On the Feedback Solutions of Differential ... - Semantic Scholar
On the Feedback Solutions of Dierential Oligopoly Games with Hyperbolic Demand Curve and Capacity Accumulation Luca Lambertini
∗,§
Arsen Palestini
]
*Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy; fax +39-051-2092664 [email protected] ENCORE, University of Amsterdam Roeterstraat 11, WB1018 Amsterdam, The Netherlands ] MEMOTEF, Sapienza University of Rome Via del Castro Laurenziano 9, 00161 Rome, Italy; [email protected] July 24, 2013
Abstract To safeguard analytical tractability and the concavity of objective functions, the vast majority of models belonging to oligopoly theory relies on the restrictive assumption of linear demand functions. Here we lay out the analytical solution of a dierential Cournot game with hyperbolic inverse demand, where rms accumulate capacity over time à la Ramsey. The subgame perfect equilibrium is characterised via the Hamilton-Jacobi-Bellman equations solved in closed form both on in nite and on nite horizon setups. To illustrate the applicability of our model and its implications, we analyse the feasibility of horizontal mergers in both static and dynamic settings, and nd appropriate conditions for their pro tability under both circumstances. Static pro tability of a merger implies dynamic pro tability of the same merger. It appears that such a demand structure makes mergers more likely to occur than they would on the basis of the standard linear inverse demand. JEL Classi cation: C73, L13 Keywords: capacity, dierential game, Markov-perfect equilibrium, HamiltonJacobi-Bellman equation, horizontal mergers
1
Introduction
Most of the existing literature on oligopoly theory (either static or dynamic) assumes linear inverse demand functions, as this, in addition to simplifying calculations, also ensures both concavity and uniqueness of the equilibrium, which, in general, wouldn't be warranted in presence of convex demand systems (see [16] and [10],
inter alia).
How-
ever, the use of linear demand function is in sharp contrast with the standard microeconomic approach to consumer behaviour, where the widespread adoption of Cobb-Douglas preferences (or their log-linear a ne transformation) yields hyperbolic inverse demand
1
functions.
The same applies to the so-called quasi-linear utility function, concave in
consumption and linear in money, that again yields a convex demand system. Indeed, both preference structures share the common property of producing isoelastic demand
1
functions.
Furthermore, such demand functions have been employed in rent-seeking games, where important contributions have been provided by [7] and [32], which showed the strict relations between the rent-seeking games and the Cournot oligopoly models. The isoelastic demand function has been widely analyzed in static settings (duopoly, [24], triopoly, [25]), devoting a particular attention to the stability of equilibrium and the possible chaotic implications (see also [26] and [27]). In fact, this is sometimes openly referred to in the eld of industrial organization, where researchers mention the opportunity of dealing with non-linear demand curves,
2 Additionally, the econo-
and then promptly leave it aside for the sake of tractability.
metric approach to demand theory has produced the highest eorts in building up a robust approach to the estimation of non-linear individual and market inverse demand functions, yielding a large empirical evidence in this direction.
3
A relevant market demand structure was proposed by Anderson and Engers at the beginning of the 90s ([1]), which deserves some discussion: quantity of a homogeneous good and 1
p(Q)
calling
Q
the aggregate
the inverse demand function, they assumed
the form
p(Q) = (a − Q) α
price).
α: α > 1, concave if α < 1, linear if α = 1, as is outlined by the following Figures.
to generalize the linear case (a
>0
is the market reservation
Such inverse demand leads to dierent scenarios depending on parameter
convex if
p6
p6
1
1
p(Q) = (a − Q) α
p(Q) = (a − Q) α
Q
Figure 1.
Anderson and Engers inverse demand functions for
α>1
and
Q α 0.
On the supply side, production entails a total cost parameter measuring marginal production cost. Cournot-Nash; therefore, rm
i
chooses
qi
Ci = cqi , where c > 0 is a constant
Market competition takes place
so as to maximise pro ts
à la
πi = (p(Q) − c) qi .
This entails that the following rst order conditions must be satis ed (given their form, it is not necessary to assume interior solutions):
aQ−i ∂πi = 2 −c=0 ∂qi (qi + Q−i ) where
Q−i ≡
(4)
P
j6=i qj . The associated second order condition:
∂ 2 πi 2aQ−i =− 3 0 for all i = 1, . . . , N . Then, imposing the symmetry condition qi = qj = q , one obtains the individual Cournot-Nash equilibrium a (N − 1) a CN output q = , yielding pro ts π CN = 2 . If the N rms were operating N 2c N a ∗ ∗ . under perfect competition, then p = c and therefore q = Nc
is always met when on all outputs, i.e.
It is apparent that the above solutions (i.e., both the Cournot-Nash equilibrium and the perfectly competitive equilibrium) are well-de ned and feasible for all
c > 0.
In the remainder of the paper, we will turn our attention to a dierential game where the demand structure is the same as here.
We will separately investigate the in nite
horizon and nite horizon cases.
2.2 Feedback solutions of the dierential game on an in nite horizon t ∈ [0 , +∞) , and which is served by N qi (t) ∈ (0, +∞) de ne the quantity sold by à la Cournot, the demand function at time t being:
We are going to consider a market existing over rms producing a homogeneous good. Let rm
i
at time
t.
Firms compete
a p (t) = , Q (t)
Q (t) =
N X
qi (t) .
(6)
i=1
Ci (t) = cqi (t) , where C > 0. ki (t) ∈ [0, +∞) over time. i at time t, we assume that ki
In order to produce, rms bear linear instantaneous costs
Moreover, they must accumulate capacity or physical capital If we denote with
yi (t)
aects the production of
the output produced by rm
yi
in the sense that
5
∂yi > 0. ∂ki
Capital accumulates as a result
of intertemporal relocation of unsold output
yi (t)−qi (t).6
This can be interpreted in two
ways. The rst consists in viewing this setup as a corn-corn model, where unsold output is reintroduced in the production process. This interpretation is admittedly unrealistic (although systematically mentioned in the whole macroeconomic debate stemming from Ramsey [28]) as it implies that the nal good and the capital input are homogeneous. The second consists in thinking, a bit more realistically, of a two-sector economy where there exists an industry producing the capital input which can be traded against the nal good at a price equal to one (for further discussion, see [5]), this channel being used to nance investment projects as an alternative to borrowing from the banking sector (which otherwise should be explicitly modelled). In a nutshell, the Ramsey dynamics is a blackbox hiding a market where rms sell part of their production at a xed price for fundraising aimed at nancing their growth. Unlike the standard macroeconomic approach to growth models in a Ramsey fashion, here we will allow for the presence of an instantaneous cost of holding installed capacity. This cost will be refer to
ki (t)
∂Γi ∂ki
as the
Γi (ki (t)),
possibly asymmetric across rms. In the remainder, we will
as a measure of the
i-th
opportunity cost of a unit of capacity.
state variable subject to the following dynamic constraints:
· dki (t) ≡ k i (t) = Gi (ki (t)) − qi (t) dt k (t ) = k > 0 i 0 i0 where rm's
We will employ
,
(7)
Gi (ki (t)) is a C 2 (R+ ) function aecting the growth dynamics of capital. strategic variable is qi (t), while the i-th rm's state variable is ki (t).
The i-th
A standard assumption in dynamic economic models concerns the kind of discounting: since the game takes place in a unique market, all rms discount pro ts at a same constant rate
ρ ≥ 0,
possibly depending on monetary factors such as the expected rate
of currency depreciation. The problem of rm
i
is to choose the output level
qi (t)
so as to maximise its own
discounted ow of pro ts (from now on, we will omit time arguments whenever possible):
Z
∞
max Ji (ki0 , t0 ) ≡
qi ∈R+
[(p (Q(τ ), τ ) − c) qi (τ ) − Γi (ki (τ ))] e−ρ(τ −t0 ) dτ =
t0
Z
∞
= t0
"
a PN
i=1 qi (τ )
!
#
− c qi (τ ) − Γi (ki (τ )) e−ρ(τ −t0 ) dτ.
(8)
We are going to adopt the dynamic programming approach, relying on the Hamilton-
7
Jacobi-Bellman equations , generally considered as the most powerful tool in dynamic Game Theory and in a large number of economic models (both in discrete and continuous time). The issue to be tackled is the determination of the
i-th
Vi (ki0 , t0 ),
i.e. the optimal value of
objective functional (8), over the set of all feasible paths
qi : [t0 , ∞) −→ R.
When the involved time horizon is in nite, the optimal value function does not depend on the initial time, but only on the initial state.
In both cases, each optimal value
6 Of course, capacity decumulates whenever y (t) − q (t) ≤ 0. i i 7 In the 50s Richard Bellman extended the Hamilton-Jacobi theory,
century to describe and solve problems in Classical Mechanics.
6
which was developed in the 19th
function can be achieved as the solution of a Hamilton-Jacobi-Bellman (HJB) equation (for the complete derivation, see [12], Chapter 3). Call
ki0 = k .
Denoting with
Vi (k)
i-th
the
optimal value function for (8), the
Hamilton-Jacobi-Bellman (HJB) system of equations reads as follows:
( ρVi (k) = max qi
for all
i = 1, . . . , N .
!
) ∂Vi − c qi − Γi (k) + (Gi (k) − qi ) , PN ∂k i=1 qi a
Note that because
rst order partial derivatives of
Vi
Ji
only depends on the
i-th
(9)
capital, in (9) the
with respect to the remaining state variables do not
appear. To proceed with the analytical solution of the feedback problem, we are going to determine the symmetric Nash equilibrium of the game. Suitable symmetry assumptions are commonly employed both in Static and in Dierential Game Theory, in that the players are identical rms, having an identical initial capital endowment and identical
8
payo structures . This leads us to introduce suitable symmetry conditions from the beginning of the procedure: one is
qi = qj
for all
i 6= j, saying that the equilibrium quantity
must be symmetric across all rms. The assumption of symmetry across capitals states that, from the standpoint of a generic rm
i,
the rivals' capacities (and therefore also
their weights in the value function) must be equal when the respective growth dynamics and cost structures are equal.
Proposition 1. Assuming symmetry across all variables, the HJB equation of the problem is given by: a V (k) Γ(k) − N 2 ∂V =ρ + , ∂k G(k) G(k)
Proof.
Maximizing the r.h.s. of (9) with respect to
aQ−i 2
(qi + Q−i )
−c−
qi
(10)
yields:
∂Vi = 0, ∂k
(11)
then, by assuming symmetry on the relevant variables and functions, i.e. q1 = . . . = qN = q , V1 = . . . = VN = V , Γ1 (·) = . . . = ΓN (·) = Γ(·), G1 (·) = . . . = GN (·) = G(·), ∗ we have that (11) yields the following expression for the optimal strategy q :
q∗ =
a(N − 1) , ∂V N2 +c ∂k
(12)
which must be plugged into (9) to achieve:
∂V N + c ∂k ρV (k) = − c N −1
a(N − 1) ∂V a(N − 1) G(k) − −Γ(k)+ ⇐⇒ ∂V ∂V ∂k 2 2 +c +c N N ∂k ∂k
8 Note that in such a problem some asymmetric elements might be admitted when they do not change the HJB dramatically, for example growth rates or marginal cost parameters, but that goes beyond the scope of our paper
7
∂V a N +c − 1) ∂V ∂V ∂k a(N − Γ(k) + ⇐⇒ ρV (k) = G(k) − ⇐⇒ ∂V ∂V ∂k ∂k 2 2 N N +c +c ∂k ∂k ∂V ∂V a ∂V N ⇐⇒ ρV (k) + Γ(k) − G(k) = + c − (N − 1) ⇐⇒ ∂V ∂k ∂k ∂k N2 +c ∂k ∂V ∂V a ⇐⇒ ρV (k) + Γ(k) − G(k) = + c ⇐⇒ ∂V ∂k ∂k N2 +c ∂k a ∂V V (k) Γ(k) − N 2 ⇐⇒ =ρ + . ∂k G(k) G(k)
Corollary 2. (10) admits the following family of solutions in any interval properly contained in the set {k ∈ R+ | G(k) 6= 0}: e+ V ∗ (k) = C
Z
k
a R N 2 e− s G(s)
Γ(s) −
ρ dτ G(τ )
Rk
ds e
ρ ds G(s)
,
(13)
where Ce is a constant depending on the initial conditions of (10). Expression (13) is useful to characterize the standard cases. In particular, when the capital's production function is linear and it does not involve xed costs in absence of capital, it suggests us the following result:
Proposition 3. If G(k) is linear, G(0) = 0 and Γ(k) is an m-th degree polynomial in k , then one solution of (10) is an m-th degree polynomial in k as well. P l Proof. By assumption, call G(k) = αk and Γ(k) = m l=0 βl k . Replacing such functions in (13) yields:
e+ V ∗ (k) = C
Z
k
Z e+ = C ρ α
e = Ck +
l=0
βl sl −
"Z
Pm
k
αs
Pm
l=0
βl sl −
k
a R N 2 e− s
αs
ρ ατ
dτ
ds e
Rk
ρ αs ds
=
a ρ 2 N s− α ds k αρ =
m
ρ 1X a − ρ −1 βl sl− α −1 − s α α αN 2
!
# ρ
ds k α =
l=0 ρ
e α+ = Ck
m X l=0
βl a kl + , αl − ρ ρN 2
hence the solution corresponding to the choice
k. 8
e=0 C
is an
m-th
degree polynomial in
By plugging the solution into (12), it follows that:
Corollary 4. If the assumptions of Proposition 3 hold, the optimal feedback strategy is given by: q ∗ (k) =
N2
a(N − 1) . lβl l−1 k +c l=1 αl − ρ
(14)
Pm
From expression (14) it is immediate to deduce that rms'optimal strategy is a decreasing function of the initial capital, and it is feasible (positive) irrespective of the parameters, and consequently of the industry under examination.
2.3 Feedback solutions of the dierential game on a nite horizon Before investigating the nite horizon case, we should recall that in such cases sometimes the objective functional to be maximized is
Ji
plus a scrap value.
The scrap value
depends on the terminal value of the state, and can be seen as a kind of terminal prize or a nal product of the accumulation of the state (see several examples in [12]). Even if we are not going to consider any scrap value to avoid further technical complications, it can suggest a further interpretation for a corn-corn model. Provided that capital and output are proportional, the scrap value can represent a part of unsold output, which can be employed in a subsequent game starting after time
T,
as the initial condition for
the new capital to be accumulated. On a nite horizon
[t, T ],
where
0 ≤ t < T < ∞,
9
the HJB system of equations of
our problem takes the following form :
∂Vi − + ρVi (k) = max qi ∂t where
Vi
depends on both
k
(
) ∂Vi (Gi (k) − qi ) , − c qi − Γi (k) + PN ∂k i=1 qi !
a
(15)
and initial time t. Dierently from the in nite horizon case,
we must additionally take into account the transversality conditions on all
Vi :
lim Vi (k, t) = 0.
(16)
t−→T
Proposition 5. If G(k) is linear, G(0) = 0 and Γ(k) is an m-th degree polynomial in k , then the system (15) admits the following solution: −β0 + V (k, t) =
Proof.
ρ
a m X N 2 [1 − eρ(t−T ) ] + l=1
As in Proposition 3, call
G(k) = αk
and
βl [1 − e(ρ−αl)(t−T ) ]k l . αl − ρ
Γ(k) =
Pm
l=0
βl k l .
(17)
The maximization
of the r.h.s. of (15) yields:
aQ−i (qi + Q−i )
2
−c−
∂V = 0, ∂k
(18)
9 The dierence with respect to equation (9) in the in nite horizon case consists in the presence of the rst partial order derivative with respect to time.
9
then, by assuming symmetry on the relevant variables and functions, i.e. q1 = . . . = qN = q , V1 = . . . = VN = V , Γ1 (·) = . . . = ΓN (·) = Γ(·), G1 (·) = . . . = GN (·) = G(·), ∗ we have that (18) yields the following expression for the optimal strategy q :
q∗ =
a(N − 1) , ∂V 2 N +c ∂k
(19)
which must be plugged into (15) to achieve (the steps are analogous to those in Proposition 3, so we omit them):
m
−
X ∂V (k, t) a ∂V (k, t) + ρV (k, t) + − 2 = 0. βl k l − αk ∂t ∂k N
(20)
l=0
We guess a function of the following kind for
V (k, t):
m X
V (k, t) =
Al (t)k l ,
(21)
l=0
Al (t) ∈ C 1 ([t, T ]) and the transversality 0, 1, . . . , m. Plugging (21) into (20), we obtain: where
−
m X
A˙ l (t)k l + ρ
l=0
m X
Al (t)k l −
l=0
conditions are
m
m
l=0
l=0
Al (T ) = 0
for all
l =
X X a l + β k − αk lAl (t)k l−1 = 0, l N2
subsequently, all the coe cients of the powers of
k
are supposed to vanish, giving rise
to the following dynamic system:
a −A˙ 0 (t) + ρA0 (t) − 2 + β0 = 0 N ˙ −A1 (t) + ρA1 (t) − αA1 (t) + β1 = 0 . . . ˙ −Am (t) + ρAm (t) − mαAm (t) + βm = 0
.
By employing the transversality conditions, we achieve the following unique solutions:
−β0 + A0 (t) = Al (t) = for all
l = 1, . . . , m.
ρ
a N 2 [1 − eρ(t−T ) ],
βl [1 − e(ρ−αl)(t−T ) ], αl − ρ
Finally, substituting the found solutions in (21), we obtain the
optimal value function in closed form:
−β0 + V (k, t) =
ρ
a m X N 2 [1 − eρ(t−T ) ] + l=1
10
βl [1 − e(ρ−αl)(t−T ) ]k l . αl − ρ
(22)
Corollary 6. If the assumptions of Proposition 5 hold, the optimal feedback strategy is given by: q ∗ (k, t) = N
Pm 2
l=1
From (23) we can note that
a(N − 1) . lβl (ρ−αl)(t−T ) l−1 [1 − e ]k +c αl − ρ
q ∗ (k, T ) = q CN ,
(23)
meaning that the optimal strategy
tends to the Cournot-Nash equilibrium at nite time irrespective of parameters and of the initial capital endowment. A direct comparison between the equilibrium quantities will be carried out in the next Section.
3
Applications
3.1 The Cournot-Ramsey game In this well-known example, in order to produce, rms must accumulate capacity or physical capital
ki (t) over time.
We chose to consider the kinematic equations for capital
accumulation as in Ramsey ([28]), i.e. the following dynamic constraints:
(· k i (t) = Aki (t) − qi (t) − δki (t) ki (0) = ki0 > 0
t
and
δ>0
Ak
version of the
denotes the output produced by rm
Ramsey model, where
A=
i
(24)
the decay rate of capital, equal across rms. I.e., this is the familiar
where
Aki (t) = yi (t)
, at time
denotes
∂yi > 0 represents the constant growth rate at which output ∂ki
is produced as capital gets accumulated, essentially a measure of output productivity. The related cost will be
Γi (t) = bki (t),
with
b ≥ 0,
representing the aforementioned
opportunity cost of a unit of capacity. Because
Gi (ki (t)) = αki (t) = (A − δ)ki (t),
the previous Section
β0 = 0,
we can apply all the results collected in
10 . In particular, if we posit the following:
β1 = b,
β2 = . . . = βm = 0,
α = A − δ,
then the application of formulas (13), (14), (22) and (23) respectively entail:
•
In nite horizon: V ∗ (k) = q ∗ (k) =
10 Since
b a k+ , A−δ−ρ ρN 2
a(N − 1)(A − δ − ρ) . N 2 [b + c(A − δ − ρ)]
the dierential game at hand is a linear state one, i.e. one where, if we call Hi (·) the i-th
rm's Hamiltonian function, we have that:
∂ 2 Hi (·) ∂ 2 Hi (·) = = 0 ∂qi ∂kj ∂kj2
for all i, j = 1, . . . , N, (for more
on linear state games, see [12](ch. 7), inter alia), the open-loop equilibrium is subgame perfect as it coincides with the feedback equilibrium q∗ (k, t) yielded by the HJB equation. Moreover, this would hold true also in the more general case where yi (t) = Gi (ki (t)) , with G0i (ki (t)) > 0 and G00i (ki (t)) ≤ 0. That is, state-linearity is not necessary to yield subgame perfection in a Cournot-Ramsey game. For more on this issue, see [4] and [6].
11
•
Finite horizon:11 V ∗ (k, t) =
b[1 − e(ρ−A+δ)(t−T ) ] a ρ(t−T ) [1 − e ] + k, ρN 2 A−δ−ρ
q ∗ (k, t) =
a(N − 1)(A − δ − ρ) . N 2 [b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ)]
In order to ensure the feasibility (i.e., the positivity) of such strategies, we need suitable parametric assumptions. In the next three Propositions, we will choose
A
as
the crucial parameter, so as to refer all conditions to the level of output productivity.
Proposition 7. In the in nite horizon case, if one of the following conditions: 1. A > ρ + δ , b c
2. δ < A < − + ρ + δ , holds, then q∗ (k) is feasible. Proof.
A > δ to ensure accumulation of k˙ i (t) would be negative at all t). If A > ρ + δ , the numerator and the q ∗ (k) are both positive, implying its feasibility. On the other hand, if
In particular, in both cases we have to assume
capital (otherwise denominator of
b δ < A < − +ρ+δ , the numerator and the denominator are both negative, so feasibility c is ensured as well.
Moreover, here
q ∗ (k)
is constant whereas
k ∗ (t)
grows unbounded in that
A > δ.
We
can easily compare the optimal output with the one in the Cournot-Nash static setup:
Proposition 8. In the in nite horizon case, we have that: b c
1. if A > − + ρ + δ , then qCN > q∗ (k); b c
2. A < − + ρ + δ , then q∗ (k) > qCN . Proof.
It su ces to evaluate the dierence between optimal quantities:
q ∗ (k) − q CN =
a(N − 1)(A − δ − ρ) a(N − 1) ab(N − 1) − =− 2 . N 2 [b + c(A − δ − ρ)] N 2c N c[b + c(A − δ − ρ)]
In the nite horizon case, the situation is dierent and we need to establish a time interval over which
q ∗ (k, t) is feasible.
t = T the optimal strategy q(k, T ) = q CN .
However, note that at
coincides with the Cournot-Nash optimal strategy:
11 As an ancillary observation, it is worth noting that here, since the feedback optimal strategy coincides with the open-loop one and the strategic contributions cannot be distinguished under symmetry, the co-state variable at initial time appearing in the open-loop formulation of the game, which we omit here for brevity, can be appropriately considered as a shadow price of an additional unit of capacity, while, in general, this is true only of the partial derivative of the value function at initial state (for more on this aspect, see [3]).
12
Proposition 9. In the nite horizon case we have that: b c
1. If δ < A < − + ρ + δ , q∗ (k, t) is feasible for each t ∈ [0, T ). b c
2. If A > − + ρ + δ , q∗ (k, t) is feasible for each t ∈ (et, T ), where e t=T+
Proof. If
hc i 1 ln (A − ρ − δ) + 1 . ρ−A+δ b
We are going to consider the two dierent cases:
A < ρ + δ,
then the numerator is negative, hence we have to ensure that the
denominator is negative too:
b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ) < 0 ⇐⇒ . . . ⇐⇒
If the r.h.s. is
c ⇐⇒ e(ρ−A+δ)(t−T ) > (A − δ − ρ) + 1. b b ∗ negative, i.e. A < − + ρ + δ , q (k, t) is positive for c
all
t ∈ [0, T ).
If the
r.h.s. is positive, then:
t−T > hence if we call
e t=T +
hc i 1 ln (A − ρ − δ) + 1 , q ∗ (k, t) > 0 ρ−A+δ b
Subsequently, consider of
hc i 1 ln (A − ρ − δ) + 1 , ρ−A+δ b over
(e t, T ).
A > ρ + δ , we have to prove the positivity of the denominator
q ∗ (k, t): b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ) > 0 ⇐⇒ . . . ⇐⇒ hc i 1 ⇐⇒ t − T > ln (A − ρ − δ) + 1 , ρ−A+δ b
then
q ∗ (k, t) > 0
(e t, T ), meaning b A > − + ρ + δ. c
over
the time interval is
that the relevant condition for the restriction of
The next Figures sketch the behaviour of the optimal strategy in nite horizon, showing the dierence between its possible domains
(e t, T ]
and
[0, T ]
in compliance
with Proposition 9. As can be seen, the strategy is de ned over the whole interval only if
A
[0, T ]
is su ciently small, i.e. if the expansion of capital does not grow too fast.
13
q 6
q ∗ (k, t) q CN =
Figure 3.
If
a(N − 1) N 2c
•
e t b A > − + ρ + δ , q ∗ (k, t) c
T
t
is decreasing and it reaches
q CN
at
T.
q CN
at
T.
q 6
q ∗ (k, t) q CN =
Figure 4.
If
a(N − 1) N 2c
•
b A < − + ρ + δ , q ∗ (k, t) c
T
t
is decreasing and it reaches
Now we are going to focus our attention on the dynamic behaviour of the optimal capacity. If we substitute
q ∗ (k, t)
in the dynamic constraint (24), we can also achieve
the expression of the optimal state
k ∗ (t):
· a(N − 1)(A − δ − ρ) k (t) = (A − δ)k (t) − i i 2 N [b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ)] ki (0) = ki0
14
,
whose unique solution is given by:
k ∗ (t) =
k0 −
a(N − 1)(A − δ − ρ) N2
Z
t
0
e−(A−δ)s ds e(A−δ)t . b(1 − e(ρ−A+δ)(s−T ) ) + c(A − δ − ρ) (25)
Now the joint feasibility of
q ∗ (t)
and
k ∗ (t)
can be evaluated:
Proposition 10. Under the same assumptions as in Proposition 9, if and if a(N − 1)(A − δ − ρ) k0 > N2
Z
t
0
q ∗ (t)
is feasible
e−(A−δ)s ds b(1 − e(ρ−A+δ)(s−T ) ) + c(A − δ − ρ)
for all t ∈ (0, T ], then k∗ (t) is feasible as well. Proof.
It immediately follows from the positivity of
q ∗ (t)
The next Proposition provides the exact expression of
and from the expression (25).
k ∗ (t):
Proposition 11. The optimal state of the Cournot-Ramsey game is given by the following function: a(N − 1)(A − δ − ρ) e−(A−δ−ρ)T −ρt k ∗ (t) = k0 − (e − 1)+ N2 ρ −(A−δ−ρ)T b + c(A − δ − ρ) e − (b + c(A − δ − ρ))e−(A−δ−ρ)t + log e(A−δ)t . ρe−2(A−δ−ρ)T e−(A−δ−ρ)T − (b + c(A − δ − ρ)) (26)
Proof.
The explicit calculation of (25) needs the calculation of the related integral:
Z I(t) = 0 where
C1 = b + c(A − δ − ρ) Z
I(t) = 0
t
and
t
e−(A−δ)s ds, C1 − C2 e(ρ−A+δ)s
C2 = e−(ρ−A−δ)T .
We have that:
t Z (−C2 )e−(A−δ−ρ)s + C1 − C1 1 e−ρs C1 t 1 ds = + ds = C2 ρ 0 C2 0 eρs (C1 − C2 e(ρ−A−δ)s ) (−C2 )eρs (C1 − C2 e(ρ−A+δ)s ) C1 1 −ρt e −1 + = ρC2 C2
Then, applying the change of variable
dx ds = − , ρx
t
Z 0
1 ds. eρs (C1 − C2 e(ρ−A−δ)s )
x = e−ρs ,
leading to the change of dierential
we obtain:
1 −ρt C1 I(t) = e −1 − ρC2 ρC2 =
1 −ρt 1 e −1 + ρC2 ρC2
e−ρt
Z 1
Z
15
dx C1 − C2 x
1
e−ρt
A−δ−ρ ρ
dx 1−
C2 C1 x
A−δ−ρ ρ
ds =
ds =
=
1 −ρt 1 e −1 + ρC2 ρC2
Z
∞ X C2
1
e−ρt k=0
∞ 1 −ρt 1 X = e −1 + ρC2 ρC2
C2 C1
k
=
∞ 1 1 −ρt C1 X e −1 + ρC2 ρC2 C2 l=1
=
C2 C1
l
x
A−δ−ρ ρ
A−δ−ρ ρ
k ds =
k+1 1
k+1
k=0
C1
x
=
e−ρt
l −
∞ C1 X C2 l=1
C2 −(A−δ−ρ)t e C1 l
l =
1 −ρt C1 C1 C1 −(A−δ−ρ)t e − log 1 − −1 + + log 1 − e = ρC2 ρC22 C2 C2 1 −ρt C1 C2 − C1 e−(A−δ−ρ)t = e −1 + log . ρC2 ρC22 C2 − C1
(27)
Finally, plugging (27) into (25) yields the complete expression (26)of the optimal capital:
a(N − 1)(A − δ − ρ) I(t) e(A−δ)t = k ∗ (t) = k0 − N2 a(N − 1)(A − δ − ρ) e−(A−δ−ρ)T −ρt (e − 1)+ = k0 − N2 ρ −(A−δ−ρ)T b + c(A − δ − ρ) e − (b + c(A − δ − ρ))e−(A−δ−ρ)t + log e(A−δ)t . ρe−2(A−δ−ρ)T e−(A−δ−ρ)T − (b + c(A − δ − ρ))
Given (26), we can evaluate
12 the time interval :
k ∗ (T ),
i.e. the terminal value of capital at the end of
a(N − 1)(A − δ − ρ) a(N − 1)(A − δ − ρ) k (T ) = k0 − I(T ) e(A−δ)T = − 1 − eρT + N2 ρN 2 b + c(A − δ − ρ) 1 − [b + c(A − δ − ρ)] + −(3(A−δ)−2ρ)T log + k0 e(A−δ)T . (28) e 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T ∗
If we call
T,
π ∗ = π(q ∗ (T ), k ∗ (T ))
the pro t function evaluated at the terminal instant
we are able to compare it with the pro t function
π CN = π(QCN )
evaluated at the
steady state in the static Cournot problem as shown in Subsection 2.1. Namely, we have that:
a a − c qi∗ (T ) − bki∗ (T ) = − cq ∗ (T ) − bk ∗ (T ) = N qi∗ (T ) N a a(N − 1)(A − δ − ρ) = 2 − b k0 e(A−δ)T − (1 − eρT )+ N N2 b + c(A − δ − ρ) 1 − [b + c(A − δ − ρ)] + −(3(A−δ)−2ρ)T log . ρe 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T
π∗ =
(29)
12 We omit the most tedious calculations, reminding the readers that all of them are available upon request to the authors.
16
Proposition 12. Under the hypotheses of Propositions 9 and 10, if q∗ (t) and k∗ (t) are both feasible at all t ∈ [0, T ], then the Cournot-Nash equilibrium pro t is larger than the Ramsey-Cournot equilibrium level at the terminal instant. Proof.
It su ces to consider the dierence:
π ∗ − π CN =
a a − bk ∗ (T ) − 2 = −bk ∗ (T ) N2 N
which is strictly negative by the feasibility of
k ∗ (t)
at all instants, meaning that the
Cournot-Nash equilibrium pro t exceeds the Ramsey-Cournot terminal pro t.
Remark 13. It is worth noting that comparing the two optimal strategies in the static and in the dynamic cases, one immediately sees that the presence of capital accumulation in the dynamic game plays a key role in opening the way towards a solution to the indeterminacy issue aecting the static game as the marginal production cost c of the consumption good drops to zero. Essentially, if c = 0, no solution exists for the static game if no strategy space is compact, whereas in the dierential game with capacity accumulation q∗ (k, t) is well-de ned and feasible under suitable parametric conditions even when the marginal cost is zero, both over nite and in nite horizons. Propositions 8 and 12 suggest that at the subgame perfect equilibrium of the dynamic game the representative rm may produce either more or less but she earns higher pro ts when she plays the Cournot-Nash equilibrium of the static game, irrespective of the levels of marginal cost, opportunity cost, intensity of capacity accumulation growth and intertemporal discount rate. Having characterised the subgame perfect equilibrium of the dierential game, we can now proceed to the analysis of its application to horizontal mergers.
3.2 Horizontal mergers To illustrate the advantages of our approach to the feedback solution of the dierential oligopoly game
à la
Ramsey, we illustrate here its applicability to the analysis of the
private pro tability of a horizontal merger, and its welfare appraisal. As is well known, a lively debate has taken place on this topic from the 1980's, based upon static oligopoly models. A thorough overview of it is outside the scope of the present paper, and it will su ce to recollect a few essential aspects.
Examining
a Cournot industry with constant returns to scale, in [29] it is shown that a large proportion of the population of rms has to participate in the merger in order for the latter to be pro table. In particular, a striking result of their analysis is that, in the triopoly case, bilateral mergers are never pro table. Enriching the picture by allowing for the presence of convex variable costs and xed costs, one may nd a way out of this puzzle (in particular, see [23] and [14]). Now take the static Cournot game and examine the incentive for to merge horizontally, out of the initial
N.
M > 1
rms
Before proceeding, we deem necessary to
clearly de ne what is meant here as horizontal merger by a subset of the population of rms: the merger creates a single rm whose property is symmetrically distributed across the owners of the previously independent rms, and the resulting merged entity has a single control as well as a single state. This view of the merger can be justi ed on several grounds.
To begin with, our assumptions on product homogeneity imply
that productive capital is also homogeneous across rms, and therefore the merging
17
process involves that indeed all previously independent plans are homogenized into a single entity. Additionally, our approach to the modelling of the merger is in line with the antitrust norms currently adopted both in the EU and in North America (US and Canada), establishing that the rms proposing a merger shall reduce their collective capacity endowment lest they attain a dominant position after the merger has taken place (see [8] and ch.
5 in [22], among others).
Of course, the capacity and output
of the rm resulting from the merger are necessarily larger than those associated to a generic rm if the merger does not occur, but this is simply the natural consequence of the decrease in the population of rms.
Therefore, we are going to compare the
performance of a rm in two games, alternatively characterized by
N
or
N −M +1
rms, all else equal. After the merger (if it does take place), there remain call
π CN (j)
N − M + 1 rms. If we j rms, without
the pro t of a rm in the Cournot static game among
distinguishing the original ones from those generated by the merger, we can prove the following:
Proposition 14. In the static Cournot game with hyperbolic inverse demand, the √ merger is pro table if and only if N < M + M . Proof.
Pro tability holds when
π CN (N − M + 1) a a > π CN (N ) ⇐⇒ 2 > N 2 ⇐⇒ M M (N − M + 1) ⇐⇒ N 2 > M (N − M + 1)2 ⇐⇒ . . . ⇐⇒ N 2 − 2M N − M (1 − M ) < 0, √ √ which entails N ∈ M − M , M + M . Since N ≥ M , the necessary and su cient √ M. condition boils down to N < M + It is easily checked that, contrary to [29], if
N = 3
and
M = 2,
the merger is
pro table. On the other hand, if we consider the terminal outcome of the dierential game over nite horizon, in compliance with the above notation, we can assess the pro t incentive scheme for an
M - rm merger in the dynamic framework too, sticking to the assumption
of one state and one control variable per rm. As for the assumptions of Proposition 9, ensuring the feasibility of the optimal strategy on
[0, T ],
we can state the following:
Proposition 15. If a horizontal merger of M rms is pro table in the Cournot static b game, then if δ < A < − + δ + ρ, the same merger is pro table in the Cournot-Ramsey c game on the horizon [0, T ] as well. Proof.
Provided that the net revenue at the equilibrium of the Cournot static game
is
and the net revenue of the same game when a merger of
a N2
a , (N − M + 1)2
M
rms occurs is
the pro tability of a merger in the Cournot-Ramsey game is measured
by
1 M
a ∗ − bkN −M +1 (T ) (N − M + 1)2
18
>
a ∗ − bkN (T ), N2
(30)
where we called
N −l
kl∗ (T )
the capital at time
T
under circumstances where a merger of
rms took place. In order to simplify the notation, call
g(T ) =
1 − eρT +
b + c(A − δ − ρ) e−(3(A−δ)−2ρ)T
log
1 − [b + c(A − δ − ρ)] 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T
as in (28). The inequality (30) becomes:
a b a(N − M )(A − δ − ρ) (A−δ)T − g(T ) + k e > − 0 M (N − M + 1)2 M (N − M + 1)2 a(N − 1)(A − δ − ρ) a (A−δ)T g(T ) + k0 e , > 2 −b − N N2 which amounts to
N 2 − M (N − M + 1)2 1 (A−δ)T − bk e − 1 + 0 M N 2 (N − M + 1)2 M N −1 N −M − > 0. +ba(A − δ − ρ)g(T ) M (N − M + 1)2 N2
a
(31)
M - rm merger is pro table M > 1, whereas the remaining
By Proposition 14, the rst term of (31) is positive when the in the static framework, the second term is positive for all term is positive for strategy over
b A < − +δ+ρ < δ+ρ c
[0, T ]
(the condition ensuring feasibility of the
by Proposition 9) if and only if the quantity
N −M M (N −M +1)2
−
N −1 N 2 is
negative. In fact, we have that:
N −M N −1 N −M 1 1 − = − + 2 < 2 2 2 M (N − M + 1) N M (N − M + 1) N N
M (N − M + 1)2 .
Hence, this completes the
proof. Taken together, these facts entail that the interval wherein the
M - rm
merger is
pro table may be the same in the dynamic setup and in the static one, given that the measure of output productivity
A
is small enough.
The examination of the welfare consequences of a merger is omitted, as it goes without saying that any merger would diminish social welfare, both in the static as well as in the dynamic setting. This is trivially due to the fact that the damage caused to
13
consumer surplus always outweighs the increase in industry pro ts.
13 In line of principle, a merger could allow for some reduction in the total opportunity costs for the industry, giving rise to a possible e ciency defense argument (see [14]). Although we omit the related calculations for brevity, it is quickly checked that this never outweighs the loss in consumer surplus necessarily generated by any merger. Hence, in this model the e ciency argument cannot be advocated to justify the merger itself.
19
4
Concluding remarks
Most of the existing literature on oligopoly theory postulates the presence of linear demand functions, this being quite speci c in itself and, in general, at odd with empirical evidence.
With this in mind, we have constructed a dynamic oligopoly model based
on a non-linear demand.
In particular, we have characterised the subgame perfect
equilibrium of a dynamic Cournot game with inverse hyperbolic demand and costly capacity accumulation, showing that the feedback solution, coincident with the openloop one, is subgame perfect. We have fully carried out the calculation of the optimal value functions and of the strategies of the dierential game subject to a Ramseytype dynamic constraint.
Then, to illustrate the applicability of our framework, we
have employed the model to analyse the feasibility of horizontal mergers in both static and dynamic settings, nding out appropriate parametric conditions under which the pro tability of a merger in a static game implies the pro tability in a dynamic game as well. Possible future developments of our ndings consist in the analysis of the feedback information structure of further dierential oligopoly games endowed with a hyperbolic inverse demand function, possibly in presence of more complex dynamic constraints dealing, e.g., with issues related to international trade, environmental and resource economics and R&D activities.
Acknowledgements We would like to thank Davide Dragone, Mauro Rota, Alessandro Tampieri, Stefan Wrzaczek and the audience at the EURO-INFORMS Joint International Conference in Rome for stimulating discussions and insightful suggestions.
We are also grateful to
three anonymous referees who greatly helped us improve the paper with their precious comments. The usual disclaimer applies.
20
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23
∗,§
Arsen Palestini
]
*Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy; fax +39-051-2092664 [email protected] ENCORE, University of Amsterdam Roeterstraat 11, WB1018 Amsterdam, The Netherlands ] MEMOTEF, Sapienza University of Rome Via del Castro Laurenziano 9, 00161 Rome, Italy; [email protected] July 24, 2013
Abstract To safeguard analytical tractability and the concavity of objective functions, the vast majority of models belonging to oligopoly theory relies on the restrictive assumption of linear demand functions. Here we lay out the analytical solution of a dierential Cournot game with hyperbolic inverse demand, where rms accumulate capacity over time à la Ramsey. The subgame perfect equilibrium is characterised via the Hamilton-Jacobi-Bellman equations solved in closed form both on in nite and on nite horizon setups. To illustrate the applicability of our model and its implications, we analyse the feasibility of horizontal mergers in both static and dynamic settings, and nd appropriate conditions for their pro tability under both circumstances. Static pro tability of a merger implies dynamic pro tability of the same merger. It appears that such a demand structure makes mergers more likely to occur than they would on the basis of the standard linear inverse demand. JEL Classi cation: C73, L13 Keywords: capacity, dierential game, Markov-perfect equilibrium, HamiltonJacobi-Bellman equation, horizontal mergers
1
Introduction
Most of the existing literature on oligopoly theory (either static or dynamic) assumes linear inverse demand functions, as this, in addition to simplifying calculations, also ensures both concavity and uniqueness of the equilibrium, which, in general, wouldn't be warranted in presence of convex demand systems (see [16] and [10],
inter alia).
How-
ever, the use of linear demand function is in sharp contrast with the standard microeconomic approach to consumer behaviour, where the widespread adoption of Cobb-Douglas preferences (or their log-linear a ne transformation) yields hyperbolic inverse demand
1
functions.
The same applies to the so-called quasi-linear utility function, concave in
consumption and linear in money, that again yields a convex demand system. Indeed, both preference structures share the common property of producing isoelastic demand
1
functions.
Furthermore, such demand functions have been employed in rent-seeking games, where important contributions have been provided by [7] and [32], which showed the strict relations between the rent-seeking games and the Cournot oligopoly models. The isoelastic demand function has been widely analyzed in static settings (duopoly, [24], triopoly, [25]), devoting a particular attention to the stability of equilibrium and the possible chaotic implications (see also [26] and [27]). In fact, this is sometimes openly referred to in the eld of industrial organization, where researchers mention the opportunity of dealing with non-linear demand curves,
2 Additionally, the econo-
and then promptly leave it aside for the sake of tractability.
metric approach to demand theory has produced the highest eorts in building up a robust approach to the estimation of non-linear individual and market inverse demand functions, yielding a large empirical evidence in this direction.
3
A relevant market demand structure was proposed by Anderson and Engers at the beginning of the 90s ([1]), which deserves some discussion: quantity of a homogeneous good and 1
p(Q)
calling
Q
the aggregate
the inverse demand function, they assumed
the form
p(Q) = (a − Q) α
price).
α: α > 1, concave if α < 1, linear if α = 1, as is outlined by the following Figures.
to generalize the linear case (a
>0
is the market reservation
Such inverse demand leads to dierent scenarios depending on parameter
convex if
p6
p6
1
1
p(Q) = (a − Q) α
p(Q) = (a − Q) α
Q
Figure 1.
Anderson and Engers inverse demand functions for
α>1
and
Q α 0.
On the supply side, production entails a total cost parameter measuring marginal production cost. Cournot-Nash; therefore, rm
i
chooses
qi
Ci = cqi , where c > 0 is a constant
Market competition takes place
so as to maximise pro ts
à la
πi = (p(Q) − c) qi .
This entails that the following rst order conditions must be satis ed (given their form, it is not necessary to assume interior solutions):
aQ−i ∂πi = 2 −c=0 ∂qi (qi + Q−i ) where
Q−i ≡
(4)
P
j6=i qj . The associated second order condition:
∂ 2 πi 2aQ−i =− 3 0 for all i = 1, . . . , N . Then, imposing the symmetry condition qi = qj = q , one obtains the individual Cournot-Nash equilibrium a (N − 1) a CN output q = , yielding pro ts π CN = 2 . If the N rms were operating N 2c N a ∗ ∗ . under perfect competition, then p = c and therefore q = Nc
is always met when on all outputs, i.e.
It is apparent that the above solutions (i.e., both the Cournot-Nash equilibrium and the perfectly competitive equilibrium) are well-de ned and feasible for all
c > 0.
In the remainder of the paper, we will turn our attention to a dierential game where the demand structure is the same as here.
We will separately investigate the in nite
horizon and nite horizon cases.
2.2 Feedback solutions of the dierential game on an in nite horizon t ∈ [0 , +∞) , and which is served by N qi (t) ∈ (0, +∞) de ne the quantity sold by à la Cournot, the demand function at time t being:
We are going to consider a market existing over rms producing a homogeneous good. Let rm
i
at time
t.
Firms compete
a p (t) = , Q (t)
Q (t) =
N X
qi (t) .
(6)
i=1
Ci (t) = cqi (t) , where C > 0. ki (t) ∈ [0, +∞) over time. i at time t, we assume that ki
In order to produce, rms bear linear instantaneous costs
Moreover, they must accumulate capacity or physical capital If we denote with
yi (t)
aects the production of
the output produced by rm
yi
in the sense that
5
∂yi > 0. ∂ki
Capital accumulates as a result
of intertemporal relocation of unsold output
yi (t)−qi (t).6
This can be interpreted in two
ways. The rst consists in viewing this setup as a corn-corn model, where unsold output is reintroduced in the production process. This interpretation is admittedly unrealistic (although systematically mentioned in the whole macroeconomic debate stemming from Ramsey [28]) as it implies that the nal good and the capital input are homogeneous. The second consists in thinking, a bit more realistically, of a two-sector economy where there exists an industry producing the capital input which can be traded against the nal good at a price equal to one (for further discussion, see [5]), this channel being used to nance investment projects as an alternative to borrowing from the banking sector (which otherwise should be explicitly modelled). In a nutshell, the Ramsey dynamics is a blackbox hiding a market where rms sell part of their production at a xed price for fundraising aimed at nancing their growth. Unlike the standard macroeconomic approach to growth models in a Ramsey fashion, here we will allow for the presence of an instantaneous cost of holding installed capacity. This cost will be refer to
ki (t)
∂Γi ∂ki
as the
Γi (ki (t)),
possibly asymmetric across rms. In the remainder, we will
as a measure of the
i-th
opportunity cost of a unit of capacity.
state variable subject to the following dynamic constraints:
· dki (t) ≡ k i (t) = Gi (ki (t)) − qi (t) dt k (t ) = k > 0 i 0 i0 where rm's
We will employ
,
(7)
Gi (ki (t)) is a C 2 (R+ ) function aecting the growth dynamics of capital. strategic variable is qi (t), while the i-th rm's state variable is ki (t).
The i-th
A standard assumption in dynamic economic models concerns the kind of discounting: since the game takes place in a unique market, all rms discount pro ts at a same constant rate
ρ ≥ 0,
possibly depending on monetary factors such as the expected rate
of currency depreciation. The problem of rm
i
is to choose the output level
qi (t)
so as to maximise its own
discounted ow of pro ts (from now on, we will omit time arguments whenever possible):
Z
∞
max Ji (ki0 , t0 ) ≡
qi ∈R+
[(p (Q(τ ), τ ) − c) qi (τ ) − Γi (ki (τ ))] e−ρ(τ −t0 ) dτ =
t0
Z
∞
= t0
"
a PN
i=1 qi (τ )
!
#
− c qi (τ ) − Γi (ki (τ )) e−ρ(τ −t0 ) dτ.
(8)
We are going to adopt the dynamic programming approach, relying on the Hamilton-
7
Jacobi-Bellman equations , generally considered as the most powerful tool in dynamic Game Theory and in a large number of economic models (both in discrete and continuous time). The issue to be tackled is the determination of the
i-th
Vi (ki0 , t0 ),
i.e. the optimal value of
objective functional (8), over the set of all feasible paths
qi : [t0 , ∞) −→ R.
When the involved time horizon is in nite, the optimal value function does not depend on the initial time, but only on the initial state.
In both cases, each optimal value
6 Of course, capacity decumulates whenever y (t) − q (t) ≤ 0. i i 7 In the 50s Richard Bellman extended the Hamilton-Jacobi theory,
century to describe and solve problems in Classical Mechanics.
6
which was developed in the 19th
function can be achieved as the solution of a Hamilton-Jacobi-Bellman (HJB) equation (for the complete derivation, see [12], Chapter 3). Call
ki0 = k .
Denoting with
Vi (k)
i-th
the
optimal value function for (8), the
Hamilton-Jacobi-Bellman (HJB) system of equations reads as follows:
( ρVi (k) = max qi
for all
i = 1, . . . , N .
!
) ∂Vi − c qi − Γi (k) + (Gi (k) − qi ) , PN ∂k i=1 qi a
Note that because
rst order partial derivatives of
Vi
Ji
only depends on the
i-th
(9)
capital, in (9) the
with respect to the remaining state variables do not
appear. To proceed with the analytical solution of the feedback problem, we are going to determine the symmetric Nash equilibrium of the game. Suitable symmetry assumptions are commonly employed both in Static and in Dierential Game Theory, in that the players are identical rms, having an identical initial capital endowment and identical
8
payo structures . This leads us to introduce suitable symmetry conditions from the beginning of the procedure: one is
qi = qj
for all
i 6= j, saying that the equilibrium quantity
must be symmetric across all rms. The assumption of symmetry across capitals states that, from the standpoint of a generic rm
i,
the rivals' capacities (and therefore also
their weights in the value function) must be equal when the respective growth dynamics and cost structures are equal.
Proposition 1. Assuming symmetry across all variables, the HJB equation of the problem is given by: a V (k) Γ(k) − N 2 ∂V =ρ + , ∂k G(k) G(k)
Proof.
Maximizing the r.h.s. of (9) with respect to
aQ−i 2
(qi + Q−i )
−c−
qi
(10)
yields:
∂Vi = 0, ∂k
(11)
then, by assuming symmetry on the relevant variables and functions, i.e. q1 = . . . = qN = q , V1 = . . . = VN = V , Γ1 (·) = . . . = ΓN (·) = Γ(·), G1 (·) = . . . = GN (·) = G(·), ∗ we have that (11) yields the following expression for the optimal strategy q :
q∗ =
a(N − 1) , ∂V N2 +c ∂k
(12)
which must be plugged into (9) to achieve:
∂V N + c ∂k ρV (k) = − c N −1
a(N − 1) ∂V a(N − 1) G(k) − −Γ(k)+ ⇐⇒ ∂V ∂V ∂k 2 2 +c +c N N ∂k ∂k
8 Note that in such a problem some asymmetric elements might be admitted when they do not change the HJB dramatically, for example growth rates or marginal cost parameters, but that goes beyond the scope of our paper
7
∂V a N +c − 1) ∂V ∂V ∂k a(N − Γ(k) + ⇐⇒ ρV (k) = G(k) − ⇐⇒ ∂V ∂V ∂k ∂k 2 2 N N +c +c ∂k ∂k ∂V ∂V a ∂V N ⇐⇒ ρV (k) + Γ(k) − G(k) = + c − (N − 1) ⇐⇒ ∂V ∂k ∂k ∂k N2 +c ∂k ∂V ∂V a ⇐⇒ ρV (k) + Γ(k) − G(k) = + c ⇐⇒ ∂V ∂k ∂k N2 +c ∂k a ∂V V (k) Γ(k) − N 2 ⇐⇒ =ρ + . ∂k G(k) G(k)
Corollary 2. (10) admits the following family of solutions in any interval properly contained in the set {k ∈ R+ | G(k) 6= 0}: e+ V ∗ (k) = C
Z
k
a R N 2 e− s G(s)
Γ(s) −
ρ dτ G(τ )
Rk
ds e
ρ ds G(s)
,
(13)
where Ce is a constant depending on the initial conditions of (10). Expression (13) is useful to characterize the standard cases. In particular, when the capital's production function is linear and it does not involve xed costs in absence of capital, it suggests us the following result:
Proposition 3. If G(k) is linear, G(0) = 0 and Γ(k) is an m-th degree polynomial in k , then one solution of (10) is an m-th degree polynomial in k as well. P l Proof. By assumption, call G(k) = αk and Γ(k) = m l=0 βl k . Replacing such functions in (13) yields:
e+ V ∗ (k) = C
Z
k
Z e+ = C ρ α
e = Ck +
l=0
βl sl −
"Z
Pm
k
αs
Pm
l=0
βl sl −
k
a R N 2 e− s
αs
ρ ατ
dτ
ds e
Rk
ρ αs ds
=
a ρ 2 N s− α ds k αρ =
m
ρ 1X a − ρ −1 βl sl− α −1 − s α α αN 2
!
# ρ
ds k α =
l=0 ρ
e α+ = Ck
m X l=0
βl a kl + , αl − ρ ρN 2
hence the solution corresponding to the choice
k. 8
e=0 C
is an
m-th
degree polynomial in
By plugging the solution into (12), it follows that:
Corollary 4. If the assumptions of Proposition 3 hold, the optimal feedback strategy is given by: q ∗ (k) =
N2
a(N − 1) . lβl l−1 k +c l=1 αl − ρ
(14)
Pm
From expression (14) it is immediate to deduce that rms'optimal strategy is a decreasing function of the initial capital, and it is feasible (positive) irrespective of the parameters, and consequently of the industry under examination.
2.3 Feedback solutions of the dierential game on a nite horizon Before investigating the nite horizon case, we should recall that in such cases sometimes the objective functional to be maximized is
Ji
plus a scrap value.
The scrap value
depends on the terminal value of the state, and can be seen as a kind of terminal prize or a nal product of the accumulation of the state (see several examples in [12]). Even if we are not going to consider any scrap value to avoid further technical complications, it can suggest a further interpretation for a corn-corn model. Provided that capital and output are proportional, the scrap value can represent a part of unsold output, which can be employed in a subsequent game starting after time
T,
as the initial condition for
the new capital to be accumulated. On a nite horizon
[t, T ],
where
0 ≤ t < T < ∞,
9
the HJB system of equations of
our problem takes the following form :
∂Vi − + ρVi (k) = max qi ∂t where
Vi
depends on both
k
(
) ∂Vi (Gi (k) − qi ) , − c qi − Γi (k) + PN ∂k i=1 qi !
a
(15)
and initial time t. Dierently from the in nite horizon case,
we must additionally take into account the transversality conditions on all
Vi :
lim Vi (k, t) = 0.
(16)
t−→T
Proposition 5. If G(k) is linear, G(0) = 0 and Γ(k) is an m-th degree polynomial in k , then the system (15) admits the following solution: −β0 + V (k, t) =
Proof.
ρ
a m X N 2 [1 − eρ(t−T ) ] + l=1
As in Proposition 3, call
G(k) = αk
and
βl [1 − e(ρ−αl)(t−T ) ]k l . αl − ρ
Γ(k) =
Pm
l=0
βl k l .
(17)
The maximization
of the r.h.s. of (15) yields:
aQ−i (qi + Q−i )
2
−c−
∂V = 0, ∂k
(18)
9 The dierence with respect to equation (9) in the in nite horizon case consists in the presence of the rst partial order derivative with respect to time.
9
then, by assuming symmetry on the relevant variables and functions, i.e. q1 = . . . = qN = q , V1 = . . . = VN = V , Γ1 (·) = . . . = ΓN (·) = Γ(·), G1 (·) = . . . = GN (·) = G(·), ∗ we have that (18) yields the following expression for the optimal strategy q :
q∗ =
a(N − 1) , ∂V 2 N +c ∂k
(19)
which must be plugged into (15) to achieve (the steps are analogous to those in Proposition 3, so we omit them):
m
−
X ∂V (k, t) a ∂V (k, t) + ρV (k, t) + − 2 = 0. βl k l − αk ∂t ∂k N
(20)
l=0
We guess a function of the following kind for
V (k, t):
m X
V (k, t) =
Al (t)k l ,
(21)
l=0
Al (t) ∈ C 1 ([t, T ]) and the transversality 0, 1, . . . , m. Plugging (21) into (20), we obtain: where
−
m X
A˙ l (t)k l + ρ
l=0
m X
Al (t)k l −
l=0
conditions are
m
m
l=0
l=0
Al (T ) = 0
for all
l =
X X a l + β k − αk lAl (t)k l−1 = 0, l N2
subsequently, all the coe cients of the powers of
k
are supposed to vanish, giving rise
to the following dynamic system:
a −A˙ 0 (t) + ρA0 (t) − 2 + β0 = 0 N ˙ −A1 (t) + ρA1 (t) − αA1 (t) + β1 = 0 . . . ˙ −Am (t) + ρAm (t) − mαAm (t) + βm = 0
.
By employing the transversality conditions, we achieve the following unique solutions:
−β0 + A0 (t) = Al (t) = for all
l = 1, . . . , m.
ρ
a N 2 [1 − eρ(t−T ) ],
βl [1 − e(ρ−αl)(t−T ) ], αl − ρ
Finally, substituting the found solutions in (21), we obtain the
optimal value function in closed form:
−β0 + V (k, t) =
ρ
a m X N 2 [1 − eρ(t−T ) ] + l=1
10
βl [1 − e(ρ−αl)(t−T ) ]k l . αl − ρ
(22)
Corollary 6. If the assumptions of Proposition 5 hold, the optimal feedback strategy is given by: q ∗ (k, t) = N
Pm 2
l=1
From (23) we can note that
a(N − 1) . lβl (ρ−αl)(t−T ) l−1 [1 − e ]k +c αl − ρ
q ∗ (k, T ) = q CN ,
(23)
meaning that the optimal strategy
tends to the Cournot-Nash equilibrium at nite time irrespective of parameters and of the initial capital endowment. A direct comparison between the equilibrium quantities will be carried out in the next Section.
3
Applications
3.1 The Cournot-Ramsey game In this well-known example, in order to produce, rms must accumulate capacity or physical capital
ki (t) over time.
We chose to consider the kinematic equations for capital
accumulation as in Ramsey ([28]), i.e. the following dynamic constraints:
(· k i (t) = Aki (t) − qi (t) − δki (t) ki (0) = ki0 > 0
t
and
δ>0
Ak
version of the
denotes the output produced by rm
Ramsey model, where
A=
i
(24)
the decay rate of capital, equal across rms. I.e., this is the familiar
where
Aki (t) = yi (t)
, at time
denotes
∂yi > 0 represents the constant growth rate at which output ∂ki
is produced as capital gets accumulated, essentially a measure of output productivity. The related cost will be
Γi (t) = bki (t),
with
b ≥ 0,
representing the aforementioned
opportunity cost of a unit of capacity. Because
Gi (ki (t)) = αki (t) = (A − δ)ki (t),
the previous Section
β0 = 0,
we can apply all the results collected in
10 . In particular, if we posit the following:
β1 = b,
β2 = . . . = βm = 0,
α = A − δ,
then the application of formulas (13), (14), (22) and (23) respectively entail:
•
In nite horizon: V ∗ (k) = q ∗ (k) =
10 Since
b a k+ , A−δ−ρ ρN 2
a(N − 1)(A − δ − ρ) . N 2 [b + c(A − δ − ρ)]
the dierential game at hand is a linear state one, i.e. one where, if we call Hi (·) the i-th
rm's Hamiltonian function, we have that:
∂ 2 Hi (·) ∂ 2 Hi (·) = = 0 ∂qi ∂kj ∂kj2
for all i, j = 1, . . . , N, (for more
on linear state games, see [12](ch. 7), inter alia), the open-loop equilibrium is subgame perfect as it coincides with the feedback equilibrium q∗ (k, t) yielded by the HJB equation. Moreover, this would hold true also in the more general case where yi (t) = Gi (ki (t)) , with G0i (ki (t)) > 0 and G00i (ki (t)) ≤ 0. That is, state-linearity is not necessary to yield subgame perfection in a Cournot-Ramsey game. For more on this issue, see [4] and [6].
11
•
Finite horizon:11 V ∗ (k, t) =
b[1 − e(ρ−A+δ)(t−T ) ] a ρ(t−T ) [1 − e ] + k, ρN 2 A−δ−ρ
q ∗ (k, t) =
a(N − 1)(A − δ − ρ) . N 2 [b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ)]
In order to ensure the feasibility (i.e., the positivity) of such strategies, we need suitable parametric assumptions. In the next three Propositions, we will choose
A
as
the crucial parameter, so as to refer all conditions to the level of output productivity.
Proposition 7. In the in nite horizon case, if one of the following conditions: 1. A > ρ + δ , b c
2. δ < A < − + ρ + δ , holds, then q∗ (k) is feasible. Proof.
A > δ to ensure accumulation of k˙ i (t) would be negative at all t). If A > ρ + δ , the numerator and the q ∗ (k) are both positive, implying its feasibility. On the other hand, if
In particular, in both cases we have to assume
capital (otherwise denominator of
b δ < A < − +ρ+δ , the numerator and the denominator are both negative, so feasibility c is ensured as well.
Moreover, here
q ∗ (k)
is constant whereas
k ∗ (t)
grows unbounded in that
A > δ.
We
can easily compare the optimal output with the one in the Cournot-Nash static setup:
Proposition 8. In the in nite horizon case, we have that: b c
1. if A > − + ρ + δ , then qCN > q∗ (k); b c
2. A < − + ρ + δ , then q∗ (k) > qCN . Proof.
It su ces to evaluate the dierence between optimal quantities:
q ∗ (k) − q CN =
a(N − 1)(A − δ − ρ) a(N − 1) ab(N − 1) − =− 2 . N 2 [b + c(A − δ − ρ)] N 2c N c[b + c(A − δ − ρ)]
In the nite horizon case, the situation is dierent and we need to establish a time interval over which
q ∗ (k, t) is feasible.
t = T the optimal strategy q(k, T ) = q CN .
However, note that at
coincides with the Cournot-Nash optimal strategy:
11 As an ancillary observation, it is worth noting that here, since the feedback optimal strategy coincides with the open-loop one and the strategic contributions cannot be distinguished under symmetry, the co-state variable at initial time appearing in the open-loop formulation of the game, which we omit here for brevity, can be appropriately considered as a shadow price of an additional unit of capacity, while, in general, this is true only of the partial derivative of the value function at initial state (for more on this aspect, see [3]).
12
Proposition 9. In the nite horizon case we have that: b c
1. If δ < A < − + ρ + δ , q∗ (k, t) is feasible for each t ∈ [0, T ). b c
2. If A > − + ρ + δ , q∗ (k, t) is feasible for each t ∈ (et, T ), where e t=T+
Proof. If
hc i 1 ln (A − ρ − δ) + 1 . ρ−A+δ b
We are going to consider the two dierent cases:
A < ρ + δ,
then the numerator is negative, hence we have to ensure that the
denominator is negative too:
b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ) < 0 ⇐⇒ . . . ⇐⇒
If the r.h.s. is
c ⇐⇒ e(ρ−A+δ)(t−T ) > (A − δ − ρ) + 1. b b ∗ negative, i.e. A < − + ρ + δ , q (k, t) is positive for c
all
t ∈ [0, T ).
If the
r.h.s. is positive, then:
t−T > hence if we call
e t=T +
hc i 1 ln (A − ρ − δ) + 1 , q ∗ (k, t) > 0 ρ−A+δ b
Subsequently, consider of
hc i 1 ln (A − ρ − δ) + 1 , ρ−A+δ b over
(e t, T ).
A > ρ + δ , we have to prove the positivity of the denominator
q ∗ (k, t): b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ) > 0 ⇐⇒ . . . ⇐⇒ hc i 1 ⇐⇒ t − T > ln (A − ρ − δ) + 1 , ρ−A+δ b
then
q ∗ (k, t) > 0
(e t, T ), meaning b A > − + ρ + δ. c
over
the time interval is
that the relevant condition for the restriction of
The next Figures sketch the behaviour of the optimal strategy in nite horizon, showing the dierence between its possible domains
(e t, T ]
and
[0, T ]
in compliance
with Proposition 9. As can be seen, the strategy is de ned over the whole interval only if
A
[0, T ]
is su ciently small, i.e. if the expansion of capital does not grow too fast.
13
q 6
q ∗ (k, t) q CN =
Figure 3.
If
a(N − 1) N 2c
•
e t b A > − + ρ + δ , q ∗ (k, t) c
T
t
is decreasing and it reaches
q CN
at
T.
q CN
at
T.
q 6
q ∗ (k, t) q CN =
Figure 4.
If
a(N − 1) N 2c
•
b A < − + ρ + δ , q ∗ (k, t) c
T
t
is decreasing and it reaches
Now we are going to focus our attention on the dynamic behaviour of the optimal capacity. If we substitute
q ∗ (k, t)
in the dynamic constraint (24), we can also achieve
the expression of the optimal state
k ∗ (t):
· a(N − 1)(A − δ − ρ) k (t) = (A − δ)k (t) − i i 2 N [b(1 − e(ρ−A+δ)(t−T ) ) + c(A − δ − ρ)] ki (0) = ki0
14
,
whose unique solution is given by:
k ∗ (t) =
k0 −
a(N − 1)(A − δ − ρ) N2
Z
t
0
e−(A−δ)s ds e(A−δ)t . b(1 − e(ρ−A+δ)(s−T ) ) + c(A − δ − ρ) (25)
Now the joint feasibility of
q ∗ (t)
and
k ∗ (t)
can be evaluated:
Proposition 10. Under the same assumptions as in Proposition 9, if and if a(N − 1)(A − δ − ρ) k0 > N2
Z
t
0
q ∗ (t)
is feasible
e−(A−δ)s ds b(1 − e(ρ−A+δ)(s−T ) ) + c(A − δ − ρ)
for all t ∈ (0, T ], then k∗ (t) is feasible as well. Proof.
It immediately follows from the positivity of
q ∗ (t)
The next Proposition provides the exact expression of
and from the expression (25).
k ∗ (t):
Proposition 11. The optimal state of the Cournot-Ramsey game is given by the following function: a(N − 1)(A − δ − ρ) e−(A−δ−ρ)T −ρt k ∗ (t) = k0 − (e − 1)+ N2 ρ −(A−δ−ρ)T b + c(A − δ − ρ) e − (b + c(A − δ − ρ))e−(A−δ−ρ)t + log e(A−δ)t . ρe−2(A−δ−ρ)T e−(A−δ−ρ)T − (b + c(A − δ − ρ)) (26)
Proof.
The explicit calculation of (25) needs the calculation of the related integral:
Z I(t) = 0 where
C1 = b + c(A − δ − ρ) Z
I(t) = 0
t
and
t
e−(A−δ)s ds, C1 − C2 e(ρ−A+δ)s
C2 = e−(ρ−A−δ)T .
We have that:
t Z (−C2 )e−(A−δ−ρ)s + C1 − C1 1 e−ρs C1 t 1 ds = + ds = C2 ρ 0 C2 0 eρs (C1 − C2 e(ρ−A−δ)s ) (−C2 )eρs (C1 − C2 e(ρ−A+δ)s ) C1 1 −ρt e −1 + = ρC2 C2
Then, applying the change of variable
dx ds = − , ρx
t
Z 0
1 ds. eρs (C1 − C2 e(ρ−A−δ)s )
x = e−ρs ,
leading to the change of dierential
we obtain:
1 −ρt C1 I(t) = e −1 − ρC2 ρC2 =
1 −ρt 1 e −1 + ρC2 ρC2
e−ρt
Z 1
Z
15
dx C1 − C2 x
1
e−ρt
A−δ−ρ ρ
dx 1−
C2 C1 x
A−δ−ρ ρ
ds =
ds =
=
1 −ρt 1 e −1 + ρC2 ρC2
Z
∞ X C2
1
e−ρt k=0
∞ 1 −ρt 1 X = e −1 + ρC2 ρC2
C2 C1
k
=
∞ 1 1 −ρt C1 X e −1 + ρC2 ρC2 C2 l=1
=
C2 C1
l
x
A−δ−ρ ρ
A−δ−ρ ρ
k ds =
k+1 1
k+1
k=0
C1
x
=
e−ρt
l −
∞ C1 X C2 l=1
C2 −(A−δ−ρ)t e C1 l
l =
1 −ρt C1 C1 C1 −(A−δ−ρ)t e − log 1 − −1 + + log 1 − e = ρC2 ρC22 C2 C2 1 −ρt C1 C2 − C1 e−(A−δ−ρ)t = e −1 + log . ρC2 ρC22 C2 − C1
(27)
Finally, plugging (27) into (25) yields the complete expression (26)of the optimal capital:
a(N − 1)(A − δ − ρ) I(t) e(A−δ)t = k ∗ (t) = k0 − N2 a(N − 1)(A − δ − ρ) e−(A−δ−ρ)T −ρt (e − 1)+ = k0 − N2 ρ −(A−δ−ρ)T b + c(A − δ − ρ) e − (b + c(A − δ − ρ))e−(A−δ−ρ)t + log e(A−δ)t . ρe−2(A−δ−ρ)T e−(A−δ−ρ)T − (b + c(A − δ − ρ))
Given (26), we can evaluate
12 the time interval :
k ∗ (T ),
i.e. the terminal value of capital at the end of
a(N − 1)(A − δ − ρ) a(N − 1)(A − δ − ρ) k (T ) = k0 − I(T ) e(A−δ)T = − 1 − eρT + N2 ρN 2 b + c(A − δ − ρ) 1 − [b + c(A − δ − ρ)] + −(3(A−δ)−2ρ)T log + k0 e(A−δ)T . (28) e 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T ∗
If we call
T,
π ∗ = π(q ∗ (T ), k ∗ (T ))
the pro t function evaluated at the terminal instant
we are able to compare it with the pro t function
π CN = π(QCN )
evaluated at the
steady state in the static Cournot problem as shown in Subsection 2.1. Namely, we have that:
a a − c qi∗ (T ) − bki∗ (T ) = − cq ∗ (T ) − bk ∗ (T ) = N qi∗ (T ) N a a(N − 1)(A − δ − ρ) = 2 − b k0 e(A−δ)T − (1 − eρT )+ N N2 b + c(A − δ − ρ) 1 − [b + c(A − δ − ρ)] + −(3(A−δ)−2ρ)T log . ρe 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T
π∗ =
(29)
12 We omit the most tedious calculations, reminding the readers that all of them are available upon request to the authors.
16
Proposition 12. Under the hypotheses of Propositions 9 and 10, if q∗ (t) and k∗ (t) are both feasible at all t ∈ [0, T ], then the Cournot-Nash equilibrium pro t is larger than the Ramsey-Cournot equilibrium level at the terminal instant. Proof.
It su ces to consider the dierence:
π ∗ − π CN =
a a − bk ∗ (T ) − 2 = −bk ∗ (T ) N2 N
which is strictly negative by the feasibility of
k ∗ (t)
at all instants, meaning that the
Cournot-Nash equilibrium pro t exceeds the Ramsey-Cournot terminal pro t.
Remark 13. It is worth noting that comparing the two optimal strategies in the static and in the dynamic cases, one immediately sees that the presence of capital accumulation in the dynamic game plays a key role in opening the way towards a solution to the indeterminacy issue aecting the static game as the marginal production cost c of the consumption good drops to zero. Essentially, if c = 0, no solution exists for the static game if no strategy space is compact, whereas in the dierential game with capacity accumulation q∗ (k, t) is well-de ned and feasible under suitable parametric conditions even when the marginal cost is zero, both over nite and in nite horizons. Propositions 8 and 12 suggest that at the subgame perfect equilibrium of the dynamic game the representative rm may produce either more or less but she earns higher pro ts when she plays the Cournot-Nash equilibrium of the static game, irrespective of the levels of marginal cost, opportunity cost, intensity of capacity accumulation growth and intertemporal discount rate. Having characterised the subgame perfect equilibrium of the dierential game, we can now proceed to the analysis of its application to horizontal mergers.
3.2 Horizontal mergers To illustrate the advantages of our approach to the feedback solution of the dierential oligopoly game
à la
Ramsey, we illustrate here its applicability to the analysis of the
private pro tability of a horizontal merger, and its welfare appraisal. As is well known, a lively debate has taken place on this topic from the 1980's, based upon static oligopoly models. A thorough overview of it is outside the scope of the present paper, and it will su ce to recollect a few essential aspects.
Examining
a Cournot industry with constant returns to scale, in [29] it is shown that a large proportion of the population of rms has to participate in the merger in order for the latter to be pro table. In particular, a striking result of their analysis is that, in the triopoly case, bilateral mergers are never pro table. Enriching the picture by allowing for the presence of convex variable costs and xed costs, one may nd a way out of this puzzle (in particular, see [23] and [14]). Now take the static Cournot game and examine the incentive for to merge horizontally, out of the initial
N.
M > 1
rms
Before proceeding, we deem necessary to
clearly de ne what is meant here as horizontal merger by a subset of the population of rms: the merger creates a single rm whose property is symmetrically distributed across the owners of the previously independent rms, and the resulting merged entity has a single control as well as a single state. This view of the merger can be justi ed on several grounds.
To begin with, our assumptions on product homogeneity imply
that productive capital is also homogeneous across rms, and therefore the merging
17
process involves that indeed all previously independent plans are homogenized into a single entity. Additionally, our approach to the modelling of the merger is in line with the antitrust norms currently adopted both in the EU and in North America (US and Canada), establishing that the rms proposing a merger shall reduce their collective capacity endowment lest they attain a dominant position after the merger has taken place (see [8] and ch.
5 in [22], among others).
Of course, the capacity and output
of the rm resulting from the merger are necessarily larger than those associated to a generic rm if the merger does not occur, but this is simply the natural consequence of the decrease in the population of rms.
Therefore, we are going to compare the
performance of a rm in two games, alternatively characterized by
N
or
N −M +1
rms, all else equal. After the merger (if it does take place), there remain call
π CN (j)
N − M + 1 rms. If we j rms, without
the pro t of a rm in the Cournot static game among
distinguishing the original ones from those generated by the merger, we can prove the following:
Proposition 14. In the static Cournot game with hyperbolic inverse demand, the √ merger is pro table if and only if N < M + M . Proof.
Pro tability holds when
π CN (N − M + 1) a a > π CN (N ) ⇐⇒ 2 > N 2 ⇐⇒ M M (N − M + 1) ⇐⇒ N 2 > M (N − M + 1)2 ⇐⇒ . . . ⇐⇒ N 2 − 2M N − M (1 − M ) < 0, √ √ which entails N ∈ M − M , M + M . Since N ≥ M , the necessary and su cient √ M. condition boils down to N < M + It is easily checked that, contrary to [29], if
N = 3
and
M = 2,
the merger is
pro table. On the other hand, if we consider the terminal outcome of the dierential game over nite horizon, in compliance with the above notation, we can assess the pro t incentive scheme for an
M - rm merger in the dynamic framework too, sticking to the assumption
of one state and one control variable per rm. As for the assumptions of Proposition 9, ensuring the feasibility of the optimal strategy on
[0, T ],
we can state the following:
Proposition 15. If a horizontal merger of M rms is pro table in the Cournot static b game, then if δ < A < − + δ + ρ, the same merger is pro table in the Cournot-Ramsey c game on the horizon [0, T ] as well. Proof.
Provided that the net revenue at the equilibrium of the Cournot static game
is
and the net revenue of the same game when a merger of
a N2
a , (N − M + 1)2
M
rms occurs is
the pro tability of a merger in the Cournot-Ramsey game is measured
by
1 M
a ∗ − bkN −M +1 (T ) (N − M + 1)2
18
>
a ∗ − bkN (T ), N2
(30)
where we called
N −l
kl∗ (T )
the capital at time
T
under circumstances where a merger of
rms took place. In order to simplify the notation, call
g(T ) =
1 − eρT +
b + c(A − δ − ρ) e−(3(A−δ)−2ρ)T
log
1 − [b + c(A − δ − ρ)] 1 − [b + c(A − δ − ρ)]e(A−δ−ρ)T
as in (28). The inequality (30) becomes:
a b a(N − M )(A − δ − ρ) (A−δ)T − g(T ) + k e > − 0 M (N − M + 1)2 M (N − M + 1)2 a(N − 1)(A − δ − ρ) a (A−δ)T g(T ) + k0 e , > 2 −b − N N2 which amounts to
N 2 − M (N − M + 1)2 1 (A−δ)T − bk e − 1 + 0 M N 2 (N − M + 1)2 M N −1 N −M − > 0. +ba(A − δ − ρ)g(T ) M (N − M + 1)2 N2
a
(31)
M - rm merger is pro table M > 1, whereas the remaining
By Proposition 14, the rst term of (31) is positive when the in the static framework, the second term is positive for all term is positive for strategy over
b A < − +δ+ρ < δ+ρ c
[0, T ]
(the condition ensuring feasibility of the
by Proposition 9) if and only if the quantity
N −M M (N −M +1)2
−
N −1 N 2 is
negative. In fact, we have that:
N −M N −1 N −M 1 1 − = − + 2 < 2 2 2 M (N − M + 1) N M (N − M + 1) N N
M (N − M + 1)2 .
Hence, this completes the
proof. Taken together, these facts entail that the interval wherein the
M - rm
merger is
pro table may be the same in the dynamic setup and in the static one, given that the measure of output productivity
A
is small enough.
The examination of the welfare consequences of a merger is omitted, as it goes without saying that any merger would diminish social welfare, both in the static as well as in the dynamic setting. This is trivially due to the fact that the damage caused to
13
consumer surplus always outweighs the increase in industry pro ts.
13 In line of principle, a merger could allow for some reduction in the total opportunity costs for the industry, giving rise to a possible e ciency defense argument (see [14]). Although we omit the related calculations for brevity, it is quickly checked that this never outweighs the loss in consumer surplus necessarily generated by any merger. Hence, in this model the e ciency argument cannot be advocated to justify the merger itself.
19
4
Concluding remarks
Most of the existing literature on oligopoly theory postulates the presence of linear demand functions, this being quite speci c in itself and, in general, at odd with empirical evidence.
With this in mind, we have constructed a dynamic oligopoly model based
on a non-linear demand.
In particular, we have characterised the subgame perfect
equilibrium of a dynamic Cournot game with inverse hyperbolic demand and costly capacity accumulation, showing that the feedback solution, coincident with the openloop one, is subgame perfect. We have fully carried out the calculation of the optimal value functions and of the strategies of the dierential game subject to a Ramseytype dynamic constraint.
Then, to illustrate the applicability of our framework, we
have employed the model to analyse the feasibility of horizontal mergers in both static and dynamic settings, nding out appropriate parametric conditions under which the pro tability of a merger in a static game implies the pro tability in a dynamic game as well. Possible future developments of our ndings consist in the analysis of the feedback information structure of further dierential oligopoly games endowed with a hyperbolic inverse demand function, possibly in presence of more complex dynamic constraints dealing, e.g., with issues related to international trade, environmental and resource economics and R&D activities.
Acknowledgements We would like to thank Davide Dragone, Mauro Rota, Alessandro Tampieri, Stefan Wrzaczek and the audience at the EURO-INFORMS Joint International Conference in Rome for stimulating discussions and insightful suggestions.
We are also grateful to
three anonymous referees who greatly helped us improve the paper with their precious comments. The usual disclaimer applies.
20
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