Exercise 14.17. Consider the one-sector Schumpeterian model in discrete time. Suppose as in the model in Section 14.2 that consumers are risk neutral, there is no population growth, and the final good sector has the production function given by (14.25). Assume that the R&D technology is such that LR > 0 workers employed in research will necessarily lead to an innovation, and the number of workers used in research simply determines the quality of the innovation via the function λ (LR), which is strictly increasing, continuously differentiable, strictly concave and satisfies the Inada conditions.

(1) Define an equilibrium in this economy. (2) Characterize the BGP and specify restrictions on parameters so that the transversality condition is satisfied. (3) Compare the BGP growth rate to the Pareto optimal growth rate of the economy. Show that the size of innovations is always too small relative to the size of innovations in the Pareto optimal allocation.