Sunk costs, windows of profit opportunities, and the dynamics of entry
Georg Götz∗
Department of Economics, University of Vienna, BWZ - Bruenner Str. 72, 1210 Vienna,
Austria
Abstract: This article adds two elements to a standard model of monopolistic competition:
First, the number of potential entrants is limited in each period and increases only over time.
Second, the potential entrants differ with respect to the consumers’ valuation of the variant
they could offer. The resulting simple model exhibits a rich dynamic structure covering the
product life cycle, a path dependent equilibrium and the traditional textbook case of entry. The
welfare analysis confirms the view that there cannot be too much entry. Even entry of
'inefficient' firms improves welfare.
Keywords: Industry evolution, product life cycle, path dependence
JEL-Classification No.: L10
∗
Tel.: +43-1-4277-37466; fax +43-1-4277-37498, E-mail address: [email protected] (Georg Götz)
-2-
1.
Introduction
Empirical studies of the process of entry and industry evolution have identified various
characteristics which are not captured by standard textbook models of entry (see, for instance,
the detailed study by Geroski, 1991). One of these features is the joint occurrence of exit and
entry. Another is the fact that some firms stay in an industry for a long time while other firms
exit quite soon after they have entered. This article presents a model in which both of these
features can be equilibrium outcomes. The model formalizes the idea of windows of profit
opportunities (see Geroski, 1991). Firms that cannot survive in the long run may nevertheless
be willing to bear sunk entry costs because of profit opportunities for a sufficient amount of
time.
The model’s key assumptions are derived from two further empirical observations:
First, in the early stages of many industries, the number of potential entrants is limited in
each period of time and increases only over time. Klepper and Graddy (1990) attribute this
limitation to the fact that firms need a certain expertise to enter an industry. The relevant
knowledge is available only to agents who for instance, are experienced in related
technologies. An example of the importance of the relevant experience is given by Klepper
and Simons (2000). They show that prior knowledge from the production of radios was
decisive for entry into the television receiver industry in the early stages of that industry.
Klepper and Simons (2000) argue that ‘radio producers were well positioned early to learn
about technological developments in TVs’ (p. 1003). Therefore, radio producers would
‘qualify earlier as a potential entrant’ (ibid., emphasis added). For the laser industry, Sleeper
(1998) finds that entry often occurs by spin-offs from existing laser firms. This, and the
findings from TVs, suggests that entry requires a prior period of learning about the industry.
This necessity limits the number of potential entrants at each point in time.
-3Second, entrants often differ with respect to their capabilities, i.e., they are
heterogeneous. In the management literature, the question seems not to be whether the initial
allocation of firms’ capabilities differs, but to what extent firms are able to change despite
organizational inertia. There is considerable debate on the scope for firms to adapt their
capabilities.1
Together with sunk costs, limitations on the number of potential entrants and
heterogeneity yield a simple model in which the process of entry can take on a wide range of
patterns. Depending on the sequence of the number and type of potential entrants, the degree
of difference between the firm types, and the magnitudes of sunk and fixed costs, various
patterns of industry evolution can be distinguished. First, a pattern can arise which I call the
product life cycle as it resembles the patterns described by Klepper (1996). Second, path
dependence may result in the sense that the long run composition of the number and types of
firms depends on the sequence of potential entrants. Finally, the ‘traditional’ case may arise in
which only 'efficient' firms enter the market.
The starting point of the model is the market structure of monopolistic competition.
Monopolistic competition is formalized here using a variant of Dixit and Stiglitz (1977) and
Spence (1976). The ‘large group’ assumption (Chamberlin, 1933) underlying this market
structure ensures virtually no strategic interaction among firms. Nevertheless, individual firms
have monopoly power due to producing a special brand of a differentiated product. Features
such as tractability - even though there is imperfect competition - led to widespread use of this
market structure (e.g., Grossman and Helpman, 1991). Monopolistic competition is
particularly well suited for the discussion of entry dynamics and industry evolution for two
further reasons. First, sunk costs, which are quite important for the pattern of industry
1
On the discussion see, for example, the Special issue on ‘The evolution of firm capabilities’ in the
-4evolution (see, for instance, Dixit, 1989, and Lambson, 1992), seem to be substantial in
markets for differentiated products (see, e.g., Sutton, 1991).
Second, the theoretical argument that supports the assumption of an unlimited number
of potential entrants does not apply to Chamberlinian monopolistic competition. This
argument is based on the notion of ‘imitative entry’ meaning that a firm just copies the
product and the strategy of an incumbent firm. If imitative entry were possible, the number of
potential entrants should not be limited. However, as Geroski (1991, p. 178) argues,
empirically product differentiation establishes barriers to entry which are especially high for
imitative entrants, but which may not be a major impediment for the entry of innovative firms.
The Chamberlinian monopolistic competition model captures this argument since Bertrand
competition among the producers of the differentiated product precludes imitative entry. The
agents who have ideas for new products and find their niches in the market are, nevertheless,
able to enter. As I shall assume below, the number of these 'smart' agents will not be large
enough to immediately drive profits down to zero.
Turning to normative issues, the model allows the evaluation of economic policy in a
field in which governments seem to be particularly active. Various political initiatives indicate
that policymakers view entry to happen both on an insufficient scale and too slowly. The
German Federal Ministry of Economics and Technology, for instance, supports programs on
entrepreneurship.2 The aim of these initiatives is to develop an 'entrepreneurial culture' leading
to an increased number of business start-ups. While it is not straightforward how such
programs affect the number of potential entrants, it is obvious that some potential entrants for
whom entry is not profitable without support, may enter if they get subsidies. The model
shows that such a governmental activity increases welfare at the margin even in the case that
Strategic Management Journal, 2000. Of particular interest is Cockburn et al., 2000.
-5is the most unfavorable one for the politicians, namely the path dependence case: the policy
increases the number of 'inefficient' firms and thereby reduces the long run number of
'efficient' firms.
In general, my welfare analysis largely confirms the results for the static model of
monopolistic competition derived by Dixit and Stiglitz (1977). Assuming that a social planner
cannot change the number of potential entrants, I find that the number of firms entering the
market in a decentralized solution is too small compared to both first- and second-best welfare
measures. The result contrasts with the excess entry result of Mankiw and Whinston (1986).
The difference stems from the fact that product differentiation plays an important role in my
model. Entry increases consumers' utility even if prices and quantities consumed do not
change. The reason is that consumers value the increase in variety. Again, this feature seems
to be particularly important when dealing with innovative entry.
In the literature a number of explanations for the characteristics of the entry process are
offered. One approach is based on the notion of entry as a selection mechanism (Jovanovic,
1982). Contrary to the firms in my paper, Jovanovic assumes that firms are not sure about
some of their characteristics. By entering the market, they learn about their capabilities and based on the new knowledge - they decide to either stay in the market or exit soon.
Uncertainty plays a decisive role. No entry of 'wrong' firms would occur if they knew their
type.
Uncertainty also plays a key role in Lambson (1992) and Jovanovic and MacDonald
(1994). In these models uncertainty relates to variables external to firms. Shocks to input
prices or to demand can explain simultaneous entry and exit in Lambson (1992). The random
arrival of exogenous innovations leads to a life cycle pattern of industry evolution in
2
An overview of the programs is given on the Ministry's web page at http://www.bmwi.de/.
-6Jovanovic and MacDonald (1994). Both models assume perfect competition with an unlimited
number of potential entrants. Due to sunk costs, Lambson's (1992) model also exhibits path
dependence for certain parameter values.
My model shows that many of the stylized facts of industry evolution can be explained
without relying on uncertainty. While the model is best thought of as being complementary to
the above-mentioned approaches, it highlights the importance of the number of potential
entrants in an environment of sunk costs. Absence of uncertainty is also a feature of Petrakis
et al. (1997) and Petrakis and Roy (1999). In their perfect competition models, a shakeout may
occur as well. The reason is yet another limitation, namely a restriction on the optimum scale
of production which is removed only over time. Due to learning by doing (Petrakis et al.
1997) and cost reducing R&D (Petrakis and Roy, 1999), respectively, firms adjust to their
long run output level only over time. As firms grow although the demand schedule is given,
some firms may have to exit due to sharply decreasing prices. While this mechanism can
explain shakeouts, both models do not exhibit simultaneous entry and exit. Furthermore, no
entry occurs after the first period.
A paper that shares some of the key elements of my model is Klepper (1996). Klepper
assumes that firms differ with respect to their innovative capabilities and that the number of
potential entrants is limited. The Klepper model differs from mine in that it assumes price
taking, myopic firms and allows for additional elements such as process innovations and
adjustment costs relating to the expansion of the output. Furthermore, Klepper does not carry
out a welfare analysis. Compared to the complexity of Klepper’s model, my model makes it
possible to derive new points concerning entry and exit in a simple way.
The assumption of a limited number of potential entrants is also made in Klepper and
Graddy (1990). In this model of price-taking firms with different costs, capacity constraints,
-7which can only gradually be removed, are decisive for the evolution of the industry. Situations
like the path dependence case cannot arise in this model of homogeneous goods producers.
The remainder of the paper is organized as follows. Section 2 and 3 present the model
and its dynamic equilibrium, respectively. In Section 4, first- and second-best welfare results
are derived. Section 5 concludes.
2.
The model
The model describes the evolution of a market for a differentiated product in discrete
time. The industry consists of a continuum of firms. For simplicity I refer to the measure of
firms as the ‘number of firms’. Each of the firms active in the industry produces a single,
unique variety of the differentiated product. Therefore, the number of firms equals the number
of brands available to the consumers.
The consumers
Consumers maximize utility over an infinite horizon. Their intertemporal preferences
are assumed to be identical. Preferences of a consumer take on the form
e

ACt 
∞
U = ∑ t = 0 Rt  yt +
,
e


with 0 < e < 1 .
(1)
Here yt denotes the quantity of a numeraire and A and e are (positive) parameters. The
discount factor R is related to the rate of time preference r as follows: R ≡ 1 (1 + r ) . As is
clear from the notation, the consumer’s rate of time preference is assumed to be equal to the
interest rate r. The index Ct describes the sub-utility from the consumption of the
differentiated product. Ct is of the Dixit-Stiglitz (1977) type and is defined as
Ct =
( ∫ ( b ( j ) x ( j, t ) )
nt
0
α
dj
)
1α
with 0 < α < 1 .
-8The quantity of variety j which is consumed at date t is denoted as x(j,t). The factor b(j)
is where heterogeneity enters the model. Different values of b imply different consumer
valuations of the respective varieties. The factor b(j) captures the notion that the entrepreneurs'
ideas are not all alike in terms of the value they convey to the consumer. I assume that
potential entrants differ with respect to the consumers’ valuations of the variety they could
produce. As in Yarrow (1985), the differences may be interpreted as differences in quality or
in design.3 By the former interpretation, the specification would mean that a vertical
dimension of product differentiation is added to the Dixit-Stiglitz model of horizontal product
differentiation. Note that due to the two dimensions of product differentiation, low quality
products are demanded even if different qualities sell for the same price. The latter
interpretation, relying on differences in design, captures the notion that different varieties are
not equally appealing to consumers. The varieties may differ along characteristics like flavor
or image conveyed by the product. Regarding the Dixit-Stiglitz utility function as a social
welfare function representing a population of heterogeneous consumers,4 different brands may
attract a different number of buyers. Again, demand for various brands may well differ for
identical prices if the characteristics of the varieties differ. Cigarettes and candy-bars are
examples which come into mind.
The budget constraint of the consumer reads as
∑
∞
t =0
(
nt
)
R t yt + ∫ p ( j , t ) x ( j , t ) dj = I ,
0
(2)
where I denotes the consumer’s total discounted income and p(j,t) is the price of variety j in
time t. I furthermore assume that income is large compared to the expenditure for the
3
Yarrow (1985) also uses a quasi-linear upper tier utility function together with the Dixit-Stiglitz index.
4
A thorough discussion on a possible micro-foundation for such a social welfare function is beyond the
scope of the paper. For a micro-foundation and further discussion see Anderson et al. (1992).
-9differentiated product. This ensures that the consumer will demand the numeraire yt. For
simplicity, I assume that the number (that is the measure) of consumers is 1. The aggregate
demand is therefore identical to the demand of the single consumer considered above. With
the above assumptions, one gets the instantaneous demand function x(i,t) for variety i at time t
(for the derivation see, for instance, Grossman and Helpman, 1991, Chapter 3):
x ( i, t ) = b ( i )
α (1−α )
p ( i, t )
−1 (1−α )
1 (1− e )
A
(∫
nt
0
b( j)
α (1−α )
p ( j, t )
−α (1−α )
dj
)
( e −α ) ( α−eα )
.
(3)
This demand function differs in two respects from the Dixit-Stiglitz type demand functions
typically used, for instance, in growth theory (see Grossman and Helpman 1991). First, it does
not depend on the consumers’ total income. This is due to the quasi-linear instantaneous
utility function. Following standard dynamic Industrial Organization models, the demand
function is the same in every period. Demand changes only if prices (or the number of firms)
change. Second, demand for different varieties differs even for identical prices, if the factors
b(j) differ.
The demand function (3) is isoelastic with the elasticity of demand σ = −1/ (1 − α ) . σ
also describes the (constant) elasticity of substitution between two varieties. Actions of rivals
that result in price changes enter the demand function through the integral. Following Dixit
and Stiglitz (1977), I define a quality adjusted price index Pt for the consumption index Ct.
This gives further insight into the properties of the utility function. With the definition
Pt ≡
(∫
nt
0
b( j)
α (1−α )
p ( j, t )
−α (1−α )
dj
)
−(1−α ) α
,
(4)
total per-period expenditure on the differentiated goods can be written as follows:
nt
(
∫ p ( j, t ) x ( j, t ) dj = A
1 1− e )
0
Pt
− e (1− e )
.
(5)
This equation gives the following relationship between the consumption and the price index:
1 (1− e )
Ct = A
Pt
−1 (1− e )
.
(6)
- 10 Equation (6) describes a standard isoelastic demand function. The elasticity ε of the
consumption index with respect to the price index can now be calculated as ε = − 1 (1 − e ) . As
the elasticity is greater than 1 (in absolute terms), the expenditures on the differentiated
product will increase if P decreases. In order for the varieties of the differentiated product to
be substitutes rather than complements, the elasticity of substitution between two varieties (σ)
must be greater (in absolute terms) than ε. Therefore, I assume that α > e .
The firms
The technology in the differentiated goods industry is as follows. Firms that are entering
the industry have to bear sunk costs S. These sunk costs may be interpreted as either product
development or advertisement costs for the special brand or as an investment in specialized
equipment that cannot be recovered in case of exit. In addition to the sunk costs that arise only
once, there are costs to be incurred in each period in which the firm is active. These costs are
assumed to be composed of constant marginal costs c and a fixed cost F. Both c and F are
identical across firms and constant over time.
In the Chamberlin model all firms are price setters. The isoelastic demand function
gives rise to a simple mark-up pricing rule for given marginal costs c (see Dixit and Stiglitz,
1977, and Grossman and Helpman, 1991). For the profit-maximizing price p one gets
p = c α.
(7)
Here the firm index has been dropped because all firms face identical demand elasticities and
have identical marginal costs. Therefore all firms will charge the same price. The demand for
the different firms' varieties will, however, differ if the valuations b differ. Rather than
charging a price premium, firms offering the higher valued brands use their competitive
advantage entirely for output expansion. The technical reason for this result is that the
specification of firm heterogeneity does not change the elasticity of demand. This property fits
- 11 well with the above interpretation that the lower valued varieties attract a smaller number of
consumers from the heterogeneous population.5 Interpreting the differences among firms as
different qualities, in general one would expect that higher qualities sell for higher prices (see,
e.g., Anderson and de Palma, 2001). This feature could be integrated into the model by using a
variable elasticity of substitution (VES) utility function rather than the CES consumption
index used here.6 However, this would complicate the calculations considerably without, as I
expect, changing the main results. Therefore, I stay with the simpler CES version.
As regards heterogeneity among firms, for simplicity, I assume that only two different
types of firms exist, a low (L-type firms) and a high (H-type firms) valuation type. The
consumers’ valuation for the low type is normalized to unity, the valuation for the H-type
firms is a constant b > 1. Given this specification, consumers and a social planner would,
ceteris paribus, prefer to have only H-type firms. Therefore, these firms might be called
'efficient' firms as I did in the introduction.
Denoting the number of active firms of each type by nL and nH, respectively, operating
profits πi for each type i = L, H can be determined as a function of the number and type of a
firm's rivals.
π L ( nH ( t ) , nL ( t ) ) = c
− e (1− e )
1 (1− e )
A
( n (t ) + b
L
α (1−α )
nH ( t )
)
( e −α ) ( α−eα )
α
e (1− e )
(1 − α )
(8)
and
π H ( nH ( t ) , nL ( t ) ) = b
α (1−α ) − e (1− e )
c
1 (1− e )
A
( n (t ) + b
L
α (1−α )
nH ( t )
)
( e −α ) ( α− eα )
α
e (1− e )
(1 − α ) .
(9)
In the derivation of these profit functions the pricing rule (7) has been used. From the
operating profits fixed costs must be deducted in order to get the per-period profits. (8) and (9)
5
An industry which seems to exhibit this pattern is the U.S. cigarette industry. Roberts and Samuelson
(1988) report that different firms charge the same prices, nevertheless market shares are different.
6
For a recent contribution using the VES specification, see Elberfeld and Götz (2002).
- 12 demonstrate the importance of the assumption that e < α. Otherwise each firm’s (operating)
profit would increase if the number of firms increases.
3.
The equilibrium of the model
The equilibrium industry evolution is presented in a number of propositions below. The
propositions show how industry evolution depends on the parameters of the model, especially
on the sequences {k L ( t ) , k H ( t )} of the number and types of potential entrants. Here kL(t)
(kH(t)) denotes the number of L-type (H-type) potential entrants arriving at the market in
period t, where t ∈ ô0. An important assumption about the sequences is that they are common
knowledge as of the time of market birth. Industry evolution may go through several stages.
For easy reference I introduce the following symbols for periods in which the described events
take place.
ta, the period in which the last entry of L-type firms occurs.
tb, the period in which the first exit of L-type firms occurs.
tc, the period in which the last exit of L-type firms occurs.
td, the period in which the last entry of an H-type firm occurs.
In the description of industry evolution, I start with the long run. As long run I denote a
situation in which no further entry or exit occurs. The interesting question here is under what
conditions only ‘efficient’ firms will be active in the long run. The answer follows from a
comparison of operating profits of H-type and L-type firms, respectively. From (8) and (9),
one obtains
π H ( nH , nL ) = b
α (1−α )
πL ( nH , nL ) .
(10)
- 13 For L-type firms to be active in the long run, they must at least be able to cover their fixed
costs in each period. Thus, the inequality π L ( nH , nL ) ≥ F must hold. Therefore, we obtain a
lower bound for the discounted profits ΠH an H-type firm receives if L-type firms are active:
b
ΠH ≥
α (1−α )
F−F
−S.
1− R
(11)
The sign of the expression on the r.h.s of (11) determines the long run properties of the model.
Therefore, I assume:
(b
Assumption 1 (A1):
α (1−α )
)
−1 F
1− R
>S
(12)
From the above reasoning it is clear that L-type firms cannot be active in the long run if A1
holds. The number of H-type firms would otherwise always be increasing.7
If no L-type firms are active in the long run, the long run number of H-type firms, nH ,
is determined by the zero profit condition
π H ( nH , 0 ) − F
=S.
1− R
(13)
We obtain
nH =
(1 − α )( ) ( ) bαe ( α−e) .
( α−eα ) ( α−e )
eα α− e
c ( ) ( S (1 − R ) + F )
α ( α− e )
A
α
eα ( α− e )
α− eα
α− e
(14)
Under A1 all potential H-type entrants will enter the market until the number of H-type active
firms is nH. In more formal terms, the evolution of the number of H-type firms is determined
by the sequence
nH ( t ) = min
*
7
the case.
{∑
t
T =0
k H (T ) , nH
} for all t ∈ ô .
0
(15)
Of course, this requires that ΣTt=0 kH(t) grows without bound as T approaches infinity. I assume this to be
- 14 The last entry of H-type firms occurs in the period in which the cumulated number of potential
H-type entrants becomes greater than nH. This condition determines td.
Proposition 1 summarizes the above discussion.
Proposition 1: If A1 holds, only H-type firms will be active in the long run. The long
run number of firms is nH. All potential H-type entrants enter the market up to the
period in which the cumulated number of potential entrants becomes greater than nH.
This period td is uniquely determined by the sequence of potential entrants.
From Proposition 1 it is clear that the existence of L-type firms can only be a transitory
phenomenon under A1. Whether L-type potential entrants become active in the industry
depends on how wide the windows of profit opportunities are. Here the evolution of the
number of H-type firms, i.e. nH ( t ) , is of crucial importance. Suppose that the ‘first’ potential
*
L-type entrants arrive in the market at period t ≥ 0. If a single (i.e. a small measure) L-type
entrant were to enter its profit would be
( (
) )
Π L = ∑ ic=t Ri −t π L nH ( i ) , 0 − F − S ,
t −1
(
*
(16)
)
where tc is the first period at which π L nH ( t ) , 0 − F < 0 . L-type entrants will enter the
*
market if Π L > 0 . To distinguish different scenarios of industry evolution, I state the
respective condition as an assumption.
Assumption 2 (A2):
∑
tc −1
i =t
( (
) )
Ri −t π L nH ( i ) , 0 − F > S ,
*
A2 will hold if the speed at which H-type potential entrants firms arrive in the market is low.
In this case the number of H-type firms increases slowly over time leaving room for the L-type
firms to recover their sunk costs.
- 15 Proposition 2: If A1 and A2 hold, entry of L-type firms will occur but eventually all L-
type firms will exit. NL, the total number of L-type firms entering the market, and the
dates of last entry, ta, of first exit, tb, and of last exit, tc, of L-type firms are uniquely
determined.
Proof: See Appendix.
An interesting corollary to Proposition 2 follows from the fact that
π L ( nH , nL ) = F
(17)
between tb and tc. As the number of H-type firms increases according to nH ( t ) , L-type firms
*
must exit for (17) still to hold. From (8) it follows that nL must change according to the
derivative dnL dnH = −b α (1−α ) in order to satisfy (17). The derivative is clearly smaller than
minus 1; producing more output than L-type firms, an efficient H-type entrant drives out more
than ‘one’ less efficient L-type firm. Thus, we obtain
Proposition 3: The total number of firms is decreasing between tb and tc.
The pattern of industry evolution that arises in situations to which Propositions 1 to 3
apply is the ‘product life cycle’. The simulation presented in Figure 1 gives an example of
industry evolution in this case.
Figure 1 about here
In the early stages of the product’s life, that is until ta, entry of all types of firms occurs.
Between ta and tb the number of firms still increases, but only H-type firms enter. From tb on,
entry and exit takes place simultaneously. The number of L-type firms decreases faster than
the number of H-type firms increases. Until tc is reached, the total number of firms is falling.
- 16 However, note that - as can easily be shown - the price index Pt, the consumption index Ct,
n
and a properly defined output index X t ≡ ∫ x ( j , t ) dj are constant between tb and tc. Entry of
0
H-type firms then carries on until td.8 In the long run, only H-type firms survive.
The above description demonstrates that my model can account for the two empirical
features of entry mentioned in the introduction, namely, the joint occurrence of exit and entry
and the fact that firms differ with respect to their lifespan. In the simulation, exit and entry
occur in 1/3 of the time in which transition to the long run takes place. The life-span of L-type
firms varies between some 40 and 105 periods. The model also exhibits the phenomenon of a
shakeout. The driving force for the shakeout is the replacement of less efficient firms by more
efficient entrants. Note that after the shake–out phase the total number of firms starts growing
again. In Figure 1, growth even leads to the result that the long run number of firms coincides
with the maximum number of firms ever active in the industry. This, however, need not be the
case as Figure 2 below shows. What is a consequence of the model, however, is that entry
goes on after the shakeout. While this feature contradicts patterns of industry shakeout
described by Klepper (1996), it fits with new findings by Carroll and Hannan (2000). For
several industries, e.g. automobile manufacturers in France, they find that entry of new firms
and products takes place after a consolidation phase. In his review of Carroll and Hannan
(2000), Jovanovic (2001) argues that the list of industry-life-cycle facts might need a
modification to include a stage of secondary entry.
Now I turn to the case where A1 is not satisfied. The discussion of A1 has shown that an
L-type firm which has made the sunk investment will never exit the industry in this case. The
operating profits πL will never fall short of the fixed costs F. Whether L-type firms find it
worthwhile to enter in the first place, and how many L-type firms will enter the industry
8
td > tc follows from the fact that nH > nH as determined in (28) by construction.
- 17 depends crucially on how fast the number of H-type firms increases. Thus, industry evolution
exhibits path dependence in the following sense. The long run number and composition of
types of firms is affected by the sequence of potential entrants. Of course, for industry
evolution the number of both types of firms is important. Even if the number of potential Htype entrants is small for a large number of periods such that an L-type entrant could break
even, entry of L-type firms could not occur if there were no potential L-type entrants.
Proposition 4 deals with the path dependence case.
Proposition 4: If A1 does not hold but A2 does, L-type firms will enter but never exit.
The long run number NH of H-type firms depends on the active number of L-type firms,
NL, via the function
N H = nH −
N
b
L
α (1−α )
where N L ∈ [ 0, nL ) and (18)
,
(
)( )
1− α)
(
.
nL =
( α− eα ) ( α− e )
eα α− e
c ( ) ( S (1 − R ) + F )
α ( α− e )
A
α
eα ( α− e )
α− eα
α− e
(19)
NL and the dates ta and td exist and are unique.
Proof: see Appendix.
Note that the path dependence result implies that industry structure is asymmetric even
in the long run. Initial differences form the basis of long run heterogeneity. Proposition 4 also
shows that ‘bad’, early entrants deter ‘good’ but late potential entrants from entering in the
path dependence case. Producers of low value goods, however, can never deter entry of high
value good producers completely (the derivative of (18) with respect to nL is smaller than 1
andnL < nH). Some H-type producers are always able to break even because of higher
operating profits but identical investment costs compared to L-type firms. As the discussion of
- 18 A2 and the following proposition show the ‘deterrence’ result depends crucially on the speed
at which potential H-type firms arrive in the market.
Proposition 5: If A2 does not hold, L-type firms will never enter and industry evolution
is described by nH ( t ) .
*
Proof: trivial.
Proposition 5 describes the ‘traditional’ case: the ‘good’ entrants deter the ‘bad’
potential entrants. Trivially, this case applies in a situation in which the traditional assumption
of an unlimited number of potential entrants (of the H-type) is satisfied. The last entry would
then take place in the period of the market birth. The traditional case arises also if the number
of potential (H-type) entrants is limited but increases so fast over time that L-type firms are
unable to cover their sunk costs. The windows of profit opportunities are not wide enough for
L-type firms under these circumstances.
The above propositions show how various factors shape industry evolution. One key
factor is the sequence of the number of potential (H-type) entrants. It determines whether A2
is satisfied and thus whether one departs from the textbook case of entry or not. I examine the
impact of different sequences of potential entrants on industry evolution in simulations below.
Before doing that, however, it is instructive to look at the factors underlying A1. Given that
A2 holds, these factors are decisive for the long run properties of the model. The respective
variables determine whether existence of L-type firms is a transitional or a long run
phenomenon. As it turns out, the effects of these variables are quite intuitive. First, consider
the degree by which the valuations of the varieties differ. It is straightforward that the path
dependence case is more likely to apply, if these differences in the valuations are small (i.e. b
is close to 1).
- 19 Next, consider the effect of the parameter α. It determines the degree of substitutability
between varieties as well as the demand elasticity. Path dependence is more likely to occur if
α is small. Small values of α imply limited substitutability among varieties and low demand
elasticities. Even though valued higher by consumers, the market ‘share’ gains of H-type
products are limited. We obtain the intuitive result that L-type firms survive in the long run if
the degree of (exogenous) product differentiation is high as it is with small values of α.
Finally, look at fixed and sunk costs. Path dependence applies if sunk costs S are large
compared to fixed costs F. Note that the path dependence case requires sunk costs to be
positive. If there are no sunk costs, the product life cycle or the textbook case will result. This
follows from the fact that per-period profits of H-type firms, i.e. their operating profits minus
fixed costs, are always greater than the respective numbers for the L-type firms. As regards
fixed costs, notice that A1 could not be satisfied if fixed costs were absent. Thus, with zero
fixed costs, we obtain path dependence. The per-period profits of inefficient firms cannot
become negative. Therefore, an L-type firm, which once became active, can never be driven
out of the market.
A1 also sheds light on how much better the next generation of a product must be in
order to completely drive out the state of the art product. To see this, interpret the producers of
the next generation as H-type potential entrants, which arrive at the market well after market
birth. A1 shows how much better (in terms of b) the next generation must be in order to gain
the whole market. It can therefore be thought of as a formalization of the popular ‘10X rule’.
Used by venture capitalists, this rule of thumb states that a new product should be backed only
if it is (perceived to be) 10 times better than the product it is supposed to replace. If the 10X
rule does not apply, the product will fail to replace the old product. An example for a
'successful' product is the CD, which replaced the vinyl record. Failures where the digital
audio tape (DAT) and the digital compact cassette (DCC). Both innovations didn’t pass the
- 20 10X test; they failed to replace the CD. (See The Economist, 28 September 1996). One might
want to add that the new technologies have, nevertheless, gained some market share. This is
exactly what the model predicts.
I conclude this section with two examples that highlight the impact of the sequence of
potential entrants on industry evolution. Figure 2 depicts industry evolution in the product life
cycle case (A1 holds).
Figure 2 about here
The long run number of firms is, of course, the same in all three cases. When this number is
reached depends, however, on how fast potential H-type entrants arrive in the market. If twice
the number arrives every period, it takes half the time to reach the steady state. As a
consequence, a smaller number of L-type firms enters the market. A smaller number of
potential L-type entrants (i.e. a smaller kL(t)) leads, ceteris paribus, to less entry. Finally, note
that the shakeout might be drastic. In the case (kH (t), kL (t)) = (1,20), 200 L- type firms enter
and later exit. Thus, more than 40% of the industry population exits.
The second example concerns the path dependence case. Here I examine what happens if
potential H-type entrants start to arrive in the market after its birth. Thus, H-type firms can be
thought of as producers of the next generation of a product. In figure 3, I vary the period x at
which the first potential H-type entrants arrive at the market. Arrival of the next generation
ranges between market birth (x = 1) and 12.5 years (x = 150). The relation between the pattern
of industry evolution and the arrival date of the next generation is clear-cut. Later arrival leads
to increases in both the (long run) number of L-type firms and the total number of firms. The
number of H-type firms decreases. As a result, the share of L-type firms in total industry
population grows from close to 0 to more than 50%. Figure 3 also shows that a given increase
- 21 in the time-span till the arrival of the next generation has different impacts depending on what
the date of arrival is in the first place. The change in the number of firms is much less drastic
when going from x = 120 to x = 150 compared to the situation where x, for instance, is
increased from 30 to 60. Put differently, the effects of delaying and moving forward the
introduction of the next generation are asymmetric. Primarily due to discounting, advancing
the next generation has a greater impact on the pattern of industry evolution.
Figure 3 about here
4.
Welfare analysis
As mentioned in the introduction, policy makers seem to become particularly active
when it comes to supporting entry of new firms. Different arguments are stated for such a
policy, e.g. labor market effects or the development of a more dynamic and innovative
economy. Irrespective of the reasons for supporting entry, there seems to be a political
consensus that more entry is better in general.9 My model gives the opportunity to evaluate
such a policy in a framework where a binding constraint on the extent of entry exists. Due to
this feature government policy is likely to pick ‘losers’, i.e. inefficient firms. The welfare
analysis of this section will give an answer on how to judge such a policy.
The welfare analysis assumes that the social planner cannot change the number of
potential entrants. In order to address welfare issues, I derive the consumers’ indirect utility
function. Using (2) in (1), one gets
9
Of course, there are exceptions especially then it comes to entry of foreign firms in ‘sensitive’ sectors.
See, for instance, the position of the Monetary Authority of Singapore regarding entry into the banking sector in
Singapore (www.mas.gov.sg).
- 22  ACt e

nt
∞
− ∫ p ( j , t ) x ( j , t ) dj  .
U = I + ∑ t =0 R t 
0
 e

(20)
I define CS, the consumers’ surplus from the differentiated good, as
 A1 (1−e ) Pt − e (1−e )
− e 1− e 
1 1− e
CS ≡ U − I = ∑ t =0 R 
− A ( ) Pt ( )  .


e


∞
t
(21)
In this expression I have used (5) and (6). It can be simplified to give
∞
 1 − e 1 (1− e ) − e (1− e) 
CS = ∑ t = 0 R t 
A
Pt

 e

(22)
As a benchmark, I mention the result for the traditional case (Proposition 5). From Dixit
and Stiglitz (1977) it is immediately apparent that the number of firms which are eventually
active in the market solution is too small compared to the first-best social optimum. The
respective optimization problem
Max CS − nH ( S + F (1 − R ) )
nH
(23)
yields
nHFB = α
−α ( α− e )
nH > nH ,
(24)
where nHFB is the first-best number of H-type firms. As equation (24) shows the extent of the
deviation of the market solution from the planner solution depends on the mark-up the firms
charge.
While the first-best result is a useful benchmark, it seems sensible for many questions of
economic policy to apply measures that do not rely on marginal cost pricing. Following
Spence (1976), two constrained social optimization problems may be distinguished.
1. The planner cannot influence the conduct, i.e. the price setting behavior of the firms.
However, the planner can choose the number of firms. To implement such a policy
either entry subsidies or entry fees are necessary. This criterion was used, for
instance, by Mankiw and Whinston (1986).
- 23 2. The planner optimizes prices subject to a zero profit constraint. Using this criterion,
Dixit and Stiglitz (1977) come up with the conclusion that the market solution under
monopolistic competition is second-best.
Here I focus on the first criterion as it seems to be more relevant for the policy measures I
want to evaluate than the more restrictive second criterion. Contrary to the assumption
underlying the latter criterion, governments actually seem to subsidize entry out of the general
budget.
Again the derivation of the second-best result is straightforward. The planner’s
optimization problem consists of the maximization of the sum of the consumers’ surplus and
the profits of the firms. In the traditional case with only H-type firms active it reads
 π (n ) − F

Max CS + nH  H H
−S.
nH
1− R


(25)
The result for the second-best number of firms, nHSB , is
n
SB
H
 1 − αe 
= nH 
 α (1 − e ) 


( α−αe ) ( α− e )
FB
H
=n α
αe ( α− e )
 1 − αe 


 (1 − e ) 
( α−αe ) ( α− e )
> nH .
(26)
Thus, the underentry result extends also to the constrained optimum. Using a less restrictive
criterion than the break-even constraint of Dixit and Stiglitz (1977) yields a case for entry
subsidization, even if a government cannot change the pricing behavior of firms. Obviously,
the consumer surplus effect associated with of an additional entrant dominates the business
stealing effect.
As far as the relation between first-best and second-best number of firms is concerned, a
formal result is not available. Calculations for specific parameters indicate that the first-best
number is greater. Note that the second-best number of firms approaches the first-best as e
goes to 0. The respective outputs of the firms differ, however. This is clear from the fact that
with e = 0 aggregate expenditure on the differentiated products is constant. Given the number
- 24 of firms, the prices are higher in the second-best case, therefore quantities must be smaller. In
general, the second-best solution trades-off the efficiency loss due to a sub-optimal scale with
the welfare gain due to increased variety of products.
The variety effect also explains the difference to the excess entry result derived by
Mankiw and Whinston (1986) for a homogeneous good oligopoly. In the Dixit-Stiglitz model,
entry takes on the form of innovative entry implying that the range of differentiated products
increases. This leads to welfare effects which go beyond a simple increase in competition.
There is a caveat regarding the underentry result. Anderson et al. (1995) show that even
in the case of product differentiation and love of variety, the market solution may exhibit too
many firms from a social point of view. A detailed discussion of the empirical relevance of
the different specifications is beyond the scope of this paper. I only want to make two points
why the policy conclusions drawn on the basis of my model are relevant. First, the underentry
result corresponds to the above mentioned ‘conventional wisdom’ of policy makers. Second,
the result is consistent with Geroski (1991) stating that one can hardly imagine that there is
too much entry. Geroski’s argument is based on the view that consumers’ needs are diverse.
More entry, especially of different types, is likely to improve the fit produced by the market
selection process.
Now I turn to welfare in the path dependence case. The welfare analysis is performed by
evaluating the effects of a small policy. That is, I calculate the derivative of the social welfare
function with respect to the number of both H-type and L-type firms. Evaluation of the
respective derivative at the market solution tells us whether a social planner wants to have
more or fewer firms of both types. Thus, starting from the market solution the planner decides
both whether she wants to have more entry of L-type firms in period ta and whether the
number of H-type firms should still increase in period td. Proposition 6 shows that the planner
actually chooses more firms of both types than the market solution. Note, however, that the
- 25 trade-off between the first-best numbers of the both types is the same as in the market
solution: If the L-type firms come in faster so that NL increases, NH decreases by
(1 b
α (1−α )
) ∆N
L
(see (18)).
Proposition 6: In the path dependence case (Proposition 4 applies), the number of both
H-type and L-type firms is greater in the first-best welfare maximizing solution than in the
market solution.
Proof: See Appendix.
The next proposition examines a kind of a ‘worst case scenario’: The government
subsidizes entry of L-type firms even though NH decreases as a result of the increase in NL.
Thus, the government picks ‘inefficient’ firms.
Proposition 7: Starting from the market solution of the path dependence case
(Proposition 4 applies) and taking the pricing behavior of the firms as given, a marginal
increase in the number of L-type firms increases welfare.
Proof: See Appendix.
Proposition 7 shows that even this policy intervention increases welfare (at least as long
as it is small). The policy does not have welfare effects in the long run (from td onwards) as
the change in the number and composition of firms is such that neither the profits nor the
utility of the consumers is changed. The short run gains in consumers' surplus obviously more
than compensate for the reduced profits of the firms between ta to td. The result not only
confirms Geroski’s (1991) abovementioned statement that there is hardly too much entry.
From the perspective of the above analysis, one would like to add that this is even true for the
- 26 entry of low quality firms. As mentioned in the introduction, this result is driven by the fact
that even 'inefficient' firms increase the variety of products available to the consumers.
The final welfare issue I want to address deals with the amount of ‘transitory entry’ by
L-type firms in the product life cycle.
Proposition 8: Starting from the market solution of the product life cycle case
(Propositions 1 and 2 apply) and taking the pricing behaviour of the firms as given, a
marginal increase in the number of L-type firms increases welfare.
Proof: See Appendix.
The result is analogous to that in the path dependence case. From a second-best
perspective, it is optimal to encourage more firms to enter than the number that would enter in
an unregulated market. Rather than ‘closing the windows of profit opportunities’, policy
should promote entry by L-type firms. The loss of sunk investments, which is due to the future
exit, is more than compensated by the short-term gains.
5.
Conclusions
In this paper, two elements have been added to a standard model of monopolistic
competition: First, the number of potential entrants is limited in each period and increases
only over time. Second, the potential entrants differ with respect to the consumers’ valuation
of the variant they could offer. It has been shown that the resulting simple model exhibits a
rich dynamic structure covering cases like the product life cycle, a path dependent equilibrium
and the traditional textbook case of entry. The model shows that adding a time dimension to a
static model does not merely yield transitional dynamics but may well have long-term
consequences if sunk costs exist.
- 27 At this point a remark on the type of heterogeneity present in the model is in order. In
general, firms could differ along various dimensions apart from consumers’ valuations of the
product. An obvious candidate for differences among firms are costs, where marginal as well
as fixed and sunk costs could be the source of heterogeneity. The results of the model
regarding industry evolution should carry over to these cases as well. It is no problem to deal
with cost asymmetries in the present framework. The reason for choosing differences in
consumers’ valuation is that –in my view- these are most likely to endure in the long run
whereas cost differences might be eroded over time by imitation. Ultimately, what type of
heterogeneity is more long-lived is an empirical question which still remains unanswered (see,
e.g., Cockburn et al. (2000)).
Regarding policy conclusions, the welfare analysis confirms Geroski’s opinion that one
can hardly imagine that there is too much entry (Geroski, 1991). Even subsidizing the entry of
'inefficient' firms improves welfare if these firms add variety to the consumers' choice. As
discussed in some detail in section 4, the welfare results are sensitive to the functional
specification of the model. This is different as far as the positive analysis is concerned. The
derived patterns of industry evolution depend on general factors, namely product
differentiation, sunk costs, and heterogeneity of firms, rather than the functional specification.
While the model is a perfect foresight model, from the point of view of (most) potential
entrants the rationality requirements are weaker. For H-type firms, it is sufficient to know
whether they break even if no further firms entered the market. That is, they do not have to
care about future entry. L-type firms must have an idea whether their window of profit
opportunities is wide enough. If future entry is sufficiently slow, they are able to break even.
While the model does not allow for uncertainty on the rate at which new entrants arrive, the
basic intuition should carry over to the case where the sequence of the number and types of
future potential entrants is random, but drawn from a distribution which is known. A bigger
- 28 problem seems to be the question whether and how this sequence should and could be
endogenized. Here spin-offs from incumbent firms as well as entry from industries with
relevant experience seem to be important determinants. While research on this topic has
begun (see Klepper and Simons, 2000), the question of how entrepreneurs are ‘produced’ by
an economy is far from being answered. Like the introduction of additional elements like
innovations into the framework presented here, this question seems to be a promising
direction for future research and a problem for which policy makers are badly in need of a
solution.
Acknowledgements
Earlier versions of this paper were presented at the EARIE conference in Copenhagen in
1998 and at the European Economic Association Congress in Berlin in 1998. I wish to thank
seminar participants, Gerhard Clemenz, Dennis Mueller, Gerhard Sorger, Dylan Supina, the
referees, and the Editor Simon Anderson for helpful comments and discussions.
Appendix
Proof of Proposition 2: The statement on entry and exit follows immediately from A2
and from the discussion of Proposition 1. The last exit of low value good producers occurs in
the period in which the condition
π L ( nH , 0 ) ≤ F
(27)
holds for the first time. This yields:
nH ≥
α ( α− e )
A
α
c
eα ( α− e )
eα ( α− e )
(1 − α )(
F(
α− eα ) ( α− e )
α− eα ) ( α− e )
From nH ( t ) , tc follows.
*
NL and ta are determined by the zero-profit condition
.
(28)
- 29 -
∑
∑
ta −1
t =0
tb −1
i = ta
( (
) )
R i −ta π L nH ( i ) , N L − F = S
*
and by
(29)
k L ( t ) = nL ( ta − 1) < N L ≤ ∑ ta=0 k L ( t ) .
t
Existence and uniqueness of NL and ta follow from A2 (discounted profits must exceed S for ta
= 0 and nL = 0) and from the fact that the l.h.s. of (29) is strictly decreasing in NL. In (29), the
fact is used that per-period profits of L-type firms are 0 from tb onwards. tb is the earliest date
at which
(
)
π L nH ( t ) , N L ≤ F .
*
(30)
The existence of a finite tb for all values of NL follows from the fact that all L-type firms
eventually exit.
Q.E.D.
Proof of Proposition 4: The statement on entry and exit follows from the discussion in
the text. The statement on the long run number NH follows from the zero-profit condition for
H-type firms for the case when N L ∈ [ 0, nL ] L-type firms entered the market. This condition
reads as follows:
πH ( N H , N L ) − F
=S.
1− R
(31)
Using (9) and (14), it can be seen that the function (18) satisfies the identity (31) in the
relevant interval. The upper bound of the interval nL is the long run number of firms if there
were only L-type entrants. It is calculated in the same way as nH. Note that this number is
never reached if there are potential H-type entrants.
NL is determined by the condition
∑
td −1
i = ta
(
)
R i −ta π L ( nH ( i ) , N L ) − F + R td −ta
πL ( N H , N L ) − F
=S,
1− R
(32)
- 30 where nH ( t ) = ∑ T =0 k H (T ) and NH is defined by (18). The dates ta and td are determined by
t
the conditions
∑
∑
ta −1
t =0
td −1
t =0
k L ( t ) = nL ( ta − 1) < N L ≤ ∑ ta=0 k L ( t )
t
and
k H ( t ) = nH ( td − 1) < N H ≤ ∑ td= 0 k H ( t ) .
t
The proof of existence and uniqueness is analogous to the proof of the respective part
Proposition 2 and is therefore omitted. Note that πL(NH,NL) does not change when NL changes
because of the associated change in NH.
Q.E.D
Proof of Proposition 6: The first-best welfare measure reads
e(1−α ) ( α−αe )
t −1
 1 − e 1 (1−e )
α 1−α
− e 1− e 
W = ∑ td=t R t −ta 
A
N L + b ( ) nH ( t )
c ( )+
a
 e

e
1
−α
α−α
e
(
)
(
)
1− e
1 1− e
α 1−α
− e 1− e
A ( ) NL + b ( ) NH
c ( )−
+ R td − ta
(1 − R ) e
(
(
)
)
− ( N L − nL ( ta − 1) ) S − N L F (1 − R )
(33)
+ ∑ td=t Rt −ta {− nH ( t ) F − k H ( t ) S } −
t −1
a
−R
td −ta
(N
H
)
F (1 − R ) + ( N H − nH ( td − 1) ) S .
As changes are absent in the periods before ta the respective expressions have been dropped.
The first two lines in (33) capture the consumers’ surplus with marginal cost pricing. The
remaining terms give the costs. Note that it is assumed in this function that L-type firms never
stop producing. The planner maximizes welfare by choosing NL and NH. The derivative with
respect to NL is
 (1 − α ) 1 (1−e )
∂W
t −1
α 1−α
= ∑ td=t Rt −ta 
A
N L + b ( ) nH ( t )
a
∂N L
 α
(
+R
td − ta
)
( e −α ) ( α−αe )
c
− e (1− e )

+

(1 − α ) A1 (1−e) N + bα (1−α) N ( e−α ) ( α−αe) c − e (1−e) − S + F 1 − R .
(
))
(
( L
H )
(1 − R ) α
(34)
This expression is now evaluated at the market solution using the zero profit condition for the
L-type firms, which defines NL. The zero profit condition can be written as (see (32))
- 31 -
∑
td −1
t = ta
+R
R t −ta c
td − ta
c
− e (1− e )
− e (1− e )
(N
1 (1− e )
A
(N
1 (1− e )
A
L
L +b
+b
α (1−α )
α (1−α )
NH
nH ( t )
)
)
( e −α ) ( α− eα )
( e −α ) ( α− eα )
(1 − α ) +
(1 − α ) (1 − R ) = α
− e (1− e )
( S + F (1 − R ) )
(35)
Using (35) in (34) yields:
∂W
∂N L
(
= α
marketsolution
−1 (1− e )
)
− 1 ( S + F (1 − R ) ) > 0
(36)
Taking the derivative of W with respect to NH, results in:
(1 − α ) A1 (1−e) N + bα (1−α) N
∂W
= R td − ta
L
H
∂N H
(1 − R ) α
(
)
e(1−α ) ( α−αe ) −1
b
α (1−α ) − e (1− e )
c
−
(37)
− Rtd −ta ( F (1 − R ) + S )
The zero profit condition for H-type firms can be written as (see (31))
c
− e (1− e )
1 (1− e )
A
(N
L
+b
α (1−α )
NH
)
( e −α ) ( α−eα )
(1 − α ) (1 − R ) = α − e (1−e) ( S + F (1 − R ) ) b−α (1−α ) .(38)
Using (38) in (37) one obtains the same expression as in (36), namely
∂W
∂N H
(
= α
marketsolution
−1 (1− e )
)
− 1 ( S + F (1 − R ) ) > 0 .
(39)
As welfare is strictly concave in NH and NL (as can be derived from (34) and (37)), the firstbest number of both L-type and H-type firms is greater than the number of such firms active in
the market solution.
Q.E.D.
Proof of Proposition 7: Welfare W is measured as the sum of the consumers’ surplus
and of aggregated profits of L-type (ΠLT) and H-type firms (ΠHT). One obtains the following
expression:
- 32 e(1−α ) ( α−αe )

α (1−α )
t − ta  1 − e 1 (1− e )
+
R
A
N
b
n
t
p − e (1− e )  +
(
)
L
H

t = ta
 e

W=
e(1−α ) ( α−αe )
1− e
+ R td − ta
A1 (1−e ) N L + b α (1−α ) N H
p − e (1−e ) +
(1 − R ) e
∑
(
td −1
(
)
(
)
+ ∑ td=t Rt −ta N L π L ( nH ( t ) , N L ) − F − ( N L − nL ( ta − 1) ) S +
t −1
a
+R
N L ( π L ( N H , N L ) − F ) (1 − R ) +
td − ta
{
(

 T
 ΠL


 T
 ΠH

}
)
+ ∑ td=t Rt −ta nH ( t ) π H ( nH ( t ) , N L ) − F − k H ( t ) S +
t −1
a
(



 CS


)
+ Rtd −ta N H S − ( N H − nH ( td − 1) ) S
)
(40)
In this expression, the fact has been used that the profits of the last entrants of the H-type must
be equal to the sunk costs. The welfare effects of a small change in NH can be calculated from
the derivative
 (1 − α ) 1 (1−e )
t −1
∂W
α 1−α
A
N L + b ( ) nH ( t )
= ∑ td=t Rt −ta 
a
∂N L
 α
(
)
( e −α ) ( α−αe )
p

+

∂N H  − e (1−e )
+
p
∂N L 
− e (1− e )
(1 − α ) A1 (1−e) N + bα (1−α) N ( e−α ) ( α−αe) 1 + bα (1−α)
( L

H )
(1 − R ) α

∂π L ( nH ( t ) , N L )
t −1
t −1
+ ∑ t =t R t −t ( π L ( nH ( t ) , N L ) − F ) + ∑ t =t R t −t N L
−S +
+ R td − ta
d
d
a
a
a
a
+ R td −ta ( π L ( N H , N L ) − F ) (1 − R ) + Rtd −ta N L
+ ∑ td=t R t −ta nH ( t )
t −1
a
∂π H ( nH ( t ) , N L )
∂N L
∂π L ( N H , N L )
∂N L
(41)
(1 − R ) +
∂N L
In this derivative, it has been taken into account that the change in NL will induce
changes in both the long run number of H-type firms and in the profits of the firms. The
following relations help to simplify (41). First, the zero profit condition for the L-type firms
(32). Second, equation (18), which gives the long run trade-off between L-type and H-type
firms. That derivative ( ∂N H ∂N L ) is −b
−α (1−α )
, the respective term in brackets cancels out as
well as ∂π L ( N H , N L ) ∂N L . Using the derivatives of the profit functions (8) and (9), and
performing some more manipulations, yields the following simple expression:
- 33 -
(1 − α ) 1 − α A1 (1−e) N + bα (1−α) n t
t −1
∂W
= ∑ td=t Rt −ta
L
H ( )
a
∂N L
α 1− e
(
)
( e −α ) ( α−αe )
c
− e (1− e )
α
e (1− e )
This expression is unambiguously greater than zero. That proves the proposition.
(42)
Q.E.D.
Proof of Proposition 8: Welfare is described by
t −1
 1 − e 1 (1−e )
α 1−α
W = ∑ tb=t Rt −ta 
A
N L + b ( ) nH ( t )
a
 e
(
(
)
e(1−α ) ( α−αe )
)
p
− e (1− e )
+ ∑ tb=t R t −ta N L π L ( nH ( t ) , N L ) − F − ( N L − nL ( tb − 1) ) S +
t −1
a
{
(

+

(43)
}
)
+ ∑ tb=t R t −ta nH ( t ) π H ( nH ( t ) , N L ) − F − k H ( t ) S + const
t −1
a
Note that the periods before ta, the last entry of an L-type firm, and from tb, the first exit of an
L-type firm, onwards do not matter here. Nothing changes in this periods. Taking the
derivative of (43) with respect to NL and performing the same steps like in the proof of
proposition 4, one gets an expression, which is equivalent to (41).
 (1 − α ) 1 (1− e )
t −1
∂W
A
N L + bα (1−α ) nH ( t )
= ∑ tb=t R t −ta 
a
∂N L
 α
(
+ ∑ tb=t R t −ta N L
t −1
∂π L ( nH ( t ) , N L )
a
+ ∑ tb=t R t −ta nH ( t )
t −1
a
∂N L
)
( e −α ) ( α−αe )

p − e (1−e )  +

+
(44)
∂π H ( nH ( t ) , N L )
∂N L
Further simplifications give
(1 − α ) 1 − α A1 (1−e) N + bα (1−α ) n t
t −1
∂W
= ∑ tb=t R t −ta
L
H ( )
a
∂N L
α 1− e
(
This expression is positive.
)
( e −α ) ( α−αe )
c
− e (1− e )
α
e (1− e )
(45)
Q.E.D.
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Number of firms
175
150
125
100
75
50
25
ta
30 t
b
60
90
tc 120
150
td180
t
Figure 1: Industry evolution, if A1 and A2 hold.
The parameters are α = ½, e = 1/5, b =3/2, A =10, c = 1, R =124/125, F = 1/6, S = 1, kH (t) = 1, kL (t) = 20.10
10
Note that the periods should be thought of as months. In this case the yearly interest rate is about 10%.
Notice also that I assumed the number of potential entrants to be the same in each period.
Number of firms
500
kH ,kL 2,20
kH ,kL 1,20
400
300
kH ,kL 1,1
200
100
100
200
300
400
t
Figure 2: Industry evolution with different sequences (kH (t), kL (t)) of potential entrants (A1 and A2 hold).
The parameters are α = ½, e = 1/4, b =3/2, A =50, c = 1, R =124/125, F = 1, S = 60.
Number of firms
350
300
x1
x30
x60
x90
x120 x150
250
200
150
100
50
50
100
150
200
t
Figure 3: Industry evolution in the path dependence case: first arrival of potential H-type entrants at period x.
The parameters are α =½, e =1/4, b =3/2, A =50, c =1, R =124/125, F =1, S =120. kH (t ≥ x) = kL =5, kH (t < x) =0.