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Answer the questions below. There are 25 regular points, and, in addition, there is an optional extra credit question at the end, worth an additional 5 points. The Bay Area has a history of producing innovations at a much faster rate than other cities in the U.S. Is the Bay Area’s population therefore likely “too small”, “too big”, or “just right”, relative to its utility-maximizing size? Why? (4 pts)

Suppose a factory can produce a shirt for the equivalent cost of 3 loaves of bread, and a household can produce a shirt for the equivalent cost of 10 loaves of bread. The factory is located in a rural area with a uniform population density. It costs the equivalent of 0.5 loaves of bread for a household to make a one mile trip to or from the factory (of course they have to travel both directions!)

Assuming the consumer price equals the factory’s cost of production, what will be the radius of the factory’s market area? (3 pt)

Now suppose the factory develops an innovation that allows it to produce a shirt for the equivalent of 1 loaf of bread. What is the new radius of the factory’s market area? (3 pt) 3. A nation has its population of 10 million living entirely within two cities. Initially, migration between the two cities is prohibited. The relationships between population (in millions), daily labor income, and daily commuting costs in each city are given by the two following tables.

Suppose that, initially, City A’s population is 4 million and City B’s population is 6 million. Then, the national government allows free migration between the two cities. a) After migration, what will be the equilbrium population of City A and City B? (8 pts) b) Will this equilibrium be stable, and if so, why? (2 pts)

Instead of allowing free migration between the two cities, is there a way to assign populations to the two cities in way that maximizes both cities’ utility above the equilibrium level? If so, what would be the population of City A and City B in that case? (2 pts)

The first fundamental theorem of welfare economics is that a competitive equilibrium leads to a Pareto-optimal allocation of resources. Does the theorem hold in this case? If not, why not? (3 pts)

EXTRA CREDIT: In the previous question, urbanization economies caused income to increase with city size, but at a decreasing rate. How would the outcome be different if City A’s urbanization economies caused its income to increase at an increasing rate (above 1 million population)? Why might this assumption be unrealistic in real life? (5 pts) hint: start with the growth rate in income from 1-2 million, and then pick any increasingly higher growth rate for the larger populations, and see what happens.

Courtney from Study Moose

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