Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 235

1
2 x k .t/ C

c k f.Q.t//
2f 0 .Q.t//

x k .t C 1/ D .k D 1;:::;N/: (5.64)

This N-dimensional dynamical system in the state variables x k can be reduced to
a one-dimensional dynamical system in the total quantity Q.t/ by summing up the
equations (5.64) to yield

1
2 Q.t/ C

Nf.Q.t/
2f 0 .Q.t//

Q.t C 1/ D (5.65)

where D P kD1 c k . The dynamic equation (5.65) for the aggregate production
includes the number of players N as a parameter. Therefore, we can investigate the
effects of this parameter on the dynamics of the global production. It is trivial to see
that if .x 1 ;:::; x N / is a steady state of the disaggregated dynamical system (5.64),

then Q D P kD1 x k is a steady state of the aggregated dynamical system (5.65). In
particular, if . x 1 ;:::; x N / is a Nash equilibrium, then it is a fixed point of (5.64) and
consequently it corresponds to a fixed point of (5.65). However, the converse is not
true in general because a fixed point Q of (5.65) can correspond to several different
arrangements of .x 1 ;:::;x N /, that do not correspond to fixed points of (5.64). If
we consider the model (5.53) with N firms and isoelastic inverse demand function

1
Q ;

p D f.Q/ D (5.66)

then the dynamical system (5.64) becomes

2 x k .t/ C Q.t/ c k Q 2 .t/ .k D 1;:::;N/;

1

x k .t C 1/ D (5.67)

and the one-dimensional map (5.65) that describes the time evolution of the aggre-
gated output Q.t/ becomes

1
2 Œ1 C N Q.t/Q.t/;

Q.t C 1/ D (5.68)

where D P kD1 c k . This is a quadratic one-dimensional map which is topologi-
cally conjugate to the standard logistic map x.t C 1/ D x.t/.1 x.t// through
the linear homeomorphism Q D x.1 C N/= and with the parameters related by
D .1 C N/=2. The time evolution of the aggregate production can be deduced
from well-known properties of the logistic map (see e.g., Devaney (1989)). In par-
ticular, here we are interested in the role of the integer parameter N.Firstof
all, we notice that the dynamics of (5.68) converge to the positive steady state
Q D .1 C N 2/= provided that N

5, corresponding to the well known con-

dition 3. The convergence is monotone if N 3, whereas it exhibits damped

oscillations if 4 5. With 6 firm s we have D 3:5; hence we have stable

oscillations of period 4 since >1 C

N

p 6. The case of N D 7 competitors gives rise