Learning-by-Doing, Organizational Forgetting, and Industry Dynamics

Learning-by-Doing, Organizational Forgetting, and Industry Dynamics
Learning-by-Doing, Organizational Forgetting,
and Industry Dynamics∗
David Besanko†
Ulrich Doraszelski‡
Yaroslav Kryukov§
Mark Satterthwaite¶
December 21, 2008
Abstract
Learning-by-doing and organizational forgetting have been shown to be important
in a variety of industrial settings. This paper provides a general model of dynamic
competition that accounts for these economic fundamentals and shows how they shape
industry structure and dynamics. Previously obtained results regarding the dominance
properties of firms’ pricing behavior no longer hold in this more general setting. We
show that forgetting does not simply negate learning. Rather, learning and forgetting
are distinct economic forces. In particular, a model with learning and forgetting can give
rise to aggressive pricing behavior, market dominance, and multiple equilibria, whereas
a model with learning alone cannot.
∗
We have greatly benefitted from the comments and suggestions of the Editor, Steve Berry, and two
anonymous referees. We are also indebted to Lanier Benkard, Luis Cabral, Jiawei Chen, Stefano Demichelis,
Michaela Draganska, Ken Judd, Pedro Marin, Ariel Pakes, Michael Ryall, Karl Schmedders, Chris Shannon, Kenneth Simons, Scott Stern, Mike Whinston, and Huseyin Yildirim as well as the participants of
various conferences. Guy Arie and Paul Grieco provided excellent research assistance. Besanko and Doraszelski gratefully acknowledge financial support from the National Science Foundation under Grant No.
0615615. Doraszelski further benefitted from the hospitality of the Hoover Institution during the academic
year 2006/07. Kryukov thanks the General Motors Center for Strategy in Management at Northwestern’s
Kellogg School of Management for support during this project. Satterthwaite acknowledges gratefully that
this material is based upon work supported by the National Science Foundation under Grant No. 0121541.
†
Kellogg School of Management,
Northwestern University,
Evanston,
IL 60208,
dbesanko@kellogg.northwestern.edu.
‡
Department of Economics, Harvard University, Cambridge, MA 02138, doraszelski@harvard.edu.
§
Department of Economics, Northwestern University, Evanston, IL 60208, kryukov@northwestern.edu.
¶
Kellogg School of Management,
Northwestern University,
Evanston,
IL 60208,
msatterthwaite@kellogg.northwestern.edu.
1
Introduction
Empirical studies provide ample evidence that the marginal cost of production decreases
with cumulative experience in a variety of industrial settings. This fall in marginal cost is
known as learning-by-doing. More recent empirical studies also suggest that organizations
can forget the know-how gained through learning-by-doing due to labor turnover, periods
of inactivity, and failure to institutionalize tacit knowledge.1 Organizational forgetting has
been largely ignored by the theoretical literature. This is problematic because Benkard
(2004) shows that organizational forgetting is essential to explain the dynamics in the
market for wide-bodied airframes in the 1970s and 1980s.
In this paper we build on the computational Markov-perfect equilibrium framework of
Ericson & Pakes (1995) to analyze how the economic fundamentals of learning-by-doing
and organizational forgetting interact to determine industry structure and dynamics.2 We
account for organizational forgetting in Cabral & Riordan’s (1994) (C-R) seminal model of
learning-by-doing.3 This seemingly small change has surprisingly large effects. Dynamic
competition with learning and forgetting is akin to racing down an upward moving escalator.
As long as a firm makes sales sufficiently frequently so that the gain in know-how from
learning outstrips the loss in know-how from forgetting, it moves down its learning curve
and its marginal cost decreases. However, if sales slow down or come to a halt, perhaps
because of its competitors’ aggressive pricing, then the firm slides back up its learning
curve and its marginal cost increases. This cannot happen in the C-R model. Due to this
qualitative difference, organizational forgetting leads to a rich array of pricing behaviors
and industry dynamics that the existing literature neither imagined nor explained.
It is often said that learning-by-doing promotes market dominance because it gives a
more experienced firm the ability to profitably underprice its less experienced rival and
therefore shut out the competition in the long run. As Dasgupta & Stiglitz (1988) explain:
. . . firm-specific learning encourages the growth of industrial concentration.
To be specific, one expects that strong learning possibilities, coupled with vigorous competition among rivals, ensures that history matters . . . in the sense that
1
See Wright (1936)HIRS:52, DeJong (1957), Alchian (1963), Levy (1965), Kilbridge (1962), Hirschmann
(1964), Preston & Keachie (1964), Baloff (1971), Dudley (1972), Zimmerman (1982), Lieberman (1984),
Gruber (1992), Irwin & Klenow (1994), Jarmin (1994), Pisano (1994), Bohn (1995), Hatch & Mowery
(1998), Thompson (2001), and Thornton & Thompson (2001) for empirical studies of learning-by-doing and
Argote, Beckman & Epple (1990), Darr, Argote & Epple (1995), Benkard (2000), Shafer, Nembhard &
Uzumeri (2001), and Thompson (2003) for organizational forgetting.
2
Dynamic stochastic games and feedback strategies that map states into actions date back at least to
Shapley (1953). Maskin & Tirole (2001) provide the fundamental theory showing how many subgame perfect
equilibria of these games can be represented consistently and robustly as Markov perfect equilibria.
3
Prior to the infinite-horizon price-setting model of C-R, the literature had studied learning-by-doing
using finite-horizon quantity-setting models (Spence 1981, Fudenberg & Tirole 1983, Ghemawat & Spence
1985, Ross 1986, Dasgupta & Stiglitz 1988, Cabral & Riordan 1997).
2
if a given firm enjoys some initial advantages over its rivals it can, by undercutting them, capitalize on these advantages in such a way that the advantages
accumulate over time, rendering rivals incapable of offering effective competition
in the long run . . . (p. 247)
However, if organizational forgetting “undoes” learning-by-doing, then forgetting may be a
procompetitive force and an antidote to market dominance through learning. Two reasons
for suspecting this come to mind. First, to the extent that the leader has more to forget
than the follower, forgetting should work to equalize differences between firms. Second,
because forgetting makes improvements in competitive position from learning transitory, it
should make firms reluctant to invest in the acquisition of know-how through price cuts. We
reach the opposite conclusion: Organizational forgetting can make firms more aggressive
rather than less aggressive. This aggressive pricing behavior, in turn, puts the industry on
a path towards market dominance.
In the absence of organizational forgetting, the price that a firm sets reflects two goals.
First, by winning a sale, the firm moves down its learning curve. This is the advantagebuilding motive. Second, the firm prevents its rival from moving down its learning curve.
This is the advantage-defending motive. But in the presence of organizational forgetting
bidirectional movements through the state space are possible, and this opens up new strategic possibilities for building and defending advantage. By winning a sale, a firm makes itself
less vulnerable to forgetting by creating a “buffer stock” of know-how. By winning and denying its rival a buffer stock, the firm also makes its rival more vulnerable to forgetting. Because organizational forgetting reinforces the advantage-building and advantage-defending
motives in this way, it can create strong incentives to cut prices in order to win the sale.
Organizational forgetting is thus a source of aggressive pricing behavior.
While the existing literature has mainly focused on the dominance properties of firms’
pricing behaviors, we find that these properties are neither necessary nor sufficient for market dominance in our more general setting. We therefore go beyond the existing literature
and directly examine the industry dynamics that firms’ pricing behaviors imply. We find
that organizational forgetting is a source of—not an antidote to—market dominance. If
forgetting is sufficiently weak, then asymmetries may arise but cannot persist. As in the
C-R model, learning-by-doing operates as a ratchet: firms inexorably—if at different rates—
move towards the bottom of their learning curves where cost parity obtains. If forgetting is
sufficiently strong, then asymmetries cannot arise in the first place because forgetting stifles
investment in learning altogether. But, for intermediate degrees of forgetting, asymmetries
arise and persist. Even extreme asymmetries akin to near-monopoly are possible. This is
because, in the presence of organizational forgetting, the leader can use price cuts to delay
or even stall the follower in moving down its learning curve.
3
Organizational forgetting is also a source of multiple equilibria. If the inflow of know-how
into the industry due to learning is substantially smaller than the outflow of know-how due to
forgetting, then it is virtually impossible that both firms reach the bottom of their learning
curves. Conversely, if the inflow is substantially greater than the outflow, then it is virtually
inevitable that they do reach the bottom. In both cases, the primitives of the model tie
down the equilibrium. This is no longer the case if the inflow roughly balances the outflow.
If both firms believe that they cannot profitably coexist at the bottom of their learning
curves, then both cut their prices in the hope of acquiring a competitive advantage early
on and maintaining it throughout. However, if both firms believe that they can profitably
coexist, then neither cuts its price, thereby ensuring that the anticipated symmetric industry
structure emerges. Consequently, in addition to the degree of forgetting, the equilibrium
by itself is an important determinant of pricing behavior and industry dynamics.
Our finding of multiplicity is important for two reasons. First, to our knowledge, all
applications of Ericson & Pakes’s (1995) framework have found a single equilibrium. Pakes
& McGuire (1994) (P-M) indeed held that “nonuniqueness does not seem to be a problem”
(p. 570). It is therefore striking that we obtain up to nine equilibria for some parameterizations. Second, we show that multiple equilibria in our model arise from firms’ expectations
regarding the value of continued play. Being able to pinpoint the driving force behind multiple equilibria is a first step towards tackling the multiplicity problem that plagues the
estimation of dynamic stochastic games and inhibits the use of counterfactuals in policy
analysis.4
In sum, learning-by-doing and organizational forgetting are distinct economic forces.
Forgetting, in particular, does not simply negate learning. The unique role forgetting plays
comes about because it makes bidirectional movements through the state space possible.
Thus the interaction of learning and forgetting can give rise to aggressive pricing behavior,
long-run market dominance of varying degrees, and multiple equilibria.
We emphasize that organizational forgetting is not unique in leading to long-run market
dominance. As we show in Section 8, a model of learning-by-doing that incorporates shutout elements such as a choke price or entry and exit may also do so, much in the way
Dasgupta & Stiglitz (1988) describe. Following C-R we exclude shut-out elements from our
basic model for two reasons. First, the interaction between learning and forgetting is subtle
and generates an enormous variety of interesting, even surprising, equilibria. Isolating it is
therefore useful theoretically. Second, from an empirical viewpoint, as with Intel and AMD
we may see an apparently stable hierarchy of firms with differing market shares, costs, and
profits. Our model can generate such an outcome, while shut-out model elements favor more
extreme outcomes where either one firm dominates or all firms compete on equal footing.
4
See Ackerberg, Benkard, Berry & Pakes (2007) and Pakes (2008) for a discussion of the issue.
4
We also make two methodological contributions. First, we point out a weakness of the
P-M algorithm, the major tool for computing equilibria in the literature following Ericson
& Pakes (1995). Specifically, we prove that our dynamic stochastic game has equilibria
that the P-M algorithm cannot compute. Roughly speaking, in the presence of multiple
equilibria, “in between” two equilibria that it can compute there is one equilibrium it cannot.
This severely limits its ability to provide a complete picture of the set of solutions to the
model.
Second, we propose a homotopy or path-following algorithm. The algorithm traces
out the equilibrium correspondence by varying the degree of forgetting. This allows us to
compute equilibria that the P-M algorithm cannot compute. We find that the equilibrium
correspondence contains a unique path that starts at the equilibrium of the C-R model.
Whenever this path bends back on itself and then forward again, there are multiple equilibria. In addition, the equilibrium correspondence may contain one or more loops that cause
additional multiplicity. To our knowledge, our paper is the first to describe in detail the
structure of the set of equilibria of a dynamic stochastic game in the tradition of Ericson &
Pakes (1995).
The organization of the remainder of the paper is as follows. Sections 2 and 3 describe
the model specification and our computational strategy. Section 4 provides an overview
of the equilibrium correspondence. Section 5 analyzes industry dynamics and Section 6
characterizes the pricing behavior that drives it. Section 7 describes how organizational
forgetting can lead to multiple equilibria. Section 8 undertakes a number of robustness
checks. Section 9 summarizes and concludes.
Throughout the paper in presenting our findings we distinguish between results, which
are established numerically through a systematic exploration of a subset of the parameter
space, and propositions, which hold true for the entire parameter space. If a proposition
establishes a possibility through an example, then the example is presented adjacent to the
proposition. If the proof of a proposition is deductive, then it is contained in the Appendix.
2
Model
For expositional clarity we focus on the basic model of an industry with two firms and
neither entry nor exit; the Online Appendix outlines the general model. Our basic model
is the C-R model with organizational forgetting added and, to allow for our computational
approach, specific functional forms for demand and cost.
Firms and states. We consider a discrete-time, infinite-horizon dynamic stochastic game
of complete information played by two firms. Firm n ∈ {1, 2} is described by its state
5
en ∈ {1, . . . , M }. A firm’s state indicates its cumulative experience or stock of know-how.
By making a sale, a firm can add to its stock of know-how. Following C-R, we use a period
just long enough for a firm to make a sale.5 As suggested by the empirical studies of Argote
et al. (1990), Darr et al. (1995), Benkard (2000), Shafer et al. (2001), and Thompson (2003)
we account for organizational forgetting. Accordingly, the evolution of firm n’s stock of
know-how is governed by the law of motion
e0n = en + qn − fn ,
where e0n and en is firm n’s stock of know-how in the subsequent and current period,
respectively, the random variable qn ∈ {0, 1} indicates whether firm n makes a sale and
gains a unit of know-how through learning-by-doing, and the random variable fn ∈ {0, 1}
indicates whether firm n loses a unit of know-how through organizational forgetting.
At any point in time, the industry is characterized by a vector of firms’ states e =
(e1 , e2 ) ∈ {1, . . . , M }2 . We refer to e as the state of the industry. We use e[2] to denote the
vector (e2 , e1 ) constructed by interchanging the stocks of know-how of firms 1 and 2.
Learning-by-doing. Firm n’s marginal cost of production c(en ) depends on its stock of
know-how en through a learning curve
(
c(en ) =
κeηn
if
1 ≤ en < m,
κmη
if
m ≤ en ≤ M,
where η = log2 ρ for a progress ratio of ρ ∈ (0, 1]. Marginal cost decreases by 100(1 − ρ)
percent as the stock of know-how doubles, so that a lower progress ratio implies a steeper
learning curve. The marginal cost of production at the top of the learning curve, c(1), is
κ > 0 and, as in C-R, m represents the stock of know-how at which a firm reaches the
bottom of its learning curve.6
Organizational forgetting. We let ∆(en ) = Pr(fn = 1) denote the probability that
firm n loses a unit of know-how through organizational forgetting. We assume that this
probability is nondecreasing in the firm’s experience level. This has several advantages.
First, experimental evidence in the management literature suggests that forgetting by individuals is an increasing function of the current stock of learned knowledge (Bailey 1989).
5
A sale may involve a single unit or a batch of units (e.g., 100 aircraft or 10,000 memory chips) that are
sold to a single buyer.
6
While C-R take the state space to be infinite, i.e., M = ∞ in our notation, they make the additional
assumption that the price that a firm charges does not depend on how far it is beyond the bottom of its
learning curve (p. 1119). This is tantamount to assuming, as we do, that the state space is finite.
6
Second, a direct implication of ∆ (·) increasing is that the expected stock of know-how in
the absence of further learning is a decreasing convex function of time.7 This phenomenon,
known in the psychology literature as Jost’s second law, is consistent with experimental
evidence on forgetting by individuals (Wixted & Ebbesen 1991). Third, in the capital-stock
model employed in empirical work on organizational forgetting, the amount of depreciation
is assumed to be proportional to the stock of know-how. Hence, the additional know-how
needed to counteract depreciation must increase with the stock of know-how. Our specification has this feature. However, unlike the capital-stock model, it is consistent with a
discrete state space.8
The specific functional form we employ is
∆(en ) = 1 − (1 − δ)en ,
where δ ∈ [0, 1] is the forgetting rate.9 If δ > 0, then ∆(en ) is increasing and concave
in en ; δ = 0 corresponds to the absence of organizational forgetting, the special case C-R
analyzed. Other functional forms are plausible, and we explore one of them in the Online
Appendix.
Demand. The industry draws its customers from a large pool of potential buyers. In
each period, one buyer enters the market and purchases the good from one of the two firms.
The utility that the buyer obtains by purchasing good n is v − pn + εn where pn is the price
of good n, v is a deterministic component of utility, and εn is a stochastic component that
captures the idiosyncratic preference for good n of this period’s buyer. Both ε1 and ε2 are
unobservable to firms and are independently and identically type 1 extreme value distributed
with location parameter 0 and scale parameter σ > 0. The scale parameter governs the
degree of horizontal product differentiation. As σ → 0, goods become homogeneous.
The buyer purchases the good that gives it the highest utility. Given our distributional
assumptions the probability that firm n makes the sale is given by the logit specification
n
exp( v−p
σ )
Dn (p) = Pr(qn = 1) = P2
v−pk
k=1 exp( σ )
=
1
,
−n
1 + exp( pn −p
)
σ
where p = (p1 , p2 ) is the vector of prices and p−n denotes the price the other firm charges.
Demand effectively depends on differences in prices because we assume, as do C-R, that the
7
See the Online Appendix for a proof.
See Benkard (2004) for an alternative approximation to the capital-stock model.
9
One way to motivate this functional form is to imagine that the stock of know-how is dispersed among
a firm’s workforce. In particular, assume that en is the number of skilled workers and that organizational
forgetting is the result of labor turnover. Then, given a turnover rate of δ, ∆(en ) is the probability that at
least one of the en skilled workers leaves the firm.
8
7
buyer always purchases from one of the two firms in the industry. In Section 8 we discuss
the effects of including an outside good in the specification.
State-to-state transitions. From one period to the next, a firm’s stock of know-how
moves up or down or remains constant depending on realized demand qn ∈ {0, 1} and
organizational forgetting fn ∈ {0, 1}. The transition probabilities are
(
Pr(e0n |en , qn )
=
1 − ∆(en )
if
e0n = en + qn ,
∆(en )
if
e0n = en + qn − 1,
where, at the upper and lower boundaries of the state space, we modify the transition
probabilities to be Pr(M |M, 1) = 1 and Pr(1|1, 0) = 1, respectively. Note that a firm can
increase its stock of know-how only if it makes a sale in the current period, an event that
has probability Dn (e); otherwise it runs the risk that its stock of know-how decreases.
Bellman equation. We define Vn (e) to be the expected net present value of firm n’s cash
flows if the industry is currently in state e. The value function Vn : {1, . . . , M }2 → [−V̂ , V̂ ],
where V̂ is a sufficiently large constant, is implicitly defined by the Bellman equation
Vn (e) = max Dn (pn , p−n (e))(pn − c(en )) + β
pn
2
X
Dk (pn , p−n (e))V nk (e),
(1)
k=1
where p−n (e) is the price charged by the other firm in state e, β ∈ (0, 1) is the discount
factor, and V nk (e) is the expectation of firm n’s value function conditional on the buyer
purchasing the good from firm k ∈ {1, 2} in state e as given by
V n1 (e) =
V n2 (e) =
eX
1 +1
e2
X
e01 =e1
e02 =e2 −1
e1
X
eX
2 +1
Vn (e0 ) Pr(e01 |e1 , 1) Pr(e02 |e2 , 0),
(2)
Vn (e0 ) Pr(e01 |e1 , 0) Pr(e02 |e2 , 1).
(3)
e01 =e1 −1 e02 =e2
The policy function pn : {1, . . . , M }2 → [−p̂, p̂], where p̂ is a sufficiently large constant,
specifies the price pn (e) that firm n sets in state e.10 Let hn (e, pn , p−n (e), Vn ) denote the
maximand in the Bellman equation (1). Differentiating it with respect to pn and using the
properties of logit demand we obtain the first-order condition (FOC):
0=
10
³
´
1
∂hn (·)
= Dn (pn , p−n (e)) σ − (pn − c(en )) − βV nn (e) + hn (·) .
∂pn
σ
In what follows we assume that p̂ is chosen large enough to not constrain pricing behavior.
8
Differentiating hn (·) a second time yields
´ 1
1 ∂hn (·) ³
∂ 2 hn (·)
=
2D
(p
,
p
(e))
−
1
− Dn (pn , p−n (e)).
n n −n
∂p2n
σ ∂pn
σ
If the FOC is satisfied, then
∂ 2 hn (·)
∂p2n
= − σ1 Dn (pn , p−n (e)) < 0. hn (·) is therefore strictly
quasi-concave in pn , so that the pricing decision pn (e) is uniquely determined by the solution
to the FOC (given p−n (e)).
Equilibrium. In our model firms face identical demand and cost primitives. Asymmetries
between firms arise endogenously from the effects of their pricing decisions on realized
demand and organizational forgetting. Hence, we focus attention on symmetric Markov
perfect equilibria (MPE). In a symmetric equilibrium the pricing decision taken by firm
2 in state e is identical to the pricing decision taken by firm 1 in state e[2] , i.e., p2 (e) =
p1 (e[2] ), and similarly for the value function. It therefore suffices to determine the value
and policy functions of firm 1. We define V (e) = V1 (e) and p(e) = p1 (e) for each state e.
Further, we let V k (e) = V 1k (e) denote the conditional expectation of firm 1’s value function
and Dk (e) = Dk (p(e), p(e[2] )) denote the probability that the buyer purchases from firm
k ∈ {1, 2} in state e.
Given this notation, the Bellman equation and FOC can be expressed as
Fe1 (V∗ , p∗ )
∗
= −V (e) +
D1∗ (e) (p∗ (e)
− c(e1 )) + β
2
X
∗
Dk∗ (e)V k (e) = 0,
(4)
k=1
∗
Fe2 (V∗ , p∗ ) = σ − (1 − D1∗ (e)) (p∗ (e) − c(e1 )) − βV 1 (e) + β
2
X
∗
Dk∗ (e)V k (e) = 0,
(5)
k=1
where we use asterisks to denote an equilibrium. The collection of equations (4) and (5) for
all states e ∈ {1, . . . , M }2 can be written more compactly as

1
F(1,1)
(V∗ , p∗ )

 F 1 (V∗ , p∗ )
(2,1)

F(V , p ) = 
..

.

2
F(M,M ) (V∗ , p∗ )
∗
∗




 = 0,


(6)
where 0 is a (2M 2 ×1) vector of zeros. Any solution to this system of 2M 2 equations in 2M 2
unknowns V∗ = (V ∗ (1, 1), V ∗ (2, 1), . . . , V ∗ (M, M )) and p∗ = (p∗ (1, 1), p∗ (2, 1), . . . , p∗ (M, M ))
is a symmetric equilibrium in pure strategies. A slightly modified version of Proposition 2
in Doraszelski & Satterthwaite (2008) establishes that such an equilibrium always exists.
9
Baseline parameterization. Since our focus is on how learning-by-doing and organizational forgetting affect pricing behavior, and the industry dynamics this behavior implies,
we explore the full range of values for the progress ratio ρ and the forgetting rate δ. To do
so, we fix the remaining parameters to their baseline values given below. We specify a grid
of 100 equidistant values of ρ ∈ (0, 1]. For each of them, we use the homotopy algorithm
described in Section 3 to trace the equilibrium as δ ranges from 0 to 1. Typically this
entails solving the model for a few thousand intermediate values of δ. If an important or
interesting property is true for each of these systematically computed equilibria, then we
report it as a result. In Section 8 we then vary the values of the parameters other than ρ
and δ in order to discuss their influence on the equilibrium and demonstrate the robustness
of our conclusions.
While we explore the full range of values for ρ and δ, we note that most empirical estimates of progress ratios are in the range of 0.7 to 0.95 (Dutton & Thomas 1984). However, a
very steep learning curve, with ρ much less than 0.7, may also capture a practically relevant
situation. Suppose the first unit of a product is a hand-built prototype and the second unit
is a guinea pig for organizing the production line. After this point the gains from learningby-doing are more or less exhausted and the marginal cost of production is close to zero.11
Benkard (2000) and Argote et al. (1990) have found monthly rates of depreciation ranging
from 4 to 25 percent of the stock of know-how. In the Online Appendix we show how to
map these estimates that are based on a capital-stock model of organizational forgetting
into in our specification. The implied values of the forgetting rate δ fall below 0.1.
In our baseline parameterization we set M = 30 and m = 15. The marginal cost at the
top of the learning curve κ is equal to 10. For a progress ratio of ρ = 0.85, this implies that
the marginal cost of production declines from a maximum value of c(1) = 10 to a minimum
value of c(15) = . . . = c(30) = 5.30. For ρ = 0.15, we have the case of a hand-built prototype
where the marginal cost of production declines very quickly from c(1) = 10 over c(2) = 1.50
and c(3) = 0.49 to c(15) = . . . = c(30) = 0.01.
Turning to demand, we set σ = 1 in our baseline parameterization. To illustrate, in the
Nash equilibrium of a static price-setting game (obtained by setting β = 0 in our model),
the own-price elasticity of demand ranges between −8.86 in state (1, 15) and −2.13 in state
(15, 1) for a progress ratio of ρ = 0.85. The cross-price elasticity of firm 1’s demand with
respect to firm 2’s price is 2.41 in state (15, 1) and 7.84 in state (1, 15). For ρ = 0.15 the
own-price elasticity ranges between −9.89 and −1.00 and the cross-price elasticity between
1.00 and 8.05. These reasonable elasticities suggest that the results reported below are not
artifacts of extreme parameterizations.
11
To avoid a marginal cost of close to zero, shift the cost function c(en ) by τ > 0. While introducing a
component of marginal cost that is unresponsive to learning-by-doing shifts the policy function by τ , the
value function and the industry dynamics remain unchanged.
10
We finally set the discount factor to β =
1
1.05 .
It may be thought of as β =
ζ
1+r ,
where
r > 0 is the per-period discount rate and ζ ∈ (0, 1] is the exogenous probability that the
industry survives from one period to the next. Consequently, our baseline parameterization
corresponds to a variety of scenarios that differ in the length of a period. For example,
it corresponds to a period length of one year, a yearly discount rate of 5 percent, and
certain survival. Perhaps more interestingly, it also corresponds to a period length of one
month, a monthly discount rate of 1 percent (which translates into a 12.68 percent annual
discount rate), and a monthly survival probability of 0.96. To put this—our focal scenario—
in perspective, technology companies such as IBM and Microsoft had costs of capital in the
range of 11 to 15 percent per annum in the late 1990s. Further, an industry with a monthly
survival probability of 0.96 has an expected lifetime of 26.25 months. Thus this scenario is
consistent with a pace of innovative activity that is expected to make the current generation
of products obsolete within two to three years.
3
Computation
In this section we first describe a novel algorithm for computing equilibria of dynamic
stochastic games that is based on the homotopy method.12 Then, we turn to the P-M
algorithm that is the standard means for computing equilibria in the literature following
Ericson & Pakes (1995). We show that it is inadequate for characterizing the set of solutions
to our model although it remains useful for obtaining a starting point for the homotopy
algorithm. A reader who is more interested in the economic implications of learning and
forgetting may skip ahead to Section 4 after reading the first part of this section that
introduces the homotopy algorithm by way of an example.
3.1
Homotopy algorithm
Our goal is to explore the graph of the equilibrium correspondence as the forgetting rate δ
and the progress ratio ρ vary:
F−1 = {(V∗ , p∗ , δ, ρ)|F(V∗ , p∗ ; δ, ρ) = 0, δ ∈ [0, 1], ρ ∈ (0, 1]} ,
(7)
where F(·) is the system of equations (6) that defines an equilibrium and we make explicit
that it depends on δ and ρ (recall that we hold fixed the remaining parameters). The graph
F−1 is a surface, or set of surfaces, that may have folds. Our homotopy algorithm explores
12
See Schmedders (1998, 1999) for an application of the homotopy method to general equilibrium models
with incomplete asset markets and Berry & Pakes (2007) for an application to estimating demand systems.
11
this graph by taking slices of it for given values of ρ:
F−1 (ρ) = {(V∗ , p∗ , δ)|F(V∗ , p∗ ; δ, ρ) = 0, δ ∈ [0, 1]} .
(8)
The homotopy algorithm starts from a single equilibrium that has already been computed
and traces out an entire path of equilibria in F−1 (ρ) by varying δ. The homotopy algorithm
is therefore also called a path-following algorithm and δ the homotopy parameter.
Example. An example helps explain how the homotopy algorithm works. Consider the
equation F (x; δ) = 0, where
F (x; δ) = −15.289 −
δ
+ 67.500x − 96.923x2 + 46.154x3 .
1 + δ4
(9)
Equation (9) implicitly relates an endogenous variable x to an exogenous parameter δ.
Figure 1 graphs the set of solutions F −1 = {(x, δ)|F (x; δ) = 0, δ ∈ [0, 1]}. There are multiple
solutions at δ = 0.3, namely x = 0.610, x = 0.707, and x = 0.783. Finding them is trivial
with the graph in hand, but even for this simple case the graph is less than straightforward
to draw. Whether one solves F (x; δ) = 0 for x taking δ as given or for δ taking x as given,
the result is a correspondence, not a function.
The homotopy method introduces an auxiliary variable s that indexes each point on the
graph, starting at point A for s = 0 and ending at point D for s = s̄. The graph is then just
the parametric path given by a pair of functions (x(s), δ(s)) satisfying F (x(s); δ(s)) = 0 or,
equivalently, (x(s), δ(s)) ∈ F −1 . While there are infinitely many such pairs, a simple way
to select a member of this family is to differentiate F (x(s); δ(s)) = 0 with respect to s:
∂F (x(s); δ(s)) 0
∂F (x(s); δ(s)) 0
x (s) +
δ (s) = 0.
∂x
∂δ
(10)
This differential equation in two unknowns x0 (s) and δ 0 (s) must be satisfied in order to
remain “on path.” One possible approach for tracing out the path in F −1 is to solve equation
(10) for the ratio
x0 (s)
δ 0 (s)
∂F (x(s);δ(s))/∂δ
= − ∂F
(x(s);δ(s))/∂x that indicates the direction of the next step along
the path from s to s + ds. This approach, however, fails because the ratio switches from
+∞ to −∞ at points such as B in Figure 1. So instead of solving for the ratio, we simply
solve for each term of the ratio. This insight implies that the graph F −1 in Figure 1 is the
solution to the following system of differential equations:
x0 (s) =
∂F (x(s); δ(s))
,
∂δ
δ 0 (s) = −
∂F (x(s); δ(s))
.
∂x
(11)
These are the so-called basic differential equations for our example. They reduce the task of
12
tracing out the set of solutions to solving a system of differential equations. Given an initial
condition this can be done with a variety of methods (see Chapter 10 of Judd 1998). If δ = 0,
then F (x; δ) = 0 is easily solved for x = 0.5, thereby providing an initial condition (point
A in Figure 1). From there the homotopy algorithm uses the basic differential equations
to determine the next step along the path. It continues to follow—step-by-step—the path
until it reaches δ = 1 (point D). In our example, the auxiliary variable s is decreasing from
point A to point D. Therefore, whenever δ 0 (s) switches sign from negative to positive (point
B), the path is bending backward and there are multiple solutions. Conversely, whenever
the sign of δ 0 (s) switches back from positive to negative (point C), the path is bending
forward.
Returning to our model of learning and forgetting, let x = (V∗ , p∗ ) denote the 2M 2
endogenous variables. Our goal is to explore F−1 (ρ), a slice of the graph of the equilibrium
correspondence. Proceeding as in our example, a parametric path is a set of functions
(x(s), δ(s)) ∈ F−1 (ρ). Differentiating F(x(s); δ(s), ρ) = 0 with respect to s yields the
necessary conditions for remaining on path:
∂F(x(s); δ(s), ρ) 0
∂F(x(s); δ(s), ρ) 0
x (s) +
δ (s) = 0,
∂x
∂δ
where
∂F(x(s);δ(s),ρ)
∂x
0
is the (2M 2 × 2M 2 ) Jacobian, x0 (s) and
∂F(x(s);δ(s),ρ)
∂δ
(12)
are (2M 2 × 1)
vectors, and δ (s) is a scalar. This system of 2M 2 differential equations in the 2M 2 + 1
unknowns x0i (s), i = 1, . . . , 2M 2 , and δ 0 (s) has a solution that obeys the basic differential
equations
µµ
yi0 (s)
= (−1)
i+1
det
∂F(y(s); ρ)
∂y
¶ ¶
,
i = 1, . . . , 2M 2 + 1,
(13)
−i
where y(s) = (x(s), δ(s)) and the notation (·)−i is used to indicate that the ith column is
removed from the (2M 2 × 2M 2 + 1) Jacobian
∂F(y(s);ρ)
.
∂y
Note that equation (13) reduces to
equation (11) if x is a scalar instead of a vector. Garcia & Zangwill (1979) and Chapter 2 of
Zangwill & Garcia (1981) prove that the basic differential equations satisfy the conditions
in equation (12).
The homotopy method requires that F(y; ρ) is continuously differentiable with respect
to y and that the Jacobian
∂F(y;ρ)
∂y
has full rank at all points in F−1 (ρ). To appreciate the
∂F(y(s);ρ)
∂y
(2M 2 × 2M 2 )
importance of regularity, note that if the Jacobian
has less than full rank at some
point y(s), then the determinants of all its
submatrices are zero. Hence,
according to the basic differential equations (13),
yi0 (s)
= 0 for i = 1, . . . , 2M 2 + 1, and the
algorithm stalls. On the other hand, with regularity in place, the implicit function theorem
ensures that F−1 (ρ) consists only of continuous paths; paths that suddenly terminate,
13
endless spirals, branch points, isolated equilibria, and continua of equilibria are ruled out
(see Chapter 1 of Zangwill & Garcia 1981).
While our assumed functional forms ensure continuous differentiability, we have been
unable to establish regularity analytically. Indeed, we have numerical evidence suggesting
that regularity can fail. In practice, failures of regularity are not a problem as long as they
are confined to isolated points. Because our algorithm computes just a finite number of
points along the path, it is extremely unlikely to hit an irregular point.13 We refer the
reader to Borkovsky, Doraszelski & Kryukov (2008) for a fuller discussion of this issue and
a step-by-step guide to solving dynamic stochastic games using the homotopy method.
As Result 1 in Section 4 shows, we have always been able to trace out a path in F−1 (ρ)
that starts at the equilibrium for δ = 0 and ends at the equilibrium for δ = 1. Whenever this
“main path” folds back on itself, the homotopy algorithm automatically identifies multiple
equilibria. This makes it well-suited for models like ours that have multiple equilibria.
Nevertheless, the homotopy algorithm cannot be guaranteed to find all equilibria.14
The slice F−1 (ρ) may contain additional equilibria that are off the main path. These
equilibria form one or more loops (see Result 1 in Section 4). We have two intuitively
appealing but potentially fallible ways to try and identify additional equilibria. First, we use
a large number of restarts of the P-M algorithm, often trying to “propagate” equilibria from
“nearby” parameterizations. Second, and more systematically, just as we can choose δ as
the homotopy parameter while keeping ρ fixed, we can choose ρ while keeping δ fixed. This
allows us to “crisscross” the parameter space in an orderly fashion by using the equilibria on
the δ-slices as initial conditions to generate ρ-slices. A ρ-slice must either intersect with all
δ-slices or lead us to an additional equilibrium that, in turn, gives us an initial condition to
generate an additional δ-slice. We continue this process until all the ρ- and δ-slices “match
up”(for details see Grieco 2008).
3.2
Pakes & McGuire (1994) algorithm
While the homotopy method has advantages in finding multiple equilibria, it cannot stand
alone. The P-M algorithm (or some other means for solving a system of nonlinear equations)
is necessary to compute a starting point for our homotopy algorithm.
Recall that V2 (e) = V1 (e[2] ) and p2 (e) = p1 (e[2] ) for each state e in a symmetric equilibrium and it therefore suffices to determine V and p, the value and policy functions of
firm 1. The P-M algorithm is iterative. An iteration cycles through the states in some
predetermined order and updates V and p as it progresses from one iteration to the next.
13
Our programs use Hompack (Watson, Billups & Morgan 1987, Watson, Sosonkina, Melville, Morgan &
Walker 1997) written in Fortran 90. They are available from the authors upon request.
14
Unless the system of equations that defines them happens to be polynomial; see Judd & Schmedders
(2004) for some early efforts along this line.
14
The strategic situation firms face in setting prices in state e is similar to a static game if
the value of continued play is taken as given. The P-M algorithm computes the best reply
of firm 1 against p(e[2] ) in this game and uses it to update the value and policy functions
of firm 1 in state e.
More formally, let h1 (e, p1 , p(e[2] ), V) be the maximand in the Bellman equation (1)
after symmetry is imposed. The best reply of firm 1 against p(e[2] ) in state e is given by
G2e (V, p) = arg max h1 (e, p1 , p(e[2] ), V)
p1
(14)
and the value associated with it is
G1e (V, p) = max h1 (e, p1 , p(e[2] ), V).
p1
(15)
Write the collection of equations (14) and (15) for all states e ∈ {1, . . . , M }2 as

G1(1,1) (V, p)

 G1 (V, p)
(2,1)

G(V, p) = 
..

.

2
G(M,M ) (V, p)




.


(16)
Given an initial guess x0 = (V0 , p0 ), the P-M algorithm executes the iteration
xk+1 = G(xk ),
k = 0, 1, 2, . . . ,
until the changes in the value and policy functions of firm 1 are deemed small (or a failure
to converge is diagnosed).
The P-M algorithm does not lend itself to computing multiple equilibria. To identify
more than one equilibrium (for a given parameterization of the model), it must be restarted
from different initial guesses. But different initial guesses may or may not lead to different
equilibria. This, however, still understates the severity of the problem. Whenever our
dynamic stochastic game has multiple equilibria, the P-M algorithm cannot compute a
substantial fraction of them even if an arbitrarily large number of initial guesses are tried.
The problem is this. The P-M algorithm continues to iterate until it reaches a fixed
point x = G (x). A necessary condition for convergence is local stability
³ of ´the fixed point.
∂G(x)
2
2
Consider the (2M ×2M ) Jacobian ∂x at the fixed point and let % ∂G(x)
be its spectral
∂x
³
´
radius.15 The fixed point is locally stable under the P-M algorithm if % ∂G(x)
< 1, i.e., if
∂x
15
Let A be an arbitrary matrix and ς(A) the set of its eigenvalues. The spectral radius of A is %(A) =
max {|λ| : λ ∈ ς(A)}.
15
all eigenvalues are within the complex unit circle. Given local stability, the P-M algorithm
converges provided
³ that´ the initial guess is close (perhaps very close) to the fixed point.
Conversely, if % ∂G(x)
≥ 1, then the fixed point is unstable and the P-M algorithm
∂x
cannot compute it. The following proposition identifies a subset of equilibria that the P-M
algorithm cannot compute.
0
−1
Proposition
1 Let (x(s),
¶ δ(s)) ∈ F (ρ) be a parametric path of equilibria. (i) If δ (s) ≤ 0,
µ
¯
¯
then % ∂G(x(s))
≥ 1 and the equilibrium x(s) is unstable under the P-M algorithm.
¯
∂x
(δ(s),ρ)
(ii) Moreover, the equilibrium x(s) remains unstable with either dampening or extrapolation
applied to the P-M algorithm.
Part (i) of Proposition 1 establishes that the P-M algorithm cannot compute equilibria
on any part of the path for which δ 0 (s) ≤ 0. Whenever δ 0 (s) switches sign from positive
to negative, the main path connecting the equilibrium at δ = 0 with the equilibrium at
δ = 1 bends backward and multiple equilibria arise. Conversely, whenever the sign of δ 0 (s)
switches back from negative to positive, the main path bends forward. Hence, for a fixed
forgetting rate δ (s), in between two equilibria for which δ 0 (s) > 0 lies a third—necessarily
unstable—equilibrium for which δ 0 (s) ≤ 0. Similarly, a loop has equilibria for which δ 0 (s) > 0
and equilibria for which δ 0 (s) ≤ 0. Consequently, if we have multiple equilibria (for a given
parameterization of the model), then the P-M algorithm can compute at best 1/2 to 2/3 of
them.
Dampening and extrapolation are often applied to the P-M algorithm in the hope of
improving its likelihood or speed of convergence. The iteration
xk+1 = ωG(xk ) + (1 − ω)xk ,
k = 0, 1, 2, . . . ,
is said to be dampened if ω ∈ (0, 1) and extrapolated if ω ∈ (1, ∞). Part (ii) of Proposition
1 establishes the futility of these attempts.16
The ability of the P-M algorithm to provide a reasonably complete picture of the set of
solutions to the model is limited beyond the scope of Proposition 1. Our numerical analysis
indicates that the P-M algorithm cannot compute equilibria on some part of the path for
which δ 0 (s) > 0:
Proposition 2 Let
∈¶
F−1 (ρ) be a parametric path of equilibria. Even if δ 0 (s) >
µ (x(s), δ(s))
¯
¯
≥ 1 so that the equilibrium x(s) is unstable under the
0 we may have % ∂G(x(s))
¯
∂x
(δ(s),ρ)
P-M algorithm.
16
Dampening and extrapolation may, of course, still be helpful in computing equilibria for which δ 0 (s) > 0.
16
In the Online Appendix we prove Proposition 2 by way of an example and illustrate equilibria of our model that the P-M algorithm cannot compute.
As is well-known, not all Nash equilibria of static games are stable under best reply dynamics (see Chapter 1 of Fudenberg & Tirole 1991).17 Since the P-M algorithm incorporates
best reply dynamics, it is reasonable to expect that this limits its usefulness. In the Online
Appendix we argue that this is not the case. More precisely, we show that, holding fixed
the value of continued play, the best reply dynamics are contractive and therefore converge
to a unique fixed point irrespective of the initial guess. The value function iteration also
is contractive holding fixed the policy function. Hence, each of the two building blocks of
the P-M algorithm “works.” What makes it impossible to obtain a substantial fraction of
equilibria is the interaction of value function iteration with best reply dynamics.
The P-M algorithm is a pre-Gauss-Jacobi method. The subsequent literature has instead sometimes used a pre-Gauss-Seidel method (Benkard 2004, Doraszelski & Judd 2004).
Whereas a Gauss-Jacobi method replaces the old guesses for the value and policy functions
with the new guesses at the end of an iteration after all states have been visited, a GaussSeidel method updates after each state. This has the advantage that “information” is
used as soon as it becomes available (see Chapters 3 and 5 of Judd 1998). We have been
unable to prove that Proposition 1 carries over to this alternative algorithm. We note,
however, that the Stein-Rosenberg theorem (see Proposition 6.9 in Section 2.6 of Bertsekas
& Tsitsiklis 1997) asserts that for certain systems of linear equations, if the Gauss-Jacobi
algorithm fails to converge, then so does the Gauss-Seidel algorithm. Hence, it does not
seem reasonable to presume that the Gauss-Seidel variant of the P-M algorithm is immune
to the difficulties the original algorithm suffers.
4
Equilibrium correspondence
This section provides an overview of the equilibrium correspondence. In the absence of
organizational forgetting, C-R establish uniqueness of equilibrium. Theorem 2.2 in C-R
extends to our model:
Proposition 3 If organizational forgetting is either absent (δ = 0) or certain (δ = 1), then
there is a unique equilibrium.18
The cases of δ = 0 and δ = 1 are special in that they ensure that movements through the
state space are unidirectional. When δ = 0, a firm can never move “backward” to a lower
17
More generally, in static games, Nash equilibria of degree −1 are unstable under any Nash dynamics,
i.e., dynamics with rest points that coincide with Nash equilibria, including replicator and smooth fictitious
play dynamics (Demichelis & Germano 2002).
18
Proposition 3 pertains to both symmetric and asymmetric equilibria.
17
state and when δ = 1, it can never move “forward” to a higher state. Hence, backward
induction can be used to establish uniqueness of equilibrium (see Section 7 for details). In
contrast, when δ ∈ (0, 1), a firm can move in either direction. These bidirectional movements
break the backward induction and make multiple equilibria possible:
Proposition 4 If organizational forgetting is neither absent (δ = 0) nor certain (δ = 1),
then there may be multiple equilibria.
Figure 2 proves the proposition and illustrates the extent of multiplicity. It shows the
number of equilibria that we have identified for each combination of progress ratio ρ and
forgetting rate δ. Darker shades indicate more equilibria. As can be seen, we have found
up to nine equilibria for some values of ρ and δ. Multiplicity is especially pervasive for
forgetting rates δ in the empirically relevant range below 0.1.
In dynamic stochastic games with finite actions, Herings & Peeters (2004) have shown
that generically the number of MPE is odd. While they consider both symmetric and asymmetric equilibria, in a two-player game with symmetric primitives such as ours, asymmetric
equilibria occur in pairs. Hence, their result immediately implies that generically the number of symmetric equilibria is odd in games with finite actions. Figure 2 suggests that this
carries over to our setting with continuous actions.
In order to understand the geometry of how multiple equilibria arise we take a close look
at the slices of the graph of the equilibrium correspondence that our homotopy algorithm
computes.
Result 1 The slice F−1 (ρ) contains a unique path that connects the equilibrium at δ = 0
with the equilibrium at δ = 1. In addition, F−1 (ρ) may contain (one or more) loops that
are disjoint from this “main path” and from each other.
Figure 3 illustrates Result 1. To explain this figure, recall that an equilibrium consists
of a value function V∗ and a policy function p∗ and is thus an element of a high-dimensional
space. To succinctly represent it, we proceed in two steps. First, we use p∗ to construct
the probability distribution over next period’s state e0 given this period’s state e, i.e., the
transition matrix that characterizes the Markov process of industry dynamics. We compute
the transient distribution over states in period t, µt (·), starting from state (1, 1) in period 0.
This tells us how likely each possible industry structure is in period t, given that both firms
began at the top of their learning curves. In addition, we compute the limiting (or ergodic)
distribution over states, µ∞ (·).19 The transient distributions capture short-run dynamics
and the limiting distribution captures long-run (or steady-state) dynamics.
19
Let P be the M 2 × M 2 transition matrix. The transient distribution in period t is given by µt = µ0 Pt ,
where µ0 is the 1 × M 2 initial distribution and Pt the tth matrix power of P. If δ ∈ (0, 1), then the Markov
process is irreducible because logit demand implies that the probability moving forward is always nonzero.
18
Second, we use the transient distribution over states in period t, µt (·), to compute the
expected Herfindahl index
Ht =
X¡
¢
D1∗ (e)2 + D2∗ (e)2 µt (e).
e
The time path of H t summarizes the implications of learning and forgetting for industry
dynamics. If the industry evolves asymmetrically, then H t > 0.5. The maximum expected
Herfindahl index
H∧ =
max
t∈{1,...,100}
Ht
is a summary measure of short-run industry concentration. The limiting expected Herfindahl index H ∞ , computed using µ∞ (·) instead of µt (·), is a summary measure of long-run
industry concentration. If H ∞ > 0.5, then an asymmetric industry structure persists.
In Figure 3 we visualize F−1 (ρ) for a variety of progress ratios by plotting H ∧ (dashed
line) and H ∞ (solid line). As can be seen, multiple equilibria arise whenever the main path
folds back on itself. Moreover, there is one loop for ρ ∈ {0.75, 0.65, 0.55, 0.15, 0.05}, two
loops for ρ ∈ {0.85, 0.35}, and three loops for ρ = 0.95, thus adding further multiplicity.
Figure 3 is not necessarily a complete picture of the equilibria to our model. As discussed
in Section 3.1, no algorithm is guaranteed to find all equilibria. We do find all equilibria
along the main path and we have been successful in finding a number of loops. But other
loops may exist because, in order to trace out a loop, we must somehow compute at least
one equilibrium on the loop, and doing so is problematic.
Types of equilibria. Despite the multiplicity, the equilibria of our game exhibit the
four typical patterns shown in Figure 4.20 The parameter values are ρ = 0.85 and δ ∈
{0, 0.0275, 0.08}; they represent the median progress ratio across a wide array of empirical
studies combined with the cases of no, low, and high organizational forgetting. One should
recognize that the typical patterns, helpful as they are in understanding the range of behaviors that can occur, lie on a continuum and thus morph into each other in complicated
ways as we change the parameter values.
The upper left panel of Figure 4 is typical for what we call a flat equilibrium without
well (ρ = 0.85, δ = 0). The policy function is very even over the entire state space. In
particular, the price that a firm charges in equilibrium is fairly insensitive to its rival’s
stock of know-how. The upper right panel shows a flat equilibrium with well (ρ = 0.85,
That is, all its states belong to a single closed communicating class and the 1 × M 2 limiting distribution
µ∞ solves the system of linear equations µ∞ = µ∞ P. If δ = 0 (δ = 1), then there is also a single closed
communicating class, but its sole member is state (M, M ) ((1, 1)).
20
The value functions corresponding to the policy functions in Figure 4 can be found in the Online
Appendix where we also provide tables of the value and policy functions for ease of reference.
19
δ = 0.0275). While the policy function remains even over most of the state space, price
competition is intense during the industry’s birth. This manifests itself as a “well” in the
neighborhood of state (1, 1).
The lower left panel of Figure 4 exemplifies a trenchy equilibrium (ρ = 0.85, δ = 0.0275).
The parameter values are the same as for the flat equilibrium with well, thereby providing
an instance of multiplicity.21 The policy function is uneven and exhibits a “trench” along
the diagonal of the state space. This trench starts in state (1, 1) and extends beyond the
bottom of the learning curve in state (m, m) all the way to state (M, M ). Hence, in a
trenchy equilibrium, price competition between firms with similar stocks of know-how is
intense but abates once firms become asymmetric. Finally, the lower right panel illustrates
an extra-trenchy equilibrium (ρ = 0.85, δ = 0.08). The policy function has not only a
diagonal trench but also trenches parallel to the edges of the state space. In these sideways
trenches the leader competes aggressively with the follower.
Sunspots. For a progress ratio of ρ = 1 the marginal cost of production is constant at
c(1) = . . . = c(M ) = κ, and there are no gains from learning-by-doing. It clearly is an
equilibrium for firms to disregard their stocks of know-how and set the same prices as in the
Nash equilibrium of a static price-setting game (obtained by setting β = 0). Since firms’
marginal costs are constant, so are the static Nash equilibrium prices. Thus, we have an
extreme example of a flat equilibrium with p∗ (e) = κ + 2σ = 12 and V ∗ (e) =
all states e ∈
σ
1−β
= 21 for
{1, . . . , M }2 .
Figure 2 shows that, in case of ρ = 1, there are two more equilibria for a range of
forgetting rates δ below 0.1. Since the state of the industry has no bearing on the primitives,
we refer to these equilibria as sunspots. One of the sunspots is a trenchy equilibrium while
the other one is, depending on δ, either a flat or a trenchy equilibrium. In the trenchy
equilibrium the industry evolves towards an asymmetric structure where the leader charges a
lower price than the follower and enjoys a higher probability of making a sale. Consequently,
the net present value of cash flows to the leader exceeds that to the follower. The value
in state (1, 1), however, is lower than in the static Nash equilibrium, i.e., V ∗ (1, 1) < 21.22
This indicates that value is destroyed as firms fight for dominance.
The existence of sunspots and the fact that these equilibria persist for ρ ≈ 1 suggests
that the concept of MPE is richer than one may have thought. Besides describing the
21
As can be seen in the upper right panel of Figure 3, the main path in F−1 (0.85) bends back on itself at
δ = 0.0275, and there are three equilibria for slightly lower values of δ and only one for slightly higher values.
This particular parameterization (if not the pattern of behavior it generates) is therefore almost nongeneric
in that it approximates the isolated occurrence of an even number of equilibria. Due to the limited precision
of our homotopy algorithm, we have indeed been unable to find a third equilibrium.
22
For example, if δ = 0.0275, then V ∗ (28, 21) = 25.43 and p∗ (28, 21) = 12.33 for the leader, V ∗ (21, 28) =
22.39 and p∗ (21, 28) = 12.51 for the follower, and V ∗ (1, 1) = 19.36.
20
physical environment of the industry the state serves as a summary of the history of play:
A larger stock of know-how indicates that—on average—a firm has won more sales than
its rival, with the likely reason being that the firm has charged lower prices. Hence, by
conditioning their current behavior on the state, firms implicitly condition on the history
of play. The difference with a subgame perfect equilibrium is that there firms have the
entire history of play at their disposal whereas here they have but a crude indication of it.
Nevertheless, “barely” payoff-relevant state variables (such as firms’ stocks of know-how if
ρ ≈ 1) open the door for bootstrap-type equilibria as familiar from repeated games to arise
in Markov-perfect settings.
In sum, accounting for organizational forgetting in a model of learning-by-doing leads
to multiple equilibria and a rich array of pricing behaviors. In the next section we explore
what these behaviors entail for industry dynamics, both in the short run and in the long
run.
5
Industry dynamics
Figures 5 and 6 display the transient distribution in period 8 and 32, respectively, and
Figure 7 displays the limiting distribution for our four typical cases.23 In the flat equilibrium
without well (ρ = 0.85, δ = 0, upper left panels), the transient and limiting distributions
are unimodal. The most likely industry structure is symmetric. For example, the modal
state is (5, 5) in period 8, (9, 9) in period 16, (17, 17) in period 32, and (30, 30) in period
64. Turning from the short run to the long run, the industry is sure to remain in state
(30, 30) because with logit demand a firm always has a positive probability of making a sale
irrespective of its own price and that of its rival so that, in the absence of organizational
forgetting, both firms must eventually reach the bottom of their learning curves.24 In short,
the industry starts symmetric and stays symmetric.
By contrast, in the flat equilibrium with well (ρ = 0.85, δ = 0.0275, upper right panels)
the transient distributions are first bimodal and then unimodal as is the limiting distribution. The modal states are (1, 8) and (8, 1) in period 8, (4, 11) and (11, 4) in period 16,
(9, 14) and (14, 9) in period 32, but the modal state is (17, 17) in period 64 and the modal
states of the limiting distribution are (24, 25) and (25, 24). Thus, as time passes, firms end
up competing on equal footing. In sum, the industry evolves first towards an asymmetric
structure and then towards a symmetric structure. As we discuss in detail in the Section 6,
the well serves to build, but not to defend, a competitive advantage.
23
To avoid clutter, we do not graph states that have probability of less than 10−4 .
The absence of persistent asymmetries is not an artifact of our functional forms. C-R point out that it
holds true as long as the support of demand is unbounded (see their Assumption 1(a) and footnote 6 on p.
1118).
24
21
While the modes of the transient distributions are more separated and pronounced in
the trenchy equilibrium (ρ = 0.85, δ = 0.0275, lower left panels) than in the flat equilibrium
with well, the dynamics of the industry are similar at first. Unlike in the flat equilibrium
with well, however, the industry continues to evolve towards an asymmetric structure. The
modal states are (14, 21) and (21, 14) in period 64 and (21, 28) and (28, 21) in the limiting
distribution. Although the follower reaches the bottom of its learning curve in the long run
and attains cost parity with the leader, asymmetries persist because the diagonal trench
serves to build and to defend a competitive advantage.
In the extra-trenchy equilibrium (ρ = 0.85, δ = 0.08, lower right panels) the sideways
trench renders it unlikely that the follower ever makes it down from the top of its learning
curve. The transient and limiting distributions are bimodal, and the most likely industry
structure is extremely asymmetric. The modal states are (1, 7) and (7, 1) in period 8, (1, 10)
and (10, 1) in period 16, (1, 15) and (15, 1) in period 32, and (1, 19) and (19, 1) in period 64,
and (1, 26) and (26, 1) in the limiting distribution. In short, one firm acquires a competitive
advantage early on and maintains it with an iron hand.
Returning to Figure 3, the maximum expected Herfindahl index H ∧ (dashed line) and
the limiting expected Herfindahl index H ∞ (solid line) highlight the fundamental economics
of organizational forgetting. If forgetting is sufficiently weak (δ ≈ 0), then asymmetries may
arise but cannot persist, i.e., H ∧ ≥ 0.5 and H ∞ ≈ 0.5. Moreover, if asymmetries arise in
the short run, they are modest. If forgetting is sufficiently strong (δ ≈ 1), then asymmetries
cannot arise in the first place, i.e., H ∧ ≈ H ∞ ≈ 0.5 because forgetting stifles investment
in learning altogether.25 But, for intermediate degrees of forgetting, asymmetries arise and
persist. These asymmetries can be so pronounced that the leader is virtually a monopolist.
Since the Markov process of industry dynamics is irreducible for δ ∈ (0, 1), the follower
must eventually overtake the leader. The limiting expected Herfindahl index H ∞ may be a
misleading measure of long-run industry concentration if such leadership reversals happen
frequently. This, however, is not the case: Leadership reversals take a long time to occur
when H ∞ is high. To establish this, define τ (e1 , e2 ) to be the first-passage time into the set
{(ẽ1 , ẽ2 )|ẽ1 ≤ ẽ2 } if e1 ≥ e2 or {(ẽ1 , ẽ2 )|ẽ1 ≥ ẽ2 } if e1 ≤ e2 . That is, τ (e) is the expected
time it takes the industry to move from state e below (or on) the diagonal of the state
space, where firm 1 leads and firm 2 follows, to state ẽ above (or on) it, where firm 1 follows
and firm 2 leads. Taking the average with respect to the limiting distribution yields the
summary measure
τ∞ =
X
τ (e)µ∞ (e).
e
For the trenchy and extra-trenchy equilibria τ ∞ = 295 and τ ∞ = 83, 406, respectively, indi25
We further document this investment stifling in the Online Appendix.
22
cating that a leadership reversal takes a long time to occur. Hence, the asymmetry captured
by H ∞ persists. In the Online Appendix we plot the expected time to a leadership reversal
τ ∞ in the same format as Figure 3. Just like H ∞ , τ ∞ is largest for intermediate degrees of
forgetting. Moreover, τ ∞ is of substantial magnitude, easily reaching and exceeding 1, 000
periods. Asymmetries are therefore persistent in our model because the expected time until
the leader and the follower switch roles is (perhaps very) long.
We caution the reader that the absence of persistent asymmetries for small forgetting
rates δ in Figure 3 may be an artifact of the finite size of the state space (M = 30 in our
baseline parameterization). Given δ = 0.01, say, ∆(30) = 0.26 and organizational forgetting
is so weak that the industry is sure to remain in or near state (30, 30). This eliminates
bidirectional movements sufficiently completely so as to restore the backward induction
logic that underlies uniqueness of equilibrium for the extreme case of δ = 0 (see Proposition
3). We show in the Online Appendix that, increasing M , whilst holding fixed δ, facilitates
persistent asymmetries as the industry becomes more likely to remain in the interior of the
state space. Furthermore, as emphasized in Section 1, shut-out model elements can give
rise to persistent asymmetries even in the absence of organizational forgetting. We explore
this issue further in Section 8.
To summarize, contrary to what one might expect, organizational forgetting does not
negate learning-by-doing. Rather, as can be seen in Figure 3, over a range of progress ratios
ρ above 0.6 and forgetting rates δ below 0.1, learning and forgetting reinforce each other.
Starting from the absence of both learning (ρ = 1) and forgetting (δ = 0), a steeper learning
curve (lower progress ratio) tends to give rise to a more asymmetric industry structure just
as a higher forgetting rate does. In the next section we analyze the pricing behavior that
drives these dynamics.
6
Pricing behavior
Re-writing equation (5) shows that firm 1’s price in state e satisfies
p∗ (e) = c∗ (e) +
σ
,
1 − D1∗ (e)
(17)
where the virtual marginal cost,
c∗ (e) = c(e1 ) − βφ∗ (e),
(18)
equals the actual marginal cost, c(e1 ), minus the discounted prize, βφ∗ (e), from winning the
current period’s sale. The prize, in turn, is the difference in the value of continued play to
23
∗
∗
firm 1 if it wins the sale, V 1 (e), versus if it loses the sale, V 2 (e):
∗
∗
φ∗ (e) = V 1 (e) − V 2 (e).
(19)
Note that, irrespective of the forgetting rate δ, the equilibrium of our dynamic stochastic
game reduces to the static Nash equilibrium if firms are myopic. Setting β = 0 in equations
(17) and (18) gives the usual FOC for a static price-setting game with logit demand:
p† (e) = c(e1 ) +
σ
1 − D1† (e)
,
(20)
where Dk† (e) = Dk (p† (e), p† (e[2] )) denotes the probability that, in the static Nash equilibrium, the buyer purchases from firm k ∈ {1, 2} in state e. Thus, if β = 0, then p∗ (e) = p† (e)
¡
¢
and V ∗ (e) = D1† (e) p† (e) − c(e1 ) for all states e ∈ {1, . . . , M }2 .
6.1
Price bounds
Comparing equations (17) and (20) shows that equilibrium prices p∗ (e) and p∗ (e[2] ) coincide
with the prices that obtain in a static Nash equilibrium with costs equal to virtual marginal
costs c∗ (e) and c∗ (e[2] ). Static Nash equilibrium prices are increasing in either firm’s cost
(Vives 1999, p. 35). Therefore, if both firms’ prizes are nonnegative, static Nash equilibrium
prices are an upper bound on equilibrium prices, i.e., if φ∗ (e) ≥ 0 and φ∗ (e[2] ) ≥ 0, then
p∗ (e) ≤ p† (e) and p∗ (e[2] ) ≤ p† (e[2] ).
A sufficient condition for φ∗ (e) ≥ 0 for each state e is that the value function V ∗ (e)
is nondecreasing in e1 and nonincreasing in e2 . Intuitively, it should not hurt firm 1 if it
moves down its learning curve and it should not benefit firm 1 if firm 2 moves down its
learning curve. While neither we nor C-R have succeeded in proving it, our computations
show that this intuition is valid in the absence of organizational forgetting:
Result 2 If organizational forgetting is absent (δ = 0), then p∗ (e) ≤ p† (e) for all e ∈
{1, . . . , M }2 .
Result 2 highlights the fundamental economics of learning-by-doing: As long as improvements in competitive position are valuable, firms use price cuts as investments to achieve
them.
We complement Result 2 by establishing a lower bound on equilibrium prices in states
where at least one of the two firms has reached the bottom of its learning curve:
Proposition 5 If organizational forgetting is absent (δ = 0), then (i) p∗ (e) = p† (e) =
p† (m, m) > c(m) for all e ∈ {m, . . . , M }2 and (ii) p∗ (e) > c(m) for all e1 ∈ {m, . . . , M }
and e2 ∈ {1, . . . , m − 1}.
24
Part (i) of Proposition 5 sharpens Theorem 4.3 in C-R by showing that once both firms
have reached the bottom of their learning curves, equilibrium prices revert to static Nash
levels. To see why, note that, given δ = 0, the prize reduces to φ∗ (e) = V ∗ (e1 + 1, e2 ) −
V ∗ (e1 , e2 + 1). But beyond the bottom of their learning curves, firms’ competitive positions
can neither improve nor deteriorate. Hence, as we show in the proof of the proposition,
V ∗ (e) = V ∗ (e0 ) for all e, e0 ∈ {m, . . . , M }2 , so that the advantage-building and advantagedefending motives disappear, the prize is zero, and equilibrium prices revert to static Nash
levels. This rules out trenches penetrating into this region of the state space. Trenchy and
extra-trenchy equilibria therefore cannot arise in the absence of organizational forgetting.
Part (ii) of Proposition 5 restates Theorem 4.3 in C-R for the situation where the leader
but not the follower has reached the bottom of its learning curve. The leader no longer has
an advantage-building motive but continues to have an advantage-defending motive. This
raises the possibility that the leader uses price cuts to delay the follower in moving down its
learning curve. The proposition shows that there is a limit to how aggressively the leader
defends its advantage: below-cost pricing is never optimal in the absence of organizational
forgetting.
The story changes dramatically in the presence of organizational forgetting. The equilibrium may exhibit soft competition in some states and price wars in other states. Consider
the trenchy equilibrium (ρ = 0.85, δ = 0.0275). The upper bound in Result 2 fails in state
(22, 20) where the leader charges 6.44 and the follower charges 7.60, significantly above its
static Nash equilibrium price of 7.30. The follower’s high price stems from its prize of −1.04.
This prize, in turn, reflects that, if the follower wins the sale, then the industry most likely
moves to state (22, 21) and thus closer to the brutal price competition on the diagonal of
the state space. Indeed, the follower’s value function decreases from 20.09 in state (22, 20)
to 19.56 in state (22, 21). To avoid this undesirable possibility, the follower charges a high
price. The lower bound in Proposition 5 fails in state (20, 20) where both firms charge 5.24
as compared to a marginal cost of 5.30. The prize of 2.16 makes it worthwhile to price
below cost even beyond the bottom of the learning curve because “in the trench” winning
the current sale confers a lasting advantage.
This discussion provides the instances that prove the next two propositions.
Proposition 6 If organizational forgetting is present (δ > 0), then we may have p∗ (e) >
p† (e) for some e ∈ {1, . . . , M }2 .
Figure 8 illustrates Proposition 6 by plotting the share of equilibria that violate the upper
bound in Result 2.26 Darker shades indicate higher shares. As can be seen, the upper
26
To take into account the limited precision of our computations, we take the upper bound to be violated
if p∗ (e) > p† (e) + ² for some e ∈ {1, . . . , M }2 , where ² is positive but small. Specifically, we set ² = 10−2 , so
25
bound continues to hold if organizational forgetting is very weak (δ ≈ 0) and possibly also
if learning-by-doing is very weak (ρ ≈ 1). Apart from these extremes (and a region around
ρ = 0.45 and δ = 0.25), at least some, if not all, equilibria entail at least one state where
equilibrium prices exceed static Nash equilibrium prices.
Taken alone, Proposition 6 suggests that organizational forgetting makes firms less aggressive. This makes sense: After all, why invest in improvements in competitive position
when they are transitory? But organizational forgetting can also be a source of aggressive
pricing behavior:
Proposition 7 If organizational forgetting is present (δ > 0), then we may have p∗ (e) ≤
c (m) for some e1 ∈ {m, . . . , M } and e2 ∈ {1, . . . , M }.
Figure 9 illustrates Proposition 7 by plotting the share of equilibria in which a firm prices
below cost even though it has reached the bottom of its learning curve. Note that Figure 9
is a conservative tally of how often the lower bound in Proposition 5 fails because the lower
bound in part (i) already fails if the leader charges less than its static Nash equilibrium price,
not less than its marginal cost. In sum, the leader may be more aggressive in defending
its advantage in the presence of organizational forgetting than in its absence. The most
dramatic expression of this aggressive pricing behavior are the diagonal trenches that are
the defining feature of trenchy and extra-trenchy equilibria.
6.2
Wells and trenches
This section develops intuition as to how wells and trenches can arise. Our goal is to provide
insight as to whether equilibria featuring wells and trenches are economically plausible and,
at least potentially, empirically relevant.
Wells. A well, as seen in the upper right panel of Figure 4, is a preemption battle that
firms at the top of their learning curves fight. Consider our leading example of a flat
equilibrium with well (ρ = 0.85, δ = 0.0275).27 Table 1 details firms’ competitive positions
at various points in time assuming that firm 1 leads and firm 2 follows. Having moved down
the learning curve first, the leader has a lower cost and a higher prize than the follower.
In the modal state (8, 1) in period 8 the leader therefore charges a lower price and enjoys
a higher probability of making a sale. In time the follower also moves down the learning
curve and the leader’s advantage begins to erode (see modal state (11, 4) in period 16) and
that if prices are measured in dollars, then the upper bound must be violated by more than a cent. Given
that the homotopy algorithm solves the system of equations up to a maximum absolute error of about 10−12 ,
Figure 8 therefore almost certainly understates the extent of violations.
27
As a point of comparison, we provide details on firms’ competitive positions at various points in time
for our leading example of a flat equilibrium without well (ρ = 0.85, δ = 0) in the Online Appendix.
26
period
0
8
16
32
64
∞
modal
state
(1,1)
(8,1)
(11,4)
(14,9)
(17,17)
(25,24)
cost
10.00
6.14
5.70
5.39
5.30
5.30
prize
6.85
3.95
1.16
0.36
-0.01
-0.01
leader
price prob.
5.48
0.50
7.68
0.81
7.20
0.62
7.16
0.53
7.31
0.50
7.30
0.50
value
5.87
22.99
20.08
20.06
20.93
21.02
cost
10.00
10.00
7.22
5.97
5.30
5.30
prize
6.85
2.20
1.23
0.64
-0.01
-0.00
follower
price
5.48
9.14
7.68
7.27
7.31
7.30
prob.
0.50
0.19
0.38
0.47
0.50
0.50
value
5.87
5.34
11.48
17.30
20.93
21.02
Table 1: Cost, prize, price, probability of making a sale, and value. Flat equilibrium with
well (ρ = 0.85, δ = 0.0275).
state
(21, 20)
(21, 21)
(22, 20)
(28, 21)
(20, 20)
cost
5.30
5.30
5.30
5.30
5.30
prize
3.53
2.14
3.22
-0.13
2.16
leader
price
5.57
5.26
6.44
7.63
5.24
prob.
0.72
0.50
0.76
0.55
0.50
value
21.91
19.79
23.98
25.42
19.82
cost
5.30
5.30
5.30
5.30
5.30
prize
0.14
2.14
-1.04
-0.71
2.16
follower
price prob.
6.54
0.28
5.26
0.50
7.60
0.24
7.81
0.45
5.24
0.50
value
19.56
19.79
20.09
22.37
19.82
Table 2: Cost, prize, price, probability of making a sale, and value. Trenchy equilibrium
(ρ = 0.85, δ = 0.0275).
eventually vanishes completely (see the modal state (17, 17) in period 64). The prizes reflect
this erosion. The leader’s prize is higher than the follower’s in state (8, 1) (3.95 versus 2.20)
but lower in state (11, 4) (1.16 versus 1.23). Although the leadership position is transitory,
it is surely worth having. Both firms use price cuts in state (1, 1) in the hope of being the
first to move down the learning curve. In the example, the prize of 6.85 justifies charging the
price of 5.48 that is well below the marginal cost of 10. The well is therefore an investment
in building competitive advantage.
More abstractly, a well is the outcome of an auction in state (1, 1) for the additional
future profits—the prize—that accrue to the firm that makes the first sale and acquires
transitory industry leadership. As equations (17)–(19) show, a firm’s price in equilibrium is
virtual marginal cost marked up. Virtual marginal cost, in turn, accounts for the discounted
prize from winning the current period’s sale and, getting to the essential point, the prize is
the difference in the value of continued play if the firm rather than its rival wins.
Diagonal trenches. A diagonal trench, as seen in the lower panels of Figure 4, is a
price war between symmetric or nearly symmetric firms. Extending along the entire diagonal of the state space, a diagonal trench has the curious feature that the firms compete
fiercely—perhaps pricing below cost—even when they both have exhausted all gains from
27
learning-by-doing. Part (i) of Proposition 5 rules out this type of behavior in the absence
of organizational forgetting.
Like a well, a diagonal trench serves to build a competitive advantage. Unlike a well, a
diagonal trench also serves to defend a competitive advantage, thereby rendering it (almost)
permanent: The follower recognizes that to seize the leadership position it would have to
“cross over” the diagonal trench and struggle through another price war. Crucially this
price war is a part of a MPE and, as such, a credible threat the follower cannot ignore.
The logic behind a diagonal trench has three parts. If a diagonal trench exists, then
the follower does not contest the leadership position. If the follower does not contest the
leadership position, then being the leader is valuable. Finally, to close the circle of logic,
if being the leader is valuable, then firms price aggressively on the diagonal on the state
space in a bid for the leadership position, thus giving rise to the diagonal trench. Table
2 illustrates this argument by providing details on firms’ competitive position in various
states for our leading example of a trenchy equilibrium (ρ = 0.85, δ = 0.0275).
Part 1: Trench sustains leadership. To see why the follower does not contest the leadership position consider a state such as (21, 20) where the follower has almost caught up with
the leader. Suppose the follower wins the current sale. In this case the follower may leapfrog
the leader if the industry moves against the odds to state (20, 21). However, the most likely
possibility, with a probability of 0.32, is that the industry moves to state (21, 21). Due to
the brutal price competition “in the trench” the follower’s expected cash flow in the next
period decreases to −0.02 = 0.50×(5.26−5.30) compared to 0.34 = 0.28×(6.54−5.30) if the
industry had remained in state (21, 20). Suppose, in contrast, the leader wins. This completely avoids sparking a price war. Moreover, the most likely possibility, with a probability
of 0.32, is that the leader enhances its competitive advantage by moving to state (22, 20). If
so, the leader’s expected cash flow in the next period increases to 0.87 = 0.76 × (6.44 − 5.30)
compared to 0.20 = 0.72 × (5.57 − 5.30) if the industry had remained in state (21, 20). Because winning the sale is more valuable to the leader than to the follower, the leader’s prize
in state (21, 20) is almost 25 times larger than the follower’s and the leader underprices the
follower. As a consequence, the leader defends its position with a substantial probability of
0.79. In other words, the diagonal trench sustains the leadership position.
Part 2: Leadership generates value. Because the leader underprices the follower, over
time the industry moves from state (21, 20) to (or near) the modal state (28, 21) of the
limiting distribution. Once there the leader underprices the follower (7.63 versus 7.81)
despite cost parity and thus enjoys a higher probability of making a sale (0.55 versus 0.45).
The leader’s expected cash flow in the current period is therefore 0.55 × (7.63 − 5.30) = 1.27
as compared to the follower’s of 0.45 × (7.81 − 5.30) = 1.14. Because the follower does not
contest the leadership position, the leader is likely to enjoy these additional profits for a
28
state
(26, 1)
(26, 2)
(26, 3)
..
.
cost
5.30
5.30
5.30
..
.
prize
6.43
6.21
5.16
..
.
leader
price
8.84
7.48
6.94
..
.
prob.
0.90
0.88
0.85
..
.
value
53.32
46.31
40.23
..
.
cost
10.00
8.50
7.73
..
.
prize
0.12
0.21
0.27
..
.
follower
price
11.00
9.44
8.65
..
.
prob.
0.10
0.12
0.15
..
.
value
2.42
2.55
2.78
..
.
(26, 7)
(26, 8)
(26, 9)
(26, 10)
5.30
5.30
5.30
5.30
3.04
2.33
1.17
0.16
6.14
5.99
6.24
6.83
0.73
0.66
0.51
0.40
24.76
21.86
19.84
19.09
6.34
6.14
5.97
5.83
0.58
1.08
1.71
1.96
7.15
6.64
6.29
6.44
0.27
0.34
0.49
0.60
4.16
4.78
6.07
8.08
Table 3: Cost, prize, price, probability of making a sale, and value. Extra-trenchy equilibrium (ρ = 0.85, δ = 0.08).
long time (recall that τ ∞ = 295). Hence, being the leader is valuable.
Part 3: Value induces trench. Because being the leader is valuable, firms price aggressively on the diagonal on the state space in a bid for the leadership position. The prize
is 2.16 in state (20, 20) and 2.14 in state (21, 21) and justifies charging a price of 5.24 and
5.26, respectively, even through all gains from learning-by-doing have been exhausted. This
gives rise to the diagonal trench. Observe that this argument applies at every state on
the diagonal because, no matter where on the diagonal the firms happen to be, winning
the current sale confers a lasting advantage. The trench therefore extends along the entire
diagonal of the state space.
All this can be summed up in a sentence: Building a competitive advantage creates the
diagonal trench that defends the advantage and creates the prize that makes it worthwhile
to fight for the leadership position. A diagonal trench is thus a self-reinforcing mechanism
for gaining and maintaining market dominance.
Sideways trenches. A sideways trench, as seen in the lower right panel of Figure 4, is a
price war between very asymmetric firms. This war is triggered when the follower starts to
move down its learning curve. Table 3 provides details on firms’ competitive positions in
various states for our leading example of an extra-trenchy equilibrium (ρ = 0.85, δ = 0.08).
The sideways trench is evident in the decrease in the leader’s price from state (26, 1) to
state (26, 8) and the increase from state (26, 8) to state (26, 10). Note that the follower has
little chance of making it down its learning curve as long as the probability of winning a sale
is less than the probability of losing a unit of know-how through organizational forgetting.
While D2∗ (26, 1) = 0.10 > 0.08 = ∆(1), we have D2∗ (26, 2) = 0.12 < 0.15 = ∆(2) and
D2∗ (26, 3) = 0.15 < 0.22 = ∆(3). Hence, the leader can stall the follower at the top of its
29
learning curve and, indeed, the modal state of the limiting distribution is (26, 1).
The additional future profits stemming from the leader’s ability to stall the follower are
the source of its large prize in state (26, 1). In state (26, 2) the prize is almost as large
because by winning a sale the leader may move the industry back to state (26, 1) in the
next period. The leader’s prize falls as the follower moves further down its learning curve
because it takes progressively longer for the leader to force the follower back up its learning
curve and because the lower cost of the follower makes it harder for the leader to do so. In
the unlikely event that the follower crashes through the sideways trench in state (26, 8), the
leader’s prize falls sharply. At the same time the follower’s prize rises sharply as it turns
from a docile competitor into a viable threat.
A sideways trench, like a diagonal trench, is a self-reinforcing mechanism for gaining
and maintaining market dominance. But whereas a diagonal trench is about fighting an
imminent threat, a sideways trench is about fighting a distant threat. One can think of a
sideways trench as an endogenously arising mobility barrier in the sense of Caves & Porter
(1977) or the equilibrium manifestation of former Intel CEO Andy Grove’s dictum “Only
the paranoid survive.”
In sum, the four types of equilibria that we have identified in Section 4 give rise to
distinct yet plausible pricing behaviors and industry dynamics. Rather than impeding
aggressive behavior, organizational forgetting facilitates it. In its absence, the equilibria
are flat either with or without well depending on the progress ratio. Generally speaking,
organizational forgetting is associated with “trenchier” equilibria, more aggressive behavior,
and more concentrated industries both in the short run and in the long run.
6.3
Dominance properties
Traditional intuition suggests that learning-by-doing leads by itself to market dominance
by giving a more experienced firm the ability to profitably underprice its less experienced
rival. This enables the leader to widen its competitive advantage over time, thereby further
enhancing its ability to profitably underprice the follower. C-R formalize this idea with
“two concepts of self-reinforcing market dominance” (p. 1115): An equilibrium exhibits
increasing dominance (ID) if p∗ (e)−p∗ (e[2] ) < 0 whenever e1 > e2 and increasing increasing
dominance (IID) if p∗ (e)−p∗ (e[2] ) is decreasing in e1 . If ID holds, the leader charges a lower
price than the follower and therefore enjoys a higher probability of making a sale. If IID
holds, the price gap between the firms widens with the length of the lead.28
28
Athey & Schmutzler’s (2001) notion of weak increasing dominance describes the relationship between
players’ states and their actions in dynamic games with deterministic state-to-state transitions and coincides
with the notion of ID in C-R. Similar notions have also been used by Vickers (1986) and Budd, Harris &
Vickers (1993) in dynamic investment games.
30
In the absence of organizational forgetting, Theorem 3.3 in C-R shows that ID and IID
hold provided that the discount factor β is sufficiently close to 1 (or, alternatively, to 0, see
Theorem 3.1 in C-R). Our computations show that β =
1
1.05
in our baseline parameterization
suffices.
Result 3 If organizational forgetting is absent (δ = 0), then IID holds. Thus, ID holds.
Even if an equilibrium satisfies ID and IID it is not clear that the industry is inevitably
progressing towards monopolization. If the price gap between the firms is small, then the
impact of ID and IID on industry structure and dynamics may be trivial.29 In such a
scenario, the leader charges a slightly lower price than the follower and this gap widens a
bit over time. However, with even a modest degree of horizontal product differentiation,
the firms still split sales more or less equally and thus move down the learning curve in
tandem. This is exactly what happens when δ = 0. For example, the flat equilibrium
without well (ρ = 0.85, δ = 0) satisfies IID and thus ID, but with a maximum expected
Herfindahl index of 0.52 the industry is essentially a symmetric duopoly at all times. More
generally, as Figure 3 shows, in the absence of organizational forgetting asymmetries are
modest if they arise at all. Although ID and IID hold, the maximum expected Herfindahl
index across all equilibria is 0.67 (attained at ρ = 0.65). Hence, ID and IID are not sufficient
for economically meaningful market dominance.
ID and IID are also not necessary for market dominance. The extra-trenchy equilibrium
(ρ = 0.85, δ = 0.08), for example, violates ID and thus IID. Yet, the industry is likely
to be a near-monopoly at all times. More generally, while the empirical studies of Argote
et al. (1990), Darr et al. (1995), Benkard (2000), Shafer et al. (2001), and Thompson (2003)
warrant accounting for organizational forgetting in a model of learning-by-doing, doing so
may cause ID and IID to fail.
Proposition 8 If organizational forgetting is present (δ > 0), then IID may fail. Also ID
may fail.
In the absence of organizational forgetting, C-R have already shown that ID and IID may
fail for intermediate values of β (see their Remark C.5 on p. 1136). Result 3 and Proposition
8 make the comparative dynamics point that ID and IID may hold when δ = 0 but fail
when δ > 0 (holding fixed the remaining parameters). Figure 10 illustrates Proposition 8
by plotting the share of equilibria that violate IID (upper panel) and ID (lower panel). As
can be seen, all equilibria fail to obey IID unless forgetting or learning is very weak. Even
29
Indeed, C-R show in their Theorem 3.2 that p∗ (e) → p† (m, m) for all e ∈ {1, . . . , M }2 as β → 1, i.e.,
both firms price as if at the bottom of their learning curves. This suggests that the price gap may be small
for “reasonable” discount factors.
31
leading example
preemption battle (well)
price war triggered by imminent threat (diagonal trench)
price war triggered by distant
threat (sideways trench)
short-run market dominance
long-run market dominance
dominance properties
flat eqbm.
without
well
ρ = 0.85,
δ=0
no
no
flat eqbm.
with well
trenchy
eqbm.
ρ = 0.85,
δ = 0.0275
yes
no
ρ = 0.85,
δ = 0.0275
no
yes
extratrenchy
eqbm.
ρ = 0.85,
δ = 0.08
no
yes
no
no
no
yes
no
no
yes
no
yes
no,
mostly
yes
yes,
modest
no,
mostly
yes
yes,
extreme
no,
mostly
Table 4: Pricing behavior and industry dynamics.
violations of ID are extremely common, especially for forgetting rates δ in the empirically
relevant range below 0.1.
Of course, we do not argue that the concepts of ID and IID have no place in the
analysis of industry dynamics. Caution, however, is advisable. Since ID and IID are neither
necessary nor sufficient for market dominance, making inferences about the evolution of the
industry on their basis alone may be misleading.
6.4
Summary
Table 4 summarizes the broad patterns of pricing behavior and industry dynamics. Acknowledging that the know-how gained through learning-by-doing can be lost through organizational forgetting is important. Generally speaking, organizational forgetting is associated
with “trenchier” equilibria, more aggressive behavior, and more concentrated industries
both in the short run and in the long run. Moreover, the dominance properties of firms’
pricing behavior can break down in the presence of organizational forgetting.
The key difference between a model with and without organizational forgetting is that
in the former a firm can move both forward to a higher state and backward to a lower state.
This possibility of bidirectional movement enhances the advantage-building and advantagedefending motives. By winning a sale, a firm makes itself less, and its rival more, vulnerable
to organizational forgetting. This can create strong incentives to cut prices. Rather than
impeding it, organizational forgetting therefore facilitates aggressive pricing as manifested
in the trenchy and extra-trenchy equilibria that we have identified.
32
7
Organizational forgetting and multiple equilibria
While the equilibrium is unique if organizational forgetting is either absent (δ = 0) or certain
(δ = 1), multiple equilibria are common for intermediate degrees of forgetting. Surprisingly,
for some values of ρ and δ, the equilibria range from “peaceful coexistence” to “trench
warfare.” Consequently, in addition to primitives of learning-by-doing and organizational
forgetting, the equilibrium by itself is an important determinant of pricing behavior and
industry dynamics.
Why do multiple equilibria arise in our model? To explore this question, think about the
strategic situation faced by firms in setting prices in state e. The value of continued play
to firm n is given by the conditional expectation of its value function, V n1 (e) and V n2 (e),
as defined in equations (2) and (3). Holding the value of continued play fixed, the strategic
situation in state e is akin to a static game. If the reaction functions in this game intersect
more than once, then multiple equilibria arise. On the other hand, if they intersect only
once irrespective of the value of continued play, then we say that a model satisfies statewise
uniqueness.
Proposition 9 Statewise uniqueness holds.
Not surprisingly the proof of Proposition 9 relies on the functional form of demand. This
is reminiscent of the restrictions on demand (e.g., log-concavity) that Caplin & Nalebuff
(1991) set forth to guarantee uniqueness of Nash equilibrium in their analysis of static
price-setting games.
Given that the model satisfies statewise uniqueness, multiple equilibria must arise from
firms’ expectations regarding the value of continued play. To see this, consider again state
e. The intersection of the reaction functions constitutes a Nash equilibrium in prices in a
subgame in which firm n believes that its value of continued play is given by V n1 (e) and
V n2 (e). If firms have rational expectations, i.e., if the conjectured value of continued play
is actually attained, then these prices constitute an equilibrium of our dynamic stochastic
game. In our model, taking the value of continued play as given, the reaction functions
intersect only once because we have statewise uniqueness, but there may be more than
one value of continued play that is consistent with rational expectations. In this sense
multiplicity is rooted in the dynamics of the model.
The key driver of multiplicity is organizational forgetting. Dynamic competition with
learning and forgetting is like racing down an upward-moving escalator. Unless a firm
makes sales at a rate that exceeds the rate at which it loses know-how through forgetting,
its marginal cost is bound to increase. The inflow of know-how into the industry is one
unit per period whereas in expectation the outflow in state e is ∆(e1 ) + ∆(e2 ). Consider
33
state (e, e), where e ≥ m, on the diagonal of the state space at or beyond the bottom of
the learning curve. If 1 ¿ 2∆(e), then it is impossible that both firms reach the bottom
of their learning curves and remain there. Knowing this, firms have no choice but to price
aggressively. The result is trench warfare as each firm uses price cuts to push the state
to its side of the diagonal and keep it there. If, however, 1 À 2∆(e), then it is virtually
inevitable that both firms reach the bottom of their learning curves, and firms may as well
price softly. In both cases, the primitives of the model tie down the equilibrium.
This is no longer the case if 1 ≈ 2∆(e), setting the stage for multiple equilibria as diverse
as peaceful coexistence and trench warfare. If firms believe that they cannot peacefully
coexist at the bottom of their learning curves and that one firm will come to dominate
the market, then both firms will cut their prices in the hope of acquiring a competitive
advantage early on and maintaining it throughout. This naturally leads to trench warfare
and market dominance. If, however, firms believe that they can peacefully coexist at the
bottom of their learning curves, then neither firm cuts its price. Soft pricing, in turn,
ensures that the anticipated symmetric industry structure actually emerges. A back-of-theenvelope calculation is reassuring here. Recall that m = 15 and M = 30 in our baseline
parameterization and observe that 1 = 2∆(15) implies δ ≈ 0.045, 1 = 2∆(20) implies
δ ≈ 0.034, and 1 = 2∆(30) implies δ ≈ 0.023. This range of forgetting rates, for which the
inflow of know-how approximately equals the outflow, is indeed where multiplicity prevails
(see again Figure 2).
A sufficient condition for uniqueness of equilibrium in a dynamic stochastic game with a
finite state space is that the model satisfies statewise uniqueness and the movements through
the state space are unidirectional. Statewise uniqueness precludes players’ actions from
giving rise to multiple equilibria and unidirectional movements preclude their expectations
from doing so. The proof of Proposition 3 illustrates the power of this sufficient condition.
Specifically, if δ = 0 in our game, then a firm can never move backward to a lower state.
Hence, once the industry reaches state (M, M ), it remains there forever, so that the value
of future play in state (M, M ) coincides with the value of being in this state ad infinitum.
In conjunction with statewise uniqueness, this uniquely determines the value of being in
state (M, M ). Next consider states (M − 1, M ) and (M, M − 1). The value of future play in
states (M − 1, M ) and (M, M − 1) depends on the value of being in state (M, M ). Statewise
uniqueness ensures that firms’ prices in states (M −1, M ) and (M, M −1) as well as the value
of being in these states are uniquely determined. Continuing to work backwards establishes
that the equilibrium is unique.
34
8
Robustness checks
We have conducted extensive robustness checks regarding our specification of the discount
factor, demand (product differentiation, outside good, and choke price), learning-by-doing,
and entry and exit. In the interest of brevity, we confine ourselves here to pointing out
if and how our results regarding aggressive pricing behavior (wells and trenches), market
dominance (persistent asymmetries), and multiple equilibria change with the specification
of the model. A more detailed discussion can be found in the Online Appendix along with
some further checks (frequency of sales, organizational forgetting) that we omit here.
8.1
Discount factor
Two extreme cases merit discussion. First, as β → 0 and firms become more myopic, the
wells and trenches vanish and we obtain a flat equilibrium without well. In the limit of β = 0
equation (20) implies that the equilibrium of our dynamic stochastic game reduces to the
static Nash equilibrium irrespective of the forgetting rate δ. Second, as β → 1, the wells and
trenches deepen: More patient firms have a stronger incentive to cut prices in the present
in order to seize the leadership position in the future. In addition to the four typical cases
in Figure 4 we obtain other types of equilibria with more complex patterns of trenches. The
fact that high discount factors exacerbate the multiplicity problem is hardly surprising in
light of the folk theorems for repeated games (Friedman 1971, Rubinstein 1979, Fudenberg
& Maskin 1986).
8.2
Demand
Product differentiation. A higher degree of horizontal product differentiation σ lowers
the extent to which firms interact strategically with each other. As σ → ∞, firms eventually
become monopolists and have no incentive to cut prices in order to acquire or defend a
competitive advantage. As a result, the equilibrium is unique and the industry evolves
symmetrically.
Outside good. As in C-R we assume that the buyer always purchases from one of the
two firms in the industry. Allowing the buyer to instead choose an alternative made from a
substitute technology (outside good) implies that the price elasticity of aggregate demand for
the two competing firms is no longer zero. If the outside good is made sufficiently attractive,
then in state (1, 1) the probability that either firm wins the one unit of demand that is
available each period becomes small unless they price aggressively—below marginal cost—
against the outside good. Making it down from the top of its learning curve consequently
requires a firm to incur substantial losses in the near term. In the long term, however,
35
fighting one’s way down the learning curve has substantial rewards because the outside
good is a much less formidable competitor to a firm at the bottom of its learning curve than
to a firm at the top.
If the discount factor is held fixed at its baseline value, then even a moderately attractive outside good sufficiently constrains firms’ pricing behavior so that we no longer have
sunspots for a progress ratio of ρ = 1. If we further increase the attractiveness of the outside
good, then the rewards from fighting one’s way down the learning curve become too far off
in the future to justify the required aggressive pricing with its attendant near-term losses.
In the ensuing equilibrium the price that a firm charges is fairly insensitive to its rival’s
stock of know-how because the outside good is the firm’s main competitor and wins the
sale most periods. As a result the inflow of know-how into the industry through learning
is much smaller than the outflow through forgetting. This implies that the equilibrium is
unique and entails both firms being stuck at the top of their learning curves. Trenchy and
extra-trenchy equilibria disappear.
But if the discount factor is increased as the outside good is made increasingly attractive,
then the near-term losses of fighting one’s way down the learning curve do not overwhelm
the long-term rewards from doing so. Firms price aggressively in trenchy and extra-trenchy
equilibria. Therefore, provided the discount factor is sufficiently close to one, the presence
of an economically significant price elasticity of aggregate demand does not seem to change
the variety and multiplicity of equilibria in any fundamental way.
Choke price. As in C-R our logit specification for demand ensures that a firm always
has a positive probability of making a sale and, in the absence of forgetting, must therefore
eventually reach the bottom of its learning curve. This precludes long-run market dominance
to occur in the absence of organizational forgetting.
Suppose instead that the probability that firm n makes a sale is given by a linear specification. Due to the choke price in the linear specification, a firm is able to surely deny its
rival a sale by pricing sufficiently aggressively. Given a sufficiently low degree of horizontal
product differentiation, firms at the top of their learning curves fight a preemption battle.
The industry remains in an asymmetric structure as the winning firm takes advantage of
the choke price to stall the losing firm at the top of its learning curve. In other words, the
choke price is a shut-out model element that can lead to persistent asymmetries even in the
absence of organizational forgetting.
8.3
Learning-by-doing
Following C-R we assume that m < M represents the stock of know-how at which a firm
reaches the bottom of its learning curve. In a bottomless learning specification with m =
36
M , we obtain other types of equilibria in addition to the four typical cases in Figure 4.
Particularly striking is the plateau equilibrium. This equilibrium is similar to a trenchy
equilibrium except that the diagonal trench is interrupted by a region of very soft price
competition. On this plateau both firms charge prices well above cost. This “cooperative”
behavior contrasts markedly with the price war of the diagonal trench.
8.4
Entry and exit
We assume that at any point in time there is a total of N firms, each of which can be either
an incumbent firm or a potential entrant. Once an incumbent firm exits the industry, it
perishes and a potential entrant automatically takes its “slot” and has to decide whether
or not to enter.
Organizational forgetting remains a source of aggressive pricing behavior, market dominance, and multiple equilibria in the general model with entry and exit. The possibility of
exit adds another component to the prize from winning a sale because, by winning a sale,
a firm may move the industry to a state in which its rival is likely to exit. But if the rival
exits, then it may be replaced by an entrant that comes into the industry at the top of
its learning curve or it may not be replaced at all. As a result, pricing behavior is more
aggressive than in the basic model without entry and exit. This leads to more pronounced
asymmetries both in the short run and in the long run.
Because entry and exit are shut-out model elements, asymmetries can arise and persist
even in the absence of organizational forgetting (see the Online Appendix for a concrete
example). Entry and exit may also give rise to multiple equilibria as C-R have already
shown (see their Theorem 4.1).
9
Conclusions
Learning-by-doing and organizational forgetting have been shown to be important in a
variety of industrial settings. This paper provides a general model of dynamic competition
that accounts for these economic fundamentals and shows how they shape industry structure
and dynamics. We contribute to the numerical analysis of industry dynamics in two ways.
First, we show that there are equilibria that the P-M algorithm cannot compute. Second,
we propose a homotopy algorithm that allows us to describe in detail the structure of the
set of solutions to our model.
In contrast to the present paper, the theoretical literature on learning-by-doing has
largely ignored organizational forgetting. Moreover, it has mainly focused on the dominance
properties of firms’ pricing behavior. By directly examining industry dynamics, we are
able to show that ID and IID may not be sufficient for economically meaningful market
37
dominance. By generalizing the existing models of learning, we are able to show that these
dominance properties break down with even a small degree of forgetting. Yet, it is precisely
in the presence of organizational forgetting that market dominance ensues both in the short
run and in the long run.
Our analysis of the role of organizational forgetting reveals that learning and forgetting
are distinct economic forces. Forgetting, in particular, does not simply negate learning. The
unique role played by organizational forgetting comes about because it makes bidirectional
movements through the state space possible. As a consequence, a model with forgetting can
give rise to aggressive pricing behavior, market dominance, and multiple equilibria, whereas
a model without forgetting cannot.
Diagonal and sideways trenches are part and parcel to the self-reinforcing mechanisms
that lead to market dominance. Since the leadership position is aggressively defended, firms
fight a price war to attain it. This provides all the more reason to aggressively defend the
leadership position because if it is lost, then another price war ensues. This seems like a
good story to tell. Our computations show that this is not just an intuitively sensible story
but also a logically consistent one that—perhaps—plays out in real markets.
Appendix
Proof of Proposition 1. Part (i): The basic differential equations (13) set
µ
¶
∂F(x(s); δ(s), ρ)
0
δ (s) = det
.
∂x
The Jacobian ∂F(x(s);δ(s),ρ)
is a (2M 2 × 2M 2 ) matrix and therefore has an even number of
∂x
eigenvalues. Its determinant is the product of its eigenvalues. Hence, if δ 0 (s) ≤ 0, then there
exists at least one real nonnegative eigenvalue. (Suppose to the contrary that all eigenvalues
are either complex or real and negative. Since the number of complex eigenvalues is even,
so is the number of real eigenvalues. Moreover, the product of a conjugate pair of complex
eigenvalues is positive, as is the product of an even number of real negative eigenvalues.)
To relate the P-M algorithm to our homotopy algorithm, let (x(s), δ(s)) ∈ F−1 (ρ) be a
parametric path of equilibria. We show in the Online Appendix that
¯
∂F(x(s); δ(s), ρ)
∂G(x(s)) ¯¯
=
+ I,
(21)
¯
∂x
∂x
(δ(s),ρ)
where I denotes the (2M 2 × 2M 2 ) identity matrix.
The proof is completed by recalling a basic result from linear algebra: Let A be an
arbitrary matrix and ς(A) its spectrum. Then ς(A + I) = ς(A) + 1 (see Proposition A.17 in
Appendix A of Bertsekas & Tsitsiklis 1997). Hence, because ∂F(x(s);δ(s),ρ)
∂x
¯ has at least one
∂G(x(s)) ¯
real nonnegative eigenvalue, it follows from equation (21) that
has at least
¯
∂x
(δ(s),ρ)
38
µ
one real eigenvalue equal to or bigger than unity. Hence, %
¯
∂G(x(s)) ¯
¯
∂x
(δ(s),ρ)
¶
≥ 1.
Part (ii): Consider the iteration xk+1 = G̃(xk ) = ωG(xk ) + (1 − ω)xk , where ω > 0.
Using equation (21) its Jacobian at (x(s), δ(s)) ∈ F−1 (ρ) is
¯
¯
∂ G̃(x(s)) ¯¯
∂G(x(s)) ¯¯
∂F(x(s); δ(s), ρ)
+ I.
=ω
+ (1 − ω)I = ω
¯
¯
¯
∂x
∂x
∂x
(δ(s),ρ)
(δ(s),ρ)
µ
As before it follows that %
¯
∂ G̃(x(s)) ¯
¯
∂x
(δ(s),ρ)
¶
≥ 1.
Proof of Proposition 3. We rewrite the Bellman equations and FOCs in state e as
¡
¡
¢¢
V1 = D1 (p1 , p2 ) p1 − c(e1 ) + β V 11 − V 12 + βV 12 ,
(22)
¡
¡
¢¢
V2 = D2 (p1 , p2 ) p2 − c(e2 ) + β V 22 − V 21 + βV 21 ,
(23)
¡
¡
¢¢
σ
− p1 − c(e1 ) + β V 11 − V 12 ,
(24)
0=
D2 (p1 , p2 )
¡
¡
¢¢
σ
0=
− p2 − c(e2 ) + β V 22 − V 21 ,
(25)
D1 (p1 , p2 )
where, to simplify the notation, Vn is shorthand for Vn (e), V nk for V nk (e), pn for pn (e),
etc. and we use the fact that D1 (p1 , p2 ) + D2 (p1 , p2 ) = 1.
Case (i): First suppose δ = 0. The proof proceeds in a number of steps. In step 1,
we establish that the equilibrium in state (M, M ) is unique. In step 2a, we assume that
there is a unique equilibrium in state (e1 + 1, M ), where e1 ∈ {1, . . . , M − 1}, and show
that this implies that the equilibrium in state (e1 , M ) is unique. In step 2b, we assume
that there is a unique equilibrium in state (M, e2 + 1), where e2 ∈ {1, . . . , M − 1}, and
show that this implies that the equilibrium in state (M, e2 ) is unique. By induction, steps
1, 2a, and 2b establish uniqueness along the upper edge of the state space. In step 3,
we assume that there is a unique equilibrium in states (e1 + 1, e2 ) and (e1 , e2 + 1), where
e1 ∈ {1, . . . , M −1} and e2 ∈ {1, . . . , M −1}, and show that this implies that the equilibrium
in state (e1 , e2 ) is unique. Hence, uniqueness in state (M −1, M −1) follows from uniqueness
in states (M, M − 1) and (M − 1, M ), uniqueness in state (M − 2, M − 1) from uniqueness in
states (M − 1, M − 1) and (M − 2, M ), etc. Working backwards gives uniqueness in states
(e1 , M − 1), where e1 ∈ {1, . . . , M − 1}. This, in turn, gives uniqueness in states (e1 , M − 2),
where e1 ∈ {1, . . . , M − 1}, etc.
Step 1: Consider state e = (M, M ). From the definition of the state-to-state transitions
in Section 2, we have
V 11 = V 12 = V1 ,
V 21 = V 22 = V2 .
Imposing these restrictions and solving equations (22) and (23) for V1 and V2 , respectively,
39
yields
D1 (p1 , p2 )(p1 − c(e1 ))
,
1−β
D2 (p1 , p2 )(p2 − c(e2 ))
V2 =
.
1−β
V1 =
(26)
(27)
Simplifying equations (24) and (25) yields
σ
− (p1 − c(e1 )) = F1 (p1 , p2 ),
D2 (p1 , p2 )
σ
0=
− (p2 − c(e2 )) = F2 (p1 , p2 ).
D1 (p1 , p2 )
0=
(28)
(29)
The system of equations (28) and (29) determines equilibrium prices. Once we have established that there is a unique solution for p1 and p2 , equations (26) and (27) immediately
ascertain that V1 and V2 are unique.
Let p\1 (p2 ) and p\2 (p1 ) be defined by
F1 (p\1 (p2 ), p2 ) = 0,
F2 (p1 , p\2 (p1 )) = 0
and set F (p1 ) = p1 − p\1 (p\2 (p1 )). The p1 that solves the system of equations (28) and (29)
is the solution to F (p1 ) = 0, and this solution is unique provided that F (p1 ) is strictly
monotone. The implicit function theorem yields
³
´³
´
∂F2
1
−
− ∂F
∂p2
∂p1
.
F 0 (p1 ) = 1 − ∂F1
∂F2
∂p1
∂p2
Straightforward differentiation shows that
³
´
1
1 (p1 ,p2 )
− ∂F
−D
∂p2
D2 (p1 ,p2 )
=
= D1 (p1 , p2 ) ∈ (0, 1),
∂F1
− D2 (p11 ,p2 )
∂p1
´
³
2
2 (p1 ,p2 )
− ∂F
−D
∂p1
D1 (p1 ,p2 )
=
= D2 (p1 , p2 ) ∈ (0, 1).
∂F2
− D1 (p11 ,p2 )
∂p
2
It follows that F 0 (p1 ) > 0.
Step 2a: Consider state e = (e1 , M ), where e1 ∈ {1, . . . , M − 1}. We have
V 12 = V1 ,
V 22 = V2 .
Imposing these restrictions and solving equations (22) and (23) for V1 and V2 , respectively,
40
yields
D1 (p1 , p2 )(p1 − c(e1 ) + βV 11 )
,
1 − βD2 (p1 , p2 )
D2 (p1 , p2 )(p2 − c(e2 ) − βV 21 ) + βV 21
V2 =
.
1 − βD2 (p1 , p2 )
V1 =
(30)
(31)
Substituting equations (30) and (31) into equations (24) and (25) and dividing through by
1−β
1
1−βD2 (p1 ,p2 ) and 1−βD2 (p1 ,p2 ) , respectively, yields
¢
(1 − βD2 (p1 , p2 ))σ ¡
− p1 − c(e1 ) + βV 11 = G1 (p1 , p2 ),
(1 − β)D2 (p1 , p2 )
¢
(1 − βD2 (p1 , p2 ))σ ¡
0=
− p2 − c(e2 ) − β(1 − β)V 21 = G2 (p1 , p2 ).
D1 (p1 , p2 )
0=
(32)
(33)
The system of equations (32) and (33) determines equilibrium prices as a function of V 11
and V 21 . These are given by V1 (e1 + 1, M ) and V2 (e1 + 1, M ), respectively, and are unique
by hypothesis. As in step 1, once we have established that there is a unique solution for p1
and p2 , equations (30) and (31) immediately ascertain that, in state e = (e1 , M ), V1 and
V2 are unique.
Proceeding as in step 1, set G(p1 ) = p1 − p\1 (p\2 (p1 )), where p\1 (p2 ) and p\2 (p1 ) are defined
by G1 (p\1 (p2 ), p2 ) = 0 and G2 (p1 , p\2 (p1 )) = 0, respectively. We have to show that G(·) is
strictly monotone. Straightforward differentiation shows that
³
´
D1 (p1 ,p2 )
1
− ∂G
− (1−β)D
∂p2
D1 (p1 , p2 )
2 (p1 ,p2 )
=
=
∈ (0, 1),
∂G1
1−βD2 (p1 ,p2 )
1 − βD2 (p1 , p2 )
− (1−β)D2 (p1 ,p2 )
∂p1
³
´
2
2 (p1 ,p2 )
− ∂G
− (1−β)D
∂p1
(1 − β)D2 (p1 , p2 )
D1 (p1 ,p2 )
=
∈ (0, 1).
=
∂G2
1−βD2 (p1 ,p2 )
1 − βD2 (p1 , p2 )
−
∂p
2
D1 (p1 ,p2 )
It follows that G0 (p1 ) > 0.
Step 2b: Consider state e = (M, e2 ), where e2 ∈ {1, . . . , M − 1}. We have
V 11 = V1 ,
V 21 = V2 .
The argument is completely symmetric to the argument in step 2a and therefore omitted.
Step 3: Consider state e = (e1 , e2 ), where e1 ∈ {1, . . . , M − 1} and e2 ∈ {1, . . . , M − 1}.
The system of equations (24) and (25) determines equilibrium prices as a function of V 11 ,
V 12 , V 21 , and V 22 . These are given by V1 (e1 + 1, e2 ), V1 (e1 , e2 + 1), V2 (e1 + 1, e2 ), and
V2 (e1 , e2 + 1), respectively, and are unique by hypothesis. As in step 1, once we have
established that there is a unique solution for p1 and p2 , equations (22) and (23) immediately
ascertain that, in state e = (e1 , e2 ), V1 and V2 are unique.
Let H1 (p1 , p2 ) and H2 (p1 , p2 ) denote the RHS of equation (24) and (25), respectively.
Proceeding as in step 1, set H(p1 ) = p1 − p\1 (p\2 (p1 )), where p\1 (p2 ) and p\2 (p1 ) are defined
41
by H1 (p\1 (p2 ), p2 ) = 0 and H2 (p1 , p\2 (p1 )) = 0, respectively. We have to show that H(·) is
strictly monotone. Straightforward differentiation shows that
³
´
1
1 (p1 ,p2 )
− ∂H
−D
∂p2
D2 (p1 ,p2 )
=
= D1 (p1 , p2 ) ∈ (0, 1),
∂H1
− D2 (p11 ,p2 )
∂p1
³
´
2
2 (p1 ,p2 )
− ∂H
−D
∂p1
D1 (p1 ,p2 )
= D2 (p1 , p2 ) ∈ (0, 1).
=
∂H2
− D1 (p11 ,p2 )
∂p
2
It follows that H 0 (p1 ) > 0.
Case (ii): Next suppose δ = 1. A similar induction argument as in the case of δ = 0
can be used to establish the claim except that in the case of δ = 1 we anchor the argument
in state (1, 1) rather than state (M, M ).
Proof of Proposition 5. Part (i): Consider the static Nash equilibrium. The FOCs in
state e are
p†1 (e) = c(e1 ) +
p†2 (e) = c(e2 ) +
σ
1−
D1 (p†1 (e), p†2 (e))
1−
D2 (p†1 (e), p†2 (e))
σ
,
(34)
.
(35)
Equations (34) and (35) imply p†1 (e) > c(e1 ) and p†2 (e) > c(e2 ) and thus in particular
p† (m, m) > c(m). In addition, p† (e) = p† (m, m) because c(e1 ) = c(e2 ) = c(m) for all
e ∈ {m, . . . , M }2 .
Turning to our dynamic stochastic game, suppose that δ = 0. The proof of part (i)
proceeds in a number of steps, similar to the proof of Proposition 3. In step 1, we establish
that equilibrium prices in state (M, M ) coincide with the static Nash equilibrium. In step
2a, we assume that the equilibrium in state (e1 +1, M ), where e1 ∈ {m, . . . , M −1}, coincides
with the equilibrium in state (M, M ) and show that this implies that the equilibrium in
state (e1 , M ) does the same. In step 2b, we assume that the equilibrium in state (M, e2 +1),
where e2 ∈ {m, . . . , M − 1}, coincides with the equilibrium in state (M, M ) and show that
this implies that the equilibrium in state (M, e2 ) does the same. In step 3, we assume
that the equilibrium in states (e1 + 1, e2 ) and (e1 , e2 + 1), where e1 ∈ {m, . . . , M − 1} and
e2 ∈ {m, . . . , M − 1}, coincides with the equilibrium in state (M, M ) and show that this
implies that the equilibrium in state (e1 , e2 ) does the same. Also similar to the proof of
Proposition 3, we continue to use Vn as shorthand for Vn (e), V nk for V nk (e), pn for pn (e),
etc.
Step 1: Consider state e = (M, M ). From the proof of Proposition 3, equilibrium prices
are determined by the system of equations (28) and (29). Since equations (28) and (29)
are equivalent to equations (34) and (35), equilibrium prices are p1 = p†1 and p2 = p†2 .
42
Substituting equation (28) into (26) and equation (29) into (27) yields equilibrium values
σD1 (p1 , p2 )
,
(1 − β)D2 (p1 , p2 )
σD2 (p1 , p2 )
.
V2 =
(1 − β)D1 (p1 , p2 )
V1 =
(36)
(37)
Step 2a: Consider state e = (e1 , M ), where e1 ∈ {m, . . . , M − 1}. Equilibrium prices
are determined by the system of equations (32) and (33). Given V 11 = V1 (e1 + 1, M ) =
V1 (M, M ) and V 21 = V2 (e1 + 1, M ) = V2 (M, M ), it is easy to see that, in state e = (e1 , M ),
p1 = p1 (M, M ) and p2 = p2 (M, M ) are a solution. Substituting equation (32) into (30)
and equation (33) into (31) yields equilibrium values V1 = V1 (M, M ) and V2 = V2 (M, M )
as given by equations (36) and (37).
Step 2b: Consider state e = (M, e2 ), where e2 ∈ {m, . . . , M − 1}. The argument is
completely symmetric to the argument in step 2a and therefore omitted.
Step 3: Consider state e = (e1 , e2 ), where e1 ∈ {m, . . . , M −1} and e2 ∈ {m, . . . , M −1}.
Equilibrium prices are determined by the system of equations (24) and (25). Given V 11 =
V1 (e1 + 1, e2 ) = V1 (M, M ), V 12 = V1 (e1 , e2 + 1) = V1 (M, M ), V 21 = V2 (e1 + 1, e2 ) =
V2 (M, M ), and V 22 = V2 (e1 , e2 + 1) = V2 (M, M ), it is easy to see that, in state e = (e1 , e2 ),
p1 = p1 (M, M ) and p2 = p2 (M, M ) are a solution. Substituting equation (24) into (22)
and equation (25) into (23) yields equilibrium values V1 = V1 (M, M ) and V2 = V2 (M, M )
as given by equations (36) and (37).
Part (ii): We show that p2 (e) > c(m) for all e1 ∈ {1, . . . , m − 1} and e2 ∈ {m, . . . , M }.
The claim follows because p∗ (e) = p2 (e[2] ).
The proof of part (ii) proceeds in two steps. In step 1, we establish that the equilibrium
price of firm 2 in state (e1 , M ), where e1 ∈ {1, . . . , m − 1}, exceeds c(m). In step 2, we
extend the argument to states in which firm 2 has not yet reached the bottom of its learning
curve. We proceed by induction: Assuming that the equilibrium in state (e1 , e2 + 1), where
e1 ∈ {1, . . . , m−1} and e2 ∈ {m, . . . , M −1}, coincides with the equilibrium in state (e1 , M ),
we show that the equilibrium in state (e1 , e2 ) does the same.
Step 1: Consider state e = (e1 , M ), where e1 ∈ {1, . . . , m − 1}. From the proof of
Proposition 3, equilibrium prices are determined by the system of equations (32) and (33).
In equilibrium we must have Vn (e) ≥ 0 for all e ∈ {1, . . . , M }2 because a firm can always
set price equal to cost. Hence, V 21 ≥ 0 and equation (33) implies p2 > c(m).
For reference in step 2 note that substituting equation (32) into (30) and equation (33)
into (31) yields equilibrium values
σD1 (p1 , p2 )
,
(1 − β)D2 (p1 , p2 )
σD2 (p1 , p2 ) + βD1 (p1 , p2 )V 21
.
V2 =
D1 (p1 , p2 )
V1 =
(38)
(39)
Step 2: Consider state e = (e1 , e2 ), where e1 ∈ {1, . . . , m − 1} and e2 ∈ {m, . . . , M − 1}.
Equilibrium prices are determined by the system of equations (24) and (25). Assuming
V 12 = V1 (e1 , e2 +1) = V1 (e1 , M ) and V 22 = V2 (e1 , e2 +1) = V2 (e1 , M ) as given by equations
43
(38) and (39) in step 1, equations (24) and (25) collapse to equations (32) and (33). Hence,
in state e = (e1 , e2 ), p1 = p1 (e1 , M ) and p2 = p2 (e1 , M ) are a solution. Further substituting
equation (24) into (22) and equation (25) into (23) yields equilibrium values V1 = V1 (e1 , M )
and V2 = V2 (e1 , M ) as given by equations (38) and (39).
Proof of Proposition 9. We rewrite the FOCs in state e as
¡
¡
¢¢
σ
− p1 − c(e1 ) + β V 11 − V 12 ,
D2 (p1 , p2 )
¡
¡
¢¢
σ
0=
− p2 − c(e2 ) + β V 22 − V 21 ,
D1 (p1 , p2 )
0=
(40)
(41)
where, to simplify the notation, V nk is shorthand for V nk (e), pn for pn (e), etc. and we use
the fact that D1 (p1 , p2 ) + D2 (p1 , p2 ) = 1. The system of equations (40) and (41) determines
equilibrium prices. We have to establish that there is a unique solution for p1 and p2
irrespective of V 11 , V 12 , V 21 , and V 22 .
Let H1 (p1 , p2 ) and H2 (p1 , p2 ) denote the RHS of equation (40) and (41), respectively.
Proceeding as in step 3 of the proof of Proposition 3, set H(p1 ) = p1 − p\1 (p\2 (p1 )), where
p\1 (p2 ) and p\2 (p1 ) are defined by H1 (p\1 (p2 ), p2 ) = 0 and H2 (p1 , p\2 (p1 )) = 0, respectively.
We have to show that H(·) is strictly monotone. Straightforward differentiation shows that
³
´
1
1 (p1 ,p2 )
− ∂H
−D
∂p2
D2 (p1 ,p2 )
=
= D1 (p1 , p2 ) ∈ (0, 1),
1
∂H1
−
D
(p
,p
)
∂p
2 1 2
³ 1 ´
∂H2
D2 (p1 ,p2 )
− ∂p1
− D1 (p1 ,p2 )
=
= D2 (p1 , p2 ) ∈ (0, 1).
∂H2
− D1 (p11 ,p2 )
∂p
2
It follows that H 0 (p1 ) > 0.
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48
0.95
0.9
0.85
D
0.8
C
x
0.75
0.7
0.65
B
0.6
0.55
A
0.5
0.45
0
0.1
0.2
0.3
0.4
0.5
δ
0.6
Figure 1: Homotopy example.
0.7
0.8
0.9
1
0.1
1
3
5
7
9
0.2
0.3
ρ
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.01
0.03
0.05
0.1
δ
0.2
Figure 2: Number of equilibria.
1
ρ = 0.85
1
0.7
0.7
H∞, H∧
H∞, H∧
ρ = 0.95
1
0.55
0.51
0.55
0.51
0.5
0.5
0
0.03
0.1
δ
0.3
1
0
0.03
1
0.7
0.7
0.55
0.51
0.3
1
0.3
1
0.3
1
0.55
0.5
0
0.03
0.1
δ
0.3
1
0
0.03
ρ = 0.55
0.1
δ
ρ = 0.35
1
1
0.7
0.7
H∞, H∧
H∞, H∧
1
0.51
0.5
0.55
0.51
0.55
0.51
0.5
0.5
0
0.03
0.1
δ
0.3
1
0
0.03
ρ = 0.15
0.1
δ
ρ = 0.05
1
1
0.7
0.7
H∞, H∧
H∞, H∧
0.3
ρ = 0.65
1
H∞, H∧
H∞, H∧
ρ = 0.75
0.1
δ
0.55
0.51
0.55
0.51
0.5
0.5
0
0.03
0.1
δ
0.3
1
0
0.03
0.1
δ
Figure 3: Limiting expected Herfindahl index H ∞ (solid line) and maximum expected
Herfindahl index H ∧ (dashed line).
12
10
8
6
4
2
0
1
Flat Eqbm. with Well (ρ=0.85, δ=0.0275)
p*(e1,e2)
*
p (e1,e2)
Flat Eqbm. without Well (ρ=0.85, δ=0)
5
10
15
20
e2
25
30
30
25
20
15
10
5
1
12
10
8
6
4
2
0
1
10
15
20
*
p (e1,e2)
e2
25
30
30
25
20
15
e1
15
20
e2
10
25
30
30
25
20
15
10
5
1
e1
Extra−trenchy Eqbm. (ρ=0.85, δ=0.08)
p*(e1,e2)
5
10
e1
Trenchy Eqbm. (ρ=0.85, δ=0.0275)
12
10
8
6
4
2
0
1
5
5
1
12
10
8
6
4
2
0
1
5
10
15
20
e2
25
30
30
25
20
15
10
5
1
e1
Figure 4: Policy function p∗ (e1 , e2 ). Marginal cost c(e1 ) (solid line in e2 = 30-plane).
Flat Eqbm. without Well (ρ=0.85, δ=0)
Flat Eqbm. with Well (ρ=0.85, δ=0.0275)
0.1
µ (e1,e2)
0.1
1
2
µ8(e ,e )
0.15
0.05
8
0.05
0
30
25
20
15
10
5
1
e2
1
5
10 15
e1
20
25
0
30
25
20
15
10
30
e2
5
10 15
e1
20
25
30
µ (e1,e2)
0.15
2
0.05
8
1
1
Extra−trenchy Eqbm. (ρ=0.85, δ=0.08)
Trenchy Eqbm. (ρ=0.85, δ=0.0275)
0.1
µ8(e ,e )
5
1
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
20
25
30
0.1
0.05
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
20
25
30
Figure 5: Transient distribution over states in period 8 given initial state (1, 1).
Flat Eqbm. with Well (ρ=0.85, δ=0.0275)
0.02
0.06
0.015
µ (e1,e2)
0.08
0.04
0.01
32
1
2
µ32(e ,e )
Flat Eqbm. without Well (ρ=0.85, δ=0)
0.02
0
30
25
20
15
10
5
1
e2
1
5
10 15
e1
20
25
0.005
0
30
25
20
15
10
30
e2
Trenchy Eqbm. (ρ=0.85, δ=0.0275)
5
1
1
5
10 15
e1
20
25
30
Extra−trenchy Eqbm. (ρ=0.85, δ=0.08)
0.04
0.04
µ (e1,e2)
0.02
32
1
2
µ32(e ,e )
0.03
0.01
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
20
25
30
0.02
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
20
25
30
Figure 6: Transient distribution over states in period 32 given initial state (1, 1).
Flat Eqbm. without Well (ρ=0.85, δ=0)
Flat Eqbm. with Well (ρ=0.85, δ=0.0275)
1
0.02
2
µ∞(e ,e )
0.5
0.01
1
µ∞(e1,e2)
0.015
0
30
25
20
15
10
5
1
e2
1
5
10 15
e1
20
25
0.005
0
30
25
20
15
10
30
e2
1
5
10 15
e1
20
25
30
Extra−trenchy Eqbm. (ρ=0.85, δ=0.08)
0.02
0.04
0.015
0.03
µ (e1,e2)
0.01
0.02
∞
µ∞(e1,e2)
Trenchy Eqbm. (ρ=0.85, δ=0.0275)
5
1
0.005
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
20
25
30
0.01
0
30
25
20
15
10
e
2
5
1
1
5
10 15
e1
Figure 7: Limiting distribution over states.
20
25
30
0.1
0.2
0.000
0.333
0.667
1.000
0.3
ρ
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.01
0.03
0.05
0.1
δ
0.2
1
Figure 8: Share of equilibria with p∗ (e) > p† (e) for some e ∈ {1, . . . , M }2 .
0.000
0.143
0.200
0.286
0.333
0.400
0.429
0.571
0.600
0.667
0.714
0.800
1.000
0.1
0.2
0.3
ρ
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.01
0.03
0.05
0.1
δ
0.2
1
Figure 9: Share of equilibria with p∗ (e) ≤ c (m) for some e1 ∈ {m, . . . , M } and e2 ∈
{1, . . . , M }.
0.1
0.2
0.000
0.333
0.667
0.800
1.000
0.3
ρ
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.01
0.03
0.05
0.1
δ
0.2
1
0.2
1
0.1
0.2
0.3
0.000
0.333
0.667
0.800
0.857
1.000
ρ
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.01
0.03
0.05
0.1
δ
Figure 10: Share of equilibria violating IID (upper panel) and share of equilibria violating
ID (lower panel).
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