Discussion Papers In Economics
And Business

Firm Size Distribution and Variable Elasticity of Demand

Yuichiro Matsumoto
Discussion Paper 19-02

March 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANFirm Size Distribution and Variable Elasticityof Demand∗
Yuichiro Matsumoto
March 14, 2019

Abstract
Prior studies suggest that a Pareto distribution of the firm’s productivity
distribution is difficult to replicate the observed log standard deviation offirm sales. These studies are based on constant elasticity preferences, which
entail too low log sales deviation. The present study shows that, in contrast
to constant elasticity cases, the log standard deviation is too high in variableelasticity cases. To match the observed sales dispersion, one must set a
Pareto tail parameter relatively higher values.

JEL Classification Codes: L10; L11; L13
Keywords: Firm Size Distribution, Pareto Distribution, Variable Elasticity ofSubstitution∗Osaka University email:u847103e@ecs.osaka-u.ac.jp I have no relevant or material financial
interests that relate to the research described in this paper.

11Introduction

Prior theoretical studies using constant elasticity of substitution preferences, faceddifficulty in replicating the observed log standard deviation of sales. This specification implies the log standard deviation is lower than one (Ghironi, 2005) but
the observed value is 1.67 (Bernard et al., 2003).
This study examines whether the introduction of variable elasticity preferences
can explain the observed patterns of firm size distribution. When demand elasticity increases with price, small firms face highly elastic demand, indicating thatsales are highly sensitive to productivity within small firms. Thus, the introduction of variable elasticity increases sales dispersion within small firms.
Section 2 describes the model, which is based on Arkolakis et al. (2018). Section 3 shows the calculated values of the log standard deviation of sales. Section
4 concludes.

2Model

L units of homogeneous households have the utilityZ
U=u(q(ω))dω,ω∈Ω

u(q) =

ρ−1ρ
(q + α) ρ ,ρ−1ρ, α > 12(1)where Ω denotes the set of differentiated goods. Utility maximization implies

u0 (q(ω)) = λp(ω),(2)

where λ is the marginal utility of income. We assume a choke price P such that

P =u0 (0)< ∞,λwhich is satisfied if α > 0. Each household purchases goods ω, if and only ifp(ω) < P . The demand function can be written aspq = D( ),Ppp
D( ) ≡ α(( )−ρ − 1).
PP(3)This expression is useful because in some class of preferences, including direct
additive, translog (Feenstra, 2003), and implicit additive preferences (Dotsey andKing, 2005; Kimball, 1995; Klenow and Willis, 2016), the demand can be written as q(v) = QD(p(v)/P ), where Q and P denote the aggregate variables andnormalization of P ensures that P is a choke price (Arkolakis et al., 2018; Fally,2019). Thus, we can conduct similar exercises with other demand specifications,although we use the special functional form of D(·) here.1
The demand elasticity only depends on p/P :(p/P ) ≡ −1log q
D0 (p/P ) p(p/P )−ρ
=−=ρ.
log p
D(p/P ) P(p/P )−ρ − 1

Arkolakis and Morlacco (2017) review the functional forms of D(·).

3(4)

Demand elasticity increases with price. This property is referred to as Marshall’s
Second Law of Demand (Mrázová and Neary, 2017) and taken as a plausible case(Krugman, 1979).

2.1Firms

Each good is served by a monopolistic firm, whose labor productivity follows
a Pareto distribution with support on [ψmin , ∞): G(ψ) = 1 − (ψ/ψmin )−k . A
Pareto distribution fits well with the observed productivity (Corcos et al., 2011)
and is consistent with Zipf’s law of the right tail of firms (Axtell, 2001). Prior
theoretical studies also use this assumption (Arkolakis et al., 2018; Behrens et al.,2014; Chaney, 2008; Matsumoto, 2017, 2018). The parameter k is interpreted
as an inverse measure of the heterogeneity of productivity: higher k implies lessvariability of productivity.
Each firm chooses the price to maximize profit as follows

π(c) = max(p − c)q(p, Q, P )L,(5)where c ≡ 1/ψ denotes unit labor requirement. Wage is taken as numeraire. Profit
maximization impliesp=(p/P )c.
(p/P ) − 1(6)(mĉ),(mĉ) − 1(7)We can rewrite (6) asm=4

where m ≡ p/c is the markup and ĉ ≡ c/P is the relative efficiency of the firm.
Firms exit if ĉ > 1. Because (·) is an increasing function, the right hand side isdecreasing in m and ĉ. Thus, this equation uniquely determines the value of m
and m = m(ĉ) is increasing in ĉ.
The distribution of ĉ doesn’t depend on P :2

P(c/P < ĉ|ĉ < 1) = ĉk ,(9)where P(c/P < ĉ|ĉ < 1) denotes the probability of c/P < ĉ conditional on ĉ < 1.
This insensitivity is derived from the self-similarity of a Pareto distribution.
The sales arer(c) = LP ĉm(ĉ)D(m(ĉ)ĉ).

(10)

Thus, the log standard deviation of sales isΣ≡pV(log(ĉm(ĉ)D(m(ĉ)ĉ)).

(11)

where V(·) is the variance conditional on ĉ ≤ 1.
2

This equality can be derived as follows:P(c1111< ĉ|ĉ < 1) = P(< ψ| < ψ) = (1 − G( ))/(1 − G( )) = ĉk .
PP ĉ
PP ĉ
P

5(8)

3

Sales Dispersion

3.1

Constant Elasticity

Before evaluating the variable elasticity cases, it is useful to analyze the constantelasticity cases (i.e. α = 0). The demand function becomes q(ω) = (λp(ω))−ρ .
Thus, markups and revenues becomep(ψ) =

ρ 1,
ρ−1ψr(ψ) = Aψ ρ−1 ,ρ 1−ρ) . Sales and employment are distributed according towhere A ≡ λ−ρ ( ρ−1Pareto distributions with a tail index k/(ρ − 1):kr
P(r(ψ) ≤ r|r(ψ) ≥ r) = 1 − ( )− ρ−1rkl
P(l(ψ) ≥ l|l(ψ) ≥ l) = 1 − ( )− ρ−1l(11) becomes
Σ=pρ−1V(log(ĉ1−ρ )) =.
k

Empirically, this value is about 1.67. However, in the constant elasticity case, it
is difficult to set the value of k and ρ to replicate the observed standard deviation.
Indeed, prior studies impose the restriction k/(ρ − 1) < 1 to ensure that theaverage profit is finite. In addition, inference from top firm distribution suggests

6k/(ρ − 1) = 1.06 (Luttmer, 2007). Thus, Σ must be lower than one, which isproblematic.
Σ increases with ρ because the sales become sensitive to productivity: higher=demand elasticity ρ implies that sales are highly elastic to productivity: ∂ ∂loglogr(ψ)ψρ − 1. It also decreases with k because productivity becomes more homogeneous.
As shown below, a similar intuition can be applied to the case of variable elasticity.

3.2

Variable Elasticity

To calculate the value of the log standard deviation Σ, one must choose the valueof k and ρ.3 We choose ρ = 2.85, which is estimated by Arkolakis et al. (2018).
We use three values of k. First, we choose k = 1.96 to match Zipf’s law ofthe right tail of firms. Precisely, because the demand is asymptotically CES
function—i.e. D(p/P ) ≈ α(p/P )−ρ for sufficiently low p— the formula that
employments of the top firms obey a Pareto distribution with a tail index k/(ρ −1) is applicable here. To match the value reported by Luttmer (2007), we setk/(ρ − 1) = 1.06, which implies k = 1.96. This value is also roughly consistent with Corcos et al. (2011), which is based on the firm level TFP and with thevalue inferred from the industrial TFP (Kang, 2017). The second and the thirdvalues are based on trade elasticity: the trade literature suggests that the value ofk equals trade elasticity (see Arkolakis, Costinot, and Rodrı́guez-Clare (2012)).
The estimated values are within 5 to 10. Thus, we choose k = 5.0 and k = 10.0.
Table 1 denotes the calculated values of Σ. As in constant elasticity cases,
3The value of α does not affect Σ.

7

Table 1: Standard Deviation of Log Sales ΣkρΣ
1.96 2.85 2.69
5.0 2.85 1.89
10.0 2.85 1.75
the rise in k reduces the log standard deviation. In contrast to constant elasticity
cases, the calculated value is higher than the value observed in reality (Σ = 1.67).
However, the value is reasonably close to the observed value if k is sufficiently
high.
Variable elasticity implies higher standard deviations because less productive
firms face highly elastic demand. As in constant elasticity cases, elastic demand
implies higher sales dispersion.

4Conclusion

We have studied how variable markups affect the distribution of firm size. Variable elasticity can potentially explain the observed sales dispersion, but in general, the calculated value tends to be too high. There are two possible extensions.
First, it is interesting to include advertising costs in the model, which entails defacto variable elasticity of demand even if the preference is CES. Arkolakis (2010)shows that the CES trade model with advertising costs can explain many observed
patterns: e.g. small firms significantly expand export sales after trade liberalization. How the existence of advertising costs change the firm size distribution isa potentially interesting question. The second extension is to examine whether8this specification is compatible with other features of firm size distribution. The
primary conclusion of this study is that variable elasticity indicates higher sales
dispersion within small firms. This feature can affect other dimensions of firm
size distribution.

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