Economic Models - Convex Optimization

112 Chirstophe Deissenberg and Pavel Ševčík
The first order maximization condition with respect to t a and t yields
for the dynamically optimal values*
t a∗ = (p − c x) + [c x (3 − 2π) + 1](1 + c x ) 2
, (21)
(1 + 3c x )[βλ(1 − π) + K]
with
t ∗ = (p − c x) + 2c x (1 + c x )π[c x (βλ(1 + 3c x )(1 − π) + 1)]
(1 + 3c x )[βπ(1 − π) + K]
K = 1 − 6c 3 x β2 λ 2 (1 − π) 2 π + 2c x [2 + 2βλ(1 − π)π]
+ cx 2 [3 − 2π + βλ(1 − π)(3 − 2βλ(1 − π)π)]. (22)
According to Pontriagin’s maximum principle, an optimal solution
(π(τ), (λ)) has to satisfy the state dynamics (6), evaluated at (t a∗ ,t ∗ ) plus:
˙λ = ρλ − ∂H(ta∗ (π, λ), t ∗ (π, λ), π, λ)
(23)
∂π
0 = lim
τ→∞ e−ρτ λ(τ). (24)
Due to the highly non-linear structure of the optimization problem an
explicit derivation of the optimal (π(τ), λ(τ)) seems infeasible. Hence, we
used Mathematica r○ to carry out a numerical analysis of the canonical
system Eqs. (6)–(23) in order to shed light on the optimal solution.
4.2. Numerical Results
Depending upon the parameter values, the model can have either two stable
equilibria separated by a so-called Skiba point, or a unique stable equilibrium.We
discuss first the more interesting case of multiple equilibria, before
carrying a summary of sensitivity analysis and briefly addressing the question
of the bifurcations, leading to a situation with a situation with unique
equilibrium. Unless and otherwise specified, the results presented pertain
to the reference parameter configuration c x = 0.6, p = 2, δ = 0.00025 that
underlies Fig. 1, together with ρ = 1, 0.0015. These are robust, with respect
to sufficiently small parameter variations: The same qualitative insights
would be obtained for a compact set of alternative parameter configurations.
From the onset, let us stress the fundamental mechanism that underlies
most of the results. Ceteris paribus, R would like to face as many Bs as
possible, since dφ/dπ >0. If the prediction costs δ are sufficiently high,
the profit difference g NB$ − g B$ see Eq. (18), will be negative at the current