# Consider a linear city composed of two different jurisdictions. The two extremes of the city are…

Consider a linear city composed of two different jurisdictions. The two extremes of the city are -½  and ½  , and the city center is located at 0. Jurisdiction 1 is located to the left of the city center and jurisdiction 2 to the right. The city is inhabited by a continuum of individuals that are uniformly distributed with a total mass of 1. Each jurisdiction has a population of ½. The median voter of jurisdiction 1 is located at  ¼, while the median voter of jurisdiction 2 is located at ¼. Each jurisdiction has a job center, located respectively at -1/3 and 1/3. Wages are exogenous and equal to λ > 1 in jurisdiction 1 and 1 in jurisdiction 2. There are ad valorem taxes on wages t1 and t2 paid to the jurisdiction where work is undertaken. Individuals incur a commuting cost equal to ½ per unit of distance to their workplace. Jurisdictions are run by majority-elected governments that maximize net income. Net income is the sum of the median voter’s net wage plus fiscal revenue per capita, [1 – ti]wi + tjNjwj when the median voter in j works in jurisdiction i, with Nj denoting the number of people working in j.

a. Suppose that each individual works in the jurisdiction where she gets the highest after-tax wage net of commuting costs. Compute the number of workers N1, N2 in each jurisdiction as a function of the wage taxes t1 and t2.

b. Suppose that in equilibrium jurisdiction 2’s median voter works in jurisdiction 1. Find the Nash equilibrium in taxes. Do the jurisdictions tax or subsidize wages? Explain briefly. Give a condition on g under which this equilibrium exists.

c. Suppose now that in equilibrium jurisdiction 2’s median voter works in jurisdiction 2. Find the Nash equilibrium in taxes. Does either jurisdiction subsidize the wage? Why or why not? Give a condition on g such that this equilibrium exists. Compare your answer to that of part b. Interpret the difference.