the value function associated with strategy baL is bounded. Hence we can apply the sufficiency
conditions in Theorem 3 of Rowat’s (2007) to demonstrate its subgame perfectness.
Applying the above sufficiency conditions and using the cornered strategy a =1,wecan
construct more subgame perfect equilibrium strategies. As shown in Lemma 2 of Appendix
C, the range of the state variable Z where the cornered strategy a =1 remains as an optimal
one (i.e., (∂pi∕∂ai) Z > V0 (Z)) depends on the chosen value of c3 in (C6). In particular, the
cornered strategy a =1 remains subgame perfect over the domain (0, ∞) if the constant c3
is larger than the infimum of the set of c3 satisfying inequality (C7) in Appendix C (see Fig.
3). When c3 is smaller than that infimum, more of the a4-families of strategies patched by
strategy a =1 will be subgame perfect as c3 increases to that infimum, as shown in Fig.3.
In Fig.1 (i.e., when aE < 1) the strategy a4 patched by the cornered strategy a =1is
qualified as a subgame perfect equilibrium strategy over the domain (0, ∞) if c3 is greater
than that infimum. If this is a case, the linear MP strategy ends up with the highest level of
steady state durable goods stock. On the other hand, in Fig.2 the patched strategy ^4 is still
qualified if c3 is greater than that critical value. In addition, the strategies left off the strategy
^1 which is tangent to the constraint a = 1 and then connected with the strategy a = 1 at
that tangent point in Fig.2 are also subgame perfect. As a result, there are non-linear MP
strategies which entail higher levels of durable goods stock in the steady state such as Z as
compared to that supported by the linear strategy aL.
We summarize with the following theorem:
Proposition 1 Consider the dynamic game defined by (5) and assume that the constant of
integration c3 is chosen so as to satisfy inequality (C7) in Appendix C. Symmetric Nash
equilibria in continuos and asymptotically stable Markov perfect strategies must satisfy the
following:
(i)If aE 5 1, thus r (ρ + δ)(n - 1) 5 [1 - r (n - 1)] δ, then there exist uncountably many
non-linear Markov perfect strategies and the unique linear Markov perfect equilibrium strategy,
coupled with strategy a(Z) = 1, that are globally defined over the entire domain of a state space,
leading to steady state equilibria ranging over (0,ZS];and
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