This exercise shows that entry barriers typically lead to multiple equilibrium wages in dynamic models. Consider the following two-period model. The production function is given by (23.2), and the distribution of entrepreneurial talent is given by a continuous cumulative distribution function G(a). There is an entry cost into entrepreneurship equal to b at each date, and each entrepreneur hires one worker (and does not work as a worker himself ). Total population is equal to 1.
(a) Ignore the second period and characterize the equilibrium wage and determine which individuals become entrepreneurs. Show that the equilibrium is unique.
(b) Now introduce the second period, and suppose that all agents discount the future at the rate β. Show that there are multiple equilibrium wages in the second period and as a result, multiple equilibrium wages in the initial period.
(c) Suppose that a fraction ε of all agents die in the second period and are replaced by new agents. New agents have to pay the entry cost into entrepreneurship if they want to become entrepreneurs. Suppose that their talent distribution is also given by G(a). Characterize the equilibrium in this case and show that it is unique.
(d) Consider the limiting equilibrium in part c with ε → 0. Explain why this limit leads to a unique equilibrium while there are multiple equilibria at ε = 0.