Consider the model with labor in Section 19.4. Suppose that countries can invest to create new varieties of products. Suppose that if a particular firm creates such a variety, it becomes the monopolist and can charge a markup equal to the monopoly price to all consumers in the world, until this variety is destroyed endogenously, which happens at the exponential rate δ > 0.
(a) Show that the optimal monopoly price for a firm in country j at time t is given by pj (t) = εrj (t)/(ε - 1). Interpret this equation.
(b) Suppose that a new variety can be created by using 1/η units of labor. Show how this changes the labor market clearing condition, and specify the free-entry condition.
(c) Define a world BGP as an equilibrium in which all countries grow at the same rate. Show that such an equilibrium exists and is uniquely defined. Explain the economic forces that lead to the existence of such a "stable" equilibrium. [Hint: show that in this BGP, the number of varieties that each country produces is constant.]
(d) What is the effect of an increase in the discount rate ρ on the number of varieties that a country produces? Interpret this result.
(e) Discuss informally how the analysis and the results would be modified if new products were produced using a combination of labor and capital.