Economics of Migration and Job Search: Problem Set 3
Suphanit Piyapromdee
1. Suppose workers prefer low s locations but firms prefer high s locations, what would
the Roback model predict in terms of relative wages and rents between city 1 and
2 where s1 < s2 ? Use a diagram, explain the intuition and give an example of such
amenities.
Solution: In this case, we have the firm’s cost curve in city 2 lies above that of city
1. Since city 2 has higher s, for firms to be indifferent between city 1 and 2, it must
be that factor prices are also higher in city 2. For workers, as they prefer city 1 given
the same factor prices, the indifference curve of city 2 has to lie below city 1’s. In
equilibrium, wage in city 2 has to be higher than wage in city 1, while relative rents
are ambiguous. This is because workers prefer city 1, but firms prefer city 2. w
and r have to be such that the indifferent conditions hold for both firms and workers.
Because a higher wage attracts workers and discourages businesses, wage in city 2
is unambiguously higher. On the other hand, high rents discourage both firms and
workers to locate in the area. So r may be higher or lower in city 2. Industrial waste
disposal facility can be an example. Such facilities can reduce firms’ cost but those
may be undesirable for residents.
Case 1 Case 2
r r
V(w,r,s1) = V0 V(w,r,s1) = V0
V(w,r,s2) = V0 V(w,r,s2) = V0
r2 C(w,r,s2) = 1 r1
r1
r2
C(w,r,s1) = 1
C(w,r,s2) = 1
C(w,r,s1) = 1
w1 w2 w1 w2
w w
2. Now allow each individual to have idiosyncratic preference shocks. The utility of
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Suphanit Piyapromdee
1. Suppose workers prefer low s locations but firms prefer high s locations, what would
the Roback model predict in terms of relative wages and rents between city 1 and
2 where s1 < s2 ? Use a diagram, explain the intuition and give an example of such
amenities.
Solution: In this case, we have the firm’s cost curve in city 2 lies above that of city
1. Since city 2 has higher s, for firms to be indifferent between city 1 and 2, it must
be that factor prices are also higher in city 2. For workers, as they prefer city 1 given
the same factor prices, the indifference curve of city 2 has to lie below city 1’s. In
equilibrium, wage in city 2 has to be higher than wage in city 1, while relative rents
are ambiguous. This is because workers prefer city 1, but firms prefer city 2. w
and r have to be such that the indifferent conditions hold for both firms and workers.
Because a higher wage attracts workers and discourages businesses, wage in city 2
is unambiguously higher. On the other hand, high rents discourage both firms and
workers to locate in the area. So r may be higher or lower in city 2. Industrial waste
disposal facility can be an example. Such facilities can reduce firms’ cost but those
may be undesirable for residents.
Case 1 Case 2
r r
V(w,r,s1) = V0 V(w,r,s1) = V0
V(w,r,s2) = V0 V(w,r,s2) = V0
r2 C(w,r,s2) = 1 r1
r1
r2
C(w,r,s1) = 1
C(w,r,s2) = 1
C(w,r,s1) = 1
w1 w2 w1 w2
w w
2. Now allow each individual to have idiosyncratic preference shocks. The utility of
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