Exercise 15.29. * Consider an economy with a constant population and risk neutral consumers discounting the future at the rate r. Utility is defined over the final good, which is d using unskilled labor with the production function y (ν, t) = l(ν, t). Assume that when there exist n goods in the economy and m goods can be produced using unskilled labor, we have n˙ (t) = bnXn (t) and m˙ (t) = bmXm (t) where Xn (t) and Xm (t) are expenditures on R&D to invent new goods and to transform existing goods to be produced by unskilled labor. A firm that invents a new good becomes the monopolist producer, but can be displaced by a new monopolist who finds a way of producing the good using unskilled labor. (1) Denote the unskilled wage by w (t) and the skilled wage by v (t). Show that, as long as v (t) is sufficiently larger than w (t), the instantaneous profits of a monopolist producing skill-intensive and labor-intensive goods are πh (t) = 1 ε − 1 v (t) H n (t) − m (t) and πl (t) = 1 ε − 1 w (t)L m (t) where L is the total supply of unskilled labor and H is the total supply of skilled labor. Interpret these equations. Why is the condition that v (t) is sufficiently larger than w (t) necessary? (2) Define a balanced growth path as an allocation where n and m grow at the same rate g (and output and wages grow at the rate g/ (ε − 1)). Assume moreover that a firm that undertakes R&D to replace the skill-intensive good has an equal probability of replacing any of the existing n − m skill-intensive goods. Show that the balanced growth path has to satisfy the following condition vH (1 − μ) (r + λ − (1 − μ) λ/μ) = wL (r − (1 − μ) λ/μ) μ where μ ≡ m/n and λ ≡ m/˙ (n − m) = gμ/ (1 − μ). [Hint: Note that a monopolist producing a labor-intensive good will never be replaced, and its profits will grow at the rate g (because equilibrium wages are growing). A monopolist producing a skillintensive good faces a constant flow rate of being replaced, and while it survives, its profits grow at the rate g.] (3) Using consumer demands over varieties (i.e., the fact that y (ν, t) /y (ν0 , t) = (p (ν, t) /p (ν0 , t))−1/ε), characterize the balanced growth path level of μ. What is the effect of an increase in H/L on μ? Interpret.