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Open-loop and feedback models of dynamic oligopoly
International
Journal
of Industrial
Organization
11 (1993) 369-389.
Open-loop and feedback dynamic oligopoly
North-Holland
models
of
Larry S. Karp and Jeffrey M. Perloff* Department of Agricultural and Resource Economics, University of California, Berkeley, CA 94720, USA Final version
received August
1992
A simple method of estimating the degree of oligopoly power in dynamic markets using either open-loop or feedback (subgame perfect) strategies is developed. New theoretical results are presented for families of oligopoly models. Differences between discrete-time and continuoustime models and open-loop and feedback models are demonstrated. The effects of increasing the number of firms or adjustment costs are examined.
1. Introduction
We develop a feedback, oligopolistic dynamic game model that can be used to estimate the degree of market power in markets with nonlinear adjustment costs in output, investment, or prices. The model nests various well-known market structures in a larger family. Although the primary use for this model is in empirical work, it also provides a simple method for comparing open-loop and (subgame perfect) feed-back equilibria for a given degree of market power. This model can be used to analyze relatively easily the effects of increasing the number of firms and the costs of adjustment on the equilibrium trajectory and steady state. Dynamic models are appropriate where there are substantial adjustment costs in prices, training, or in capital accumulation, or where there is learning over time. For example, these adjustment factors are important in the rice and coffee export markets according to the parameter estimates and statistical tests of both open-loop and feedback models in Karp and Perloff (1989, 1993). By describing the theoretical characteristics of the model, we provide a basis for interpreting our existing empirical results and for assessing the suitability of the model for other empirical applications. Our previous papers discuss the mechanics of empirical implementation of the model. Correspondence to: J.M. Perloff, Department of Agricultural University of California, Berkeley, CA, USA. *We thank Drew Fudenberg, Stanley Reynolds, and participants of California at Berkeley, BAGEL, and the University of Michigan 0167-7187/93/$06.00
0
1993-Elsevier
Science
Publishers
and
Resource
Economics,
at seminars at the University for useful suggestions.
B.V. All rights
reserved
370
L.S. Karp and J.M. Perloff; Open-loop and feedback models of dynamic oligopoly
For reasons of tractability, many theoretical and empirical dynamic oligopoly models assume that firms use open-loop strategies that are not subgame perfect. Recently there have been a number of papers that estimate subgame percent equilibria.’ By comparing the paths and the steady-state equilibria of families of open-loop and subgame perfect, feedback dynamic oligopoly models, we show where the open-loop models are the same as feedback models and where they deviate substantially.’ The families of open-loop and feedback oligopoly models are nested by using an index of oligopoly power that meets two criteria. First, the index includes three leading market structures: perfect competition, the noncooperative Nash-Cournot equilibria obtained from a limiting finite horizon game, and the cartel solution when firms are identical. Second, the index can be estimated easily. The index is designed to describe rather than explain the market outcome. A given markup may be consistent with a number of different situations. For example, a non-cooperative equilibrium may be identical to the perfect competition equilibrium (as in a price-setting game with homogeneous products) or the cartel solution, or it may lie between those two extremes. The multiplicity of subgame perfect equilibria in supergames (a set of results described as folk theorems) is well known. Similarly, subgame perfect equilibria for dynamic games (also known as state-space games) are typically not unique. One reason for the lack of uniqueness stems from the use of punishment or trigger strategies. Dynamic games with trigger strategies have been constructed by, inter alia, Cave (1987) and Haurie and Pohjohla (1987). Even if attention is restricted to differentiable Markov strategies, the equilibrium is typically not unique in an infinite horizon differential game. Tsutsui and Mino (1990) provide an example in a game with a single state variable where the non-uniqueness is due to an incomplete transversality condition. Because even with differentiability and the strong refinement of Markov perfection, there is no guarantee of uniqueness, there is no way to identify the underlying game by observing market outcomes. A practical alternative is to develop an index, such as the one presented here, to describe dynamic equilibria. Our oligopoly index is ordinal rather than cardinal. There is no unambiguous meaning to statements such as ‘a dynamic equilibrium lies halfway between monopoly and perfect competition’. Cardinal measures, however, ‘Empirical studies based on repeated game models with trigger strategies, estimated using switching regressions, include: Porter (1983), Lee and Porter (1984), and Hajivassiliou (1989). Baker (1989) tests whether the oligopoly markup in a repeated game falls after an unexpected decline in demand. Slade (1987) finds evidence that small deviations from cooperation invite small levels of retaliation, rather than the use of trigger strategies. Roberts and Samuelson (1988) estimate an approximation to a subgame perfect equilibrium in a state-space game. 2Fudenberg and Levine (1988) give other types of conditions under which the open-loop and feedback equilibria are approximately the same with many players. See also Fershtman (1987).
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
371
can be inferred using our index and specific information from the problem. For example, the index and the exogenous parameters determine a steadystate quantity, price, and consumer surplus. These levels can be expressed as percentages of the perfect competition, Nash-Cournot, or cartel equilibrium levels. In this way the index can be used to determine whether an observed equilibrium is close to or far from one of the well-known equilibria. We show that, for a given market structure, the feedback model leads to more competitive behavior than the open-loop model. Using other models, Fudenberg and Tirole (1986) point out that the greater competitiveness in feedback models stems from a preemptive incentive that is absent in openloop equilibria. Of course, with different types of models, feedback equilibria may be less competitive [Fudenberg and Tirole (1984, 1986)]. In our model agents choose quantities, and there is a strategic incentive to increase these in the current period. The linear-quadratic structure has frequently been used in oligopoly models. For example, Fershtman and Kamien (1987) and Reynolds (1987) use dynamic linear-quadratic models to compare the steady states for a Nash-Cournot model for a fixed number of firms.3 We extend their work in three ways. We study a family of oligopoly models, analyze both the adjustment paths and the steady states, and examine the effects of an increase in the number of firms. In section 2 we describe our model and provide definitions. In the next section, the index of market structure is explained, and the difference between open-loop and feedback equilibria is discussed. In section 4 the chief features of the resulting equilibria are described. Comparative dynamics are discussed in section 5. Section 6 contains comments about empirical implementation. Our conclusions are in section 7. 2. Definitions
and the model
We start with a discrete-time model in which the length of a period is E. As ~40, the continuous-time model is obtained. Most of our analysis is based on the continuous-time model. The industry consists of n+ 1 firms where nz 1. At time t, firm i decides how much to produce in the current period, qit, or, equivalently, by how much to change its output, UitE= 4it - qi,t-E, where Uit is a rate. Firm i incurs a quadratic cost of adjustment,
3Hansen et al. (1985) also use the dynamic linear-quadratic model to study various open-loop models as well as the open-loop and feedback Stackelberg model. Van der Ploeg (1987) compares the open-loop and feedback steady states in a natural resources setting with general functional forms.
372
L.S. Karp and J.M. Perloff, Open-loop andfeedback
models of dynamic oligopoly
and a quadratic cost of production,
(e,,+eQ> 2
q.E
If’
In period t, the linear demand curve facing firm i is ?I+1
Pit=Ui-b
1 qjt.
(1)
j=l
Firm i’s revenue in period t is pitqit&.Given an instantaneous interest rate of discount rate is eerE, and the objective of firm i is to maximize its discounted stream of profits: r, the one-period
For expositional simplicity, we set ui= a for all i and assume Oi =0 and Oi=O. The last equality implies that adjustment costs are minimized when there is no adjustment. As a result, under the assumption of open-loop behavior, the steady-state levels of output for the three leading market structures (price-taking, non-cooperative Nash-Cournot and cartel with identical firms) are equal to their static analogues. This equality holds for general cost and revenue functions and not simply the quadratic ones assumed here. The ith firm’s objective (2) is written in a matrix form as
fl e-ri’-lle[ Ue:(q,_,+U,E)-_(qt-,+ute)‘Ki(q,-,+ut&)-_fu;Siut
c,(2a)
1
where ei is the ith unit (column) vector, e is a (column) vector of l’s, Si = eiei6, and Ki= b(eei+ eie’) + f3eie$ That is, the (i, i) element of Ki is 2b + 8 and the other elements of the ith row and column equal b. As s-0, this expression approaches m
I[
e If aeiq, - iq;Kiqr - &Siut
0
1
dt.
CW
We assume that qir is unconstrained. If price is negative, firms would prefer to buy rather than sell; a negative value of q represents purchases. 3. Two families of equilibria We consider two families of equilibria: open loop and feedback, which assume different information sets. Agents’ information set under the open-
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
313
loop structure consists of calendar time and the value of the state vector at the beginning of the game. In the feedback structure, the information set consists of calendar time and the value of the current state vector.4 In our model, the state vector at t is the lagged value qf-E. Strategies map the information set in actions in each period. An open-loop (feedback) equilibrium is a Nash equilibrium in open-loop (feedback) strategies. In a feedback equilibrium, players choose state-contingent policy rules, and the resulting equilibrium is subgame perfect. In each period of this game, the firms take the current state (lagged output) and the mechanism for determining future behavior as given. There are many such equilibria, but we restrict attention to linear decision rules. Linear rules are a natural choice because they can be derived as the limit of finite horizon models as the horizon goes to infinity. In an open-loop equilibrium, each firm chooses a sequence of controls. In our model, the changes are in its own output. The controls are a function of time and the initial state. The feedback equilibrium is given by a set of decision rules - not by a set of trajectories, as in the open-loop model. There are two ways to compare the two equilibria. First, we can compare the trajectory under the equilibrium feedback rules to the open-loop equilibrium. Alternatively, we can directly compare the open-loop controls written in ‘feedback form’ (that is, as a function of the current state) to the equilibrium feedback rules. The second alternative is more appealing for both theoretical and empirical reasons. Thus, we use the feedback form of the open-loop equilibrium to compare the equilibria under the two information structures outside the steady state. Our procedure involves the estimation of decision rules. The form of these decision rules is the same for both information structures. The inferences made from the estimated decision rules regarding the index of market power depends on the assumed information structure. In both the open-loop and feedback models, the rules depend on the lagged state. This similarity is significant because it means that, in estimating the open-loop equilibrium, it is not necessary to determine the initial condition when the trajectory was chosen, which is not feasible. An economic interpretation of the feedback form of the open-loop equilibrium is that firms respond to surprises that are caused by additive random shocks from nature. This equilibrium is not subgame perfect5 It is 4More generally, the information set can include the entire history of play, as with the closedloop information structure. When actions can be conditioned on history, trigger strategies of the type described by the Folk Theorems can be constructed. sThe open-loop equilibrium is a Nash equilibrium in trajectories. The feedback form of these trajectories is not a Nash equilibrium in decision rules. That is, if firm i takes the feedback form of firm j’s open-loop trajectory as given, the feedback form of firm i’s open-loop trajectory is not a best response.
374
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
appropriate to compare this version of the open-loop model to the feedback model because the two equilibria differ only in the assumptions one makes about firms’ rationality (and not about firms’ ability to respond to new conditions ex post). 3.1. Static analog In most empirical static models of oligopoly, aggregate or firm-level data are used to estimate a parameter, v, that reflects the markup of price over marginal cost. In this approach, given demand equation (1) for a homogeneous product, firm i’s effective marginal revenue curve (the marginal revenue given the degree of market power actually exercised) is MRi(Vi)=p+(l
+nVi)p’4i=p,--(1-nvj)bqi.
For simplicity, suppose there is a firms, MC= MC,= 13, which could variables such as weather, and that estimates the demand curve, and
common constant marginal cost for all be a function of various exogenous Vi= v for all firms. To determine v, one equilibrium equations for each firm,
MR,(v) = MC = 8:
Dividing the coefficient on the qi term by the estimate of b from the demand curve, subtracting one and dividing by n gives an estimate of v. The markup between price and marginal cost, p- MC=(l +nv)bq, is proportional to 1+ nv. For example, if v= - l/n, there is no markup (MR =p= MC); whereas, if v= 1, the monopoly markup is observed: MR=p+p’((n+l)q,)=p+p’Q=MC,
where Q=(n+ 1)q is aggregate output. Intermediate solutions, such as the Nash-Cournot (v = 0), are also possible. In the empirical literature on static models, the parameter v often is interpreted as a constant conjectural variation: v-dqj/dq,. We prefer the neutral interpretation that v represent a gap between marginal cost and price _ a measure of market power - rather than a conjectural variation. 3.2. Dynamic index An index v can be similarly defined for the dynamic games. Again, the index can be given the neutral interpretation as a measure of market power, v, or it can be defined as v = auit/aujt for i# j for all t. For simplicity in what
L.S. Karp and J.M. Perlofi Open-loop and feedback models of dynamic oligopoly
375
follows, we consider only the symmetric case where v is independent of i and j. The relation between v and the price-marginal cost markup is not as simple in the dynamic model as in the static model. In the dynamic model, for all market structures, the equilibrium markup depends on the shadow value of the state, which is endogenous. For example, under perfect competition, price does not equal marginal cost, except at the steady state.6 In an open-loop equilibrium each firm maximizes (2) given an initial condition, the index z), and an assumed path for its rivals. The parameter v enters the first-order condition for firm i’s problem - the maximization of firm i’s Hamiltonian with respect to its control - in a manner analogous to the static problem. The solution for the continuous time problem is described in the appendix. The feedback equilibrium is obtained by the simultaneous solution to the n + 1 dynamic programming equations:7 J,(q, -,) = max
aelq,-_q;Kiqt-~ujSiu,
wt
) E+e-”
Ji(q,)] >
(3)
where Ji is the present value of player i’s profits when the lagged output is qf_Er and qt =qt_e+p,~. Because the strategies are linear, Ji is quadratic. The parameter v enters the first-order condition to the maximization of (3) in a manner analogous to the static problem [see eq. (AS) in the appendix]. 4. Steady states and paths We now compare the open-loop and feedback equilibria emphasizing the effect of the parameter v on the equilibria. We define Q: as aggregate output at time t, where the superscript refers to the solution to the open-loop (0) and feedback (f) equilibria. If the initial output is the same for all firms, qO= (eQ,,)/(n + l), so Q: is the solution to
for k= o,f. The parameters yi and pi depend on the parameters in the firm’s objective functions, the number of firms, and the index v. In the feedback ‘In the static model, the index u contains no information that is not also contained in the markup. In the dynamic model, on the other hand, the index u provides a summary statistic for the equilibrium trajectory of the markup. ‘In infinite horizon games, there typically exist many subgame perfect equilibria. We avoid the problem of non-uniqueness by considering the equilibrium strategies that result from the game with finite horizon T and letting T-co.
376
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
equilibrium, yf and pf are obtained by aggregating the firms’ equilibrium decision rules. In the open-loop equilibrium, y” and p0 are obtained by aggregating the ‘feedback form’ of the open-loop equilibrium trajectory. By comparing the two linear differential equations (4), we can compare the open-loop and feedback equilibria both at and outside the steady state. We now make two assumptions for use in comparing the two equilibria. Assumption
1.
Assumption 2.
Stationary feedback rules exist.
The vector q, generated by the equilibrium converges for arbitrary qO.
feedback rules,
The proofs of the following propositions, given in the appendix, are based on comparison of the systems of equations that define yi and pi: Proposition 1. Zf Assumption 1 holds, a sufficient condition for the open-loop and feedback equilibria to be identical is v = 1 (cartel with symmetric firms) or v = - l/n (price takers). Proposition 2. Zf Assumptions 1 and 2 hold and behavior is Nash-Cournot (v=O), output is smaller in the open-loop equilibrium and converges to its steady state more rapidly than in the feedback equilibrium.
Based on simulation results reported below, the condition given in Proposition 1 appears to be necessary. However, because Proposition 1 only establishes sufficiency, we cannot prove that the comparison in Proposition 2 also holds for u #O. Extensive simulation, however, supports the conjecture that the result does hold for v ~0. Because the feedback Nash-Cournot equilibrium is farther from the monopoly solution than is the open-loop Nash-Cournot equilibrium, the feedback solution is relatively procompetitive. Fershtman and Kamien (1987) and Reynolds (1987) show that the Nash-Cournot feedback steady state is greater than the open-loop steady state. Proposition 2 generalizes their results by comparing the entire equilibrium path. The intuition is that capacity discourages rivals’ investment under the feedback assumption. Therefore, firms have a greater incentive to invest today as a means of preempting their rivals’ future investment, so they develop larger capacities and hence larger output levels. This incentive is absent in the open-loop equilibrium. In fig. l(a), the feedback and open-loop steady-state outputs are graphed as a function of v where n= 1 (duopoly), a =250, b = 10, 6 = 5, p =0.95, E= 1, and 8=0 (the ‘base’ parameters). Fig. l(b) shows the difference between the open-loop and feedback steady states is largest when the market is nearly
L.S. Karp and J.M. Perlofi Open-loop and feedback models of dynamic oligopoly
377
6’ -1
-0.6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.6
1
V
0.e
(‘4
0.5
7 i
0.4
e:,
0.3
n
0.2
C
a 0.1 0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.8
0.8
1
V
Fig. 1. (a) Feedback
and
open-loop steady-state output. (b) Difference open-loop steady-state output.
between
feedback
and
competitive at v z -0.7.8 Based on these simulation results, the bias from incorrectly assuming firms use open-loop strategies is less serious when the outcome of these strategies is between the Nash-Cournot and collusive equilibria than when it lies between the Nash-Cournot and competitive equilibria. To illustrate the relative rates of adjustment, we use a Nash-Cournot duopoly model. With the base parameters, the Nash-Cournot steady state is 8.5 in the open-loop model and 9.1 in the feedback model. That is, where 0=0 (marginal production cost, net of adjustment cost, is constant), the feedback steady state is 7% higher than the open-loop steady state. By *The estimated market structure parameters for both the rice export market [Karp and Perloff (1989)] and the coffee export market [Karp and Perloff (1993)] are such that the steadystate differences between the feedback and open-loop outputs are nearly maximized; however, in percentage terms, these the differences in the steady state are small.
378
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly Table 1 Trajectories
for Nash-Cournot
models.
Period
Open-loop 8=0
Feedback 0=0.7917
0 1 2 3 4 5 6
1.000 7.435 8.369 8.476 8.490 8.491 8.492
1.000 7.552 8.374 8.477 8.490 8.491 8.492
Parameters: a= 250, n=l, 6=5, and ~=l.
b = 10,
p=O.95,
choosing different parameter values for the open-loop and feedback models, the steady states can be made equal. For example, if we use the base parameters with 6 =0 in the open-loop model, but change 0 to 0.79 in the feedback model, both steady states equal 8.5. Thus, with a slightly different short-run marginal cost curve, the two models can produce the same steady state. These two models can be distinguished empirically by determining the slope of the marginal cost curve, 8, or observing the adjustment paths. Table 1 shows that, although the open-loop with 8=0 and the feedback model with 8 =0.79 produce the same steady state, adjustment is more rapid in the feedback model. In the first period, output in the feedback model is 1.6% higher than in the open-loop model. Note that this faster rate of adjustment does not contradict Proposition 2 because, in this example, the open-loop and feedback models have different values of 0. To show the trajectories more clearly for the four leading models, we increase 6 to 150 in fig. 2. As shown in fig. 2, both the paths and the steady states vary with v and the type of strategy used. 5. Number of firms and adjustment costs As the number of firms, n+ 1, increases, the equilibrium trajectories change. By setting 9 =0 and normalizing so that 6 =(n + l)c, where c >O is constant, the price-taking and collusive equilibria are invariant to n. As n becomes large, the adjustment cost for each firm becomes infinite so each firm makes only infinitesimal adjustments and captures only an infinitesimal share of the market. Thus: Proposition 3. Given 0=0 and the normalization 6 =(n + l)c, the open-loop and feedback Nash-Cournot equilibria converge to the competitive equilibrium as n-tco.
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
379
14 Price-taker 12 10 0 u *
8
~: Cartel
6
10
20
15
Time Period Fig. 2. Trajectories
for four models.
Table 2 The effect of increasing the number of Nash-Cournot on industry output.
Monopoly
Price-taking Parameters: 6=2.5 (n+ I).
firms
Number of firms (n + 1)
Open-loop
Feedback
1 2 3 4 5 10 15 00
12.74 16.98 19.11 20.38 21.23 23.16 23.88 25.48
12.74 17.48 19.92 21.35 22.21 24.16 24.14 25.48
a = 250,
b = 10,
/S= 0.95,
0=0,
s=l,
and
This proposition can be proved by examining the equations that determine the stationary control rules. The following is a heuristic proof. From Proposition 2, the open-loop and feedback equilibria are identical for u= - l/n, which goes to 0 as n+co. The open-loop model is a static game and, in a static game, the Nash-Cournot equilibrium converges to the competitive equilibrium as n--t cc. The relation between the Nash-Cournot steady-state output and the number of firms, given the normalization 6 =(n + l)c, is shown in table 2 for the base parameters. The Nash-Cournot feedback output increases more than does the Nash-Cournot open-loop output as the number of firms
380
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
increases. Thus, not only is the feedback equilibrium relatively procompetitive compared with the open-loop equilibrium, but the feedback equilibrium is also more sensitive to changes in the number of firms. We have shown that as n-+cc and 6 is adjusted to leave the price-taking equilibrium unchanged, the open-loop and feedback Nash-Cournot steady states converge to the static (competitive) equilibrium. Thus, there are two senses in which the static model is nested in the dynamic model, because the model is also static if 6 =O. This nesting has obvious implications for empirical work. For example, we may want to test joint hypotheses regarding market structure and the importance of dynamics. However, although 6 = 0 yields a static model, it does not necessarily follow that the dynamic equilibrium is close to the static equilibrium for 6 close to 0. Indeed, Reynolds (1987), using a continuous-time, linear-quadratic model of costly adjustment, shows that the steady-state feedback output in the limiting game as 6-O is greater than the steady-state open-loop (and the static) output level. The equilibrium is discontinuous at 6 =O. Reynolds (1991) shows that the steady-state feedback output in the limiting game as 6+cc equals the steady-state open-loop (and the static) output level. Thus, the static equilibrium is nested within the steady state of our dynamic in exactly the ‘opposite sense’ that one might expect. We obtain the static equilibrium for large, but not for small, adjustment costs. What lies behind this result? The open-loop steady state equals the static equilibrium for all values of 6, due to our assumption that adjustment costs are minimized when adjustment is 0 (i.e. at the steady state).g Therefore, it is only necessary to explain why the feedback and open-loop steady states converge as 6+co but do not converge as 6-+0. This explanation turns on the intuition for the difference between the open-loop and feedback equilibria: only in the feedback model is there a preemptive incentive. If 6 is very large, firm i’s rival finds it costly to respond to changes in firm i’s output so firm i’s preemptive incentive is small, and the two steady states are similar. If, on the other hand, 6 is very small, firm i’s rival can easily respond to changes in firm i’s output so the preemptive incentive is greater, and the two steady states are dissimilar. Reynolds’ result may appear to have unfortunate implications for the empirical testing of the model because it suggests that dynamics are important (at least in determining the steady state) where adjustment costs, which are the reason for the dynamics, are unimportant. Because the principal objective of this paper is to provide a foundation for using this model to estimate dynamic market structure, it is important to assess the likely empirical significance of this apparent paradox. ‘Reynolds does not make this assumption, than our interpretation of them.
so his conclusions
are. stated
somewhat
differently
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
381
Table 3 Deviations
of the feedback steady states from static equilibria various values of 6 and E.
for
6 E
0.5
0.01
0.001
0.000001
0.005 0.1 0.5
0.036 0.032 0.019
0.035 0.015 0.001
0.023 0.000037 0.00000015
0.001 0.0000037 0.0000002
Parameters:
a = b = 1, p = 0.95, 0 = 0, E= 1, and TE = 50.
To this end, we use a discrete-time, Nash-Cournot (v=O) duopoly model to simulate the differences between the feedback and open-loop steady-states for various combinations of 6 and E. From Proposition 2, we know that this difference between the feedback and the open-loop steady-state outputs, d(6,s), is positive. Table 3 gives values of d(b,s) for four values of 6 and three values of &.I0 These simulations illustrate two results. First, for the values of E we chose, the value of A is non-monotonic in 6 and can be made arbitrarily close to 0 by choosing 6 sufficiently small. Second, for a small positive difference, A*, the function ~*(A*,E) is decreasing in E, where ~*(A*,E) is the largest value of 6 such that A(6, E)< A* for all 6 < 6*(A*, E). Based on the first simulation result, it appears that the discrete-time feedback model does approach the static model as 6 approaches 0. By making 6 sufficiently small, the equilibrium output level is arbitrarily close to the steady state in a single period (i.e. adjustment is rapid), and the steady state can be made arbitrarily close to the static equilibrium output. This result is reassuring for empirical purposes (where a discrete-time model is the natural one) because it means that dynamics have an important effect on the equilibrium if and only if the source of dynamics is significant. Based on the second simulation result, for the feedback steady state to be within a given distance from the static equilibrium requires a smaller 6, the smaller is E. This result provides a way to reconcile Reynolds’ analytic result and our first simulation result. Reynolds obtained his result using a continuous-time game, whereas our simulations are from a discrete-time model. Although the discrete-time model does not appear to contain a discontinuity at 6 =O, the simulations illustrate that the steady state is more sensitive to 6 the closer we get to continuous time (i.e. as s-0).
“‘In stages obtain a = b=
this comparison, as we diminish the length of each stage (E), we increase the number of (T) so that the horizon (TE) remains constant at 50. This horizon is long enough to stationary feedback rules, which were then used to calculate the steady state. Because 1, the static equilibrium is l/3.
382
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
6. Additional empirical considerations
There is an alternative approach to estimating an index of market power in a dynamic oligopoly model. The open-loop equilibrium can be obtained by solving a control problem, so the index of market power can be obtained by estimating the parameters of that control problem.” There is an extensive literature on estimating such problems, and it is worih recognizing that this literature has potential uses in empirical industrial organization. Given a control problem of the form J(q)
=
max 7 ert ae’q u
$q’Kq
0
s.t. 4 = U,
q.
-
6
2 u’u
1 dt
(5)
given,
where K = k, ee’ + (k, - k,)l, K positive-semidefinite and 6 > 0, we can show that aggregate output in the control problem and in the open-loop equilibrium to the game are the same if and only if k, +nk, =(2+n(v+ l))b+Q. In addition, if k,=(2 + nu)b+ 0 and k, = b, the levels of output for the two problems are the same even if the n+ 1 firms have different initial levels of output.‘* Hansen et al. (1985) used the relation between the open-loop equilibrium and the solution to a control problem to study the Nash-Cournot (v=O) model. If firms are price takers (V= - l/n), the integrand in eq. (5) is social surplus, which illustrates the well-known result that the competitive equilibrium can be obtained by solving the social planner’s problem. Where firms are identical and collusive (u= l), in the control problem that matches the open-loop game, k. = (2 + n)b + 8, kl = b; whereas, in the control problem for the monopolist, k, =2b+8, kl =2b. Therefore, the collusive game gives rise to the monopoly solution only if all firms produce equal quantities. Analogously, in static games, the value u = 1 produces the monopoly solution only in a symmetric equilibrium. We have estimated the value of u under the assumptions of open-loop and feedback behavior. In our theoretical discussion, it is convenient to assume that firms (and their initial levels of q) are identical so that we can aggregate the decision rules into the scalar eq. (5). This assumption is not reasonable for empirical work. Instead, we estimate the decision rules as function of own and rivals’ lagged output, rather than as a function of lagged aggregate output. In addition we relax the assumption of identical firms by allowing the intercepts of the demand, cost of production, and cost of adjustment “To do so requires the open-loop assumption. See Slade (1990) for a general obtaining oligopoly equilibria by solving maximization problems. “These assertions are proven in an earlier version of this paper.
discussion
of
LX Karp and J.M. Perlox Open-loop and feedback models of dynamic oligopoly
383
equations, to be firm specific and time dependent. For tractability, we maintain the assumption that the parameter v and demand and adjustment cost slopes (b and 6) are the same for all firms and stationary. This implies that the intercepts of the firm’s decision rules (either the feedback decision rules or the feedback form of the open-loop equilibria) are firm specific and time dependent, but that the coefficients on own and rivals’ lagged output are stationary and symmetric across firms. By estimating the latter coeffrcients we are able to infer the value of u, under either the open-loop or feedback assumption. 7. Summary and conclusions Our model provides a simple method of estimating the degree of oligopoly power in dynamic markets under the assumption the firms use either openloop or feedback (subgame perfect) strategies. This method, which is analogous to those used with static games, allows the data to indicate the degree of market power without specifying the exact nature of the game being played by the firms. If firms are either collusive or perfectly competitive, the open-loop and feed-back paths and steady states are the same. Based on extensive simulations and estimates for two actual markets, even for intermediate market structures, incorrectly assuming that open-loop strategies are used results in only a small downward bias on the market structure parameter. Given an estimate of the market parameter, the slope of the demand curve, and the cost-of-adjustment parameter, it is straightforward to calculate the equilibrium price, consumer surplus, and industry profit, either at a point or as a discounted future stream. These values can then be compared with the corresponding price-taking and collusion values to obtain a cardinal measure of the closeness of the observed market structure to perfect competition or cartel, as illustrated in Karp and Perloff (1989, 1993) for the international rice and coffee export markets. There are a number of possible empirical and methodological extensions. Allowing the market structure parameter to differ across firms is straightforward in principle, but it is computationally difficult if there are more than two firms. Similarly, the inclusion of additive random terms following an ARMA process can be accommodated by augmenting the state vector, which would permit the estimation of market structure with rational expectations about exogenous terms. Furthermore, in principle, a second state variable for stocks could be added. Appendix: Proofs of Propositions
1 and 2
We begin by proving a lemma in which we show that the parameters
pk
384
L.S. Karp
and J.M. P&off, Open-loop and feedback models of dynamic oligopoly
are the negative root of a quadratic function and yk is a function of pk, k = o, f. We then prove the propositions by comparing the solutions to these equations. As in the text, superscripts are used to distinguish between open loop and feedback equilibria. Subscripts i and j are used to identify firms i and j; all other subscripts are used to identify elements from matrices. Lemma. Under Assumptions 1 and 2, the parameters pk and yk (k= o, f) used in eq. (5) satisfy the restrictions that pk is the negative root of (p”)2-r6p0-6{b(2+n(v+1))+8}=0
(A.la)
(pf)2-r~pf-6{b(2+n(v+1))+8}=n(v).
(A.lb)
and
where _q(v)=^ -(np:(l-v+nv-nv2)+n2p,z(l+nv-v2-nv3)+n3z2(v+v2+ nv2 - 2vL - nv’)) and p1 and z are parameters in a firm’s value function: these parameters are defined in eqs. (A.6a)+A.6c) below. 7he parameters yk are given by (n + 1)a “=(r8-p”)
(A.2a)
and yf =
(n+ 1)(a+ 4(v)) (74-p’)
’
(A.2b)
where 4(v) =
n(p,+nvz)h(l-v)(l+nv) 6
and h is the parameter in firm i’s value function, which equals the marginal effect on firm i’s present discounted value of profits of a change in qj when all firms’ stock levels are 0 (as discussed below in more detail). A.1. Proofs of (AJa)
and (A.2a)
Define ViE R” + 1 as a column vector with 1 in the ith position and v in every other position. Firm i behaves as if au/aui=tTi and its objective is to maximize eq. (2b). The open-loop current value Hamiltonian for firm i, yiq.,is yi”i= aeiq -+q’Kiq -+u’Siu +&u, where ii is the shadow value of the state q for firm i and Ki is defined below eq. (2a) in the text. The necessary conditions for an interior solution are
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
38.5
(A.3a) and
as _ ~ = lqli -
(A.3b)
r&) = I?;(- aei + &q).
&li
(A denotes dJ/dt). The first-order conditions (A.3a) and (A.3b) are obtained as the limiting form as s-0 of the first-order conditions to the discrete openloop problem. We try a solution of the form &=h,+Hiq, where hi is a n+ 1 column vector and Hi is an n + 1 x n+ 1 matrix. For a steady-state control rule, hi=Ai=O for all i. We denote the (endogenous) elements of Hi and hi as follows. For the matrix Hi: the (i, i) element is Hi(i, i) = p,,; Hi(i, j) = pr and Hi(j, i) = fil for all j# i; finally, H,(1, s) =z for all I, s#i. For the vector hi: the ith element is hi(i) = h, and all other elements are h,(j) = fi for j# i. Since we assume that agents are symmetric and we restrict attention to symmetric equilibria, each agents’ value function is a permutation of i’s value function. Substituting the linear expression for li into (A.3a) and using the notation described above, and then rearranging, implies h+nuC
1 (Po+nuj&ji+(pl+nUZ)
ui=F+j
[
C
j#i
qj
1
(A.3c)
.
Next, we stack the n+ 1 necessary conditions of eq. (A.3c) to obtain ZJ=(h* + N*q)/6, where h* =(/I + nvh) e and H* is a matrix with elements: H*(i, i) =po + nujl, H*(Z,s) =pl +nuz, for all Z#s (i.e. for all off-diagonal elements). By substituting this solution into eq. (A.3b) together with fii=O, I;i=O, we obtain the systems
and r&$hi = iT:(a6ei+ Hih*).
These two systems give three independent functions, p. + nup,, p1 + nvz, and h + nvh:
equations for the three unknown
-r6(po+nv~,)=6((2+nu)b+8)-(po+nu~,)2-n(p, -~6(pI+nuz)=6b-2(po+n~,)(p,+nuz)-(n-1)(p,+nuz)2, and
+nz)‘,
(A.4a) (A.4b)
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
(h+nv&)=
a6
(A.4c)
(r6-jo-nvjl--@-n2vz)’
We define p” = p0 + nvj, + np, + n2vz. Using this definition, the equilibrium control rule (A.~c), and the assumption of symmetry (so that qi=qj), implies that p”/6 is the coefficient on aggregate output Q in eq. (4) of the text. To obtain the equation that defines p”, multiply both sides of eq. (A.4b) by n and add it to eq. (A.4a). This equation together with the definition of p” gives us eq. (A.la). We define y”=(n+ 1) (h+nvt%); multiply both sides of (A.4c) by n+ 1 and use (A.3c) to obtain eq. (A.2a). A.2. Proofs of(A.lb)
and (A.2b)
In order to avoid a proliferation of notation, and also to emphasize the similarity between the open-loop and feedback equilibria, we use many of the same symbols used above. For example, Hi and hi will continue to represent a matrix and vector which determine (together with the state vector 4) firm i’s vector of shadow values; similarly, pO, pl, h and h^are elements of that matrix and vector. Since the open-loop and feedback equilibria differ except in special cases, the value of those elements will differ in the two equilibria. The stationary dynamic programming equation for firm i is r(yi+hiq+$q’Hiq)=max
ui
aeiq-sq’K,q--~u~+(hi+Hiq)‘u
1.
The parentheses on the left-hand side gives Ji(q), which is firm i’s value function; the parameter yi and the elements of hi and Hi are endogenous. We know that the value function is quadratic for a finite horizon game. Because we are interested in the limiting form of that game as the horizon approaches infinity, we restrict attention to quadratic functions for the infinite horizon game. The necessary condition for an interior maximum is
(A.9 Substituting eq. (A.5) into the dynamic programming coefficients, we obtain the system 1 (1;;
0=rbHi+6Ki+Hii$t7:Hi-Hi
O=r6hi+6aei-Hi&i$hi+
1 (;I:
equation and equating
eJ.H. 1 ’ J J, (1;:
H3.e’. JJJ)’
h.+H.
I(;::
c
H.5.e:
J J J)
e.fi’.h. JJJ)’
Hi,
L.S. Karp and J.M. Perlo& Open-loop and feedback models of dynamic oligopoly
387
We now try a solution of the form Hi(i, i)=po, Hi(i, j)=Hi(j, i)=p, for and Hi(s, 1)=z for s#i, l#i; hi(i) =h, hi(j) =h for j#i. By substituting this trial solution into the system above, we obtain
j#i,
O=r6p, +6(2b + e) -pi -(2n-(nv)Q:
-221t%!p1z,
(A.6a)
O=r6p,+6b-nvp:-(n-l)p:-22p,p,
+((nu)2pl-(no= r6z-p:
-2zp,
l)nvp, -np,
-n2zu)z,
- 2(n- l)zp, - 2nvz(p, +(n-
(A.6b) 1)z) +(nuz)2,
0= -r6h+6a+(h+nvh3np,(l_v)+(p,+np,v)h+1z(p,+nzv)t;
(A.6c) (A.7a)
and o= -r6~+(h+nuL)nz(l-u)+(p,+nplu)~ +(p,+nzu)(h+(n-l)Q.
(A.7b)
In deriving system (A.6) we use Assumption 1, which implies that the system (PO,pi, i)’ is stable. [To obtain system (A.6), we set the vector of time derivatives (pO,bl,i)’ equal to 01. Assumption 1 is not needed for the open-loop equilibrium because that equilibrium can be obtained by solving a control problem for which standard results guarantee stability of the Ricatti system that leads to eq. (A.4). Similar results for the feedback game have, to our knowledge, not been obtained, so Assumption 1 is required. We define pf = p0 + n(1 + u)pl + n2uz. By multiplying eqs. (A.6b) and (A.6c) by $1 +v) and n2u, respectively, and adding them to eq. (A.6a), we obtain eq. (a.lb). We define yf =(n + 1) (h+nu@. By multiplying eq. (A.7b) by nu and adding it to eq. (A.7a), we obtain eq. (A.2b). 0 A.3. Proof of Proposition I Proposition 1. If Assumption 1 holds, a s@cient condition for the open-loop and feedback equilibria to be identical is v= 1 (cartel with symmetric firms) or v= - l/n (price takers). Proof. This proposition follows immediately from comparison of eqs. (A.la) and (A.lb), and of eqs. (A.2a) and (A.2b), and because q(u)=~#~(v)=O for u=-l/n or v=l. 0 A.4. Proof of Proposition 2 Proposition 2.
If Assumptions 1 and 2 hold and behavior is Nash-Cournot
388
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
(u=O), output is smaller in the open-loop equilibrium and converges steady state more rapidly than in the feedback equilibrium. Proof.
to its
We now show that r(0) < 0 < NO).
(A4
Given this inequality, eqs. (A.la), (A.lb), (A.2a), and (A.2b), and the definition of p” and pf, Proposition 2 follows immediately. We first establish that pl(pl + nz) >O so that ~(0) ~0. By comparing eqs. (A.la) and (A.lb), we find that @‘O implies that an increase in the initial level of qj would cause firm i to want to begin with a higher level of sales. However, for large enough qj, price is negative and firm i would prefer to begin with a lower level of qi; hence, p1 O. Because pl, p,, ~0, p0 0. Therefore, p1 + nz < 0 for all n, and ~(0) ~0 as stated. To complete the proof, rewrite eq. (A.7) for u = 0 as
rd-_p0--nk [ -h +nz)
rb--pO-I;Pnl-l)p,l[~]~[~]~
Using our previous results, all elements of this matrix are positive so h and k must have the opposite sign. To establish the second inequality in eq. (A@, we need to show only that 6~0. Using eqs. (A.7a) and (A.7b) and the results of the previous paragraph, h and 6 must have the opposite sign. Because h= 8Ji(0)/8qi, h = Mi(0)/aqj, j# i, if h t0 c 6 then, if all firms begin with 0 sales so that initial price is positive, firm i would prefer to begin with negative sales and have its rival(s) begin with positive sales. Because this result must be false, we conclude h > 0 > E, so C/J(O) > 0, which completes the proof of Proposition 2. 0 References Baker, J.B., 1989, Identifying cartel policing under uncertainty: Journal of Law and Economics 32, part 2, S41Sl6.
The US steel industry
1933-1939,
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Cave, J., 1987, Long term competition in a dynamic game: The cold fish war, Rand Journal of Economics 18, 596-610. Fershtman, C., 1987, Identification of a class of differential games for which the open loop is a degenerate feedback Nash equilibrium, Journal of Optimization Theory and Applications 55, 217-231. Fershtman, C. and M.R. Kamien, 1987, Dynamic duopolistic competition with sticky prices, Econometrica 55, 1151-1164. Fudenberg, D. and D. Levine, 1988, Open-loop and closed-loop equilibria in dynamic games with many players, Journal of Economic Theory 44, 1-18. Fudenberg, D. and J. Tirole, 1984, The fat-cat effect, the puppy-dog ploy, and the lean and hungry look, American Economic Review 72,361-366. Fudenberg, D. and J. Tirole, 1986, Dynamic models of oligopoly (Harwood Academic Publishers, London). Hajivassiliou, V.A., 1989, Measurement errors in switching regressions models: With applications to price-fixing behavior, Cowles Foundation for Research in Economics Working Paper. Hansen, L.P., D. Epple and W. Roberds, 1985, Linear-quadratic duopoly models of resource depletion, in: T.J. Sargent, ed., Energy foresight and strategy (Resources for the Future, Washington, DC). Haurie, A. and M. Pohjohla, 1987, Efficient equilibria in a differential game of capitalism, Journal of Economic Dynamics and Control 11, 65-78. Karp, L.S. and J.M. Perloff, 1989, Dynamic oligopoly in the rice export market, Review of Economics and Statistics 71, 462470. Karp, L.S. and J.M. Perloff, 1993, A dynamic model of oligopoly in the coffee export market, American Journal of Agricultural Economics 75,448457. Lee, L.-F. and R.H. Porter, 1984, Switching regression models with imperfect sample separation information - with an application on cartel stability, Econometrica 52, 391-418. Porter, R.H., 1983, A study of cartel stability: The joint executive committee, 188G1886, Bell Journal of Economics and Management Science 14,301-314. Reynolds, S.S., 1987, Capacity investment, preemption and commitment in an infinite horizon model, International Economic Review 28, 69-88. Reynolds, S.S., 1991, Dynamic oligopoly with capacity adjustment costs, Journal of Economic Dynamics and Control 15, 491-514. Roberts, M.J. and L. Samuelson, 1988, An empirical analysis of dynamic, nonprice competition in an oligopolistic industry, Rand Journal of Economics 19, 200-220. Slade, M.E., 1987, Interlirm rivalry in a repeated game: An empirical test of tacit collusion, Journal of Industrial Economics 35, 499-516. Slade, M.E., 1990, What does an oligopoly maximize? Working paper, University of British Columbia. Tsutsui, S. and K. Miho, 1990, Nonlinear strategies in dynamic duopolistic competition with sticky prices, Journal of Economy Theory 52, 136161. Van der Ploeg, F., 1987, Inefficiency of credible strategies in oligopolistic resource markets with uncertainty, Journal of Economic Dynamics and Control 11, 123-145. ’
Journal
of Industrial
Organization
11 (1993) 369-389.
Open-loop and feedback dynamic oligopoly
North-Holland
models
of
Larry S. Karp and Jeffrey M. Perloff* Department of Agricultural and Resource Economics, University of California, Berkeley, CA 94720, USA Final version
received August
1992
A simple method of estimating the degree of oligopoly power in dynamic markets using either open-loop or feedback (subgame perfect) strategies is developed. New theoretical results are presented for families of oligopoly models. Differences between discrete-time and continuoustime models and open-loop and feedback models are demonstrated. The effects of increasing the number of firms or adjustment costs are examined.
1. Introduction
We develop a feedback, oligopolistic dynamic game model that can be used to estimate the degree of market power in markets with nonlinear adjustment costs in output, investment, or prices. The model nests various well-known market structures in a larger family. Although the primary use for this model is in empirical work, it also provides a simple method for comparing open-loop and (subgame perfect) feed-back equilibria for a given degree of market power. This model can be used to analyze relatively easily the effects of increasing the number of firms and the costs of adjustment on the equilibrium trajectory and steady state. Dynamic models are appropriate where there are substantial adjustment costs in prices, training, or in capital accumulation, or where there is learning over time. For example, these adjustment factors are important in the rice and coffee export markets according to the parameter estimates and statistical tests of both open-loop and feedback models in Karp and Perloff (1989, 1993). By describing the theoretical characteristics of the model, we provide a basis for interpreting our existing empirical results and for assessing the suitability of the model for other empirical applications. Our previous papers discuss the mechanics of empirical implementation of the model. Correspondence to: J.M. Perloff, Department of Agricultural University of California, Berkeley, CA, USA. *We thank Drew Fudenberg, Stanley Reynolds, and participants of California at Berkeley, BAGEL, and the University of Michigan 0167-7187/93/$06.00
0
1993-Elsevier
Science
Publishers
and
Resource
Economics,
at seminars at the University for useful suggestions.
B.V. All rights
reserved
370
L.S. Karp and J.M. Perloff; Open-loop and feedback models of dynamic oligopoly
For reasons of tractability, many theoretical and empirical dynamic oligopoly models assume that firms use open-loop strategies that are not subgame perfect. Recently there have been a number of papers that estimate subgame percent equilibria.’ By comparing the paths and the steady-state equilibria of families of open-loop and subgame perfect, feedback dynamic oligopoly models, we show where the open-loop models are the same as feedback models and where they deviate substantially.’ The families of open-loop and feedback oligopoly models are nested by using an index of oligopoly power that meets two criteria. First, the index includes three leading market structures: perfect competition, the noncooperative Nash-Cournot equilibria obtained from a limiting finite horizon game, and the cartel solution when firms are identical. Second, the index can be estimated easily. The index is designed to describe rather than explain the market outcome. A given markup may be consistent with a number of different situations. For example, a non-cooperative equilibrium may be identical to the perfect competition equilibrium (as in a price-setting game with homogeneous products) or the cartel solution, or it may lie between those two extremes. The multiplicity of subgame perfect equilibria in supergames (a set of results described as folk theorems) is well known. Similarly, subgame perfect equilibria for dynamic games (also known as state-space games) are typically not unique. One reason for the lack of uniqueness stems from the use of punishment or trigger strategies. Dynamic games with trigger strategies have been constructed by, inter alia, Cave (1987) and Haurie and Pohjohla (1987). Even if attention is restricted to differentiable Markov strategies, the equilibrium is typically not unique in an infinite horizon differential game. Tsutsui and Mino (1990) provide an example in a game with a single state variable where the non-uniqueness is due to an incomplete transversality condition. Because even with differentiability and the strong refinement of Markov perfection, there is no guarantee of uniqueness, there is no way to identify the underlying game by observing market outcomes. A practical alternative is to develop an index, such as the one presented here, to describe dynamic equilibria. Our oligopoly index is ordinal rather than cardinal. There is no unambiguous meaning to statements such as ‘a dynamic equilibrium lies halfway between monopoly and perfect competition’. Cardinal measures, however, ‘Empirical studies based on repeated game models with trigger strategies, estimated using switching regressions, include: Porter (1983), Lee and Porter (1984), and Hajivassiliou (1989). Baker (1989) tests whether the oligopoly markup in a repeated game falls after an unexpected decline in demand. Slade (1987) finds evidence that small deviations from cooperation invite small levels of retaliation, rather than the use of trigger strategies. Roberts and Samuelson (1988) estimate an approximation to a subgame perfect equilibrium in a state-space game. 2Fudenberg and Levine (1988) give other types of conditions under which the open-loop and feedback equilibria are approximately the same with many players. See also Fershtman (1987).
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
371
can be inferred using our index and specific information from the problem. For example, the index and the exogenous parameters determine a steadystate quantity, price, and consumer surplus. These levels can be expressed as percentages of the perfect competition, Nash-Cournot, or cartel equilibrium levels. In this way the index can be used to determine whether an observed equilibrium is close to or far from one of the well-known equilibria. We show that, for a given market structure, the feedback model leads to more competitive behavior than the open-loop model. Using other models, Fudenberg and Tirole (1986) point out that the greater competitiveness in feedback models stems from a preemptive incentive that is absent in openloop equilibria. Of course, with different types of models, feedback equilibria may be less competitive [Fudenberg and Tirole (1984, 1986)]. In our model agents choose quantities, and there is a strategic incentive to increase these in the current period. The linear-quadratic structure has frequently been used in oligopoly models. For example, Fershtman and Kamien (1987) and Reynolds (1987) use dynamic linear-quadratic models to compare the steady states for a Nash-Cournot model for a fixed number of firms.3 We extend their work in three ways. We study a family of oligopoly models, analyze both the adjustment paths and the steady states, and examine the effects of an increase in the number of firms. In section 2 we describe our model and provide definitions. In the next section, the index of market structure is explained, and the difference between open-loop and feedback equilibria is discussed. In section 4 the chief features of the resulting equilibria are described. Comparative dynamics are discussed in section 5. Section 6 contains comments about empirical implementation. Our conclusions are in section 7. 2. Definitions
and the model
We start with a discrete-time model in which the length of a period is E. As ~40, the continuous-time model is obtained. Most of our analysis is based on the continuous-time model. The industry consists of n+ 1 firms where nz 1. At time t, firm i decides how much to produce in the current period, qit, or, equivalently, by how much to change its output, UitE= 4it - qi,t-E, where Uit is a rate. Firm i incurs a quadratic cost of adjustment,
3Hansen et al. (1985) also use the dynamic linear-quadratic model to study various open-loop models as well as the open-loop and feedback Stackelberg model. Van der Ploeg (1987) compares the open-loop and feedback steady states in a natural resources setting with general functional forms.
372
L.S. Karp and J.M. Perloff, Open-loop andfeedback
models of dynamic oligopoly
and a quadratic cost of production,
(e,,+eQ> 2
q.E
If’
In period t, the linear demand curve facing firm i is ?I+1
Pit=Ui-b
1 qjt.
(1)
j=l
Firm i’s revenue in period t is pitqit&.Given an instantaneous interest rate of discount rate is eerE, and the objective of firm i is to maximize its discounted stream of profits: r, the one-period
For expositional simplicity, we set ui= a for all i and assume Oi =0 and Oi=O. The last equality implies that adjustment costs are minimized when there is no adjustment. As a result, under the assumption of open-loop behavior, the steady-state levels of output for the three leading market structures (price-taking, non-cooperative Nash-Cournot and cartel with identical firms) are equal to their static analogues. This equality holds for general cost and revenue functions and not simply the quadratic ones assumed here. The ith firm’s objective (2) is written in a matrix form as
fl e-ri’-lle[ Ue:(q,_,+U,E)-_(qt-,+ute)‘Ki(q,-,+ut&)-_fu;Siut
c,(2a)
1
where ei is the ith unit (column) vector, e is a (column) vector of l’s, Si = eiei6, and Ki= b(eei+ eie’) + f3eie$ That is, the (i, i) element of Ki is 2b + 8 and the other elements of the ith row and column equal b. As s-0, this expression approaches m
I[
e If aeiq, - iq;Kiqr - &Siut
0
1
dt.
CW
We assume that qir is unconstrained. If price is negative, firms would prefer to buy rather than sell; a negative value of q represents purchases. 3. Two families of equilibria We consider two families of equilibria: open loop and feedback, which assume different information sets. Agents’ information set under the open-
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
313
loop structure consists of calendar time and the value of the state vector at the beginning of the game. In the feedback structure, the information set consists of calendar time and the value of the current state vector.4 In our model, the state vector at t is the lagged value qf-E. Strategies map the information set in actions in each period. An open-loop (feedback) equilibrium is a Nash equilibrium in open-loop (feedback) strategies. In a feedback equilibrium, players choose state-contingent policy rules, and the resulting equilibrium is subgame perfect. In each period of this game, the firms take the current state (lagged output) and the mechanism for determining future behavior as given. There are many such equilibria, but we restrict attention to linear decision rules. Linear rules are a natural choice because they can be derived as the limit of finite horizon models as the horizon goes to infinity. In an open-loop equilibrium, each firm chooses a sequence of controls. In our model, the changes are in its own output. The controls are a function of time and the initial state. The feedback equilibrium is given by a set of decision rules - not by a set of trajectories, as in the open-loop model. There are two ways to compare the two equilibria. First, we can compare the trajectory under the equilibrium feedback rules to the open-loop equilibrium. Alternatively, we can directly compare the open-loop controls written in ‘feedback form’ (that is, as a function of the current state) to the equilibrium feedback rules. The second alternative is more appealing for both theoretical and empirical reasons. Thus, we use the feedback form of the open-loop equilibrium to compare the equilibria under the two information structures outside the steady state. Our procedure involves the estimation of decision rules. The form of these decision rules is the same for both information structures. The inferences made from the estimated decision rules regarding the index of market power depends on the assumed information structure. In both the open-loop and feedback models, the rules depend on the lagged state. This similarity is significant because it means that, in estimating the open-loop equilibrium, it is not necessary to determine the initial condition when the trajectory was chosen, which is not feasible. An economic interpretation of the feedback form of the open-loop equilibrium is that firms respond to surprises that are caused by additive random shocks from nature. This equilibrium is not subgame perfect5 It is 4More generally, the information set can include the entire history of play, as with the closedloop information structure. When actions can be conditioned on history, trigger strategies of the type described by the Folk Theorems can be constructed. sThe open-loop equilibrium is a Nash equilibrium in trajectories. The feedback form of these trajectories is not a Nash equilibrium in decision rules. That is, if firm i takes the feedback form of firm j’s open-loop trajectory as given, the feedback form of firm i’s open-loop trajectory is not a best response.
374
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
appropriate to compare this version of the open-loop model to the feedback model because the two equilibria differ only in the assumptions one makes about firms’ rationality (and not about firms’ ability to respond to new conditions ex post). 3.1. Static analog In most empirical static models of oligopoly, aggregate or firm-level data are used to estimate a parameter, v, that reflects the markup of price over marginal cost. In this approach, given demand equation (1) for a homogeneous product, firm i’s effective marginal revenue curve (the marginal revenue given the degree of market power actually exercised) is MRi(Vi)=p+(l
+nVi)p’4i=p,--(1-nvj)bqi.
For simplicity, suppose there is a firms, MC= MC,= 13, which could variables such as weather, and that estimates the demand curve, and
common constant marginal cost for all be a function of various exogenous Vi= v for all firms. To determine v, one equilibrium equations for each firm,
MR,(v) = MC = 8:
Dividing the coefficient on the qi term by the estimate of b from the demand curve, subtracting one and dividing by n gives an estimate of v. The markup between price and marginal cost, p- MC=(l +nv)bq, is proportional to 1+ nv. For example, if v= - l/n, there is no markup (MR =p= MC); whereas, if v= 1, the monopoly markup is observed: MR=p+p’((n+l)q,)=p+p’Q=MC,
where Q=(n+ 1)q is aggregate output. Intermediate solutions, such as the Nash-Cournot (v = 0), are also possible. In the empirical literature on static models, the parameter v often is interpreted as a constant conjectural variation: v-dqj/dq,. We prefer the neutral interpretation that v represent a gap between marginal cost and price _ a measure of market power - rather than a conjectural variation. 3.2. Dynamic index An index v can be similarly defined for the dynamic games. Again, the index can be given the neutral interpretation as a measure of market power, v, or it can be defined as v = auit/aujt for i# j for all t. For simplicity in what
L.S. Karp and J.M. Perlofi Open-loop and feedback models of dynamic oligopoly
375
follows, we consider only the symmetric case where v is independent of i and j. The relation between v and the price-marginal cost markup is not as simple in the dynamic model as in the static model. In the dynamic model, for all market structures, the equilibrium markup depends on the shadow value of the state, which is endogenous. For example, under perfect competition, price does not equal marginal cost, except at the steady state.6 In an open-loop equilibrium each firm maximizes (2) given an initial condition, the index z), and an assumed path for its rivals. The parameter v enters the first-order condition for firm i’s problem - the maximization of firm i’s Hamiltonian with respect to its control - in a manner analogous to the static problem. The solution for the continuous time problem is described in the appendix. The feedback equilibrium is obtained by the simultaneous solution to the n + 1 dynamic programming equations:7 J,(q, -,) = max
aelq,-_q;Kiqt-~ujSiu,
wt
) E+e-”
Ji(q,)] >
(3)
where Ji is the present value of player i’s profits when the lagged output is qf_Er and qt =qt_e+p,~. Because the strategies are linear, Ji is quadratic. The parameter v enters the first-order condition to the maximization of (3) in a manner analogous to the static problem [see eq. (AS) in the appendix]. 4. Steady states and paths We now compare the open-loop and feedback equilibria emphasizing the effect of the parameter v on the equilibria. We define Q: as aggregate output at time t, where the superscript refers to the solution to the open-loop (0) and feedback (f) equilibria. If the initial output is the same for all firms, qO= (eQ,,)/(n + l), so Q: is the solution to
for k= o,f. The parameters yi and pi depend on the parameters in the firm’s objective functions, the number of firms, and the index v. In the feedback ‘In the static model, the index u contains no information that is not also contained in the markup. In the dynamic model, on the other hand, the index u provides a summary statistic for the equilibrium trajectory of the markup. ‘In infinite horizon games, there typically exist many subgame perfect equilibria. We avoid the problem of non-uniqueness by considering the equilibrium strategies that result from the game with finite horizon T and letting T-co.
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
equilibrium, yf and pf are obtained by aggregating the firms’ equilibrium decision rules. In the open-loop equilibrium, y” and p0 are obtained by aggregating the ‘feedback form’ of the open-loop equilibrium trajectory. By comparing the two linear differential equations (4), we can compare the open-loop and feedback equilibria both at and outside the steady state. We now make two assumptions for use in comparing the two equilibria. Assumption
1.
Assumption 2.
Stationary feedback rules exist.
The vector q, generated by the equilibrium converges for arbitrary qO.
feedback rules,
The proofs of the following propositions, given in the appendix, are based on comparison of the systems of equations that define yi and pi: Proposition 1. Zf Assumption 1 holds, a sufficient condition for the open-loop and feedback equilibria to be identical is v = 1 (cartel with symmetric firms) or v = - l/n (price takers). Proposition 2. Zf Assumptions 1 and 2 hold and behavior is Nash-Cournot (v=O), output is smaller in the open-loop equilibrium and converges to its steady state more rapidly than in the feedback equilibrium.
Based on simulation results reported below, the condition given in Proposition 1 appears to be necessary. However, because Proposition 1 only establishes sufficiency, we cannot prove that the comparison in Proposition 2 also holds for u #O. Extensive simulation, however, supports the conjecture that the result does hold for v ~0. Because the feedback Nash-Cournot equilibrium is farther from the monopoly solution than is the open-loop Nash-Cournot equilibrium, the feedback solution is relatively procompetitive. Fershtman and Kamien (1987) and Reynolds (1987) show that the Nash-Cournot feedback steady state is greater than the open-loop steady state. Proposition 2 generalizes their results by comparing the entire equilibrium path. The intuition is that capacity discourages rivals’ investment under the feedback assumption. Therefore, firms have a greater incentive to invest today as a means of preempting their rivals’ future investment, so they develop larger capacities and hence larger output levels. This incentive is absent in the open-loop equilibrium. In fig. l(a), the feedback and open-loop steady-state outputs are graphed as a function of v where n= 1 (duopoly), a =250, b = 10, 6 = 5, p =0.95, E= 1, and 8=0 (the ‘base’ parameters). Fig. l(b) shows the difference between the open-loop and feedback steady states is largest when the market is nearly
L.S. Karp and J.M. Perlofi Open-loop and feedback models of dynamic oligopoly
377
6’ -1
-0.6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.6
1
V
0.e
(‘4
0.5
7 i
0.4
e:,
0.3
n
0.2
C
a 0.1 0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.8
0.8
1
V
Fig. 1. (a) Feedback
and
open-loop steady-state output. (b) Difference open-loop steady-state output.
between
feedback
and
competitive at v z -0.7.8 Based on these simulation results, the bias from incorrectly assuming firms use open-loop strategies is less serious when the outcome of these strategies is between the Nash-Cournot and collusive equilibria than when it lies between the Nash-Cournot and competitive equilibria. To illustrate the relative rates of adjustment, we use a Nash-Cournot duopoly model. With the base parameters, the Nash-Cournot steady state is 8.5 in the open-loop model and 9.1 in the feedback model. That is, where 0=0 (marginal production cost, net of adjustment cost, is constant), the feedback steady state is 7% higher than the open-loop steady state. By *The estimated market structure parameters for both the rice export market [Karp and Perloff (1989)] and the coffee export market [Karp and Perloff (1993)] are such that the steadystate differences between the feedback and open-loop outputs are nearly maximized; however, in percentage terms, these the differences in the steady state are small.
378
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly Table 1 Trajectories
for Nash-Cournot
models.
Period
Open-loop 8=0
Feedback 0=0.7917
0 1 2 3 4 5 6
1.000 7.435 8.369 8.476 8.490 8.491 8.492
1.000 7.552 8.374 8.477 8.490 8.491 8.492
Parameters: a= 250, n=l, 6=5, and ~=l.
b = 10,
p=O.95,
choosing different parameter values for the open-loop and feedback models, the steady states can be made equal. For example, if we use the base parameters with 6 =0 in the open-loop model, but change 0 to 0.79 in the feedback model, both steady states equal 8.5. Thus, with a slightly different short-run marginal cost curve, the two models can produce the same steady state. These two models can be distinguished empirically by determining the slope of the marginal cost curve, 8, or observing the adjustment paths. Table 1 shows that, although the open-loop with 8=0 and the feedback model with 8 =0.79 produce the same steady state, adjustment is more rapid in the feedback model. In the first period, output in the feedback model is 1.6% higher than in the open-loop model. Note that this faster rate of adjustment does not contradict Proposition 2 because, in this example, the open-loop and feedback models have different values of 0. To show the trajectories more clearly for the four leading models, we increase 6 to 150 in fig. 2. As shown in fig. 2, both the paths and the steady states vary with v and the type of strategy used. 5. Number of firms and adjustment costs As the number of firms, n+ 1, increases, the equilibrium trajectories change. By setting 9 =0 and normalizing so that 6 =(n + l)c, where c >O is constant, the price-taking and collusive equilibria are invariant to n. As n becomes large, the adjustment cost for each firm becomes infinite so each firm makes only infinitesimal adjustments and captures only an infinitesimal share of the market. Thus: Proposition 3. Given 0=0 and the normalization 6 =(n + l)c, the open-loop and feedback Nash-Cournot equilibria converge to the competitive equilibrium as n-tco.
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
379
14 Price-taker 12 10 0 u *
8
~: Cartel
6
10
20
15
Time Period Fig. 2. Trajectories
for four models.
Table 2 The effect of increasing the number of Nash-Cournot on industry output.
Monopoly
Price-taking Parameters: 6=2.5 (n+ I).
firms
Number of firms (n + 1)
Open-loop
Feedback
1 2 3 4 5 10 15 00
12.74 16.98 19.11 20.38 21.23 23.16 23.88 25.48
12.74 17.48 19.92 21.35 22.21 24.16 24.14 25.48
a = 250,
b = 10,
/S= 0.95,
0=0,
s=l,
and
This proposition can be proved by examining the equations that determine the stationary control rules. The following is a heuristic proof. From Proposition 2, the open-loop and feedback equilibria are identical for u= - l/n, which goes to 0 as n+co. The open-loop model is a static game and, in a static game, the Nash-Cournot equilibrium converges to the competitive equilibrium as n--t cc. The relation between the Nash-Cournot steady-state output and the number of firms, given the normalization 6 =(n + l)c, is shown in table 2 for the base parameters. The Nash-Cournot feedback output increases more than does the Nash-Cournot open-loop output as the number of firms
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
increases. Thus, not only is the feedback equilibrium relatively procompetitive compared with the open-loop equilibrium, but the feedback equilibrium is also more sensitive to changes in the number of firms. We have shown that as n-+cc and 6 is adjusted to leave the price-taking equilibrium unchanged, the open-loop and feedback Nash-Cournot steady states converge to the static (competitive) equilibrium. Thus, there are two senses in which the static model is nested in the dynamic model, because the model is also static if 6 =O. This nesting has obvious implications for empirical work. For example, we may want to test joint hypotheses regarding market structure and the importance of dynamics. However, although 6 = 0 yields a static model, it does not necessarily follow that the dynamic equilibrium is close to the static equilibrium for 6 close to 0. Indeed, Reynolds (1987), using a continuous-time, linear-quadratic model of costly adjustment, shows that the steady-state feedback output in the limiting game as 6-O is greater than the steady-state open-loop (and the static) output level. The equilibrium is discontinuous at 6 =O. Reynolds (1991) shows that the steady-state feedback output in the limiting game as 6+cc equals the steady-state open-loop (and the static) output level. Thus, the static equilibrium is nested within the steady state of our dynamic in exactly the ‘opposite sense’ that one might expect. We obtain the static equilibrium for large, but not for small, adjustment costs. What lies behind this result? The open-loop steady state equals the static equilibrium for all values of 6, due to our assumption that adjustment costs are minimized when adjustment is 0 (i.e. at the steady state).g Therefore, it is only necessary to explain why the feedback and open-loop steady states converge as 6+co but do not converge as 6-+0. This explanation turns on the intuition for the difference between the open-loop and feedback equilibria: only in the feedback model is there a preemptive incentive. If 6 is very large, firm i’s rival finds it costly to respond to changes in firm i’s output so firm i’s preemptive incentive is small, and the two steady states are similar. If, on the other hand, 6 is very small, firm i’s rival can easily respond to changes in firm i’s output so the preemptive incentive is greater, and the two steady states are dissimilar. Reynolds’ result may appear to have unfortunate implications for the empirical testing of the model because it suggests that dynamics are important (at least in determining the steady state) where adjustment costs, which are the reason for the dynamics, are unimportant. Because the principal objective of this paper is to provide a foundation for using this model to estimate dynamic market structure, it is important to assess the likely empirical significance of this apparent paradox. ‘Reynolds does not make this assumption, than our interpretation of them.
so his conclusions
are. stated
somewhat
differently
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
381
Table 3 Deviations
of the feedback steady states from static equilibria various values of 6 and E.
for
6 E
0.5
0.01
0.001
0.000001
0.005 0.1 0.5
0.036 0.032 0.019
0.035 0.015 0.001
0.023 0.000037 0.00000015
0.001 0.0000037 0.0000002
Parameters:
a = b = 1, p = 0.95, 0 = 0, E= 1, and TE = 50.
To this end, we use a discrete-time, Nash-Cournot (v=O) duopoly model to simulate the differences between the feedback and open-loop steady-states for various combinations of 6 and E. From Proposition 2, we know that this difference between the feedback and the open-loop steady-state outputs, d(6,s), is positive. Table 3 gives values of d(b,s) for four values of 6 and three values of &.I0 These simulations illustrate two results. First, for the values of E we chose, the value of A is non-monotonic in 6 and can be made arbitrarily close to 0 by choosing 6 sufficiently small. Second, for a small positive difference, A*, the function ~*(A*,E) is decreasing in E, where ~*(A*,E) is the largest value of 6 such that A(6, E)< A* for all 6 < 6*(A*, E). Based on the first simulation result, it appears that the discrete-time feedback model does approach the static model as 6 approaches 0. By making 6 sufficiently small, the equilibrium output level is arbitrarily close to the steady state in a single period (i.e. adjustment is rapid), and the steady state can be made arbitrarily close to the static equilibrium output. This result is reassuring for empirical purposes (where a discrete-time model is the natural one) because it means that dynamics have an important effect on the equilibrium if and only if the source of dynamics is significant. Based on the second simulation result, for the feedback steady state to be within a given distance from the static equilibrium requires a smaller 6, the smaller is E. This result provides a way to reconcile Reynolds’ analytic result and our first simulation result. Reynolds obtained his result using a continuous-time game, whereas our simulations are from a discrete-time model. Although the discrete-time model does not appear to contain a discontinuity at 6 =O, the simulations illustrate that the steady state is more sensitive to 6 the closer we get to continuous time (i.e. as s-0).
“‘In stages obtain a = b=
this comparison, as we diminish the length of each stage (E), we increase the number of (T) so that the horizon (TE) remains constant at 50. This horizon is long enough to stationary feedback rules, which were then used to calculate the steady state. Because 1, the static equilibrium is l/3.
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
6. Additional empirical considerations
There is an alternative approach to estimating an index of market power in a dynamic oligopoly model. The open-loop equilibrium can be obtained by solving a control problem, so the index of market power can be obtained by estimating the parameters of that control problem.” There is an extensive literature on estimating such problems, and it is worih recognizing that this literature has potential uses in empirical industrial organization. Given a control problem of the form J(q)
=
max 7 ert ae’q u
$q’Kq
0
s.t. 4 = U,
q.
-
6
2 u’u
1 dt
(5)
given,
where K = k, ee’ + (k, - k,)l, K positive-semidefinite and 6 > 0, we can show that aggregate output in the control problem and in the open-loop equilibrium to the game are the same if and only if k, +nk, =(2+n(v+ l))b+Q. In addition, if k,=(2 + nu)b+ 0 and k, = b, the levels of output for the two problems are the same even if the n+ 1 firms have different initial levels of output.‘* Hansen et al. (1985) used the relation between the open-loop equilibrium and the solution to a control problem to study the Nash-Cournot (v=O) model. If firms are price takers (V= - l/n), the integrand in eq. (5) is social surplus, which illustrates the well-known result that the competitive equilibrium can be obtained by solving the social planner’s problem. Where firms are identical and collusive (u= l), in the control problem that matches the open-loop game, k. = (2 + n)b + 8, kl = b; whereas, in the control problem for the monopolist, k, =2b+8, kl =2b. Therefore, the collusive game gives rise to the monopoly solution only if all firms produce equal quantities. Analogously, in static games, the value u = 1 produces the monopoly solution only in a symmetric equilibrium. We have estimated the value of u under the assumptions of open-loop and feedback behavior. In our theoretical discussion, it is convenient to assume that firms (and their initial levels of q) are identical so that we can aggregate the decision rules into the scalar eq. (5). This assumption is not reasonable for empirical work. Instead, we estimate the decision rules as function of own and rivals’ lagged output, rather than as a function of lagged aggregate output. In addition we relax the assumption of identical firms by allowing the intercepts of the demand, cost of production, and cost of adjustment “To do so requires the open-loop assumption. See Slade (1990) for a general obtaining oligopoly equilibria by solving maximization problems. “These assertions are proven in an earlier version of this paper.
discussion
of
LX Karp and J.M. Perlox Open-loop and feedback models of dynamic oligopoly
383
equations, to be firm specific and time dependent. For tractability, we maintain the assumption that the parameter v and demand and adjustment cost slopes (b and 6) are the same for all firms and stationary. This implies that the intercepts of the firm’s decision rules (either the feedback decision rules or the feedback form of the open-loop equilibria) are firm specific and time dependent, but that the coefficients on own and rivals’ lagged output are stationary and symmetric across firms. By estimating the latter coeffrcients we are able to infer the value of u, under either the open-loop or feedback assumption. 7. Summary and conclusions Our model provides a simple method of estimating the degree of oligopoly power in dynamic markets under the assumption the firms use either openloop or feedback (subgame perfect) strategies. This method, which is analogous to those used with static games, allows the data to indicate the degree of market power without specifying the exact nature of the game being played by the firms. If firms are either collusive or perfectly competitive, the open-loop and feed-back paths and steady states are the same. Based on extensive simulations and estimates for two actual markets, even for intermediate market structures, incorrectly assuming that open-loop strategies are used results in only a small downward bias on the market structure parameter. Given an estimate of the market parameter, the slope of the demand curve, and the cost-of-adjustment parameter, it is straightforward to calculate the equilibrium price, consumer surplus, and industry profit, either at a point or as a discounted future stream. These values can then be compared with the corresponding price-taking and collusion values to obtain a cardinal measure of the closeness of the observed market structure to perfect competition or cartel, as illustrated in Karp and Perloff (1989, 1993) for the international rice and coffee export markets. There are a number of possible empirical and methodological extensions. Allowing the market structure parameter to differ across firms is straightforward in principle, but it is computationally difficult if there are more than two firms. Similarly, the inclusion of additive random terms following an ARMA process can be accommodated by augmenting the state vector, which would permit the estimation of market structure with rational expectations about exogenous terms. Furthermore, in principle, a second state variable for stocks could be added. Appendix: Proofs of Propositions
1 and 2
We begin by proving a lemma in which we show that the parameters
pk
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L.S. Karp
and J.M. P&off, Open-loop and feedback models of dynamic oligopoly
are the negative root of a quadratic function and yk is a function of pk, k = o, f. We then prove the propositions by comparing the solutions to these equations. As in the text, superscripts are used to distinguish between open loop and feedback equilibria. Subscripts i and j are used to identify firms i and j; all other subscripts are used to identify elements from matrices. Lemma. Under Assumptions 1 and 2, the parameters pk and yk (k= o, f) used in eq. (5) satisfy the restrictions that pk is the negative root of (p”)2-r6p0-6{b(2+n(v+1))+8}=0
(A.la)
(pf)2-r~pf-6{b(2+n(v+1))+8}=n(v).
(A.lb)
and
where _q(v)=^ -(np:(l-v+nv-nv2)+n2p,z(l+nv-v2-nv3)+n3z2(v+v2+ nv2 - 2vL - nv’)) and p1 and z are parameters in a firm’s value function: these parameters are defined in eqs. (A.6a)+A.6c) below. 7he parameters yk are given by (n + 1)a “=(r8-p”)
(A.2a)
and yf =
(n+ 1)(a+ 4(v)) (74-p’)
’
(A.2b)
where 4(v) =
n(p,+nvz)h(l-v)(l+nv) 6
and h is the parameter in firm i’s value function, which equals the marginal effect on firm i’s present discounted value of profits of a change in qj when all firms’ stock levels are 0 (as discussed below in more detail). A.1. Proofs of (AJa)
and (A.2a)
Define ViE R” + 1 as a column vector with 1 in the ith position and v in every other position. Firm i behaves as if au/aui=tTi and its objective is to maximize eq. (2b). The open-loop current value Hamiltonian for firm i, yiq.,is yi”i= aeiq -+q’Kiq -+u’Siu +&u, where ii is the shadow value of the state q for firm i and Ki is defined below eq. (2a) in the text. The necessary conditions for an interior solution are
L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
38.5
(A.3a) and
as _ ~ = lqli -
(A.3b)
r&) = I?;(- aei + &q).
&li
(A denotes dJ/dt). The first-order conditions (A.3a) and (A.3b) are obtained as the limiting form as s-0 of the first-order conditions to the discrete openloop problem. We try a solution of the form &=h,+Hiq, where hi is a n+ 1 column vector and Hi is an n + 1 x n+ 1 matrix. For a steady-state control rule, hi=Ai=O for all i. We denote the (endogenous) elements of Hi and hi as follows. For the matrix Hi: the (i, i) element is Hi(i, i) = p,,; Hi(i, j) = pr and Hi(j, i) = fil for all j# i; finally, H,(1, s) =z for all I, s#i. For the vector hi: the ith element is hi(i) = h, and all other elements are h,(j) = fi for j# i. Since we assume that agents are symmetric and we restrict attention to symmetric equilibria, each agents’ value function is a permutation of i’s value function. Substituting the linear expression for li into (A.3a) and using the notation described above, and then rearranging, implies h+nuC
1 (Po+nuj&ji+(pl+nUZ)
ui=F+j
[
C
j#i
qj
1
(A.3c)
.
Next, we stack the n+ 1 necessary conditions of eq. (A.3c) to obtain ZJ=(h* + N*q)/6, where h* =(/I + nvh) e and H* is a matrix with elements: H*(i, i) =po + nujl, H*(Z,s) =pl +nuz, for all Z#s (i.e. for all off-diagonal elements). By substituting this solution into eq. (A.3b) together with fii=O, I;i=O, we obtain the systems
and r&$hi = iT:(a6ei+ Hih*).
These two systems give three independent functions, p. + nup,, p1 + nvz, and h + nvh:
equations for the three unknown
-r6(po+nv~,)=6((2+nu)b+8)-(po+nu~,)2-n(p, -~6(pI+nuz)=6b-2(po+n~,)(p,+nuz)-(n-1)(p,+nuz)2, and
+nz)‘,
(A.4a) (A.4b)
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
(h+nv&)=
a6
(A.4c)
(r6-jo-nvjl--@-n2vz)’
We define p” = p0 + nvj, + np, + n2vz. Using this definition, the equilibrium control rule (A.~c), and the assumption of symmetry (so that qi=qj), implies that p”/6 is the coefficient on aggregate output Q in eq. (4) of the text. To obtain the equation that defines p”, multiply both sides of eq. (A.4b) by n and add it to eq. (A.4a). This equation together with the definition of p” gives us eq. (A.la). We define y”=(n+ 1) (h+nvt%); multiply both sides of (A.4c) by n+ 1 and use (A.3c) to obtain eq. (A.2a). A.2. Proofs of(A.lb)
and (A.2b)
In order to avoid a proliferation of notation, and also to emphasize the similarity between the open-loop and feedback equilibria, we use many of the same symbols used above. For example, Hi and hi will continue to represent a matrix and vector which determine (together with the state vector 4) firm i’s vector of shadow values; similarly, pO, pl, h and h^are elements of that matrix and vector. Since the open-loop and feedback equilibria differ except in special cases, the value of those elements will differ in the two equilibria. The stationary dynamic programming equation for firm i is r(yi+hiq+$q’Hiq)=max
ui
aeiq-sq’K,q--~u~+(hi+Hiq)‘u
1.
The parentheses on the left-hand side gives Ji(q), which is firm i’s value function; the parameter yi and the elements of hi and Hi are endogenous. We know that the value function is quadratic for a finite horizon game. Because we are interested in the limiting form of that game as the horizon approaches infinity, we restrict attention to quadratic functions for the infinite horizon game. The necessary condition for an interior maximum is
(A.9 Substituting eq. (A.5) into the dynamic programming coefficients, we obtain the system 1 (1;;
0=rbHi+6Ki+Hii$t7:Hi-Hi
O=r6hi+6aei-Hi&i$hi+
1 (;I:
equation and equating
eJ.H. 1 ’ J J, (1;:
H3.e’. JJJ)’
h.+H.
I(;::
c
H.5.e:
J J J)
e.fi’.h. JJJ)’
Hi,
L.S. Karp and J.M. Perlo& Open-loop and feedback models of dynamic oligopoly
387
We now try a solution of the form Hi(i, i)=po, Hi(i, j)=Hi(j, i)=p, for and Hi(s, 1)=z for s#i, l#i; hi(i) =h, hi(j) =h for j#i. By substituting this trial solution into the system above, we obtain
j#i,
O=r6p, +6(2b + e) -pi -(2n-(nv)Q:
-221t%!p1z,
(A.6a)
O=r6p,+6b-nvp:-(n-l)p:-22p,p,
+((nu)2pl-(no= r6z-p:
-2zp,
l)nvp, -np,
-n2zu)z,
- 2(n- l)zp, - 2nvz(p, +(n-
(A.6b) 1)z) +(nuz)2,
0= -r6h+6a+(h+nvh3np,(l_v)+(p,+np,v)h+1z(p,+nzv)t;
(A.6c) (A.7a)
and o= -r6~+(h+nuL)nz(l-u)+(p,+nplu)~ +(p,+nzu)(h+(n-l)Q.
(A.7b)
In deriving system (A.6) we use Assumption 1, which implies that the system (PO,pi, i)’ is stable. [To obtain system (A.6), we set the vector of time derivatives (pO,bl,i)’ equal to 01. Assumption 1 is not needed for the open-loop equilibrium because that equilibrium can be obtained by solving a control problem for which standard results guarantee stability of the Ricatti system that leads to eq. (A.4). Similar results for the feedback game have, to our knowledge, not been obtained, so Assumption 1 is required. We define pf = p0 + n(1 + u)pl + n2uz. By multiplying eqs. (A.6b) and (A.6c) by $1 +v) and n2u, respectively, and adding them to eq. (A.6a), we obtain eq. (a.lb). We define yf =(n + 1) (h+nu@. By multiplying eq. (A.7b) by nu and adding it to eq. (A.7a), we obtain eq. (A.2b). 0 A.3. Proof of Proposition I Proposition 1. If Assumption 1 holds, a s@cient condition for the open-loop and feedback equilibria to be identical is v= 1 (cartel with symmetric firms) or v= - l/n (price takers). Proof. This proposition follows immediately from comparison of eqs. (A.la) and (A.lb), and of eqs. (A.2a) and (A.2b), and because q(u)=~#~(v)=O for u=-l/n or v=l. 0 A.4. Proof of Proposition 2 Proposition 2.
If Assumptions 1 and 2 hold and behavior is Nash-Cournot
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L.S. Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
(u=O), output is smaller in the open-loop equilibrium and converges steady state more rapidly than in the feedback equilibrium. Proof.
to its
We now show that r(0) < 0 < NO).
(A4
Given this inequality, eqs. (A.la), (A.lb), (A.2a), and (A.2b), and the definition of p” and pf, Proposition 2 follows immediately. We first establish that pl(pl + nz) >O so that ~(0) ~0. By comparing eqs. (A.la) and (A.lb), we find that @‘
rd-_p0--nk [ -h +nz)
rb--pO-I;Pnl-l)p,l[~]~[~]~
Using our previous results, all elements of this matrix are positive so h and k must have the opposite sign. To establish the second inequality in eq. (A@, we need to show only that 6~0. Using eqs. (A.7a) and (A.7b) and the results of the previous paragraph, h and 6 must have the opposite sign. Because h= 8Ji(0)/8qi, h = Mi(0)/aqj, j# i, if h t0 c 6 then, if all firms begin with 0 sales so that initial price is positive, firm i would prefer to begin with negative sales and have its rival(s) begin with positive sales. Because this result must be false, we conclude h > 0 > E, so C/J(O) > 0, which completes the proof of Proposition 2. 0 References Baker, J.B., 1989, Identifying cartel policing under uncertainty: Journal of Law and Economics 32, part 2, S41Sl6.
The US steel industry
1933-1939,
LX Karp and J.M. Perloff, Open-loop and feedback models of dynamic oligopoly
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