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# Interface Urban Forest Management in an Urbanizing Landscape

## Abstract and Keywords

Although forests located near urban areas are a small fraction of the forest cover, a good understanding of the extent to which —wildland-urban interface (WUI) forest conversion affects local economies and environmental services can help policy-makers harmonize urban development and environmental preservation at this interface, with positive impact on the welfare of local communities. A growing part of the forest resource worldwide has come under urban influence, both directly (i.e., becoming incorporated into the interface or located at the interface with urban areas) and indirectly (as urban uses and values have come to dominate more remote forest areas). Yet forestry has been rather hesitant to recognize its urban mandate. Even if the decision to convert land at the WUI (agriculture, fruit, timber, or rural use) into an alternative use (residential and commercial development) is conditional on the relative magnitude and timing of the returns of alternative land uses, urban forestry is still firmly rooted in the same basic concepts of traditional forestry. This in turn neglects features characterizing this type of forestland, such as the urban influences from increasingly land-consumptive development patterns. Moreover, interface timber production-allocated land provides public goods that otherwise would be permanently lost if land were converted to an irreversible use. Any framework discussing WUI optimal rotation periods and conversion dates should then incorporate the urban dimension in the forester problem. It must reflect the factors that influence both urban and forestry uses and account for the fact that some types of land use conversion are irreversible.

The goal is to present a framework that serves as a first step in explaining the trends in the use and management of private land for timber production in an urbanizing environment. Our framework integrates different land uses to understand two questions: given that most of the WUI land use change is irreversible and forestry at this interface differs from classic forestry, how does urban forestry build upon and benefit from traditional forestry concepts and approaches? In particular, what are the implications for the Faustmann harvesting strategy when conversion to an irreversible land use occurs at some point in the future?

The article begins with a short background on the worldwide trend of forestland conversion at the WUI, focusing mostly on the case of developed countries. This provides a context for the theoretical framework used in the subsequent analysis of how urban factors affect regeneration and conversion dates. The article further reviews theoretical models of forest management practices that have considered either land sale following clear-cutting or a switch to a more profitable alternative land use without selling the land. A brief discussion on the studies with a generalization of the classic Faustmann formula for land expectation value is also included. For completeness, comparative statics results and a numerical illustration of the main findings from the private landowner framework are included.

# Introduction

Urbanization is a worldwide trend. The causes of urbanization are often debated, but commonly cited factors include land use policies, transportation infrastructure, socioeconomic factors, and population growth. In most parts of the world, urban land is expanding faster than urban populations. Urban populations are actually expected to double from 2.6 billion in 2000 to 5 billion in 2030, while urban areas are forecasted to triple within the same period (Seto, Güneralp, & Hutyra, 2012; UN, 2014). Thus, urbanization’s contribution to land use change emerges as an important sustainability concern. In addition, due to the rapid pace of urbanization, people’s preferences for green areas at the wildland–urban interface (WUI), the area where houses and wildland vegetation meet or intermingle, have increased and changed.1 Even though forests located in the immediate outskirts of urban areas may represent a small fraction of the forest cover of a region, they are of high importance because they supply many goods and services (e.g., recreation, wildlife, and timber products) to urban residents. The generally positive impact of these interface forests is already documented on property values (Luttik, 2000; Tyrvainen, 2001; Tyrväinen & Miettinen, 2000).

The WUI is widespread in the United States and in many other parts of the world, including Argentina, Australia, France, and South Africa (Radeloff et al., 2018). For instance, the WUI in the United States grew rapidly from 1990 to 2010 both in number of new houses (41% growth) and land area (33% growth), making it the fastest-growing land use type in the conterminous United States (Radeloff et al., 2018). The vast majority of new WUI areas were the result of new housing (97%), not related to an increase in wildland vegetation. As such, many forests and other working lands have now become interface land uses and subject to the pressures and impacts of close proximity to urban development and roads.

New Hampshire is an example of the fate of working forests at the WUI.2 New Hampshire is dominated by forested land, which covers more than three-quarters of the total land area. Moreover, because most of the state’s forests are within a three-hour drive from Boston and a five-hour drive from New York City, the entire state is considered to be in general at the WUI. Most forestland in New Hampshire is privately owned. Rarely compensated for the natural services their forests provide, landowners can derive income from the sustainable sale of forest products, including higher-valued saw logs and lower-valued firewood and wood chips. Yet New Hampshire interface working forests have gradually declined since the early 2000s (Foster et al., 2017; Thompson, Plisinski, Olofsson, Holden, & Duveneck, 2017). Conversion to more valuable land uses such as residential and commercial uses is one of the sources of such losses (Foster et al., 2017; Thompson et al., 2017). This in turn has resulted in fragmented forested parcels of fewer than 500 acres in a status that does not assure their retention as working forests, despite the local and regional incentives and efforts already in place to retain working landscapes.

One reason for the conversion of working landscapes at the WUI is that the public good nature of most of the ecological and environmental services provided by these interface landscapes distorts the market price of those lands below its social value. Land within convenient driving distance of urban areas has greater value in urban uses than in forestry. Hence, as development pressure increases at the WUI, active forest management is curtailed in areas adjacent to built-up areas, as the New Hampshire case illustrates. Premature harvesting also occurs in anticipation of development, which results in the fragmentation of the forested landscape and losses of biotic and abiotic services provided by forestland. Higher costs for forest management activities from smaller forested tracts due to forest parcelization are another emerging concern. Thus, higher and better possibilities with conflicting values at the WUI interface combined with the uncertainty of the regulatory future, and changing timber market conditions, all lead to more landowners opting to sell their WUI working forests.

Despite these growing concerns with the current increasing trend of WUI forestland conversion, most research on forest management decisions has not examined the effects of urbanization and urban growth pressure on private landowners’ decision-making at this interface. This is surprising given that much of the change from these forested landscapes into non-forestry uses results from the decisions of private landowners. As the urban population continues to grow and place development pressures on working forests, changing forested landscapes into urban uses, it is becoming more important to understand where conversion from forestland to residential and commercial uses occurs, what factors are associated with the conversion, and how to slow down the rate of conversion through innovative landowner and governmental programs.

The purpose of this article is to use the private landowner land use decision framework as a basis for understanding the following two questions: given that most of the WUI land use change is irreversible and forestry at this interface differs from classic forestry, how does urban forestry build upon and benefit from traditional forestry concepts and approaches? In particular, what are the implications for the Faustmann harvesting strategy when conversion to an irreversible land use occurs at some point in the future?3

The inclusion of rotation-end land sale or of a change to a more profitable post-harvest land use have seldom been examined in the optimal rotation literature. Yet the fact that urban rents change over time alters the nature of the timber decision problem. Within such a context, and in contrast to the traditional Faustmann setup, the value of forestland must then reflect not only the current use but also the possible future alternative use of the land, with impacts on forest management practices. Therefore, this article complements the existing, and scarce, literature on urban forestry by providing insights on how urban rents, urban rent growth, and conversion costs on the alternative use side, and interest rates, stumpage prices, and regeneration costs on the current use side, affect WUI forest management decisions in a certain world.

Furthermore, the framework presented here is general enough to accommodate future research that includes both an alternative use to forestry (such as agriculture or fruit) other than just residential use as well as environmental benefits from forested land. Our focus in this article is nevertheless to illustrate the integration of alternative land uses into the private WUI developer setup for the case of forestry and urban use. Conversion to an urban use, say residential or commercial development, is an interesting scenario because of its irreversible nature. Moreover, once land is converted to an irreversible use as opposed to a reversible use (say agriculture or fruit), the public good services provided by the prior use are permanently lost. Future work can then expand our framework to include non-timber amenities from working urban forests into the landowner’s objective function and further examine how forest management decisions in such a context would be adapted.

The article begins with a short background on the worldwide trend of forestland conversion at the WUI, focusing mostly on the case of developed countries. This provides a context for the framework used in the subsequent analysis of how urban factors affect regeneration and conversion dates. We review existing theoretical models of forest management practices that have considered either land sale following clear-cutting or a switch to a more profitable alternative land use without selling the land. A brief discussion on the studies with a generalization of the classic Faustmann formula for land expectation value is also included. Next, the new private landowner framework when a land use change occurs in the future is presented and the solution of the private landowner in this context is also described, highlighting how the optimal harvesting strategy changes from the traditional and generalized Faustmann setups. For completeness, comparative statics results and a numerical illustration of the main findings are included. Finally, the article offers concluding thoughts and some avenues for future research.

# Forest Management and Land Use Change

## Current Projections of Forest Conversion in the Developed World

A major concern with the current conversion of forests in close proximity to urban areas is that the forests being lost to the alternative land uses often have much higher values ecologically than the marginal agricultural or grazing lands that are moving into forest. Forests at the interface also provide important outdoor recreation environments to urban residents living in close proximity and often attract several thousand visits per year (Konijnendijk, 1999, 2003; Moigneu, 2001; Rydberg, 1998).4 Moreover, they can also act as a nature-based solution for sustainable urbanization and climate change adaptation in cities (European Commission, 2015). Furthermore, and in contrast to the past, today’s urban forestlands are rapidly being converted to irreversible uses such as housing, commercial use, and associated infrastructures. In North America, where the percentage of total population living in urban areas is already high (78%), the forecasts show a near doubling of urban land cover by 2030 (Seto et al., 2012). For example, the South of the United States has already experienced an aggregate net loss of forestland of almost 14 million acres since the early 1960s, a consequence of both urban expansion and some conversion of forests to agriculture (USDA/Forest Service, 2008).

In the case of Europe, forests are already fragmented towards a mixed landscape mosaic pattern where forest is intermingled with urban uses and agriculture (Europe & Unece, 2015). Landscapes with poorly connected woodlands represent more than 60% of the European Union (EU) territory. Nearly 73% of the European population lives in cities and this proportion is expected to reach 82% by 2050 (UN, 2014). Between 1990 and 2006, urban areas in Europe grew by 15,000 km2, an increase of half the size of Belgium within 16 years (Nilsson, Bernhard, & Nielsen, 2011). From 2000 to 2006, Europe lost 1120 km2 per year of natural or semi-natural areas (of which, on average, almost 50% was arable or cultivated land) to development. There is also a high probability that large areas (approx. 77,500 km2) of the European continent will be or have been converted to urban areas between 2000 and 2030 (EEA, 2011). Hence, Europe is expected to have one of the highest percentages (33%) of Alliance for Zero Extinction (AZE) species to be affected by urban expansion by 2030 (Seto et al., 2012). Moreover, a trend towards continued de-concentration and urban sprawl is a common phenomenon throughout Europe (EEA, 2016).

Population and personal incomes growth are primary forces driving changes in the urban forest landscape to urbanization (Alig, Plantinga, Haim, & Todd, 2010; Konijnendijk, 1997). By increasing the demand for land in residential use and therefore its value, these forces create pressure for urban foresters to either sell or convert their land to urban use. Factors such as new transport systems and even land use planning policies have nevertheless also stimulated development from the edge of the cities to formerly remote rural areas. Thus, the role of urbanization in the change of forestry should not be overlooked.

On the other hand, rising prices for wood products tend to increase the relative rents associated with keeping the land in urban forest rather than converting it to an alternative use. Yet revealed behavior by landowners in fast rapidly urbanizing areas indicates that the values for residential uses are higher than the values for forestry and agriculture (Alig, Kline, & Lichtenstein, 2004).5 Consequently, many private landowners are increasingly under financial pressure to sell their parcels for development (Butler, 2007). Hence, traditional timber management is expected to become a transitional use on the WUI in urbanizing regions, as at some point in the future those lands are irreversibly converted. In such a context, the traditional Faustmann approach, which applies in rural and remote lands, has to be adapted to include the features that characterize the forestlands at the WUI, with impact on management practices.

## A Brief Review of Previous Contributions

Very few theoretical studies have investigated the implications for the Faustmann strategy of forestland conversion over time. Only three studies have examined this issue (Burgess & Ulph, 2001; Cunha-e-Sá & Franco, 2017; McConnell, Daberkov, & Hardie, 1983). McConnell et al. (1983) determine the “approximately” optimal harvesting strategy of a forester who maximizes the present discounted value net revenue from a single site when timber prices and regeneration costs vary exogenously and agricultural rents remain constant over time.6 Burgess and Ulph (2001) use a forest land use option model to allow for the conversion of forestland between alternative management options over time to explain the ongoing process of deforestation in the tropics.

More specifically, McConnell et al. (1983) deal with the possibility of shifting from forestry to agriculture and vice versa, while Burgess and Ulph (2001) focus on the switch between alternative valued tree crops over time. Therefore, none of these previous frameworks is suitable to understand current trends in forestland conversion in areas with strong urban growth pressure, where land conversion is irreversible. Yet both studies provide important insights on how harvesting decisions are affected when evolving prices are considered, allowing for varying optimal rotation lengths over time.

Klemperer and Farkas (2001) also examine the impacts on optimal timber rotations when future land use changes. The study assumes, nevertheless, that landowners project a post-harvest change in land use with a land value equal to or exceeding today’s Faustmann value and that this future land value is independent of their assumed values for annual costs, annual revenues, stumpage prices, and income taxes. In this context, it is shown that incorporation of high market valuations of timberland in the optimal harvest condition derived from the Faustmann formulation results in significantly reduced optimal rotation ages. Moreover, the impact of changes in the preceding variables on optimal rotations may be substantially greater, and sometimes in the opposite direction, compared with the Faustmann case.

In contrast to Klemperer and Farkas (2001), the framework developed in this article studies the case where the original landowner can switch to a more profitable post-harvest land use without selling the land and further considers that the value of the alternative land use changes over time.

In contrast to McConnell et al. (1983) and Burgess and Ulph (2001), the framework presented here also considers that the alternative land use to forestry is residential use and that timber prices and regeneration costs, even though they change over time, are not required to evolve in a specific manner from timber crop to timber crop. This in turn allows us not only to provide much richer results but also to discuss how changes in urban rents affect forest management practices and explain the conversion of private working forests in an urbanizing setup.

Cunha-e-Sá and Franco (2017) have developed a model of a forest owner operating in a small open-city environment where the rent for urban use rises over time, reflecting an upward trend in households’ income. In order to accommodate the land use change at some point in the future, the model includes two types of harvests: regeneration cuts and conversion cuts. Regeneration cuts are timber cuts that harvest the current stand while providing for regeneration of the subsequent stand, while a conversion cut harvests the current stand with no provision for regenerating a future stand and land is converted into urban use. Within this framework, the authors examine how unanticipated changes in nearby preserved open space and alternative development constraints affect an individual private landowner’s decisions regarding both regeneration harvests and conversion dates.

In contrast to Cunha-e-Sá and Franco (2017), this article’s setup presents a generalized Faustmann land expectation value (LEV) that accommodates alternative uses to forestry, allowing for all the parameters in both the forest and urban sectors to change over time. Presenting the setup in terms of LEVs has the advantage of making the setup more comparable to existing generalized Faustmann setups that focus only on management of trees for timber (e.g., Chang, 1998; Chang & Gadow, 2010).

Though Cunha-e-Sá and Franco (2017) contribute to a better understanding of how existing urban laws and best management practices provide some context for delivering social goods, benefits, and services from forest management, they do not explore how changes in market factors affect private management decisions including timber management and land use change. Therefore, this article complements Cunha-e-Sá and Franco (2017) by developing a rigorous analysis of the marginal and non-marginal impacts of changes in the urban and forest factors, such as stumpage prices, regeneration costs, and interest rates, on optimal harvesting dates and optimal rotation lengths.

Some studies have already developed a generalized Faustmann land expectation value expression to allow for variable rotation lengths by considering that stumpage prices, stand volumes, and regeneration costs change over time without imposing a specific pattern (see, e.g., Chang, 1998, 2014; Chang & Gadow, 2010).7 While these studies also provide good insights for non-constant rotation lengths over time, even without alternative uses to forestry, the framework is not suitable to study management of the timber resource in the WUI interface. The reason rests on the fact that forestry at this interface is a temporary use, and the existing generalized Faustmann setups assume a land expectation value at the beginning of a particular timber crop based on the expected present net worth of future timber harvests assuming that forestland cannot or will never be converted to an alternative use. Thus, these previous generalized Faustmann setups are suitable to examine forest management decisions in more remote areas but not decisions on forestland at the WUI interface. Like the generalized Faustmann setup, this article’s framework also develops a setup based on the land expectation value at the beginning of a particular timber crop. However, it differs from the previous generalized Faustmann settings, because it considers the value of convertible forestland, that is, land that is currently in forestry but may be converted to residential use (that is, an irreversible use) at some future date, denoted as the conversion cut date. Thus, the value of the asset held by the forest owner derives both from the returns currently realized from forestry production as well as from the returns expected from urban uses in the future. To simplify our discussion, landowners in the current period have perfect knowledge of the expected returns from residential use over time. Given that the decision to convert such land depends on the relative magnitude and timing of returns of alternative uses, appropriate examination of current timber management practices at the WUI must also reflect these factors.

# The Framework

## Assumptions

Assume a forest owner holding a plot of bare land of fixed size $L¯$. The entire plot is under forestry at $t=0$ but it will be converted to residential use after $0 rotation cycles. Let $t$ be the calendar time. Denote switching costs,$S$, as the cost per unit of land of switching from timber production to residential use. Development is irreversible as conversion from residential use back to forestry is economically infeasible.

Let $Ti$ denote the rotation length of the timber stand or the age of the trees at the $ith$ harvest (regeneration cut), for $i=1,...,K−1$, and $TK$ the rotation length of the timber stand at the $Kth$ rotation, that is, the last rotation before land is irreversibly converted (conversion cut). Thus, $t=D=∑i=1KTi$ stands for the conversion cut date.

Stumpage prices, planting costs, and interest rates differ from timber crop to timber crop. Let $pi$ represent the stumpage price at the $ith$ timber crop, $ci$ is the regeneration cost for the $ith$ timber crop, and $v(t)$ is a strictly concave production function of wood per unit of land as a function of the age of the current stand.

Finally, denote $R$ as the annual residential land rent until the end of the $Kth$ rotation period and $μ$ the inflation rate of this annual rate after land use conversion occurs at the end of the $Kth$ rotation. The interest rate associated with $ith$ time interval or timber crops is given by $ri$ and $r>μ>0$ is the interest rate after land use conversion occurs. For simplicity it is assumed that the interest rate after conversion remains constant over time.

## The Landowner’s Problem

### Developable Forestland

When forestland is converted to residential use immediately after $0 harvests, the landowner receives the discounted forest rents until the time of conversion plus the discounted residential rents thereafter, net the discounted conversion costs. This suggests that urban rents are assumed not to be high enough in early periods so that land is immediately converted at time zero. Therefore, there are $0 time intervals of length $T1,...,TK$ that correspond to the lengths of the $K$ timber crops before conversion occurs. Thus, in a certain world, the landowner chooses a forest management plan ${T1,...,TK}$ to maximize the value of its developable land:8

$Display mathematics$
(1)

where $LEV1$ denotes the land expectation value at the beginning of the first timber crop.9 Normalizing $L¯$ to unity, and rearranging (1), yields

$Display mathematics$
(2)

where $LEV2$ is defined as:10

$Display mathematics$
(3)

Generalizing, it is obtained

$Display mathematics$
(4)

with

$Display mathematics$
(5)

Therefore, (4) is consistent with the generalized Faustmann land expectation value setup for the determination of the optimal harvest ages (Chang, 1998). Yet this equation also represents the value of developable forestland, that is, land currently under forestry but that will be converted to residential use at some future date, $D$. Other than including the present value of the forest returns at the end of all $K$ timber crops, it also incorporates the present value of the land expectation value immediately after conversion. This contrasts with the generalized Faustmann framework (namely Chang, 1998), where the land expectation value at the beginning of the second rotation assumes that land remains under forestry forever and thus

$Display mathematics$
(6)

While the generalized Faustmann setup frees the classic Faustmann setup from the assumption that stumpage prices, stand volumes, regeneration costs, and harvest ages remain constant over time, it still does not account for future returns from alternative uses to forestry. Yet the certainty of forestry returns truncates the set of possible returns to convertible forestland.

Note also that in (4), conversion to an alternative land use is optimal at some future finite date $D$ as the real income growth rate,$μ−r$, after $D$ is such that the value of future timber harvests plus the value of future conversion to residential use, discounted to the present, are higher than the value of an immediate land use change. Therefore, there is a $0 and a set of ${T1*,...,TK*}$ for which $D*=∑i=1K*Ti*$, such that the following condition must hold:11

$Display mathematics$
(7)

with $V*(T1,...,T∞)$ the optimal solution of the corresponding generalized Faustmann model without endogenous conversion. Moreover, the land expectation value is no longer a constant from timber crop to timber crop. This implies that the optimal rotation lengths are not constant, as in the classic Faustmann model, nor do they evolve in a specific way.12

### The Case Where $K*=2$

As conversion takes place at some arbitrary future date, the problem defined by equation (4) can be solved for any finite number of rotation periods. For exposition purposes, only two are considered in the subsequent theoretical analysis, though such assumption will be relaxed in the simulation exercised. However, the conclusions would not change if instead a different value for $K$ was considered.13

In this case, the landowner’s problem consists of choosing the management plan${T1,T2}$ that maximizes the present value of convertible forestland, as follows:

$Display mathematics$
(8)

where:14

$Display mathematics$
(9)

Let ${T1*,T2*}$ represent the landowner’s optimal timber rotation lengths. The necessary conditions for an interior local maximum at ${T1*,T2*}$, after some simplifications, are:

$Display mathematics$
(10)

$Display mathematics$
(11)

at ${T1*,T2*}$.15 Once ${T1*,T2*}$ are determined, the optimal conversion date to residential use can be determined as

$Display mathematics$
(12)

Equation (10) defines the optimal condition for the first timber rotation length. Since land is replanted after the first rotation, (10) also defines the optimal condition for the regeneration cut date. The left-hand side (LHS) of (10) is the marginal benefit from waiting one more year and consists of the extra amount of money earned because of a larger stand volume from the first timber crop. The right-hand side (RHS) is the marginal cost of waiting one more year. The cost of waiting comprises the forgone interest returns from harvesting immediately, that is, the cost of holding the trees plus the cost of holding the land. When the LHS is lower (greater) than the RHS, one should harvest the timber stand (should wait one more year). When the RHS equals the LHS, the optimal harvest age for the first timber crop is reached.

Equation (11) is the optimal conversion cut condition. A parcel is converted to residential use when the net benefit of postponing conversion one year equals the net cost from postponing conversion. The net benefit includes the gain in stumpage value from added timber growth net the interests forgone from delaying second harvest timber revenues one year. The net cost represents the value of residential land rents net the switching costs savings that accrue from postponing the switch to residential use one year.

It is interesting to note that the optimal harvest age for the first timber crop is a function of its own stand value and the land expectation value immediately after harvest, but the optimal harvest age for the second timber crop does not depend on the harvest age decision for the previous tree crop.

The second-order conditions can be expressed as

$Display mathematics$
(13)

since $p1[∂2v(T1)∂T12−r1∂v(T1)∂T1]<0$ and $p2[∂2v(T2)∂T22−r2∂v(T2)∂T2]<0$ by concavity of the stand growth function. Moreover, $∂2LEV1∂T1∂T2=∂2LEV1∂T2∂T1=0$ and the determinant of the second-order Hessian matrix is positive

$Display mathematics$
(14)

Thus, the second-order conditions of the problem assure that ${T1*,T2*}$ is a global maximum.

# Determinants of Forest Management at the WUI

In this section, with the equilibrium conditions established, the impacts of parameter changes on harvesting cuts and rotation lengths within the context of our framework are derived, assuming that an optimal interior solution exists for two timber crops before conversion. In the numerical exercise (see the section “Numerical Example”) this assumption is relaxed.

## Urban Side Determinants

The results obtained from changes in urban forces, namely in $R$, $μ$, and $S$ on forest management decisions, are as follows:16

$Display mathematics$
(15)

$Display mathematics$
(16)

implying that

$Display mathematics$
(17)

where the signs indicate the direction of the shift in timber cuts and conversion date as parameters change. Note that the changes in these parameters only affect the cost of holding the land by changing the present value of the alternative use of land. Therefore, for a given parameter change, the impact on both rotation periods is the same, that is, either increasing or decreasing.

According to (15), (16), and (17), reductions in the optimal regeneration cut date and in the optimal conversion cut date associated with larger annual residential rent at the end of the second rotation and/or greater inflation rate of the annual rate after conversion, and decreasing conversion costs, suggest that landowners reduce active forest management of their land as urbanization progresses. Note that regeneration cuts signal intentions to keep the land in forestry while conversion cuts signal the opposite. Higher residential rents increase the profitability of land in residential use, making it more costly to postpone timber harvesting cuts. Therefore, a higher opportunity cost of delaying the future timber crop would lead to a younger harvest age for the first timber crop.

Similarly, in the case of the second rotation, if the value of the alternative use is higher it is costly to postpone the second harvest. Therefore, conversion is optimally anticipated. Finally, a decrease in conversion costs works qualitatively in a similar fashion by increasing the opportunity cost of delaying timber harvesting.

The preceding results are thus complementary of the comparative static analysis of a change in the land expectation value at the beginning of the next rotation on the current cutting cycle under the generalized Faustmann setup without conversion. Intuitively,$LEV2$ can be higher as a result of a change in future forest production factors such as lower interest rate in the future, lower future regeneration costs, or higher future stumpage prices, but also due to changes in relevant parameters regarding the future value of the alternative use. Regardless of the reason, their impact on the current timber crop, $T1*$, can be examined through the impact of $LEV2$. The overall conclusion is that a higher $LEV2$ will shorten the current cutting cycle, $T1*$. However, because an alternative use to forestry is not accounted for in the generalized Faustmann model (see, e.g., Chang, 1998), additional sources other than the future forest production factors that may also affect the land expectation value at the beginning of the second timber crop are not examined.

## Forest Side Determinants

Next, the impact on the cutting cycles and conversion cut date as a result of changes in the parameters of the forest side (current use) are examined.

### Changes in $c1$ and $c2$

A higher regeneration cost in the future timber crop, $c2$, postpones the regeneration cut, that is, the current timber crop as it decreases $LEV2$. As the cost is constant it is worth postponing harvest, decreasing the present value of those costs. In contrast, following the increase in $c2$ there is no impact on the second harvesting cut (future timber crop), as it does not affect the second rotation period decision. Therefore, tenure in forestry is increased. Yet an increase in the initial planting cost as a sunk cost, $c1$, has no impact on harvesting cuts while reducing the present value of the forest investment. These results are established in Appendix A, where it is shown that

$Display mathematics$
(18)

$Display mathematics$
(19)

implying that

$Display mathematics$
(20)

It is worth mentioning that (18) and (19) are also similar to the results reported earlier in Chang (1998).

### Changes in $p1,$ and $p2$

In the classical Faustmann model a higher stumpage price would lead to an unambiguous decrease in all optimal regeneration cut dates, that is, to a decrease in $TF$ (see (A1) in Appendix A). A higher stumpage price increases the profitability of harvesting, making it more costly to leave the stand standing one more year. In the presence of a potential land use change at rotation-end, when stumpage prices are allowed to vary from crop to crop, the results obtained on the current timber crop are again similar to those reported in Chang (1998).

As shown in Appendix A,

$Display mathematics$
(21)

$Display mathematics$
(22)

implying that

$Display mathematics$
(23)

From (21) it is possible to conclude that an increase in the stumpage price at the regeneration cut, $p1$, postpones the conversion cut date while it has no impact on the length of the second rotation. Thus, conversion is postponed due to a longer rotation cycle for the first timber crop, suggesting that convertible forestland with highly valued tree species are converted later. In contrast, from (22), an increase in the stumpage price at the second and last rotation cycle,$p2$, shortens the regeneration cut date, as it increases the timber revenues from the second timber crop, thus increasing the cost of holding the land. That is, $LEV2$ increases. However, it also postpones the second harvesting cut date. As the opportunity cost of holding the land in forest at the end of this rotation does not change, the increase in the stumpage price is compensated by an increase in the rotation period. Thus, the timber volume increment from leaving the trees standing is smaller. In this case, a higher stumpage price at the second rotation will have an ambiguous effect on the optimal conversion cut date, depending on which effect dominates.

### Change in $r1$, $r2$ and $r$

Finally, to examine the impact of higher interest rates associated with each rotation period, and after conversion, it is shown in Appendix A that

$Display mathematics$
(24)

$Display mathematics$
(25)

$Display mathematics$
(26)

implying that

$Display mathematics$
(27)

Therefore, a higher current interest rate leads to a younger harvest age while a higher interest rate for a future timber crop would lead to an older harvest age for the current timber crop, as $LEV2$ decreases.

These results of the change in current and future interest rates on rotation lengths also agree with those presented in Chang (1998), but are in contrast to those from the traditional Faustmann model.

In the traditional Faustmann model, the opportunity cost of land is the discounted value of future rotations. If harvest in a rotation is delayed, the harvests in all future rotations will also be delayed, which reduces the present value of future rotations. From (A1) in Appendix A, an increase in the interest rate reduces the Faustmann optimal harvest age as the opportunity cost of delaying harvest increases. The land expectation value is also constant from timber crop to timber crop since it is assumed that stumpage price, stand volume, regeneration cost, and the interest rate would repeat themselves from harvest to harvest. Therefore, if the interest rate rises, the Faustmann optimal rotation length declines since land devoted to forestry becomes less valuable.

It is also worth mentioning that the differing impact of changes in current and future interest rates on the optimal conversion cut date is particularly noteworthy in the context of the setup of the current article. Note that the impact of a higher interest rate for the second timber crop on the optimal conversion cut date is ambiguous because of its opposing effects on the optimal cutting cycles (see (25)). Consequently, the impact on conversion date depends on which effect dominates. In contrast, an increase in the interest rate associated with the first timber crop hastens the conversion of the land to residential use, as $LEV2$ decreases.

Finally, an increase in the interest rate after conversion, $r$, increases both rotation lengths and, therefore, postpones the optimal conversion cut date. An increase in $r$ decreases the cost of postponing conversion, and hence the cost of maintaining the land under forestry, as the cost of holding the land decreases.

If, after the second rotation cycle, residential development is more valuable than forestry, the land expectation value at the beginning of the second crop increases. This effect is, nevertheless, lower the larger the interest rate after conversion, which reduces the land expectation value at the beginning of the second crop ($LEV2$). Since the interest rate that applies after conversion is constant, a higher $r$ would lower the land expectation value at the beginning of the second rotation period and thus unambiguously lengthens the harvest age for the first timber crop. A similar argument applies to explain the impact of a higher $r$ on the length of the second rotation cycle. Therefore, the forester postpones both harvesting cut dates. Note that this would not necessarily be the case if the interest rate $r$ was the same as in previous periods.

# Numerical Example

This section develops a forestry simulation model to assess the length of the rotation periods, number of optimal rotations and conversion date, and their behavior in terms of the parameters of the model. Consistent with the analytical setup, the landowner is assumed to own a parcel of bare land, which will be converted to an urban use at some arbitrary future time. At time $0$, let the returns to developed use be $388 per acre with an annual growth rate of 3.436% and the conversion costs be$1600 per acre. Furthermore, conversion costs do not change over time. It is further assumed that the stumpage price per cubic foot of timber is $10, the total regeneration costs are$35 per cubic foot of timber, and the landowner’s discount rate is set to 7.377%. In the benchmark, both stumpage prices and regeneration costs don’t change over time. However, the latter assumptions are relaxed when performing sensitivity analysis of the optimal solution. Initial planting costs, which are sunk costs, are set at $100 per acre.17 ## Benchmark Case: Faustmann Versus Developable Forestland Table 1 in Appendix B presents the Faustmann solution under the benchmark parameters as well as the expectation land value if conversion to urban use occurs immediately. The second column in Table 2 in Appendix B illustrates the benchmark privately optimal solution when the landowner chooses her or his forest management plan while taking into account the existence of an alternative use to their parcel of land, that is, the optimal management plan of developable forestland. The Faustmann setup investigates the optimal harvesting strategy for successive timber crops under the assumptions that stumpages prices, regeneration costs, and timber growth are predictable and remain constant over time, and disregards the existence of alternative uses to forestry. As a result, timber production is the perpetual use for land and rotation lengths are constant over time. Under the benchmark parameters, the optimal rotation length suggested by the Faustmann model is 10 years, yielding a total site expectation value of$6229 per acre.

On the other hand, when the landowner takes into account the alternative use to forestry and decides the land management plan that maximizes the value of developable forestland, it is no longer optimal to keep the land perpetually in timber production. In particular, Table 2 in Appendix B shows that it is optimal to convert the parcel to urban use after two rotations. Even though all the parameters in the forest sector remain constant over time and at the same values as those used to solve the Faustmann model, both optimal rotation lengths are lower than the optimal Faustmann rotation length in the developable forestland model.18 Specifically, the optimal first rotation length is nine years while the second rotation length drops to five years. As a result, transition to urban use occurs after 14 years and the land expectation value in this case is $9577 per acre. Therefore, considering an alternative use to forestry with positive returns over time creates the possibility that site rents under different uses change over time and that there may be a change in land use as predicted by the theoretical setup. It is worth mentioning that, under the benchmark calibration, the land expectation value when land is immediately converted equals$8245 per acre, which is higher than the value at time zero of pure forestland, which is land that remains under forestry forever (the Faustmann LEV value is $6229 per acre). However, immediate conversion is still suboptimal. The value of convertible forestland amounts to$9577 per acre. This implies that, even under certainty about future returns to urban use, it may be optimal for the landowner to delay her or his decision to convert in order to realize returns to forestry until they are exceeded by urban returns net of conversion costs, even if urban value already exceeds the Faustmann LEV value. Such a scenario is illustrated for the benchmark parameters, by showing that it is optimal to wait 14 years before land transits to urban use.

Therefore, when the value of alternative uses to forestry change over time, the nature of the timber problem changes because the possibility to changes in land use from timber to this alternative use have to be taken into account. This, in turn, has important implications for the management of forestland located at the WUI. As the results obtained have shown, the classic Faustmann setup is not suitable to study forest management decisions at the WUI. This interface has been of particular interest in the policy arena because it is characterized by expansion of residential and other irreversible urban uses onto forest landscapes in a manner that threatens the ecological (e.g., water quality and wildlife habitat) and socioeconomic value of urban forests.

## Sensitivity Analysis

A sensitivity analysis of the privately optimal solution while accounting for the alternative use was also conducted to understand how marginal and non-marginal changes in the parameters of the model would affect such solution. The values were changed below and above the reference values used in Table 2 in Appendix B. Columns 3–10 in Table 2 in Appendix B present the results for the marginal changes, while Table 3 in Appendix B illustrates the results for non-marginal changes in the parameters.

In Table 2 in Appendix B, a number of standard results shown by various authors such as Klemperer and Farkas (2001), Burgess and Ulph (2001), and McConnell et al. (1983) are confirmed as well as the new results derived in the present article. The marginal changes of the parameters do not change the optimal number of harvests before the switch, which remains equal to two. Yet the optimal rotation lengths and optimal conversion date change according to the derived analytical results from the section “Determinants of Forest Management at the WUI.”

As expected, the regeneration cost for the current timber crop, as a sunk cost, has no impact on the optimal harvesting plan of the landowner, while reducing the present value of the forest investment. Higher regeneration costs (even if marginal) for any future timber crop, on the other hand, lead to a lower LEV and increase the harvest age for the current crop. Since costs are constant, postponing the current harvest decreases the present value of those costs. This occurs because the increase in future regeneration costs reduces the profitability at the margin of the current rotation extension. In contrast, it does not affect the future rotation. As a result, the landowner also postpones the conversion to urban use.19

Regarding the impact of marginal changes in current and future stumpage prices, Table 2 in Appendix B reveals that a one-time decrease in the current stumpage price level (that is, the stumpage price in the first rotation) shortens the harvest age for the current stand while leading to an older harvest age for the future stand. Anticipated higher stumpage prices make future rotations more profitable so it is not surprising that the optimal harvest date for the second rotation is delayed in time. Note, however, that under the chosen parameter values, the changes in the optimal rotation lengths are such that the net impact on the optimal conversion date is still to switch after 14 years. On the other hand, a higher future stumpage price also leads to a younger harvest age for the current timber crop, as it increases the opportunity cost of holding the land since timber revenues increase in the future. However, as the opportunity cost of holding the forestland at the end of the future rotation does not change, the increase in the future stumpage price is compensated by an increase in the rotation period. Thus, the timber volume increment from leaving the trees standing is smaller. In this case, a higher stumpage price at a future rotation will have an ambiguous effect on the optimal conversion cut date, depending on which effect dominates. Yet, in this case, the net changes in the optimal rotation lengths lead to postponing the conversion of the parcel to urban use.

A higher interest rate for the current timber crop or a lower interest rate for the future timber crop would shorten the harvest age of the current timber crop while lengthening the harvest age for the future stand. The impact on the optimal conversion date depends, however, on the net impact and magnitudes of these two countervailing effects on the optimal rotation lengths. For example, if the interest rate for the future timber crop decreases from 0.07377 to 0.07, then the values in Table 2 in Appendix B suggest that the optimal rotation age for the current timber crop decreases in two years while the optimal rotation age for the future stand increases in four years. This, in turn, yields a higher land expectation value relative to the benchmark in the total amount of $9666 per acre and to transit to urban use after 15 years. On the other hand, if the interest rate for the current timber crop increases from 0.07377 to 0.075, then Table 2 in Appendix B shows that the changes in the optimal rotation ages for the current and future timber crops offset each other, leading to no change in the optimal conversion date relative to the benchmark case. Yet the land expectation value decreases to$31 per acre.

Finally, as expected, a marginal increase in the conversion costs per acre or in the future discount rate, or a marginal decrease in the urban rent at time zero or in the expected rate of growth of urban rents, postpones the conversion process by bringing forward the date of conversion and decreases the optimal LEV value. This suggests that any policy that increases conversion costs or affects urban returns or the growth of urban rents can postpone the conversion of forestland into urban uses and help preserve ecosystem services from standing trees, which are reduced when harvest occurs. Examples of such ecosystem services include recreation, habitat, carbon storage, species diversity, and scenic beauty. Since these services are mostly public good amenities, private landowners do not consider them when making their private management decisions. However, as shown in Cunha-e-Sá and Franco (2017), because development constraints such as minimum lot zoning or growth moratoria create an incentive to delay both regeneration and conversion cut dates, these policies can protect, even if only temporarily, such ecosystem services. But once forestland is irreversibly converted to residential use at some point in the future, there is not only a one-time loss of an ecosystem product, but also a disruption of ecosystem processes, which affects the capacity for the production of future ecosystem services such as timber production, water quality, wildlife, and carbon sequestration (Smail & Lewis, 2009).

Table 3 in Appendix B shows the results of the sensitivity analysis for non-marginal changes in the parameters. While the qualitative results on optimal rotation lengths and optimal conversion date from marginal changes in the parameters carry over to non-marginal changes, the impact on the optimal number of rotations before the switch is particularly noteworthy. The results in Table 3 in Appendix B highlight that when urban returns per acre are extremely high or the expected future stumpage prices are very low, it is optimal to hasten the conversion of forestland into urban use. In particular, it may even be optimal to convert the parcel immediately ($K*=0$). On the other hand, when conversion costs become really prohibitive or the growth rate for urban rents is extremely slow or stagnant (when the urban rent at time zero is already low), then it is optimal to keep the land in forestry in perpetuity as it is optimal never to make the switch at any future date ($K=∞$). For instance, take the case where the growth rate is set to zero. In this scenario, $LEV1(K=0)=3660$ < $LEV1(K=1)=5577$ < $LEV1(K=2)=6149$ per acre < $LEV(K=∞)=6229$ per acre. This latter LEV corresponds to the Faustmann case where there is no switch, so the optimal number of rotations is indefinitely large. Once again, the Faustmann solution is a special case of more general setup.

# Conclusions

Worldwide urban growth and urbanization have significant potential to influence urban forests through increased risk of forestland conversion, incidences of wildfire, and spread of invasive species. Even though there is a large and rich set of studies in forestry economics that have focused on forest management practices, remarkably little theoretical work on how urbanization affects forest management practices and deforestation at the WUI exists. Yet interface forests face increasing conversion pressure in metropolitan areas due to sprawling trends. These trends pose a serious threat to the ecosystem services derived from forested landscapes. A better understanding of how current and alternative uses influence WUI forest harvesting and conversion can thus help policy-makers harmonize urban development and environmental preservation towards a more sustainable urbanization with positive impact on the welfare of local populations.

This article has shed some light on this issue using a framework that accounts for the transitional feature of timber management at the WUI and examining how changes in residential rents, timber prices, and input costs affect regeneration and conversion cut dates. For simplicity, we have abstracted from non-timber benefits from standing urban forests and from the possibility of alternative reversible land uses such as agriculture or fruit. Yet our framework is general enough to accommodate such extensions in future research.

The results presented here reveal that in the presence of a land use change at rotation-end, the impact of a one-time change in stumpage price or in the interest rate on optimal rotations cannot be generalized as in the Faustmann case. Reductions in the optimal regeneration cut date and in the optimal conversion cut date associated with increasing urban rents and decreasing conversion costs suggest that landowners reduce active forest management of their land as urbanization progresses. Furthermore, planting more valuable tree species in areas under urban growth pressure may optimally delay conversion. Similarly, policies that increase conversion costs can prevent forestland conversion where urban sprawl is a stressing problem. Finally, if investing in forestry is an inflationary hedge, the increase in long-run interest rates also preserves forestland from early conversion (Miranda, 1989).

Future work could examine how regeneration and conversion cut dates change when urban rents are driven by uncertainties in the housing market. One could also think about a learning-by-doing approach, since timber production is well known but urban use is new to the private land owner. Another possible extension would be to include fire risk. In spite of these caveats, the current framework provides a useful step in explaining the trends in the use and management of private land for timber production in an urbanizing environment.

# Acknowledgment

This work was funded by Fundação para a Ciência e a Tecnologia (UID/ECO/00124/2019, UIDB/00124/2020 and Social Sciences DataLab, PINFRA/22209/2016), POR Lisboa and POR Norte (Social Sciences DataLab, PINFRA/22209/2016).

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# Appendix A

The impacts of a higher stumpage price, higher interest rate, and higher planting costs on $TF$ (optimal constant rotation period) are respectively:20

$Display mathematics$
(A1)

2. 2. Generalized Faustmann Model

Totally differentiating (13) and (14) with respect to the optimal rotation lengths and the parameter of interest $(θ)$, dividing the resulting system of equations by $dθ$, and applying Cramer’s rule yields

$Display mathematics$
(A2)

$Display mathematics$
(A3)

Where

$Display mathematics$
(A4)

Given that $VT1T1<0,VT2T2<0$, and $|H|=VT1T1VT2T2>0$, for $VT1θ<0$, $dT1*dθ<0$, while for $VT1θ>0$, $dT1*dθ>0$, and similarly for $dT2*dθ$.

Urban Side Determinants: Deriving (15), (16), and (17)

For $θ=R,μ,S$ it is obtained $VT1R=−r1e−(r1T1*+r2T2*)r−μ<0$, $VT2R=−r2e−(r1T1*+r2T2*)r−μ<0$, $VT1μ=−r1e−(r1T1*+r2T2*)R(r−μ)2<0$, $VT2μ=−r2e−(r1T1*+r2T2*)R(r−μ)2<0$, $VT1S=r1e−(r1T1*+r2T2*)>0$, and $VT2S=r2e−(r1T1*+r2T2*)>0$.

Forest Side Determinants: Deriving (18)(27)

1. (i) $c1,c2$

By differentiating totally (13) and (14) with respect to $c2$, and using it again, it is obtained

$Display mathematics$
(A5)

$Display mathematics$
(A6)

Hence,

$dT1*dc2=−VT1c2VT2T2|H|>0,dT2*dc2=0$, implying that $dD*dc2>0.$

Moreover,

$Display mathematics$
(A7)

implying that

$Display mathematics$
(A8)

2. (ii) $p1,p2$

By differentiating totally (13) with respect to $p1$, and using it again, it is obtained

$Display mathematics$
(A9)

Hence,

$Display mathematics$
(A10)

By differentiating (14) with respect to$p1$, it is obtained

$Display mathematics$
(A11)

$Display mathematics$
(A12)

By differentiating (14) with respect to $p2$ and using it again, it can be shown that

$Display mathematics$
(A13)

Thus,

$Display mathematics$
(A14)

3. (iii) $r1,r2,r$,

By differentiating totally (13) with respect to $r1$, and using it again, yields

$Display mathematics$
(A15)

By differentiating totally (14) with respect to $r1$, it is obtained

$Display mathematics$
(A16)

By differentiating (13) with respect to $r2$ it is obtained

$Display mathematics$
(A17)

By differentiating (14) with respect to $r2$ and using it again, it can be shown that

$Display mathematics$
(A18)

Finally, differentiating (13) and (14) with respect to $r$, yields

$Display mathematics$
(A19)

$Display mathematics$
(A20)

Therefore, it is obtained

$Display mathematics$
(A21)

$Display mathematics$
(A22)

$Display mathematics$
(A23)

implying that

$Display mathematics$
(A24)

# Appendix B

Table 1. Faustmann Solution and Land Expectation Value if Land Is Immediately Converted

 $c1=100$ $p=10$ $c2=35$ $r=0.07377$ $T*=TF=10$ $K*=∞$ $LEV1*=V(TF)=6,229$ $LEV1*(K*=0)=8,245$

Table 2. Comparative Statics Results for the Convertible Land Model Results for Marginal Changes in the Benchmark Parameters

Bench. Case

$r1$

$r2$

$p1(/ft3)$

$p2(/ft3)$

$c1(/acre)$

$c2(/acre)$

$r$

$μ$

$R(0)(/acre)$

$S(/acre)$

$r1$

0.07377

0.075

0.07377

0.0737

0.0737

0.0737

0.07377

0.0737

0.0737

0.0737

0.0737

$r2$

0.07377

0.0737

0.07

0.0737

0.0737

0.0737

0.07377

(1) 0.07377

(2) 0.07377

(3) 0.07377

(4) 0.07377

(5) $p1$

(6)

10

10

10

9.9

10

10

10

10

10

10

10

$p2$

10

10

10

10

10.2

10

10

10

10

10

10

$c1$

100

100

100

100

100

200

100

100

100

100

100

$c2$

35

35

35

35

35

35

74

35

35

35

35

$r$

0.07377

0.07377

0.07377

0.07377

0.7377

0.07377

0.07377

0.08

0.07377

0.07377

0.07377

$μ$

0.03436

0.034366

0.03436

0.03436

0.03436

0.03436

0.03436

0.03436

0.02

0.03436

0.03436

$R(0)$

388

388

388

388

388

388

388

388

388

350

388

$S$

1600

1600

1600

1600

1600

1600

1600

1600

1600

1600

1 800

$T1*$

9

8

7

8

8

9

10

9

11

10

9

$T2*$

5

6

9

6

7

5

5

8

11

10

6

$D*$

14

14

16

14

15

14

15

17

22

20

15

$K*$

2

2

2

2

2

2

2

2

2

2

2

$LEV1*$

$9577$9546

$9666$9545

$9590$9477

$9545$8402

$7245$8337

$9497 Table 3. Comparative Statics Results for the Convertible Land Model Results for Non-Marginal Changes in the Benchmark Parameters Bench. Case $r1$ $r2$ $p1(/ft3)$ $p2(/ft3)$ $c1(/acre)$ $c2(/acre)$ $r$ $μ$ $R(0)(/acre)$ $S(/acre)$ $r1$ 0.07377 0.07 0.07377 0.0737 0.0737 0.0737 0.07377 0.0737 0.0737 0.0737 0.0737 $r2$ 0.07377 0.0737 0.06 0.0737 0.0737 0.0737 0.07377 0.07377 0.07377 0.07377 0.07377 $p1$ 10 10 10 15 10 10 10 10 10 10 10 $p2$ 10 10 (7) 10 (8) 10 (9) 15 (10) 10 (11) 10 (12) 10 (13) 10 (14) 10 (15) 10 (16) $c1$ (17) 100 100 100 100 100 200 100 100 100 100 100 $c2$ 35 35 35 35 35 35 10 35 35 35 35 $r$ 0.07377 0.07377 0.07377 0.07377 0.7377 0.07377 0.07377 0.07 0.07377 0.07377 0.07377 $μ$ 0.03436 0.034366 0.03436 0.03436 0.03436 0.03436 0.03436 0.03436 0 0.03436 0.03436 $R(0)$ 388 388 388 388 388 388 388 388 388 1 000 388 $S$ 1600 1600 1600 1600 1600 1600 1600 1600 1600 1600 120 000 $T1*$ 9 13 0 15 0 9 11 11 10 0 10 $T2*$ 5 5 10 10 $D*$ 14 13 0 (18) 15 (19) 0 (20) 14 (21) 11 (22) 11 (23) ∞ (24) 0 (25) ∞ $K*$ 2 1 0 1 0 2 1 1 $LEV1*$$9577

$9758$10,409

$11,628$11,605

$9477$9740

$10,570$6229

$25,056$6229

## Notes:

(1.) Urban forestry encompasses forests in and near urban areas, but also other tree resources and associated vegetation such as urban parks as well as the vast tree resource in gardens and on other private land. For the purpose of this article, the focus is only on the management of urban forests located at the edge of the building environment, that is, the urban–wildland interface, as woodlands in and close to urban areas are strongly affected by the urbanization process. Other aspects of urban forestry related to urban greenery and urban tree management are also important but are beyond the scope of this article.

(2.) Working forests are forests that are actively managed to generate revenue from multiple sources, including sustainably produced timber and other ecosystem services, but are also threatened by urbanization in the next decade.

(3.) The Faustmann management is an even-aged management in which all trees in an area are harvested at one time, called clear-cutting, or in several cuttings over a short time to produce stands that are all the same age or nearly so. This management method is typically applied to shade-intolerant conifers and hardwoods. Two-thirds of European forests are even-aged (Europe & FAO, 2015).

(4.) Konijnendijk (2003) nevertheless provides evidence that in absolute terms urban forest resources are significant, covering millions of hectares (ha) in Europe. The share of urban forests in the overall forest resource is also shown to be higher in the more urbanized parts of Western Europe. For example, Berlin, Germany, owns around 27,000 ha of nearby forests, the Greater Paris region, France, has almost 80,000 ha of urban forests, and St. Petersburg, Russia, manages around 142,000 ha of forest greenbelt. In the case of the United States, the southern part has a high portion of forests near metropolitan areas (USDA/Forest Service, 2008).

(5.) Dwyer, Nowak, Noble, and Sisinni (2000) found that the South of the United States had the most cities with forests within 50 miles than any other part of the United States. The highest rural land prices are found in these counties, which bring about a corresponding increase in the costs of producing timber there. Because of this, selling urban forests at the interface for real estate can be more profitable for both industry and NIPF (non-industrial private forests) owners than timber production. Case studies conducted by American Forests (2002) also show that forest cover for four metropolitan areas, namely Atlanta, Chattanooga, Houston, and Roanoke and Fairfax County, a county near Washington, DC, declined by over 585,000 acres due to rapid urban expansion over a 24-year period.

(6.) Armstrong and Philips (1989) also develop a theoretical framework to determine the optimal timing of land use change from timber production to agriculture when the parcel of land supports a productive stand of timber. In contrast to McConnell et al. (1983), the authors assume that the landowner starts with a stand of trees of a particular age (so they relax the bare land assumption), and the goal of the landowner is to determine the age of timber harvest (and therefore the timing of land use conversion) under this scenario. It is also assumed that stumpage prices, regeneration costs, and agricultural rent remain constant over time. Armstrong and Philips (1989) show that failure to separate forest bare land values from the productive value of standing timber can bias decisions towards immediate conversion. Yet the authors do not examine the implications for the traditional Faustmann strategy of converting forestland into an alternative use at some point in the future or how evolving prices may affect forest management practices.

(7.) Chang (1998) develops the generalized Faustmann model for even-aged management, and Chang and Gadow (2010) extend the theory of the model of Chang (1998) to the case of uneven-aged management. Both studies ignore the possibility to convert forestland to an alternative use (say agriculture or urban use) at some point in the future. Chang (2014) mentions the possibility of alternative uses to forestry, yet the problem of endogenous conversion to an irreversible alternative use is not formalized, and thus not treated analytically.

(8.) In addition, the time interval after conversion to an irreversible use beginning at $D=∑i=1KTi$ and lasting forever can be considered as$K+1$, that is, from $[D,+∞[$. In this case, $LEVK+1$ denotes the land expectation value immediately after the land has been converted, that is, at the beginning of the $i=K+1$ time interval. This implies that the net returns from conversion are immediately received.

(9.) According to the assumptions presented in subsection “Assumptions, ” $Rr−μ=∫D∞Re(μ−r)(t−D)dt$.

(10.) Alternatively, it is possible to write$LEVK+1=Rr−μ−S$. In this case, when forestland is converted to residential use immediately after the $i=K$ time interval (harvest), the landowner receives the net value of the stream of residential land returns from the date of conversion onward. This is different from all expectation values before conversion, where timber revenues are also included.

(11.) For simplicity, the annual residential land rent $R$ is constant until $t=D$. After conversion, the urban rent growth rate is $μ>0$. If it is optimal to convert at time 0, then LEV at time 0 is given by $Rr−μ−S$.

(12.) Cunha-e-Sá and Franco (2017) have shown that it is optimal to adjust forest management practices when conversion to residential use occurs at some finite future date. In fact, if stumpage prices, regeneration cuts, and discount factors are constant over time, it is optimal to deviate from Faustmann’s by shortening the rotation lengths. This is due to the increase in the relative opportunity costs of later harvests. This is in contrast to McConnell et al. (1983), where without imposing additional structure to the problem, it was not possible to derive the relation between the optimal rotation ages.

(13.) An endogenous number of rotation cycles can be imposed without changing the results. Since the focus is on the sequence of harvest dates given a number of timber harvests, rather than on the number of timber harvests itself, two rotation cycles are assumed to make the problem more tractable and easier to interpret. In addition, given discounting and the long production period in forestry, the impacts of increases in the alternative use rent are likely more important on earlier rotation lengths.

(14.) Alternatively, it is possible to define $LEV3=Rr−μ−S$. Note that $LEV3$ only depends on the parameters from the urban side.

(15.) Note that the right-hand side of (14) may also be stated as $r2LEV3$.

(16.) In the numerical exercise (section “Numerical Example”) the direction of changes in the cutting cycles and conversion cut date as a result of changes in the parameters from the forest side described in the model is further examined, and the results are contrasted with the effects obtained in the traditional Faustmann and generalized Faustmann setups. All mathematical proofs are presented in Appendix A.

(17.) The wood production functions is based on Tietenberg and Lewis (2012) and is given by $v(t)=40t+3.1t2−0.01t3$, where $v(t)$ is the volume per cubic feet and $t$ is the age of the stand in years. The simulations were performed using the Knitro optimization software with AMPL, version 20140711.

(18.) Cunha-e-Sá and Franco (2017) present the theoretical proof of this result.

(19.) See Appendix A.

(20.) These results are well known and are not covered in detail here (see, e.g., Hartwick & Olewiler, 1998.). In the Faustmann setup the interest rate is constant over time and is assumed as equal to $r$.