Homeowners association vs. leasehold

Abstract

Homeowners association (HOA) and leasehold (such as shopping mall) are two important types of urban institutions that have been developing very fast worldwide. The model developed in this paper is built on Hart–Moore model on cooperative vs. outside ownership by incorporating a distinctive feature of urban land use: the transaction and consumption of land is bundled with that of collective goods. The focus is placed on the ex post pricing efficiency of providing collective goods. The impacts of endogenous outside market and the capitalization of collective goods into land price are also discussed. Findings suggest that HOAs are likely to be located in more competitive markets such as the suburbs. Rich communities may prefer HOA while leasehold is more common for poor communities. Leasehold also becomes more efficient when the capitalization effect is weaker.

Appendix

Proposition 8

When t rises, HOA becomes less efficient and leasehold becomes more attractive if its effect on outsiders’ reservation price p ^* offsets that on the number of outsiders included.

Proof

If the transaction cost of trading and consuming the collective good in the outside market, t, increases, then the number of people who do not consume in the outside market will increase and the payoff to those who do consume in the outside market, w _i , will decrease, leading to a general increase of net payoff a _i , as specified in (1). In the case of leasehold, J in the first term of the RHS of (9) increases and the second term in the RHS of (9) decreases. So, the change of S ^o in (9) is not certain. But, S ^oo will certainly increase as specified in (10). Therefore, the outside owner will become more inclined to choose the second strategy, although as pointed out by Hart and Moore (1998), the second strategy usually does not yield the first-best because some insiders are excluded and no outsiders are included, i.e., there is unused capacity. In the case of HOA, P ^* decreases in (3) given the increase of t, resulting in the increase of $\hat{P}$ according to (11). Then, the first term in the RHS of (13) decreases. However, the change of the second term in the RHS of (13) is not certain since p ^* decreases but the number of outsiders increases. If it is assumed that the two effects offset each other, then on net S ^c will decrease. In summary, when t rises, leasehold becomes more efficient while HOA becomes less efficient.$\square$

Proposition 9

An increase in C _r is likely to make leasehold more efficient if it results in more insiders included, and make HOA less efficient if it results in a sharp reduction of outsiders.

Proof

When the price difference between the outside market and current compound, C _r , increases, the number of people who do not consume at the outside market will increase and the payoff to those who do consume at the outside market, w _i , will decrease, resulting in a general increase of net payoff a _i , as specified in (1). In the case of leasehold, the change of J in the first term on the RHS of (9) is uncertain, depending on the distribution of u _i . If J increases, then the first term also increases and the second term will probably increase given that the magnitude of C _r change is usually big. Hence, S ^o will increase if the number of insiders increases. Besides, the increase of J also means that S ^oo will increase as well given that a _i does not decline. In the case of HOA, P ^* increases in (3) given the rise of C _r , leading to the decline of $\hat{P}$, as specified in (11). Combined with the rise of a _i , the decrease of $\hat{P}$ means that the number of insiders will increase and, consequently, the first term on the RHS of (13) increases. However, the change of the second term in the RHS of (13) is uncertain because P ^* increases but the number of outsiders decreases. It is then accurate to say that S ^c will decrease if the number of outsiders is greatly reduced. In summary, a rise in C _r is likely to make leasehold more efficient if it results in more insiders, and make HOA less attractive in the case of a sharp reduction in outsiders.$\square$

Proposition 10

Assuming t small enough, if C ^o _r  = 0 and C ^c _r  > 0, S ^c becomes bigger than when C ^c _r  = 0 as long as u _i  ≥ π ^* + C ^c _r  + t holds for those insiders who quit from the group. Or vice versa if u _i  < π ^* + C ^c _r  + t. If C ^c _r is big enough, $\hat{P}$ will be very small and almost all insiders remain inside the HOA. Then the first-best can be achieved in the HOA as long as there are enough insiders whose valuation of the collective good is bigger than π ^* + C ^c _r .

Proof

Given I individuals, P ^* increases by C ^c _r since P ^*c = P ^*o + C ^c _r . From (11) we have $\hat{P}^{c} \le \hat{P}^{o}$. Besides, given (4) some individuals’ net payoff a _i may increase. Thus the number of individuals satisfying $a_{i} \ge \hat{P}$ will increase. From (1) if w _i  > 0 then u _i has to be big enough, since for small u _i their w _i is zero. So, if u _i  ≥ π ^* + C ^c _r  + t then S ^cc ≥ S ^co and HOA becomes more efficient.

Vice versa, if u _i  < π ^* + C ^c _r  + t, then S ^cc < S ^co and HOA become less efficient with higher C ^c _r .

If C ^c _r is big enough, then from (11) $\hat{P}$ will be very small. So, almost every insider will remain. Now obviously S ^c ≥ S ^oo. As to the comparison between S ^c and S ^o, let us consider two extreme situations. First, if all insiders’ valuation u _i is higher than π ^* + C ^c _r  + t then (16) becomes S ^c = I · (π ^* + C ^c _r  + t) − F and (14) becomes S ^o = I · (π ^* + C ^o _r  + t) − F. Obviously now S ^c ≥ S ^o. Second, if all insiders’ valuation is lower than π ^* + C ^o _r  + t then we have $S^{c} = \sum_{i - 1}^{I} u_{i} - F$ and S ^o = I · (π ^* + C ^o _r  − t) – F assuming t is small enough. Now S ^c < S ^o. So if the distribution of u _i is between these two extremes, the comparison between S ^c and S ^o largely depends on the tradeoff between “high-value” individuals and C ^c _r . If there are enough “high-value” individuals, HOA can achieve the first-best.$\square$

Proposition 11

Assuming t small enough, C ^o _r  = 0 and C ^c _r  < 0, S ^cc ≤ S ^co if u _i  ≥ π ^* + C ^c _r  + t for those insiders who remain inside the community and the total revenue from membership sale does not increase. If C ^c _r is low enough, $\hat{P}$ will be close to P ^* and the first-best can be achieved in HOA.

Proof

If C ^o _r  = 0 and C ^c _r  < 0, then we know from (3) that P ^* decreases.

$$P^{*c} = P^{*o} + C_{r}^{c}$$

P ^*c, P ^*o denote P ^* when C ^c _r equals non-zero and zero, respectively. From (11) we have $\hat{P}^{c} \ge \hat{P}^{o}$. Meanwhile, it can be seen in (4) some individuals’ net payoff a _i will decrease. Thus, the number of individuals satisfying $a_{i} \ge \hat{P}$ will decrease and the number of those who quit from the community (and whose vacant places are sold to outsiders) will increase. If u _i  ≥ π ^* + C ^c _r  + t for those who remain inside the community, then the first term on the RHS of (16) decreases. If the total revenue from selling membership, i.e., the second term, does not increase, then S ^cc ≤ S ^co and HOA becomes less efficient than when no price difference exists (here S ^cc and S ^co denote S ^c when C ^c _r equals non-zero and zero, respectively).

If C ^c _r is low enough, then $\hat{P}$ will increase so much as to be close to P ^* and many more insiders will quit from the community and be replaced by outsiders. Then the first-best can be achieved in HOA in the sense that there is no inefficient inclusion.$\square$

Proposition 12

Assuming t small enough, if C ^o _r  < 0 and C ^c _r  = 0, S ^o becomes smaller than when C ^o _r  = 0 and leasehold becomes less efficient. If C ^o _r is low enough, almost all insiders remain inside the community and the first-best can be achieved in leasehold.

Proof

Since C ^o _r  < 0 and C ^c _r  = 0, we know from (3) that P ^* decreases (relative to when C ^o _r  = 0) and, then, less “low-value” insiders will be replaced by outsiders in the first strategy of the outside owner in leasehold. The first term on the RHS of (14) increases while the second term decreases.

Let us consider two extreme scenarios. First, assume insiders are concentrated in the lower end of the range of u _i . Then, no insider is changed and the first term in (14) does not change while the second term decreases. So, S ^o decreases and S ^oo remains the same in the second strategy of the outside owner. In a word, leasehold becomes less efficient.

In the second scenario, assume that the insiders’ u _i are all close to P ^* when C ^o _r  = 0 and they remain inside the community. Then the first term in (14) increases while the second term becomes zero. Since for those people in the middle of the distribution π ^* − t ≥ u _i (assuming t small enough), the decrease of the second term is bigger than the increase of the first term. On net, S ^o decreases. But, S ^oo increases. However, S ^o ≥ S ^oo for whatever value of P set above P ^*, which is equal to π ^* − t when C ^o _r  = 0. In a word, leasehold becomes less efficient.

Since leasehold becomes less efficient in both two extreme scenarios, it is safe to say that leasehold becomes less efficient for scenarios between the two extremes.

If C ^o _r is low enough, then P ^* will become so low that almost all insiders will remain inside the community and none be replaced by outsiders. Then, S ^o = S ^oo (if P is set close to P ^*). The first-best can be achieved in leasehold.$\square$

Proposition 13

Assuming t small enough, if C ^o _r  > 0 and C ^c _r  = 0, especially if IC ^o _r  ≥ ∑  ^I _i=1 U _i , then leasehold becomes more efficient. If C ^o _r is big enough, almost all insiders will be replaced by outsiders and HOA becomes an attractive alternative.

Proof

If C ^o _r  > 0 and C ^c _r  = 0, it is known from (3) that P ^* increases (relative to when C ^o _r  = 0) and more “low-value” insiders will be replaced by outsiders in the first strategy of the outside owner. The first term on the RHS of (14) decreases while the second term increases.

Consider two extreme scenarios. First, if no insiders change position in the sense that all remaining ones are “high-value” ones, then the first term in (14) does not change and the second term increases due to the rise of P ^*. Hence, S ^o increases and S ^oo remains the same. Put another way, leasehold becomes more efficient in this scenario.

In the second scenario, if the insiders are all low-value ones and they are all replaced by outsiders due to the increase of P ^*, then the first term on the RHS of (14) becomes zero and the second term increases. If $IC_{r}^{o} \ge \sum_{i = 1}^{I} U_{i}$ , net effect will be an increase of S ^o. S ^oo is now equal to –F. Obviously, S ^o ≥ S ^oo.

Given the same results in the above two extreme scenarios, it is safe to conclude that if IC ^o _r  ≥ ∑  ^I _i=1 U _i , leasehold generally becomes more efficient than when C ^o _r  = 0.

If C ^o _r is big enough, P ^* will be so high that almost all insiders will be replaced by outsiders in the ownership structure of leasehold. Then there is maximum “inefficient exclusion” in leasehold and S ^o > S ^oo. Hence, HOA becomes an attractive alternative.$\square$

Proposition 14

When the capitalization effect decreases (β increases), leasehold becomes more efficient if the lower limit of P is a binding constraint. If the revenue from membership sale in HOA does not decrease, HOA may also become more efficient.

Proof

In the second-best world of HOA, since $P^{*} \ge \hat{P}$ as in (12), we know from (18) that

$$\left( {1 + \tfrac{1}{\beta }} \right)\pi^{*} - \frac{{\hat{P}}}{\beta } - t \ge \hat{P}$$

Rearranging terms yields

$$\hat{P} \le \pi^{*} - \frac{\beta }{1 + \beta }t$$

(32)

In the system of equations of (11) and (18), if β increases, then P ^* increases and $\hat{P}$ decreases. On the other hand, the number of people satisfying $a_{i} < \hat{P}$ decreases while those satisfying $a_{i} \ge \hat{P}$ increases. The first term on the RHS of (25) does not decrease. If the revenue from membership sale [the second term on the RHS of (25)] does not decrease due to rising P ^*, then S ^c increases.

In the second-best world of leasehold, the owner’s profit is JP. Since P ≥ P ^*, substituting (18) into this condition yields

$$P \ge \pi^{*} - \frac{\beta }{1 + \beta }\left( {P^{*} + t} \right)$$

If β increases, the lower limit of P decreases. If this is a binding constraint, or put another way, if some insiders are distributed around the lower limit of P, then P will also decrease, resulting in more insiders satisfying a _i ≥ P. a _i increases and, consequently, S ^oo increases.$\square$