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### 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

Zero Profit Bid-Rent Functions for Asparagus and Broccoli

R(u)

Outer envelope is the land rent

price function

Ra

Rb

distance

to market (u)

u2

u1

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

- Slope of Bid-Rent:
- R/ u= -tB/T
- Locational equilibrium
- R*T = -tB*u

- (PBB - C)/T

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

- Slope of Bid-Rent
- R/ u= -m*W*A/T
- Locational Equilib:
- (R)*T= -m*W*A*u

- (PAA - C)/T

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.

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

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

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.

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

Retail

O

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

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