In the question that follows, n refers to the number of people rather than a fraction of the population. In the land of Pampa, living in the countryside gives you a fixed payoff of 100 (Pampa has lots of land), while living in a city gives you a payoff that first increases with the number of people living in the city (agglomeration), and then declines after the number of people goes above a certain threshold (congestion). Let us write this payoff as r = 20n - n²/2, where n is the number of city dwellers in that particular city. (a) Let N be the total population in Pampa. If only one city can exist in the entire country, trace out the set of equilibria (i.e., population allocations between countryside and city) as N varies from 0 to infinity. (b) Now suppose that new cities can come up, each yielding exactly the same payoff function as above. Focus on the equilibrium in each case with the maximum possible city dwellers, and explain how this equilibrium will move with the overall population N.
In the question that follows, n refers to the number of people rather than a fraction of the population. In the land of Pampa, living in the countryside gives you a fixed payoff of 100 (Pampa has lots of land), while living in a city gives you a payoff that first increases with the number of people living in the city (agglomeration), and then declines after the number of people goes above a certain threshold (congestion). Let us write this payoff as r = 20n - n²/2, where n is the number of city dwellers in that particular city. (a) Let N be the total population in Pampa. If only one city can exist in the entire country, trace out the set of equilibria (i.e., population allocations between countryside and city) as N varies from 0 to infinity. (b) Now suppose that new cities can come up, each yielding exactly the same payoff function as above. Focus on the equilibrium in each case with the maximum possible city dwellers, and explain how this equilibrium will move with the overall population N.
Chapter1: Introducing The Economic Way Of Thinking
Section: Chapter Questions
Problem 7SQ
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In the question that follows, n refers to the number of people rather than a fraction of the population. In the land of Pampa, living in the countryside gives you a fixed payoff of 100 (Pampa has lots of land), while living in a city gives you a payoff that first increases with the number of people living in the city (agglomeration), and then declines after the number of people goes above a certain threshold (congestion). Let us write this payoff as r = 20n - n²/2, where n is the number of city dwellers in that particular city.
(a) Let N be the total population in Pampa. If only one city can exist in the entire country, trace out the set of equilibria (i.e., population allocations between countryside and city) as N varies from 0 to infinity.
(b) Now suppose that new cities can come up, each yielding exactly the same payoff function as above. Focus on the equilibrium in each case with the maximum possible city dwellers, and explain how this equilibrium will move with the overall population N.
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