Discussion Papers In Economics
And Business

Endogenous Sunk Cost, Scale Economies,
and Market Concentration
Yuichiro Matsumoto

Discussion Paper 18-20

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Sunk Cost, Scale Economies,
and Market Concentration
Yuichiro Matsumoto

Discussion Paper 18-20

July 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Sunk Cost, Scale Economies, andMarket Concentration∗
Yuichiro Matsumoto†
July 12, 2018

Abstract
This paper offers a theoretical explanation of the recent sales concentration in the U.S. economy. The model is based on in-house R&D, which is
involved in scale economies. An R&D subsidy helps the expansion of larger
firms and allows them to take higher markups. Thus, it induces a concentrated market structure.

JEL Classification Codes: D61; L11; L13; L16; O3
Keywords: Endogenous Sunk Cost; Firm Size Distribution; Heterogenous
Firms; Markup; Quality Upgrading∗

I am grateful to the seminar participants at Osaka University, especially to Hirokazu Ishise,
Shuhei Kitamura, Takeshi Murooka, and Yosuke Yasuda for their helpful comments. This research
has no relevant interests.
†
Osaka University email:u847103e@ecs.osaka-u.ac.jp1

1

Introduction

The structure of the U.S. market has changed. Firms have increased markup rates1(De Loecker and Eeckhout, 2017) and sales are concentrated in a limited numberof firms (Autor et al., 2017a,b).2 An increase in the markup and sales concentration seem correlated: the markup is increasing in sectors with an increasing shareof mega-firms employing more than 10,000 workers (Hall, 2018).
I offer a theoretical explanation for these phenomena. The theory is basedon scale economies attributable to in-house R&D. Since 1990, the R&D subsidy has increased (Impullitti, 2010), which stimulates R&D investments. The
resource concentrates on the small number of firms because R&D is involved inscale economies.3 Support for this hypothesis is that the increase in industry concentration is positively correlated with the increase in patent intensity (Autor et al.
(2017b)).
This paper works with the hypothesis that scale economies attributable to
R&D are a source of the recent concentration. This view is highly indebted tothe concept of endogenous sunk cost (Shaked and Sutton, 1983; Ellickson, 2007;
Etro, 2014). However, most studies on endogenous sunk cost use the homogeneous firm model, which is inappropriate for the present purpose because I am
interested in the distribution of sales across firms. Thus, I use the theoretical1

One skeptical view is taken by Traina (2018). However, many studies agree on the increasein markups (see, e.g., Gutiérrez and Philippon (2018)).
2
Comin and Mulani (2006) also report that the volatility of firm-level sales growth is increasing.
3
See, e.g., Atkeson and Burstein (2010), Dasgupta and Stiglitz (1980), Dhingra (2013), andVives (2008) for studies involving in-house R&D.

2

model offered by Antoniades (2015), who embedded endogenous sunk costs into
the heterogeneous firm model of Melitz and Ottaviano (2008). He examines thepatterns of international trade, whereas I focus on market concentration.
Compared with previous studies examining the consequence of the increase inmarkups (Barkai, 2016; Eggertsson et al., 2018), the contribution of the present
study is to offer an explanation of the cause of the change in market conditions.
These previous studies assume an exogenous decrease of substitutability acrossgoods, which can be interpreted as a less competitive market, and obtain its implication for factor returns. However, the assumption counterfactually predicts
the decrease in sales concentration (see appendix). Lower substitutability makes
sales less sensitive to productivity, which reduces sales dispersion. In contrast,
the present study can simultaneously explain the increase in markup and sales
concentration.
The increase in markups affects resource allocation.4 The implication of thisfact for efficiency in the present model is ambiguous because technology is endogenous. On the one hand, R&D investment increases markup dispersion, whichstrengthens misallocation. On the other hand, it reduces misallocation by allowing large firms, which undersupply goods, to expand. It is worth noting that
this ambiguity is consistent with the two empirical observations of Baqaee andFarhi (2017): (i) allocative efficiency is increasing; (ii) firms that charge a higher
markup is expanding. If technology is exogenous, these observations seem para4

Since the study of Hsieh and Klenow (2009), resource misallocation has become an important
topic in economics; see, e.g., Dhingra and Morrow (2012), Edmond et al. (2015), Epifani and
Gancia (2011), and Peters (2013) for studies involved in misallocation attributable to markups.

3

doxical because allocative efficiency improves when markup distortion becomeslarger. However, these observations are compatible (although an examination of
the efficiency properties of the model is beyond the scope of this paper) with theview in the present paper.
Although this study only examines the cause of concentration, concentration isconsidered to be related to the recent increase in earnings inequality because eachfirm’s sales are positively correlated with the wages that the firm pays (Helpman
et al., 2017).5 Indeed, Berlingieri et al. (2017) report that the increase in wage
dispersion is mainly attributed to the increasing inequality of performance acrossfirms. Gabaix and Landier (2008) and Edmans et al. (2008) show theoretically andempirically that firm size is positively correlated with CEO compensation. The
present study complements these studies by theoretically examining a potential
cause of inequality across firms.
The question of this study is what firms gain from a change in market conditions. In this sense, this study is related to studies involved in the economics of superstars, which examined how market conditions—technological change (Rosen,1981) or international trade (Manasse and Turrini, 2001)—affect the distributionof economic rent. This paper also relates studies of biased technical change6 ,
which examines how market conditions change the earnings distributions acrossworkers, whereas I focus on the distributional issues across firms.
5

Some theoretical studies associate profit with entrepreneurial income (Jones and Kim, 2014;
Nocke, 2006; Pokrovsky et al., 2014).
6
See, e.g., Acemoglu (1998), Burstein and Vogel (2017), Epifani and Gancia (2008), Galorand Moav (2000), Krusell et al. (2000), Matsumoto (2018), Matsuyama (2007), and Parro (2013).

4

Section 2 exposits the model. Section 3 applies the model to examine theeffect of R&D cost subsidies on concentration. Section 4 concludes the paper.

2

Model

2.1

Consumers

Specification of the demand side is a variant of Ottaviano et al. (2002). Consumers have identical utility over homogeneous goods q(0) and a continuum ofdifferentiated goods indexed by v ∈ Ω:Zγ(α + z(v))q(v)dv −U = q(0) +
2
v∈Ω

Z

Z
η
q (v)dv − (q(v)dv)2 (1)
2
v∈Ω
v∈Ω
2

where z(v) is the quality of v. I normalize the homogeneous goods price and thewage to one. Each agent has the budget constraintZ
p(v)q(v)dv = h − T + Π

p(0)q(0) +v∈Ω

where p(v) is the price of v. h is the efficiency units of labor supply of eachconsumer.7 Wage is normalized to one. Homogeneous goods are competitively
supplied and its labor productivity is one, which implies that the homogeneous
goods price is one (p(0) = 1). T is the lump sum tax seized by the governmentand Π is the firm’s profit distributed to each household.
7

To avoid a corner solution, I assume that h is sufficiently high. Whenever both goods are
consumed, the resource constraint does not need to be considered explicitly.

5The demand of each consumer isp(v) = α + z(v) − γq(v) − ηQ

where Q ≡

2.2

R
v∈Ω

(2)

q(v)dv denotes total differentiated goods consumption.

Firms

Following the assumption made by Melitz and Ottaviano (2008), the distributionof a firm’s labor productivity c follows a Pareto distribution with support on [0, 1]:G(c) ≡ P[labor productivity ≤ c] = ck ,

0≤c≤1

(3)

For simplicity, I assume k > 2. The firm, which supplies one good monopolistically, maximizes profits:8θπ(c) = (p(c) − c)y(c) − (1 − τθ ) z 2 (c)2(4)

where y ≡ qL is the output and L is the number of consumers. The term 2θ z 2
represents the sunk R&D cost for quality upgrading and τθ represents the R&D
subsidy rate.9 To maintain the concavity of the profit maximization problem, I
assume 2γ(1 − τθ )θ > L.
8
9

I use v and c for interchangeable indexes of goods.
R
R&D subsidies are paid by lump sum tax for consumers: v∈Ω τθ2θ z 2 (v)dv = T L.

6

By profit maximization,
∂π
=0:
∂q
∂π
=0:
∂z

p(c) − c = γq(c)(5)(1 − τθ )θz(c) = y(c)(6)

Because R&D is involved in scale economies, larger output induces stronger R&D
efforts, which is roughly consistent with Cohen and Klepper (1996) and Blundell
et al. (1999).
Firms exit if and only if π(c) < 0: this is equivalent toc < cm ≡ α − ηQ

I assume that the exit cutoff cm is lower than 1 in equilibrium.10 Inverse demand(2) can be rewritten asp(c) = cm + z(c) − γq(c)(7)(5), (6), and (7) allow us to rewrite z as a function of cm :z(c) = λ(cm − c)(8)where λ ≡ (2γ(1 − τθ )θ/L − 1)−1 > 0 is quality intensity. So far, I use λ asa measure of R&D subsidy because λ is increasing in τθ . Correspondingly, one
10

A sufficient condition for cm < 1 is α < 1.

7

obtains prices and quantities:1p(c) = ((1 + λ)cm + (1 − λ)c))2(1 + λ)L(cm − c)y(c) = q(c)L =
2γ

(9)
(10)

The main interest of this study is how the reduction in λ affects the distribution of
profit π(c) and sales S(c) ≡ p(c)y(c).
(1 + λ)L(cm − c)24γ
(1 + λ)LS(c) =((1 + λ)cm + (1 − λ)c)(cm − c)4γπ(c) =

2.3

(11)
(12)

Measures of Market Condition

I use six measures to evaluate the market condition. The first measure is theaverage markup:11p(c)E[|c ≤ cm ] ≡
c

Z cm
0

p(c)
dG(c|c ≤ cm )c

(13)

where G(c|c ≤ cm ) = ( ccm )k denotes the cumulative distribution conditional onc ≤ cm . This measure is related to the resource misallocation across sectors(differentiated goods and homogeneous goods sectors). The second measure isthe variance of the markup, which is relevant for evaluating the misallocation
11
Antoniades (2015) uses the absolute markup E[p(c) − c|c ≤ cm ], but the markup measure I
use is more appropriate for comparing the result with empirical studies.

8within the sector:

V[

p(c)
p(c)
p(c)
|c ≤ cm ] ≡ E[(− E[|c ≤ cm ])2 |c ≤ cm ]ccc

The third and fourth measures are the profits of the largest firms and the average
profit:(1 + λ)L 2
cm
4γ(1 + λ)LE[π(c)|c ≤ cm ] =cm 22γ(k + 1)(k + 2)π(0) =(14)(15)

The fifth and sixth measures are the profit and sales inequalities:
cm − c1 2π(c1 ))=(π(c2 )
cm − c2
S(c1 )(1 + λ)cm + (1 − λ)c1 cm − c1=
S(c2 )(1 + λ)cm + (1 − λ)c2 cm − c2

(16)
(17)

A similar measure of inequality is often used in the assignment model (Sampson,
2014; Grossman and Helpman, 2018).
Because these measures depend on cm , considering how cm is determined isneeded. However, this is not the case for the markup distribution, which depends

9only on λ:12

E[

p(c)
2k − 1 + λ|c ≤ cm ] =,c2(k − 1)V[p(c)(1 + λ)2 k|c ≤ cm ] =c4(k − 2)(k − 1)2

Thus, both the average markup and the markup dispersion increase as the R&D|c ≤ cm ]/dλ > 0 and dV[ p(c)|c ≤ cm ]/dλ > 0.
subsidy increases dE[ p(c)cc3

R&D Subsidy and Concentration

To concentrate on the effect of in-house R&D, I consider the case that the entry is
restricted: the number of potential firms N is fixed. Nonetheless, the main resultcontinues to hold even if the free entry of new firms is allowed (see appendix).
The total output isZ cmq(c)dG

Q = N E[q(c)] = N

(18)

0

Because cm = α − ηQ, one obtains

cm = α −η(1 + λ)N k+1c2γ(k + 1) m

(19)

12

Two effects induced by the reduction in cm affect the distribution of the markup. First, themarkup decreases because each firm faces stronger competition. Second, the markup increases
because low markup firms exit. Nonetheless, they are exactly offset when the cost distribution is
Pareto. The markup distribution depends only on λ:
P(

p(c)
1+λ
≥ µ|c ≤ cm ) = ()kc
2µ − (1 − λ)for µ ≥ 1. Intuitively, the self-similarity of the distribution of c ensures the result (See also
Arkolakis et al. (2018)).

10

ck+1
α − η(1+λ)N
2γ(k+1) m

cm

cm
Figure 1: Effect of the increase in λ(19) determines unique values of cm because the left-hand side is increasing in cm ,whereas the right-hand side is decreasing in cm . As shown in Figure 1, the increasein λ reduces the cutoff cm . Intuitively, the increase in the R&D subsidy would hurt
small firms by strengthening competition, and these unproductive firms must exitfrom the market.
From (19), one can obtain the effect of R&D subsidies. The average and the
top firm profits increase by the direct effect of R&D subsidies and decrease bystrengthening competition:dE[π(c)|c ≤ cm ]d log cmd log π(0)==1+2d log(1 + λ)d log(1 + λ)d log(1 + λ)11

The second effect dominates because d log cm /d log(1+λ) > −1/(k+1).13 Thus,
R&D subsidies increase the average profit and the profit of the most productive
firms.
Because large firms gain and small firms lose, within-firm inequality also increases:dπ(c1 )
1
1
dcm
log() = 2(−)>0dλπ(c2 )
cm − c1 cm − c2 dλS(c1 )
2cm (c2 − c1 )
d
log(
)= Q
dλ
S(c2 )
i=1,2 ((1 + λ)cm + (1 − λ)ci )

(20)

1
dcm
1 − λ2Q−Q)+(c2 − c1 )(dλi=1,2 (cm + ci + λ(cm − ci ))i=1,2 (cm − ci )>0(21)
for c1 < c2 .14 The increase in R&D subsidies induces a more concentrated market
13

By the total derivative of (19), one obtains
cm d log cm = −η(1 + λ)N k+1
cm(d log(1 + λ) + (k + 1)d log cm )2γ(k + 1)

This equation can be rewritten as
0>

1
d log cm1=−>−d log(1 + λ)k+1
k + 1 + 2γ(k+1) cm −kη(1+λ)N14

To obtain the inequality in (21), it is sufficient to showQ

1 − λ2
1
<Q
(c
+
c
+
λ(c
−
c
))
(c
i
m
i
i=1,2 mi=1,2 m − ci )This is indeed satisfied for all λ ≥ 0 because (i) this inequality is satisfied at λ = 0 and (ii) theleft-hand side is decreasing in λ whenever it takes a positive value.

12structure.

4

Conclusion

This paper examines the implication of endogenous sunk costs for the recent concentration phenomena. Because R&D is involved in scale economies, subsidy
works in favor of large firms and disproportionately expands the sales and profits
of large firms.

References
Acemoglu, Daron, “Why do new technologies complement skills? Directed technical change and wage inequality,” Quarterly Journal of Economics, 1998, 113(4), 1055–1089.
Antoniades, Alexis, “Heterogeneous firms, quality, and trade,” Journal of International Economics, 2015, 95 (2), 263–273.
Arkolakis, Costas, Arnaud Costinot, Dave Donaldson, and Andrs RodrguezClare, “The elusive pro-competitive effects of trade,” Review of Economic
Studies, 2018, p. rdx075.
Atkeson, Andrew and Ariel Tomas Burstein, “Innovation, firm dynamics, andinternational trade,” Journal of political economy, 2010, 118 (3), 433–484.

13

Autor, David, David Dorn, Lawrence Katz, Christina Patterson, and John
Van Reenen, “Concentrating on the fall of the labor share,” American Economic Review, 2017, 107 (5), 180–85.
,

,

,

, and

, “The fall of the labor share and the rise of superstar firms,”

NBER Working Paper, 2017.
Baqaee, David Rezza and Emmanuel Farhi, “Productivity and misallocation in
general equilibrium.,” NBER Working Paper, 2017.
Barkai, Simcha, “Declining labor and capital shares,” Working Paper, Universityof Chicago, 2016.
Berlingieri, Giuseppe, Patrick Blanchenay, and Chiara Criscuolo, “The great
divergence (s),” 2017.
Bernard, Andrew, Jonathan Eaton, Bradford Jensen, and Samuel Kortum,“Plants and productivity in international trade,” American Economic Review,
2003, 93 (4), 1268–1290.
Blundell, Richard, Rachel Griffith, and John Van Reenen, “Market share, market value and innovation in a panel of British manufacturing firms,” Review of
Economic Studies, 1999, 66 (3), 529–554.
Burstein, Ariel and Jonathan Vogel, “International trade, technology, and theskill premium,” Journal of Political Economy, 2017, 125 (5), 1356–1412.

14

Cohen, Wesley and Steven Klepper, “Firm size and the nature of innovation
within industries: the case of process and product R&D,” Review of Economics
and Statistics, 1996, pp. 232–243.
Comin, Diego and Sunil Mulani, “Diverging trends in aggregate and firm volatility,” Review of Economics and Statistics, 2006, 88 (2), 374–383.
Dasgupta, Partha and Joseph Stiglitz, “Industrial structure and the nature ofinnovative activity,” Economic Journal, 1980, 90 (358), 266–293.
De Loecker, Jan and Jan Eeckhout, “The rise of market power and the macroeconomic implications,” NBER Working Paper, 2017.
Dhingra, Swati, “Trading away wide brands for cheap brands,” American Economic Review, 2013, 103 (6), 2554–2584.
and John Morrow, “Monopolistic competition and optimum product diversity
under firm heterogeneity,” 2012.
Edmans, Alex, Xavier Gabaix, and Augustin Landier, “A multiplicative modelof optimal CEO incentives in market equilibrium,” The Review of Financial
Studies, 2008, 22 (12), 4881–4917.
Edmond, Chris, Virgiliu Midrigan, and Daniel Yi Xu, “Competition, markups,
and the gains from international trade,” American Economic Review, 2015, 105(10), 3183–3221.

15

Eggertsson, Gauti, Jacob Robbins, and Ella Getz Wold, “Kaldor and pikettys
facts: The rise of monopoly power in the United States,” NBER Working Paper,
2018.
Ellickson, Paul B, “Does Sutton apply to supermarkets?,” The RAND Journal of
Economics, 2007, 38 (1), 43–59.
Epifani, Paolo and Gino Gancia, “The skill bias of world trade,” Economic Journal, 2008, 118 (530), 927–960.
and, “Trade, markup heterogeneity and misallocations,” Journal of Inter-national Economics, 2011, 83 (1), 1–13.
Etro, Federico, “Some thoughts on the Sutton approach,” Journal of Economics,
2014, 112 (2), 99–113.
Gabaix, Xavier and Augustin Landier, “Why has CEO pay increased so
much?,” Quarterly Journal of Economics, 2008, 123 (1), 49–100.
Galor, Oded and Omer Moav, “Ability-biased technological transition, wage
inequality, and economic growth,” Quarterly Journal of Economics, 2000, 115(2), 469–497.
Ghironi, Fabio and Marc Melitz, “International Trade and Macroeconomic Dynamics with Heterogeneous Firms,” Quarterly Journal of Economics, 2005, 120(3), 865–915.

16

Grossman, Gene and Elhanan Helpman, “Growth, trade, and inequality,”
Econometrica, 2018, 86 (1), 37–83.
Gutiérrez, Germán and Thomas Philippon, “How EU Markets Became More
Competitive Than US Markets: A Study of Institutional Drift,” NBER Working
Paper, 2018.
Hall, Robert, “New evidence on the markup of prices over marginal costs and therole of mega-firms in the US economy,” NBER Working Paper, 2018.
Helpman, Elhanan, Oleg Itskhoki, Marc-Andreas Muendler, and Stephen
Redding, “Trade and inequality: From theory to estimation,” Review of Economic Studies, 2017, 84 (1), 357–405.
Hsieh, Chang-Tai and Peter J Klenow, “Misallocation and manufacturing TFP
in China and India,” Quarterly Journal of Economics, 2009, 124 (4), 1403–
1448.
Impullitti, Giammario, “International competition and US R&D subsidies: A
quantitative welfare analysis,” International Economic Review, 2010, 51 (4),1127–1158.
Jones, Charles and Jihee Kim, “A Schumpeterian model of top income inequality,” NBER Working Paper, 2014.
Krusell, Per, Lee Ohanian, José-Vı́ctor Rı́os-Rull, and Giovanni Violante,“Capital-skill complementarity and inequality: A macroeconomic analysis,”Econometrica, 2000, 68 (5), 1029–1053.
17

Manasse, Paolo and Alessandro Turrini, “Trade, wages, and superstars,” Journal of International Economics, 2001, 54 (1), 97–117.
Matsumoto, Yuichiro, “Nonlinear effects of the transport costs on the wage inequality,” Discussion Papers in Economics and Business, Osaka University,
2018.
Matsuyama, Kiminori, “Beyond icebergs: towards a theory of biased globalization,” Review of Economic Studies, 2007, 74 (1), 237–253.
Melitz, Marc and Gianmarco Ottaviano, “Market size, trade, and productivity,”
Review of Economic Studies, 2008, 75 (1), 295–316.
Nocke, Volker, “A gap for me: Entrepreneurs and entry,” Journal of the European
Economic Association, 2006, 4 (5), 929–956.
Ottaviano, Gianmarco, Takatoshi Tabuchi, and Jacques-François Thisse,“Agglomeration and trade revisited,” International Economic Review, 2002,
pp. 409–435.
Parro, Fernando, “Capital-skill complementarity and the skill premium in aquantitative model of trade,” American Economic Journal: Macroeconomics,
2013, 5 (2), 72–117.
Peters, Michael, “Heterogeneous mark-ups, growth and endogenous misallocation,” 2013.

18

Pokrovsky, Dmitry, Kristian Behrens, and Evegeny Zhelobodko, “Market
size, entrepreneurship, and income inequality,” CEPR Discussion Paper, 2014.
Rosen, Sherwin, “The economics of superstars,” American Economic Review,
1981, 71 (5), 845–858.
Sampson, Thomas, “Selection into trade and wage inequality,” American Economic Journal: Microeconomics, 2014, 6 (3), 157–202.
Shaked, Avner and John Sutton, “Natural oligopolies,” Econometrica, 1983,
pp. 1469–1483.
Traina, James, “Is Aggregate Market Power Increasing? Production Trends Using Financial Statements,” Stigler Center New Working Paper Series, 2018.
Vives, Xavier, “Innovation and competitive pressure,” Journal of Industrial Economics, 2008, 56 (3), 419–469.

A

CES Demand and Exogenous Decrease in Substitutability

Representative households have the utility
Z
U =(

ρ−1

ρ

y(v) ρ dv) ρ−1

v∈Ω

19

where ρ > 1 denotes the elasticity of substitution across goods. By utility maximization, the demand isp(v)−ρ
y(v) = 1−ρ E,PZ

p(v)1−ρ dv

P ≡v

where E denotes the total expenditure. Each monopolistic firm has heterogeneous
labor productivity c. A firm’s profit is p(c)y(c) − cy(c) and profit maximization
implies
p=

ρ
c,
ρ−1

py =

E
P 1−ρ

(

ρ
c)1−ρ
ρ−1

Wage is normalized to one. In this specification, the decrease in ρ implies higher
d pmarkups ( dρ< 0). However, it also implies a decrease in the sales inequality:cdp(c1 )y(c1 )
log
> 0,
dρ
p(c2 )y(c2 )c1 < c2

Another measure of sales dispersion entails the same result. To see this, supposethat c follows the Parato cumulative distribution function with support on [0, 1]:G(c) = ck

The standard deviation of the log of sales—this measure is used by Bernard et al.
(2003) and Ghironi and Melitz (2005)—isppV[log(p(c)y(c))] = (ρ − 1) V[log c] = (ρ − 1)/k20

This measure decreases as ρ decreases. Thus, although the decrease in ρ is compatible with the recent increase in markups, it counterfactually predicts a reduction
in sales dispersion.

B

Free Entry

Entrants pay fE units of the labor cost and, after paying fE , draws c from a Pareto
distribution: G(c) = ck . Free entry ensures that the expected post entry profit
equals the entry cost:P(c ≤ cm )E[π(c)|c ≤ cm ] = fE

(B.1)

This condition implies

cm = (

2γ(k + 1)(k + 2)fE 1/(k+2))(1 + λ)L(B.2)

Correspondingly, the elasticity of the cutoff is
d log cm1=−d log(1 + λ)
k+2

21

(B.3)

Thus, all of the results in this paper remain intact:d log π(0)dE[π(c)|c ≤ cm ]
2
=
≥1−
>0
d log(1 + λ)d log(1 + λ)k+2π(c1 )
1
1
dcm
d
log(
) = 2(−)>0dλπ(c2 )
cm − c1 cm − c2 dλdS(c1 )log()>0
dλS(c2 )
for c1 < c2 .

22