Discussion Papers In Economics
And BusinessEﬃcient policy with ﬁrm heterogeneityand variable markups
Atsushi Tadokoro

Discussion Paper 20-17

January 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Eﬃcient policy with ﬁrm heterogeneity and variablemarkups∗
Atsushi Tadokoro †January 2021

Abstract
This study analyzes policy instruments to make the market outcome achieve theeﬃcient outcome in a model of monopolistic competition with ﬁrm heterogeneity andvariable markups. In this model, ﬁrm heterogeneity and the markup pricing create distortions in the market equilibrium. Ad valorem and per-unit production taxes/subsidies
work diﬀerently to improve these distortions. I show that by combining an ad valorem
production tax and a per-unit production subsidy, these distortions are removed without adopting a ﬁrm-speciﬁc tax/subsidy.

Keywords: Variable markups, Misallocation, Firm heterogeneity, Monopolistic competition
JEL Classiﬁcation: D4, D6, F1, L0, L1∗

I thank Koichi Futagami and Kazuhiro Yamamoto for helpful comments and suggestions. All remaining
errors are my own.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043,
JAPAN; e-mail address: sifurywhizagbal@gmail.com

1

Introduction

In the canonical models of monopolistic competition with constant elasticity of substitution(CES) preferences, which generate constant markups, opposing externalities related to the
ineﬃciency of product variety oﬀset each other. That is, the market equilibrium is eﬃcientwhen there is no other sector than the monopolistically competitive one. Therefore, there
is no room for welfare-improving policy intervention, which reduces opportunities to address
the many distortion-related problems originally associated with monopolistic competition.1
In this study, I examine the eﬀects of policy intervention on welfare based on the model of
Melitz and Ottaviano (2008), who incorporate endogenous markups by introducing the linear
demand system into the model of Melitz (2003). The presence of endogenous markups aﬀects
the eﬃciency of resource allocation. Based on the closed economy setting of the MelitzOttaviano model, Nocco et al. (2014) show that, due to ﬁrm heterogeneity and variablemarkups, the market outcome fails to be eﬃcient in terms of product selection, product mix,
and product variety.2 The purpose of the present study is to provide policy instruments tocorrect these distortions and make the market outcome achieve the eﬃcient outcome.
Regarding the eﬀectiveness of policy instruments, Nocco et al. (2014) characterize thepolicy tools that policymakers can use to make the market outcome achieve the eﬃcient
outcome. Under an unconstrained choice of tools, Nocco et al. (2014) show that the market
can achieve the ﬁrst-best outcome through a ﬁrm-speciﬁc per-unit production tax/subsidy, alump-sum entry tax, and a lump-sum tax on consumers. When the ﬁrm-speciﬁc production
subsidy/tax is unavailable, they consider a second-best scenario in which a per-unit production subsidy is oﬀered to all ﬁrms and ﬁnanced by a lump-sum tax on consumers. Therefore,
according to Nocco et al. (2014), in order to achieve the ﬁrst-best outcome, policymakers
need to determine taxes/subsidies based on the level of productivity (marginal cost) of eachﬁrm.
The present study shows, however, policymakers do not need to have any way of observing
the level of productivity of each ﬁrm to make the market outcome achieve the ﬁrst-best outcome: the ﬁrst-best outcome can be decentralized through an uniform ad valorem production
tax, a uniform per-unit production subsidy, and a lump-sum tax on consumers. Ad valorem
1

As Dhingra and Morrow (2019) show, the market outcome is ﬁrst-best under CES preferences, withdemand-side elasticity determining how resources are misallocated.
2
Using the multi-country setting of Melitz and Ottaviano (2008), Nocco et al. (2019) also show that free
trade allocation of resources fails to be eﬃcient.

1and per-unit taxes/subsidies play diﬀerent roles on each distortion. As a result, combining
an ad valorem production tax and a per-unit production subsidy can create a tax/subsidy
system similar to a ﬁrm-speciﬁc production tax/subsidy, which enables the ﬁrst-best resource
allocation to be achieved without a ﬁrm-speciﬁc tax/subsidy.
The rest of the paper is organized as follows. Section 2 brieﬂy presents the model. Section
3 derives the market and the eﬃcient outcomes, and compares the outcomes. Section 4
characterizes the eﬃcient policy that can be implemented to attain eﬃciency at the market
equilibrium. Section 5 concludes.

2

Model

Following the closed economy of Melitz and Ottaviano (2008), consider an economy with
two sectors and one production factor, which is labor. Labor L is inelastically suppliedby consumers. The preferences of consumers are deﬁned over a continuum of diﬀerentiatedvarieties of goods and a homogeneous good. The diﬀerentiated goods are indexed by ω ∈Ω and the homogeneous good is chosen as the numeraire. All consumers share the same
preference and each consumer maximizes the following utility function:∫
U = q0c + αγ
q (ω)dω −
2
Ω
c

∫

η
q (ω) dω −
2
Ω
c

)2

(∫

2

c

q (ω)dω,(1)Ωsubject to the budget constraint∫q0c +p(ω)q c (ω)dω = I,Ωwhere Ω is the set of all available diﬀerentiated good varieties; q0c and q c (ω) are the individual
consumption levels of the numeraire good and each variety ω, respectively; p(ω) is the priceof variety ω; and I is the income. The parameters α, η, and γ are positive constants. A lower
γ indicates that the diﬀerentiated varieties become closer substitutes and in the limit case ofγ = 0, consumers care only about the total amount of diﬀerentiated goods they consume. η
represents the degree of non-separability. When η equals zero, the utility function becomesseparable across the diﬀerentiated varieties. I assume that the consumers have positive
demand for the numeraire good (q0c > 0).
2

Perfect competition prevails in the homogeneous good market. Production of one unitof a homogeneous good requires one unit of labor input. In the diﬀerentiated goods sector,
each variety is produced by a monopolistically competitive ﬁrms. To enter the market, aﬁrm pays a ﬁxed cost fe > 0—a sunk labor requirement to design a new variety—and draw
its marginal cost c—the unit labor requirement. I assume that the unit labor requirement cfollows Pareto distribution:(c ∼ G(c) =

c
cM

)θ
, c ∈ [0, cM ], θ ≥ 1,

(2)

where G(c) is the cost distribution, θ is an index of the dispersion of the cost, and cM is theupper bound of the cost.

3

Market and eﬃcient outcomes

3.1

Market outcome

The labor market and the market of the homogeneous good are perfectly competitive, andthe homogeneous good is chosen as the numeraire. Thus, the wage becomes one.
Using the ﬁrst-order conditions for utility maximization, the inverse demand for eachvariety ω is given byp(ω) = α − γq c (ω) − ηQc∀ω ∈ Ω∗ ,where Ω∗ ⊂ Ω represents the subset of varieties in which q c (ω) > 0, and Qc ≡

(3)
∫
Ω∗

q c (ω)dω isthe total consumption of all diﬀerentiated goods by individuals. By integrating both sides ofEq. (3) over Ω∗ , I obtainQc =N(α − p̄),γ + ηN(4)∫
where N is the number of consumed varieties, and p̄ = (1/N ) Ω∗ p(ω)dω is the average priceof consumed varieties. Using Eqs. (3) and (4), I obtain the following market demand forvariety ω, q(ω):q(ω) = Lq c (ω) =

3

L max
(p
− p(ω)),
γ

wherepmax ≡

γα + ηN p̄
γ + ηN(5)

represents the threshold price at which demand for a variety is driven to zero. Note thatEq. (3) implies pmax
≤ α.
i
In the diﬀerentiated good sector, when a variety is produced by a ﬁrm with unit labor
requirement c, the proﬁt maximization problem for the ﬁrm is given bymax (p(c) − c) q(c),s.t.q(c) =

L max
(p
− p(c)),
γ

where p(c) and q(c) denote the price and quantity set by a ﬁrm with marginal cost c, respectively. The proﬁt-maximizing price and quantity are1
pm (c) = (pmax + c),2
q m (c) =

L max
(p
− c),
2γ

where “m” labels the equilibrium variables.
Next, I deﬁne the cost cutoﬀ. Let cm be the upper bound of the cost for ﬁrms to earn
positive operating proﬁts at the market equilibrium:
cm = sup{c : π(c) > 0} = pmax .

(6)

As described in Melitz and Ottaviano (2008), the cost cutoﬀ represents the toughness ofcompetition in the market. Assume that cM is suﬃciently high to be greater than cm .
Using the cost cutoﬀ, I obtain the price, quantity, proﬁt, and markup of a ﬁrm with cost

4

c.
1
pm (c) = (cm + c),2
L m
q m (c) =(c − c),2γL mπ m (c) =(c − c)2 ,
4γ
pm (c)1 cmµm (c) =
= (
+ 1).
c
2 c

(7)
(8)

Lower-cost ﬁrms set lower prices but also higher markups. This generates distortions because
more productive ﬁrms do not pass on their entire cost advantage to households by raising
their markups and end up selling too little, while less productive ﬁrms end up oversupplying
the market.

∫ cm

π m (c)dG(c)−fe . Due to free entry condition,∫ cm
an entrant’s expected proﬁt is oﬀset by the sunk entry cost so that 0 π m (c)dG(c) = fe .
Prior to entry, the expected ﬁrm proﬁt is0

Given Eqs. (2) and (8), this free entry condition determines the cost cutoﬀ:∫ cm
π m (c)dG(c) = fe0[2γ(θ + 1)(θ + 2)cθM fe
⇔c =
L

1
] θ+2

m

.

(9)

Substituting Eq. (6) into Eq. (5), the number of varieties can be expressed in terms ofthe cost cutoﬀ.
γα + ηN p̄
γ + ηN2γ(θ + 1) α − cm
⇔ Nm =
,
η
cm
cm =

where
1
p̄ =
G(c)

∫ cm
p(c)dG(c) =02θ + 1 mc .
2(θ + 1)

(10)

(11)

The number of entrants is then given by Nem = N m /G(cm ).
Substituting Eqs. (10) and (11) into Eq. (4), the total consumption level of the diﬀeren-5tiated varieties, Qm , can be written by
Qm ≡ LQc =L(α − cm ).
η

(12)Finally, substituting Eqs. (7) and (10) into (1), the welfare level at the market equilibrium
can be expressed in terms of the cost cutoﬀ.
W

3.2

m

(
)
L
θ+1 m
m
= LU = L + (α − c ) α −
c
.
2η
θ+2

Eﬃcient outcome

A benevolent social planner maximizes welfare, taking as given the endowment of labor (L);the production functions of the two goods; and the mechanism determining each variety’s
unit labor requirement c—as a random draw from the distribution G(c) after fe units of labor
have been allocated to the design of a variety.
The planner’s problem is given by
max W = LU

(13)

q0 ,q(c),Ne

subject to the resource constraint:∫ cM
q0 + Necq(c)dG(c) + fe Ne = L,

(14)

0

where q0 = Lq0c . Substituting Eq. (14) into Eq. (13), the planner’s problem can be rewrittenbymax W = L − fe Ne − Ne

q(c),Ne

∫ cM

∫ cM

cq(c)dG(c) + αNe
q(c)dG(c)0( ∫ cM
)2
∫ cM
η
γ
2
Ne
q(c) dG(c) −Neq(c)dG(c) .
−
2L
2L
0
0
0

6

(15)

The ﬁrst order conditions are given by∂WLη= 0 ⇔ q o (c) = (α − Qo − c),
∂q(c)
γ
L
L
⇔ q o (c) = (co − c),
γ

(16)
(17)

∂W
=0
∂Ne
( ∫ cM
)2
∫ cM
∫ cM
∫ cM
η
γ
2
q(c) dG(c) − Neq(c)dG(c) = 0,
⇔ −fe −
cq(c)dG(c) + α
q(c)dG(c) −
2L 0
L
0
0
0
(18)
∫c
where “o” labels ﬁrst-best optimum variables, Qo ≡ Neo 0 M q o (c)dG(c) is the optimal total
output level of diﬀerentiated varieties, andco ≡ α −

η o
Q
L

(19)

represents the optimal cost cutoﬀ. Integrating Eq. (16), I obtain
Qo =

θ o
LN o
(α −
c ).
o
γ + ηNθ+1(20)

Substituting Eq. (20) into Eq. (19) givesγ(θ + 1) α − co
.
η
co

(21)

No
γ(θ + 1)cM θ α − co=.
G(co )
η
coθ+1

(22)

No =
The number of entrants is thenNeo =Substituting Eq. (21) into Eq. (20), I obtainQo =L(α − co ).
η

7

(23)

Therefore, substituting Eqs. (17) and (22) into Eq. (18), the optimal cost cutoﬀ is determined by[γ(θ + 1)(θ + 2)cM θ fe
c =
L

1
] θ+2

o

.

(24)

Finally, substituting Eqs. (17) and (22) into (15), the welfare level at the ﬁrst-best optimum can be expressed in terms of the cost cutoﬀ.
Wo = L +3.3

L(α − co )2 .
2ηMarket failure

I compare the market outcome with the eﬃcient outcome. As Nocco et al. (2014) show, themarket outcome departs from the eﬃcient outcome in terms of product selection, productmix, and product range.
The ineﬃciency of product selection is characterized by the gap between the cost cutoﬀs
in Eqs. (9) and (24):1co − cm = −(2 θ+2 − 1)co < 0,which implies co < cm . Thus, varieties with c ∈ [co < cm ] is ineﬃciently supplied. Intuitively,
more productive ﬁrms set higher markups in the market equilibrium, which leaves room forless productive ﬁrms to survive, albeit ineﬃciently.
The ineﬃciency of product mix is characterized by the gap between the output level ofeach ﬁrm with marginal cost c in Eqs. (7) and (17):1 ≥ 0 c ∈ [0, (2 − 2 θ+2
)
)co ]
1
o
m
o
θ+2
(2 − 2 )c − c,q (c) − q (c) =12γ
 < 0 c ∈ ((2 − 2 θ+2 )co , cm ]L (which implies q o (c) ≥ (<)q m (c) for a ﬁrm with lower (higher) c. Accordingly, in the market
equilibrium, more productive ﬁrms end up selling too little, while less productive ﬁrms end up
oversupplying the market. More productive ﬁrms do not pass on their entire cost advantage
to consumers by raising their markups, and they end up selling below the eﬃcient output
level, which causes less productive ﬁrms to oversupply the market.

8The ineﬃciency of product variety is characterized by the gap between the number ofvarieties in Eqs. (10) and (21):

)
( θ+1

o

[
(
)
]
θ+2 − 1
≥
0
c
≥
2
α
θ+1
γ(θ + 1)αNo − Nm =1 − 2 θ+2 − 1 o
( θ+1
) ,
η
c 
 < 0 co < 2 θ+2 − 1 αwhich implies that the number of varieties at the market equilibrium is smaller (larger) thanthe ﬁrst-best when α is small (large) or co is large (small). Although the direction of departurefrom the eﬃcient number of varieties in the market outcome depends on relationship between
parameters, the total output level of diﬀerentiated varieties in the market equilibrium is lower
than that in the ﬁrst-best. From Eqs. (12) and (23), the gap between the total output levelis
Qo − Qm =L m(c − co ) > 0.
η

At the end of this section, I summarize in the following lemma how the market outcome
departs from the eﬃcient outcome.
Lemma 1. Compared to the ﬁrst-best outcome, in the market outcome, the cost cutoﬀ is abovethat in the ﬁrst-best—ineﬃcient product selection; the output level of more (less) productiveﬁrms is below (above) that in the ﬁrst-best—ineﬃcient product mix; the number of varieties)( θ+1
is below (above) that in the ﬁrst-best if and only if co ≥ (<) 2 θ+2 − 1 α—ineﬃcient productvariety.

4

Eﬃcient policy

In this section, I characterize the policy tools through which the market outcome can achieve
the eﬃcient outcome without the use of ﬁrm-speciﬁc taxes/subsidies. The eﬃcient outcome can be decentralized through an ad valorem production tax and a per-unit production
subsidy—these are uniform across ﬁrms—accompanied by a lump-sum tax on consumers.

9

4.1

Policy tools

Each consumer maximizes the utility function Eq. (1) subject to budget constraint∫p(ω)q c (ω)dω = 1 − T,

q0c +
Ω

where T is a lump-sum tax (subsidy) on consumers if positive (negative) and assume that
1 − T > 0. In the same way as in Section 3.1, the market demand for variety ω is given by
Eq. (3).
In the diﬀerentiated good sector, the proﬁt maximization problem for a ﬁrm with unit
labor requirement c is given by(max π(c) =

)
p(c)
L
− (tu + c) q(c) s.t. q(c) = (pmax − p(c)),
1 + taγ

where p(c) is the taxes/subsidies-inclusive price, and ta > −1 and tu are ad valorem and perunit production taxes (subsidies) if positive (negative), respectively. The proﬁt maximizing
price and quantity arepg (c) =q g (c) =1 + ta pmax(
+ tu + c),
2
1 + taL(1 + ta ) pmax(
− tu − c),2γ
1 + ta(25)

where “g” labels equilibrium variables under the introduction of policy tools by a government.
From Eq. (25), the cost cutoﬀ cg is
cg =

pmax
− tu .
1 + ta(26)

Using the cost cutoﬀ, I obtain the price, quantity, proﬁt, and markup for a ﬁrm with cost c:
1 + ta g(c + 2tu + c),
2
L(1 + ta ) g
q g (c) =(c − c),2γ
L(1 + ta ) g
π g (c) =(c − c)2 ,
4γ
pg (c)
1 + ta cg 2tuµg (c) =
=
( +
+ 1).
c
2
c
c
pg (c) =10(27)

Due to the free entry condition, the expected proﬁt is given as∫ cg
π g (c)dG(c) = fe0[2γ(θ + 1)(θ + 2)cθM fe⇔ cg =
L(1 + ta )

1
] θ+2

.

(28)

Note that from Eqs. (27) and (28) a per-unit production tax dose not aﬀect the cost cutoﬀ
and the output level of each ﬁrm.
Using Eq. (26), rewrite Eq. (5) to obtain the number of varieties as a function of the cost
cutoﬀ:
Ng =
where
1
p̄ =
G(cg )

2γ(θ + 1) α − (1 + ta )(cg + tu ),η
(1 + ta )cg
(

∫ cg
g

p (c)dG(c) = (1 + ta )0)2θ + 1 g
c + tu .
2(θ + 1)

(29)

(30)

Thus, the number of entrant is given Neg = N g /G(cg ).
From Eqs. (4), (29), and (30), the total consumption level of the diﬀerentiated varieties,
Qg , can be written byQg =L(α − (1 + ta )(cg + tu )) .
η

Finally, substituting Eqs. (27) and (29) into (1), the welfare level can be expressed interms of the cost cutoﬀ.
[
(
)]
L
θ+1 g
g
W = LU = L − LT +[α − (1 + ta )(c + tu )] α − (1 + ta )c + tu .
2η
θ+2
g

4.2

Decentralization

Using these policy instruments—an ad valorem production tax, a per-unit production tax,
and a lump-sum tax on consumers—I prove the following proposition.
Proposition 1. The eﬃcient outcome can be achieved though the following policies:• an ad valorem production tax t∗a = 1;
• a per-unit subsidy t∗u = − c2 ;
o

11

o

o

)c
• a limp-sum tax on consumers T ∗ = (α−c,
η(θ+2)

where t∗a , t∗u , and T ∗ are the eﬃcient ad valorem tax, per-unit tax, and lump-sum tax on
consumers, respectively.
By combining an ad valorem production tax and a per-unit production subsidy, the ﬁrstbest outcome can be achieved without adopting a ﬁrm-speciﬁc tax/subsidy. This is because
a combination of these policy instruments can create a tax/subsidy system similar to a ﬁrmspeciﬁc per-unit production tax/subsidy shown by Nocco et al. (2014).3 In fact, in thepresent study, the net production tax imposed on one unit of good produced by a ﬁrm withcost c when ta = t∗a and tu = t∗u is1t∗a g
p (c) + t∗u = (c − co ),∗1 + ta2(31)

which implies that the net production tax for one unit of good is increasing in the cost c,
being negative (non positive) for more productive ﬁrms with c ∈ [0, co ] and positive for lessproductive ﬁrms with c ∈ (co , cM ]. Eq. (31) is similar in form to the ﬁrm-speciﬁc per-unit
OS
production tax used by Nocco et al. (2014) to achieve the ﬁrst-best outcome: tN(c) = c−co .
u

4.3

The role of the ad velorem and per-unit taxes

To better understand the characteristics of the eﬃcient policy indicated by Proposition 1,
I examine the role of ad valorem and per-unit production taxes on distortions in terms of
product selection, product mix, and product variety.
An ad valorem production tax is needed to improve the product selection and the productmix distortions. Diﬀerentiating Eqs. (28) and (27) with respect to ta , I obtain

dcg
cg
=−
< 0,
dta
(θ + 2)(1 + ta )dq g (c)
L
=
dta
2γ

(


)


θ+1 g
≥ 0 c ∈ [0, θ+2
c ]
θ+1 g
,
c −c

θ+2
 < 0 c ∈ ( θ+1 cg , cg ]θ+2which indicates that an increase in ta decreases the cost cutoﬀ and increases (decreases) the
output level of more (less) productive ﬁrms. Accordingly, from Lemma 1, an increase in thead valorem production tax improve the product selection and the product mix distortions
3

See Section 5 of their paper

12when ta is suﬃciently low. It is straightforward to show that these distortions are removedby the eﬃcient ad valorem tax: cg = co and q g (c) = q o (c) when ta = t∗a . An increase in an
ad valorem production tax also aﬀects the number of varieties. Diﬀerentiating Eq. (29) withrespect to ta , I obtaindN g2γ(θ + 1) ((θ + 1)α + (1 + ta )tu )
=−
< 0.
dta
η(θ + 2)(1 + ta )2 cg
An increase in ta decreases the number of varieties. When the eﬃcient ad valorem tax is
imposed, the number of varieties is below that in the ﬁrst-best: N g < N o when ta = t∗a andtu = 0. Thus, while the eﬃcient ad valorem tax can remove the product selection and the
product mix distortions, it causes the product variety distortion due to an excessive reduction
in ﬁrm entry.
A per-unit production subsidy is then needed to to improve the product variety distortion.
An increase in a per-unit production tax aﬀects the product variety distortion without changing the cost cutoﬀ and the output level: dcg /dtu = 0 and dq g (c)/dtu = 0.4 Diﬀerentiating
Eq. (29) with respect to tu , I obtain
2γ(θ + 1)
dN g
=−
< 0.
dtu
ηcg
A decrease in tu increases the number of varieties. Thus, the eﬃcient per-unit subsidy removes
the product variety distortion when ta = t∗a : N g = N o when tu = t∗u and ta = t∗a .

5Conclusion

This study analyzes the eﬃcient policy based on the closed economy setting of Melitz andOttaviano (2008), a model of monopolistic competition with ﬁrm heterogeneity and variablemarkups. In this model, the market outcome departs from the ﬁrst-best outcome in terms
of product selection, product mix, and product variety due to markup pricing.
The conclusions of this study are summarized as follows. The ﬁrst-best outcome can bedecentralized through an ad valorem production tax and a per-unit production subsidy—
4

In a version of this model that regulates the entry of ﬁrms, both ad valorem and per-unit taxes/subsidies
aﬀect the cost cutoﬀ, the output level, and the number of varieties. Even in the case, ﬁrst-best outcome canbe achieved by combining these taxes/subsidies without adopting a ﬁrm-speciﬁc tax/subsidy.

13these are uniform across ﬁrms—accompanied by a lump-sum tax on consumers. This result
is diﬀerent from that of Nocco et al. (2014), who characterize the ﬁrst-best policy as aﬁrm-speciﬁc per-unit production tax/subsidy, a lump-sum entry tax, and a lump-sum tax
on consumers. When the per-unit production subsidy cannot be diﬀerentiated across ﬁrms,
they consider the second-best scenario. In the present study, I show that policymakers do
not need to have any way of observing the level of productivity of each ﬁrm to make themarket outcome achieve the ﬁrst-best outcome. Ad valorem and per-unit taxes/subsidies
play diﬀerent roles on each distortion. As a result, combining an ad valorem production tax
and a per-unit production subsidy can create a tax/subsidy system similar to a ﬁrm-speciﬁc
per-unit production tax/subsidy, which enables eﬃcient resource allocation to be achieved
without a ﬁrm-speciﬁc tax/subsidy.

References
Dhingra, S., Morrow, J., 2019. Monopolistic competition and optimum product diversity
under ﬁrm heterogeneity. Journal of Political Economy, 127 (1), 196–232.
Melitz, M.J., 2003. The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica, 71 (6), 1695–1725.
Melitz, M.J., Ottaviano, G.I.P., 2008. Market size, trade, and productivity. Review of Economic Studies, 75 (1), 295–316.
Nocco, A., Ottaviano, G.I.P., Salto, M., 2014. Monopolistic competition and optimum
product selection. American Economic Review, 104 (5), 304–309.
Nocco, A., Ottaviano, G.I.P., Salto, M., 2019. Geography, competition, and optimal multilateral trade policy. Journal of International Economics, 120, 145–161.

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