Bobos in Paradise
Gilles Saint-Paul
PSE, TSE, NYUAD
December 15, 2013
Gilles Saint-Paul PSE, TSE, NYUAD ()
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December 15, 2013
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The model
There are three categories of workers:
H1 Skilled workers specialized in the new economy "Bobos".
Endowed with one unit of specialized skilled (type 1) labor.
H2 Skilled workers specialized in the old economy, "Cadres".
Endowed with one unit of (type 2) skilled labor.
L Unskilled workers, "workers", who are specialized in services.
Each endowed with one unit of unskilled labor.
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Goods
There are …ve goods:
A generic consumption good which is imported (price = 1)
Two exported goods:
A new economy good, 1 (price = 1)
y1 = a1 h1 ,
An old economy good, 2, (price = 1)
y2 = min(a2 h2 , t ),
Services, S.
yS = aS lS ,
Workers must be physically located in the residence of the customer.
Housing: at any time, one unit of land can be converted into one
unit of housing at no cost.
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Geography
There are two areas
The center (C), or "city", …xed size X in terms of land..
max(H1 , H2 ) < X < 2 min(H1 , H2 ).
Land is usable for housing or as an input for good 2.
A …xed amount of additional land is allocated to public goods:
urban amenities
transportation infrastructures.
The periphery (P), or "suburb", with an in…nite supply of land.
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Timing
Overlapping generations of agents who live for two periods.
Each cohort’s size is equal to one.
People can borrow and lend at r .
In period 1 of their life, people
decide where to locate (city vs. suburb),
purchase housing
In period 2, people
vote on the level of amenities in the city,
work, sell their home, and consume.
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Utility
All agents have the same utility
ln c + β ln cS
δφ + α,
c =generic consumption good,
cS = services
δ =1 if commutes
α = amenity level, equal to αC in the city and zero in the suburb.
Each individual needs exactly one unit of housing
Indirect utility
V (R, pSj , δ, α) = (1 + β) ln R
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β ln pSj
δφ + α
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Who would commute?
Workers may commute to o¤er their service to consumers not located
in their area.
Cadres may commute to a production site in the other area, as land
itself is not moveable.
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The amenity/commuting cost trade-o¤
In the city, public decisions over amenities involve reallocating
available space for public goods
Trade-o¤ between amenities and commuting costs.
φ = φ(αC ), φ0 > 0.
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The "welfare state"
Instead of working and consuming, people can live in autarky
They get a reservation utility equal to ū.
Due to housing costs, only possible in suburb
I will limit the analysis to a regime where the constraint u
binding for workers only.
ū is
Consequently, only a fraction of the total mass of workers L will be
employed, and the others will be "unemployed".
An increase in ū can be interpreted as an increase in welfare bene…ts.
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Solving the problem
I will solve the problem in three steps:
The consumption and production decisions of all three social classes
when old, as a function of their residential location
The locational decisions of the young of the three classes, as a
function of expected amenity levels and house prices
The voting decision of urban households on amenities, as a function
of the sociological composition of the city
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Consumption and production: Income
Class
Workers in city
Workers in suburb
Bobos in city
Bobos in suburb
Cadres in city
Cadres in suburb
Table 1: Income
Work in city
qt +1 (1 + r )qt + pSCt aS
pSCt aS
qt +1 (1 + r )qt + a1
a1
qt +1 (1 + r )qt + a2 (qt +1
a2 (qt +1 q1t++r2 )
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q t +2
1 +r )
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Class
Workers in city
Workers in suburb
Bobos in city
Bobos in suburb
Cadres in city
Cadres in suburb
Work in suburb
qt +1 (1 + r )qt + pSPt aS
pSPt aS
qt +1 (1 + r )qt + a1
a1
qt +1 (1 + r )qt + a2
a2
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Consumption and production: Utility (Steady State)
Class
Work in city
W.in city
(1+β) ln(pSCt aS rq ) β ln pSCt + αC
W. in suburb (1+β) ln(pSCt aS ) β ln pSPt φ
B. in city
(1 + β) ln(a1 rq ) β ln pSCt + αC
B. in suburb (1 + β) ln a1 β ln pSPt φ
C.in city
(1 + β) ln(a2 rq rq/(1 + r )) β ln pSCt + αC
C. in suburb (1 + β) ln(a2 rq/(1 + r )) β ln pSPt φ
Table 2 – Utilities
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Class
W.in city
W. in suburb
B. in city
B. in suburb
C.in city
C. in suburb
Work in suburb
(1 + β) ln(qt +1 (1 + r )qt + pSPt aS ) β ln pSCt + αC
(1+β) ln(pSPt aS ) β ln pSPt
(1 + β) ln(qt +1 (1 + r )qt + a1 ) β ln pSCt + αC φ
(1 + β) ln a1 β ln pSPt
(1 + β) ln(qt +1 (1 + r )qt + a2 ) β ln pSCt + αC φ
(1 + β) ln a2 β ln pSPt
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φ
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Ruling out some con…gurations: A. Old economy takes
place outside
It is never optimal for cadres to work in the city:
(1 + β )
ln(qt +1 (1 + r )qt + a2 )
ln(qt +1 (1 + r )qt + a2 (qt +1
q t +2
1 +r ))
> φ,
In steady state, this is equivalent to
a2
a2
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rq
r
r 21 +
+r q
φ
> e 1+β .
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Ruling out some con…gurations: B. Working class is partly
unemployed
Workers are at their reservation utility and do not live in the city
This pins down the price of services in the suburbs:
ln pSPt = ū
(1 + β) ln aS ,
as well as in the city:
ln pSCt = ln pSPt +
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φ
.
1+β
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Ruling out some con…gurations: C. Working class cannot
a¤ord the city.C
We also need that
(1 + β) ln(qt +1
(1 + r )qt + pSCt aS )
β ln pSCt + αC < ū.
In steady state, this is equivalent to
ln (pSC aS
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rq ) < ū
β ln aS
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αC
βφ
+
.
1 + β (1 + β )2
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Two main regimes
Regime I – All the cadres live in the city, and bobos are indi¤erent
city vs suburb
. The cadres are a political majority in the city and they …x the
amenity level.
The bobos are pivotal in the determination of house prices
Regime II – All bobos live in the city and cadres are indi¤erent.
Bobos political majority, cadres pivotal for house prices
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Regime I
Bobos indi¤erence condition
(1 + β) ln(qt +1 (1 + r )qt + a1 )
= (1 + β) ln a1 β ln pSPt .
β ln pSCt + αC
House prices determination in steady state:
a1
q=
(1 exp λ(αC , φ)) = qA ,
r
where
λ ( αC , φ ) =
αC (1 + β )
βφ
(1 + β )2
.
λ() =net cost of living in the city if one does not commute
For this to be possible, we need
αC >
Gilles Saint-Paul PSE, TSE, NYUAD ()
βφ
.
1+β
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Comments
House prices are larger, the greater the amenity level and the smaller
the commuting cost.
The bobos in the city su¤er indirectly from commuting costs due to
more expensive services
House prices are higher, the greater the bobo’s productivity.
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Political decisions in regime I: House prices taken as given
Utility of cadres
(1 + β) ln(qt +1
(1 + r )qt + a2 )
β ln pSCt + αC
φ
Decision is made
after they have picked their residence and paid the house price qt ,
before selling back their real estate to the new generation.
As long as amenities are not a state variable,
qt +1 will depend on the amenities set by the new generation.
Therefore amenity level independent of qt +1 and qt .
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Political decisions in regime I: E¤ect on service prices
internalized
. On the other hand, pSC goes up with transportation costs.,
Therefore, the cadres set amenities by maximizing
αC
φ ( αC )
β ln pSCt ,
Solution αA , φA
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When is regime I an equilibrium?
Utility of the cadres
(1 + β) ln(a2
rq )
β ln pSCt + αA
φA .
This must be larger than their utility from residing in the suburb:
a2
1
>
a1
1
exp(λA )
= zA > 1,
exp(µA )
where
λA
µA
= λ ( αA , φA ),
(1 + 2β)φA αA (1 + β)
=
= µ ( αA , φA ) > λA .
(1 + β )2
µ() =net cost of living in the city if one commutes
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Employment in regime I: City
Total income in the city is
YC = a2 H2 + a1 (X
H2 )
rqX .
Total service demand in the city
CSC = β
a 2 H2 + a 1 ( X H2 )
(1 + β)pSC
rqX
.
Number of service workers who work in the city is
LC
a2 H2 + a1 (X H2 ) rqX
(1 + β)pSC aS
φA
a2 H2 + a1 (X H2 ) rqA X
= β
exp(
)
(1 + β)pSP aS
1+β
= β
= LAC
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Employment in regime I: Periphery
Total income in the periphery
YP = a1 (H1 + H2
Again LP =
LP
=
β
pSP aS
β
YP
1 + β a S p SP
X ) + pSC aS LC + pSP aS LP ,
, i.e.
β
( a 2 H2 + a 1 ( X
1+β
H2 )
rqA X ) + a1 (H1 + H2
X)
= LAP .
∂(LP + LC )/∂a1 > 0,
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Main properties of regime I
An equilibrium in regime I prevails if aa21 is larger than a critical
threshold, which is itself greater than 1.
Other things equal, Bobos are willing to pay more to live in the city
than cadres, because the former do not have to commute.
In regime I, the price of housing goes up with productivity in the new
economy sector.
In regime I, employment of unskilled workers goes up with
productivity in the new economy.
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Regime II
Indi¤erence condition for the cadres
(1 + β) ln(a2
rq )
β ln pSCt + αC
φ = (1 + β) ln a2
β ln pSPt
House prices are now equal to
q=
a2
(1
r
exp µ(αC , φ)).
For this regime to be possible, we need that
αC >
Gilles Saint-Paul PSE, TSE, NYUAD ()
1 + 2β
φ.
1+β
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Regime II
The utility of the bobos is
(1 + β) ln(qt +1
They maximize
(1 + r )qt + a1 )
β ln pSCt + αC .
β ln pSCt + αC , or equivalently
αC
βφ(αC )/(1 + β).
Solution: αB , φB
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When is regime II an equilibrium?
Utility of the bobos
(1 + β) ln(a1
rq )
β ln pSCt + αB .
For regime II to prevail, this must be larger than their utility from
residing in the suburb:
1
a2
<
a1
1
exp(λB )
= zB > 1,
exp(µB )
where
λB
µB
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= λ ( αB , φB ),
= µ ( αB , φB ) > λB
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Multiple equilibria
LEMMA – λA > λB and µB > µA
Proof – The optimization problem of the pivotal voter in each regime
implies that µ(α, φ) is minimum in regime I, while λ(α, φ) is minimum in
regime II.
Corollary – zB > zA
Expectation of winning the election makes it more valuable for the
winning group to locate in the city
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Comparing the two regimes
Proposition 1 – Amenities, and therefore commuting costs and the price of
services in the city, are larger in regime II.
Proposition 2 – House prices are larger in regime II if and only if
1
a2
>
a1
1
e λA
= z̃ 2 [zA , zB ]
e µB
Expectations of a bobo win raise the bobos willingness to pay to live in
city, but reduce that of the cadres.
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Comparing unskilled employment levels
Three e¤ects:
The price of services is higher in the city in the regime where the
bobos are politically decisive. This tends to reduce unskilled
employment.
If the cost of housing (rq ) is larger in regime B than in regime A, this
depresses the demand for services in the city. The converse holds if
the cost of housing falls.
In regime B, the bobos who live in the suburbs are replaced by cadres.
In the zone where a switch may occur, the latter are richer than the
former.
Demand for services goes up in suburbs and down in city
But former e¤ect stronger because services more expensive in city to
begin with.
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