Análisis funcional de la morfología de la mano

Now suppose there are J cities, J ∈ Z+. Note first that the same logic behind the instability of the symmetric equilibrium in the two-city economy implies that no two cities have the same set of skill-intensive varieties in any stable equilibrium. Without loss of generality, cities can be thus ranked in such a way that the measure of the set of varieties, {Ω_j}J_j=1, is monotonically increasing in j.6 I start with the equilibria without this strict ranking of cities, however, to illustrate why such equilibria are unstable.

1.4.1 Unstable Equilibria without Strict Ranking of Cities

There are equilibria in which some cities have the same set of skill-intensive varieties. Without strict ranking, {Ω_j}J_j=1 is merely non-decreasing in j. The logic of symmetric

6

Thus, the subscript j indicates the position of a city in a particular equilibrium, not the identity of the city.

equilibrium in the two-city economy carries over to the case of J > 2. Suppose for some j, Φj = Φj+1, i.e., there are two cities that have the same number of team leaders and

all skill types are equally represented. Then it implies that for some j, Cj(s)

Cj+1(s)

= wj wj+1

= 1,

∀s ∈ [0, 1] . These two cities have the same cost of producing each good s. For a positive measure of goods, the consumers would be indifferent between buying from jth or (j + 1)th city. Thus it implies the same argument as in the case of J = 2 that this class of equilibria is unstable.

1.4.2 Stable Multi-City Equilibrium

The stability of equilibrium requires that no two cities share the same set of endogenous variable, Ω_j. Cities can be ranked such that {Ωj}J_j=1 is strictly increasing in j.

Comparative Advantage

From (1.2.11), the relative cost of producing good s between the jth and the (j + 1)th cities can be written as

Cj(s) Cj+1(s) = wj wj+1 ! Φj Φj+1 !−γ(s) ,

which is increasing in s for any j = 1, 2, ..., J − 1. In other words, a city with more or better team leaders (a larger Φ) has comparative advantage in higher-indexed goods, which rely more heavily on skill-intensive components. Wage rates {wj}J_j=1 also adjust

in equilibrium so that each city becomes the lowest cost producer of a positive measure of goods. This implies that there is a sequence {Sj}J −1_j=1, which can summarize the

specialization pattern of final goods. The sequence {S_j}J −1_j=1 is defined by

Cj(Sj) Cj+1(Sj) = wj wj+1 ! Φj Φj+1 !−γ(Sj) = 1, j = 1, 2, ..., J − 1,

Figure 1.3: Comparative Advantage and Patterns of Specialization in the Multiple-City World 1 Sj-1 Sj 0 1 s Cj-1(s)/Cj(s) Cj(s)/Cj+1(s)

City j produces and exports

which is increasing in j. It implies that the borderline sector, S_j, could be produced and exported by either jth or (j + 1)th city because the cost of producing the good Sj is

the same in both cities.

As shown in Figure 1.3, the final tradable goods are partitioned into J subintervals of positive measure such that the city j becomes the lowest cost producer of s ∈ (Sj−1, Sj) .

The equilibrium wages can be written as wj wj+1 = Φj Φ_j+1 !γ(Sj) < 1 j = 1, 2, ..., J − 1.

Therefore, the wage sequence, {wj}J_j=1 is also increasing in j. That is, more productive

city has a higher wage rate.

Individual Behavior

With J > 2 cities, there is a population of workers located in each city. In equilibrium, housing prices adjust to make each worker indifferent between locating in any of these

J cities: wj wj+1 = Rj Rj+1 !1−α .

Therefore, housing price sequence {R_j}J_j=1 is also increasing in j, i.e., more productive cities have higher housing prices. The following Lemma 3 is the multiple-city version of Lemma 1.

Lemma 3. (Occupational Choice) In any stable equilibrium with strict ranking of

cities, the productivity cutoff for being a team leader is the same across all cities. That is, ϕ∗_j = ϕ∗_j+1 = ϕ∗, for j = 1, 2, ..., J − 1.

The intuition behind this result is the same as in the two-city case. A city with a higher w_j has a proportionally higher π_j(ϕ) , due to the Dixit-Stiglitz demand of skill- intensive components and CRS at the level of producing each variety. The cutoff ϕ∗, which equates w_j and π_j(ϕ), does not depend on j.

Lemma 4. A team leader with productivity ϕ can always get higher income by locating

to the more productive city. That is, for all ϕ, πj+1(ϕ) > πj(ϕ), for j = 1, 2, ..., J − 1.

This means that given her productivity ϕ, a team leader can get the highest income if she locates in the J th city. In equilibrium, however, not all team leaders locate there, because the J th city also has the highest housing price. Therefore, there exists a sequence, nϕ∗∗_j oJ −1

j=1 such that ϕ

∗∗

j > ϕ∗ ∀j = 1, 2, ..., J − 1 and team leaders with

ϕ∗∗_j are indifferent between residing in jth and (j + 1)th city. The cutoff nϕ∗∗_j oJ −1

j=1 is defined as πj  ϕ∗∗_j  R1−α_j = πj+1  ϕ∗∗_j  R1−α_j+1 , j = 1, 2, ..., J − 1.

Proposition 3. In any stable equilibrium with strict ranking of cities, the hetereoge-

neous individuals are partitioned into J + 1 intervals. Individuals with ϕ < ϕ∗ become workers and reside in every city. All individuals with ϕ > ϕ∗ become team leaders. (1) Team leaders with ϕ such that ϕ∗ < ϕ < ϕ∗∗₁ reside in the 1st city. (2) Team leaders with ϕ such that ϕ∗∗_j < ϕ < ϕ∗∗_j+1 reside in the (j + 1)th city. (3) And team leaders with ϕ such that ϕ > ϕ∗∗_J reside in the J th city.