Here we used primes to denote the value of a given variable in the next period.
Proposition 2.1. For each в there exists a unique, bounded and continuous function W. The value function W is strictly concave and differentiable almost everywhere, and the decision rules are continuous. single-valued functions.
Proof. See appendix.
There is an explicit analytical solution to the value function W which is described in the appendix.
Lemma 2.2 (Specialization). The economy has a comparative advantage in good a: if we start the economy with Ма > 0 and Mb = 0 we will not observe entry in sector b; the economy will remain specialized in the production of good a.
Proof. See Lemmas 6.1 and 6.2 in the appendix.
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The Steady State Set
In the steady state of this economy the number of firms in both sectors remains constant. There are multiple (Me, Мь) combinations consistent with the steady state. These combinations can be characterized by studying the firm entry and exit decisions for both sectors of the economy.
A pair (М0} Мъ) belongs to the steady state set if it satisfies two properties: (i) the value of firms in both sectors is non-negative (Vu > 0, г = a, 6), which implies that there is no incentive to exit; and (ii) the value of existent firms is lower than the cost of entry (Vit < ф(1 -f r*), i = a, 6), which implies that there is no incentive to enter.