Zhichao currently has a job in a small city (S) with cheap housing but he has been offered a higher paying job in a big city (B) with expensive housing. His utility function over housing (good x) and all other goods (good y) is U(x,y)=.5*ln(x)+.5*ln(y). In the small city, the price of housing is pX S =2, the price of all other goods is pY S =1, and his small city job gives him income IS =200. In the big city the price of housing is pX B =16, the price of all other goods is the same as in the small city (pY S =pY B =1), and his big city job would pay him income IB =640.
a. On one single graph, show Zhichao’s budget constraint in the small city and his budget constraint in the big city.
b. Solve for Zhichao’s optimal consumption of housing and all other goods in the small city.
c. Now solve for Zhichao’s optimal consumption of housing and all other goods in the big city.
d. Will Zhichao take the big city job or stay in the small city? Explain in 1-2 sentences.
e. We will now try to answer this question more systematically. Define Zhichao’s utility function as U(x,y)=α*ln(x)+(1-α)*ln(y). In the small city, let the price of housing, the price of all other goods, and Zhichao’s income be pX S , pY S =1, and IS . In the big city these are pX B , pY B =1, and IB . Note that we are still setting the price of all other goods equal in the two cities, pY S =pY B =1. If the price of housing is higher in the big city, how much higher should Zhichao’s income be so that he is indifferent between living in the big city and the small city? Your solution should be an algebraic expression showing how IB relates to IS .