Urban location, housing equilibrium, and retiree
migration∗
Paolo Giovanni Piacquadio†
University of Oslo
Nicholas Sheard‡
Aix-Marseille University
September 2013
Abstract
Cities provide dense labor markets and favorable employment opportunities. To access
these markets, city residents pay a premium on housing and often commute long distances.
Despite the costs of living, not all households residing in the city include workers, in large
part because of retirees who remain in the city even though they no longer value the proximity to employment. We study the effect that the presence of retirees has on the efficiency
of the market allocation of housing in a city. Although scarcity is priced on the market, the
equilibrium outcome has too many retirees in the city, due to an unpriced external effect
that retirees impose on workers by staying. We evaluate the potential of various policies
to correct the externality and find that a combination of transfer and property taxes can
achieve an efficient outcome.
Keywords: Housing locations, Migration, Overlapping generations, Efficiency
JEL classification: D91, R23, R31
∗ Thanks
to Jan Brueckner, Pierre-Philippe Combes, and Yves Zenou.
of Economics and ESOP, University of Oslo.
‡ Aix-Marseille University (Aix-Marseille School of Economics), CNRS, and EHESS, 2 rue de
la Charité, 13236 Marseille cedex 02, France.
E-mail: [email protected].
Website:
https://sites.google.com/site/nicholassheard/.
† Department
1
1
Introduction
The allocation of residents to housing locations within a city is a fundamental issue in urban
economics. It has long been established that the instantaneous market allocation of housing
is efficient, under broad conditions, as the scarcity of land near employment is reflected in
the relative house price bids of individuals. Housing markets are not temporally independent,
however, as individuals have a tendency to remain in the same locations due for instance to an
attachment to their homes and the cost of establishing social networks. As a result many retired
people live on valuable land in cities, which implies some combination of longer commutes and
higher house prices for the working-age population. Given the high overall cost of commuting
it is therefore important to understand how effective the market is in allocating land between
working and retired people. Are there potential efficiency gains from relocating households
within or outside the city? And if so, would it be possible to achieve such gains through policy?
As societies age, these questions are destined to become increasingly important.
We investigate whether the market allocation of housing is efficient when individuals have
a preference for staying in their homes upon retirement and decide endogenously whether to
do so. We show that the market allocates housing inefficiently in this case: though scarcity is
priced, individuals receive no payment for vacating their houses upon retirement, which leaves
a social benefit unpriced in the market. We then assess the potential of a range of different
policies to improve overall welfare. Finally, we study the effects of a shock to employment
in the city on existing residents’ decisions about whether to move away upon retirement, on
efficiency, and on equality within and between generations.
The analysis is based on an overlapping-generations model in which individuals decide
where to buy a house at the beginning of their working lives and then whether they will stay
in the house or move to a retirement community outside of the city when they retire. Moving
away implies lower overall housing costs, allowing higher consumption, but induces a utility
cost of relocation associated with distance from their family and the need to establish new
social networks. The resulting structure of the city is generally of an inner zone of ‘movers’,
who intend to sell their houses upon retirement and move away, surrounded by an outer zone
of ‘stayers’, who intend to live in the same locations when they retire. Sufficiently small cities
are comprised exclusively of stayers, as even central locations are cheap relative to the cost of
2
relocation. Cities comprised exclusively of movers are also possible, if the reservation value of
land exceeds the consumption equivalent of the cost of relocating. We assume that all residents
are homeowners, but show that the main results continue to hold if we allow for a rental market.
The market allocation of housing in the model is inefficient in the steady state, due to an
unpriced housing-market externality imposed by agents who stay in the city when they retire.
By occupying scarce urban land, an agent who stays in the same house for retirement prevents
a worker from living in that house and having a shorter commute to work. As urban land is
scarce, the retiree pays more to remain in the same house than to live in the outside retirement
community, but the amount paid does not fully reflect the costs from increased commuting time
imposed on others and so there are too many stayers in the decentralized equilibrium for it to be
efficient. Two factors combine to create the inefficiency. On the one hand, the different levels
of consumption of adult movers and stayers impose a rigidity in the second-period housing
decision. On the other hand, workers in the next period cannot make bids to current workers for
the right to buy before these housing and consumption decisions are taken.
We consider the introduction of a tax and subsidy scheme to correct for the externality.
The scheme is composed of an ad valorem property tax (levied in each period), an ad valorem
transfer tax (levied on the purchase of a house), and a lump-sum subsidy to each adult resident.
A combination of the two taxes, balanced by the lump-sum subsidy, is sufficient to achieve an
efficient allocation of housing. The optimal policy is a positive property tax and a negative
transfer tax, which are of equal magnitude and thus offset each other exactly for adult agents
but represent a positive tax on retired residents, with the proceeds distributed by a positive
lump-sum subsidy. The tax and subsidy scheme constitutes a net intergenerational transfer
from retired ‘stayers’ to all adults that increases the incentives for retirees to migrate out of
the city. The levels of the taxes and subsidies do not depend on the residential location within
the city and the information sufficient to implement the policy is simply the interest rate: no
information is required on the magnitude of the relocation cost or the intentions of agents in
particular locations to migrate away upon retirement.
We then analyze the effects of a productivity shock in the city. An increase in productivity
increases local wages, which in turn increases the city population and the prices bid for houses.
This has a positive effect on the wealth of homeowners. The increase in wealth may induce
3
an intended mover to stay in the city or induce an intended stayer to move, depending on how
the wealth increase compares with the increased cost of holding the house for an additional
period. We present conditions under which each of these changes would occur. The effects
of a negative productivity shock are similar, except that intended movers may be induced to
stay whereas no intended stayers decide to move. Productivity shocks affect the welfare levels
of existing homeowners to different degrees depending on where they live, which presents the
restoration of equality in outcomes as a potential policy goal. We show that it is possible to introduce incentive-compatible policies in response to the shock that either bring about efficiency
or equality amongst retirees, but that it is not possible to simultaneously achieve both efficiency
and equality.
This paper contributes to the literature in several ways. The first contribution is theoretical:
we present a simple and intuitive model for endogenously-determined relocation at retirement.
The theoretical literature about retirees’ migration is focused on factors determining the choice
of destination (Graves and Knapp, 1988). Englund (1986) modeled relocation decisions during the individual’s lifetime in order to estimate the effects of different types of property taxes
on housing demand, but did not address retirement or its implications for city structure. Another strand of literature models the effects of retirees on systems of cities rather than on city
structure. Gaigné and Thisse (2009) find that cities specialize in either production or housing
the elderly and that retirees provide a dispersive force, whereas Grafeneder-Weissteiner and
Prettner (2013) find that retirees increase agglomeration. Sato (2007) presents an overlappinggenerations model to study geography and fertility rates, but does not model retirees’ migration.
The second contribution is to identify a novel source of inefficiency in the allocation of
urban residential land: that of the unpriced social benefit of vacating land upon retirement. This
inefficiency arises unambiguously from the model as a consequence of minimal and intuitive
assumptions. Furthermore, we are able to evaluate the effectiveness of various policies for
correcting the inefficiency.
In keeping with the theoretical literature, the empirical literature on retirees’ migration is
largely concerned with factors determining their choice of destination. Retirees are more likely
to migrate to places with nicer weather (Rappaport, 2007), better amenities (Clark and Hunter,
1992; Chen and Rosenthal, 2008), and less attractive conditions for firms and therefore less
4
competition for land (Gabriel and Rosenthal, 2004). Retirees also migrate to be near family
members or return to their country of origin (Gobillon and Wolff, 2011). The model presented
here generates testable predictions about how retirees’ migration will respond to factors in their
adult locations. Namely, (1) there should be more migration from larger, more expensive cities,
(2) controlling for demographic factors such as lifetime income, there should be more migration
from central neighborhoods where housing is expensive, and (3) the rate of out-migration by
retirees may depend on recent changes in the city’s fortunes.
The remainder of this paper is arranged as follows. The model is presented in Section 2.
The derivation of the house price equilibrium is detailed in Section 3. The steady state city
structure and the analysis of the efficiency of the spatial equilibrium, along with the evaluation
of different policy alternatives, are presented in Section 4. The dynamics of the model are
analyzed in Section 5, with a description of agents’ behavior and policy evaluation. The final
section presents concluding remarks.
2
Model
The model is a of monocentric city in which all employment occurs at the arbitrarily-defined
center and households reside on the surrounding land. There is also a retirement community
outside of the city to which individuals may relocate when they retire. Locations in the city are
defined by their distance r ∈ R+ from the city center, as space is otherwise homogeneous in
nature and effect. The number of housing units available at distance r is defined by the density
function h (r) : R+ → R+ . The reservation value of land in the city is ν ∈ R+ , which measures
the opportunity cost of alternative uses such as agriculture.
Time is defined as an infinite stream of discrete periods t ∈ T ≡ {0, 1, ...}. Agents live for
three periods: childhood, adulthood, and old age. Children do not play an active role and live
with their parents. Adults work in the city center commute the distance r from their homes,
which reduces the amount of time they have to work. Old agents are retired and either remain
in the city or move away to live in the external retirement community. Adult agents are denoted
by the superscript a, old agents by the superscript o.
The remainder of this section outlines the model in detail.
5
2.1
Households
An adult agent at time t ∈ T supplies labor inelastically and earns a wage wt ∈ R+ per unit of
time worked. At the beginning of the adult period, the agent decides where in the city to live.
The distance r from the city center to the house determines the commuting time τ (r), where
τ 0 > 0. The adult has a single unit of time to spend on work and commuting and thus earns
labor income (1 − τ (r)) wt . The price of the house at location r and time t is described by the
function pt (r) : R+ → R+ .
The same agent will be old at time t + 1 and will have retired. At that time the agent decides
between staying in the same house or relocating to the retirement community. Housing in
the retirement community is costless, which allows an agent who moves away to enjoy higher
consumption. On the other hand the agent has a preference for staying in the same community
and pays a relocation cost in utility terms of φ ∈ R+ .
o , respectively.
The agent’s consumption levels when adult and old are denoted cta and ct+1
The well-being of the agent is described by:
o
Uta = u (cta ) + βUt+1
(2.1)
o
o
o
Ut+1
= u ct+1
− I rt+1
φ
(2.2)
o
The term I rt+1
is an indicator function that assumes value 0 if the agent stays in the same
o = r a , and value 1 if the agents relocates, in which case
house for retirement, in which case rt+1
t
o 6= r a . As commuting is irrelevant to retirees and housing in the retirement community is
rt+1
t
costless, any agent who relocates moves out of the city. We assume that the per-period utility
function u is twice differentiable, strictly increasing, strictly concave, and satisfies the Inada
conditions.1
The budget constraint of the adult requires the labor income to cover the consumption cta ,
the housing price at the chosen location pt (rta ), and savings sta :
cta + pt (rta ) + sta ≤ (1 − τ (rta )) wt
1 Differentiability,
strict concavity, and the Inada conditions are introduced for computational simplicity.
6
(2.3)
The agent’s wealth when old consists of the savings of the previous period, on which interest
has accumulated at the exogenous rate i, and the value pt+1 (rta ) of the house purchased at the
o , depends on the relocation decision.
beginning of the adult period. Consumption when old, ct+1
If the agent decides to relocate when old, the wealth carried over from the adult period is spent
o = r a , the
entirely on consumption. If the agent decides to remain in the same house, so rt+1
t
house will be sold only at the end of the old period of life and the agent can consume its
o ). This gives rise
discounted value by contracting a corresponding debt (the negative saving st+1
to two further budget constraints, one to hold at the beginning of the old period of life and one
to hold at death:
o
o
o
ct+1
+ pt+1 rt+1
+ st+1 ≤ pt+1 (rta ) + (1 + i) sta
(2.4)
o
o
pt+2 rt+1
+ (1 + i) st+1
≥0
(2.5)
The lifetime budget constraint is found by aggregating the budget constraints (2.3), (2.4),
and (2.5):
(1 − τ (rta )) wt =
{z
}
|
Labor income
1 o
pt+1 (rta )
cta +
ct+1 + pt (rta ) −
1{z
+i }
|
{z 1 + i }
|
Consumption cost
Housingcost adult
o
o
pt+1 rt+1
pt+2 rt+1
+
−
1+i
(1 + i)2
|
{z
}
Housing cost old
(2.6)
Independent of their location decisions, the adult agents maximize lifetime utility by spreading consumption over their lifetimes according to the following condition, which is obtained by
substituting (2.2) into (2.1):
o
u0 (cta ) = β (1 + i) u0 ct+1
2.2
(2.7)
Firms
The production sector is characterized by perfect competition on labor and product markets. All
firms are located in the center of the city, to where their employees commute. Firms employ Lt
7
units of labor as their sole input and produce according to the production function Ft (Lt ) = At Lt ,
where At is the productivity factor at time t that is common to all local firms. The output is
transported costlessly and sold for unit price. Perfect competition ensures that labor is paid its
marginal product, so at each t ∈ T the wage wt is simply:
wt = At
2.3
(2.8)
Migration
The population and spatial dimension of the city are endogenously determined by migration
flows. Agents may migrate at two stages in their lives: when they reach adulthood and upon
retirement.
When agents begin their working lives, they choose which city to live in based on the maximization of lifetime utility. We assume that any agent could obtain lifetime utility Ū by living
in another location outside of the city. We further assume that migration at the beginning of
adulthood is not costly, so at each t ∈ T a necessary equilibrium condition is that each adult in
the city have anticipated lifetime utility equal to Ū.
At retirement, the agent decides whether to stay in the same location or to move away to a
retirement community. As stated above, migration at retirement has a utility cost φ that reflects
the attachment to place and the costs of establishing new social networks in the new location.
We refer to those who intend to migrate away upon retirement as ‘movers’ and those who intend
to remain in the same location upon retirement as ‘stayers’.
We use the superscripts M and S to denote the movers and stayers, respectively. Using this
notation, the conditions for the migration equilibrium of movers and stayers are the following:
a,M
o,M
u ct
+ β u ct+1 − β φ = Ū
(2.9)
o,S
u cta,S + β u ct+1
= Ū
(2.10)
Let gta (r) : R+ → R+ and gto (r) : R+ → R+ be the density functions of agents by location r. The feasibility constraint gta (r) + gto (r) ≤ h (r) must hold for each r. By defini-
8
tion, the population size of the city is Pt =
´∞
0
(gta (r) + gto (r)) dr. As each adult has one unit
of time and the agent living at r spends τ (r) units of time commuting, the labor supply is
´∞
Lt = 0 (1 − τ (r)) gta (r) dr.
3
House price equilibrium
3.1
Consumption levels
Let us define the lifetime consumption cost C as the minimal expenditure necessary to obtain
the reservation utility Ū, when measured in terms of first-period wealth. The relative price of
consumption when adult and old is simply the return on savings 1 + i. The lifetime consumption
costs of movers and stayers are therefore the following:
o,M
ct+1
a,M
ct +
(
M
C ≡ min
o,M
cta,M ,ct+1
1+i
a,M
o,M
+ β u ct+1 − β φ = Ū
u ct
co,S o,S
cta,S + t+1 u cta,S + β u ct+1
= Ū
1+i
(
CS ≡ min
o,S
cta,S ,ct+1
)
(3.1)
)
.
(3.2)
We denote the consumption bundle that minimizes the movers’ expenditure given lifetime
utility Ū to be ca,M , co,M and the equivalent bundle for the stayers to be ca,S , co,S . By
1 o,S
1 o,M
c
and CS ≡ ca,S + 1+i
c .
definition CM ≡ ca,M + 1+i
The following lemma highlights two fundamental consequences of the optimality conditions
(2.7), (2.9), and (2.10) in terms of the agents’ expenditure on consumption.
Lemma 1. In each period t ∈ T and independent of housing location, all movers consume the
bundle ca,M , co,M that satisfies (2.7) and (2.9) and all stayers consume the bundle ca,S , co,S
that satisfies (2.7) and (2.10). Since φ > 0, movers attain higher consumption in each period:
ca,M > ca,S and co,M > co,S . Consequently, CM > CS .
Figure 1 represents the consumption decisions of the two types of agents in the Cartesian
space, with consumption when adult on the horizontal axis and consumption when old on the
vertical axis. Consider the generation who are adults in period t ∈ T . The indifference curves
labelled Ū + β φ and Ū represent the consumption sequences that yield the reservation utility Ū
for movers and stayers, respectively, according to (2.9) and (2.10). Naturally, movers require a
9
higher level of consumption to obtain Ū because of the cost of migration of β φ utils. Given the
interest rate i, the relative price of consumption in the two periods is fixed. Thus the minimal
expenditure to obtain Ū is achieved by the sequences ca,M , co,M and ca,S , co,S for movers
and stayers, respectively. Using consumption at time t as the numeraire, the level of expenditure
necessary to achieve these consumption sequences are CM and CS .
Figure 1: Movers’ and stayers’ consumption levels.
3.2
House prices
For each t ∈ T , let us define the temporary house price equilibrium at t as the prices pt that
emerge from the market allocation of housing when agents have expectations pt+1 and pt+2
about future prices. The (intertemporal) house price equilibrium is the sequence of prices
{pt }t∈T such that, for each t ∈ T , pt is a temporary house price equilibrium at t in which agents
correctly anticipate future prices. In the following, we compute such equilibria.
We define the house price bid to be the price that adult agents would pay for a house at
location r at time t. The house price bids of the movers and stayers are denoted ptM (r) and ptS (r),
respectively. The (temporary) equilibrium house price at location r at time t is denoted pt (r)
10
and is the maximum of the house price bids of the two types of agents – for given anticipated
future prices pt+1 (r) and pt+2 (r) – and the reservation value of land ν:2
pt (r) = max
h
ptM (r) , ptS (r) , ν
i
(3.3)
Before deriving the formal expressions for ptM (r) and ptS (r), we show graphically how
these house price bids are determined. Figure 2 includes the budget sets from Figure 1 but
for clarity the indifference curves and consumption bundles have been removed. The labor
income (1 − τ (r)) wt of an adult living at location r is represented on the horizontal axis. We
define the adult-period resources to be the labor income less the purchase price of the house
at the beginning of the adult period, which we denote aM ≡ (1 − τ (r)) wt − ptM (r) and aS ≡
(1 − τ (r)) wt − pts (r). Agents anticipate that selling the house after one or two periods will
yield old-period resources of pt+1 (r) or
pt+2 (r)
1+i .
The budget lines represent the combinations of
adult- and old-period resources that fund lifetime consumption expenditure of CM or CS .
The house price bids of movers and stayers for the house at location r at time t must exactly
satisfy the migration conditions (2.9) and (2.10), so the discounted lifetime resources available
for consumption are CM for a mover and CS for a stayer. The resulting house price bids ptM (r)
and ptS (r) can thus be read from Figure 2. The movers’ bid ptM (r) implies resources of aM in
the adult period and the sale of the house upon retirement yields pt+1 (r), which combine at a
point on the budget curve CM . The stayers’ bid ptS (r) implies resources of aS in the adult period
and the sale of the house after retirement yields
pt+2 (r)
1+i ,
which combine at a point on the budget
curve CS .
2 The
price of each house on the market is always determined by the bids of the adult agents. Any price lower
than pt (r) would be inferior to the bids of adult agents. The house cannot be sold for more than pt (r) as no adults
would bid such a price and if the existing resident is not willing to sell the house for pt (r) then no transaction
occurs.
11
Figure 2: Movers’ and stayers’ house price bids
We now formally derive the equilibrium prices. Since consumption levels are determined
by the optimal distribution of consumption across time (2.7) and by the migration equilibrium
conditions (2.9) and (2.10), we can use the budget constraint (2.6) to derive the price each type
of agent would offer at each location. The constraints for movers and stayers are:
CM
CS
pt+1 rta,M
= 1 − τ rta,M wt + pt rta,M −
|
{z
} |
{z 1 + i }
Labor income
Housing cost adult
(3.4)
a,S
pt+2 rt+1
= 1 − τ rta,S wt + pt rta,S −
(1 + i)2
|
{z
}
|
{z
}
Labor income
Housing cost adult and old
(3.5)
The house price bids ptM (r) and ptS (r) are found by rearranging (3.4) and (3.5):
12
ptM (r) =
pt+1 (r)
+ (1 − τ (r)) wt −CM
1+i
ptS (r) =
pt+2 (r)
2
(1 + i)
(3.6)
+ (1 − τ (r)) wt −CS
(3.7)
The equilibrium price function is found by substituting the house price bids (3.6) and (3.7)
into the expression for the equilibrium house price (3.3):
"
pt+1 (r)
pt+2 (r)
pt (r) = max
+ (1 − τ (r)) wt −CS , ν
+ (1 − τ (r)) wt −CM ,
1+i
(1 + i)2
#
(3.8)
Computing the equilibrium prices according to (3.8) for each r yields the temporary house
price equilibrium at time t. The combination of these temporary house price equilibria for all t
is the intertemporal house price equilibrium.
4
Steady state analysis
In this section we analyze the properties of the steady state of our economy. Let w be the steady
state wages and p (r) be the steady state price at location r. Considering that in the steady state
an agent of each type will eventually sell the house to an agent of the same type, we can find
the steady state house price bids by setting the house prices to be constant in (3.6) and (3.7):
1+i (1 − τ (r)) w −CM
i
(4.1)
i
(1 + i)2 h
(1 − τ (r)) w −CS
i (2 + i)
(4.2)
pM (r) =
pS (r) =
Substituting the house price bids (4.1) and (4.2) into (3.3) yields the equilibrium house prices
in the steady state:
"
2h
i
1+i M (1 + i)
S
p (r) = max
(1 − τ (r)) w −C ,
(1 − τ (r)) w −C , ν
i
i (2 + i)
#
(4.3)
Note that, in terms of distance from the city center, the movers’ house price bid curve is
13
steeper than the stayers’ house price bid curve. This is because the two types of agents value
relative proximity to the city center differently: both commute for a single period but stayers
remain in the city for two, so stayers value proximity less per period of residence than movers
do. On the other hand, stayers spend less on consumption (as CS < CM ) to obtain the reservation
utility Ū, so the location at which they would bid zero is further from the city center than the
equivalent location for movers.
Since households are able to purchase a house only if they bid more than the land value
ν, a necessary and sufficient condition for the city to be inhabited is that the house price (4.3)
adjacent to the city center exceed this value:
"
#
2h
i
1+i (1
+
i)
max
w −CM ,
w −CS > ν
i
i (2 + i)
(4.4)
The condition for the efficient outcome to have at least one individual is trivially identical to
(4.4), as the condition that either the marginal mover or the marginal stayer would pay a positive
price for a house adjacent to the city center is the same as the condition that one of these agents
generates a positive surplus. In the remainder of this section we assume that condition (4.4)
holds, so that the city population is positive.
4.1
City structure
Let r̄M and r̄S be the maximum distances at which a mover or a stayer, respectively, outbid the
reservation land value ν while spending enough on consumption to obtain utility Ū. Formally,
r̄M satisfies pM r̄M = ν and r̄S satisfies pS r̄S = ν so these maximum distances assume the
following values:
+
i
1
M
−1
ν +C
r̄ = τ
1−
w 1+i
M
"
r̄S = τ −1
1
1−
w
i (2 + i)
2
(1 + i)
(4.5)
!!#+
ν +CS
(4.6)
Recall that the movers’ house price bid curve is steeper than the stayers’ house price bid
curve. The two curves therefore either do not intersect within the city or intersect at a unique
distance, with movers submitting the higher bids at all nearer locations and stayers submit14
ting the higher bids at all more distant locations. This results in a city structure in which all
movers reside within a certain distance from the city center and all stayers reside at more distant locations. We define the most distant location in which movers reside to be the equilibrium
mover-stayer threshold, which we denote r̂EQ . The mover-stayer threshold is effectively zero
if the stayers outbid the movers at the center and is effectively equal to the maximum movers’
distance r̄M if the movers outbid the stayers at that location. For an interior solution, the equilibrium mover-stayer threshold is the distance at which the house price bids of movers and stayers
are equal. The following expression for r̂EQ can therefore be derived from the house price bids
(4.1) and (4.2):
r̂EQ =




r̄M




if r̄M ≥ r̄S
τ −1 1 − w1 (2 + i)CM − (1 + i)CS






0
if r̄M < r̄S and pM (0) > pS (0)
(4.7)
if pM (0) ≤ pS (0)
Three types of city structure are possible. The first is a movers-only city, which has only
working-age residents. This type of city arises when the reservation value of land is so high
that staying for an additional period in any location, given that the residents must outbid ν, is
necessarily more costly in utility terms than the cost of relocation. That is, when CM − CS ≤
i
ν.
(1+i)2
The mixed city has some but not all agents migrating away upon retirement. The inner part
of the city is inhabited exclusively by adults: as it would be too expensive to remain in the same
location when old, these agents relocate when retired. The suburbs are inhabited by agents who
remain in the same locations when retired and are therefore populated by both adult and old
agents. The mixed city arises when both CM −CS >
i
ν,
(1+i)2
so the value of the land is not too
high, and w > (2 + i)CM − (1 + i)CS , so movers outbid stayers at the center. In the following
sections we focus our attention on this type of city, as the presence of both types of agents is
realistic and highlights the issues we wish to investigate.
The stayers-only city is a relatively small city in which all residents remain in their houses
when they retire. This type of city arises when wages are low relative to relocation costs, so
house prices are low enough in all parts of the city that it is not worthwhile to relocate upon
retirement. Formally, the conditions for this type of city to arise are that CM −CS >
15
i
ν
(1+i)2
and
w ≤ (2 + i)CM − (1 + i)CS .
Figure 3 represents the house price bids and resulting city structure for a mixed city. The
horizontal axis represents the distance r to the housing location; the vertical axis represents
the house prices and reservation land value. The downward-sloping curves pM (r) and pS (r)
represent the house price bids of the movers and stayers, respectively. The horizontal line ν
represents the reservation value of land. The dotted lines p̃M (r) and p̃S (r) represent the house
price bids that would leave the existing old agent in each location indifferent about whether to
stay in the house for retirement, which are derived in Appendix A.
As discussed above, the movers’ house price bid curve is steeper than the stayers’ house
price bid curve as the movers do not distribute the scarcity premium over the two periods of
life. The bid curves intersect at the threshold distance r̂EQ : movers make the higher bids for
locations between the city center and r̂EQ while stayers make the higher bids for locations
between r̂EQ and the edge of the city. The outer limit of the city r̄S is determined by the house
price bids and the reservation value of land, as a location is occupied if and only if the house
price bid is higher than ν. The prices that leave the old agents indifferent about whether to
stay in their houses for retirement are below the market prices for movers and above the market
prices for stayers, so both types of agents make time-consistent retirement relocation decisions,
a general result that is proven in Appendix A.
16
Figure 3: House price bids and city structure.
We now outline some comparative statics in the city structure. An increase in wages leads
to a larger city, as higher incomes allow individuals to make higher house price bids at all
locations. However, a feature of the model is that the house price at the mover-stayer threshold
r̂ is independent of wages. The house price at r̂EQ can be derived from (4.1) and (4.2) by setting
the house price bids of the two types of agents to be equal:
(1 + i)2 p r̂EQ =
CM −CS
i
(4.8)
Therefore, an increase in w shifts the movers’ and the stayers’ bid curves up such that the
distance from the city center to their intersection r̂EQ increases without changing p r̂EQ .
An increase in the reservation utility Ū would lead to a smaller city, as a larger amount of
consumption would be required by each individual and the house price bids would therefore be
lower. An increase in the relocation cost φ leads to a city with fewer movers, as such a change
would decrease the house price bids of the movers but leaves the house price bids of the stayers
unchanged.
17
4.2
Efficient housing allocation
We now define the efficient locations of agents in the city for their adult lives and retirement.
Due to migration this exercise is not trivial. Migration imposes the requirement that all agents
obtain the reservation utility Ū and endogenizes the size of the population. Both features are
problematic for the application of Pareto efficiency.
We define efficiency as follows. We first introduce a fictitious agent who appropriates the
net wealth created by the adults in the city, above that required for them to obtain the reservation
utility Ū over their lifetimes and the rental price of the land they occupy in the city.3 The net
wealth can be thought of as the total income that a city can make by renting a certain amount
of land around the city center, assigning the houses to the agents (for one or two periods),
allocating the reservation consumption levels to the agents, and appropriating the remaining
labor income. Then, for any given number of adult agents in the city Pa ∈ R+ , we define
the efficient allocation of housing as the locations and types of agents (movers or stayers) that
maximize the net wealth of the fictitious agent.
We begin by outlining two fundamental features of the efficient housing allocation. The
first is that the city is dense: no land between the city center and the location of the most distant
city resident is left vacant or used for anything other than housing city residents. To understand
why this must be so, consider what the contrary would imply. For any non-dense city there
would be an agent residing at some r while a unit of housing at some r0 < r was vacant. The net
wealth of the city would trivially be increased by relocating such an agent from r to r0 , which
would decrease commuting time and thereby increase the labor supply without imposing any
additional cost. Any such allocation could therefore by definition not maximize net wealth. A
formal proof of this is given in the appendix.
The second fundamental feature of the efficient housing allocation is that there is some
threshold distance r̂ within which all movers are located and beyond which all of the stayers are
located. This strict sorting follows reasoning similar to that for the city density. If the agents
were not sorted in this way, then there would be some mover in a more distant location than a
stayer. Upon retirement a stayer continues to occupy a location in the city that may therefore
not be occupied by an adult of the following generation. The more distant an agent’s adult
3 We
consider the welfare of all agents who reside in the city while adults over their entire lifetimes, whether
they remain in the city or move away when old.
18
location, the more time is spent commuting and the less income is generated. Net wealth would
therefore be increased by switching the locations of these two agents. See the appendix for a
formal proof.
What remains is to define the numbers of movers and stayers in the city, for the total adult
population Pa . Given the density and strict sorting of movers and stayers this is equivalent to
defining the optimal mover-stayer threshold r̂∗ ∈ R+ , the most distant location at which movers
reside and beyond which all stayers reside in the efficient housing allocation. To identify the
optimal mover-stayer threshold we impose the density and strict sorting conditions and find
the net wealth for any given mover-stayer threshold r̂ ∈ R+ , then identify the value of r̂ that
maximizes the net wealth. The city edge r̄ ∈ R+ is defined as the most distant location at which
a city resident lives. We assume that any part of the area occupied by stayers has even shares of
adult and old agents.4 As the city is dense and divided at the mover-stayer threshold r̂, the city
edge r̄ is thus determined by the mover-stayer threshold r̂ to be the value that exactly satisfies the
´ r̄
´ r̂
feasibility constraint Pa = 0 h (r) dr + 12 r̂ h (r) dr.5 Using this notation for the mover-stayer
threshold and the city edge, the net wealth can be stated as:
ˆ
Ω ≡
r̂ (1 − τ (r)) w −CM h (r) dr
|0
{z
}
Net
earnings
of
movers
ˆ
i
1 r̄ h
+
(1 − τ (r)) w −CS h (r) dr
2
{z
}
| r̂
ˆ r̄ Net earnings of stayers
i
−
νh (r) dr
0 1+i
|
{z
}
Land rental cost
(4.9)
The derivative of the net wealth (4.9) with respect to r̂ is:
1
∂Ω
i
M
S
= h (r̂) − C −C + (τ (r̄) − τ (r̂)) w +
ν
∂ r̂
2
1+i
(4.10)
The three terms inside the square brackets in (4.10) represent the various effects of the shift
in r̂ on the net wealth. The density h (r̂) converts the shift in the distance to the mover-stayer
4 If
an area occupied by stayers was initially occupied only by adult stayers, then in the subsequent periods
it would oscillate between generations of old and new agents. To ensure that the shares of the two concurrent
generations are equal, we assume that half of the housing was occupied by old agents in the initial period.
5 This constraint implies that r̄ changes with r̂ according to ∂ r̄ = − h(r̂) .
∂ r̂
h(r̄)
19
threshold into the number of additional movers that this shift implies, so we can consider the
terms inside the square brackets as the effects of the number of movers being increased by one.
The term CM −CS represents the additional cost of providing Ū to a single mover rather than to
a stayer, as consumption must be higher to compensate for the cost of moving away. The term
1
2 (τ (r̄) − τ (r̂)) w
represents the saving in commuting costs of half an agent living at the mover-
stayer threshold when adult rather than at the edge of the city, as a single new adult mover
at r̂ implies half a stayer less at the same location and half a stayer less at r̄. The term
i
1+i ν
represents the saving in land costs from renting one housing unit less, as converting a mover to
a stayer reduces the old-age population by one.
The saving
i
1+i ν
from providing one unit less of housing and the cost CM − CS of pro-
viding the reservation utility to one additional agent are both independent of r̂. On the other
hand, the saving in commuting costs
1
2 (τ (r̄) − τ (r̂)) w
is continuous and decreasing in r̂, as
a larger number of movers implies a more distant mover-stayer threshold and a nearer city
edge. Therefore, the net wealth (4.9) is differentiable and strictly concave in r̂, so its value
is maximized by setting r̂ either at one of its limits or such that the first derivative of the net
wealth function (4.10) is equal to zero. This gives rise to the following proposition, in which
to simplify notation we use Ψ (r̂) = 21 (τ (r̄) − τ (r̂)) w for the commuting cost differential and
i
ϒ (r̂) = − CM −CS + 12 (τ (r̄) − τ (r̂)) w + 1+i
ν for the overall benefit of an additional mover.
Proposition 1. For a given number of adult residents Pa , the optimal mover-stayer threshold r̂∗
is:
r̂∗ =




r̄




if
i
1+i ν
≥ CM −CS
i
ϒ−1 (0) if Ψ (0) + 1+i
ν ≥ CM −CS >






i
0
if Ψ (0) + 1+i
ν < CM −CS
i
1+i ν
(4.11)
The three possible city types correspond to the types that arise in the market equilibrium.
The first type is the movers-only city, which arises when the benefit of providing one less unit
of housing is greater than the additional cost of funding consumption for a mover. In this case,
the benefit of having an additional mover in the city is positive even if the benefit from reduced
commuting time is zero, so it is optimal for all residents of the city to be movers. This condition
does not depend on the size of the city, but a simple comparison of the reservation value of land
20
and the money value of the relocation cost evaluated at Ū.
The mixed city arises when locating a mover adjacent to the city center increases net wealth
but locating a mover at the city edge would reduce net wealth. The number of movers is set
such that the benefit of adding a mover would be zero. In such a city there is an inner zone of
movers surrounded by an outer zone of stayers.
The stayers-only city arises when the benefit of having a mover adjacent to the city center is
negative, as the cost of providing the higher consumption for a mover exceeds the savings from
providing one less unit of housing and from the difference in commuting costs between the city
edge and the city center. In this case it is optimal for all city residents to be stayers.
4.3
Inefficiency of the decentralized equilibrium
In this section we compare the efficient allocation of housing with the allocation that arises
in the decentralized equilibrium. This exercise is facilitated by the common basic structure
of the market equilibrium and the efficient outcome: both involve dense cities divided by a
mover-stayer threshold r̂ into an inner zone of movers and an outer zone of stayers. To evaluate
the efficiency of the decentralized equilibrium, it therefore suffices to assess the efficiency of
the equilibrium mover-stayer threshold r̂EQ . For cities that are of the movers-only type in the
equilibrium and efficient outcomes, the equilibrium outcome is trivially efficient as the agents
are all movers and occupy the same set of locations in the two cases. Likewise with cities that
are of the stayers-only type in both cases, as the equilibrium and efficient outcomes feature
stayers in the same set of locations.
For mixed cities, however, the equilibrium outcome is not efficient. To evaluate the efficiency of the equilibrium in the mixed city, we assess whether the net wealth (4.9) can be
increased by changing the distance to the mover-stayer threshold. As (4.9) is concave in r̂, this
is a sufficient test for efficiency. Substituting the house price bids of movers (4.1) and stayers
(4.2) into the change in the net wealth with respect to the mover-stayer threshold r̂ (4.10) and
evaluating the resulting expression at r̂EQ yields the following inequality:
1 i2
1 EQ M
∂ Ω S
EQ
=
ih
r̂
C
−C
+
h
r̂
ν >0
∂ r̂ r̂=r̂EQ 2
2
(1 + i)2
(4.12)
>From (4.12) it is evident that in a mixed city the equilibrium mover-stayer threshold r̂EQ is
21
too near the city center to achieve efficiency. Net wealth (4.9) would be increased by shifting out
the mover-stayer threshold and leaving the number of adults constant, so the market equilibrium
has too few movers to achieve efficiency. This reflects the effect that a decision to stay in one’s
house for retirement has on other agents, which is not properly priced on the market.
A movers-only city arises for a wider range of parameters in the efficient outcome than in
the market equilibrium, where stayers do not consider the external effect of occupying land by
discounting its cost. This is consistent with the finding that the mixed city has too few movers
in the decentralized equilibrium. For the parameters
i
ν
(1+i)2
< CM − CS ≤
i
1+i ν,
the efficient
outcome is a movers-only city but the equilibrium outcome is a mixed city.
The city has a positive population of stayers when the reservation value of land is less than
the money value of the relocation cost evaluated at Ū, which makes it optimal for the agent
located at the outer limit of the city to be a stayer. Given that this condition holds, whether the
city has only stayers or both movers and stayers depends on its size: below a certain population
of adult agents the city with have only stayers; above that population the city will have both
movers and stayers. Consistent with the finding that the market generates too few movers to
attain efficiency, the minimum adult population for which the city has at least one mover is
lower for the efficient outcome than for the market equilibrium.6
The nature of the market failure can be understood by considering the house values over
the lifetimes of the agents. Consider the schedule of house prices by location shown in Figure
3. The stayers would be willing to pay more to remain in their houses for retirement than they
paid one period earlier, so a stayer who occupies a house implicitly outbids two generations of
potential residents for the same location. On the other hand, the house prices and commuting
costs are continuous in distance from the city center. On either side of the threshold distance
r̂EQ reside agents who occupy housing units of practically the same value and have the same
expected lifetime utility, but the stayer occupies the location for two periods whereas the mover
occupies the location for only one period. The externality arises from the lack of compensation
6 The
adult population Pa defined by the market – which we also use to assess the efficient outcome – is
increasing in the wage w. To show that the threshold population is lower for the efficient outcome is therefore
equivalent to showing that the threshold wage is lower. In the market equilibrium, the stayers-only city arises
i
when wEQ < (2 + i)CM − (1 + i)CS . In the efficient outcome, it arises when 21 τ (r̄) w + 1+i
ν < CM − CS . By
substituting in the equilibrium price (4.2) evaluated at the city edge r̄, which must be equal to ν, this condition
2
becomes w∗ < 2CM − CS − i 2 ν < (2 + i)CM − (1 + i)CS = wEQ where the latter inequality follows from the
(1+i)
condition
i
1+i ν
< CM −CS for the presence of stayers.
22
for moving out and freeing up the location for the following generation of adult residents.
4.4
Optimal policy
In this section, we study how a planner could increase efficiency through the implementation
of policy. In practice it would likely be infeasible for the planner to elicit information about
whether individual residents plan to migrate away when they retire or to levy taxes or give
subsidies that vary by neighborhood, so we consider policies that do not require information
on specific individuals or residents of particular locations. Thus we study the application of a
proportional property tax θ (paid in each period), a proportional acquisition tax ζ (paid only
when the house is purchased), and a lump sum T that is paid to the adult agents. The budget
constraint of the planner, which we assume must be balanced in each period, is therefore:
ˆ
ˆ
∞
a
T g (r) dr =
0
ˆ
∞
a
o
∞
θ p (r) (g (r) + g (r)) dr +
0
ζ p (r) ga (r) dr
(4.13)
0
The steady state house price bids (4.1) and (4.2) become:
pM
T (r) =
pST
(r) =
i
+θ +ζ
1+i
−1
i (2 + i)
2+i
θ +ζ
+
2
1+i
(1 + i)
(1 − τ (r)) w −CM + T
!−1
h
i
S
(1 − τ (r)) w −C + T
(4.14)
(4.15)
The equilibrium mover-stayer threshold given in (4.7) for a mixed city becomes:
r̂TEQ = τ −1
"
#!
2
ζ
1
(1
+
i)
1−
(2 + i)CM − (1 + i)CS +
CM −CS − T
w
i + (1 + i) θ
(4.16)
The house price at the new equilibrium mover-stayer threshold is:
p r̂TEQ =
(1 + i)2 M
C −CS
i + (1 + i) θ
(4.17)
The maximum distances at which the agents bid at least the land value ν (4.5) and (4.6)
become:
23
r̄TM
=τ
−1
i
1 M
1−
C −T +
+θ +ζ ν
w
1+i
"
1
1−
CS − T +
w
r̄TS = τ −1
i (2 + i)
2+i
+
θ +ζ
2
1+i
(1 + i)
(4.18)
! #!
(4.19)
ν
We now have all the necessary ingredients to investigate which policies, if any, bring about
efficiency as an equilibrium housing allocation. We proceed by calculating the restrictions on
the tax and subsidy rates that are necessary for the efficient outcome to arise for each of the
three types of city, then evaluate whether these can be combined to constitute a coherent tax and
subsidy scheme.
To begin with, recall that a movers-only city is efficient when CM −CS ≤
i
1+i ν,
whereas the
same type of city arises in the decentralized housing allocation only when the more restrictive
i
ν is satisfied. The condition for a movers-only city to arise
(1+i)2
is that r̄TM ≥ r̄TS . Observing (4.18) and (4.19) it is clear that whether
requirement that CM − CS ≤
as an equilibrium outcome
the city is movers-only is independent of both the subsidy and the transfer tax. Furthermore, the
following rate of property tax, derived by comparing CM − CS ≤
i
1+i ν
with (4.18) and (4.19),
ensures that the market outcome is a movers-only city if and only if this is efficient:
θ=
i2
1+i
(4.20)
Consider now the case of mixed cities. Any mixed city that arises in a decentralized housing allocation but that efficiency requires to be movers-only is corrected by the property tax in
(4.20). Cities that are mixed in the efficient outcome have too few movers, so the mover-stayer
threshold must be shifted out by the policy for efficiency to be achieved. From (4.11), the effi
i
cient mover-stayer threshold in a mixed city satisfies CM −CS − 12 (τ (r̄) − τ (r̂∗ )) w − 1+i
ν=
0. The solution to this equation for the policy-adjusted equilibrium mover-stayer threshold
(4.16) and city edge (4.19), taking the property tax (4.20) as given, yields the unique rate of
transfer tax that achieves the efficient outcome for mixed cities:
ζ =−
i2
1+i
(4.21)
The transfer tax (4.21) is identical in magnitude but opposite in sign to the property tax
24
(4.20), so they offset each other exactly in the adult period but old residents pay a positive rate
of tax. Again, the transfer tax has no influence on whether a city contains only movers, so the
rate in (4.21) is by definition consistent with the correction of movers-only cities. Equations
(4.20) and (4.21) determine the lump-sum subsidy T as the residual from (4.13):
´∞
T=
i2
p (r) go (r) dr
0 1+i
´∞
a
0 g (r) dr
(4.22)
The lump-sum subsidy is effectively an intergenerational transfer from old to adult agents,
as it redistributes funds from the taxes that are zero for adults and positive for old residents. The
level of the lump-sum subsidy has no bearing on whether a city is of the movers-only type or
on the location of the mover-stayer threshold.
The remaining case is of cities that are comprised entirely of stayers in both the decentralized and efficient outcomes. The housing allocation in this type of city does not require any
correction to achieve efficiency, so no additional policy instruments are necessary. Furthermore,
the policy instruments introduced above do not interfere with the efficient outcome in this case.
The tax and subsidy scheme outlined in this section is coherent and is sufficient to correct
for the externalities arising from endogenous retiree migration. The scheme has the attractive
properties of being parsimonious in terms of the information required by the policymaker and
simple to implement. Furthermore, it is not necessary to adjust the tax rates for the size of the
city or even the magnitude of the relocation cost.
4.5
The new house price equilibrium
Figure 3 illustrates the effect of the optimal policies discussed above on the steady state house
price equilibrium. The house price bids of the movers and stayers in the decentralized equilibrium are the curves labeled pM (r) and pS (r). The equivalent house price bids with the
S
application of policy are represented by the curves labeled pM
T (r) and pT (r).
The proportional acquisition subsidy ζ and the proportional property tax θ are equal in
magnitude but opposite in sign. The two therefore offset each other exactly for the movers, who
only stay in the house for one period. The policy shifts the movers’ house price bid curve up by
the amount of the lump-sum transfer T , but as the acquisition subsidy and property tax offset
each other the slope of the curve does not change.
25
As stayers pay the property tax in two periods, the net amount of the acquisition subsidy and
property tax is a positive proportional tax on the purchase price of the house. The house price
bids of stayers are therefore reduced proportionally by these two policy instruments, which
reduces the slope of the bid curve from
i(2+i)
(1+i)2
to
i(4+i)
.
(1+i)2
In addition, the curve is shifted up by
the amount of the lump-sum transfer T .
The equilibrium mover-stayer threshold is shifted out by the policy instruments to r̂TEQ = r̂∗ ,
which is necessarily larger than r̂EQ .
The budget of the planner is represented by the shaded areas on Figure 3, which are the
differences between the house price bids with and without the tax and subsidy scheme for
the type of agent who resides at each location. The budget is balanced when the sum of the
areas above and below the original house price bids, weighted by the density of the city at the
corresponding distance, are equal.
Figure 4: House price equilibrium in the steady state with the application of the tax and subsidy scheme.
26
5
Dynamics
In this section, we study the effects of a permanent and unexpected shock to local productivity
on the housing market equilibrium. A shock to local productivity affects wages and thereby
house price bids, which affects both new residents and existing homeowners. We begin by
analyzing how the productivity shock affects individuals’ location decisions, then evaluate the
potential for policy intervention to achieve an equitable outcome.
5.1
Retirees’ migration decisions
Consider an agent who begins the old period of life at t + 1. The wealth available to the agent
when old is the sum of the market value of the house purchased when adult pt+1 (rta ) and a given
amount of savings (possibly negative) (1 + i) sta . The newly-retired agent has the choice either
to move to a retirement community and spend the entire wealth on consumption or to stay in the
1
pt+2 (rta ) and consume the remainder.7 Formally the agent’s
same house at cost pt+1 (rta ) − 1+i
budget constraint is obtained by aggregating (2.4) and (2.5):
1
o
o
o
≤ pt+1 (rta ) + (1 + i) sta
ct+1
+ pt+1 rt+1
−
pt+2 rt+1
|
{z
}
1
+
i
|
{z
}
old wealth
old housing cost
(5.1)
Given the utility function (2.2), the optimal choice of the retiree will depend on whether
moving, which yields utility u (pt+1 (rta ) + (1 + i) sta ) − φ , is preferred to staying, which yields
1
pt+2 (rta ) + (1 + i) sta .
utility u 1+i
A consequence of the first-period consumption decisions that agents make based on anticipated house prices is that if house prices do not vary too much, then the moving-staying
decisions of existing residents are not affected by the shock and retirees only adjust consumption to the changed budget possibilities. To demonstrate this result, we introduce the concept
of a neighborhood around anticipated house prices of size ε ∈ R+ . We say that house prices
p0 ≡ {pt0 (r)}t∈T are in the ε-neighborhood of p ≡ {pt (r)}t∈T if |pt (r) − pt0 (r)| < ε for each
r ∈ R+ and t ∈ T .
7 Recall
that although the retiree could move to any other house in the city, that would not be optimal as the
agent no longer commutes to work and so proximity to the center would no longer be valuable.
27
Proposition 2. Let p ≡ {pt (r)}t∈T be a house price equilibrium and rta be the location optimally selected by some agent at t. There exists an ε-neighborhood of p, such that at t + 1 the
agent does not revise his relocation decision. That is, for each p0 in the ε-neighborhood of p:
u (pt+1 (rta ) + (1 + i) sta ) − φ
1
pt+2 (rta ) + (1 + i) sta
≷u
1+i
implies
u
0
(rta ) + (1 + i) sta
pt+1
1 0
a
a
−φ ≷ u
p (r ) + (1 + i) st .
1 + i t+2 t
Proof. See Appendix.
The idea is related to the result about the time-consistency of location decisions shown in
Appendix A. At the beginning of the adult period of life, agents anticipate whether they will
move away or stay in their houses for retirement and set their consumption levels ca,M or ca,S
accordingly. As demonstrated in Appendix A, revising the decision about whether to relocate
upon retirement given the adult level of consumption would yield a strictly lower level of oldage utility. The corollary is that there must exist some level of perturbation to house prices that,
whether positive or negative, is sufficiently small that changing the relocation decision would
not yield a higher level of utility.
The result in Proposition 2 may thus be expressed with the two following inequalities. For
a mover living at rta , substituting the old-age budget constraint (5.1) into the inequality between
the old-age utility levels when moving or staying yields:
o,M
u c
1
o
−φ > u
pt+2 rt+1
− pt+1 (rta ) + co,M
1+i
(5.2)
The equivalent expression for a stayer living at rta is:
u
pt+1 (rta ) −
o,S
1
o
pt+2 rt+1 + c
− φ < u co,S
1+i
(5.3)
The inequalities (5.2) and (5.3) can be read from Figure 3 as the differences between the
equilibrium prices and the prices that leave old agents indifferent about whether to relocate.
We next derive the sufficient conditions for a shock to induce changes in agents’ retirement
28
relocation decisions and investigate which agents are affected.
5.2
Shock size and switching decisions
Let us assume that at time t the economy is at the steady state, so that the house price equilibrium
is determined by (4.3). Local productivity in the steady state is equal to A, so that wages at time
t are wt = A. The optimal savings sta are determined by the respective optimal consumption
level for each type of agent and the budget constraints (2.4) and (2.5):
sta (rta ) =




1
1+i
co,M − pM (rta )



1
1+i
1 S a
co,S − 1+i
p (rt )
if 0 ≤ rta ≤ min r̄M , r̂EQ
(5.4)
if max r̂EQ , 0 ≤ rta ≤ r̄S
a )) A −CM
where the equilibrium house price bids in the initial steady state are pM (rta ) = 1+i
(1
−
τ
(r
t
i
(1+i)2 a
S
S
a
and p (rt ) = i(2+i) (1 − τ (rt )) A −C .
Let the new firm productivity be A≥t+1 ≡ A + ∆A, where ∆A is the size of the shock. Since
the shock is permanent, from time t + 1 the house price equilibrium directly jumps to the new
steady state with wages w≥t+1 = A + ∆A, so the value of the house at location rta becomes:

h
i

EQ

1+i
a
M
a
M

if 0 ≤ rt ≤ min r̄≥t+1 , r̂≥t+1

i (1 − τ (rt )) (A + ∆A) −C



h
i
2 p≥t+1 (rta ) = (1+i) (1 − τ (rta )) (A + ∆A) −CS if max r̂EQ , 0 ≤ rta ≤ r̄S
≥t+1
≥t+1
i(2+i)






ν
otherwise
(5.5)
EQ
where r̂≥t+1
is the distance at which the new steady state house price bids of movers and stayers
M
S
are equal and r̄≥t+1
and r̄≥t+1
are the locations at which movers and stayers bid exactly the land
value ν. We use ∆p (rta ) ≡ p≥t+1 (rta ) − p (rta ) to denote the change in the equilibrium house
price.
We can now use (5.4) to determine at which locations, if any, the retirees who were adult at
time t are indifferent between moving and staying following the productivity shock. For agents
who originally planned to move away, the indifference condition is:
o,M
u ∆p (r) + c
1
i M
∆p (r) −
p (r) + co,M
−φ = u
1+i
1+i
29
(5.6)
For agents who originally planned to stay, the indifference condition is:
u ∆p (r) +
1
i S
o,S
o,S
−φ = u
p (r) + c
∆p (r) + c
1+i
1+i
(5.7)
Substituting the saving function (5.4) into the consistency conditions (5.2) and (5.3) yields:
o,M
u c
i M
o,M
p (r) + c
−φ > u −
1+i
(5.8)
i S
o,S
u
p (r) + c
− φ < u co,S
1+i
(5.9)
The equations (5.6) and (5.7) and inequalities (5.8) and (5.9) allow us to study the effects of
positive and negative productivity shocks on retiree migration choices, which are the subject of
the two following propositions.
Proposition 3. If the utility function u (·) is sufficiently concave, then for the intended mover
h
i
at each r ∈ 0, r̂tEQ there exists some positive productivity shock that would induce the agent
to stay. If the utility function u (·) is not too concave, then for the intended stayer at each
h
i
r ∈ r̂tEQ , r̄tS there exists some positive productivity shock that would induce the agent to move.
For both types, the minimum required productivity shocks are increasing in the distance from
the city center.
Proof. See Appendix D.
When the city receives a positive productivity shock, house prices increase as workers in
the city can earn higher wages. Retirees thus experience a capital gain, as the houses they own
become more valuable than they had anticipated. The additional wealth can be used to increase
consumption, to adjust the relocation decision, or both. Proposition 3 reveals that if the welfare
increase from additional consumption reduces sufficiently rapidly compared to the welfare cost
of relocating (sufficiently concave utility function), then a mover would decide to stay after a
large capital gain. Conversely, if the welfare increase from additional consumption does not
marginally decrease (not too concave utility function), then a stayer would decide to move after
a large capital gain.
The last point that Proposition 3 highlights concerns the relationship between the decisions
30
h
i
of retirees at different locations. If an intended mover at r ∈ 0, r̂tEQ decides to stay, it must be
the case that all intended movers closer to the city center would also decide to stay. The reason
is that, for a given shock, the capital gain is decreasing with distance (since it is proportional to
the house prices). Thus, if the capital gain is sufficient for the retiree at r to decide to stay, then
all retirees living closer to the center – who each experience a larger capital gain – would also
be willing to give up some amount of consumption in order to stay. Similarly, if an intended
h
i
stayer at r ∈ r̂tEQ , r̄tS decides to move, it must be the case that all intended stayers closer to the
city center would also decide to move.
We now consider the implications of negative productivity shocks. The main difference is
that with a negative productivity shock, retirees may not be able to repay the debt or may not be
able to afford to remain in the same house. Bankruptcy may occur if an agent borrows money to
purchase a house, with the intention of using the proceeds of its sale to fund consumption and to
repay the loan, but then the house price drops so drastically that they are no longer able to repay
the loan. We assume that if a retiree’s wealth is negative, so that (1 + i) sta + p≥t+1 (rta ) < 0, the
retiree simply has zero consumption when old and is obliged to move. The house price need not
in fact decrease this drastically for the retiree to be forced to move out, as the retiree’s wealth
may be reduced to some small positive value that is less than the cost of holding the house for
1
p≥t+1 (rta ) < 0.
the period of retirement, which occurs when (1 + i) sta + 1+i
h
i
Proposition 4. For the intended mover at each r ∈ 0, r̂tEQ there exists some negative produc
1
tivity shock that would induce the agent to stay if u co,M − pM (r) + 1+i
ν > u co,M − pM (r) + ν −
h
i
1
φ and co,M − pM (r) + 1+i
ν ≥ 0. For the intended stayer at each r ∈ r̂tEQ , r̄tS there exists some
1 S
negative productivity shock that would induce the agent to move if either co,S − 1+i
p (r) +
1
1+i ν
< 0 or the utility function u (·) is sufficiently concave. The magnitude of the minimum
required productivity shock is decreasing in the distance from the city center for the movers but
increasing in the distance from the city center for the stayers.
Proof. See Appendix D.
To understand what Proposition 4 implies for the intended movers, consider a large negative
productivity shock that hits the economy and reduces the value of the house owned by the retiree
at r to the minimal level ν. This capital loss has two effects: a wealth effect, due to the decreased
value of the house, and a substitution effect, due to the reduced price of staying an additional
31
1
period. When the intended mover is not made bankrupt, formally co,M − pM (r) + 1+i
ν ≥ 0,
and the reservation value of land is not too small, the substitution effect dominates and does so
independently of the form of the utility function. The retiree would therefore gain more utility
1
ν .
from staying, that is u co,M − pM (r) + ν − φ < u co,M − pM (r) + 1+i
Proposition 4 outlines two cases in which a stayer would decide to move. The first case
is where the retiree at r is no longer able to afford the cost of living in the same house for an
additional period. The second case is where the drop in consumption due to the capital loss is
so costly in utility terms that mitigating it by moving out, despite the utility cost φ , is preferred.
For this to hold, the utility function u (·) must be sufficiently concave.
5.3
Horizontal equity and redistribution policies
The above analysis describes the role of the unexpected capital gains or losses that result from
a productivity shock on the relocation decisions of retirees. In this section, we address the
heterogeneity of these gains and losses across locations as a motivation for policy intervention.
When there is a productivity shock in the city, the effect on each retiree’s wealth depends on
the value of the house that he or she purchased as an adult. Since house prices are decreasing
in the distance from the city center, the absolute shock to retirees’ wealth will therefore also
be decreasing with distance. Formally, the net wealth of the agents at rta is equal to pt+1 (rta ) +
(1 + i) sta and the variation in the net wealth is proportional to the working time that can be
supplied by the adult residents at rta , which is the unit of time that each agent has to supply less
the time spent commuting. As the area occupied by movers is comprised entirely of workingage adults while the area occupied by stayers contains equal shares of adults and retirees, the
change in net wealth also varies by a larger amount for movers than for stayers to reflect the
larger share of movers who value proximity to the city center. The relevant multipliers are
for movers and
(1+i)2
i(2+i)
1+i
i
for stayers. This result is stated in the following lemma.8
Lemma 2. The closer to the city center, the larger is the gain (loss) of a positive (negative)
productivity shock.
We appeal to the principle of horizontal equity as a moral justification for redistributing
8 This
result is straightforward to show by taking the derivative of the prices in (5.5) with respect to the change
in productivity.
32
gains or losses. Following Mirrlees (1972), we propose to establish horizontal equity by redistributing the gains or losses amongst the existing residents and thus adopt a maximin social
welfare function. This is a Rawlsian type of welfare function, according to which the social
welfare is the lowest utility level experienced by any agents. The idea behind the redistribution
is that retirees are not responsible for the productivity shock and it was not possible to anticipate that it would occur, so the effect on the retirees’ welfare should be equalized and therefore
independent of the initial housing location.
The first-best redistribution policy is the one that exactly equalizes the ex-post lifetime wellbeing of the retirees who are hit by the productivity shock. This policy would assign a transfer
to each agent such that a uniform utility level U ∆A is equally attained by all intended movers
and stayers, independent of their ex-post relocation choice, and the transfers sum to zero.
The transfer scheme is characterized by the transfer amount µ (r) that is paid to the retiree
at location r. We will be more precise about the functional form of µ (r) later. The following
conditions are necessary for the transfer scheme µ (·) to achieve the first-best outcome. Firstly,
h
i
for intended movers, who live at locations r ∈ 0, r̂tEQ :


∆p (r) + co,M + µ (·)
u
−φ,


∆A
U
u ca,M + β max 
=
|{z}
| {z }
i
p≥t+1 (r) + co,M + µ (·)
u ∆p (r) − 1+i
total utility
adult utility |
{z
}
old-age utility
(5.10)
h
i
And for intended stayers, who live at locations r ∈ r̂tEQ , r̄tS :

 u
u ca,S + β max 
| {z }
adult utility |
i
1
o,S + µ (·)
1+i ∆p (r) + 1+i p≥t+1 (r) + c
1
u 1+i
∆p (r) + co,S + µ (·)
{z
old-age utility

−φ, 
=
∆A
U
|{z}
total utility
}
(5.11)
The lifetime-utility conditions (5.10) and (5.11) can be understood as follows. Intended
movers who indeed move can use the capital gain ∆p (r), net of the transfer µ (·), for additional
consumption. An intended mover who decides to stay would incur the cost
i
1+i p≥t+1 (r)
that
corresponds to selling the house a period later and is evaluated at post-shock prices. Intended
33
stayers who indeed stay would increase consumption by using the capital gain ∆p (r) net of
i
∆p (r) and the transfer µ (·). An intended stayer
the cost of staying an additional period − 1+i
who decides to move would also obtain a consumption benefit
i
1+i p≥t+1 (r)
that corresponds to
the advantage of selling the house a period earlier. Retirees are free to choose their relocation
behavior and so their decisions must satisfy the concept of strict incentive compatibility (see
Mailath, 1987).9
In the following, we first analyze the case of a small productivity shock, which corresponds
to the case described in Proposition 2 in which the relocation decisions of agents are not affected.10
Proposition 5. Let p (r) be a steady state house price equilibrium and ∆A be a permanent
productivity shock at t + 1 such that at the new steady state equilibrium prices p≥t+1 no retiree
at t + 1 revises his or her relocation decision. In this case, horizontal equity, efficiency, and
strict incentive compatibility are achieved by the transfer scheme µ : R+ × {M, S} → R that
satisfies:
1) i) Movers’ transfer levels: µ (r; M) = −∆p (r) + λ M ;
1
∆p (r) + λ S ;
ii) Stayers’ transfer levels: µ (r; S) = − 1+i
2) Utility-equalization condition: β1 u ca,M +u co,M + λ M −φ = β1 u ca,S +u co,S + λ S ;
´
´
3) Balanced-budget condition: r h (r) µ (r; M) dr + r 12 h (r) µ (r; S) dr = 0.
In the case of a small shock, all retirees who decide to move or to stay under the transfer
scheme µ (·) are exactly those who had originally intended to do so.11 This implies that the
relocation decision reveals the retirees’ types – whether each was an intended mover or stayer
– and allows the planner to differentiate their transfers. This proposition should thus be interpreted as follows: when the productivity shock is sufficiently small, there exists a transfer
scheme (linear in capital gains and relocation dependent) that is equitable, efficient, and strictly
incentive compatible.
Unfortunately, these properties are lost in the case of large permanent productivity shocks.
9A
subset of allocations of consumption and housing is strictly incentive compatible if it is consistent with all
agents’ utility-maximizing decisions.
10 The proof of the result is omitted.
11 Observe that a shock is small when even the retiree who experiences the largest capital gain or loss would
not revise the initial relocation decision. Therefore, when these gains or losses are redistributed between retirees
according to µ (·), it remains true that all intended movers find it optimal to move and all intended stayers find it
optimal to stay.
34
When both intended movers and intended stayers may decide to move (or similarly to stay), a
transfer scheme that is linear in house prices and relocation dependent cannot eliminate wealth
heterogeneity within each group. The tax scheme µ (r; M) = −∆p (r) + λ M , independently of
λ M , eliminates differences across intended movers who indeed decide to move, but not across
intended stayers who decide to move. A similar argument holds for the agents who decide to
stay.
The problem is however even deeper. Assume that the planner observes the agents’ types
(these could be inferred from the agents’ saving and location decisions). Assume as well that
it is morally acceptable for the transfer scheme to discriminate between agents based on their
intended relocation decisions. Then, horizontal equity is achieved when the transfer scheme
µ̄ : R+ × {ExM, ExS} × {M, S} → R satisfies:
1) i) [for intended movers who move] µ̄ (r; intM, M) = −∆p (r) + λ intM,M ;
1
∆p (r) + λ intM,S ;
ii) [for intended movers who stay] µ̄ (r; intM, S) = − 1+i
iii) [transfer scheme for intended stayers who move] µ̄ (r; intS, M) = −∆p (r) + λ intS,M ;
1
∆p (r) + λ intS,S ;
iv) [transfer scheme for intended stayers who stay] µ̄ (r; intS, S) = − 1+i
2) [utility equalization across intended types] i) β1 u ca,M +u co,M + λ intM,M −φ = β1 u ca,S +
u co,S + λ intS,S ;
[utility equalization within intended movers] ii) u co,M + λ intM,M −φ = u co,M + λ intM,S ;
[utility equalization across intended stayers] iii) u co,S + λ intS,M − φ = u co,S + λ intS,S .
For the transfer scheme to achieve an efficient outcome, intended movers and stayers must
be optimally divided according to the relocation decision that most efficiently allows them to
achieve the maximin utility level U ∆A . However, a transfer scheme that achieves horizontal
equity cannot simultaneously generate incentives that induce efficient relocation decisions. Under the transfer scheme µ̄ described above, each intended mover is indifferent between moving
and staying, but efficiency would require the agents near the center to move away rather than
occupying particularly valuable space.
The corollary is that all transfer schemes µ̄ that generate allocations that are efficient and
strictly incentive compatible cannot achieve horizontal equity. This result is stated formally in
the following proposition.
Proposition 6. Let p (r) be a steady state house price equilibrium, let ∆A be a permanent
35
productivity shock at t + 1, and let p≥t+1 be the new steady state equilibrium prices. Then, there
are no transfer schemes µ : R+ × {intM, intS} × {M, S} → R that achieve horizontal equity,
efficiency, and strict incentive compatibility.
The planner therefore faces a trade-off between efficiency and horizontal equity in the outcomes following a productivity shock, as it is not possible to achieve both aims with a policy
that satisfies the minimal requirement of incentive compatibility.
6
Conclusions
This paper studies the effect of the presence of retirees in a city on the efficiency of the equilibrium housing allocation. Though the effects of a misallocation of city housing are potentially
large, due to the concentration of economic activity in cities and the value of residing in proximity to one’s place of work, this problem has received little attention in the literature. We
develop a model to represent the decisions of retirees to remain in the city or to move away
to a retirement community. The model we propose is a minimal and intuitive framework for
understanding private incentives and the social implications concerned with locational choices
across an individual’s lifetime.
There are five main insights of this paper. The first is that residents of more costly land
nearer the city center are more likely to move away when they retire than are suburban residents.
This follows from the trade-off between land prices and the utility cost of moving away: the
more valuable the house, the higher is the economic advantage of moving out upon retirement.
The city is therefore stratified into an inner area in which all agents move out upon retirement
and an outer area in which retirees stay in their homes. Particular parameters can lead to cities
in which all residents stay for retirement or all move away, but if both types coexist in a city
then the movers necessarily live nearer the center than the stayers.
The second main insight is that the market allocates land inefficiently when individuals
decide endogenously whether to stay in the city upon retirement. This result is unambiguous in
a city that includes some measure of both movers and stayers and arises even though the scarcity
of land is priced by the market. The inefficiency follows directly from the fact that individuals
are not compensated on the market for freeing up space when they retire, which would benefit
individuals in the subsequent generation.
36
The third main insight of the paper is that the social optimum can be achieved through a
combination of taxes and subsidies. To correct the individual incentives it is necessary to apply
both a property tax and a transfer subsidy. Since the property tax is paid in each period while the
transfer subsidy is obtained only once, these policies constitute a net transfer from retirees who
remain in the city to the local workers. This has the effect of providing an additional incentive
for individuals to move away upon retirement, thus pricing the externality imposed on others
by staying in the city.
The fourth main insight is that a change in productivity and thereby wages disproportionately affects those residents who live nearer to locations where employment is provided. This
occurs for two reasons. Firstly, residents who live nearer to jobs benefit more from a wage increase because they generally have shorter commutes and are therefore able to spend more time
working. Secondly, as more of the residents in central locations move away when they retire,
a higher proportion of the residents in these locations are of working age, which also increases
the premium from higher wages.
The fifth main insight is that policy may correct for the inefficiency or inequality that results
from a productivity shock, but may not do both. If we impose the minimal condition that
a policy be incentive compatible, it is not possible to achieve an efficient outcome that also
leaves all agents in the affected generation with an equal level of utility. However, it is possible
to achieve either one of these goals individually, or to make marginal improvements on both
fronts.
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A
Time-consistency of location decisions
This appendix demonstrates the time-consistency of agents’ location decisions over their lifetimes. That is, we demonstrate that, in the steady state and in the absence of any unanticipated
parameter changes, movers will indeed decide to move away when they retire and stayers will
indeed decide to stay in their homes.
We begin by defining the house price at retirement that would leave an agent indifferent
between moving and staying. In the period of old age, the individual maximises utility (2.2)
38
subject to the budget constraints when old (2.4) and dead (2.5). Consider a price offer p̃t+1 (r)
for the house owned by the agent at r and the market price pt+2 (r) that the individual would
receive upon death. The amount saved by the agent when adult is denoted st . Old-age consumption levels when moving (m) or staying (s) are thus:
m
ct+1
= p̃t+1 (r) + (1 + i) st
s
ct+1
=
(A.1)
1
pt+2 (r) + (1 + i) st
1+i
(A.2)
The agent is therefore indifferent between moving and staying when the price offer when
old satisfies:
1
pt+2 (r) + (1 + i) st
u ( p̃t+1 (r) + (1 + i) st ) − φ = u
1+i
A.1
(A.3)
Time-consistency of movers’ location decisions
Consider the agents who intend to move away at retirement and consume ca,M in the adult
period. The house price bid at the start of the period of retirement that leaves such an agent at r
indifferent between staying and moving away is:
M
p̃t+1
(r) = u−1
1
o,M
u
pt+2 (r) + c − pt+1 (r) + φ − co,M + pt+1 (r)
1+i
(A.4)
In the area populated by movers who were adults in period t, their house price bids must
have been higher than those of stayers, so ptM (r) > ptS (r). Therefore, from (3.6) and (3.7),
the following must be true of the relationship between the equilibrium house prices in the two
following periods:
pt+1 (r) −
h
i
1
pt+2 (r) > (1 + i) CM −CS
1+i
Substituting (A.5) into (A.4):
39
(A.5)
M
(r) < u−1
p̃t+1
i
h
S
a,M
+ φ − co,M + pt+1 (r)
u (1 + i) C − c
(A.6)
Now, u (1 + i) CS − ca,M is the old-period utility from consumption of an agent who had
the resources of a stayer in the first period (a budget sufficient to obtain Ū if allocated optimally)
but spent ca,M . As the optimal intertemporal consumption levels with the budget CS would have
been ca,S and co,S , this necessarily violates (3.2) and yields lower lifetime utility than Ū:
i
h
u ca,M + β u (1 + i) CS − ca,M < u ca,S + β u co,S = Ū
(A.7)
By definition u ca,M + β u co,M − β φ = Ū, so the inequality in (A.7) can also be expressed as:
h
i
S
a,M
u (1 + i) C − c
+ φ < u co,M
(A.8)
Substituting (A.8) into the relationship for the price that leaves the movers indifferent (A.6)
yields the following:
M
p̃t+1
(r) < pt+1 (r)
(A.9)
The relationship in (A.9) is the result that proves the time-consistency of the retirementmigration decisions made by the movers. As an intended mover would only stay in the house at
M (r), which is less than the equilibrium price
retirement were the price offer to be less than p̃t+1
pt+1 (r), the intended movers indeed move away when they retire.
A.2
Time-consistency of stayers’ location decisions
Consider the agents who intend to stay in the city for retirement and consume ca,S in the adult
period. The house price bid at the start of the period of retirement that leaves such an agent at r
indifferent between staying and moving away is:
S
p̃t+1
(r) = u−1
1
o,S
u c
+ φ − co,S +
pt+2 (r)
1+i
(A.10)
In the area populated by stayers who were adults in period t, their house price bids must
40
have been higher than those of movers, so ptM (r) < ptS (r). Therefore, from (3.6) and (3.7),
the following must be true of the relationship between the equilibrium house prices in the two
following periods:
h
i
1
M
S
pt+2 (r) < (1 + i) C −C
pt+1 (r) −
1+i
(A.11)
Substituting (A.11) into (A.10):
i
h
S
p̃t+1
(r) > u−1 u co,S + φ + pt+1 (r) − (1 + i) CM − ca,S
(A.12)
Now, (1 + i) CM − ca,S is the old-period consumption of an agent who had the resources of
a mover in the first period (a budget sufficient to obtain Ū if allocated optimally) but spent ca,S .
As the optimal intertemporal consumption levels with the budget CM would have been ca,M and
co,M , this necessarily violates (3.1) and yields lower lifetime utility than Ū:
h
i
a,S
M
a,S
u c
+ β u (1 + i) C − c
− β φ < u ca,M + β u co,M − β φ = Ū
(A.13)
By definition u ca,S + β u co,S = Ū, so the inequality in (A.13) can also be expressed as:
h
i
M
a,S
u (1 + i) C − c
< u co,S + φ
(A.14)
Substituting (A.14) into the relationship for the price that leaves the stayers indifferent
(A.12) yields the following:
S
(r) > pt+1 (r)
p̃t+1
(A.15)
The relationship in (A.15) is the result that proves the time-consistency of the retirementmigration decisions made by the stayers. As an intended stayer would only move out at retireS (r), which is greater than the equilibrium price
ment were the price offer to be greater than p̃t+1
pt+1 (r), the intended stayers indeed stay in the city when they retire.
41
B
Equivalence of home ownership and renting in the housing
market
This appendix demonstrates the equivalence of housing markets in which residences are purchased or rented. We denote the rent in period t on a house at distance r to be Rt (r). The
lifetime budget constraints of the movers and stayers, respectively, can be expressed as:
(1 − τ (r)) wt = Rt (r) +CM
(1 − τ (r)) wt = Rt (r) +
(B.1)
1
Rt+1 (r) +CS
1+i
(B.2)
The rent that an adult mover would be willing to pay is denoted RtM (r) and can be derived
directly from (B.1):
RtM (r) = (1 − τ (r)) wt −CM
(B.3)
The rent that the adult mover is willing to pay, RtM (r), is equal to the house price bid less
1
pt+1 (r), as can be seen by
the discounted sale price of the house one period later, ptM (r) − 1+i
comparing (3.6) and (B.3). The purchase and rental markets are therefore equivalent for movers,
as an investor who anticipated renting a house to a mover for RtM and required the market rate
of interest r on the capital investment would bid exactly ptM (r) for the house.
The same would apply for stayers if rents were constant for the full term of residence of a
single tenant. This can be seen by substituting the per-period rent the stayers would be willing
S
to pay, denoted Rt,t+1
(r), into (B.2):
1
RS (r) = (1 − τ (r)) wt −CS
1+
1 + i t,t+1
(B.4)
S
The rent a stayer is willing to pay, Rt,t+1
(r), is equivalent to the price a stayer is willing to
pay for the house less its discounted sale price two periods later, ptS (r) −
1
pt+2 (r),
(1+i)2
as can
be seen by comparing (3.7) and (B.4).
Were rents set by the landlord in each period, then the second period rent would be set such
that the individual is indifferent between moving and staying:
42
o,S
Rt+1
(r) = u−1
o,S
+ φ − co,S
u c
(B.5)
Equation (B.5) states that the old-age rent must equal the dollar value of the relocation cost
φ . Knowing this to be the case, a stayer is willing to pay the rent when adult that allows lifetime
consumption CS according to (B.2):
Rta,S (r) = (1 − τ (r)) wt − ca,S −
1 −1 o,S u
u c
+φ
1+i
(B.6)
o,S
Provided that it is feasible for the rents Rta,S (r) and Rt+1
(r) as defined in (B.5) and (B.6) to
be paid, the purchase and rental markets are equivalent as can be seen by comparing (3.7) and
(B.2). The rent payments would have the same discounted value as the cost of purchasing the
house and holding it for two periods. There are two potential threats to the feasibility of the
rents (B.5) and (B.6). The first is that a sufficiently high φ would imply such a high old-age
o,S
rent Rt+1
(r) that the corresponding Rta,S (r) is negative. The existence of such a case is trivial:
consider a city comprised entirely of stayers in which the φ could be arbitrarily high. However,
rational landlords would accept the negative rents, knowing that the tenants will be willing to
pay the amount when old that earns them the market return on their investments.
o,S
The second potential threat to the feasibility of Rta,S (r) and Rt+1
(r) would arise if the rent
were lower for old agents than for contemporaneous adults. If the landlord were able to evict
the tenant when old, no contracts existed to allow the landlord to commit to offering the tenant
to stay for a pre-determined price, and old agents paid lower rent than adults, then the landlord
would evict an existing old tenant and offer the house to a new adult. Anticipating this, the adult
would not be willing to rent the house for the Rta,S (r) specified in (B.6) as this would not yield
utility Ū, so the landlord would not be able to buy the house for ptS (r) and earn the market rate
of interest r.
To determine whether it is possible for the rent to be lower for old agents than for adults,
consider first that the rent paid when old is equal to the cost for a homeowner of staying in the
house when old if offered the price that leaves him indifferent about whether to stay. This can
be seen by combining (A.10) and (B.5):
43
o,S
S
Rt+1
(r) = p̃t+1
(r) −
1
pt+2 (r)
1+i
(B.7)
The amount spent on housing over the lifetime must be equal in the two types of market,
which implies a similar equivalence between the rent paid when adult and the cost for a homeowner who will face the price offer when old that leaves him indifferent about whether to stay.
This is derived substituting (B.7) into the combination of (3.7) and (B.2):
Rta,S (r) = ptS (r) −
1 S
p̃ (r)
1 + i t+1
(B.8)
Now, as the price offer that leaves a homeowner indifferent about whether to stay is higher
than the equilibrium price offer (A.15) and the equilibrium prices in the area populated by
stayers are the bids of the adult stayers, the relationships (B.7) and (B.8) can be restated as the
following inequalities:
Rto,S (r) > ptS (r) −
1 S
p (r)
1 + i t+1
(B.9)
Rta,S (r) < ptS (r) −
1 S
p (r)
1 + i t+1
(B.10)
The relationship between the rents paid by adult and old agents follows directly from (B.9)
and (B.10):
Rto,S (r) > Rta,S (r)
(B.11)
The inequality in (B.11) demonstrates that an old stayer will always be charged a higher rent
than an adult stayer, so the landlord has no incentive to evict an existing resident. Therefore,
the rents specified in (B.5) and (B.6) are feasible and even in a context where the landlord can
set the rent at any level in each period, the rental market is equivalent to a market with home
ownership.
44
C
Fundamental characteristics of the optimal allocation of
housing in the steady state
C.1
Proof of the city density requirement
Proof. Let r̄ be the most distant location at which some agent lives, so r̄ ≡ max [r ∈ R+ |ga (r) > 0].
Then, a city is dense if agents occupy all the space available inside its boundaries, i.e. if for each
´ r00 ´ r00
r0 , r00 ∈ [0, r̄] with r0 < r00 r0 ga(M) (r) + 2ga(S) (r) dr = r0 h (r) dr. By contradiction, assume
´ r00 this were not the case: there exist r0 , r00 ∈ [0, r̄] with r0 < r00 such that r0 ga(M) (r) + 2ga(S) (r) dr <
´ r00
´ 00
´ 00 a(M)
a(S) (r) dr.
0 and r 00 be e 0 00 ≡ r h (r) dr − r
g
(r)
+
2g
h
(r)
dr.
Let
the
empty
area
between
r
[r ,r ]
r0
r0
r0
Define r̄0 to be the distance that leaves a number e[r0 ,r00 ] of agents further from the center; for
´ r̄ mally it is such that r̄0 ga(M) (r) + 2ga(S) (r) dr = e[r0 ,r00 ] . Let the adult population of each
´ r̄
´ r̄
a(M)
a(S)
type in this area be P[r0 ,r00 ] ≡ r̄0 ga(M) (r) dr and P[r0 ,r00 ] ≡ r̄0 ga(S) (r) dr. When these agents are
uniformly relocated between r0 and r00 , at each r ∈ [r0 , r00 ] an additional number of adult movers
a(M)
i
P 0 00 h
[r ,r ]
a(M)
and stayers, measured by the densities ∆g
(r) ≡ e 0 00 h (r) − ga(M) (r) + 2ga(S) (r) and
[r ,r ]
∆ga(S) (r) ≡
a(S)
P 0 00
[r ,r ]
e[r0 ,r00 ]
h
i
h (r) − ga(M) (r) + 2ga(S) (r) . Then, the change to the net wealth Ω is:
´ r00
(1 − τ (r)) w −CM ∆ga(M) (r) dr+
´ r̄
− r̄0 (1 − τ (r)) w −CM ga(M) (r) dr+
´ r00
+ r0 (1 − τ (r)) w −CS ∆ga(S) (r) dr+
´ r̄
− r̄0 (1 − τ (r)) w −CS ga(S) (r) dr
ˆ r00
ˆ r̄
a(M)
=
((1 − τ (r)) w) ∆g
(r) dr −
((1 − τ (r)) w) ga(M) (r) dr+
0
0
{z r̄
}
|r
>0
ˆ r00
ˆ r̄
a(S)
+
((1 − τ (r)) w) ∆g (r) dr −
((1 − τ (r)) w) ga(S) (r) dr
0
0
|r
{z r̄
}
∆Ω =
r0
>0
which shows the contradiction and proves the result.
45
C.2
Proof of the requirement that movers and stayers be strictly sorted
by distance
Proof. At the optimal distribution of agents, whenever
´∞
0
ga(M) (r) dr > 0 and
´∞
0
ga(S) (r) dr >
0, it must be the case that movers and stayers are sequentially located: movers are located nearer
the center, while stayers are located further out. Formally,
 in the optimal allocation of movers

h (r) for each r ∈ [0, r∗ ]

∗
a(M)
and stayers there exists r ∈ R+ such that g
(r) =
.


0
otherwise
By contradiction, assume that this were not the case. Let
h
i
rM ≡ max r ∈ R+ ga(M) (r) > 0 ∈ R+
be the maximal distance at which some adult mover is located at the efficient distribution
´ rM
´ rM
ga(M) (r). Assume that 0 ga(M) (r) dr < 0 h (r) dr. By Step 1, we know that the city is
dense and, therefore, there are stayers living between 0 and rM . Let r̃ ∈ 0, rM be the distance
for which the number of stayers on [0, r̃] is equal to the number of movers on r̃, rM . Formally,
´ r̃ ´ rM
r̃ is such that 0 h (r) − ga(M) (r) dr = r̃ ga(M) (r) dr. Then, it is always possible that each
adult mover located between r̃ and rM switches his location with a stayer (adult or old) between
0 and r̃. Let ∆ga(M) and ∆ga(S) be the difference in the densities of movers and stayers due to
the switch. Then the variation in the net wealth Ω is:
´ r̃
M ∆ga(M) (r) dr
(1
−
τ
(r))
w
−C
0
´ rM
− r̃ (1 − τ (r)) w −CM ga(M) (r) dr
´ r̃
− 0 (1 − τ (r)) w −CS ga(S) (r) dr
´ rM
+ r̃ (1 − τ (r)) w −CS ∆ga(S) (r) dr
´ r̃
´ rM
= 0 (1 − τ (r)) wga(S) (r) dr − r̃ (1 − τ (r)) w 21 ga(M) (r) dr > 0
∆Ω =
and contradicts the efficiency of agent’s location.
D
Post-shock dynamics
This appendix contains the proofs of Propositions 3 and 4, which describe the behavior of
existing residents in response to positive and negative productivity shocks.
46
D.1
Positive productivity shocks and elderly migration (proof of Proposition 3)
When there is a positive productivity shock in the city, formally ∆A > 0, house prices increase
at each location within the city, so ∆p (r) > 0 for each r ∈ R+ . As the existing homeowners
will be retired by the time the wage increases, the productivity shock affects that generation
solely through the changes in house prices. Therefore, to characterize the behavior of existing
retirees in response to the shock, we analyze how a positive change in prices affects old agents’
migration decisions at each location and then analyze how prices react to the productivity shock.
We begin with the case of intended movers and then proceed to the case of intended stayers.
D.1.1
Intended movers
h
i
Consider an intended mover who lives at r ∈ 0, r̂tEQ and let (∆p)M+ (r) be the positive change
in prices at retirement that satisfies (5.6), leaving the intended mover indifferent between moving and staying. We first determine the restriction on the utility function that leads to the existence of (∆p)M+ (r).
h
i
By the mean value theorem, there must exist a consumption level cM+ ∈ co,M , (∆p)M+ (r) + co,M
such that:
u (∆p)M+ (r) + co,M − φ = u co,M − φ + u0 (·) |cM+ (∆p)M+ (r)
and, by the same argument, a consumption level cM− ∈
(D.1)
i
h
M+
i
i
M
o,M
M
o,M
− 1+i p (r) + c , (∆p) (r) − 1+i p (r) + c
such that:
1
i M
i M
1 0
M+
o,M
o,M
u
(∆p) (r) −
p (r) + c
=u −
p (r) + c
+
u (·) |cM− (∆p)M+ (r)
1+i
1+i
1+i
1+i
(D.2)
Combining (5.2) and (5.6) yields:
i
u co,M − φ − u − 1+i
pM (r) + co,M
M+
(∆p)
(r)
=
1 0
u (·) |cM− − u0 (·) |cM+ > 0
1+i
(D.3)
Since cM+ > cM− , for the intended mover to be indifferent between moving and staying
47
after the price shock (∆p)M+ (r), the marginal utility of consumption must decrease rapidly
enough. In other words, the utility function must be sufficiently concave.
Let (∆A)M+ (r) be the level of positive productivity shock that satisfies the indifference
condition (5.6) for the intended mover at r ∈ 0, r̂EQ . We can use implicit differentiation to
study how (∆A)M+ (r) varies in the neighborhood of r. This will allow us to determine which
intended movers will stay after a given shock ∆A. Differentiating (5.6) with respect to r yields:
i h
M+
0 (r) (∆A)M+ (r) + 1+i (1 − τ (r)) ∂ (∆A) (r) u0 (∆p)M+ (r) + co,M
τ
− 1+i
i
i
∂r
i h
M+
1
i
(∆p)M+ (r) − 1+i
pM (r) + co,M
= − 1i τ 0 (r) (∆A)M+ (r) + 1i (1 − τ (r)) ∂ (∆A)∂ r (r) + τ 0 (r) A u0 1+i
Since u0
M+
M+
1
i
0
o,M
M
o,M
< u 1+i (∆p) (r) − 1+i p (r) + c
(∆p) (r) + c
, the following must
hold:
∂ (∆A)M+ (r)
τ 0 (r) M+
>
A + (∆A) (r) > 0
∂r
(1 − τ (r))
The productivity shock must therefore be larger for an agent at a more distant location to
be indifferent between moving and staying. It follows that for the level of productivity shock
h
i
(∆A)M+ (r) that satisfies the indifference condition (5.6) at r ∈ 0, r̂tEQ , all intended movers
residing at locations r0 ∈ [0, r) decide to stay.
D.1.2
Intended stayers
h
i
Consider an intended stayer who lives at r ∈ r̂tEQ , r̄tS and let (∆p)S+ (r) be the positive change
in prices at retirement that satisfies (5.6), leaving the intended stayer indifferent between moving
and staying.
By the mean value theorem, there must exist a consumption level cS+ ∈
h
S+
i
i
S
o,S
1+i p (r) + c , (∆p) (r) + 1+i p
such that:
u (∆p)S+ (r) +
i S
i S
o,S
o,S
p (r) + c
−φ = u
p (r) + c
− φ + u0 (·) |cS+ (∆p)S+ (r)
1+i
1+i
(D.4)
h
i
1
and a consumption level cS− ∈ co,S , 1+i
(∆p)S+ (r) + co,S such that:
48
1
(∆p)S+ (r) + co,S
u
1+i
1 0
= u co,S +
u (·) |cS− (∆p)S+ (r)
1+i
(D.5)
Combining (5.3) and (5.7) yields:
u
i
S
o,S
1+i p (r) + c
S+
(∆p)
− φ − u co,S
=
(r)
1 0
u (·) |cS− − u0 (·) |cS+ < 0
1+i
(D.6)
Since cS+ > cS− , for the intended stayer to be indifferent between moving and staying after
the price shock (∆p)S+ (r), the marginal utility of consumption must not decrease too rapidly.
In other words, the utility function must not be too concave.
Let (∆A)S+ (r) be the level of positive productivity shock that satisfies the indifference coni
h
EQ S
dition (5.6) for the intended stayer at r ∈ r̂t , r̄t . As with the case of the intended mover,
we can use implicit differentiation to study how (∆A)S+ (r) varies in the neighborhood of r and
thus assess which intended stayers decide to move for a given positive productivity shock ∆A.
However, for the intended stayer it is necessary to distinguish between the cases in which the
new steady state price at r is determined by the bid of a young mover or the bid of a young
stayer, as expressed by (5.5).
h
i
EQ EQ
If r ∈ r̂t , r̂≥t+1 , then ∆p (r) =
1+i
i (1 − τ (r)) ∆A.
Differentiating (5.7) with respect to r
on this interval yields:
h
i S+
0 (r) (∆A)S+ (r) + 1+i (1 − τ (r)) ∂ (∆A) (r) − (1+i) τ 0 (r) A u0 (∆p)S+ (r) + i pS (r) + co,S
− 1+i
τ
i
i
1+i
∂r
(2+i)
h
i S+
(r)
S+
1
0
o,S
= −τ 0 (r) (∆A)S+ (r) + (1 − τ (r)) ∂ (∆A)
u
(∆p)
(r)
+
c
1+i
∂r
Since
u0
S+
S+
i
1
S
o,S
0
o,S
(∆p) (r) + 1+i p (r) + c
< u 1+i (∆p) (r) + c
, the following must
hold:
∂ (∆A)S+ (r)
τ 0 (r)
>
∂r
(1 − τ (r))
i (1 + i)
S+
A + (∆A) (r) > 0
(2 + i)
The productivity shock that leaves an intended stayer indifferent between moving and stayh
i
EQ
ing is therefore increasing in the distance from the city center over the interval r ∈ r̂tEQ , r̂≥t+1
.
h
i
2
EQ
If r ∈ r̂≥t+1
, r̄tS , then ∆p (r) = (1+i)
i(2+i) (1 − τ (r)) ∆A. Differentiating (5.7) with respect to r
on this interval yields:
49
h
i 2
2
S+
0 (r) (∆A)S+ (r) + (1+i) (1 − τ (r)) ∂ (∆A) (r) − (1+i) τ 0 (r) A u0 (∆p)S+ (r) + i pS (r) + co,S
− (1+i)
τ
1+i
∂r
i(2+i)
i(2+i)
(2+i)
h
i S+
(1+i) 0
(1+i)
(r)
S+
1
o,S
0
= − i(2+i)
τ (r) (∆A)S+ (r) + i(2+i)
(1 − τ (r)) ∂ (∆A)
(∆p)
(r)
+
c
u
1+i
∂r
Since
u0
S+
S+
i
1
0
S
o,S
o,S
(∆p) (r) + 1+i p (r) + c
< u 1+i (∆p) (r) + c
, the following must
hold:
∂ (∆A)S+ (r)
τ 0 (r) S+
>
A + (∆A) (r) > 0
∂r
(1 − τ (r))
(D.7)
As with the intended stayers nearer the city center, the productivity shock that leaves an
intended stayer indifferent between moving and staying is increasing over the interval r ∈
h
i
EQ
r̂≥t+1
, r̄tS . Therefore, the required productivity shock is increasing over the entire interval
h
i
r ∈ r̂tEQ , r̄tS . It follows that if there is a level of productivity shock (∆A)S+ (r) that satisfies
h
i
the indifference condition (5.6) for some r ∈ r̂tEQ , r̄tS , then all intended stayers residing at
h
locations r0 ∈ r̂tEQ , r decide to move.
D.2
Negative productivity shocks and elderly migration (proof of Proposition 4)
When there is a negative productivity shock in the city, formally ∆A < 0, house prices weakly
decrease at each location within the city, so ∆p (r) ≤ 0 for each r ∈ R+ . Again we analyze
separately how the price variation influences old agents’ migration decisions and how prices
change in response to the productivity shock. We begin with the case of intended movers and
then proceed to the case of intended stayers.
D.2.1
Intended movers
h
i
EQ
Consider an intended mover who lives at r ∈ 0, r̂t
and let (∆p)M− (r) be the negative
change in prices at retirement that satisfies (5.6), leaving the intended mover indifferent between moving and staying. To prove the first part of the proposition, it must be shown that
1
1
ν and co,M − pM (r) + 1+i
ν ≥ 0 are suffiu co,M − pM (r) + ν − φ < u co,M − pM (r) + 1+i
h
i
cient conditions for (∆p)M− (r) to exist for some r ∈ 0, r̂tEQ .
50
Consider a negative productivity shock that is sufficiently large that the house price decreases to the reservation value of land ν. At that price, the wealth of the retiree is co,M +
(∆p)M− (r) = co,M − pM (r) + ν. Given that the cost of staying is the cost of postponing the sale
of the house by one period, the amount available for consumption when old if the agent decides
1
ν. A necessary condition for staying is that the agent can afford to
to stay is co,M − pM (r) + 1+i
do so, in the sense of having a non-negative amount of wealth to spend on consumption when
1
ν ≥ 0.
old, formally co,M − pM (r) + 1+i
If the house price decreases to the reservation value of land ν, then the retiree will prefer to
1
stay if doing so provides a higher level of utility than moving, formally u co,M − pM (r) + 1+i
ν >
u co,M − pM (r) + ν − φ . By continuity of the utility function, there exists a shock such that
for a price change (∆p)M− (r), (5.6) is satisfied.
1
∂ u( 1+i
∂ u( p≥t+1 (r)−pM (r)+co,M )
p≥t+1 (r)−pM (r)+co,M )
<
, if there is a negative price change
Since
∂r
∂r
h
i
(∆p)M− (r) that satisfies the indifference condition (5.6) for some r ∈ 0, r̂tEQ , then all intended
i
movers residing at locations r0 ∈ r, r̂tEQ decide to stay.
D.2.2
Intended stayers
h
i
EQ S
Consider r ∈ r̂t , r̄t and let (∆p)S− (r) be the negative change in prices at retirement that
satisfies (5.6), leaving the intended stayer indifferent between moving and staying.
Two situations could induce the intended stayer to move. The first arises if the negative
shock is sufficiently large that the agent has negative wealth to spend on consumption when
1
old, formally co,S − 1+i
∆p (r) < 0.
The second situation depends on the concavity of the utility function. As for the case of a
positive shock, we can apply the mean value theorem to derive (D.4) and (D.5), which imply
that:
u
i
S
o,S
1+i p (r) + c
S−
(∆p)
− φ − u co,S
(r)
=
1 0
u (·) |cS− − u0 (·) |cS+ > 0
1+i
(D.8)
Note that the sign of the inequality in (D.8) is different from that in (D.6) as (∆p)S− (r) <
0. Thus, for a stayer to move after a negative shock, the utility function must be sufficiently
concave.
Furthermore, the magnitude of the negative productivity shock that leaves the intended stay51
ers indifferent between moving and staying is increasing in the distance from the city center,
∂ ∆A(r)
∂r
> 0 for any ∆A ∈ (−A, 0) as shown in (D.7). Therefore, if (5.6) holds for some
location r, then all intended stayers residing at locations r0 ∈ r, r̄tS decide to move.
since
52