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Lecture 17
Migration Equilibrium
- Utility Hills
(a) The first piece of machinery we need for understanding migration equilib- rium is “utility hills.”^1 For this, we return to the individual allocations that are feasible under equal treatment. We then trace out the maximum utility an individual could achieve as more people are added. Figure 1 presents the “hilly” case.
Figure 1
Other outcomes are certainly possible. If preferences are tilted towards Xi, then it is possible that the overall (“variable number of region”) optimum has one-person communities and no public good. The utility “hill” would be monotone decreasing in pop- ulation. If preferences are tilted towards G, then it is possible that the overall opti- mum has a single community and no private good. The utility “hill” would be monotone increasing in population.
Figure 2
(b) Formally, for Nj > 0, the “utility hill” for individual i in region j is the function V (^) ji (Nj ): V (^) ji (Nj ) = Max U(Gj , Xji ) Xji , Gj subject to: Nj Xji + Gj = fj (Nj ) For the case Nj = 0, it is common to define: V (^) ji (0) = lim Nj → 0 V (^) ji (Nj )
provided the limit exists.^2 (^1) The term comes from Marcus Berliant, who wants to know more about when they exist and
whether they look like a hill. (^2) Note that a limit may be infinite and still exist.
Page 1—Rothstein–Lecture 17–November 2006
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(c) By definition, if Nj > 0, the utility hill tells us the maximum utility person i in region j obtains when the population is Nj.^3 (d) Our construction assumes all agents are small and identical. Suppose we also assume that immigrants and original occupants must be treated the same. It now follows that V (^) ji gives the maximum utility an immigrant into region j could obtain. (e) If Nj = 0 then the interpretation is trickier. In a related context, Atkinson- Stiglitz write: [A]n individual must form a conjecture about what his utility would be if there is no one of exactly his type within the commu- nity. For instance, if there are not doctors within a community, a doctor would have to conjecture the wages that a doctor would be paid (after tax). We assume that these conjectures are correct. (p. 541) Note, however, that this problem is strictly an artifact of the assumption that individuals are small. If individuals are large then it never arises: an individual would examine V (^) ji (1), the utility he would obtain after becom- ing the sole (large) resident of region j.
- Definition of migration equilibrium Notation shift. The analysis of multi-community models requires many subscripts. It is super- fluous now to keep using i for an individual. We therefore drop this, and use i to index communities. Assume we have J possible communities (“jurisdictions”). Note that the com- munities need not be identical – they could have different amounts of the fixed factor, for example. A migration equilibrium is a vector (n 1 , ..., nJ ) such that ni ≥ 0 for all i,
∑J i=1 ni^ = N¯, and
ni > 0 ⇒ Vi(ni) ≥ Vj (nj ), all i, j
(^3) We say “obtains” and not “could obtain”: it is assumed that the resource is properly allocated
between private and public good.
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- Existence of migration equilibria. The usual reference is Ginsburgh, V., Papageorgiou, Y.Y., Thisse, J.-F., 1985, On existence and stability of spatial equilibria and steady-states, Regional Sci- ence and Urban Economics 15, 149158. They use a fixed point argument. Other approaches are possible. Rothstein (2006) uses the theorem of the max- imum to show the existence of an equilibrium (in a mobile capital model) and establish the continuity of the allocation in the characteristics.^4
- Characterization of migration equilibria.
(a) Theorem 1. Suppose we have a vector n = (n 1 , ..., nJ ) such that ni ≥ 0 for all i and ∑J i=1 ni^ =^ N. Then n is a migration equilibrium if and only if all regions i, j with ni > 0 and nj > 0 satisfy:
Vi(ni) = Vj (nj )
and all regions i, k with ni > 0 and nk = 0 satisfy:
Vi(ni) ≥ Vk (nk)
Proof. Suppose n is a migration equilibrium. Consider the case ni > 0 and nj >
- By definition of migration equilibrium we have Vi(ni) ≥ Vj (nj ) and Vj (nj ) ≥ Vi(ni). Therefore Vi(ni) = Vj (nj ) as required. Now consider the case ni > 0 and nk = 0. We have ni > 0, so by definition of migration equilibrium we have Vi(ni) ≥ Vk (nk ) as required. Now suppose we have the equality and inequality conditions. We want to show that n is a migration equilibrium. Fix ni > 0 and any region j. If nj > 0 then the equality condition gives Vi(ni) = Vj (nj ). Therefore Vi(ni) ≥ Vj (nj ). If instead nj = 0 then the inequality condition gives Vi(ni) ≥ Vj (nj ). Thus the latter holds for all j, so we have a migration equilibrium. (b) Theorem 2. Suppose we have a vector n = (n 1 , ..., nJ ) such that ni > 0 for all i and ∑J i=1 ni^ =^ N. Then n is a migration equilibrium if and only if all regions i, j satisfy: Vi(ni) = Vj (nj ) (^4) Discountinuous payoffs, shared resources, and games of fiscal competition: Existence of pure
strategy Nash Equilibrium, Journal of Public Economic Theory, forthcoming.
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- Stability and comparative statics of migration equilibrium Stability rules out counter-intuitive comparative statics. Suppose there is a technology shock in region 1 making it more productive. Or, more interestingly, suppose region 1 is systematically providing incorrect levels of the local public good and then improves itself.
Figure 4
Setting aside the question of how society would move to the new equilibrium, the implication of instability is clear: the region that improves itself must have less population in the new equilibrium. This is completely unintuitive! We usually assume stability and use whatever convenient properties it implies. We sometimes even assume something stronger than stability and justify the assumption on the basis that it gives us stability.^5
- Formal comparative statics of migration equilibrium. If we know every region is occupied in every equilibrium, then the following conditions must hold:
V 2 (n 2 ) = V 1 (n 1 )
V 3 (n 3 ) = V 1 (n 1 )
VJ (nJ ) = V 1 (n 1 )
n 1 + ... + nJ = N¯
Assuming differentiability, we can use the implicit function theorem to derive migration equilibrium functions:
n 1 (.), n 2 (.), ...nJ(.)
where each function depends on the characteristics of all regions. We can differentiate these functions with respect to the characteristics and derive comparative statics for the equilibrium populations. This is only a local result, however, if the number of regions occupied in equi- librium changes as we vary the characteristics. This is quite likely – although the situation isn’t entirely chaotic. We return to this point later in the course.
(^5) Yes, that is a little woolly. I’m just reporting the facts.
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- A trivial result. Theorem. Suppose there are exactly two regions. Both regions are identical, each utility hill has one peak and is symmetric around the peak, and you can assign everyone to the communities so they achieve the peak utility level. Then this allocation of the population is a stable migration equilibrium. Under these assumptions the utility hills (drawn as functions of N 1 ) coincide. Thus, every split of the population is a migration equilibrium. Symmetry en- sures that utility in one region after it gains population is equal to the utility in the other region after it loses population, so the equilibria are stable. If there are more than two regions, we need to think about each utility hill on its own (as a function of Nj , say). In this case an equal split of the population gives everyone the common peak level of utility, so this is a migration equilibrium. Is it stable?
- All that can go wrong.
Each utility hill has one peak, but there are two exogenously specified commu- nities that are not identical. The equilibrium that provides the highest common level of utility is unstable.
Figure 5
- Overall:
(a) Existence does not seem to be much of a problem as long as all of the basic underlying functions are continuous. (b) Uniqueness seems virtually guaranteed not to hold. (c) Stability is quite problematic.
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