Polinsky and Shavell,
Tauchen and Witte,
Pashigian
© Allen C. Goodman 2002
Polinsky and Shavell
Examine the distinction between closed and open cities when
looking at the measurement of benefits.
Take a little different tack in the modeling but with similar
results.
They use an indirect utility function:
V = V (y - T(k), p(k), a(k))
where
k = distance, y = income, T = transportation
costs, a(k) is an amenity.
We have V1 > 0, V2 < 0, V3 > 0.
What can we do with this?
V = V (y - T(k), p(k), a(k))
Within a city dV = 0.
dV/dk = -V1T´ + V2p´ +V3 a´ = 0.
Leads to p´ = (V1/V2)T´- (V3/V2)a´.
(V1/V2) = [Utility/$]/[Utility/(acres/$)] (1/Land).
We have negative price-distance function.
With an open city V is fixed at V*, so suppose there is an
increase in a(k).
For V to stay equal to V*, p(k) must rise.
What can we do with this?
Suppose we have a closed city.
V** = V (y - T(k), p(k), a(k))
V starts at V**. Suppose amenities increase everywhere but
k. V** must rise.
Since amenities haven’t improved at k, p(k) must fall relative
to elsewhere. This is an indirect effect.
Now increase a(k). p(k) must rise to maintain V**. This is
the direct effect.
Regression analysis w/ PS
Cobb-Douglas Example
U = Axaqba(k)d; a + b = 1.
 x(k) = a (y – T(k))
q(k) = b (y – T(k))/p(k)
Putting x and q into U 
V(k) = C[y-T(k)]p(k)-ba(k)d C is a constant
Solve for p(k) as:
log p(k) = (1/b) log (C/V*) + (1/b) log [Y – T(k)] + (d/b) log a(k)
= b0 + b1 log [Y – T(k)] + b2 log a(k)
Regression analysis w/ PS
log p(k) = (1/b) log (C/V*) + (1/b) log [Y – T(k)] + (d/b) log a(k)
= b0 + b1 log [Y – T(k)] + b2 log a(k)
In an open city, since V* is fixed, a change in a(k) will predict
change in log p(k).
In closed city V*  V**. Must know what happens to a(k) all
over city. Gen’l eq’m model is necessary.
SO:
Changes in aggregate land values correspond to WTP only with
an open city model.
Eq’m rent schedule will give enough information to identify
demand for a(k), all else equal; in “closed city” all else may not
be equal.
Tauchen - Witte RE externalities
Early in the term we talked about agglomeration economies.
Tauchen and Witte look at 2 different types
Interfirm contacts depend on density
Economies of actual numbers of firms
First, if you’re in any location, higher density leads to cheaper
contacts.
Second, at any given density larger # of firms makes a
difference.
As before, planning optimum and market optimum are not the
same.
Model
F identical firms, each w/ N
transactions.
Semi-net Revenue q is modeled
q = q (F); q´ > 0; q´´ < 0.
q refers to all revenue before
spatial costs are considered.
CBD is a square. No congestion.
Cost to firm at (x,y) for a contact
at (u,v) is
t (x, y, u, v) = C{|x-u| + |y-v|}
q
F
v,y
(0,0)
u,x
Model
CBD is a square. No congestion.
Cost to firm at (x,y) for a contact
at (u,v) is
t (x, y, u, v) = C{|x-u| + |y-v|}
Total Transactions Costs
T ( x, y)  N   t ( x, y, u, v)G(u, v, )dudv
v u
G(u,v) is the density of firms.
v,y
(x,y)
(u,v)
(0,0)
u,x
Model
Cost of providing office space for G firms in one unit of area
is K(G), where KG>0, KGG> 0.  at increasing rate.
Planning Problem - Maximize net value of output.
NV  Fq ( F )    T ( x, y)G( x, y)dxdy    KG( x, y)dxdy  Ls2
L = opportunity cost of land. Not important here.
Maximize above w.r.t. F and G, subject to:
F    G( x , y )dxdy
Optimum (using the calculus of variations) is:
q(F*) + F*q´ = 2 T*(x, y) + K´(G*).
Model
Optimum (using the calculus of variations) is:
q(F*) + F*q´ = 2 T*(x, y) + K´(G*).
Mgl. Semi-net Rev. Mgl. Social Costs
q(F*) + F*q´ - 2 T*(x, y) - K´(G*) = 0.
Does a market get us here?
2 market eq’m conditions
1. Zero economic profits
2. Rent for office space = Marginal cost of providing it.
1.  (x, y) = q(F) - T(x,y) - R(x,y) = 0.
2. R(x,y) = KG [G(x,y)]
q(F*) - T*(x, y) - K´(G*) = 0.
Model
Social optimum
q(F*) + F*q´ - 2 T*(x, y) - K´(G*) = 0.
Market optimum
q(F*) - T*(x, y) - K´(G*) = 0.
Two are equal ONLY if:
Fq´ = T(x,y)
Fq´ is the external benefit to other firms from a new firm.
T(x,y) is the increased communications cost to other firms
when a new firm comes in.
Rent function
You get an interesting rent
function.
Since if you move away
from CBD you’re
moving away from more
firms than you’re moving
toward, you get an
unusual rent function.
Rent
Distance
B. Peter Pashigian, Eric Gould,“Internalizing Externalities: The Pricing of Space in
Shopping Malls,”. Journal of Law and Economics, 41(1), April 1998, 115-142.
Shopping Malls
Pashigian and Gould talk about agglomeration
economies.Consumers are attracted to malls because of wellknown anchor stores.
These anchors give other stores opportunities to “free-ride” off
of the better-known stores.
Mall developers internalize these externalities by offering rent
subsidies to anchors, and charging rent premium to other mall
tenants.
Anchors pay lower rent/sq.ft. in super regional malls than in
regional malls, even though sales/sq.ft. are the same.
In contrast, sales and rent/sq.ft. are higher for other mall stores
in the super regional malls than in the regional malls.
Retail Demand functions
Pa = Da (qa, B)
(1)
Po = Do (qo, qa)
Pa, Po are prices received by anchor, other; B is reputation of
anchor.
Presumably Po/ qa > 0. Increased demand.
Each retailer’s total cost is a cost proportional to quantity
sold, plus rent paid:
Ca = caqa + Ravaqa
(2)
Co = coqo + Rovoqo
R is rent; v is space required to sell q units.
Retail Demand functions
If retail prices are competitive, price = marginal cost
Da (qa , B) = ca + Rava
(3)
Do (qo , qa)= co + Rovo
R is rent; v is space required to sell q units.
Solving, we get:
qa = g(Ra , B)
(4)
qo = h(Ro , qa) = h[Ro , g(Ra ,B)]
Developer selects Ra, Ro to maximize his/her own profits,
subject to (4).
d = (Ra - R) vaqa + (Ro - R)voqo
(5)
R = shadow price/sq.ft. of land and structures.
Retail Demand functions
Ro
 d
1
 Ro (1 
)R
Ro
Eq R
R
(6)
o o
 d
v q
1
 Ra (1 
)  R  ( Ro  R) o o
Ra
Eq R
va qa
qo
(7)
a a
LHS of each is Mgl. Rev. w.r.t. Rent. E = elasticities
RHS of (7) indicates that anchors’ MC is adjusted downward to
increase qa, and through the externality to increase the demand
for the independent store. So Ra < Ro, as
 Eqo Ro  va qa
 1
< Eqo qa (8)

 Eqa Ra  vo qo
Retail Demand functions
 Eqo Ro  va qa
 1
< Eqo qa (8)

 Eqa Ra  vo qo
RHS is assumed to be greater than or equal to 0.
In summary, developer offers a lower rent to anchor if anchor’s
rent elasticity is greater than independent’s, and may offer a lower
rent if anchor’s rental elasticity is less elastic, providing that
the size of the externality is large enough.
Testing this
Data from Urban Land Institute.
Super regional shopping center has 3 or more full line
department stores and more than 600,000 sq.ft. of gross
leasable space.
Regional malls have one or two full-line stores and not less
than 100,000 sq.ft.
They compare the typical rent paid by anchor national
department store with typical rent paid by non-anchor
national store whether in a super-regional or regional mall.
Testing this
Rent/sq.ft. for national tenant in
sup = 14.07+2.46 = 16.33.
Rent for nat’l dept. store in sup
= 14.07 - 12.66 = 1.41.
91.4% reduction!
In regional malls reduction is
84.5%.
Rent Per Square Foot
Variable
Coefficient (t-stat)
Intercept
14.07
10.1
Sup-National
2.46
1.7
Sup-National Anchor
-12.66
-3.7
Sup-Local
3.25
1.9
Sup-Local Anchor
-12.68
-1.2
Sup-Independent
8.14
4.3
Sup-Independent Anchor
-12.86
-0.9
Regional - National
-0.08
-0.1
Regional - National Anchor
-11.90
-2.8
Regional - Local
1.88
0.9
Regional - Local Anchor
-11.66
-1.1
Sample Size
405
Standard Error
42.4
Adj. R2
0.136