Private contracts for durable local public good provision

JOURNAL

OF URBAN

ECONOMICS

29, 380-402 (1991)

Private Contracts for Durable Local Public Good Provision

Department of Economics, lJniversi@ of Maryland at Baltimore County, Baltimore, Maryland 21228 Received November 3, 1988; revised August 9, 1989 This paper presents a model of imperfectly durable local public good provision in a dynamic development framework. The private provision of these goods under contracts between developers and residents of condominium and homeowners’ associations is examined. An optimal trajectory of public goods is determined and compared to time consistent contractual trajectories. This comparison is used to explain why developers typically transfer control of maintenance of these goods to residents before the development process ends. The optimal date to transfer control is determined and compared to existing contracts and recommendations in the industry literature. The optimal transfer date is nondecreasing in community size and length of the development process. 0 1991 Academic press, IIIC.

Communities governed by condominium and homeowners’ associations are often excellent examples of homogeneous communities predicted in papers by Tiebout [lo], Ellickson [3], McGuire [9], and Hamilton [4]. These papers suggest that, due to financing considerations for public services, a metropolitan area tends to stratify into communities in which residents are relatively homogeneous in wealth, public good demands, and preferences. Henderson [5] then introduced a dynamic development process in which the number of residents in the community increases over time. The homogeneity result combined with the dynamic development process is the starting point of this paper. I depart from this starting point by introducing several new aspects that are critical in a dynamic model of local public good provision. First is imperfect durability of local public goods, which is critical for examining maintenance of local public goods. Second, I introduce a documentation problem which results from the fact that the future quality of local public goods cannot be contractually specified in an enforceable manner. Then, given the documentation problem, I analyze typical contracts between a developer and residents of a condominium or homeowners’ association for the maintenance of local public goods. Two questions about these contracts are addressed. First, how does altering who determines maintenance -the developer or the residents-affect the quality of local public goods 380 0094-1190/91 $3.00 Copyright All rights

8 1991 by Academic Press, Inc. of reproduction in any form reserved.

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provided in the community over time? Second, what is the optimal date to transfer control of maintenance decisions from the developer to the residents of the community? This paper focuses on the transfer of control of maintenance decisions because it is critical in ensuring the success of newly developed communities with condominium or homeowners’ associations. Some of these communities, in which the developer retains control long into the development process, are plagued with inadequate maintenance of local public goods. On the other hand, because in the beginning of the development process the developer owns most of the lots and therefore typically pays for most of the maintenance, if the developer allows the residents to control maintenance decisions too early in the development process, the residents may force the developer to pay for inefficiently large levels of maintenance. This problem is particularly severe in the beginning of the development process when there are only a few residents living in the community and the Pareto-efficient level of maintenance is low. Thus, the issue of transfer of control of maintenance decisions is important to both developers and residents of communities that are governed by condominium and homeowners’ associations.’ Because the transfer of control of maintenance decisions is critical in ensuring the stability and success of these communities, it is also important to government agencies involved in mortgage lending such as the Federal Housing Administration and the Veterans Administration. I. COMMUNITY DEVELOPMENT In the dynamic model of community development there are N platted2 lots, where N 2 2. Lot sizes are fixed in the first period of the development process through a binding uniform scheme of land use.3 Each resident buys one lot. Initially, I assume that there is a uniform inflow of one entrant per period. Consequently, there is an N-period development process. The development process is complete when all of the lots in the community have been sold. While there is a finite N-period development process, residents are infinitely lived. The developer charges the ith resident a price, Ri, to enter the community and for the right to use the local public goods for the life of ‘According to the Urban Land Institute and Community Associations Institute [12], by 1986 over 10 million housing units were privately governed by such associations. *The size and position of each lot in the community is typically shown on a plat of the community. 3See Knapp [7, pp. 8-101 for a discussion of the legal, institutional, and economic issues regarding the uniform scheme of land use.

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the community. It is assumed that contractual arrangements are structured so that no lot owner can be excluded from using these local public goods.4 Local public goods in this community are imperfectly durable and are provided initially by the developer. Quality of the local public goods decays at the constant proportionate rate of 6 per period in the absence of maintenance. The quality of local public goods evolves over time according to the equation, Q, = 9, + (1 - S)Q,-,, where Q, is the quality of local public goods and qt, which is maintenance, must be nonnegative. It is assumed that there is a constant marginal cost, c, for providing the initial quality level, Q,, and maintenance in period t, qt. Thus, the total cost of Q, is cQi, which is paid by the developer. Similarly, the cost of maintenance in period t is cq,. The utility function of the ith entrant is uj = Cy+(l/l + r)‘-‘[QQ + xi(t)] for i = 1,. . . , N, where r is the rate of time preference, assumed to equal the interest rate.5 xi(t) is individual i’s consumption of all other goods at time t, the price of which is normalized to 1. It is assumed the utility parameter, a, satisfies 0 < a < 1. With this specification of the utility function, given the same costs of maintenance, each resident will want the same level of local public goods in a given period. This homogeneous demand for local public goods within the community is consistent with the stratification results derived from Tiebout’s model [lo]. The ith resident has a reservation utility, c, and a level of wealth, vi, which are taken as exogenous by the developer.

II. PARETO-EFFICIENT

FORMULATION

AND

SOLUTION

The solution to the developer’s optimization problem given below is Pareto-efficient because it maximizes the developer’s welfare, holding all residents’ welfare fixed at their reservation levels.

4The right to use the local public goods is tied to the lot the resident buys, not to the owner as an individual. Thus, this right is tied to the land. Klein et al. 161 and Knapp 17, pp. 25-271 provide an explanation of why this occurs. ‘This utility function was chosen to capture the essential elements in a tractable model. Lot size is exogenous in this model and has been omitted from the utility function. Introducing the fixed lot size into the utility function would not affect the results discussed in this paper.

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383

subject to

~i(&--)tp’[Q~ +xi(t)] 2 vi, i= l,...,N w,=Ri+

5 t=i

(1

xi(t) +

4t

i=l

rjt-i’

2

7”*yNy

0,

(lb) (lc)

(14

where R, and q, are vectors of length N and a~, respectively. Equation (la) is the present value of the developer’s profits from this community. Equation (lb) is the constraint that each resident must receive at least her reservation utility and (lc) is the wealth constraint of individual i. Equation (Id) is the nonnegativity constraint on maintenance. It profits the developer to charge the residents prices that reduce the residents to their reservation utility levels. Thus, (lb) holds with an equality. Given this, substituting the residents’ wealth constraints (1~) into their respective reservation utility constraints (lb) for CyZi x,(t)/(l + rYei, yields the willingness to pay function t-i Ri=

Q;

C(k) t=i

+

F-c.

Substituting the equation of motion for Q, and (2) into (la) and simplifying give the developer’s profit function N

f-l

1

7r=c

f-1

(

l

+r

1

t-1

tQP + I=N+l it

- z2(&)‘-‘c(Q,

(‘)

1+ r

NQp - cQ1

- (1 - QQt-,) + K,

where K = C~=i,(l/(l + r>>t-l<~t - v,> and the third summation is the present value of the developer’s maintenance costs. Thus, the developer’s optimization problem can be simplified to maximizing the present value of

384

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his profits, (3), subject to the nonnegativity constraint on maintenance, (Id). The Pareto-efficient solution referred to throughout the paper is Q;"

Q,'"

=

($j'-'(

= ( ,)l”a-l)(

!%)'-

+“‘-=)

for for

5 N

t

t

2 N,

(4b)

where Q,‘” is the Pareto-efficient quality of local public goods at time t. For t < N, QF” satisfies the first-order condition taQ:-’ + ((1 - S)/(l + T))C = c, which has the following economic interpretation. The sum of the current marginal benefits to the t residents in the community at time t (taQ:-‘) and th e d iscounted future marginal benefits in terms of decreased maintenance costs in period t + 1 due to Q,, which is ((1 - 6)/ (1 + T))c, is equal to the marginal cost of maintenance (~1.~ The Pareto-efficient quality of local public goods increases at an increasing rate for 1 I I I N as the number of residents receiving benefits from the local public goods increases during the development process. For t > N, the number of residents in the community is unchanging and the Pareto-efficient quality of local public goods remains constant at the steady-state level given in (4b).’ The Pareto-efficient solution incorporates the current and future benefits and costs to the developer and all residents. Because the Pareto-efficient solution maximizes the present value of the developer’s profit from the community, and because the residents pay for the quality of local public goods they expect will be provided over time at the time they enter the community, at the date the development process begins it is in the developer’s interest to guarantee that the Pareto-efficient trajectory of quality of public goods will be provided. However, it is impossible to specify contractually the quality of local public goods in an enforceable manner. Thus, neither the developer nor residents can credibly precommit to providing the future Pareto-efficient, but time inconsistent, quality of public goods. Therefore, when modeling the shared financing contract, the time consistent decisions are analyzed to determine whether this contract yields time consistent solutions that are Pareto-efficient. First the documentation problem is clarified. 6Throughout this paper, the first-order conditions are interpreted for the quality of local public goods rather than the maintenance levels. ‘The Pareto-efficient level of maintenance increases at an increasing rate during the development process, after which it remains constant.

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385

It is assumed that residents know the quality of local public goods provided each time period. The problem lies in documenting this quality in a contract that is enforceable in court. That is, the problem lies in writing quality of local public goods into the contract in a way such that contract violation can be proven in court. Next, a contract, which occurs in the market given that it is impossible to contractually specify quality of local public goods in an enforceable manner, is modeled. III. THE SHARED FINANCING CONTRACT The shared financing contract specifies that each lot owner (developer or resident) must pay a share equal to l/N of total maintenance costs each period for each lot owned in that period. As development continues and the developer sells more lots, the share of total maintenance costs paid by the developer falls and that paid by residents as a group rises. The shared financing rule is chosen because it is the typical rule in contracts between the developer and residents of a condominium or homeowners’ association. Prior to 7, which is contractually specified by the developer, the developer alone chooses the amount of maintenance. For t 2 T residents choose the amount of maintenance through a majority vote in which each resident lot owner has one vote per lot. The date, T, at which control of maintenance decisions is transferred from developer to residents is contractually specified by the developer at the beginning of the development process. I examine contracts in which the developer must relinquish control at the date of or prior to his last lot sale, that is, r I N. The model of the shared financing contract is utilized to study the trajectory of quality of local public goods over time and to determine theoretically the optimal date to transfer control of maintenance decisions from developer to residents. A. Model

The model is solved by using backward optimization to obtain time consistent solutions. I assume each agent has perfect foresight. Thus, at a given time, s, the developer maximizes the present value of remaining profits taking as given q1 and R, for all t < s. In choosing q,, the developer takes into consideration the effect of qs on the maintenance levels chosen in the future by either himself or the residents. The developer is subject to the constraints that residents receive at least their reservation utility levels and satisfy their wealth constraints. The developer is also restricted by the nonnegativity constraint on maintenance. To determine the time consistent choices of the developer under the shared financing contract, the solution technique of backward optimization

386

KIM KNAPP

is applied to

subject to e(&)‘-‘[Qf

+xi(t)]

2 F,

i = l,...,N

(5b)

t=i q:.=

2 t=i

1

xi(t) (1

+

ry

+

Ri

+

v

7-l

f-i

t;i

i=l

7.*., N,

(5~) (54

where R, and ql are vectors of length N and T - 2, respectively, and Q, is chosen by the residents as described below. In the last two summations in (5a), KN - t)/N) cq, is the fraction of the maintenance costs paid by the developer at time t (where 2 I t < N), because N - t is the number of lots the developer still owns in period t. A resident votes for maintenance, q:j, and chooses consumption of all other goods to maximize her remaining intertemporal utility function, subject to her wealth constraint, taking as given any previous decisions and that future decisions will be similarly selected. Each resident also knows that the median voter determines the actual level of maintenance from the individua1 votes. Each resident pays a fixed share l/N of maintenance costs each period she is in the community. To determine the time consistent choices by residents, the solution technique of backward optimization is applied to I-*

[QP + xi@>]

(64

subject to

M/;.=E

xi(t)

t=l (1 + fyi

1 7-l + Ri + E tFi

(

1 r-i 1 + r 1 cst

t-t + ;

2

t=7(

&

1

Glf,

9r 2 0,

i=l

,...1 N, (6~)

DURABLE

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PUBLIC

GOOD

387

PROVISION

where q:l is a vector of infinite length, t 2 r for i I T, and t 2 i for i > T. 4, is chosen by the developer as previously described. To determine the time consistent solution to the shared financing contract, the general solution technique of backward optimization is used for each possible date of transfer of control. This is complicated by the nonnegativity constraint on maintenance. With the nonnegativity constraint on maintenance, the backward optimization process yields N - 1 necessary conditions for each of the 2N-’ potential solutions for each possible date of transfer of control. Each possible solution corresponds to a different combination of zero and positive maintenance in the various periods of the development process. For a two-, three-, and four-period development process, it has been shown that the only potential solutions that satisfy all of their necessary conditions are those in which maintenance is done each period. The solution procedure and the rigorous proofs of many of my findings are lengthy, and therefore are confined to Appendixes available from the author upon request. Within this paper, I rely largely on intuitive arguments to demonstrate the validity of my findings. B. Quality

of Local

Public Goods

The time consistent solution to the shared financing contract in which maintenance is done each period is for t < 7

(74

for t 2 7,

(7b)

where the superscript D of Q, indicates that the quality level was chosen by the developer and the superscript R indicates that the quality level was chosen by the residents. In the appendixes, I prove that (7) is the appropriate solution for a two-, three-, and four-period development process. Since for N = 2, N = 3, and N = 4 it has been proven analytically8 that maintenance would be done each period under a shared financing contract, I conjecture that maintenance will also be done during each period for longer development processes. The first-order condition for t < T, i.e., when the developer chooses maintenance, is aQ:-’

+

jy+yg+=(-;+l)c.

‘The exception is Case 4a which was eliminated by computer maintenance is done in all but the second period of a four-period

(8) simulations. development

In Case 4a, process.

388

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This first-order condition implies that under the shared financing contract the developer provides a quality of public goods at time t so that the marginal benefit to the t th entrant from Q, at time t, which is a&P-‘, plus the sum of the discounted future marginal benefits’ to the developer ((c(N - t - l)/NX(l - a)/(1 + r))), the tth entrant ((c/NX(l - 6)/ (1 + r))) and the t + 1st entrant ((c/NX(l - a)/(1 + r))) equal the sum of the current marginal cost of maintenance to the developer (c(N - t)/N) and the tth entrant (c/N). Thus, the developer ignores the benefits and costs to previous entrants when choosing current maintenance. In the time consistent solution to the shared financing contract, the developer does not consider benefits or costs to previous entrants when choosing current maintenance. The reason is that at a given time, previous entrants have already paid the developer for the right to use the local public goods for the life of the community and cannot be excluded by the developer from using them. Furthermore, side payments to the developer by previous entrants are ruled out because of the documentation problem explained at the end of Section II and/or incentives for the developer to cheat in the Nth period by accepting side payments and then not doing the agreed upon level of maintenance. The first-order condition for t 2 r, i.e., when the residents choose maintenance, is aQ;-’

+ ($)(

+$+)

= 5.

(9)

Equation (9) has the following economic interpretation: at time t, each resident chooses a quality of local public goods such that her current marginal benefit (aQ;-‘1 plus the discounted future marginal benefit to her in terms of decreased maintenance costs in period t + 1, ((1 - S)/ (1 + r)Xc/N), equals the marginal cost to her (c/N). Examining the time path of quality of local public goods under the shared financing contract, we can see from (7a) that quality of public goods increases over time when the developer chooses maintenance. The reason for this can be seen by examining (8). As t increases for 1 I t I N, QP increases because future marginal benefits considered by the developer fall by less than marginal costs considered by the developer. From (7b), it can be seen that the quality of local public goods chosen by the residents is autonomous. Thus, the steady-state quality of local public + r)/(6 + r))l/(l-O) begins when the resigoods, Q: = (c/Nu)‘/‘“-“((1 dents take control of maintenance decisions. Under the solution to the ‘The future due to (2,.

marginal

benefits

are the decrease

in their

maintenance

costs in period

t + 1

DURABLE

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389

PUBLIC GOOD PROVISION

Qt

PE

R

/

Qt.

Qt

R

Qt

I I I N

1

t

FIG. 1

shared financing contract given in (71, the quality of local public goods increases until date r, at which the steady state begins. This is illustrated in Fig. 1, which also illustrates that only for N = 2 is the first best solution attainable under the shared financing contract. Figure 2 illustrates the smoothed trajectory of maintenance of local public goods under the shared financing contract, which results in the quality trajectory shown in Fig. 1. Figure 2 shows that at t = r, the

2

T

FIG.

2

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residents choose a larger than steady-state level of maintenance to increase the quality level from Q,“_, to QF. The quality comparisons between the contractual solution and the Pareto-efficient solution are summarized in (lo).”

Q,‘” = Q,” Q,“> Q:” > Q: QF = Qr”

for t = 1

(104

for 1 < t < N

(lob)

for

t2N

(1Oc)

In the first period, the developer chooses the Pareto-efficient quality of local public goods because there are no previous entrants whose costs and benefits he ignores in his time consistent decision. However, for 1 < t < N, there are previous entrants whose costs and benefits the developer ignores in his time consistent decisions. This causes him to choose a less than Pareto-efficient quality of local public goods.” Due to the shared costs of maintenance, when the residents choose the level of maintenance, they choose a level such that the present value of marginal benefits to each resident is equal to c/N or where N times the present value of the marginal benefits to a single resident equals the total marginal cost, c. During the development process, when there are vacant lots and, therefore, fewer than N residents enjoying benefits from maintenance, the residents choose a greater than Pareto-efficient level of maintenance. Once the development process is complete and no empty lots remain, the residents choose the Pareto-efficient quality of local public goods. C. Transfer

of Control

of Maintenance

Decisions

This subsection focuses first on why it may be optimal from the developer’s perspective to relinquish control of maintenance decisions to residents under a shared financing contract. Then, the optimal date, r*, to transfer control of maintenance decisions to the residents of the condominium or homeowners’ association is derived. The effects of community size, length of the development process, and tastes on r* are examined. This is followed by comparisons of the theoretical results on T* to the “The residents “To show Q;” (;)‘I(“-“(

never choose Q, and the developer > QP for 1 < < N, I must show

t

l+&-.)

> (c(N

;;

for 1 < 6 < N which reduces to showing t > (N/(N multiplying, rearranging, and canceling the common which is clearly true for 1 < t < N.

never

+ l))“(a-‘)(

chooses

Q, for I 2 N

l+#-0)

- 1 + 1)) for 1 < t < N. By cross factor f - 1, this reduces to N > t

DURABLE

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391

PROVISION

recommendations in the industry literature and to existing contractual arrangements. The results in this section hold for solutions to the shared financing contract in which maintenance is done each period. Why might it be profitable for the developer to specify contractually at the beginning of the development process that he will relinquish control of maintenance decisions at date, r, where r I N and, consequently, the developer still has lots left to sell in the community? If in the time consistent shared financing solution, for f 2 r the residents would choose a quality of local public goods that is a Pareto improvement over that which the developer would choose, then it is profitable for the developer to specify contractually that control of maintenance decisions will be transferred to the residents at 7, even if r I N. Thus, by contractually specifying that residents will be allowed to choose maintenance according to specifications in the shared financing contract, the developer indirectly guarantees a quality of local public goods which gives him greater profits than if he retains control of maintenance decisions. When this occurs, the interesting theoretical result is that due to the inability to credibly precommit future Pareto-efficient, but time inconsistent, quality levels of local public goods, it can be profitable for the developer to relinquish control of maintenance decisions before all of his lots have been sold. That is, by reducing his number of decision variables, the developer can increase his profits. 1. N-period development process. I begin by assuming sufficient conditions for q1 > 0 for all t are satisfied and determine the optimal date to transfer control of maintenance decisions for an N-period development process. In terms of Fig. 1, I determine the optimal date to move from the trajectory Q,” to the trajectory Q,“. To determine this I must compare the developer’s profits under any two possible dates-for example, between transferring control of maintenance j periods before the end of the development process and j + 1 periods before the end of the development process. To make this comparison, first insert the time consistent solution given in (7) when r = N - j and r = N - j - 1 into the developer’s profit function given by (3). Then, eliminate the common terms of the comparison and rearrange to obtain the following result: T&T

= N - j - 1) > rsr(r

= N - j)

j + 1 - (j + 2)“(‘-l’( j + 1) a/(a-1)( 1 (a/( j + 2))) (1 -a) - (j + 2) 1)
iff N > rsF(7=N-j-

iff N <

j + 1 - (j + 2)“~“-I)( j + 1) (1 - a) - (j + 2) a/(a-l)(l - (a/( j + 2)))

('la)

('lb)

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for j = 0, 1,2, . . . , N - 3, where r&r = N - j) is the developer’s profits under the shared financing contract when control of maintenance decisions is transferred at date N - j. The comparison in (lla) and (lib) shows that for a given value of a and j, if N is sufficiently large, the developer should transfer control j + 1 periods before the end of the development process rather than waiting until j periods before the end of the development process. Curves in Fig. 3 show combinations of a and N for which (11) is satisfied with an equality and the developer is consequently indifferent between transferring control in period N - j or in period N - j - 1. The results shown in Fig. 3 on the optimal date CT*) to transfer control of maintenance decisions follow from (11) and the single-peakedness of the developer’s profit function with respect to 7. As N goes to infinity, the top curve in Fig. 3 asymptotically approaches a = 1. This implies that as N goes to infinity, it is profitable to transfer control prior to the Nth period for all values of the utility parameter except a = 1. Next r* is examined for a three- and a four-period development process for which sufficient conditions for q1 > 0 for all t have been shown to be satisfied.‘* From Fig. 3 note that it can be optimal to transfer control of maintenance decisions to residents prior to the last period of the development process only for development processes longer than three periods. That is, for short development processes (N I 3), the developer should retain control of maintenance decisions prior to the end of the development process. 2. Three- and four-period development process. In a community with a three-period development process, there are two possible dates to transfer control of maintenance decisions-the second period or the third period. To determine which of these alternatives the developer chooses to specify contractually in period one, the time consistent solutions and the corresponding levels of profit are determined for both T = 2 and T = 3. The developer’s profits are then compared for these two alternatives. The one that yields maximum profit is the optimal date from the developer’s perspective to transfer control of maintenance decisions to the residents. It turns out that in a three-period model the optimal date to transfer control of maintenance decisions is the third period. The formal proof of this is provided in the Appendixes. Although in the three-period model it I2 I do not discuss the two-period development process because with a two-period development process, the issue of the transfer of control of maintenance is irrelevant, given the restriction 7 5 N. The reason is that when N = 2 the developer and the residents would choose the same level of maintenance in the only period in which control of maintenance decisions is an issue, which is the second period.

DURABLE

0

5

LOCAL

10

15

PUBLIC

20

--/ -_.I. ,’ 2

GOOD PROVISION

25

30

N J

-0 --4 _-_ --

72

_ ___ __ _ , 2 _..-.. ,3

----_ -; ----

10 14

--.---7 ----

3 11 15

FIG. 3

is never optimal to transfer control of maintenance decisions prior to the end of the development process, this is not true for communities in which the development process lasts longer than three periods. In the case of a four-period development process, for example, it is often optimal for the developer to specify at the beginning of the development process that control of maintenance decisions will be transferred to the residents in the third period.r3 Why is it sometimes optimal to transfer control of maintenance decisions prior to the end of the development process in a four-period model but never in a three-period model? The developer does not choose the Pareto-efficient levels of maintenance prior to N because he ignores benefits and costs to previous entrants when deciding current maintenance. As development continues there are more previous entrants whose costs and benefits are ignored by the developer. The residents, on the other hand do not choose the Pareto-efficient maintenance levels prior to N because they do not take into account vacant lots in the community 13This finding is formally demonstrated in the available Appendixes, where the following results are proven. Transfer of control in the third period always dominates transfer of control in the second period. For 0 < a < 0.5, transfer of control in the third period dominates transfer in the fourth and r* = 3. But for 0.5 < e < 1, transfer in the fourth period dominates transfer in the third and r* = 4.

394

KIM

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when choosing current maintenance. This becomes less serious as development continues and there are fewer vacant lots. Thus, in a development process which lasts longer than three periods, it appears that for at least some values of the utility parameter the developer chooses maintenance during the initial periods when there are fewer previous entrants and the residents choose maintenance in later periods when there are fewer vacant lots. A three-period development process is not long enough for the efficiency loss due to the developer ignoring previous entrants in a given time period of the development process to exceed the efficiency loss due to the residents not considering vacant lots in the same time period. 3. Impact of community

size, length of development process and tastes.

Two parameters of the model determine the optimal date to transfer control of maintenance decisions: N and a. The first parameter, N, determines both the length of the development process and the number of lots in the community,‘4 that is, the size of the community. a is the parameter of the utility function. The following result is derived by Harl Ryder in the available Appendixes. For a = 0.5 and odd values of N, r* = 1.5 + OSN. For a = 0.5 and even values of N, the developer is indifferent between transferring control of maintenance decisions at T* = 1 + OSN and r* = 2 + 0.5N. However, both of these are preferred to any other date of transfer of control. For a > 0.5 and values of N such that UN is not an integer, asymptotically T* = aN rounded up. For a > 0.5 and values of N such that aN is an integer, asymptotically T* = 1 + aN.15 Table 116 and this result imply that the optimal date to transfer control of maintenance decisions is nondecreasing in both a and N. Figure 4 illustrates that for the points plotted for a = 0.6, a = 0.7, a = 0.8, and a = 0.9, increasing the length of the development process by 10 periods increases the optimal date to transfer control of maintenance decisions by 1Oa periods. I7 Figure 5 shows that r * is not linear between the points plotted in Fig. 4. Figures 4 and 5 also indicate that r* is nondecreasing in a and N. The results presented thus far in this section support the proposition that the optimal date to transfer control of maintenance, r*, is nonde14This follows from the assumption in Section I that there is a uniform inflow of one entrant per period. In Subsection 4, 1 alter this assumption and discuss the effects on T*. ISNote, this is an asymptotic result for a > 0.5, so for low values of N, some of the T* in Table 1 may not follow this rule. “The results in Table 1 were derived using computer simulations of a previous form of Eq. (11) by Harl Ryder and using the graphical approach in Fig. 3 by this author. Both approaches yield the same results. “However, as shown in the Appendixes, T* is not a linear function of a and N when a < 0.5.

DURABLE

LOCAL

Optimal N

a = 0.1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 8 8 9 9 9 9 10 10 10 10 11

a = 0.2

2 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 10 11 11 11 12

a = 0.3

2 3 3 4 4 5 5 5 6 6 6 7 7 8 8 8 9 9 9 10 10 10 11 11 12 12 12 13 13

PUBLIC

GOOD

TABLE 1 Date(s) to Transfer a = 0.4

2 3 3 4 4 5 5 6 6 6 7 7 8 8 9 9 9 10 10 11 11 12 12 12 13 13 14 14 15

a = 0.5

2 3 3,4 4 4,5 5 536 6 6,7 7 7,8 8 839 9 9,lO 10 10,ll 11 11,12 12 12,13 13 13,14 14 14,15 15 15,16 16 16,17

395

PROVISION

Control a = 0.6

2 3 4 4 5 5 6 6 7 8 8 9 9 10 11 11 12 12 13 14 14 15 15 16 16 17 18 18 19

a = 0.7

2 3 4 4 5 6 6 7 8 8 9 10 10 11 12 13 13 14 15 15 16 17 17 18 19 20 20 21 22

a = 0.8

a = 0.9

2 3 4 5 5 6 7 8 9 9 10 11 12 13 13 14 15 16 17 17 18 19 20 21 21 22 23 24 25

2 3 4 5 6 7 8 9 10 10 11 12 13 14 15 16 17 18 19 19 20 21 22 23 24 25 26 27 28

creasing in both N and a. To provide an explanation for this proposition, I begin by holding a fixed at a given value but allow N to increase by small enough amounts such that T* still lies between the same curves shown in Fig. 3. For such small increases in N, r* = N - k, where k is fixed so as N increases 7* increases. In Section C.2, an explanation is given for why the developer chooses maintenance during the initial periods of the development process and for why residents choose maintenance during the “later periods” of the development process. Given the intuition behind that result, for a given number (k + 1) of later periods, T* is nondecreasing in N because, as N increases sufficiently, the date corresponding to the beginning of the later periods of the development process increases, causing T* to also increase. This can best be seen by using an

396

KIM

KNAPP

z* 100.1

0 0=.9

90’

. O=.R

80‘

*0=.7

70.

- o=

6

I = .7N I = .6N

60’ 50. 40. 30. 20. 10 0 0

10

20

30

40

50

60

70

80

90

100

FIG. 4

example and referring to Fig. 3 and Table 1. For a = 0.8 and N = 6 through 10, the residents should be allowed to choose maintenance beginning one period (k = 1) before the end of the development process. Thus, in this region of parameter space, the later periods are the last two (k + 1) periods of the development process. For a = 0.8 when N = 6 the later periods begin at t = 5 and T* = 5, when N = 7 the later periods begin at t = 6 and 7* = 6, when N = 8 the later periods begin at t = 7 and T* = 7, and when N = 9 the later periods begin at t = 8 and T* = 8.

a= .9 a= .8 a= .7 a = .6 a= .4 a= .3 ::::

0

. ...,.,,.,...,,,,,,,.,,....,., 0

10 Length

20

of the Development FIG.

Process 5

30 (N)

DURABLE

LOCAL

PUBLIC

GOOD

397

PROVISION

Furthermore, for sufficiently large increases in N, the number of later periods (the k + 1 periods before the end of the development process in which the residents should choose maintenance) increases. For example, for a = 0.8 and for 6 I N 5 10 the later periods are the last two periods of the development process, but for a = 0.8 and 41 I N I 44 the later periods are the last nine periods of the development process. The intuition behind why the number of periods of the development process in which the residents should do maintenance increases with sufficiently large increases in N is as follows. For a = 0.8 and N = 10, consider allowing residents to choose maintenance during the last nine periods of the development process, i.e., T = N - 8 = 2. This would not be desirable because the residents in the second period would choose a very large quality of public goods, which would be Pareto-efficient only if there were 10 people in the community, but actually there are only 2 people in the community. That is, the residents ignore the eight empty lots in the community at t = 2, whereas at t = 2 the developer would only ignore the benefits and costs to one previous entrant. However, for a = 0.8 and N = 44 allowing residents to choose maintenance in the last nine periods of the development process is better than allowing the developer to choose during the last nine periods, because at t = 36 the residents only ignore eight empty lots, whereas at t = 36 the developer would ignore the costs and benefits to 35 previous entrants. Thus, for sufficiently large increases in N, the number of periods of the development process in which residents should choose maintenance increases.18 A numerical example from Fig. 3 illustrates the effect of increasing N on the optimal number of periods of the development process in which residents choose maintenance and on the optimal date to transfer control of maintenance decisions, T*, for a given value of a. For a = 0.8, increasing the length of the development process from 6 to 44 periods increases the optimal number of periods of the development process in which residents should choose maintenance from 2 to 9, which causes T* to increase from T* = N - 1 = 6 - 1 = 5 to T* = N - 8 = 44 - 8 = 36. This example and in a more general way Table 1 and Fig. 3 illustrate that sufficiently large increases in community size or length of development process (N), not only increase T*, but also increase the optimal number of periods before the end of the development process in which residents should control maintenance decisions. The intuition behind the result that T* is nondecreasing in a is as follows. As a increases, QF increases more than Q,” for any t where t < N. Therefore, the level of maintenance the residents would choose at IsFrom Fig. 3 this can be seen by choosing moves to the right across several curves.

a value

of a and increasing

N such that

T*

398

KIM

KNAPP

TABLE2 OptimalTransferofControl(N

= 2 through

N = 30)

N

a = 0.1

a = 0.2

a = 0.3

a = 0.4

a = 0.6

a = 0.1

a = 0.8

a = 0.9

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.500 0.667 0.500 0.600 0.500 0.429 0.500 0.444 0.400 0.455 0.417 0.385 0.357 0.400 0.375 0.353 0.333 0.368 0.350 0.333 0.364 0.348 0.333 0.320 0.346 0.333 0.321 0.310 0.333

0.500 0.667 0.500 0.600 0.500 0.429 0.500 0.444 0.400 0.455 0.417 0.385 0.429 0.400 0.375 0.412 0.389 0.368 0.400 0.381 0.364 0.391 0.375 0.360 0.346 0.370 0.357 0.345 0.367

0.500 0.667 0.500 0.600 0.500 0.571 0.500 0.444 0.500 0.455 0.417 0.462 0.429 0.467 0.438 0.412 0.444 0.421 0.400 0.429 0.409 0.391 0.417 0.400 0.423 0.407 0.393 0.414 0.400

0.500 0.667 0.500 0.600 0.500 0.571 0.500 0.556 0.500 0.455 0.500 0.462 0.500 0.467 0.500 0.471 0.444 0.474 0.450 0.476 0.455 0.478 0.458 0.440 0.462 0.444 0.464 0.448 0.467

0.500 0.667 0.750 0.600 0.667 0.571 0.625 0.556 0.600 0.636 0.583 0.615 0.571 0.600 0.625 0.588 0.611 0.579 0.600 0.619 0.591 0.609 0.583 0.600 0.577 0.593 0.607 0.586 0.600

0.500 0.667 0.750 0.600 0.667 0.714 0.625 0.667 0.700 0.636 0.667 0.692 0.643 0.667 0.688 0.706 0.667 0.684 0.700 0.667 0.682 0.696 0.667 0.680 0.692 0.704 0.679 0.690 0.700

0.500 0.667 0.750 0.800 0.667 0.714 0.750 0.778 0.800 0.727 0.750 0.769 0.786 0.800 0.750 0.765 0.778 0.789 0.800 0.762 0.773 0.783 0.792 0.800 0.769 0.778 0.786 0.793 0.800

0.500 0.667 0.750 0.900 0.833 0.857 0.875 0.889 0.900 0.818 0.833 0.846 0.857 0.867 0.875 0.882 0.889 0.895 0.900 0.857 0.864 0.870 0.875 0.880 0.885 0.889 0.893 0.897 0.900

any r is higher for higher values of a. Since q, increases as a does, the higher a is, the longer the developer must wait for enough people in the community to benefit from the large q, to make it profitable for him to transfer control of maintenance decisions to the residents. 4. Comparison of T* to actual and institutionally recommended r. Both the actual and recommended 7 typically specify that control of maintenance decisions will be transferred to residents after a given percentage of lots or housing units are sold. Therefore, to compare theoretical results to actual and recommended 7, I subtract one from the 7* given in Table 1 and divide that by N to get the values given in Table 2. For example, if N = 5 and a = 0.1 then Table 1 indicates that r* = 4. That is, residents should begin choosing maintenance in the fourth period of the five period

399

DURABLE LOCAL PUBLIC GOOD PROVISION TABLE3 OptimalTransfer ofControl(N = 10 through N = 300by 10) N

a = 0.1

a = 0.2

a = 0.3

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

0.400 0.350 0.333 0.300 0.280 0.283 0.271 0.263 0.256 0.260 0.255 0.250 0.246 0.243 0.240 0.238 0.235 0.233 0.232 0.230 0.229 3.227 0.226 0.225 0.224 0.223 0.222 0.218 0.217 0.217

0.400 0.400 0.367 0.350 0.320 0.317 0.314 0.313 0.300 0.300 0.300 0.292 0.292 0.286 0.287 0.281 0.282 0.283 0.279 0.280 0.276 0.277 0.274 0.275 0.272 0.273 0.270 0.271 0.269 0.267

0.500 0.400 0.400 0.375 0.380 0.367 0.357 0.363 0.356 0.350 0.355 0.350 0.346 0.343 0.347 0.344 0.341 0.339 0.342 0.340 0.338 0.336 0.335 0.338 0.336 0.335 0.333 0.332 0.334 0.333

a = 0.4

0.500 0.450 0.467 0.450 0.440 0.443 0.429 0.425 0.422 0.420 0.418 0.417 0.423 0.421 0.420 0.419 0.418 0.417 0.416 0.415 0.414 0.414 0.413 0.413 0.412 0.412 0.411 0.411 0.410 0.410

a = 0.6

a = 0.7

0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600

0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700 0.700

a = 0.8

0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800

a = 0.9

0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900

development process. Thus, residents should be choosing maintenance after (4 - 1)/5 = 3/5 or 60% of the lots have been sold. Tables 2 and 3, which are based on the procedure just outlined, provide estimates of the optimal transfer of control in terms of the proportion of lots sold. Developers typically transfer control to the residents of the community when between 50 and 80% of the lots have been sold.” However, the existing range has extended from a low of 25%20 to a high of 100%. The theoretical results in Tables 2 and 3 show that for 0.5 < a I 0.8 it is optimal to transfer control when between 50 and 80% of the development "Reference [12, p. 251. 2oReference[12, p. 251.

400

KIM

KNAPP

process is complete. However, for a < 0.5, it is only occasionally optimal to wait to transfer control until over 50% of the lots have been sold. As indicated by the 21.7% figure at the bottom of the second column of Table 3, the theoretical results support a lower bound for the optimal date of transfer that is slightly below the existing lower bound of 25%. The theoretical results support a slightly wider range of optimal transfer than the existing range of 25% to 1OO%.21 Since the typical control transfer when 50 to 80% of lots have been sold is the optimal range of r when the utility parameter lies between 0.5 and 0.8, one could speculate that developers have found that the true value of the utility parameter lies in this range. 22 However, I prefer to be agnostic about what the true value of the utility parameter might be until this issue has been examined empirically. In most of the industry literature, it is recommended that the developer retain control of maintenance decisions until at least 50% of the lots have been sold. This recommendation is supported by the theoretical results for a > 0.5. The most commonly cited recommendation, one endorsed by the Federal Housing Authority and the Veterans Administration, is that control be transferred when 75% of the development process is complete. This recommendation is supported by the theoretical results when a = 0.75. Another recommendation found in the industry literature,23 is that control be transferred after two-thirds of the lots have been sold, which is theoretically supported for a = 4. Wolfe’s argument that there should not be a problem with transferring control of maintenance decisions by the time 40 to 50% of the development process is complete is also supported by the theoretical results in some regions of parameter space.24 An obvious question that arises is the following: Does altering the assumption of a uniform flow of entrants affect the optimal date to transfer control of maintenance decisions from the developer to the residents? To begin addressing this question, several examples were constructed using the model given in this paper. These examples suggest that while small deviations from a uniform flow of entrants alter the time period of the optimal transfer of control of maintenance decisions, they do not affect the optimal transfer of control when expressed as a percentage “In the last period of the development process, the developer and residents would choose the same level of maintenance. Thus, rr(r = N) = rrfr = N + 11. Theoretical examples in Table 1 and Fig. 3 in which r* = N combined with this result imply that it can be optimal to transfer control after 100% of the lots have been sold, that is when T* = N + 1. However, under these circumstances, it is equally optimal to transfer control in the Nth period. The figures in Table 3 are all based on this latter alternative. 22This type of argument was presented by Vernon Henderson. ‘“Cornish [I, p. 3871. 24Wolfe [13, p. 361.

DURABLE

LOCAL

PUBLIC

GOOD PROVISION

401

of lots sold.25 These examples provide further support for the types of rules suggested in the industry literature and by government agencies, which specify that the transfer of control of maintenance will occur after a given percentage of the lots have been sold. IV. CONCLUSION In this paper, a shared financing contract for the provision of imperfectly durable local public goods in a growing community has been modeled. The time consistent trajectory of quality of local public goods under the shared financing contract has been determined and compared to the Pareto-efficient trajectory. Only in the first period of the development process, when there are no previous entrants, and from the last period of the development process onward, when there are no empty lots, does the shared financing contract yield the Pareto-efficient quality of local public goods. This result suggests the exploration of alternative financing rules and contracting instruments as a future extension. Typically it is profitable for the developer to contractually specify that he will relinquish control of maintenance decisions to the residents before all of his lots have been sold. The theoretical results of this paper support the possibility of an even wider range of optimal specifications of transfer of control, 7, than those which actually exist. The regions of parameter space in which the typical existing specification of 7 (50 to 80%) is theoretically optimal have been characterized. The theoretical analysis of the shared financing contract, even with its simplifying assumptions, implies that the optimal transfer of control of maintenance decisions from developer to residents may be more complicated than a single 40%, 50%, two-thirds, or 75% rule. However, in various regions of parameter space the theoretical results do support these rules which are recommended in the industry literature. The sensitivity of the optimal transfer of control to the utility-function parameter suggests that an empirical study, which would generate an estimate of this parameter, would be a useful extension of this work. ACKNOWLEDGMENTS Vernon Henderson, Harl Ryder, Gilbert Skillman, David Greenburg, and an anonymous referee provided valuable comments on various versions of this paper. One of the results, as cited in the paper, was derived by Harl Ryder, to whom I am grateful. 25Exceptions to this occurred when examples were constructed in which the flow of entrants was uniform, except for one period in which there was a deluge of entrants. In this case, the optimal date of transfer was substantially affected, even when expressed as a percentage of lots sold; the later the deluge occurred, the later was the optimal date to transfer control of maintenance decisions.

402

KIM

KNAPP

REFERENCES 1. M. J. Cornish, Ed., “The Homes Association Handbook,” Urban Land Institute, Washington, DC (1964). 2. C. J. Dowden, “Community Associations: A Guide for Public Officials,” Urban Land Institute and Community Associations Institute, Washington, DC (1980). 3. B. Ellickson, Jurisdictional fragmentation and residential choice, Amer. Econom. Rrl,. hoc., 61, 334-339 (1971). 4. B. W. Hamilton, Zoning and property taxation in a system of local governments, Urban Srud., 12, 205-21 I (1975). 5. J. V. Henderson, Community development: The effects of growth and uncertainty, Amer. Econom. Rw., 70, 894-910 (1980). 6. B. Klein, R. G. Crawford, and A. A. Alchian, Vertical integration, appropriable rents and the competitive contracting process, J. Law Econom., 297-326 (1978). 7. K. Knapp, “Contractual Arrangements for the Maintenance of Imperfectly Durable Local Public Goods,” unpublished Ph.D. dissertation, Brown University (1988). 8. F. E. Kydland and E. C. Prescott, Rules rather than discretion: The inconsistency of optimal plans, J. Polit. Econom., 85, 473-491. 9. M. McGuire, Group segregation and optimal jurisdictions, J. PO/~. Econom., 82, 112-132 (1974). 10. C. Tiebout, A pure theory of local public expenditures, J. Polit. Econom., 64, 416-424 (1956). I I. Urban Land Institute and Community Associations Institute, “Financial Management of 2nd ed., Urban Land Institute and Condominium and Homeowners’ Associations,” Community Associations Institute, Washington, DC (1985). 12. Urban Land Institute and Community Associations Institute, “Creating a Community Association,” Urban Land Institute and Community Associations Institute, Washington, DC (1986). 13. D. B. Wolfe, “Condominium and Homeowner Associations That Work: On Paper and In Action,” Urban Land Institute and Community Association Institute, Washington, DC (1978).