Is higher-quality land developed earlier?

Abstract

Is higher-quality land developed earlier? To answer this question, the paper applies comparative static analysis to the Arnott–Lewis model of the transition of land from agricultural to urban use. It is shown that (i) an increase in agricultural fertility increases structural density and delays development; (ii) a decrease in land preparation cost reduces structural density and hastens development; and (iii) both an increase in amenities and a decrease in structure construction costs normally hasten but in anomalous cases can delay development.

Keywords

Land use patterns housing supply and markets housing demand

Introduction

In a typical introductory course in land economics, a distinction is made between the Ricardian model of land use, rent, and value and the corresponding von Thünen model. The Ricardian model focuses on the intrinsic properties of land, including fertility and natural amenities, which are aspects of “land quality.” The von Thünen model, in contrast, focuses on differences in accessibility. Courses in urban economics focus on the monocentric city model, which extends the von Thünen model of agricultural land use and land rent to the urban context. The monocentric city model has been analyzed exhaustively, while Ricardian differences feature prominently in hedonic valuation analysis. Remarkably, however, it appears that Ricardian differences in land have not been systematically incorporated into the monocentric city model. The aim of this paper is to at least partially remedy this oversight by addressing the following question: “Is higher-quality land developed (in urban use) earlier?” The answer to this question depends on what is meant by land quality. We consider four dimensions of quality: fertility, natural amenities, land preparation costs (so that servicing land is cheaper on higher-quality land), and structure construction costs.

We focus on a dynamic model in which structures are durable and of invariant quality. In this situation, the owner of a parcel of land that has not yet been developed in urban use faces the decisions of when and at what structural density (floor-area ratio, FAR) to convert the parcel to urban use. To simplify the model as much as possible, we assume that structures are immutable and indestructible; there is no depreciation and demolition and reconstruction does not occur. We also assume that parcels are equally accessible.¹ The answers to the paper's question can then be derived from a variant of the Arnott and Lewis (1979) model of the transition of land to urban use on a single parcel of land with growing urban floor area rent and stationary agricultural rent.

The most obvious application of the paper's result is to urban sprawl, including both scattered and leap-frog development. Suppose, for the sake of argument, that, holding accessibility fixed, in some dimension of quality, higher-quality land is developed first. Then one would observe some higher-quality but less accessible parcels being developed before some lower-quality but more accessible parcels, giving rise to what would be observed as scattered and leap-frog development. Consistent with the observation, Burchfield et al. (2006), using satellite data on US metropolitan areas to explore differences in sprawl across regions between 1976 and 1992, find that differences in microclimate, physical terrain, and access to aquifers account for 25% of the variation in metropolitan areas. Arnott encountered the question posed in the paper's title in a quite different context. Zhang and Arnott (2015) had extensive data on vacant land sales, and enquired whether, in the absence of data on land quality, it is sound to impute the land value per unit area of a developed site as the land value per unit area of a proximate vacant site with the same zoning. If higher-quality land is developed earlier, then the land value per unit area of the proximate vacant site provides only a lower bound on the land value per unit of the area of the developed site. In fact, Gedal and Ellen (2018) find that in New York City “teardown” parcels are valued higher than vacant parcels at the city level as well as when land values are compared both within census tracts and within city blocks. In a different context, Pollard (1980) developed a static housing supply model to consider the role that access to and views of local amenities had in determining housing rents and building heights. Estimates for Chicago found that both rents and building heights fell with distance from local amenities while buildings were taller in locations with better access to amenities. However, the timing of development was not considered, which is a key component of this paper. In addition, our paper focuses on variations in the quality of an amenity over space, such as a microclimate, rather than accessibility to a specific locational amenity. Saiz (2010) is also relevant in considering variation in land quality that contributes to its developability. His results indicate that metropolitan areas with a greater degree of geographical features which limit the quantity of available land, such as steeply sloped terrain or bodies of water, tend to be more expensive and have lower housing supply elasticities with respect to demand shocks. However, the focus of the paper is on how land constraints affect the housing supply elasticity at the level of an entire metropolitan area, whereas in this paper we consider how intraurban variation in land quality alters development patterns.

After deriving and discussing the comparative static results for the model, we provide a brief discussion of factors other than differences in accessibility and Ricardian differences that affect development timing: planning decisions on when to release land for urban development, density restrictions on development, structural quality, risk and differences in expectations, economies of scale in development, landowner market power, taste for privacy, availability of municipal services, redevelopment and redevelopment restrictions, and public services and taxes.

In the proceeding section we present the model. We then present the qualitative results of the comparative static exercises for the different dimensions of land quality (an extended formal analysis is provided in the online appendix for this paper and in Lopez and Arnott (2018)). We then consider additional factors affecting the timing of development. And finally we conclude.

The model

A profit-maximizing developer decides when and at what structural density to convert a unit area of land he owns from agricultural to urban use under certainty and in the absence of development restrictions. Structures are immutable and indestructible; once built, a structure cannot be modified or replaced, nor can structures be added to the site. Structural quality is fixed; construction quality is fixed and depreciation does not occur. To simplify, we assume that the interest rate, the growth rate of urban rent (per unit floor area), and the agricultural rent remain constant over time. Furthermore, we augment the Arnott–Lewis model to allow for a more general rentals function (Turnbull, 1988) as well as (fixed) land preparation (or servicing) costs. The component of land preparation costs that increases in structural density is included in the construction cost function. We employ the following notation: s

structural density (or floor-area ratio)

timing of development

quality of land, whose interpretation varies depending on the type of quality differentiation being considered

S(s,q)

current floor rent per unit area of land, which depends on s and on q only when q represents natural amenities

interest/discount rate

growth rate of floor rent

C(s, q)

current structure construction cost per unit area of land, which depends on s and on q only when q represents elements of the land that affect structure construction costs

L(q)

current land preparation (servicing) cost per unit area of land, which depends on q only when q represents elements of the land that affect land preparation costs

R(q)

agricultural rent, which depends on q only when q represents agricultural fertility

\hat{π} (s, T, q)

present value of profit per unit area of land

Note the general notation S(s, q) for the floor area rentals function. A particular form of this would be $S (s, q) = s p (q)$ , with $p' (q) > 0$ , where p(q) is rent per unit floor area, with the dependence of p on q capturing natural amenities. This particular form assumes that an improvement in natural amenities increases floor area rent by the same amount on all floors. Our more general notation allows for the possibility that an improvement in natural amenities increases floor area rent by different amounts on different floors; for example, a view of the ocean may be visible from floors above a certain level but not below it. Our more general notation also accommodates floor area rent that differs by floor independent of natural amenities due for example to vertical transportation costs (Liu et al., 2018) or dependence of floor area rent on the height of the building due for example to greater prestige associated with a taller skyscraper (Koster et al., 2014). The notation C(s, q) provides a general description of the construction cost technology. It allows for the possibility, for instance, that building on bedrock rather than sand lowers construction costs proportionally more for higher than for lower buildings (Ahlfeldt and Mcmillen, 2018).

The developer's discounted profit is

\hat{π} (s, T, q) = [\frac{S (s, q)}{(r - g)}] e^{- (r - g) T} - [(C (s, q) + L (q)] e^{- r T} + [\frac{R (q)}{r}] (1 - e^{- r T})

(1)

The first term on the right-hand side is the discounted floor rent from the land after it is developed. The second term is discounted construction costs and land preparation costs. The third term is the discounted agricultural rent from the land before it is developed.

The first-order condition with respect to structural density is

{\hat{π}}_{s} (s, T, q) = [\frac{S_{s} (s, q)}{(r - g)}] e^{- (r - g) T} - C_{s} (s, q) e^{- r T} = 0

(2)

The first term on the right-hand side is the discounted marginal revenue from structural density, and the second term is the discounted marginal cost.

The first-order condition with respect to development time is

{\hat{π}}_{_{T}} (s, T, q) = - S (s, q) e^{- (r - g) T} + r [C (s, q) + L (q)] e^{- r T} + R (q) e^{- r T} = 0

(3)

The first term on the right-hand side is the discounted marginal cost of postponing development one period, which is the discounted structure rent forgone. The sum of the second and third terms is the discounted marginal benefit of postponing development one period, which equals the discounted saving from postponing construction and land preparation costs for one year plus the discounted agricultural rent.

Also note that e^−rt appears in all the terms of equation (3). To simplify the algebra, we eliminate it from the two first-order conditions. The economic interpretation of doing so is that the profit-maximizing program is the same at whatever point in time profits are measured. Defining π (s,T,q) to be the value of profit at development time, the two first-order conditions may be written as

π_{s} (s, T, q) = [\frac{S_{s} (s, q)}{(r - g)}] e^{g T} - C_{s} (s, q) = 0

(4)

π_{_{T}} (s, T, q) = - S (s, q) e^{g T} + r [C (s, q) + L (q)] + R (q) = 0

(5)

which we refer to as FOC_s (first-order condition with respect to s) and FOC_T (first-order condition with respect to T), respectively. To simplify notation, we define

B (s, q) \equiv r [C (s, q) + L (q)] + R (q)

to be the contemporaneous benefit from postponing development for a unit of time, which equals the interest on the construction and land preparation costs, plus the agricultural rent from land. The first-order condition with respect to T can then be rewritten as

π_{_{T}} (s, T, q) = - S (s, q) e^{g T} + B (s, q) = 0

(6)

Figure 1 plots the two first-order conditions, FOC_s and FOC_T, in T – s space. Sufficient conditions for their point of intersection to characterize a local profit maximum are that the two first-order conditions be positively sloped, and that the first-order condition with respect to T is steeper than that with respect to s.²

Figure 1.

First-order conditions in T – s space. FOC_s: first-order condition with respect to s; FOC_T: first-order condition with respect to T.

Comparative static analysis

Summary of the results

In the body of the paper, we undertake a qualitative analysis based on Figure 1. The formal analysis is presented in Lopez and Arnott (2018). For each of the four dimensions of quality they consider, Table 1 records the comparative static results. Row 1 indicates whether the quality improvement shifts FOC_s up (↑), down (↓), or has no effect (=). Row 2 indicates whether the quality improvement shifts FOC_T to the right (→), to the left (←), or has no effect (=). Rows 3 and 4 give the signs of the comparative static effect on s and T. The notation, “*”, indicates that only later development at lower density can be ruled out.

Table 1.

Comparative static results.

	Increase in fertility	Decrease in servicing fixed cost	Decrease in construction cost	Increase in amenities
Shift of FOC_s (equation (4))	=	=	↑	↑
Shift of FOC_T (equation (6))	→	←	←	←
Comparative static change on s	+	–	*	*
Comparative static change on T	+	–	*	*

Note. FOC_s: first-order condition with respect to s; FOC_T: first-order condition with respect to T.

The columns correspond to different dimensions of improvement in land quality. The rows correspond to the effects of the land quality improvement on the positions of the first-order conditions (FOC_s and FOC_T) and on s and T. Later development at lower structural density can be ruled out. The intuition is that a reduction in cost or an increase in revenue should lead to an expansion in production, which occurs through either earlier or higher-density development or both. Therefore, earlier development at lower structural density, earlier development at higher structural density, and later development at higher structural density are all possible. As we shall see, it is the last result that is anomalous.

An improvement in land quality corresponds to an increase in agricultural fertility

This is a simple case to treat. An increase in agricultural fertility has no effect on construction density, s, conditional on the timing of development, FOC_s. The number of stories is chosen such that “the top story” pays for itself, and agricultural fertility affects neither the marginal revenue nor the marginal cost associated with the top story. The increase in agricultural fertility increases the marginal benefit from postponing development (which includes the agricultural rent) and has no effect on the marginal cost (the floor area rent forgone), causing the optimal development time, T, conditional on structural density, to be postponed, and hence the development timing condition, FOC_T, to shift to the right. Thus, an increase in agricultural fertility causes a later development time, T, at higher density, s.

An improvement in land quality corresponds to a decrease in the fixed cost of servicing land per unit area

This too is a simple case to treat. A decrease in the fixed cost of servicing land also has no effect on the structural density decision, conditional on development time, FOC_s. Nor does it affect the marginal cost of postponing development but rather causes the marginal benefit of postponing development to fall, bringing the optimal development time, conditional on structural density, forward, leading to the developing timing condition, FOC_T, to shift to the left. Thus, a decrease in the fixed cost of servicing land per unit area causes an earlier development time, T, at lower density, s.

An improvement in land quality corresponds to a decrease in construction costs

The profit-maximizing density of a building is such that the marginal benefit from adding a story equals the marginal construction cost. Thus, holding development timing fixed, a decrease in marginal construction cost causes profit-maximizing density to increase, which corresponds to an upward shift in the density condition, FOC_s. Optimal development time is such that the marginal benefit from postponing development, which equals agricultural rent plus amortized construction and servicing cost, equals the marginal cost, which is the rent forgone. Thus, holding structural density, s, fixed, a decrease in total construction cost causes the marginal benefit from postponing development to fall, leading optimal development time, T, to be brought forward, which corresponds to the development timing condition, FOC_T, shifting to the left.

Thus, the improvement causes FOC_s to shift up, by how much depending on the change in marginal construction cost, and FOC_T to shift left, by how much depending on the change in total construction cost. Of the four possible qualitative outcomes, the only one that can be ruled out is later development at lower density. Which of the other three occurs depends on how much FOC_s shifts up compared to how much FOC_T shifts to the left, as well as on the slope of the two curves, which in turn depend on the form of the rent function, the construction cost function, and parameter values.

A complete analysis is provided in the online appendix and Lopez and Arnott (2018). Here we focus on how the outcome is affected by the form of the reduction in construction costs. At one extreme, consider a decrease in fixed construction cost that has no effect on marginal construction costs. This has exactly the same effect as a decrease in land preparation costs, which results in earlier development at lower density. At the other extreme, consider a decrease in marginal construction cost, with no effect on total construction costs. This causes an upward shift in FOC_s and no change in FOC_T, resulting in later development at higher density.

Now consider an intermediate situation in which construction costs fall proportionally, and in which there is neither agricultural rent nor land servicing costs. From the first-order conditions, the ratio of the marginal cost of construction density to the marginal benefit of postponing development equals the ratio of the marginal benefit from construction density (the increase in rent) to the marginal cost of postponing construction (due to rent foregone). Since the latter ratio is unaffected by a reduction in construction cost of whatever form, so must be the former ratio, which requires that development density be unaffected by the fall in construction costs. This in turn implies that a proportional fall in construction costs causes earlier development. Thus, a proportional fall in construction costs results in an earlier development time, T, with structural density, s, unchanged.

Can a decrease in construction costs lead to later development and higher density? Suppose that construction on higher-quality land leads to a decrease in marginal construction costs with no effect on fixed construction costs. Further, consider a reduction in construction costs that causes very little change up to some density, and then a sharp decrease in marginal construction costs above that density. This too should lead to later development time, T, at a higher level of structural density, s. The intuition is that a developer may find it more profitable to develop a building at a higher structural density than is justified by current rental rates but will be justified by future rental rates, leading to postponement of development. Lopez and Arnott (2018) provide a numerical example in which such a change in the construction cost function does indeed lead to later development at higher density. However, partly because the example was difficult to construct and partly because a decrease in construction cost leading to a postponement in development runs against our empirical intuition, but not on the basis of data, we conjecture that this qualitative outcome is anomalous.

An improvement in land quality corresponds to an increase in amenities

The increase in amenities operates via the upward shift in the floor area rent function that it induces. Start with the extreme situation in which the increase in amenities increases total rent but not marginal rent – the rent from constructing another story. Since the change affects neither marginal revenue nor marginal cost functions, it does not change the position of the FOC_s locus. The change has no effect on the marginal benefit from postponing development but increases the marginal cost, which is the revenue forgone, so that FOC_T shifts to the left. Thus, the change causes earlier development at lower density. Consider next another extreme situation in which the increase in amenities does not affect total rent up to some density, but sharply increases marginal rent above that density. Slightly above that density, the increase in amenities causes FOC_T to shift leftward only slightly and FOC_s to move upward substantially, with the net effect of an increase in both development density and development time. Lopez and Arnott (2018) provide a numerical example of this result.³ Again, partly because the example was difficult to construct, partly because an increase in amenities leading to a postponement in development runs against our empirical intuition, but not on the basis of data, we conjecture that this qualitative outcome too is anomalous. It is not totally implausible, however. One can imagine a situation in which renters are willing to pay a premium for an apartment with a view of the ocean, which is available only above a certain height. A reduction in smog that increases this premium would have the anomalous effect.

Hybrid cases

There are, of course, hybrid cases in which proximate lots differ in more than one dimension of quality. Practically the most important is likely lots on the flatlands compared with lots on hills. The lots on the hills have better amenities – a better microclimate, less pollution, and a better view – but are more expensive to construct on. In this case, the two pure quality effects operate in opposite directions, the better amenities by themselves leading to earlier development and the higher construction costs by themselves leading to later development.

There are also hybrid cases in which lots differ in terms of both accessibility and quality. Downtowns are typically built in lowlands, whereas the prime residential real estate is in hills, Los Angeles being an example. Additionally, the relative strength of one measure of land quality over another may depend on the underlying local economy. For example, agricultural fertility may influence development decisions more in developing than in developed regions, and natural amenities less.

Other factors affecting the timing of development

In the introduction, we listed 10 factors other than land quality and accessibility that may affect the timing and density of development of a parcel. Empirical work would want to take these other factors into account. Some of the factors apply at the level of the market and hence have general equilibrium effects that operate primarily through the rent function. Other factors apply differentially over sites.

Planning restrictions affecting the timing of development

In many jurisdictions, the local authorities decide when land at different locations is to be released for development. Suppose that their decisions impose a binding restriction on the timing of development—it would be more profitable for landowners to develop earlier. In terms of our analytical apparatus, the equilibrium timing and density of development then occur at the point of intersection of FOC_s and the development timing restriction, $T =_{T}$ (where $T$ is the earliest time at which development is permitted), as shown in Figure 2(a). Then, at a particular location, all lots are developed at the same time, but higher-quality lots are developed at higher density.

Figure 2.

First-order conditions with timing and development restrictions and quality variation: (a) development timing restriction and (b) development density restriction. Note: $\bar{q} > q$ . FOC_s: first-order condition with respect to s; FOC_T: first-order condition with respect to T.

Zoning restrictions affecting density

In many jurisdictions, the local authorities impose ceilings on development density (FAR restrictions). Suppose that their decisions impose a binding restriction on the density of development—it would be more profitable for landowners to develop at higher density. Under the current analytical framework, the equilibrium timing and density of development then occur at the point of intersection of FOC_T and the development density restriction, $s = \bar{s}$ (where $\bar{s}$ is the maximum allowable density), as shown in Figure 2(b). Then, at a particular location, all lots are developed at the same density, but higher-quality lots are developed earlier.

Planning and zoning restrictions may restrict both the timing and density of development. When they are binding on both margins of choice, planning essentially replaces the market in determining the timing and density of development of the individual parcel. Finally, it bears repeating that planning and zoning restrictions, and indeed all the other factors considered, affect development timing and development, not only through their effects, holding fixed the time path of rents, which is the focus of this paper, but also through their general equilibrium effects on market-clearing rents and prices now and in the future. For example, restrictions on housing development at the level of market not only cause housing rents to rise but also, by increasing the cost of living, reduce the supply of labor, which may undermine productivity growth when agglomeration economies are present (Hilber and Vermeulen, 2014; Hsieh and Moretti, 2018).

Structural quality

Thus far we have ignored that the developer has the choice of not only when to build and at what density but also the quality of the units. The above analysis applies for any level of quality. At what qualities or over what ranges of quality construction occurs depends on the floor area rent premium the market attaches to higher quality compared to the increase in construction costs with higher quality, as addressed in the filtering literature.

Risk and expectations

The model assumes certainty. However, all economic actors involved in real estate development face risk and an incomplete set of insurance markets, and most are risk averse. The incompleteness of insurance markets means that perceived differences in risk are generally not arbitraged away, which introduces an element of randomness to all real estate development decisions. Uncertainty also affects the timing and density decisions through the option value of waiting (Capozza and Li, 2002). Furthermore, political considerations in zoning variances or planning decisions may generate additional risk which can impact development decisions.

Economies of scale in development

The model assumes that real estate development exhibits constant returns to scale. However, the scale of most real estate projects suggests that there are economies of scale up to some scale. The construction of residential subdivisions is a good example. With scale comes reduced marginal cost for permits, design, utility services, and roads (where these are provided or partially financed by the developer). Economies of scale in development cause spatial autocorrelation in the timing and density of development over neighboring parcels.

Landowner market power

At times, concern has been expressed that landowner market power affects the timing and density of development (Markusen and Scheffman, 1978), though, to our knowledge, there are no empirical studies that document this. A landowner with market power would differ from a competitive landowner in considering marginal revenue rather than price. One therefore expects that a monopoly landowner would construct at lower density, holding development time fixed, than a competitive landowner, and that whether he would construct earlier or later would depend on how the market demand elasticity for floor area would vary over time.

Taste for privacy

Through examining the spatial pattern of development at the parcel level, Irwin and Bockstael (2002) have persuasively argued that households' taste for privacy tends to result in scattered development at the metropolitan periphery. This phenomenon may affect the timing of development of the individual lot but is unlikely to be important at the aggregate level.

Municipal servicing

Rather than impose explicit restrictions, many municipalities choose to influence the timing of suburban development through deciding when to provide municipal services, especially sewerage, to an area.

Redevelopment and redevelopment restrictions

To simplify the analysis, the paper's model assumes that structures are immutable and indestructible. But redevelopment is practically important, though often subject to density restrictions in suburban areas and to more extensive restrictions in central areas. If the model were augmented to include redevelopment, the landowner would make a sequence of timing and density decisions. The same general marginal conditions would apply for the profit-maximizing development time and density for each building, though the marginal revenue from an extra story would be received over only that building's lifetime.

Public services and taxes

Public services affect development decisions through their effects on rents. Taxes may alter the profit-maximizing timing and density conditions. A tax on true economic rental income would be akin to a tax on pure profits and would not therefore affect development timing and density. However, a higher property tax rate has the same effect as a higher discount rate (Arnott, 2005), which, by giving less weight to floor area rental revenue received in the future, would, at the partial equilibrium level, cause FOC_s to shift down and FOC_T to shift to the left.

Conclusion

One of the central topics in urban economics is the spatial pattern of urban development. What determines when and at what density non-urban land (vacant, agricultural, etc.) at the metropolitan periphery will be developed in urban use? The topic has been extensively explored in the urban economics literature on a growing monocentric city with durable structures (reviewed in Brueckner, 2000). That literature focuses on general equilibrium effects and considers only differences in accessibility across lots.

Employing partial equilibrium analysis, this paper addressed a different question. Taking as given the current and future floor rent, agricultural rent, and construction cost functions in a local area, what determines differences in the timing and density of development across lots? Even more narrowly, it analyzed the effects of differences in various dimension of land quality by applying an extended version of the Arnott–Lewis model in which development time occurs when the marginal benefit of postponing development by one period equals the marginal cost, and development occurs at that density at which the discounted marginal revenue from adding a story equals the discounted marginal construction cost.

With the exception of a couple of anomalous cases, the results accord with intuition. Parcels that are more valuable in non-urban use are developed later at higher density. Parcels with better amenities and/or lower construction costs tend to be developed earlier. The anomalies indicate that the analysis is not as straightforward as intuition might suggest (which is confirmed by the formal analysis in the online appendix and in Lopez and Arnott (2018)), but the anomalousness of the anomalies support the intuition.

A section at the end of the paper listed factors other than accessibility and land quality that contribute to explaining differences in the density and timing of development across lots in a local area. To our knowledge, there have been no empirical analyses to date of the density and timing of development at the level of the individual parcel. With rapid improvements in parcel-level data, this situation will hopefully change soon.

Supplemental Material

Supplemental material for Is higher-quality land developed earlier?

Supplemental Material for Is higher-quality land developed earlier? by Juan Carlos G Lopez and Richard J Arnott in Environment and Planning B: Urban Analytics and City Science

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

Supplemental material

Supplemental material for this article is available online.

Juan Carlos G Lopez is an Assistant Professor of Economics at the University of Denver, CO, USA. His research areas are in urban and regional economics with a particular focus on land use, urban water supply and regional migration.

Richard J Arnott is a distinguished Professor of Economics at the University of California, Riverside, CA, USA. His research areas include urban economics, the economics of downtown parking and traffic congestion and on urban transportation, land use and environmental forecasting.

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