Urban and Regional Urban and Regional EconomicsEconomics

Part II: The Structure of Urban Areas

Location TheoryLocation 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 ModelThe 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.

AssumptionsAssumptions

• 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 acreProfit 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 FunctionBid 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 FunctionBid-Rent Function

R(u)

distance to market (u)

Profits are zero

Family of Bid Rent FunctionsFamily of Bid Rent Functions

R(u)

distance to market (u)

Profits increase withlower rents

Zero Profit Bid-Rent Functions Zero Profit Bid-Rent Functions for Asparagus and Broccolifor Asparagus and Broccoli

R(u)

distance to market (u)

Ra

Rb

Zero Profit Bid-Rent Functions Zero Profit Bid-Rent Functions for Asparagus and Broccolifor Asparagus and Broccoli

R(u)

distance to market (u)

Ra

Rb

Outer envelope is the land rentprice function

u1u2

Model of Land UseModel of Land Use

u1 u2A

B

Generalizing the ModelGeneralizing 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 Standard Urban Location ModelModel

• 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 Simplistic City AssumptionsAssumptions

• 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 locationManufacturers 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 ManufacturingBid-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 Convexity of Bid-Rent CurveCurve

• 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 CurveZero Profit Bid Rent Curve

R

u

Slope of Bid-Rent:

R/ u= -tB/T

Locational equilibrium

R*T = -tB*u

(PBB - C)/T

Bid Rent

Office FirmsOffice 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 FirmsBid-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 CurveZero Profit Bid Rent Curve

R

u

Slope of Bid-Rent

R/ u= -m*W*A/T

Locational Equilib:

(R)*T= -m*W*A*u

(PAA - C)/T

Bid Rent

Which is Steeper?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 CurveZero Profit Bid Rent Curve

R

u

Office Bid RentManuf. Bid Rent

Land rent function isouter envelope.

Retail firmsRetail 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 What determines WTP for Land?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 CurveZero Profit Bid Rent Curve

R

u

Office and Retail Bid Rent

Manuf. Bid Rent

Land rent function isouter envelope.

Land Use PatternsLand Use Patterns

O

Manuf.

OfficeRetail

Residential Location Residential Location ModelsModels

• 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 OptimizationConstrained 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: 2PH/u2=+(t*H/u)/H2>0

Bid Housing Price FunctionBid Housing Price Function

P

u

Slope: PH/u= -t/H

Locational Equilibium

PH*H= -t *u

Bid Rent

Family of Bid FunctionsFamily of Bid Functions

P

u

Utility falls as we moveto higher bid functions.

Why?

Which is most relevant?Related to S and D

for labor.

From Bid Housing Price to From Bid Housing Price to Housing Price GradientHousing 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 From Bid Housing Price to Bid RentBid 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 RentDerivation of Bid Rent

$

KP(u)*Q

R

u

Rent-Gradient and Rent-Gradient and Housing Price GradientHousing 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 Land Use Patterns in the Monocentric ModelMonocentric Model

O

Households

Office Retail

Manuf.

Why are householdsat most distant location?

Does SUM say anything about Does SUM say anything about Population Density?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.

SummarySummary

• 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 EvidenceRent 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 GradientHousing 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 Land Rent vs. Housing Price GradientGradient

• 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 Declining Population DensityDensity

• 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?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 RealismAlso Add other Realism

• Add in amenities/disamenities

• Add in fiscal factors– Look at Clark/Allison paper