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(Preference matching 1) Consider a society with many persons who can choose freely to live in either region 1 or region 2. It is more expensive to live in region 2: it costs c1 to live in region 1 and c2 = c1 + ∆ to live in region 2 (with ∆ > 0). Individuals differ in their incomes, denoted by y. Income takes on values between 0 and 1 and is uniformly distributed. Individuals care about the income of those living in their region. The mean income of a region j = 1; 2 is a function of the average value of y in that region, denoted by yj. An individual with income y choosing to live in region j with mean income yj derives utility net of cost of U = [1 + y][1 + j]- j. Hence richer individuals place greater value on living together with other rich residents.
a. Suppose that all individuals simultaneously make their location choices. Show that in any equilibrium (where no one wishes to move given the location choice of everyone else) both regions must be occupied if ½ < ∆ < 1, that is the cost differential is neither too high nor too low. What would happen if ∆ > 1 or ∆
b. For ½
c. Show that there exists a critical level of income y such that all individuals with higher income choose to live in region 2 and all individuals with lower income choose to live in region 1. Provide an expression for this critical income level.
d. Show that in the equilibrium with income sorting, it is possible to make everyone better o¤ by changing slightly the residential choices. [Hint: Consider a small change in the critical income y.]