Modeling Climate Policy Instruments in a Stackelberg Game with ...

26 3 MODEL DEVELOPMENT AND ANALYSIS
Because u(C, L) is concave in (C, −L) and ˙K is linear in (−C, L) it exists a local
optimal solution that can be found by derivating H H respect to C and L and
setting these derivatives equal to zero. Together with the equation of motion for
the costate variable λ H and the transversality condition first order conditions
read:
u ′ C = λ H, (14)
u ′ L = −λ H w, (15)
˙λ H = λ H (ρ + δ − ¯r). (16)
0 = lim
t→∞
λ H Ke −ρt (17)
Reaction Function Solving the differential equation for λ H (16) at given
initial value λ H,0 = λ H (0) yields:
λ H (t) = λ H,0 e R t
0 (ρH+δ−¯r(s))ds (18)
Transforming (14) and (15), gives the explicit reaction function of the household,
which depends on λ 0 , ¯r(t) and w(t): 25
C(x, t) = 1 λ 0
e − R t
0 (ρH+δ−¯r(s))ds (19)
L(x, t) = L max −
C(x, t)
w(t)
(20)
Transversality Condition Assuming lim t→∞ K > 0, applying the transversality
condition (17) to the explicit solution for λ H from Eq. 18 leads to the
condition:
lim λ H,0e R t
0 (δ−¯r(s))ds = 0, (21)
t→∞
that is, as long as the net interest rate is for t → ∞ greater than the depreciation
rate, the transversality condition is fulfilled for every λ H,0 . Otherwise, λ H,0 =
0 = u ′ C which is a contradiction to the assumption u′ C > 0. Thus, if lim t→∞ ¯r <
δ, capital stocks has to break down to zero.
Ramsey Rule As a direct consequence of substituting (14) and its derivative
respect to time in (16) the optimizing conditions state the Ramsey rule of
optimal capital saving:
¯r − δ =: ˜r = ρ H −
∂u ′ C
dt
u ′ C
= ρ H − u′′ C
u ′ Ċ (22)
C
˜r = ρ H + ηĈ (23)
where η = − u′′ C
u ′
C
C denotes the elasticity of marginal utility of consumption which
equals 1 in the case of a logarithmic utility function (12).
25 λ 0 depends on C 0 and L 0 .