Transcript Urban and Regional Economics

Urban and Regional
Economics
Part II: The Structure
of Urban Areas
Location Theory
• Firms and households can be thought of
as optimizers.
– Households make decisions to maximize their own
utility.
– Firms make decisions to maximize profits, or
minimize costs.
• This applies to locational choices as well.
– We have looked at this in regards to regional
location.
– We now turn to locational choices within urban
areas.
The Von Thunen Model
• Von Thunen first discussed the issue of
locational choice in the context of an
agricultural land use model.
• We will extend this model to investigate
locational choice of firms and
households in urban areas.
– We will get a better understandings of
economic forces operating within urban
areas.
Assumptions
• Assume that farmers produce output, q.
– Productivity per acre is constant at q.
• Markets for inputs and outputs are competitive.
• There are constant nonland inputs per acre, C.
• There are linear transportation costs to the
market.
– There is no congestion.
– Cost per lb per mile is constant at t.
• Rents per acre are R.
Profit Function per acre
=p*q-C-t*q*u - R
• Access to the market reduces transport
costs.
• Competition for land would increase the
price of land.
– This is known as the bid-rent.
• Competition for land would drive out all
profits.
Bid Rent Function
• Set profits equal to zero, and solve for
R.
=p*q-C-t*q*u - R=0
• R=p*q-C - t*q*u
• Plot this in R-u space
– Intercept: p*q-C
– Slope: dR/du=-tq
Bid-Rent Function
R(u)
Profits are zero
distance
to market (u)
Family of Bid Rent
Functions
R(u)
Profits increase with
lower rents
distance
to market (u)
Zero Profit Bid-Rent Functions
for Asparagus and Broccoli
R(u)
Ra
Rb
distance
to market (u)
Zero Profit Bid-Rent Functions
for Asparagus and Broccoli
R(u)
Outer envelope is the land rent
price function
Ra
Rb
u1
u2 distance
to market (u)
Model of Land Use
A
B
u1
u2
Generalizing the Model
• Apply to land use patterns in cities.
– Develop for firms
– Develop for households
• Start with simple model, and then add
realism.
– Amenities and disamenities
– Fiscal factors
Standard Urban Location
Model
• We will evaluate both firm and household
location models
– Firms: Choose location within city to
maximize profits.
• Generates a land rent function.
– Households: Choose location within city to
maximize utility.
• Generates a housing price function, and an
underlying land-rent function.
• Look at firms first and then households.
Simplistic City
Assumptions
• Look at a turn of the century city
• Characteristics
– Monocentric with central export node.
– Horse-drawn wagons to node for manuf.
– Workers/shoppers commute using
streetcars (hub & spoke system).
– Agglomeration economies exist for office
industry.
Manufacturers location
• Attraction to city proximity to export node.
• Produce output B with K,L,T, other inputs.
• Prices constant at PB.
–
–
–
–
Input and output markets competitive.
Cost of K,L constant at C.
Expenditure on land is R*T
Substitution possible.
• Transport prices are constant/ton/mile, t.
– Distance is u
• Look at profit function.
Bid-Rent for
Manufacturing
• Look at the profit function
 = PBB - C - t*B*u - R*T
• Competition for space drives out all
profits.
 = PBB - C - t*B*u - R*T=0
– Solve for R= (PBB - C - t*B*u)/T
R/ u= -tB/T
• Since t,B, T are positive, this is
negatively sloped.
Convexity of Bid-Rent
Curve
• Simple Von Thunen model did not allow
substitution, and this lead to constant slope
function.
• Here do allow substitutablity. Look at effect on
slope:
R/ u= -tB/T
• (slope at a point, so T cannot vary at that point)
• Now treat T usage as dependent on distance.
2R/u2= +T/u*(tB)/T2
– Since T/u>0, then 2R/u2>0
Zero Profit Bid Rent
Curve
R
(PBB - C)/T
Slope of Bid-Rent:
R/ u= -tB/T
Locational equilibrium
R*T = -tB*u
Bid Rent
u
Office Firms
• Attraction agglomeration economies.
• Consultations (A) with clients take place in CBD.
• Travel is by foot since they cannot rely on public transport
system (too irregular).
• Produce output A with K,L,T, and other inputs.
• Prices constant at PA.
–
–
–
–
Input and output markets competitive.
Cost of K,L constant at C.
Expenditure on land is R*T
Substitution possible.
• Transport prices per consultation are constant at m*W.
– m=minutes, W=wage/minute, Distance is u.
• Look at profit function
Bid-Rent for Office Firms
• Look at the profit function
 = PAA - C - m*W*A*u - R*T
• Competition for space drives out all profits.
 = PAA - C - m*W*A*u - R*T=0
– Solve for R= (PAA - C - m*W*A*u)/T
R/ u= -m*A*W/T
• Since m,W,A, and T are positive, this is
negatively sloped.
Zero Profit Bid Rent
Curve
R
(PAA - C)/T
Slope of Bid-Rent
R/ u= -m*W*A/T
Locational Equilib:
(R)*T= -m*W*A*u
Bid Rent
u
Which is Steeper?
• Since W*m for office firms, is likely
greater than t*u.
• On the other hand the ability to substitute
away from land is more difficult for
manufacturing. Thus, T is likely greater
in the manuf. sector.
• Thus, bid-rent for office is steeper.
Zero Profit Bid Rent
Curve
R
Land rent function is
outer envelope.
Manuf. Bid Rent
Office Bid Rent
u
Retail firms
• Attraction is because hub of streetcar
system drops them in CBD.
• Their markets are related to the density
of their demand, the scale economies
associated with production, and
transportation costs.
– Central Place theory determines market
size.
– Firms carve up the city into submarkets.
What determines WTP for
Land?
• Customers come to the firm to buy goods.
• Profit Function: =G*(PG-ACG)
– where G=volume of goods, P=price, AC=avg. cost.
• If P-AC is constant, then profit max. at max G.
– This is maximized at the center.
• Conclusion:
– Willingness to pay for land depends on accessibility
of land to customers, and thus it increases with
access to CBD.
• These bid rents vary by firm.
Zero Profit Bid Rent
Curve
R
Land rent function is
outer envelope.
Manuf. Bid Rent
Office and Retail
Bid Rent
u
Land Use Patterns
Office
O
Retail
Manuf.
Residential Location
Models
• Households choose locations to maximize
utility.
• Household characteristics
– Households choose between Housing (H) and
other goods (X), thus: V=(X,H) (identical tastes)
– Households work in the CBD
• Assume away decentralized employment.
– Income is constant at W.
– Commuting costs per mile are constant at t.
• Look at optimization problem:
Constrained
Optimization
• L=V(X,H)+(I-PXX-PHH-t*u)
• We will look at the First Order Condition
with respect to u:
L/ u= -PH/u*H - t) = 0
• What does binding constraint imply?
• Thus, PH/u*H - t =0 or PH/u=t/H
• In addition, given substitutability, this is
convex since:
Bid Housing Price
Function
P
Slope: PH/u= -t/H
Locational Equilibium
PH*H= -t *u
Bid Rent
u
Family of Bid Functions
P
Utility falls as we move
to higher bid functions.
Why?
Which is most relevant?
Related to S and D
for labor.
u
From Bid Housing Price to
Housing Price Gradient
• Slope: PH/u= -t/H
• The housing price gradient is simply the
percent change in housing prices
brought about by a unit change in
distance.
• Divide the numerator by PH to get:
• (PH/PH)/u= -t/(H*PH)
– What does this mean?
From Bid Housing Price
to Bid Rent
• Demand for land by households is derived
from the demand for housing.
– Thus, the bid housing price function generates a
bid-rent function.
• Book shows this using revenue & cost
function:
profit=P(u)*Q - K-R(u)*T
R(u)=(P(u)*Q - K)/T
• where P(u) is price per square foot of housing, Q=number
of square feet, K=nonland inputs, T=land inputs.
Derivation of Bid Rent
$
K
P(u)*Q
u
R
Rent-Gradient and
Housing Price Gradient
• Since the demand for land is derived from
the demand for housing, the gradients are
also related.
• (R/R)/u=1/landshare*(PH/PH)/u
• Land share = Rent exp./Housing exp.
• If land share is say 0.1, which is steeper?
• Land Rent gradient is steeper.
Land Use Patterns in the
Monocentric Model
Office
O
Retail
Manuf.
Households
Why are households
at most distant location?
Does SUM say anything
about Population Density?
• Density falls as consumption of housing
increases.
• H decreases as u decreases for two
reasons.
– Builders substitute away from land as R
increases.
– Households substitute away from housing
as PH increases.
Summary
• SUM predicts:
– Downward sloping, and convex rent gradient.
– Downward sloping, and convex housing price
gradient.
– Steeper rent gradient than housing price gradient.
– Declining density.
– Accessibility matters to households.
– Rings of activity in Monocentric city
• Lets look at some empirical evidence
Rent Gradient Evidence
• There is not a lot of evidence here since
land rent is not typically observed. That
is, there are few transactions on
undeveloped land.
– Mills shows that rent gradients are
downward sloping, and have been falling
over time.
– Chicago, 1928, rents fall about 20%/mi.
– Chicago, 1960, rents fall about 11.5%/mi.
Housing Price Gradient
• Evidence from Jerry Jackson
• Some support here.
– Housing prices fall by approximately 2.5%
per/mile.
• More later!
Land Rent vs. Housing Price
Gradient
• If land rent share is 0.1 to 0.2, then we
get the following prediction on rent
gradient:
• (R/R)/u=1/LS*(PH/PH)/u
LS=0.1 implies (R/R)/u=1/0.1*2.5=25%/mi.
LS=0.2 implies (R/R)/u=1/0.2*2.5=12.5%/mi.
• Thus, some support here.
Declining Population
Density
• There is substantial evidence here.
– McDonald(1989, Journal of Urban Economics)
has a lengthy review article on this evidence.
– Next time, I will briefly review this article
Does Accessibility
Matter?
• Jackson article suggest that the answer is yes.
• However, Bruce Hamilton published an
influential article in 1982 (JPE) that cast doubt
on the predictability of the SUM.
– Measured wasteful commuting, by looking at pop.
and employment density functions for cities.
• He found that there was 8 times more commuting taking
place than could be explained by SUM.
• Next time, we will relax model to incorporate
multicentric cities
– look at article by Bender and Hwang.
– Also, we begin looking at some real world data
Also Add other Realism
• Add in amenities/disamenities
• Add in fiscal factors
– Look at Clark/Allison paper