- Email: [email protected]
Market structure, location rents, and the land development process
JOURNAL
OF URBAN ECONOMICS n,261-277
(1988)
Market Structure, Location Rents, and the Land Development Process GEOFFREYK. TIJRNBULL Department of Economics, Louisiana State University, Baton Rouge, Louisiana 70803-6306 Received July 31,1985; revised February 3,1986
The recent trend toward larger size development projects has stimulated interest in effects of market power on the performance of urban land markets [5, 71. This interest reflects the need to comprehend how different land market structures influence urban development, since appropriate land use policies can be determined only when the positive and normative aspects of noncompetitive land markets are well understood. Mills [4] established an important normative benchmark by demonstrating that, under neoclassical assumptions, a competitive land market generates socially efficient land use patterns and development rates. Looking at the other end of the market structure spectrum, Markusen and Scheffman [2] show that a monopolistic land owner has an incentive to withhold land from development in the static long run. Generalizing their analysis to a two-period framework [3], they also find (as does Mills [5]) that market power reduces the pace of development, as more projects are postponed to later periods. Comparing the monopolistic land market to the competitive benchmark established by Mills [4], the literature reveals the intuitively appealing result that, in the absence of externalities or locational rents, there is a social cost associated with land market power. This paper investigates how market structure and location rents affect the development process by comparing the polar cases of competitive and monopolistic land markets. It extends the existing literature in two directions previously neglected. First, introduction of the development sector and the essential role it plays in the dynamics of development indicates that market power in this sector should also be examined. To this end, this paper also considers the polar cases of monopsonistic and competitive development sectors. Second, as in the standard static urban land use framework, even though land may be homogeneous in all other respects, i.e., the region extends over a featureless plane, it is heterogeneous with respect to locational characteristics. Therefore, land close to employment centers or transportation networks earns higher rents in the developed state than land less well situated. This paper also considers the impact that these 261 0094-1190/88 $3.00 Copyright 6 1988 by Academic Press. Inc. All rights of reproduction m any form reserved.
262
GEOFFREY K. TURNBULL
rents may have on the development process. By extending the analysis to cover the structure of the development sector and locational rents, we ascertain the effects that each of these factors has alone and in conjunction with each other and the land market structure. As one might suspect, the efficiency characteristics of the competitive and monopolistic land use trajectories are altered by the introduction of these factors. The paper is organized as follows. Section I outlines the model used in the analysis. Section II considers the competitive land market with and without locational rents. Section III extends the discussion to the monopolized land market with and without differential rents. Section IV considers the impact of monopsony power in the development sector in conjunction with land market monopoly. Section V contains a summary of results and conclusions. I. THE MODEL The basic framework employed in this study is a dynamic extension of the familiar static urban land use model and draws from Mussa [6]. The basic framework presumes the development market has a prehistory (for time t < 0) in some steady state and is perturbed by a demand shock at t = 0. The analysis centers upon the adjustment toward the new equilibrium. To concentrate on market structure and location rents, all analysis is conducted in the absence of distorting land taxes or externalities. Following convention, the region examined is assumed to comprise the land market, with a fixed supply of land, L, available. All land is homogeneous, except for locational characteristics (proximity to employment centers, transportation networks, etc.) which may give rise to location rents. There are two land use sectors considered, urban and nonurban. The quantity of land allocated to the urban sector at time t is l(t); the rental rate for urban land is given by the inverse demand R(I), where R’( ) < 0. The demand for nonurban land is perfectly elastic for the region, reflecting the constant reservation rent R*. There is no explicit time dependency in either R( ) or R*. Although such a time dependence can be easily introduced to reflect secular growth in the sector land demand (as in Mills [5] for the urban sector), it would only serve to obscure the basic issues examined here. In the third sector of the model, the development sector, a composite nonland input IJ is used to change nonurban land into urban land (and urban land into nonurban land when reversibility is allowed). Once developed, the land remains in that state (unless improvements are removed). The production technology is embodied in the inverse production relationship
LAND
DEVELOPMENT
263
where uj is the input required by firm j to change xi of land from nonurban to urban use. This production relationship used by each firm in the development sector is characterized as f(0) = 0; f’( ) 5 0 as xi $ 0; and f”( ) > 0. This technology exhibits the familiar properties (i.e., positive and diminishing marginal products in terms of ]xj]) and also preserves the correspondence between the static model equilibrium and the dynamic model long run equilibrium. Nonland inputs are assumed to be mobile; the region can import or export nonland resources at the constant price w. The development market supply is the horizontal sum of each firm’s marginal cost, or MC
=
Wfl(Zljxj)
=
+lx>V
0)
where x = Xjxj, (p(x) 5 0 as x $ 0, and 4’(e) > 0. This provides the supply side of the development sector; the characterization of the demand side will depend upon the land market structure. II. LAND USE UNDER COMPETITION This section considers the case where the land market and the development sector are competitive. The effects of locational rents on the development pattern can be isolated by comparing the trajectory when there are no locational rents (the CCN case) to the trajectory when there are such rents (the CCR case). The CCN case has been considered by Mills [4,5]. Nevertheless, a careful examination of the CCN trajectory in this framework is in order since it serves as the benchmark against which other trajectories will be compared. Assume agents have perfect foresight. Along the CCN trajectory, each competitive landowner selects the development time t to maximize the net returns to his plot: 7T=
I0
‘R*e-“&
+ pw)e-
‘“a!s - +(x(t))e-rr,
where r is the appropriate discount rate and l(s) is the CCN time path (with foresight, agents recognize that future development will drive down rental returns in the urban sector). Thus, dn/dt = 0 along the CCN trajectory, which implies’
W) = WN? ‘da/dt
= 0 yields, with rearrangement: [R(l(f))
- R*]e-”
= [r+(x(t))
- +‘(x(t))i(t)]e-“.
264
GEOFFREY K. TURNBULL
where h(t)
= lm[R(l(s))
- R*]e-‘(S-‘)ds
(3)
is the marginal benefit from developing land at time t. Therefore, (2) simply states that development occurs up to the point where the marginal benefit equals the marginal cost at each t. To characterize the CCN trajectory, invert (2) using the definition dl/dt = f(t) = x(t), to obtain the instantaneous development rate of
i(t) = w(t))? where I/,(O) = 0 and #‘(e) > 0 follows from (p. Take the time derivative of (3): A(t) = h(t)
- [R(l(t))
-R*].
(5)
The CCN trajectory is the solution to the set of differential equations given by (4) and (5) with an initial state I(O). Figure 1 depicts the phase portrait of the solution. In the long run equilibrium, (4) and (5) are both zero so that fi = 0; this further implies the steady-state i satisfies R(f) = R*, which is the usual urban boundy condition encountered in static formulations. The trajectory approaches 1 from the initial state along the stable manifold as indicated by the connected arrows in Fig. 1. Mills [5] provides an important result concerning the social efficiency of competitive land market trajectories. The following proposition is the parallel of his equivalence theorem. PFC~P~SITI~N
1. The CCN trajectoy is the socially eflcient trajectory
(SET). The proof of this proposition is in the appendix. This result is not surprising given the well-known social efficiency of competition in static land use models. Proposition 1 should not, however, be construed to imply that the SET for the region will be the SET for the nation’s economy as a whole. Gordon’s [l] results indicate that, in general, decentralized (regional) planning will not replicate centralized planning, and hence will not be “optimal” from the national perspective. However, extending the analysis in this paper to deal with these concerns is outside the scope of this study. Suffice it to say that the conventional efficiency benchmark employed herein pertains to the regional notion of the SET. Using s as the time index, I(s) is the land allocation satisfying this condition at time s. Since this holds for all development times along the CCN trajectory, use s as the time index in this condition, multiply through by en, and integrate with respect to s from t onward to obtain (2).
LAND DEVELOPMENT
265
FIG. 1. The CCN land allocation trajectory.
The EJects of Locational Rents under Competition Now consider the competitive regime where different plots of land earn location-specific rents (the CCR case). It is assumed that the land plots are ordered along the continuum over the interval [0, L] in decreasing order of their reservation rents so that R(1) reflects the appropriate urban use rents to I on the interval. The urban rent R(I) reflects agents’ willingness to acquire the proximity of the plot located at 1. Each landowner determines the optimal development for his plot 1 as that which maximizes net returns a
=
‘R*e-“& J
e-“ds - (p(x(t))eprt.
+
0
Notice that, with locational rents, the urban rent for a given site 1 remains R(Z) regardless of the quantity of land allocated to the urban sector at time s, l(s). Thus, the arc 1(s) does not enter the second term above and da/dt = 0 implies, after rearrangement: [R(l)
- R*l/r
= +(x(t))
- #(x(t))i.(t)/r.
(6)
The LHS of (6) is the present value of the marginal rental benefit from
266
GEOFFREY
K. TURNBULL
developing plot 1 at time t. + is the marginal cost of development while the second RHS term is the change in development costs induced by the variation in the development pace over time. With foresight, landowners recognize that the development rate decreases over time along the CCR trajectory (dlxl/dt < 0 along 1(t); see below) so that it is profitable to postpone development somewhat in order to take advantage of lower future development costs. Thus, the competitive development rate will be slower than that implied by simply equating the marginal rental benefit and cost of development, i.e., [R(I) - R*]/r = +(x(l)). Clearly, even though the time path I(t) does not affect the rental returns of the Ith plot, R(l), it does affect landowners’ decisions via changes in development cost as the development rate varies along the land use trajectory. To characterize the CCR trajectory, denote the land lot for which (6) holds at time s as r(s).* Since (6) holds for all development times s and I(s), using s as the time index in (6) multiply (6) by re’(‘-‘) and integrate with respect to s to derive the familiar condition P(f) = 44NL
(7)
where p(t)
= [OCIR(l(s))
- R*]e-‘(“-‘)A
(8)
is the benefit from developing the marginal unit of land (i.e., the last unit to be developed) at time t.3 Invert (7) and take the time derivative of (8) to obtain the equations of motion determining the CCR trajectory: i(r) = d4CLW) a(t)
= q(f)
- [R(l(t))
(9) -R*].
(10)
The phase portrait of t@s system can be portrayed as in Fig. 1. The steady-state equilibrium I satisfies R(l) = R*; the i = 0 locus coincides with the l-axis while the fi = 0 coincides with the i = 0 locus in the figure. The CCR and CCN trajectories coincide in the phase plane and we have the following result.4 ‘Solve (6) implicitly for t(r), or s(l) using the s time notation, and invert to obtain I(s). 3The divergence of p from the marginal rental benfit (in (6)) is caused by development cost changes over the solution Ltime path, i.e., p = [R(I) - R*]/r + +‘(.)i/r, so that IpI < I[R(I) - R*]/rI for all I Z I. 4The SET allocates land to maximize regional benefits (net of adjustment costs). The social planner’s objective functional (Al) in the appendix remains unaltered by the appropriability of locational rents by land owners; thus, the SET derived in the appendix pertains to the case with locational rents as well as the case without such rents.
267
LAND DEVELOPMENT
PROPOSITION 2. The CCR trajectory is the SET. This result is verified in the Appendix. In sum, locational rents do not alter the competitive land use trajectory: The CCN trajectory maximizes the present value of landusers’ surplus plus rents while the CCR trajectory maximizes the present value of rents-which include the “landusers’ surplus” of the CCN case. The only difference between these trajectories is distributional, that is, who ends up with the “landusers’ surplus”. III. LAND USE UNDER MONOPOLY When the land market is monopolized there is also only a single development demander. To concentrate on the land market structure, assume that the monopolistic land owner is integrated upstream; that is, the monopolist has internalized the development sector. In this case the development sector exhibits quasi-competitive behavior; i.e., development is determined where the marginal benefit equals marginal cost, as in the competitive development sector case. This treatment allows us to postpone all substantive discussion concerning market power in the development sector until the next section. The analysis in this section proceeds as in the previous section: First, the monopolist with the internalized development sector with no locational rents (MIN) case is examined; then the monopolist with internalized development sector with location rents (MIR) case is examined. The resulting trajectories are then compared with each other and with the CCN and CCR counterparts to ascertain the qualitative impacts of the land market structure and location rents. In the MIN case, all urbanized land earns a rent return R(l(t)). The present value of the returns to all land, net of development costs, is given by 7l= /“{ 0
R(l(t))l(t)
+ [L - l(t)]R*
- J’(‘)+(z) 0
dz)e-“dl,
where i(t) = x(t). Define the current value Hamiltonian H(I, x, 6) = R(l(t))l(t)
+ [L - l(t)] R* - Jb;(‘)Q(z) dz + 8(t)x(t)
with the costate variable 8(t) = lm{ R’(l(s))l(s)
+ R(l(s))
- R*}e-‘(‘-‘)ds.
01)
s(t) is the benefit to the monopolist of additional land in urban use. The term R’( )1 + R( ) is the marginal to R( ); that is, it is the marginal urban
268
GEOFFREY K. TURNBULL
land rent. Therefore, the benefit of further development for the monopolist is the present value (at t) of the difference between the marginal urban land rent and the marginal nonurban land rent. The necessary conditions for the MIN solution reduce to
i(t) = W(t)) i(t) = rs - [R’(f(t))f(t) + R(f(t)) - R*],
(12)
(13)
with (11) and the initial state I(0). PROPOSITION3~. The MZN trajectory is not the SET. The proof is straightforward: it is sufficient to demonstrate that the MIN long run equilibrium differs from the SET long run equilibrium. In the steady state, (12) and (13) are both zero, which implies 6 = 0 and the long run land allocation fmN satisfies zGmN)I^,IN
+ R( TM,,) - R* = 0.
(14)
The SET steady state _fulfills p(&,) - R* = 0. Given R’( ) < 0 it immediately follows that l&.nN-CI,,, which proves the proposition. This result should not be surprising; in order to keep urban rents up, the monopolistic land owner must withhold land from the urban use sector in the long run equilibrium. This is consistent with the usual monopoly result and with Markusen and Scheffman [2] for the static case. The question remains, however: does the existence of land market power have a systematic influence on the land allocation time path? Basically, the effects on the land allocation time path are dependent upon the reversibility of development. In order to show the relationship between the SET and MIN development rates, we must first show that the SET must lie everywhere above the MIN trajectory in the phase plane. To demonstrate this, initially suppose the trajectories intersect at, say, 1,; then 6, = A, (see Fig. 2). The slope of the MIN trajectory at I, is
(d/i)= {r&- [R'(hd& +R(b)- R*I}/rC1(&)05) and the slope of the SET at I, is
(h/i) = {‘A, - [R(~o)- R*]}/+(b)-
(16)
Use 8, = A, and substitute (15) into (16) to obtain (i/i)
= (8/i)
+ R’(1,)l,,/il/(X,,).
(17)
LAND DEVELOPMENT
269
FIG. 2. Comparing the MIN and the SET land allocation trajectories.
When the initial state Z(0) -CfMIN, the second RHS term in (17) is negative, so that (A//) < (8/i); i.e., the SET must be steeper than the MIN trajectory at l,, as pictured in Fig. 2. But notice if the SET intersects the MIN path from above, as it must, then the SET cannot reach l,, which is a contradiction to the SET control program. Therefore, it must be true that the MIN and SET cannot intersect: the SET must lie everywhere above the MIN trajectory when I(0) < I-. A similar analysis for the case when the initial state 1(O)> &n.r reveals that the SET must lie everywhere above the MIN trajectory for any initial state, as pictured in Fig. 3. PROPOSITION 3~. The MIN development pace is slowe! than the SET development pace at each 1 when the initial state l(0) < I,,,. The MZN developpent pace is faster than the SET developmentpace at each I when 40) ’ lSET. This proposition is easily verified using Fig. 3. If I(0) < fM, then A > 6 along their respective trajectories, which, using (4) and (12), implies iMrN < This much is consistent with intuition: the monopolist proceeds with LT. development at a slower rate than the SET. When reversibility is aIlowed and I(0) > fsrrr, however, 1x1 < 161 along their respective trajectories and, using (4) and (12), Ii,,1 > lisETl. In this
270
GEOFFREY
K. TURNBULL
FIG. 3. The MIN and SET land allocation trajectories.
case the pace of development from urban to nonurban land use is faster under monopoly than competition (SET) basically because the opportunity cost to the monopolist of not developing exceeds that to the competitive land owners. This higher opportunity cost, which reflects in part the fact that the monopolist has “further to go” to reach the steady state, justifies the faster development with its atte?dant higher costs. Finally, when I(O) falls between I,, and I,, 6 < 0 and h > 0 so that LIN < 0 while isET> 0. The Effects of Locational Rents under Monopoly With locational rents (the MIR case) the monopolist’s objective functional becomes h + /” R*dr - Ix(‘)+(z) dz)e-” dt. (18) 0 f(r) The MIR case is analogous to a perfect price discriminating monopolist. In standard static analysis, such discrimination results in a socially efficient outcome; the same holds true here. VT=
/“( 0
PROPOSITION
/“‘)R(s) 0
4. The MIR trajectory is the SET.
This proposition is easily verified. Note that the monopolist’s objective functional (18) is identical to the social planner’s functional (Al): the MIR
271
LAND DEVELOPMENT
problem is identical to the SET problem and the proposition immediately follows given the uniqueness of the SET. Unlike the competitive case, the existence of locational rents in a monopolized land market fundamentally alters the land use time path. In addition, Propositions 2 and 4 reveal that, since the CCR and MIR trajectories coincide, land market power does not matter when land earns locational rents. IV. LAND USE UNDER MONOPSONY In the last section, the monopolist behaved quasi-competitively in the development sector. In this section, the firm exercisesmonopsonistic market power in the development sector. The two casesexamined here concern the monopolist-monopsonist without location rents (MMN) and with location rents (MMR). The monopsonist faces the development sector supply, given by +. Without location rents, the objective functional becomes T = /m{R(l(t))l(t) 0
+ [L - l(t)]R*
- +(x(t))x(t)}e-“dt,
where i(t) = x(t). The appropriate Hamiltonian for this problem is H(1, x, a) = R(l(t))l(t)
+ [L - l(t)]R*
- $a(x(t))x(t)
+ a(t)x(t),
where the current value costate variable is a(t) = lw{ R’(l(s))l(s)
+ R(l(s))
- R*}e-‘(S-‘)ds.
(19
The MMN trajectory is the solution to 4) = +‘(m)m + G(W) a(t) = ra - [R’(l(t))l(t) + R(l(t))
(20) - R*]
(21)
with the initial state I(0). The rate of development at any time satisfies (20). The RHS can be interpreted as the marginal development expenditure for the monopsonist and is marginal to I#. Since a is the marginal benefit of further development, (20) requires that the monopolist-monopsonist develop up to the point where the marginal benefit equals the marginal expenditure-which is the familiar monopsony employment co+ition. As long as the marginal development expenditure is monotonic in 1, (20) can be inverted to obtain the differential equation i(t) = r(a(t)),
(22)
272
GEOFFREY K. TURNBULL
where I(O) = 0 and I’( ) > 0. The MMN trajectory is therefore the solution to the set of differential equations given by (21) and (22). PROPOSITION 5. (a) The MMN trajectory is not the SET. (b) The MMN developmtnt pace is slower than the SET pace at each I when the initial state Z(0) < I,,,. The MMN developmentpace is faster than the SET pace at each 1 when l(0) > fsET.
The proof of Proposition 5a is straightforward and follows the proof of Proposition 3a. The proof of Proposition 5b follows the procedure in the proof of Proposition 3b. Alternatively, 5b follows from the direct application of Propositions 6 and 7 below. Following the suggestedearlier proof of Proposition 3a, the steady-state land allocation under MMN is less than the SET equilibrium. The divergence of these steady states is due sojely to Athe land market power. Comparing the MMN and MIN cases, I,, = I=, so that the monopsony power only serves to alter the speed of development, not the terminal allocation. The Efects of Locational Rents under Monopsony With locational rents, the monopolist-monopsonist’s (MMR) objective functional is amended to TIT=
/om( ~‘(‘)R(s)
dr + fP*d~
- +(x(t))+)}
e-” dt.
To find the land allocation time path that maximizes 7~use the Hamiltonian
H(h x, 8) = ld’ks)
a!~+ l;,R*h
- $+(t>)x(t)
+ B(t)x(t).
The marginal benefit of development is /3(t) = lm[R(l(s))
- R*]e-‘(S-L)ds.
(23)
The MMR time path is determined by
Qt) = ww)) b(t) = r/3(t) - [R(l(t))
(24) -R*].
(25)
From this set of differential equations, the steady-state i satisfies R(f) = R*. Therefore the long run MMR equilibrium is the same as the SET
equilibrium, hence the CCN and MIR, from Propositions 3 and 4. Never-
LAND DEVELOPMENT
213
theless, the MMR trajectory out of the steady state does not exhibit this efficiency characteristic. PROPOSITION 6~.
The h4MR trajectory is not the SET.
To prove by contradiction, suppose MMR coincides with the SET: the l(t) solution arc is identical in both cases. If the arcs are identical, then A(t) = /3(t), from (2) and (23); conditions (4) and (24) then require (26)
However, $’ -C I’ so that
Or, using #(O) = I(0) = 0, 40)
< m).
(27)
Since (27) contradicts (26), the MMR trajectory cannot coincide with the SET, which proves the proposition. Notice that if we construct the phase diagram for the MMR solution in the same plane as the SET phase diagram, the 6 = 0 locus is the same as the h = 0 locus. Since the i = 0 loci are also identical, it might seem that the MMR trajectory must coincide with the SET in diagram, contrary to Proposition 6a. Even though the loci are identical in the plane, the trajectories do not coincide, however, because the contours of the stable manifolds differ for each regime, due to the difference between 4 and I’. The existence of monopsony power alters the intermediate land allocation time path, even though it does not alter the ultimate state. This should not be surprising since monopsonistic power can only be wielded when development is taking place. The question remains: just how does monopsonistic power alter the trajectory? Intuition suggeststhat is should slow the speed of development, as the monopsonist employs the services of fewer development inputs. It turns out that this is indeed the case, as the following proposition will help establish. PROPOSITION 6~. The MMR rate of developmentis slower than the SET rate of development at each 1.
To prove this result, define A as the difference between the SET and MMR trajectory slopes in phase space; taking a Taylor approximation at the steady state, at a given I we have
A = (A/i) - (p/i) = r [l/V(O) - vw91 - R’(h PW)/v(o) - wlw,
(28)
274
GEOFFREY K. TURNBULL
FIG. 4. Comparing the MMR and SET land allocation trajectories.
where n = I - i and A and /3 are evaluated along their respective trajectories. The first term in (28) is unambiguously negative. If IAl < 181l?/#‘, then the second term is unambiguously negative. In this case, the SET is steeper than the MMR trajectory, as pictured in Fig. 4 for the limiting case A = PI”/+‘. From the figure it is clear that the SET cannot lie at or below the locus /W/q’ when h > 0 and still attain long run equilibrium. Similarly, the SET cannot lie above the locus /3F’/#’ when X < 0. Therefore, it must be true that along their respective trajectories
1x1’ IBIr’N.
(29)
The pace of development along the SET is isnr = $‘A, whereas the pace along the MMR trajectory at 1 is I,, = I’& From (29), when X > 0 then Similar reasoning for X < 0 estabP ry3, that is, isET > iMm. isET lishes &m.l > &,.&, which proves the proposition. Proposition 4 states that the monopolist with locational rents and a quasi-competitive development sector replicates the SET. Proposition 6b therefore establishes that the existence of monopsony power in the development sector serves to reduce the pace of development. If the development sector were internalized, or otherwise made to behave quasi-competitively, the development rate would be socially efficient. This is the analog to the
LAND DEVELOPMENT
275
well-known result that in&ciencies induced by monopsony power can be vitiated by backward integration. Notice, however, that this only holds true when locational rents are present. In the absence of such rents, the monopolist will not replicate the SET even if the development sector is internalized (by Proposition 3). Finally, the effect of locational rents alone in the presence of monopoly-monopsony is formally summarized in the following result. PROPOSITION 7. The MMN developmentpace is slower than the MMR development pace at each 1 when the initial state l(0) < I,,,. The MMN development pace is faster than the MMR developmentpace at each 1 when 40) ’ l,,,.
This proposition can be verified using the same procedure as in Proposition 3b. This result in conjunction with Propositions 3b and 4 establishes the existence of locational rents fundamentally alters both the steady state and the pace of development undertaken when there is monopoly power in the land market. V. SUMMARY AND CONCLUSION This paper examined the effects of market power and location rents on the land use allocation process. Drawing together the main results, we find that increases in land market power may either hasten or slow land use adjustment, depending on the initial state, although it will lower the amount of land allocated to urban use in the long run. Also, increased market power in the development sector will slow the pace of development, but will not alter the ultimate land allocation. We find that location rents generally slow the land use adjustment if all sectors are competitive and such rents alter both the pace of development and the ultimate allocation if the land market is monopolistic. Further, location rents cause the competitive trajectory to diverge from the socially efficient trajectory out of the steady state, even though the long run equilibrium is not affected. Finally, location rents enhance the social efficiency of monopolized land markets by speeding up the rate of development and mitigating the monopolist’s propensity to withhold land from the urban sector. In sum, concern over land market concentration should be tempered. As this study has illustrated, a tendency toward concentration cannot, by itself, indicate an increasing divergence from the socially efficient land use trajectory. One must consider the impact of possible locational rents as well as the degree of market power in the development sector, both of which can vitiate the social inefficiency of land market power.
276
GEOFFREY K. TURNBULL
APPENDIX This Appendix outlines the proof of Proposition 1, that the CCN trajectory is socially efficient. The social planner’s objective is to maximize the present value of total land rents net of the value of resources used in the process of adjustment
PI, or J = /om{ j’(‘)R(s) 0
ds + /” R*ds - Jo”“‘+(z) dr)e+ NO
dt,
(Al)
where i(t) = x(t). The current value Hamiltonian is H(I, x, x”) = i’(‘)R(s)
ds + jc;jR*d.s - /u”“+)
dz + P(t)x(t).
The Hamiltonian is strictly concave in I and x; therefore the socially efficient trajectory (SET) is unique and is the solution to
i(t) = W(t)) k(t)
= rP(t)
(A21 - [R(l(t))
- R*]
(A3)
given the initial state 1(O). The costate variable x”(t) is the marginal social benefit of urban land at t (with the program adjusted optimally from t on). To find a convenient expression for X”(t), rewrite (A3) using s as the time index, multiply through by epr@‘) and integrate to obtain epr(‘-‘)&
=
~mr~s(s)e-‘(‘-‘)& t
-
Jm[R(l(s)) I
-
R*]e-‘(“-‘)&.
Using integration by parts on the LHS and rearranging, (A4) reduces to A’(t) = lm[R(l(s))
- R*]e-‘(S-f)ds.
(A51
(A5) resembles the CCN marginal benefit of developing a land parcel at time f (see (3)). Inverting (A2) for F(t) = +(x(t)), taking the time derivative, setting the result equal to (A3), and rearranging yields A’(t) = [R(l(t))
- R*]/r
+ c$‘(*)i(t)/r.
Clearly, h”, like p in the CCR case, is the marginal rental benefit (the LHS
LAND DEVELOPMENT
277
term in (6)) adjusted to take into account the development cost changes that occur as the development rate varies along the optimal trajectory (see footnote 3). Turning to the propositions, notice that the CCN trajectory fulfills (A2) (from (4)) and (A3) (from (5)) and is therefore the unique solution to these conditions: the CCN is the SET (Proposition 1). Likewise, the CCR trajectory fulfills (A2) (from (9)) and (A3) (from (10)): the CCR is the SET (Proposition 2). REFERENCES 1. R. H. Gordon, An optimal tax approach to fiscal federalism, Quarr. J. Econom. 48, 567-586 (1983). 2. J. R. Markusen and D. T. Scheffman, Ownership concentration and market power in urban land markets, Rev. Econom. Stud. 45,519-526 (1978). 3. J. R. Markusen and D. T. Scheffman, “The Timing of Residential Land Development: A General Eqtilibrium Approach,” J. of LIrban Econ., 5,411-24 (1978). 4. D. E. Mills, The residential land allocation process, Quart. J. Econom. 92,227-244 (1978). 5. D. E. Mills, Market power and land development timing, Lund Econom. 56,10-20 (1980). 6. M. Mussa, Dynamic adjustment in the He&her-Ohlin-Samuelson model, J. Polit. Econom. 88, 775-791 (1978). 7. M. D. Wilbum and R. M. Gladstone, “Optimizing Development Profits in Large Scale Estate Projects,” Urban Land Institute, Washington, DC (1972).
OF URBAN ECONOMICS n,261-277
(1988)
Market Structure, Location Rents, and the Land Development Process GEOFFREYK. TIJRNBULL Department of Economics, Louisiana State University, Baton Rouge, Louisiana 70803-6306 Received July 31,1985; revised February 3,1986
The recent trend toward larger size development projects has stimulated interest in effects of market power on the performance of urban land markets [5, 71. This interest reflects the need to comprehend how different land market structures influence urban development, since appropriate land use policies can be determined only when the positive and normative aspects of noncompetitive land markets are well understood. Mills [4] established an important normative benchmark by demonstrating that, under neoclassical assumptions, a competitive land market generates socially efficient land use patterns and development rates. Looking at the other end of the market structure spectrum, Markusen and Scheffman [2] show that a monopolistic land owner has an incentive to withhold land from development in the static long run. Generalizing their analysis to a two-period framework [3], they also find (as does Mills [5]) that market power reduces the pace of development, as more projects are postponed to later periods. Comparing the monopolistic land market to the competitive benchmark established by Mills [4], the literature reveals the intuitively appealing result that, in the absence of externalities or locational rents, there is a social cost associated with land market power. This paper investigates how market structure and location rents affect the development process by comparing the polar cases of competitive and monopolistic land markets. It extends the existing literature in two directions previously neglected. First, introduction of the development sector and the essential role it plays in the dynamics of development indicates that market power in this sector should also be examined. To this end, this paper also considers the polar cases of monopsonistic and competitive development sectors. Second, as in the standard static urban land use framework, even though land may be homogeneous in all other respects, i.e., the region extends over a featureless plane, it is heterogeneous with respect to locational characteristics. Therefore, land close to employment centers or transportation networks earns higher rents in the developed state than land less well situated. This paper also considers the impact that these 261 0094-1190/88 $3.00 Copyright 6 1988 by Academic Press. Inc. All rights of reproduction m any form reserved.
262
GEOFFREY K. TURNBULL
rents may have on the development process. By extending the analysis to cover the structure of the development sector and locational rents, we ascertain the effects that each of these factors has alone and in conjunction with each other and the land market structure. As one might suspect, the efficiency characteristics of the competitive and monopolistic land use trajectories are altered by the introduction of these factors. The paper is organized as follows. Section I outlines the model used in the analysis. Section II considers the competitive land market with and without locational rents. Section III extends the discussion to the monopolized land market with and without differential rents. Section IV considers the impact of monopsony power in the development sector in conjunction with land market monopoly. Section V contains a summary of results and conclusions. I. THE MODEL The basic framework employed in this study is a dynamic extension of the familiar static urban land use model and draws from Mussa [6]. The basic framework presumes the development market has a prehistory (for time t < 0) in some steady state and is perturbed by a demand shock at t = 0. The analysis centers upon the adjustment toward the new equilibrium. To concentrate on market structure and location rents, all analysis is conducted in the absence of distorting land taxes or externalities. Following convention, the region examined is assumed to comprise the land market, with a fixed supply of land, L, available. All land is homogeneous, except for locational characteristics (proximity to employment centers, transportation networks, etc.) which may give rise to location rents. There are two land use sectors considered, urban and nonurban. The quantity of land allocated to the urban sector at time t is l(t); the rental rate for urban land is given by the inverse demand R(I), where R’( ) < 0. The demand for nonurban land is perfectly elastic for the region, reflecting the constant reservation rent R*. There is no explicit time dependency in either R( ) or R*. Although such a time dependence can be easily introduced to reflect secular growth in the sector land demand (as in Mills [5] for the urban sector), it would only serve to obscure the basic issues examined here. In the third sector of the model, the development sector, a composite nonland input IJ is used to change nonurban land into urban land (and urban land into nonurban land when reversibility is allowed). Once developed, the land remains in that state (unless improvements are removed). The production technology is embodied in the inverse production relationship
LAND
DEVELOPMENT
263
where uj is the input required by firm j to change xi of land from nonurban to urban use. This production relationship used by each firm in the development sector is characterized as f(0) = 0; f’( ) 5 0 as xi $ 0; and f”( ) > 0. This technology exhibits the familiar properties (i.e., positive and diminishing marginal products in terms of ]xj]) and also preserves the correspondence between the static model equilibrium and the dynamic model long run equilibrium. Nonland inputs are assumed to be mobile; the region can import or export nonland resources at the constant price w. The development market supply is the horizontal sum of each firm’s marginal cost, or MC
=
Wfl(Zljxj)
=
+lx>V
0)
where x = Xjxj, (p(x) 5 0 as x $ 0, and 4’(e) > 0. This provides the supply side of the development sector; the characterization of the demand side will depend upon the land market structure. II. LAND USE UNDER COMPETITION This section considers the case where the land market and the development sector are competitive. The effects of locational rents on the development pattern can be isolated by comparing the trajectory when there are no locational rents (the CCN case) to the trajectory when there are such rents (the CCR case). The CCN case has been considered by Mills [4,5]. Nevertheless, a careful examination of the CCN trajectory in this framework is in order since it serves as the benchmark against which other trajectories will be compared. Assume agents have perfect foresight. Along the CCN trajectory, each competitive landowner selects the development time t to maximize the net returns to his plot: 7T=
I0
‘R*e-“&
+ pw)e-
‘“a!s - +(x(t))e-rr,
where r is the appropriate discount rate and l(s) is the CCN time path (with foresight, agents recognize that future development will drive down rental returns in the urban sector). Thus, dn/dt = 0 along the CCN trajectory, which implies’
W) = WN? ‘da/dt
= 0 yields, with rearrangement: [R(l(f))
- R*]e-”
= [r+(x(t))
- +‘(x(t))i(t)]e-“.
264
GEOFFREY K. TURNBULL
where h(t)
= lm[R(l(s))
- R*]e-‘(S-‘)ds
(3)
is the marginal benefit from developing land at time t. Therefore, (2) simply states that development occurs up to the point where the marginal benefit equals the marginal cost at each t. To characterize the CCN trajectory, invert (2) using the definition dl/dt = f(t) = x(t), to obtain the instantaneous development rate of
i(t) = w(t))? where I/,(O) = 0 and #‘(e) > 0 follows from (p. Take the time derivative of (3): A(t) = h(t)
- [R(l(t))
-R*].
(5)
The CCN trajectory is the solution to the set of differential equations given by (4) and (5) with an initial state I(O). Figure 1 depicts the phase portrait of the solution. In the long run equilibrium, (4) and (5) are both zero so that fi = 0; this further implies the steady-state i satisfies R(f) = R*, which is the usual urban boundy condition encountered in static formulations. The trajectory approaches 1 from the initial state along the stable manifold as indicated by the connected arrows in Fig. 1. Mills [5] provides an important result concerning the social efficiency of competitive land market trajectories. The following proposition is the parallel of his equivalence theorem. PFC~P~SITI~N
1. The CCN trajectoy is the socially eflcient trajectory
(SET). The proof of this proposition is in the appendix. This result is not surprising given the well-known social efficiency of competition in static land use models. Proposition 1 should not, however, be construed to imply that the SET for the region will be the SET for the nation’s economy as a whole. Gordon’s [l] results indicate that, in general, decentralized (regional) planning will not replicate centralized planning, and hence will not be “optimal” from the national perspective. However, extending the analysis in this paper to deal with these concerns is outside the scope of this study. Suffice it to say that the conventional efficiency benchmark employed herein pertains to the regional notion of the SET. Using s as the time index, I(s) is the land allocation satisfying this condition at time s. Since this holds for all development times along the CCN trajectory, use s as the time index in this condition, multiply through by en, and integrate with respect to s from t onward to obtain (2).
LAND DEVELOPMENT
265
FIG. 1. The CCN land allocation trajectory.
The EJects of Locational Rents under Competition Now consider the competitive regime where different plots of land earn location-specific rents (the CCR case). It is assumed that the land plots are ordered along the continuum over the interval [0, L] in decreasing order of their reservation rents so that R(1) reflects the appropriate urban use rents to I on the interval. The urban rent R(I) reflects agents’ willingness to acquire the proximity of the plot located at 1. Each landowner determines the optimal development for his plot 1 as that which maximizes net returns a
=
‘R*e-“& J
e-“ds - (p(x(t))eprt.
+
0
Notice that, with locational rents, the urban rent for a given site 1 remains R(Z) regardless of the quantity of land allocated to the urban sector at time s, l(s). Thus, the arc 1(s) does not enter the second term above and da/dt = 0 implies, after rearrangement: [R(l)
- R*l/r
= +(x(t))
- #(x(t))i.(t)/r.
(6)
The LHS of (6) is the present value of the marginal rental benefit from
266
GEOFFREY
K. TURNBULL
developing plot 1 at time t. + is the marginal cost of development while the second RHS term is the change in development costs induced by the variation in the development pace over time. With foresight, landowners recognize that the development rate decreases over time along the CCR trajectory (dlxl/dt < 0 along 1(t); see below) so that it is profitable to postpone development somewhat in order to take advantage of lower future development costs. Thus, the competitive development rate will be slower than that implied by simply equating the marginal rental benefit and cost of development, i.e., [R(I) - R*]/r = +(x(l)). Clearly, even though the time path I(t) does not affect the rental returns of the Ith plot, R(l), it does affect landowners’ decisions via changes in development cost as the development rate varies along the land use trajectory. To characterize the CCR trajectory, denote the land lot for which (6) holds at time s as r(s).* Since (6) holds for all development times s and I(s), using s as the time index in (6) multiply (6) by re’(‘-‘) and integrate with respect to s to derive the familiar condition P(f) = 44NL
(7)
where p(t)
= [OCIR(l(s))
- R*]e-‘(“-‘)A
(8)
is the benefit from developing the marginal unit of land (i.e., the last unit to be developed) at time t.3 Invert (7) and take the time derivative of (8) to obtain the equations of motion determining the CCR trajectory: i(r) = d4CLW) a(t)
= q(f)
- [R(l(t))
(9) -R*].
(10)
The phase portrait of t@s system can be portrayed as in Fig. 1. The steady-state equilibrium I satisfies R(l) = R*; the i = 0 locus coincides with the l-axis while the fi = 0 coincides with the i = 0 locus in the figure. The CCR and CCN trajectories coincide in the phase plane and we have the following result.4 ‘Solve (6) implicitly for t(r), or s(l) using the s time notation, and invert to obtain I(s). 3The divergence of p from the marginal rental benfit (in (6)) is caused by development cost changes over the solution Ltime path, i.e., p = [R(I) - R*]/r + +‘(.)i/r, so that IpI < I[R(I) - R*]/rI for all I Z I. 4The SET allocates land to maximize regional benefits (net of adjustment costs). The social planner’s objective functional (Al) in the appendix remains unaltered by the appropriability of locational rents by land owners; thus, the SET derived in the appendix pertains to the case with locational rents as well as the case without such rents.
267
LAND DEVELOPMENT
PROPOSITION 2. The CCR trajectory is the SET. This result is verified in the Appendix. In sum, locational rents do not alter the competitive land use trajectory: The CCN trajectory maximizes the present value of landusers’ surplus plus rents while the CCR trajectory maximizes the present value of rents-which include the “landusers’ surplus” of the CCN case. The only difference between these trajectories is distributional, that is, who ends up with the “landusers’ surplus”. III. LAND USE UNDER MONOPOLY When the land market is monopolized there is also only a single development demander. To concentrate on the land market structure, assume that the monopolistic land owner is integrated upstream; that is, the monopolist has internalized the development sector. In this case the development sector exhibits quasi-competitive behavior; i.e., development is determined where the marginal benefit equals marginal cost, as in the competitive development sector case. This treatment allows us to postpone all substantive discussion concerning market power in the development sector until the next section. The analysis in this section proceeds as in the previous section: First, the monopolist with the internalized development sector with no locational rents (MIN) case is examined; then the monopolist with internalized development sector with location rents (MIR) case is examined. The resulting trajectories are then compared with each other and with the CCN and CCR counterparts to ascertain the qualitative impacts of the land market structure and location rents. In the MIN case, all urbanized land earns a rent return R(l(t)). The present value of the returns to all land, net of development costs, is given by 7l= /“{ 0
R(l(t))l(t)
+ [L - l(t)]R*
- J’(‘)+(z) 0
dz)e-“dl,
where i(t) = x(t). Define the current value Hamiltonian H(I, x, 6) = R(l(t))l(t)
+ [L - l(t)] R* - Jb;(‘)Q(z) dz + 8(t)x(t)
with the costate variable 8(t) = lm{ R’(l(s))l(s)
+ R(l(s))
- R*}e-‘(‘-‘)ds.
01)
s(t) is the benefit to the monopolist of additional land in urban use. The term R’( )1 + R( ) is the marginal to R( ); that is, it is the marginal urban
268
GEOFFREY K. TURNBULL
land rent. Therefore, the benefit of further development for the monopolist is the present value (at t) of the difference between the marginal urban land rent and the marginal nonurban land rent. The necessary conditions for the MIN solution reduce to
i(t) = W(t)) i(t) = rs - [R’(f(t))f(t) + R(f(t)) - R*],
(12)
(13)
with (11) and the initial state I(0). PROPOSITION3~. The MZN trajectory is not the SET. The proof is straightforward: it is sufficient to demonstrate that the MIN long run equilibrium differs from the SET long run equilibrium. In the steady state, (12) and (13) are both zero, which implies 6 = 0 and the long run land allocation fmN satisfies zGmN)I^,IN
+ R( TM,,) - R* = 0.
(14)
The SET steady state _fulfills p(&,) - R* = 0. Given R’( ) < 0 it immediately follows that l&.nN-CI,,, which proves the proposition. This result should not be surprising; in order to keep urban rents up, the monopolistic land owner must withhold land from the urban use sector in the long run equilibrium. This is consistent with the usual monopoly result and with Markusen and Scheffman [2] for the static case. The question remains, however: does the existence of land market power have a systematic influence on the land allocation time path? Basically, the effects on the land allocation time path are dependent upon the reversibility of development. In order to show the relationship between the SET and MIN development rates, we must first show that the SET must lie everywhere above the MIN trajectory in the phase plane. To demonstrate this, initially suppose the trajectories intersect at, say, 1,; then 6, = A, (see Fig. 2). The slope of the MIN trajectory at I, is
(d/i)= {r&- [R'(hd& +R(b)- R*I}/rC1(&)05) and the slope of the SET at I, is
(h/i) = {‘A, - [R(~o)- R*]}/+(b)-
(16)
Use 8, = A, and substitute (15) into (16) to obtain (i/i)
= (8/i)
+ R’(1,)l,,/il/(X,,).
(17)
LAND DEVELOPMENT
269
FIG. 2. Comparing the MIN and the SET land allocation trajectories.
When the initial state Z(0) -CfMIN, the second RHS term in (17) is negative, so that (A//) < (8/i); i.e., the SET must be steeper than the MIN trajectory at l,, as pictured in Fig. 2. But notice if the SET intersects the MIN path from above, as it must, then the SET cannot reach l,, which is a contradiction to the SET control program. Therefore, it must be true that the MIN and SET cannot intersect: the SET must lie everywhere above the MIN trajectory when I(0) < I-. A similar analysis for the case when the initial state 1(O)> &n.r reveals that the SET must lie everywhere above the MIN trajectory for any initial state, as pictured in Fig. 3. PROPOSITION 3~. The MIN development pace is slowe! than the SET development pace at each 1 when the initial state l(0) < I,,,. The MZN developpent pace is faster than the SET developmentpace at each I when 40) ’ lSET. This proposition is easily verified using Fig. 3. If I(0) < fM, then A > 6 along their respective trajectories, which, using (4) and (12), implies iMrN < This much is consistent with intuition: the monopolist proceeds with LT. development at a slower rate than the SET. When reversibility is aIlowed and I(0) > fsrrr, however, 1x1 < 161 along their respective trajectories and, using (4) and (12), Ii,,1 > lisETl. In this
270
GEOFFREY
K. TURNBULL
FIG. 3. The MIN and SET land allocation trajectories.
case the pace of development from urban to nonurban land use is faster under monopoly than competition (SET) basically because the opportunity cost to the monopolist of not developing exceeds that to the competitive land owners. This higher opportunity cost, which reflects in part the fact that the monopolist has “further to go” to reach the steady state, justifies the faster development with its atte?dant higher costs. Finally, when I(O) falls between I,, and I,, 6 < 0 and h > 0 so that LIN < 0 while isET> 0. The Effects of Locational Rents under Monopoly With locational rents (the MIR case) the monopolist’s objective functional becomes h + /” R*dr - Ix(‘)+(z) dz)e-” dt. (18) 0 f(r) The MIR case is analogous to a perfect price discriminating monopolist. In standard static analysis, such discrimination results in a socially efficient outcome; the same holds true here. VT=
/“( 0
PROPOSITION
/“‘)R(s) 0
4. The MIR trajectory is the SET.
This proposition is easily verified. Note that the monopolist’s objective functional (18) is identical to the social planner’s functional (Al): the MIR
271
LAND DEVELOPMENT
problem is identical to the SET problem and the proposition immediately follows given the uniqueness of the SET. Unlike the competitive case, the existence of locational rents in a monopolized land market fundamentally alters the land use time path. In addition, Propositions 2 and 4 reveal that, since the CCR and MIR trajectories coincide, land market power does not matter when land earns locational rents. IV. LAND USE UNDER MONOPSONY In the last section, the monopolist behaved quasi-competitively in the development sector. In this section, the firm exercisesmonopsonistic market power in the development sector. The two casesexamined here concern the monopolist-monopsonist without location rents (MMN) and with location rents (MMR). The monopsonist faces the development sector supply, given by +. Without location rents, the objective functional becomes T = /m{R(l(t))l(t) 0
+ [L - l(t)]R*
- +(x(t))x(t)}e-“dt,
where i(t) = x(t). The appropriate Hamiltonian for this problem is H(1, x, a) = R(l(t))l(t)
+ [L - l(t)]R*
- $a(x(t))x(t)
+ a(t)x(t),
where the current value costate variable is a(t) = lw{ R’(l(s))l(s)
+ R(l(s))
- R*}e-‘(S-‘)ds.
(19
The MMN trajectory is the solution to 4) = +‘(m)m + G(W) a(t) = ra - [R’(l(t))l(t) + R(l(t))
(20) - R*]
(21)
with the initial state I(0). The rate of development at any time satisfies (20). The RHS can be interpreted as the marginal development expenditure for the monopsonist and is marginal to I#. Since a is the marginal benefit of further development, (20) requires that the monopolist-monopsonist develop up to the point where the marginal benefit equals the marginal expenditure-which is the familiar monopsony employment co+ition. As long as the marginal development expenditure is monotonic in 1, (20) can be inverted to obtain the differential equation i(t) = r(a(t)),
(22)
272
GEOFFREY K. TURNBULL
where I(O) = 0 and I’( ) > 0. The MMN trajectory is therefore the solution to the set of differential equations given by (21) and (22). PROPOSITION 5. (a) The MMN trajectory is not the SET. (b) The MMN developmtnt pace is slower than the SET pace at each I when the initial state Z(0) < I,,,. The MMN developmentpace is faster than the SET pace at each 1 when l(0) > fsET.
The proof of Proposition 5a is straightforward and follows the proof of Proposition 3a. The proof of Proposition 5b follows the procedure in the proof of Proposition 3b. Alternatively, 5b follows from the direct application of Propositions 6 and 7 below. Following the suggestedearlier proof of Proposition 3a, the steady-state land allocation under MMN is less than the SET equilibrium. The divergence of these steady states is due sojely to Athe land market power. Comparing the MMN and MIN cases, I,, = I=, so that the monopsony power only serves to alter the speed of development, not the terminal allocation. The Efects of Locational Rents under Monopsony With locational rents, the monopolist-monopsonist’s (MMR) objective functional is amended to TIT=
/om( ~‘(‘)R(s)
dr + fP*d~
- +(x(t))+)}
e-” dt.
To find the land allocation time path that maximizes 7~use the Hamiltonian
H(h x, 8) = ld’ks)
a!~+ l;,R*h
- $+(t>)x(t)
+ B(t)x(t).
The marginal benefit of development is /3(t) = lm[R(l(s))
- R*]e-‘(S-L)ds.
(23)
The MMR time path is determined by
Qt) = ww)) b(t) = r/3(t) - [R(l(t))
(24) -R*].
(25)
From this set of differential equations, the steady-state i satisfies R(f) = R*. Therefore the long run MMR equilibrium is the same as the SET
equilibrium, hence the CCN and MIR, from Propositions 3 and 4. Never-
LAND DEVELOPMENT
213
theless, the MMR trajectory out of the steady state does not exhibit this efficiency characteristic. PROPOSITION 6~.
The h4MR trajectory is not the SET.
To prove by contradiction, suppose MMR coincides with the SET: the l(t) solution arc is identical in both cases. If the arcs are identical, then A(t) = /3(t), from (2) and (23); conditions (4) and (24) then require (26)
However, $’ -C I’ so that
Or, using #(O) = I(0) = 0, 40)
< m).
(27)
Since (27) contradicts (26), the MMR trajectory cannot coincide with the SET, which proves the proposition. Notice that if we construct the phase diagram for the MMR solution in the same plane as the SET phase diagram, the 6 = 0 locus is the same as the h = 0 locus. Since the i = 0 loci are also identical, it might seem that the MMR trajectory must coincide with the SET in diagram, contrary to Proposition 6a. Even though the loci are identical in the plane, the trajectories do not coincide, however, because the contours of the stable manifolds differ for each regime, due to the difference between 4 and I’. The existence of monopsony power alters the intermediate land allocation time path, even though it does not alter the ultimate state. This should not be surprising since monopsonistic power can only be wielded when development is taking place. The question remains: just how does monopsonistic power alter the trajectory? Intuition suggeststhat is should slow the speed of development, as the monopsonist employs the services of fewer development inputs. It turns out that this is indeed the case, as the following proposition will help establish. PROPOSITION 6~. The MMR rate of developmentis slower than the SET rate of development at each 1.
To prove this result, define A as the difference between the SET and MMR trajectory slopes in phase space; taking a Taylor approximation at the steady state, at a given I we have
A = (A/i) - (p/i) = r [l/V(O) - vw91 - R’(h PW)/v(o) - wlw,
(28)
274
GEOFFREY K. TURNBULL
FIG. 4. Comparing the MMR and SET land allocation trajectories.
where n = I - i and A and /3 are evaluated along their respective trajectories. The first term in (28) is unambiguously negative. If IAl < 181l?/#‘, then the second term is unambiguously negative. In this case, the SET is steeper than the MMR trajectory, as pictured in Fig. 4 for the limiting case A = PI”/+‘. From the figure it is clear that the SET cannot lie at or below the locus /W/q’ when h > 0 and still attain long run equilibrium. Similarly, the SET cannot lie above the locus /3F’/#’ when X < 0. Therefore, it must be true that along their respective trajectories
1x1’ IBIr’N.
(29)
The pace of development along the SET is isnr = $‘A, whereas the pace along the MMR trajectory at 1 is I,, = I’& From (29), when X > 0 then Similar reasoning for X < 0 estabP ry3, that is, isET > iMm. isET lishes &m.l > &,.&, which proves the proposition. Proposition 4 states that the monopolist with locational rents and a quasi-competitive development sector replicates the SET. Proposition 6b therefore establishes that the existence of monopsony power in the development sector serves to reduce the pace of development. If the development sector were internalized, or otherwise made to behave quasi-competitively, the development rate would be socially efficient. This is the analog to the
LAND DEVELOPMENT
275
well-known result that in&ciencies induced by monopsony power can be vitiated by backward integration. Notice, however, that this only holds true when locational rents are present. In the absence of such rents, the monopolist will not replicate the SET even if the development sector is internalized (by Proposition 3). Finally, the effect of locational rents alone in the presence of monopoly-monopsony is formally summarized in the following result. PROPOSITION 7. The MMN developmentpace is slower than the MMR development pace at each 1 when the initial state l(0) < I,,,. The MMN development pace is faster than the MMR developmentpace at each 1 when 40) ’ l,,,.
This proposition can be verified using the same procedure as in Proposition 3b. This result in conjunction with Propositions 3b and 4 establishes the existence of locational rents fundamentally alters both the steady state and the pace of development undertaken when there is monopoly power in the land market. V. SUMMARY AND CONCLUSION This paper examined the effects of market power and location rents on the land use allocation process. Drawing together the main results, we find that increases in land market power may either hasten or slow land use adjustment, depending on the initial state, although it will lower the amount of land allocated to urban use in the long run. Also, increased market power in the development sector will slow the pace of development, but will not alter the ultimate land allocation. We find that location rents generally slow the land use adjustment if all sectors are competitive and such rents alter both the pace of development and the ultimate allocation if the land market is monopolistic. Further, location rents cause the competitive trajectory to diverge from the socially efficient trajectory out of the steady state, even though the long run equilibrium is not affected. Finally, location rents enhance the social efficiency of monopolized land markets by speeding up the rate of development and mitigating the monopolist’s propensity to withhold land from the urban sector. In sum, concern over land market concentration should be tempered. As this study has illustrated, a tendency toward concentration cannot, by itself, indicate an increasing divergence from the socially efficient land use trajectory. One must consider the impact of possible locational rents as well as the degree of market power in the development sector, both of which can vitiate the social inefficiency of land market power.
276
GEOFFREY K. TURNBULL
APPENDIX This Appendix outlines the proof of Proposition 1, that the CCN trajectory is socially efficient. The social planner’s objective is to maximize the present value of total land rents net of the value of resources used in the process of adjustment
PI, or J = /om{ j’(‘)R(s) 0
ds + /” R*ds - Jo”“‘+(z) dr)e+ NO
dt,
(Al)
where i(t) = x(t). The current value Hamiltonian is H(I, x, x”) = i’(‘)R(s)
ds + jc;jR*d.s - /u”“+)
dz + P(t)x(t).
The Hamiltonian is strictly concave in I and x; therefore the socially efficient trajectory (SET) is unique and is the solution to
i(t) = W(t)) k(t)
= rP(t)
(A21 - [R(l(t))
- R*]
(A3)
given the initial state 1(O). The costate variable x”(t) is the marginal social benefit of urban land at t (with the program adjusted optimally from t on). To find a convenient expression for X”(t), rewrite (A3) using s as the time index, multiply through by epr@‘) and integrate to obtain epr(‘-‘)&
=
~mr~s(s)e-‘(‘-‘)& t
-
Jm[R(l(s)) I
-
R*]e-‘(“-‘)&.
Using integration by parts on the LHS and rearranging, (A4) reduces to A’(t) = lm[R(l(s))
- R*]e-‘(S-f)ds.
(A51
(A5) resembles the CCN marginal benefit of developing a land parcel at time f (see (3)). Inverting (A2) for F(t) = +(x(t)), taking the time derivative, setting the result equal to (A3), and rearranging yields A’(t) = [R(l(t))
- R*]/r
+ c$‘(*)i(t)/r.
Clearly, h”, like p in the CCR case, is the marginal rental benefit (the LHS
LAND DEVELOPMENT
277
term in (6)) adjusted to take into account the development cost changes that occur as the development rate varies along the optimal trajectory (see footnote 3). Turning to the propositions, notice that the CCN trajectory fulfills (A2) (from (4)) and (A3) (from (5)) and is therefore the unique solution to these conditions: the CCN is the SET (Proposition 1). Likewise, the CCR trajectory fulfills (A2) (from (9)) and (A3) (from (10)): the CCR is the SET (Proposition 2). REFERENCES 1. R. H. Gordon, An optimal tax approach to fiscal federalism, Quarr. J. Econom. 48, 567-586 (1983). 2. J. R. Markusen and D. T. Scheffman, Ownership concentration and market power in urban land markets, Rev. Econom. Stud. 45,519-526 (1978). 3. J. R. Markusen and D. T. Scheffman, “The Timing of Residential Land Development: A General Eqtilibrium Approach,” J. of LIrban Econ., 5,411-24 (1978). 4. D. E. Mills, The residential land allocation process, Quart. J. Econom. 92,227-244 (1978). 5. D. E. Mills, Market power and land development timing, Lund Econom. 56,10-20 (1980). 6. M. Mussa, Dynamic adjustment in the He&her-Ohlin-Samuelson model, J. Polit. Econom. 88, 775-791 (1978). 7. M. D. Wilbum and R. M. Gladstone, “Optimizing Development Profits in Large Scale Estate Projects,” Urban Land Institute, Washington, DC (1972).