Quantity premia in real property markets.
We often observe in historical big cities that some districts are planned with large lot sizes and wide roads whereas other districts are developed without planning with small lot sizes and narrow roads. These districts are located within the same city, constituting a mosaic spatial pattern. In the conventional theory of urban economics (Alonso 1964), lot size is a reciprocal of population density, and should exhibit the geographical regularity of a monotonic increase with distance from the central business district (CBD). In other words, in a perfect land market, a mixture of different lot sizes at the same distance from the CBD is not supposed to occur. In reality, however, it often occurs because a change in lot size is costly. These costs are incurred by land assembly and subdivision. Existence of these costs hinders arbitrage between lots with different sizes, leading to variations in the price of land per area.
The major purpose of this paper is to examine the systematic variations of the price of land per area with respect to lot size. In doing this, we may evaluate the desirability of the districts from a viewpoint of social welfare. Social welfare is evaluated by the aggregate land price of the whole district since these benefits and costs are fully capitalized by the price of land under certain conditions according to the hedonic theory.
Both supply side and demand side conditions determine the land price. The supply side determinants are the costs of land improvements, road construction, and so forth. In general, since these costs per area are lower or constant for larger lots, the unit land price may be expected to decrease as the lot size gets large. The demand side conditions are the environmental externalities, urban design, or various aesthetic aspects which are incorporated in utility functions of consumers. Unlike the supply side, the demand side conditions may increase the unit land price because larger lot size means lower household density, and hence ensures a better environment.(1) Thus, the overall effects on the unit land price are ambiguous.
Quantity premia prevail in the land market if larger lots are proportionately more expensive. In this case, developers would try to maximize lot size, and never subdivide any lot. They would assemble as many contiguous lots as possible, resulting in lower household density. On the other hand, quantity discounts prevail if the unit land price decreases with lot size. Developers would subdivide the lots until they reach the minimum marketable size resulting in higher household density.
Whether quantity premia or discounts prevail depends on the presence of land assembly or subdivision costs (Colwell and Sirmens 1993). In studies of many American cities, quantity discounts are extensively observed. The urbanizations in the studies include the township of Ramapo (White 1988), Champaign-Urbana (Colwell and Scheu 1989), Detroit (Kowalski and Paraskevopoulos 1990), nine U.S. cities (Holway and Burby 1990), and the Southern California region (Brownstone and De Vany 1991). These studies suggest that land subdivision costs are much greater than land assembly costs, and that higher household density is more desirable. However, as will be shown later, this is not true in Japanese big cities.
The organization of the paper is as follows. In Section II, we formulate a developer's problem of determining lot sizes. In Section III, an attempt is made to solve that problem by conducting regression analysis of land price data in the Osaka metropolitan area. In Section IV, we compare the results obtained in our analysis of the land market with those appearing in the literature on condominium markets. In Section V, we further discuss the results in relation to theories of nonlinear pricing in a differentiated product market. Section VI concludes the paper.
II. OPTIMIZATION OF LOT SIZES
We assume that landowners or developers will subdivide or assemble lots so that the aggregate value of their lots is maximized if the costs of lot size change are outweighed by the increase in the land value in the absence of institutional constraints. Of so, the lot size should exhibit empirical regularities in accordance with the (time) distance from the CBD. In Japanese big cities with a long history, however, the household density (a reciprocal of lot size) distribution does not exhibit geographic regularities. It varies even within a small district according to various local conditions. It is commonly observed in Japanese cities that high-rise apartment houses are located next to detached houses. Such a housing mixture is considered to be a historical consequence of cumulative development processes under myopic foresight by developers and landowners when housing is durable (Harrison and Kain 1974). Given uncertain factors in the future land market conditions, perfect foresight behavior is not always possible, and the result has been uneven sizes of land lots within each district.
Admittedly, cities with a long development history are quite different from the cities envisioned in pure theory of urban economics. However, it is wrong to claim that the difference between reality and theory is simply ascribed to the myopic foresight behavior of developers and landowners together with the durability of buildings. Although developers do not perfectly foresee the future, they are not so totally myopic insofar as the amount of land is scarce and the price of land is very high. There must exist other important factors determining the lot size, which we seek in the subsequent analysis of this paper.
In deciding the optimal lot size, the profit-maximizing developers would carefully observe the market price function for land:
[Mathematical Expression Omitted],
where p is the price of a lot per unit area, s is the lot size, and [Mathematical Expression Omitted] is a vector of other attributes such as access to the CBD, amenity, local environments, and so forth. Provided that the specification of  is correct, the developers can compute the second derivative of the land price ([[Delta].sup.2]P/[Delta][s.sup.2], where P [equivalent to] ps) while controlling the set of locational attributes [Mathematical Expression Omitted] constant.
If the sign of [[Delta].sup.2]P/[Delta][s.sup.2] is shown to be positive, we conclude that quantity premia are prevalent in the land market. If the sign of [[Delta].sup.2]/[Delta][s.sup.2] is negative, we conclude quantity discounts. Developers and landowners will assemble or subdivide the lots until they reach an optimum lot size.(2) In either case, ignoring the indivisibility problem and institutional constraints, the currently existing lot size must be a maximizer of unit and total land price (Asami 1995). An empirical test of the sign of [[Delta].sup.2]P/[Delta][s.sup.2] is conducted in the next section.
III. EMPIRICAL ANALYSIS OF THE LAND MARKET
An empirical analysis was undertaken to see whether equation  would reveal quantity discounts or quantity premia prevail. Since the lot size s is considered to be the quantity variable in land markets, special attention is paid to the impact of the lot size on the unit land price p.
Because there is no reliable theory to specify the functional form of f in , we tried several kinds of functional forms and decided to use the following semi-logarithm form:(3)
log [p.sub.i] = [a.sub.0] + [a.sub.1][s.sub.i] + [summation of] [a.sub.k][x.sub.ki] where k = 2 to K + [e.sub.i]
for i = 1, ..., I, 
where [a.sub.k] (k = 0, ..., K) are parameters to be estimated statistically, K is the number of independent variables, [x.sub.k]'s are the independent variables, e is the residual, I is the number of observations, and subscript i is added to each variable for the sake of statistical analysis.
Since [[Delta].sup.2]P/[Delta][s.sup.2] = (2 + [a.sub.1]s)[a.sub.1]p, we test statistical significance of the sign of [a.sub.1] and conclude that:(4)
if [a.sub.1] [less than] 0, quantity discounts prevail in the land market;
if [a.sub.1] [greater than] 0, quantity premia prevail in the land market.
We used the data set from koji chika, or officially posted land prices, issued annually by the National Land Agency (1993). They are assessed values based upon recent transaction prices by real estate appraisers who are commissioned by the National Land Agency. They are collected approximately every 1 [km.sup.2] all over Japan. Collected samples are "standard" in that each shape is rectangular and each size is within a narrow range (about 100 to 300 [m.sup.2]). Such controlled data are suitable for empirical analysis.
We confined our analysis to residential districts along six major commuter railroads starting from the CBD of Osaka: Keihan Line, JR Katamachi Line, JR Hanwa Line, Nankai Line, Kintetsu Osaka Line, and Kintetsu Nara Line. A schematic map is shown in Figure 1. The Osaka metropolitan area is the second largest metropolitan area in Japan with a population of more than 10 million people.
The data set contains observations in 1993 on the unit price of lots p (in thousand yen per [m.sup.2]) and the site-specific data such as the lot size s (in [m.sup.2]) and the distance from the lot to the nearest railroad station [x.sub.2] (in m). Data on the total minutes from the nearest station to the CBD of Osaka [x.sub.3] (in minutes) are calculated by the minutes needed by using the fastest commuter trains with train changing time of five minutes, if any.
Summary statistics for the data set are given in Table 1. We know from the statistics that each variable is within a certain range since each standard deviation is not very large. In particular, the standard deviation of the lot size is small enough. This is due to the sampling of "standard" lots as mentioned above. It should be noted that the average value of unit land prices is much higher and the average lot size is much smaller than the North American standard. To put it differently, the Osaka metropolitan area is much more densely inhabited and its land market is of paramount importance.
Since the Keihan Line and the JR Katamachi Line are close and parallel with each other, we ran an OLS regression together using a dummy variable for one line. The same is true for the JR Hanwa Line and the [TABULAR DATA FOR TABLE 1 OMITTED] Nankai Line, and for the Kintetsu Osaka Line and the Kintetsu Nara Line. We also ran an OLS regression using all six lines using five dummy variables.
The results are summarized in Table 2.(5) Generally speaking, there is a good fit. The values of the regression coefficients in the first three cases are similar, and the values of [R.sup.2] (between log p and its estimate, and [TABULAR DATA FOR TABLE 2 OMITTED] between p and its estimate) are high. The t-statistics of all regression coefficients are so large that they are significantly different from zero at the 1 percent level except one.
In addition, we computed the values of skewness [[Gamma].sub.1] and the values of kurtosis [[Gamma].sub.2] in order to check the normality of residual distributions [e.sub.i]. We found that there is no reason to believe the nonnormality of residual distributions at the 5 percent level. The only exception is the skewness of [e.sub.i] in the case of "all six lines." Presumably, it is attributable to the aggregation of six lines whose land markets have different characteristics.
Putting these results together, we are inclined to justify not only our specification of equation , but also our choice of the six lines. In other words, we may say that (the logarithm of) the land price along these lines can be reasonably explained only by the two variables: the access to the CBD and the lot size.
Observing the sign of [a.sub.1] (the first row in Table 2), we recognize that it is positive and statistically significant at the 1 percent level (Keihan-Katamachi Lines, Kintetsu Osaka-Nara Lines, and all six lines) or at the 5 percent level (Hanwa-Nankai Lines). These results do support the quantity premia hypothesis in the land market.(6) This exhibits a clear contrast to various markets of consumption goods, where quantity discounts are common practices: for example, apples are often sold for one for 100 yen and three for 200 yen. We discuss the reasons for the difference in the following sections.
Before the discussion, we should mention two points. The first point is an econometric problem of multicollinearity between the independent variables in equation . According to the theory of urban economics, both the land price p and the lot size s are simultaneously determined mainly by the distance from the CBD.(7) If this is true, s would be correlated with the distance to the station [x.sub.2] and/or the time distance to the CBD [x.sub.3], which would invalidate our specification of , and hence the present analysis.
To check this problem, we computed the correlation coefficients between the lot size s and the total time from the lot to the CBD ([a.sub.2]/[a.sub.3])[x.sub.2] + [x.sub.3], using the estimates of [a.sub.2] and [a.sub.3] in Table 2. The values of respective correlation coefficients are: 0.237 in Keihan-Katamachi Lines, 0.167 in Hanwa-Nankai Lines, and 0.093 in Kintetsu Osaka-Nara Lines. The corresponding t-statistics to each correlation coefficient are: 2.56, 2.17, and 1.19, respectively. The first two values are statistically significant at the 5 percent level, but not at the 1 percent level, whereas the last one is not significant at the 5 percent level. We thus confirm that the above multicollinearity problem does not arise in our data set. In other words, the data set shows that given a distance from the CBD, large lots and small lots coexist in the Osaka metropolitan area. This is in accord with our casual observations that in Japanese cities wealthy households are located near the poor in many districts. However, the main reason for such weak correlations is the data sampling method. As mentioned above, lots are sampled so that their sizes are controlled to be within a narrow range whereas the access to the CBD has variations. That is, whereas the lot size is generally increasing (and hence, population density is decreasing) in the access to the CBD of the Osaka metropolitan area, our controlled data set does not exhibit such an empirical regularity, and hence the multicollinearity problem does not arise.
The second point to note is the difference in land market conditions between the United States and Japan. The existing literature indicate that quantity discounts dominate in most American cities whereas our result from the Osaka metropolitan area indicates that quantity premia are dominant. Needless to say, there are several different market conditions between the two countries, such as differences in population sizes, average lot sizes, city locations, and development histories. In particular, the average price of unit land in the Osaka metropolitan area is about 300 times as high as that in Ramapo while the average lot size in Ramapo is about 40 times as large as that in the Osaka metropolitan area.
These comparisons would indicate that the population density in the Osaka metropolitan area (as well as many other large metropolitan areas in Japan) is such that there is scarcely any vacant land left. Most importantly to the purpose of this paper is the implication that land assembly is very difficult. On the other hand, in Ramapo (and other American cities), land is not so scarce, but there are other binding constraints such as large lot zoning, which restricts subdivision of land. Consequently, the land market conditions between Japan and the United States are not directly comparable.(8)
What we are dealing with in this paper is differentiated scarcity. Large lots are more scarce than small ones in Japanese big cities, the reasons for which will be clarified in the subsequent sections.
IV. IRREVERSIBILITY IN REAL PROPERTY
Let us contrast the economic implications of the above results in the land market with observations in commodity markets. Spence (1977, 17) states the following: "Quantity discounts tend to be undone by resale, where resale is possible. Quantity premia are undone by repeat purchasing, where that activity is feasible and not too costly." In most commodity markets, it is not easy for an individual consumer to purchase a commodity in large quantities and then resell it to other consumers because the resale would need to be coordinated. However, it is rather easy for the same consumer to repeatedly purchase the commodity in small quantities. It thus follows that in many commodity markets quantity discounts are enforceable, but quantity premia are not.
Alternatively, we may say that quantity discounts are a direct result of the first-degree or second-degree price discrimination. Under any downward demand function, the unit price is a decreasing function of the quantity sold. This function is analogous to quantity discounts. Furthermore, economies of scale in retailing and production are another conceivable reason for quantity discounts since we expect that lower costs of distribution and production are associated with a lower unit price.
In the land market of Osaka, however, we have empirically shown that the reverse is true. Unlike ordinary commodities, repeat purchase of contiguous lots is prohibitively difficult, whereas resale by subdividing a large lot is easy because it depends only on one landowner's decision. Nonetheless, no lot would be subdivided since the unit land price is positively associated with lot size in the observed range of lot size.
On the other hand, landowners have an incentive to assemble their lots if possible because the market price of unit land becomes greater. Large lots go for more because they retain an option value of splitting them or constructing high-rise buildings. In fact, intermediary business of land assembly is sometimes conducted by developers especially when large lots for high-rise buildings are highly demanded in the business districts.
Nevertheless, land assembly is rarely done (especially in residential districts) because intermediaries have to mediate property rights between landowners in assembling lots. If one of the landowners is simply uninterested in an increase in the value of his property, then land assembly itself is not put into practice. Or, a landowner may contrive to maximize his return by not selling his lot until all the other landowners bargain away their lots to an intermediary developer (Eckart 1985). By doing so, the landowner can exploit the developer's super-normal profit from the project because in the negotiation he can gain an advantage over the developer who has already invested to purchase lots in the vicinity. Existence of such transaction costs is favorable to the landowner while unfavorable to the developer, resulting in an impediment to land assembly.
We can therefore say that lot size is characterized by irreversibility: once it is subdivided and sold to different people, it is very difficult to be assembled as it was before because of the intermediary costs. That is why the quantity premia are enforceable, but quantity discounts are not, in the densely inhabited districts of the Japanese land market.
In the case of condominium apartment markets, however, both the quantity discounts and the quantity premia are enforceable as tools for nonlinear pricing. The floor area of an apartment is usually used as a proxy for lot size s, i.e., the quantity variable. In building a condominium, a developer would determine the floor area of each apartment so that the total asset price of the condominium is maximized. Since it is newly built, the developer can freely determine the floor area of each apartment unit and its price without caring for any sociopolitical reasons inherent in the land markets mentioned above. However, once it is constructed, both assembly and subdivision of apartments are not possible. That is, unlike the land market, the irreversibility takes place in both directions in the apartment market, and hence both quantity discounts and quantity premia are enforceable.
Several recent empirical analyses of condominium markets in Japanese big cities support the quantity premia hypothesis. For example, Kato (1988) ran a hedonic regression of condominium prices using data of Saitama and Chiba prefectures in 1986, and Arima (1993) and Nakamura (1993) also ran a hedonic regression of condomium prices using data of the Osaka metropolitan area in 1986 and 1990. Although their research objectives are different from ours, we can read from their results that quantity premia prevail in the condominium apartment markets: larger apartments are proportionately more expensive.
It is worth noting that unlike ordinary commodities, the pricing of apartments is not building cost-based. If it were cost-based, then with increasing returns to scale in apartment construction the unit floor price would become lower for larger apartment houses and indicate quantity discounts. In reality, however, we have seen that quantity premia prevails in these apartment markets of dense inhabitation as well. The reason for the quantity premia in such property markets is not ascribed to the supply side conditions of scale economies in production. Nor is it ascribed to the enforceability conditions. We must seek an alternative reason.
V. NONLINEAR PRICING THEORY IN A DIFFERENTIATED MARKET
Pricing is often done strategically under imperfect competition. Imagine a situation where a monopolistic developer has many lots with different sizes. Then, this may be modeled by the theory of nonlinear pricing developed by Spence (1977) and Mussa and Rosen (1978), where a developer offers differentiated products (i.e., a variety of lot sizes) with a price-quantity schedule exercising price discrimination via quantity discounts/premia. Extending their analysis further, Maskin and Riley (1984) rigorously showed that quantity discounts are the profit-maximizing strategy in a monopolized market.
The nonlinear pricing model assumes that consumers are heterogeneous in taste [Theta]. The developer cannot observe their taste, but knows its variation, which is a uniform distribution. Assume that each consumer's utility function is additively separable. A typical one is of the form:
U(z, s) = z + [Theta]u(s), 
where z is the outside numeraire good, [Theta] is a positive parameter distributed uniformly, and u is an increasing function of lot size s. Given an income constraint, each consumer maximizes this utility. The profit-maximizing developer with limited information on the consumers' taste [Theta] offers a price-quantity schedule. Tirole (1988) showed that the price-quantity schedule is proportional to the utility-quantity function. That is, if the marginal utility is a decreasing function of quantity, then the developer exercises quantity discounts; if the marginal utility is an increasing function of quantity, then the developer exercises quantity premia. It is the consumers' demand that determines whether quantity discounts or quantity premia prevail in the case of a monopolized market.
Since quantity premia prevail in the real property markets of Japanese big cities, the marginal utility must be increasing, which is not in accord with conventional theory of utility. So, some of the model assumptions do not correspond with the actual situations in Japanese real property markets. Among the assumptions, it seems inappropriate to assume the monopoly in the real property markets. Although developers are usually local monopolists competing with their neighboring firms, their property is not perfectly protected from an influence of other real property nearby. Real property is quantitatively differentiated in that different sizes mean different goods so far as a change in their sizes is prohibitively costly. Here the differentiation is not horizontal but vertical because real property larger in size is normally preferred unanimously.
On the basis of these considerations, let us allow free entry of firms so that the market structure will be oligopolistic. In equation , suppose u is linear in s, then each indifference curve of each consumer becomes a straight line in (s, P(s)) plane, where P(s) [equivalent] p(s)s is the total expenditure on a lot. In Figure 2, the line linking A and B is an indifference curve of a consumer with taste [[Theta].sub.AB], which is represented by its slope. This consumer is indifferent between A and B in purchasing a lot. Point A represents firm A selling a lot with size [s.sub.A] and price [P.sub.A], and point B represents firm B selling a lot with size [s.sub.B] and price [P.sub.B]. It should be noted that whereas the monopolist offers a price-quantity schedule in the above model, each oligopolist chooses a single quantity with a single price.(9) It is easily confirmed that consumers with a taste parameter smaller than [[Theta].sub.AB] will self-select a lot from A while consumers with a taste parameter larger than [[Theta].sub.AB] will self-select a lot from B. The former group of consumers put less weight on s as compared to z (since is [Theta] small), whereas the reverse is true for the latter group.
We know from Figure 2 that offers from northwest of the line AB cannot secure any consumer since they give a lower utility than either A or B to any consumer with positive [Theta]. On the other hand, offers of southeast of the line AB obtain some customers. For example, suppose firm C enters the land market, offering a lot with size [s.sub.C] and price [P.sub.C]. Then, consumers with a taste parameter between [[Theta].sub.AC] and [[Theta].sub.CB] would purchase a lot from C (as a result, firms A and B lose some customers).
Likewise, further entry of firms would take place offering property southeast of line AC or CB until profits of new entrants become negative. Therefore, the overall price schedule will be increasing and convex, i.e., the quantity premia.(10) These results contrast to those of a monopolistic market.(11) The quantity premia result is also true in case that u(s) is increasing and convex. Hence, we may conclude that the quantity premia in the real property markets in densely inhabited big cities in Japan are attributed to the oligopolistic competition between developers selling lots and apartments which are vertically differentiated under the nondecreasing marginal utility in space consumption, and attributed to the irreversibility in assembling lots.
VI. CONCLUDING REMARKS
According to the theory of nonlinear pricing by Maskin and Riley (1984), quantity discounts are shown to be optimal in a monopolized market. We tested the theory by using the data set of residential lots in the Osaka metropolitan area in 1993. It revealed that it is not quantity discounts, but quantity premia that prevail in the land market. We concluded that this is due to the irreversibility in changing lot size. Once a lot is subdivided and sold to different people, it is prohibitively difficult to repurchase and reassemble because of the existence of intermediary costs. In addition, the existence of quantity premia can be explained by the setting of an oligopolistic market with vertically differentiated products under non-decreasing marginal utility of lot size.
The existence of quantity premia means higher land values per unit for larger lots. Thus, developers and landowners will maximize their lot sizes by assembling contiguous lots if possible. This leads to lower density of population, providing good exterior environments and relieving traffic congestion, and so on.
However, such a desirable state may not be attained in a free market economy due to the following two reasons. First, developers tend to sacrifice open space such as roads and parks if allowed. Being characterized as a (local) public good, such open space is likely to be underprovided due to free riders in the absence of city planning such as land use readjustment projects. Second is the nature of irreversibility in changing lot sizes. Without any land use regulations like large lot zoning, some landowners may sell a lot by the piece just for personal convenience. As time goes on, lots are subdivided piecemeal, which decreases the overall land value of the district and prevents efficient use of land.
PROOF OF CONVEXITY OF THE PRICE SCHEDULE
Following Anderson, de Palma and Thisse (1992, subsection 8.3.2), we show that the price schedule is convex.
Suppose consumers' taste [Theta] is distributed uniformly over a positive interval [[[Theta].sub.0], [[Theta].sub.j]], and the relative value of [[Theta].sub.J]/[[Theta].sub.0] is given such that J firms (developers, or sellers of their lots) are "viable" in the market. Consumers self-select among goods [s.sub.j] produced by firm j (= 1, ..., J). The utility function is the same as  except that subscript j is added to s, and the budget constraint is
y = z + P(s). [A1]
Firms play a noncooperative three-stage game. In the first stage, firms make an entry decision. In the second, they choose quantity [s.sub.j], and finally select price [P.sub.j]. Without loss of generality, let [s.sub.j] [less than or equal to] [s.sub.j] + 1 for all j = 1, ..., J - 1
The market boundary [Mathematical Expression Omitted] is determined by the condition that a marginal consumer [Mathematical Expression Omitted] is indifferent between [s.sub.j] and [s.sub.j + 1]. Namely, using  and [A1],
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Note that [s.sub.j] [not equal to] [s.sub.j + 1] for all j = 1, ..., J - 1 since otherwise firms are involved in the Bertrand price competition ending up with zero profit. Let us solve the last stage game to obtain subgame perfect equilibrium. Each profit-maximizing firm j (= 2, ..., J - 1) optimizes its price as to
maximize [Mathematical Expression Omitted]
with respect to [P.sub.j].
Computing the first-order conditions yields
[Mathematical Expression Omitted].
From the definition of [Mathematical Expression Omitted], we have
[P.sub.j + 1] - [P.sub.j]/u([s.sub.j + 1]) - u([s.sub.j]) [greater than] [P.sub.j] - [P.sub.j - 1]/u([s.sub.j]) - u([s.sub.j - 1])
for all j = 2, ..., J - 1, [A2]
given the choice of each [s.sub.j].
Since each side of [A2] represents the slope of P(u(s)), it becomes larger as u(s) gets larger. Hence, if u(s) is linearly increasing in s, then the slope of P(s) becomes steeper as s gets larger, and so P(s) is convex.
1 However, very large lot size may lead to high household density due to high-rise apartments.
2 To be more precise, the developers would maximize the aggregate value of land, which is the product of the unit land price p and the total area of the lots in the district. The latter varies due to the width of roads and the size of public space such as a park in the district.
Given the fixed amount of land in the district, the total area of the lots is greater for narrower roads and smaller parks. However, smaller open space would decrease the overall environmental value, and hence lower the unit land price. It is therefore ambiguous if the aggregate land price is higher. Such a microscale analysis is beyond the scope of this paper.
3 Note however that if one of the independent variables is the distance from the CBD, equation  implies that p is a negative exponential function of the distance. This is consistent with the theoretical and empirical analysis by Muth (1969).
4 Notice that 2 + [a.sub.1]s [greater than] 0 because [Delta]P/[Delta]s = (1 + [a.sub.2]s)p [greater than] 0.
5 Table 2 was developed after trying various kinds of explanatory variables such as area classification of zoning regulations, ratio of frontage to depth, width of a front road, levels of infrastructure, several dummy variables, and so on. In particular, contrary to our expectations, zoning regulations were not shown to affect the land price, implying that the land price in the Osaka metropolitan area would be determined mainly by economic factors, or there are no substantial zoning regulations.
6 This finding is also supported by regression analyses by Edmonds (1985) using koji chika data of the Tokyo metropolitan area in 1970 and 1975, and by Suzaki (1991) using koji chika data of the Kanagawa prefecture in 1986-88.
7 Scotchmer (1985) argues that the lot size is endogenously determined by the distance in the context of hedonic theory.
8 Or, one might generalize that as the average lot size increases, the unit land price first increases and then declines. The former may be applicable to densely inhabited big cities while the latter may be applied to sparsely inhabited small cities.
Adding a quadratic term of [s.sup.2] to the case of "all six lines," we also ran a regression using the same data set. The estimates and t-values of all coefficients were very similar to those of the last column in Table 2. The estimate of the coefficient of the additional term [s.sup.2] was -2.61 x [10.sup.-7] and its t-value was -0.632. Although statistically insignificant, the estimate is negative. This implies that as the lot size increases, the unit land price first rises and then falls in the Osaka metropolitan area.
9 Such a "bunching" of consumers was theoretically justified by Champsaur and Rochet (1990), who proved that each duopolist produces a unique quality (quantity) even if each is allowed to produce a series of quality (quantity).
10 See the Appendix for a proof of convexity.
11 The case of competitive markets may be analyzed by making the number of firms large (although competitive situations are unlikely to occur due to the existence of local monopoly in differentiated markets). For a sufficiently large number of firms, we can show that the profit of each firm goes down to zero, and linear pricing takes place in the competitive markets.
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Takatoshi Tabuchi is with the Faculty of Economics, Kyoto University. An earlier version of this paper was presented at the Workshops of Osaka University and Kyoto University and at the 1993 ARSC Meeting held at Tohoku University. The author is grateful to M. Fujita, K. Hagiwara, T. Hatta, C. Horioka, H. Ohta, G. Patchell, H. Yamada, and two anonymous referees for valuable comments and suggestions.
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|Title Annotation:||includes appendix|
|Date:||May 1, 1996|
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