PAI786: Urban Policy

1. PAI786: Urban Policy Class 3: Housing Concepts, Household Bids

2. Urban Policy: Housing Concepts Household Bids • Outline of Class • Land concepts • Housing concepts • Housing bids and locational equilibrium

3. Urban Policy: Housing Concepts Household Bids • Land Concepts • Land rent is the price for using one unit of land, say an acre, for one unit of time, say a year. • Land value is the price of buying one unit of land, again say an acre.

4. Urban Policy: Housing Concepts Household Bids • Land Concepts, 2

5. Urban Policy: Housing Concepts Household Bids • The Determination of Land Rent • Land is an input; the price of land (= annual rent) is a derived demand—derived from its role in producing an output, say Q. • In equilibrium, the price of an input equals the value of its marginal product:

6. Urban Policy: Housing Concepts Household Bids • Land Rent, Continued • Now suppose that • Qmust be shipped to a market • The distance to the market, designated u, varies across firms. • It costs $s to ship a unit of Q one mile. • The marginal product of land equals a. • Then land rent is determined by:

7. Urban Policy: Housing Concepts Household Bids • Land Rent, Continued • With a constant a, land rent is a linear function of distance, u. • But when the price of land goes down, firms are likely to substitute land for capital. • This increase in the amount of land at greater distances from the market leads to a lower MPL at greater distances • And hence to a land-rent function with a slope that gets flatter as distance from the market increases.

8. Urban Policy: Housing Concepts Household Bids Land Rent and Distance from the Market without with substitution substitution < Figure 1 > R(u) A R(u) B Marketu Market u

9. Urban Policy: Housing Concepts Household Bids • Housing Concepts • Housing is measured in units of housing services = H • H= quality-adjusted square feet. • Depends on housing characteristics (X1, X2, …) • P = the price per unit of H per year. • R = rent for a housing unit = PH. • If the unit is an apartment, R = contract rent. • If the unit is owner-occupied, R is not observed.

10. Urban Policy: Housing Concepts Household Bids • Housing Concepts, Continued • V = the value of a housing unit = the present value of the rental flow (not observed for renters). • So:

11. Urban Policy: Housing Concepts Household Bids • How Does a CBD Worker Decide Where To Live? • She compares the marginal benefit (MB) and the marginal cost (MC) of moving one mile farther from the CBD.

12. Urban Policy: Housing Concepts Household Bids • How Does a CBD Worker Decide Where To Live? (Continued) • She then keeps moving out until she comes to the location (u*) at which MB equals MC:

13. Urban Policy: Housing Concepts Household Bids Tradeoff Between Housing and Commuting Costs

14. Urban Policy: Housing Concepts Household Bids • The Twist: How Housing Prices Are Determined • Now suppose that all households are alike (an assumption to be relaxed!). Then they all pick the same u*! • This is impossible, so P{u}adjusts until people are equally satisfied no matter where they live. • This is called locational equilibrium.

15. Urban Policy: Housing Concepts Household Bids • The Twist: How Housing Prices Are Determined (Continued) • Thus, P{u}adjusts until, at all locations, • that is, until the slope of the P{u}function equals –t/H.

16. Urban Policy: Housing Concepts Household Bids • The Twist: How Housing Prices Are Determined (Continued) • Because the slope is negative, P{u} is higher closer to the CBD than it is in the suburbs. • When P{u} is high, people substitute away from housing so that H is low. • When H is low, the slope of P{u}, namely, -t/H, is high in absolute value. • It follows that P{u}is steep near the city center but flattens as one moves out toward the suburbs.

17. Urban Policy: Housing Concepts Household Bids The Bid Function for Housing (Price per Unit of Housing Services) Slope = ΔP/Δu = -t/H < Figure 3 > P(u) • CBD u ΔP Δu

18. Urban Policy: Housing Concepts Household Bids Finding the Edge of the City • Urban activities must compete with rural activities for access to land. • Suppose P* is the opportunity cost of pulling land out of agriculture and into housing. • Then urban activities will take place out to the point, say, u*, at which the price of housing exceeds P*.

19. Urban Policy: Housing Concepts Household Bids Determining the Outer Edge of the Urban Area < Figure 3A > P(u) P* • CBD u* u

20. Urban Policy: Housing Concepts Household Bids Policy Questions and Bid Functions • Some policies affect a single urban area. • If they make the area more attractive, people move in; otherwise, people move out to other areas. • These policies are analyzed with an “open” model. • Other policies affect all urban areas. • These policies do not give anyone an incentive to move out of an area. • These policies are analyzed with a “closed” model.

21. Urban Policy: Housing Concepts Household Bids The Height of the Bid Function and the Size of the Area • To understand the distinction between open and closed models, recall that we derived a formula for the slope of P{u}, not for its height. • As the height of P{u}, goes up, • The level of satisfaction in an urban area goes down, • And the population goes up.

22. Urban Policy: Housing Concepts Household Bids The Height of the Bid Function and the Size of the Urban Area < Figure 3B > P(u) • CBD u Lower Utility

23. Urban Policy: Housing Concepts Household Bids Open versus Closed Models • In an open model, one selects the height of P{u}that yields the same level of satisfaction as a household can obtain in another urban area. • At any other height, people would move in or out. • In a closed model, one selects the height of P{u} that makes the area large enough to fit all its population.

24. Urban Policy: Housing Concepts Household Bids Open versus Closed Examples • Suppose one city in a regions cleans up its air and no other city does. • The impacts are given by an open model. • People move in and housing prices go up until the higher cost of living offsets the utility gain from cleaner air! • Suppose all cities in the region clean their air. • The impacts are given by a closed model. • Nobody has an incentive to move out and utility goes up due to cleaner air.