Question2. Suppose there are two types of workers: rich (r) and poor (p). Their utility function is given by:

min{c, h},

wherecis consumption andhis housing.

Let price of consumption be 1 and letp(x) denote the price of housing at the distancexmiles. Suppose the region is one dimensional, width of the region is 2. All business district is located one edge and households located west to the business district. Population density in the residential area is 3 per square mile. Transportation cost for rich ist_r= 6 (i.e. total transportation cost ist_rx) and for poor ist_p= 1 per distance. There is another region which provides utility of 4 for rich and provides utility of 2 for poor. Let income of rich isI_rand income of poor isI_p.

(a) [5 points] Explain why at any distancex, the optimal choice ofc^?(x) andh^?(x) satisfiesc^?(x) =h^?(x).

(b) [10 points] What is the price of housing as a function ofI_r,I_p, andx? Assume thatI_r>2I_p. , first find maximum willingness to pay of poor and rich as a function ofx. Then decide for a givenx, who lives at that location. Lastly, figure out the price at

locationx.

Price of housing is

p(x) =

(c) [5 points] What is the labor supply function of rich individuals as a function ofI_r, andI_p? Keeping everything else constant, explain why it is decreasing with income of poor?

Labor supply function of rich is

LS_r(I_r, I_p) =