1. Polinsky and Shavell, Tauchen and Witte, Pashigian © Allen C. Goodman 2002

2. Polinsky and Shavell Examine the distinction between closed and open cities when looking at the measurement of benefits. Take a little different tack in the modeling but with similar results. They use an indirect utility function: V = V (y - T(k), p(k), a(k)) wherek = distance, y = income, T = transportation costs, a(k) is an amenity. We have V 1 > 0, V 2 0.

3. What can we do with this? V = V (y - T(k), p(k), a(k)) Within a city dV = 0. dV/dk = -V 1 T´ + V 2 p´ +V 3 a´ = 0. Leads to p´ = (V 1 /V 2 )T´- (V 3 /V 2 )a´. (V 1 /V 2 ) = [Utility/$]/[Utility/(acres/$)]  (1/Land). We have negative price-distance function. With an open city V is fixed at V*, so suppose there is an increase in a(k). For V to stay equal to V*, p(k) must rise.

4. What can we do with this? Suppose we have a closed city. V** = V (y - T(k), p(k), a(k)) V starts at V**. Suppose amenities increase everywhere but k. V** must rise. Since amenities haven’t improved at k, p(k) must fall relative to elsewhere. This is an indirect effect. Now increase a(k). p(k) must rise to maintain V**. This is the direct effect.

5. Regression analysis w/ PS Cobb-Douglas Example U = Ax  q  a(k)  ;  +  = 1.  x(k) =  (y – T(k)) q(k) =  (y – T(k))/p(k) Putting x and q into U  V(k) = C[y-T(k)]p(k) -  a(k)  C is a constant Solve for p(k) as: log p(k) = (1/  ) log (C/V*) + (1/  ) log [Y – T(k)] + (  ) log a(k) = b 0 + b 1 log [Y – T(k)] + b 2 log a(k)

6. Regression analysis w/ PS log p(k) = (1/  ) log (C/V*) + (1/  ) log [Y – T(k)] + (  ) log a(k) = b 0 + b 1 log [Y – T(k)] + b 2 log a(k) In an open city, since V* is fixed, a change in a(k) will predict change in log p(k). In closed city V*  V**. Must know what happens to a(k) all over city. Gen’l eq’m model is necessary. SO: Changes in aggregate land values correspond to WTP only with an open city model. Eq’m rent schedule will give enough information to identify demand for a(k), all else equal; in “closed city” all else may not be equal.

7. Tauchen - Witte RE externalities Early in the term we talked about agglomeration economies. Tauchen and Witte look at 2 different types Interfirm contacts depend on density Economies of actual numbers of firms First, if you’re in any location, higher density leads to cheaper contacts. Second, at any given density larger # of firms makes a difference. As before, planning optimum and market optimum are not the same.

8. Model F identical firms, each w/ N transactions. Semi-net Revenue q is modeled q = q (F); q´ > 0; q´´ < 0. q refers to all revenue before spatial costs are considered. F q CBD is a square. No congestion. Cost to firm at (x,y) for a contact at (u,v) is t (x, y, u, v) = C{|x-u| + |y-v|} (0,0) u,x v,y

9. Model CBD is a square. No congestion. Cost to firm at (x,y) for a contact at (u,v) is t (x, y, u, v) = C{|x-u| + |y-v|} (0,0) u,x v,y Total Transactions Costs G(u,v) is the density of firms. (x,y) (u,v)

10. Model Cost of providing office space for G firms in one unit of area is K(G), where K G >0, K GG > 0.  at increasing rate. Planning Problem - Maximize net value of output. L = opportunity cost of land. Not important here. Maximize above w.r.t. F and G, subject to: Optimum (using the calculus of variations) is: q(F*) + F*q´ = 2 T*(x, y) + K´(G*).

11. Model Optimum (using the calculus of variations) is: q(F*) + F*q´ = 2 T*(x, y) + K´(G*). Mgl. Semi-net Rev.Mgl. Social Costs q(F*) + F*q´ - 2 T*(x, y) - K´(G*) = 0. Does a market get us here? 2 market eq’m conditions 1. Zero economic profits 2. Rent for office space = Marginal cost of providing it. 1.  (x, y) = q(F) - T(x,y) - R(x,y) = 0. 2. R(x,y) = K G [G(x,y)] q(F*) - T*(x, y) - K´(G*) = 0.

12. Model Social optimum q(F*) + F*q´ - 2 T*(x, y) - K´(G*) = 0. Market optimum q(F*) - T*(x, y) - K´(G*) = 0. Two are equal ONLY if: Fq´ = T(x,y) Fq´ is the external benefit to other firms from a new firm. T(x,y) is the increased communications cost to other firms when a new firm comes in.

13. Rent function You get an interesting rent function. Since if you move away from CBD you’re moving away from more firms than you’re moving toward, you get an unusual rent function. Rent Distance

14. Shopping Malls Pashigian and Gould talk about agglomeration economies.Consumers are attracted to malls because of well- known anchor stores. These anchors give other stores opportunities to “free-ride” off of the better-known stores. Mall developers internalize these externalities by offering rent subsidies to anchors, and charging rent premium to other mall tenants. Anchors pay lower rent/sq.ft. in super regional malls than in regional malls, even though sales/sq.ft. are the same. In contrast, sales and rent/sq.ft. are higher for other mall stores in the super regional malls than in the regional malls. B. Peter Pashigian, Eric Gould,“Internalizing Externalities: The Pricing of Space in Shopping Malls,”. Journal of Law and Economics, 41(1), April 1998, 115-142.

15. Retail Demand functions P a = D a (q a, B)(1) P o = D o (q o, q a ) P a, P o are prices received by anchor, other; B is reputation of anchor. Presumably  P o /  q a > 0. Increased demand. Each retailer’s total cost is a cost proportional to quantity sold, plus rent paid: C a = c a q a + R a v a q a (2) C o = c o q o + R o v o q o R is rent; v is space required to sell q units.

16. Retail Demand functions If retail prices are competitive, price = marginal cost D a (q a, B) = c a + R a v a (3) D o (q o, q a )= c o + R o v o R is rent; v is space required to sell q units. Solving, we get: q a = g(R a, B)(4) q o = h(R o, q a ) = h[R o, g(R a,B)] Developer selects R a, R o to maximize his/her own profits, subject to (4).  d = (R a - R) v a q a + (R o - R)v o q o (5) R = shadow price/sq.ft. of land and structures.

17. Retail Demand functions LHS of each is Mgl. Rev. w.r.t. Rent. E = elasticities RHS of (7) indicates that anchors’ MC is adjusted downward to increase q a, and through the externality to increase the demand for the independent store. So R a < R o, as qoqo R RoRo

18. Retail Demand functions RHS is assumed to be greater than or equal to 0. In summary, developer offers a lower rent to anchor if anchor’s rent elasticity is greater than independent’s, and may offer a lower rent if anchor’s rental elasticity is less elastic, providing that the size of the externality is large enough.

19. Testing this Data from Urban Land Institute. Super regional shopping center has 3 or more full line department stores and more than 600,000 sq.ft. of gross leasable space. Regional malls have one or two full-line stores and not less than 100,000 sq.ft. They compare the typical rent paid by anchor national department store with typical rent paid by non-anchor national store whether in a super-regional or regional mall.

20. Testing this Rent/sq.ft. for national tenant in sup = 14.07+2.46 = 16.33. Rent for nat’l dept. store in sup = 14.07 - 12.66 = 1.41. 91.4% reduction! In regional malls reduction is 84.5%.