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) Rb Ra 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 distance to market (u) u2 u1

Model of Land Use A u1 u2 B

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.

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 Retail O 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: 2PH/u2=+(t*H/u)/H2>0

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 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

Urban and Regional Economics