Consider the model of multiple tasks presented in class. In the model, there is some final good whose production depends on tasks 1,2,..., being completed. The production function of the final good is: =min(1,2,...,)

There are L individuals in the economy. Each individual has total time T available for working. If an individual allocates units of time to a given task, she produces of the intermediate input associated with that task. The production function of each intermediate input features increasing returns to scale (i.e. >1).

3.1) What does it mean to have a production function that exhibits increasing returns to scale?

3.2) In the context of the multiple task model, why do we expect the production function of intermediate goods to exhibit increasing returns to scale?

3.3) Suppose there is only one task needed to produce (that is, M = 1). Would there be a reason for teams (cities) to exist? Why/why not?

3.4) Now suppose that M>1. Why are people going to want to work together? Would we necessarily see only one city? Why/why not?

3.5) Suppose there are 2 tasks to be performed, that is M = 2. How much of each task should be performed relative to the other task in order to optimize ? Show this on a graph.

3.6) How much time should be allocated toward each task? Why?

3.7) Suppose L= sM and s is a positive integer. How many cities will likely develop? Explain why. How does this depend on s?

3.8) What does the empirical fact that employment is not split evenly among professions in cities imply about the assumptions about the production technology? Generalize the production function =min(1,2,...,) to account for this fact while maintaining perfectly complementary inputs.