the correlation of variable x
and output growth, and ACF is the …rst order autocorrelation coe¢ cient.

500; 000 draws from the posterior mean for each of the two chains. Half of the draws
are discarded and the scale in the jumping distribution is adjusted to achieve a 25

30

percent acceptance rate for each chain. The table shows that the parameters are precisely
estimated and consistent with previous studies. The love of variety is estimated at 0:19,
which suggests a strong ampli…cation mechanism of product creation within superstars
that guarantees indeterminacy but is below the value that would be implied under theoriginal CES con…guration in Section 2, i.e. 1=(
1) = 0:25. The persistence of thepermanent technology shock is essentially zero and, consistent with a determinate realbusiness cycle model, a positive shock causes a fall in detrended output. The investment
shock is moderately persistent and as expected, raises output on impact. Finally, bothdemand shocks are highly persistent and also cause an increase in output.
The choice of priors leads to a prior predictive probability of indeterminacy of 0:50
and indicates no prior bias toward either determinacy or indeterminacy. The …rst main
result is that, through the lens of our model, the post 1990 period, is best characterized
by the indeterminacy version of our model. Speci…cally, the log data densities come in as
780:60 for the indeterminacy model versus 914:96 for its competitor (see Appendix
A.3). Thus, since data prefers the indeterminacy, and in winner takes it all fashion, thebelow will present the results for the animal spirits driven model.
Table 2 reports the theoretical second moments of the main macroeconomic aggregates and the credit spread at the posterior mean. Our admittedly small scale model
replicates the behavior of the considered U.S. macroeconomic variables quite well. The
relative volatilities and correlations are consistent with the data. The model slightly overpredicts the volatilities of output and consumption but underpredicts for hours worked.
Government expenditures and the credit spread are matched well. The sole outlier isinvestment, where the model strongly overpredicts its variance. As a result of the richer
internal propagation mechanism under indeterminacy, the autocorrelation functions areable to show persistence in the growth rates despite the lack of the many real frictions
employed in the literature.
15

Table 3: Unconditional variance decomposition (in percent)
ln
"st
"A
t
"zt
"t
"gt

Yt
Yt 1

13.18
30.26
19.23
34.80
2.53

ln

Ct
Ct 1

ln

0.38
34.74
0.69
64.12
0.08

Xt
Xt 1

20.49
18.65
29.00
28.10
3.76

ln

Gt
Gt 1

0
5.42
0
0
94.58

Ht
H

spreadt

2.81
16.46
20.21
51.29
9.23

0
0
100
0
0

ln

Variance decompositions are performed at the posterior mean.

Table 3 displays the variance decomposition which reveals the relative contributionof each of the shocks to the macroeconomic aggregates. As discussed earlier, the nonfundamental animal spirits propagate the fundamental disturbances, while also causing‡uctuations on their own. The e¤ect of "pure" animal spirits on the U.S. business cycle isnon-trivial: they drive a modest fraction of output and a sizeable portion of investment.
We see this as success as these shocks primarily stand for business’ expectations andtheir alternations between euphoric and pessimistic states. The technology shock o¤ers agood explanation of output and consumption, while the marginal e¢ ciency of investment
shock best explains investment data. In fact, its importance shrinks considerably whencompared to Justiniano et al. (2011). The preference shock is the dominant shock andexplains most of consumption, half of hours worked, and about a third of output andinvestment. Finally, the government expenditure shock is a negligible source of businesscycles.

5

Concluding remarks

Recent empirical research has highlighted the signi…cant rise of market power in the lastfew decades primarily driven by the largest (superstar) …rms. The current paper examines the impact on aggregate ‡uctuations of these multi-product superstars through theirinteraction with smaller competitors. We …nd that the rising market share of superstar…rms increases the parameter space for indeterminacy and allows for greater divergence
between markups of large and small …rms. Market share reallocation between superstars and ordinary …rms via product scope adjustments generates heterogeneous markup
dynamics across …rms and time. The endogenous ampli…cation mechanism of productcreation within superstars improves the …t of the estimated general equilibrium modeland implies animal spirits play a non-trivial role in driving U.S. business cycles.

References[1] Baqaee, D. and E. Farhi (2020): “Productivity and Misallocation in General Equilibrium“, Quarterly Journal of Economics 135, 105-163.
16

[2] Barkai, S. (2020): “Declining Labor and Capital Shares“, Journal of Finance 75,
2421–2463.
[3] Bartelsman, E., J. Haltiwanger and S. Scarpetta (2013): “Cross-Country Di¤erencesin Productivity: The Role of Allocation and Selection,“ American Economic Review
103, 305-334.
[4] Bilbiie, F., F. Ghironi and M. Melitz (2012); “Endogenous Entry, Product Variety,and Business Cycles“, Journal of Political Economy 120, 304-345.
[5] Bernard, A., S. Redding and P. Schott (2010): “Multi-product Firms and ProductSwitching“, American Economic Review 100, 70-97.
[6] Broda, C. and D. Weinstein (2010): “Product Creation and Destruction: Evidence
and Price Implications“, American Economic Review 100, 691-732.
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UCLA mimeo.
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[9] De Loecker J., J. Eeckhout and G. Unger (2020): “The Rise of Market Power andthe Macroeconomic Implications,“ Quarterly Journal of Economics 135, 561-644.
[10] Dixit, A. and J. Stiglitz (1977): “Monopolistic Competition and Optimum Product
Diversity“, American Economic Review 67, 297-308.
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Decline“, Quarterly Journal of Economics 136, 1031-1087.
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17[17] Neary, P. (2010): “Two and a Half Theories of Trade,“ The World Economy 33,
1-19.
[18] Pavlov, O. (2021): “Multi-product Firms and Increasing Marginal Costs,“ Journalof Economic Dynamics and Control 133, 104239.
[19] Pavlov, O. and M. Weder (2017): “Product Scope and Endogenous Fluctuations“,
Review of Economic Dynamics 24, 175-191.
[20] Shimomura, K and J. Thisse (2012): “Competition Among the Big and the Small“,
RAND Journal of Economics 43, 329-347.
[21] Yang, X. and B. Heijdra (1993): “Monopolistic Competition and Optimum Product
Diversity: Comment“, American Economic Review 83, 295–301.

A

Appendix

A.1

Product scopeThis Appendix derives the …rm’s optimal product scope. Product scope determination
largely follows the approach of Minniti and Turino (2013) and Pavlov and Weder (2017).
Firm j maximizes pro…ts with respect to S(j) and takes into account the e¤ect of its
product scope on its own prices, prices of other …rms, and the aggregate price index.
First we rewrite pro…ts as
p(j)
p(j)

(j) =

P Y (j)[S(j) s +f]and obtain the …rst-order condition
@ (j)
= PY
@S(j)

2

p(j)
p(j)

@ (j)
+ Y (j)
@S(j)

Then

@ (j)
(j)
=
(
@S(j)
S(j)
and for other multi-product …rms

1) (j)

p(j)
p(j)

@P
@S(j)

1 @p(j)p(j) @S(j)

1 @P
P @S(j)

@ (k)
=
@S(j)

(

1) (k)

1 @p(k)p(k) @S(j)

1 @P
P @S(j)

@ (i)
=
@S(j)

(

1) (i)

1 @p(i)p(i) @S(j)

1 @P:
P @S(j)and regular …rms

Next, rewrite the aggregate price index as
P =

M
X
i=1

p(i)1

di +

N
X
k=1

18

S(k)p(k)1!11:

s = 0:

From here, we use symmetry to simplify. After some algebra @P=@S(j) can be expressed as@Po @p(i)
s
=M
+
@S(j)
po @S(j) ps
where

(N

1)

@p(k)
@p(j)
+
@S(j) @S(j)

+

1
1

S

@p(i)
=
@S(j)

(

1)( o

1)(1

1= o ) o

@p(i)
@S(j)

po

@P
@S(j)

@p(k)
=
@S(j)

(

1)( s

1)(1

1= s ) s

@p(k)
@S(j)

ps

@P
@S(j)

1)(1

1= s ) ps

@p(j)
= ( s
@S(j)

s

(

S

1) s

@p(j)
@S(j)

ps

s

@P
@S(j)

:

@p(k)
@p(i)
@P
Putting all these together, it can then be shown that @S(j)< 0, @S(j)< 0; @S(j)
< 0,
@p(j)
@S(j)

@ (j)
@ (k)
@ (i)
@ (j)
@p(j)
@P
> 0; @S(j)> 0; @S(j)< 0; and @S(j)< 0: Finally, @S(j); @S(j); and @S(j)can be@ (j)= 0 to …nd the product scope
substituted in the …rst-order condition @S(j)S = f ( s;o ; N; M;

)

Y
s ps

where f > 0.

A.2

Dynamic entry

This Appendix presents the version of the model where the entry of mono-product …rms is
dynamic as in Bilbiie et al. (2012) and shows that indeterminacy remains. A prospective
entrant i computes their expected value
vt (i) = Et

1
X

Qt;s o;t+s (i)s=1where Qt;s is the stochastic discount factor ando;t (i) denotes pro…ts of ordinary …rms.

There is a time-to-build lag in that period t entrants begin operating in period t + 1 andthe number of …rms evolves according to
Mt = (1where

M

M )(Mt 1 + ME;t 1 )is the exogenous exit probability and ME;t is the number of entrants. Entry

occurs until the expected value, vt (i); is equal to the sunk cost of entry. To enter, fE
amount of labor needs to be hired and since labor is paid the real wage wt ; this sunk cost
is equal tovt (i) = wt fE :The production function for new …rms is thus
ME;t =
19HE;tfE

where HE;t is the amount of labor hired for the production of new …rms. In a symmetricequilibrium, a representative household enters period t with mutual fund share holdings
xt and has the budget constraint
Ct + It + vt (Mt + ME;t )xt+1 = ( o;t + vt )Mt xt + wt Ht + rt Ut Kt + N s;twheres;t are pro…ts from a constant number of superstar …rms and Ht = HE;t +Ho;t +Hs;t .

The Euler equation for share holding is thenvt = Et (1M)

Ct( o;t+1 + vt+1 ):Ct+1Imposing the equilibrium condition xt+1 = xt = 1 for all t gives
Ct + It + vt ME;t =o;t Mt + wt Ht + rt Ut Kt + N

Yt

s;t

where Yt is GDP consisting of consumption, investment in capital, and investment in new…rms. Total investment is thenXt

It + vt ME;tand the CES aggregator is now
MX

Yg;t = Ct + It =xt (i)

1

+

N
X
j=1

i=1

Yt (j)

1!

1:

Small …rms no longer have …rm-level …xed costs and the symmetric equilibrium goods
production is then1po;t Ut Ko;t Ho;t
Yg;t =
Mt o;t=1
ps;t Ut Ks;t Hs;t

ps;t N St s
N s;tps;t N f:

We calibrate the model as in Section 3 and additionally set M = 0:025: Analogous to
Figure 1, Figure A1 plots the indeterminacy and determinacy zones for di¤erent levelsof superstar market shares. Indeterminacy not only remains but now exists for a greater
range of parameters. The reason is due to the interaction between entry of ordinary …rms
and the product scope of superstars. Entry pushes the market shares and markups ofboth …rm types downwards. However, superstar …rms are able to defend their market
shares by increasing their product scopes. Since higher product scopes and more small…rms both increase e¢ ciency via the love of variety e¤ect, the wage-hours locus becomessteeper relative to the labor supply curve and indeterminacy becomes more plausible.

A.3

Determinacy model

Table 4 presents the prior and posterior distributions of the estimated determinate versionof the model. The main di¤erence here is that the animal spirits shock is no longer
20Figure A1: Dynamic entry model: indeterminacy (blue) and determinacy (orange).

available and in order to keep the number of shocks equal to observables, we introduce a
temporary technology shock, ATt , that a¤ects all …rms equally with persistence T andvariance

T : For example, the output of an ordinary …rm is now

xt (i) = ATt kt (i) [At ht (i)]1

o;t

where
ln ATt =TT
T ln At 1 + "t :Comparing the log-data densities between Tables 1 and 4, data clearly favors the indeterminate model with a strong product creation mechanism and animal spirits shocks.

A.4

Data sources

This Appendix details the source and construction of the U.S. data used in Section 4.
All data is quarterly and for the period 1990:I-2019:IV.
1. Gross Domestic Product. Seasonally adjusted at annual rates, billions of chained(2009) dollars. Source: Bureau of Economic Analysis, NIPA Table 1.1.6.
2. Gross Domestic Product. Seasonally adjusted at annual rates, billions of dollars.
Source: Bureau of Economic Analysis, NIPA Table 1.1.5.
3. Personal Consumption Expenditures, Nondurable Goods. Seasonally adjusted atannual rates, billions of dollars. Source: Bureau of Economic Analysis, NIPA Table 1.1.5.
4. Personal Consumption Expenditures, Services. Seasonally adjusted at annual rates,
billions of dollars. Source: Bureau of Economic Analysis, NIPA Table 1.1.5.
21

Table 4: Prior and posterior distributions for the determinate model
Name
b
T
A
z

g
ag
T
A
z

g
m:e:

{

Range
R+
[0,1)
[0,1)
[0,1)
[0,1)
[0,1)
[0,1)
[0,1)
R+
R+
R+
R+
R+
[0; 0:18][-20,0]Prior
Density
Normal
Beta
Beta
Beta
Beta
Beta
Beta
Beta
Inverse Gamma
Inverse Gamma
Inverse Gamma
Inverse Gamma
Inverse Gamma
Uniform
Uniform

Mean
0.075
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.1
0.1
0.1
0.1
0.1
0.09
-10

Std. Dev.
0.05
0.1
0.2
0.2
0.2
0.2
0.2
0.2
Inf
Inf
Inf
Inf
Inf
0.05
5.77

Mean
0.01
0.42
0.90
0.07
0.90
0.99
0.98
0.99
0.12
0.78
0.02
0.47
0.80
0.18
-15.03

Posterior
90% Interval[0.00,0.02][0.34,0.48][0.87,0.92][0.05,0.09][0.83,0.98][0.98,0.99][0.97,0.99][0.99,0.99][0.10,0.14][0.69,0.87][0.02,0.03][0.41,0.53][0.71,0.89][0.18,0.18][-19.20,-11.37]

This table presents the prior and posterior distributions for model parameters and shocks ofthe determinate model. Log-data density (modi…ed harmonic mean): -914.66.

5. Personal Consumption Expenditures, Durable Goods. Seasonally adjusted at annual rates, billions of dollars. Source: Bureau of Economic Analysis, NIPA Table 1.1.5.
6. Gross Private Domestic Investment, Fixed Investment, Residential. Seasonallyadjusted at annual rates, billions of dollars. Source: Bureau of Economic Analysis, NIPA
Table 1.1.5.
7. Gross Private Domestic Investment, Fixed Investment, Nonresidential. Seasonallyadjusted at annual rates, billions of dollars. Source: Bureau of Economic Analysis, NIPA
Table 1.1.5.
8. Government consumption expenditures and gross investment. Seasonally adjusted
at annual rates, billions of dollars. Source: Bureau of Economic Analysis, NIPA Table
1.1.5.
9. Nonfarm Business Hours. Index 2009=100, seasonally adjusted. Source: Bureauof Labor Statistics, Series Id: PRS85006033.
10. Civilian Noninstitutional Population. 16 years and over, thousands. Source:
Bureau of Labor Statistics, Series Id: LNU00000000Q.
11. GDP De‡ator = (2)=(1):12. Real Per Capita Output, Yt = (1)=(10):
13. Real Per Capita Consumption, Ct = [(3) + (4)]=(11)=(10):
14. Real Per Capita Investment, Xt = [(5) + (6) + (7)]=(11)=(10):15. Real Per Capita Government Expenditures, Gt = (8)=(11)=(10):16. Per Capita Hours Worked, Ht = (9)=(10):22