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1 Production with knowledge spillovers Consider an industrial sector with a large nmnber of individual producers indexed by 2' E [0, 1]. Each producer behaves competitively and chooses how much labor to employ in production. Producer 's production function is given by: 96:143.: where 1- denotes the amount of labor employed by producer 3', a E (U, 1) is a returns to scale parameter, and A is sectorwide productivity term sumJnariZing the overall level of knowledge and technology in the industry. Letting n: denotes the real wage, each producer's prots can be expressed as: in: = A3? 7 we; (51) Find producer is rst-order condition for the choice of 2". (b) What is the prot-maximizing choice of 51:? How does it depend on productivity A and the real wage w? Now imagine that the process of production generates indirect learning externalities. In other words, suppose that the level of knowledge and productivity is an increasing function of the average amount of labor employed in the sector: 8 MM (c) Using your answer from part (b), express 's choice of 6,; as a best response function. How does 181- depend on the average amount of labor 9? Are the production decisions of each producer strategic complements or substitutes? How do we know? ((1) Solve for the Nash equilibrium. leNT: notice that all producers are symmetrical.] (e) Let 8: refer to producer 92's Nash equilibrium employed labor. How does an increase in the real wage affect 2? Is it bigger or smaller than the amount obtained in part (b)? (E) Setup and solve the social planner's optimization problem. Remember that the social planner maximizes the joint prots of every producer and internalizes the knowledge externalities. (g) Would a social planner choose to employ more or less labor in this industry? How do we know? Explain the economic intuition