Beach Nourishment as a Dynamic Capital Accumulation Problem

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Beach nourishment as a dynamic capital accumulation problem

Martin D. Smith a,, Jordan M. Slott b, Dylan McNamara c, A. Brad Murray d

a Nicholas School of the Environment, Department of Economics, Box 90328, Duke University, Durham, NC 27708, USAb Sun Microsystems Laboratories, Sun Microsystems, Inc. 1 Network Drive, Burlington, MA 01803, USAc Department of Physics and Physical Oceanography, University of North CarolinaWilmington, 601 South College Road, Wilmington, NC 28403, USAd Nicholas School of the Environment and Center for Nonlinear and Complex Systems, Box 90230, Duke University, Durham, NC 27708, USA

a r t i c l e i n f o

Article history:

Received 9 March 2007

Available online 18 January 2009

JEL classification:

Q24

Q54

Keywords:

Beach nourishment

Optimal rotation

Natural capital

Erosion

Shoreline stabilization

Coastal management

a b s t r a c t

Beach nourishment is a common coastal management strategy used to combat

erosion along sandy coastlines. It involves building out a beach with sand dredged

from another location. This paper develops a positive model of beach nourishment

and generates testable hypotheses about how the frequency of nourishment

responds to property values, project costs, erosion rates, and discounting. By treating

the decision to nourish as a dynamic capital accumulation problem, the model produces

new insights about coupled economic geomorphological systems. In particular,

determining whether the frequency of nourishment increases in response to physical

and economic forces depends on whether the decay rate of nourishment sand exceeds

the discount rate.

& 2009 Elsevier Inc. All rights reserved.

1. Introduction

Most US coastlines are either moderately or severely eroding[14,22]. Both wave-driven alongshore sediment transport

and sea-level rise contribute to coastal erosion, and climate change is likely to intensify both of these forces[30]. At the

same time, more humans are living in the coastal zone[16,31]. Cost estimates of avoiding inundation from a 1 m rise in sea

level are between $270 billion and $475 billion[32]. A recent Heinz report suggests that over the next 60 years, erosion will

affect 25% of US structures within 500 ft of the coastline, and the lost property value without any further build-out on

vacant lots would be between $3.3 billion and $4.8 billion[14]. These estimates reflect diminished property values from

erosion without accounting for any potential damages to structures or lost recreational amenities that are not fullycapitalized into land values.

Trends in the coastal zone point to an inevitable conflict between coastal developments and an encroaching shoreline.

As coastal erosion takes place, residential and commercial properties as well as coastal infrastructure are threatened.

Humans can, and do, intervene in the coastal zone to defend against shoreline changes. To address erosion, coastal

managers and engineers can pursue beach nourishment,1 build hard structures like sea walls, or simply move or abandon

coastal property[17,25].

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/jeem

Journal ofEnvironmental Economics and Management

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0095-0696/$- see front matter & 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jeem.2008.07.011

Corresponding author. Fax: +1919 684 8741.

E-mail address: [email protected] (M.D. Smith).1 Nourishment is also referred to as re-nourishment or replenishment to reflect the need for repeated applications of sand if this strategy is pursued.

Journal of Environmental Economics and Management 58 (2009) 5871
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capital accumulation problem. We draw on the Faustmann and Hartman models in forest economics to cast nourishment as

an optimal rotation problem[9,13].

In order to explore the capital-theoretic nature of beach management, it is essential to incorporate the geologic response

of a shoreline to beach nourishment in a more physically accurate way than previous treatments in the economic literature. In

the absence of nourishment, beaches erode for any of several reasons, including sea-level rise [5,16], shifts in the shoreline

position on large scales caused by wave-driven sediment transport [1], and coastal structures built by humans[17]. Althoughwidely believed to cause permanent beach erosion, the effects of storm landfalls on coastal areas are only temporary, and

calm seas restore the wide beaches very soon afterwards [17]. Erosion rates can also differ greatly on beaches within the same

region[30]. To build intuition about how individual communities make nourishment decisions, however, we abstract away

from the spatial particularities of erosion and assume that the background rate of beach erosion is constant.

Beach nourishment places sand on an eroding section of beach, restoring it to some width. Beach nourishment,

therefore, creates an idealized rectangular bump in the plan-view (i.e. birds-eye view) shoreline trend (Fig. 1a, darkened

rectangle). Wave action, assuming most waves approach the shoreline nearly straight-on, tends to spread beach

nourishment sediment laterally, thus smoothing the plan-view bump. This process is known as alongshore sediment

transport. The rate of this smoothing decays exponentially over time, where the time until only half the original beach

nourishment sand volume remains is on the order of several years to one decade [6].

In the absence of beach nourishment, the profile of the nearshore seabed forms a time-averaged, concave up,

equilibrium shape (Fig. 1b, idealized as linear), resulting from a balance between forces tending to cause onshore sediment

transport (from waves) and forces tending to cause offshore sediment transport (chiefly gravity outside the zone ofbreaking waves). Beach nourishment sand is typically placed only on the dry beach and the region immediately seaward

where waves break and run-up on the beach (the surf and swash zone, respectively). The profile-view wedge created by

beach nourishment disturbs the cross-shore equilibrium profile (Fig. 1b, darkened triangle). Wave action and gravity,

however, tend to redistribute this sediment over the nearshore seabed to restore the equilibrium profile (Fig. 1b, dotted

line) on the time-scale of years. Like the alongshore smoothing of beach nourishment sand, the rate at which sand is

redistributed in the cross-shore also decays exponentially over time [6]. Taken together, the cross-shore and alongshore

nearshore response makes it appear as if nourished beaches erode faster than the background erosion rateU

With potential future increases in erosion rates, coastal property values, and scarcity of sand to nourish beaches, how

often do we expect to see communities nourishing their beaches? By choosing how often to build out beaches through

nourishment projects, coastal managers choose the depreciation rate of beach capital because the net rate of erosion is

effectively endogenous for nourished beaches. With this critical feature in the model, we analyze a representative community

that makes nourishment decisions independently of other communities. Consequently, the testable hypotheses from our

comparative static results apply to nourishment frequencies across different communities and within communitiesacross time.

3. The model

We explore beach nourishment as a dynamic capital accumulation problem. The model presents a positive analysis of

what we might expect to see if coastal managers follow a dynamically optimal capital accumulation path. We follow the

Hartman[13]approach to amenity flows from a standing stock of forest. In our case, benefits accrue as a function of the

stock of beach width. As in a forest rotation problem, this stock changes continuously over time and is reinitialized each

time a control is applied. Cutting the forest returns the stock of timber volume to zero, whereas for beach management,

nourishment implies returning the beach to some initial beach width. Our model differs from the forestry literature in that

there is no harvest benefit. All of the benefits in our model are flows (amenity and storm protection), but the time-varying

variable cost, essentially negative benefits, provide an analog to timber benefits in the Hartman model. Assuming the costsof nourishment are incurred at time zero, net benefits of a single nourishment event can be written as a function of the

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Fig. 1. Cross-shore and alongshore response of the coastline to beach nourishment: (a) Beach nourishment creates an idealized rectangular plan-view

bump in the shoreline (darkened rectangle). Wave action spreads beach nourishment sand laterally, smoothing the bump; (b) beach nourishment

creates an idealized triangular profile-view wedge, distributing the cross-shore equilibrium profile (here, idealized as linear). Wave action redistributes

beach nourishment sand to restore the profile its time-averaged equilibrium shape.

M.D. Smith et al. / Journal of Environmental Economics and Management 58 (2009) 587160


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nourishment interval T:

NBT BT CT, (1)

whereC(T) denotes the cost associated with the nourishment project, and B(T) is the benefits function. Both of these, in

turn, will depend on the dynamics of beach erosion. Unlike the FaustmannHartman problem, costs are incurred at the

beginning of a rotation because nourishment takes place then, and the benefits accrue subsequently. In the

FaustmannHartman model, costs are incurred at the end of a rotation when the forest is cut and re-seeded. Another

crucial difference is that costs in our model are a function of the rotation length. In the FaustmannHartman framework,

there are no variable costs; the costs of cutting and re-seeding are a constant amount per unit of area. In our model,

nourishment sand is a large fraction of the cost. Normalizing to a given alongshore distance, the amount of sand is

proportional the width of beach build-out. This, in turn, is a function of the length of time T that the beach has been

allowed to erode. We develop the analytical framework for costs after introducing the state equation that describes beach

erosion.3

In order to describe costs of nourishment, it is necessary to specify beach dynamics. Unlike previous economic

literature, our beach erosion dynamics capture both a background erosion rate and the exponentially decaying rate due to

beach nourishment. In the absence of nourishment, the beach erodes at a constant rate of g ft/year (Fig. 1a). Let x(t)represent the beach width at time,t, and assume beach nourishment restores the beach to some initial width, x0, reflecting

some realities in the coastal zone, namely fixed locations of beachfront houses, utility pipelines and conduits, and

transportation infrastructure. We combine the exponentially decaying, accelerated erosion rate from the separate cross-

shore and alongshore responses into a single term. We, therefore, express the beach width, x(t) as

xt 1 mx0 meytx0 gt, (2)

where 0pmp1 is the fraction of the initial beach width,x0, which erodes exponentially at rate, yX0.4 The remainder of the

beach width, (1m), erodes linearly at a rate,gX0. By inspection, x(0) x0(Fig. 1a). Differentiating Eq. (2) with respect totime yields the state equation for beach width:

_xt myeytx0 g. (3)

We now consider a series of beach nourishments that are done on a periodic basis.Fig. 2illustrates the beach width over

a 150-year time horizon with a 10-year nourishment interval (rotation length).5 Initial beach width is 100 ft, baseline

erosion is 2 ft/year, 35% of the beach decays exponentially due to the nourishment return to equilibrium profile effect, and

the nourishment decay rate is 0.10 (roughly half of the nourishment sand lost in 7 years).6 Notice that this figure appears

much like a figure that depicts the standing wood volume in a forest rotation problem. However, in the nourishment case,

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0 50 100 150

55

60

65

70

75

80

85

90

95

100

Time (years)

BeachWidth(feet)

Fig. 2. Beach width on a 10-year rotation.

3 As a positive model of beach nourishment, C(T) might only include the engineering, planning, and construction costs of a project. However, to

approach the problem normatively,C(T) would also have to include potential non-market damages to the benthic environment, sea birds, and risks to sea

turtles[11,24]. We note these critical normative dimensions of the beach management problem but leave them as topics for future research.4 The parameter y captures the half-life of a beach nourishment, typically in the range of 35 years. For empirical approaches, see [6].5 We use nourishment interval and rotation length interchangeably. When the dynamic problem is time autonomous, the optimal sequence of

nourishment intervals will be a constant interval such that the problem has the rotational feature of the FaustmannHartman model.

6 On long timescales (greater than decades, but on the order of (1m)x0/g)), m and y would be functions of absolute time that the beach has been undernourishment because un-nourished beaches surrounding the community continue to erode at the baseline rate. The positions of these un-nourished

M.D. Smith et al. / Journal of Environmental Economics and Management 58 (2009) 5871 61


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the figure is inverted. This makes sense given that the stock of beach width decays, whereas the stock of forest grows. A

rotation resets the forest volume at zero but resets the beach width at its maximum, x0.

By changing the rotation length, the manager effectively controls the average beach width and thus the flow of

amenities and storm protection. The associated use of sand depends on the baseline erosion and the exponential decay of

nourishment sand. If the latter effect is zero, then the sand used is the same on average regardless of the rotation length.

The top panel of Fig. 3 depicts this effect. The cumulative sand use paths for a 10-year rotation and a 5-year rotation

overtake each other throughout the 150-year horizon. These overlaps reflect only the discreteness of the rotations. The total

amount of sand that is lost from the system is independent of human decisions. In the bottom panel in which nourishmentsand decays exponentially, the two sand use paths diverge. Here, the 5-year rotation uses more sand over time because

more frequent nourishments mean that the beach spends more time in the steeply sloped portion of the state equation (3).

Also note that in both cases, the cumulative sand is higher in the bottom panel compared with the top panel with baseline

erosion only. Thus, human interventions in the geomorphological system increase the amount of sand lost from the

nearshore environment, highlighting what some coastal scientists perceive as the wastefulness of beach nourishment[27].

By introducing benefits of beach width, we can weigh these losses of nourishment sand against gains and explore the

circumstances under which, from the communitys perspective, it is optimal to nourish the beach or to allow the shoreline

to retreat naturally.

There are fixed and variable components of nourishment project costs [6]. Fixed costs are associated with capital

equipment needed for dredging and spreading sand as well as the costs of planning, obtaining permits, and preparing

environmental impact statements. Variable costs are a function of the amount of nourishment sand required, which is

proportional to the width of beach build-out. Since the shoreline is pinned to x0each time nourishment occurs, the amount

of sand is proportional to cumulative erosion. We can thus write the cost function as

CT c fx0 xT, (4)

wherecis the fixed cost and f the variable cost of beach sand and includes the engineering conversion from beach width to

sand volume. We assume cX0 and fX0. Substituting for x(T) and simplifying, the cost can be expressed as

CT c fmx01 eyT gT. (5)

Because beach widths change continuously but hedonic models capture values of standing capital stocks, it is necessary

to convert stocks to flows for understanding benefits of nourishment. Let G(x(t)) capture the value of the stock of beach

width. We convert this to a flow benefit using the discount rate. Though only some benefits from beaches are capitalized

into home values on the beachfront, while others accrue to beach visitors and to the broader local community, we assume

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0 50 100 150

0

100

200

300Sand Use with Baseline Erosion Only

CumulativeSandUsed

5-year Rotation

10-year Rotation

0 50 100 150

0

200

400

600

800Sand Use with Exponential Decay of Nourishment

Time

CumulativeSandUsed

5-year Rotation

10-year Rotation

Fig. 3. Sand use patterns with and without nourishment decay.



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thatG(x(t)) encompasses total community values of beach width. Thus,

BT

Z To

edtdGxt dt, (6)

whered is the discount rate, which we assume is strictly positive, and we assume G0 0; @G=@x40; @2G=@x2o0. The

curvature assumption suggests that as the beach gets wider, the marginal benefit of adding more beach width declines, and

there is some empirical support for this assumption [26,28]. Eq. (6) is similar to Hartmans[13]value of the standing stock

of forest. Instantaneous amenities are proportional to the standing stock, which is changing continuously. Computing totalamenities involves integrating the present value flow over the course of a forest rotation. The beach nourishment problem

presents two crucial differences: (1) the beach is eroding over time, not growing (as a forest) and (2) the flow conversion in

the beach problem uses the discount rate, which we assume is the same as the capitalization rate for rent into

housing price; in the forestry problem the flow conversion is a measure of non-timber benefits that may not be the same

(or related to) the capitalization rate of the forest.

As in the FaustmannHartman style forestry literature, the problem for communities is to determine how often to

nourish the beach. The choice is a sequence ofTs:

vT1; T2; T3;. . .; Tn NBT1 edT1 NBT2 e

dT2 NBT3 edTn1 NBTn, (7)

wherevis total present value, and each Tirepresents the absolute time from the beginning of the planning horizon (not the

marginal time from the previous rotation). Assuming that the instantaneous benefits function is time autonomous and the

erosion dynamics are stationary, we can write the value of an infinite nourishment rotation as an infinite geometric series:

vT X1k0

edkTNBT NBT=1 edT, (8)

whereTis the length of time since the previous rotation.

The community chooses a T* to solve the following maximization problem:

max vT BT CT=1 edT. (9)

The first-order condition is

@v T

@T B0T C0T1 edT dedTBT CT=1 edT2

set0. (10)

Optimal nourishment occurs whereT*solves Eq. (10). Two notes are in order. First, by optimal we mean optimal from the

point of view of the coastal manager of a particular location; socially optimal would need to include ecological costs of

nourishment.7

In a positive sense, one could interpret Eq. (10) to represent narrowly the net benefits of nourishment thatare capitalized into property values or values of the surrounding local community. Second, it is possible that no nourishing

is the optimal strategy. To ensure that T* is in fact optimal, one need only check that v4 0 in Eq. (9).

Multiplying Eq. (10) through by (1edT), we see that the optimal nourishment interval occurs where the difference

between marginal benefits and marginal costs of a single rotation is exactly offset by the interest payment lost on delaying

all future rotations. This can be seen from the following expression in which the right-hand side is simply the discount rate

multiplied by the present value of all rotations after the first one:

B0T C0T dBT CT=edT 1. (11)

So far, the intuition generally matches that of the classic Faustmann problem. To clarify interpretation, the Faustmann

first-order condition can be written (in our notation) as

B0T dBT c dBT c=edT 1, (12)

where costs in the Faustmann model include only fixed costs c. In Faustmann, there are no marginal costs such that theleft-hand side of (12) is different from that of (11), and the right-hand side of (12) has crather thanC(T) as in (11). Another

important difference is that the term d(B(T)c) in (12) does not appear in (11). This reflects the fact that Faustmann

generates a lumped benefit when the forest is cut, and delaying incurs lost interest on that lumped benefit. There are no

such lumped benefits in the beach problem; all benefits are continuous flows as in Hartmans value of standing forest. The

term d(B(T)c)/(edT1) is interest on future rent and is the opportunity cost of keeping land in forestry [29]. By analogy, for

beaches this term is the opportunity cost of pursuing nourishment as an erosion management strategy.

To generate results from the model, we take derivatives of the benefit and cost functions to find expressions

for the pieces of (10). Applying Liebnizs rule to the benefits function in (6), the marginal benefit of extending the

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(footnote continued)

beaches ultimately affects the alongshore sediment transport within the nourished region. We treat mandy as fixed parameters for analytical tractabilityand to develop some insights about the nourishment problem, but we acknowledge this caveat as a limitation of our model.

7 Socially optimal nourishment would also need to include spatial externalities of nourishment (both positive and negative) especially given theemerging evidence that shoreline perturbations can propagate over large spatial scales[1].



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rotation is

B0T edTdGxT edTdG1 mx0 meyTx0 gT. (13)

Thus, the marginal benefit of extending a single rotation is decreasing. The corresponding marginal cost of delaying a

rotation is

C0T fg fmyx0eyT. (14)

By inspection of (14), the marginal cost of delaying is decreasing. The cost of additional sand is constant, but theadditional sand required is decreasing over time because the total erosion rate decreases and approaches the baseline rate

as the beach returns to equilibrium profile (in both profile- and plan-view senses).

4. Comparative statics of beach nourishment rotations

We cannot analytically sign most of the comparative statics from the model described above because there are two

countervailing forces in the first-order condition (10). The optimal rotation length ( T*) balances the difference between

marginal benefits and marginal costs of a single rotation with the interest lost on delaying all future rotations. While this

property characterizes the original Faustmann formula as well, the simple form of the benefits function combined with the

assumption of only fixed costs in Faustmann leads to cancellations and signable comparative statics.

For the cost parameters, however, it is possible to sign comparative statics by assuming that the second-order condition

holds at the maximum. Because the fixed cost c is neither in the benefits function nor the marginal cost function, the

comparative static result is relatively simple.

Proposition 1. The optimal rotation length increases if the fixed cost of nourishment increases.

Proof.

dTn

dc

dedTn

1 edTn

2

,@2vTn

@T2

!.

The numerator is positive, and the denominator is negative by the second-order condition. Hence, dTn=dc40. &

In the next proposition, we explore the importance of the variable cost of nourishment on the optimal rotation length T*.

Here, the sign will neither be strictly positive nor negative, but we can sign the parts and determine the geomorphological

and economic drivers.

Proposition 2. The optimal rotation length can increase or decrease if the variable cost of nourishment increases. The sign

depends on whether the nourishment sand decay rate is higher than the discount rate, the fraction of beach width that decays

exponentially, and the baseline erosion rate g.

Proof. From Proposition 1, we know that signfdT=dfg signf@2vTn=@f@Tg. DefineG 1 edTn

2f@2vTn=@f@Tg, such

thatsignfdT=dfg signfGg. We can then write G as the sum of two terms that have known signs:

G y dmx0eydTn dedT

n

yeyTn

mx0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflffl}A

gdedTn

Tn edTn

1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflffl}B

.

We will show that the sign{A} depends only on whether y4d. First, by inspection, we see that when y d, the A term is

zero. Factoring outmx0, A40 if y deydTn dedT

n

yeyTn

40. We now use second-order Taylor approximations to

sign this expression, noting that ekT ffi1 kT k2T2=2. Thus,

Affi y y2Tn y3

2 Tn2 d d2Tn d

3

2 Tn2 y y2Tn ydTn y

3

2 Tn2

yd

2

2 Tn2 y

2dTn2 d dy d

2Tn

y2d

2 Tn2

d3

2 Tn2 yd

2Tn2

dy

2 Tn2 y d .

Thus,sign{A} sign{yd}. To determine thesign{B}, we note that atT* 0,B 0.@B=@TnjTn0 gd2

edTn

o0. Hence,B is

strictly non-positive. Putting together the pieces, whenever yod, both terms are negative, and increasing the variable cost

increases nourishment frequency. When y4d, increasing variable cost increases nourishment frequency if this difference

(between y and d) is small, baseline erosion is high, or the share of beach width that erodes exponentially ( m) is small. &

One explanation for Proposition 2 is that the variable cost influences the relative importance of fixed cost of

nourishment in determiningT*. When variable costs increase, the relative importance of fixed costs decrease, reducing the

incentive to delay. Consider for simplicity that there was only baseline erosion. Then the total variable cost (undiscounted)would be the same whether nourishment intervals were high or low because the beach is always returned to x0. In this

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simplified setting, total undiscounted fixed costs increase asT*decreases, whereas total undiscounted variable costs do not

change withT*. If there were no fixed costs, we would simply nourish continuously and always maintain the beach at x0for

maximum benefits. When we introduce exponential decay of nourishment sand, more sand is lost quickly, and the variable

costs of nourishing frequently are substantially higher.

The discount rate and exponential decay of nourishment sand play important roles in Proposition 2. In essence, the

manager endogenizes the depreciation rate of beach capital. Depreciation here is a mixture of exponential decay of

nourishment sand and linear sand loss from baseline erosion. Nourishment frequency determines the net rate of

depreciation. Consider first the effect of discounting in isolation. When the discount rate is high, the weight placed on themarginal net benefits in the short run is high relative to the weight placed on the discounted stream of future rotations. An

increase in the variable cost of sand increases the marginal cost of a rotation that must be offset by increasing the marginal

benefits. Since marginal benefits are decreasing inT, T* must decrease. When the discount rate is low, the weight on all

future rotations is high. This, in turn, is based on the difference between total benefits and total costs of a rotation. With a

low discount rate, total benefits of a rotation must increase to compensate for an increase in the total cost of a rotation due

to the variable cost increase. This requires increasing T*.

Now consider the exponential decay of sand together with the discount rate. An increase in variable cost of nourishment

will increase the implicit undiscounted cost of maintaining the capital stock. When sand decays rapidly, marginal cost of a

rotation is high but decreasing in T. So, one can lower the marginal cost by increasingT*. If the discount rate is relatively

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0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

Marginals from a Single Rotation - phi = 1,

gamma = 0.5

Rotation Length

MBMCMNBNB Rent

0 5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

40


gamma = 0.5

Rotation Length

MBMCMNBNB Rent

0 5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

40


gamma = 2

Rotation Length

MBMCMNBNB Rent

0 5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

40


gamma = 2

Rotation Length

MBMCMNBNB Rent

Fig. 4. (a) Illustration of Proposition 2

low erosion, low sand cost. (b) Illustration of Proposition 2

low erosion, high sand cost. (c) Illustration ofProposition 2high erosion, low sand cost. (d) Illustration of Proposition 2high erosion, high sand cost.



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low, we saw above that there is also an incentive to increaseT*. When sand decays slowly, marginal cost of a rotation is low

and declining more slowly in T. This dampens the incentive to increaseT*, while a high discount rate provides an incentive

to decreaseT*.

In the Faustmann model, Eq. (12) shows that optimal rotation balances future rents against marginal net benefits of a

single rotation. The beach problem has a similar feature. But in Faustmann, costs are only fixed and do not affect the

marginal net benefit of the single rotation. As such, a cost increase reduces rents, so marginal net benefits must also go

down to compensate. That can only be accomplished by extending the rotation. In our case, a variable cost increase affects

both sides of the first-order condition (11) such that both terms shift down. When background erosion is high, the shiftdown in marginal net benefits overshoots the reduction that is required to balance marginal net benefits and future rents.

So, the rotation must shrink to increase marginal net benefits and balance the first-order condition.

To clarify Proposition 2 further, considerFig. 4. The four panels represent two values of baseline erosion (high and low)

and two values of variable cost (high and low). In the first two panels, background erosion is low ( g 0.5). The increase invariable cost (fromf 1 to 2) shifts both the total net benefits of future rotations (NB rent) and the marginal net benefit

(MNB) down. The shifts are such that the optimal rotation length increases with the increased variable cost (as intuition

suggests). When variable cost increases, it decreases the rent on all future rotations. The rotation increases to reduce

marginal net benefits to compensate. This is the same process that governs the Faustmann rotation when fixed costs

increase. In the second two panels, background erosion is high (g 2.0). The increase in variable cost still shifts bothcurves, but the shifts have different consequences. Here, the shift in marginal net benefits with the higher variable cost

overcompensates for the lost rent from the cost increase. Thus, the rotation length must shrink to increase marginal net

benefits and compensate for the change in rent. The input demand for sand thus slopes upward in this case. We will see

below that this case is a consequence of fixing the extensive margin (the initial beach width x0). An analogouscounterintuitive result comes out of the basic Faustmann model. When the extensive margin (i.e. the amount of land in

forestry) is fixed in Faustmann, a timber price increase shortens the rotation length; in doing so, the average supply of

timber decreases and output supply slopes downward.

The remaining comparative static results are problematic because they involve integrals that do not have closed-form

expressions. To see this, substitute Eqs. (5), (6), (13), and (14) into Eq. (10):

@v

@T edTdG1 mx0 me

yTx0 gT fg fmyx0eyT=1 edT

dedTZ T

o

edtdG1 mx0 meytx0 gtdt c fmx01 e

yT gT

1 edT2

set0. (15)

Note that the parametersm,x0,y,g, andd all enter in both the first line and the second line, the latter being subtractedfrom the former. Moreover, these parameters are all in the integrand of the integral that is being subtracted. By Liebnizs

rule, the integral will appear in all comparative statics for these parameters. We thus resort to numerical simulations toexplore the model further. To this end, we introduce a two-parameter functional form for benefits:

BT

Z To

edtdaxtb dt. (16)

Eq. (16) conforms to our assumptions about the benefits function above (strictly positive first derivative and negative

second derivative). The parameter a can be interpreted as a base value of the beachfront property, while the parameter bcontrols the hedonic price of beach width, conditional on having a beachfront property. As before, d converts the capital

value into a flow. A particular advantage of this functional form is that static hedonic pricing models actually estimate the

beach width parameter. For example, Pompe and Rinehart [28]find b 0.2632 in a constant-elasticity hedonic model.

In the following simulations, we use the same numerical values for parameters that generated Fig. 2. For each set of

simulations, we fix all but two parameters and vary the others together. The base parameters are: b 0.25, a 200,d 0.06, c 10, and f 1. These parameters are illustrative and used to generate testable hypotheses; they do not

represent any particular community. For each combination of parameters, we numerically solve for T* in two differentways. First, we minimize (v) in Eq. (9) using Matlabs constrained minimization procedure (FMINCON) to obtain T. We use

FMINCON to constrain Tto fall between 0 and Tmax, whereTmax is defined implicitly:

xTmax 1 mx0 meyTmaxx0 gT

max0. (17)

We nest a numerical quadrature (Matlabs QUAD function) within the objective function to evaluate the integral in

Eq. (16). We then evaluate Eq. (9) at T and compare with the value of abandoning the property, which is zero by

construction:

Tn T ifvT40;

1 else:

( (18)

Second, we solve the first-order condition in Eq. (10) directly to obtain ^Tusing Matlabs FSOLVE procedure. Again we use

numerical quadrature to evaluate Eq. (16). We compute T* as in Eq. (16), substituting ^

T for ^

Tand check that the twonumerical solutions for T* are close, i.e. verify that the absolute value of the difference is less than 0.001.

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Numerical Result 1. The optimal rotation length decreases as the value of beach width increases.

Nourishment occurs more frequently when the hedonic price of beach width (b) is higher. This result is intuitive because,

ceteris paribus, a high-bcommunity would prefer a wide beach. A higher b increases both the total and marginal benefits

of beach width, and more frequent nourishment increases the average beach width.

Numerical Result 2. The optimal rotation length decreases as the base property value increases.

Communities will nourish more frequently when the base property value (a) is higher. As population pressure in thecoastal zone drives up coastal property values, holding other features of the problem constant, our model predicts thatmore beach nourishment will occur.

Numerical Result 3. The optimal rotation length decreases as the baseline erosion rate increases.

When the beach erodes faster, communities must nourish more often to keep up.

Numerical Result 4. The optimal rotation length decreases as the share of beach width subject to exponential decay increases.

When more of the beach is subject to exponential decay (m is big), a larger fraction of beach width addition fromnourishment is lost quickly, and communities must nourish more often to keep up. Keeping up with erosion sounds

intuitive, but high-m communities are getting less bang for their buck, and this suggests that Numerical Result 4 issomewhat counterintuitive. The former effect dominates because all communities will keep up with erosion as long as the

present value of beach nourishment for some T is positive. Communities that lose a lot of nourishment sand in the

adjustment to equilibrium profile essentially become addicted to nourishment as a beach stabilization strategy.The present value of nourishment exceeds the present value of abandoning the beach for all parameter value combinations,

even though the total present value of the optimal programthe solution to Eq. (9), v (T*)is lower for high-mcommunities than for low-m communities. This is no different than suggesting that two otherwise identical properties withdifferent depreciation rates will be priced differently in the market.

Numerical Result 5. The optimal rotation length decreases as the discount rate increases.

The intuition flows directly from that standard thinking in resource economics; a higher discount rate places more

weight on the near term. In rotation problems, this implies that the marginal net benefits of a single rotation receive more

weight than the present value of future rotations. The manager shrinks the rotation length to trade more net benefits in the

short run for less net benefits in the long run. Note that Figs. A1A4 all illustrate Numerical Result 5.

Numerical Result 6. The optimal rotation length increases (decreases) as the exponential decay rate of nourishment sand

increases if the decay rate is higher(lower) than the discount rate.

Fig. 6illustrates this result. The thick solid line is the 451line along whichy d. Recall that in the traditional Faustmann

problemand the interpretation due to Samuelson [29]the second term in the first-order condition is the foregone

interest payment by delaying land rent, where land rent is the discounted present value of all future rotations. A higher

discount rate means that the difference between marginal costs and marginal benefits of a single rotation must be greater

to compensate. To the left of the 451line inFig. 6, the discount rate is higher than the decay rate of nourishment sand. In

other words, the depreciation of financial capital (foregone interest) dominates the depreciation of sand capital. To balance

interest lost on land rent with marginal net benefits, the rotation length must decrease as the exponential decay rate of

sand increases. The opposite is true to the right of the 451line where depreciation of sand capital dominates. In essence, if

the discount rate exceeds the decay rate, an increase in decay rate is relatively more costly to the short run because one

places more weight on the short run with a high discount rate, and rotation length is shortened to adjust.

5. Endogenous initial widthallowing the extensive margin to change

We now consider whether our most surprising resultthat input demand for sand can slope upwards or

downwardscontinues to hold if initial beach width (x0) is a choice variable. The analytical approach to this extension

is straightforward; we now have an additional choice variable in (9) and a second first-order condition. However, x0enters

both the benefit and cost functions, making analytical propositions problematic and requiring numerical analysis.

Numerical Result 7. With endogenous initial beach width, input demand for sand can slope upwards for low variable costs but

turns downwards for high variable costs.

For the same parameter values used above, we find that with endogenous x0 an increase in variable cost still decreases

the nourishment interval. However, it also decreasesx0. Communities are nourishing more often but building the beach out

less when costs increase. When initial beach width is unconstrained, the net result is that sand use decreases when variable

cost of sand increases. That is, input demand slopes downwards. Fig. A5 (available in the online appendix) illustrates this

result for a range of values of baseline erosion. Conditional on baseline erosion (fixing a particular level on the y-axis), avariable cost increase (moving along the x-axis) leads to less sand use (a lower sand use contour).

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The generality of downward-sloping sand demand is far from clear because there are physical limits onx0, i.e. feasibility

constraints on how far out a beach can be built. When these physical constraints are put in the problem, there is a range over

which managers always choose the maximum x0. The desired (unconstrained) x0 is essentially greater than the physical

maximum. As a consequence, a variable cost increase shrinks the desired x0 but not necessarily enough to be below the

maximum feasiblex0. Because rotation length also shrinks with the increased variable cost, the amount of sand used increases.

That is, input demand can slope upwards. Fig. 7(a) illustrates this result for the case of high erosion and low cost of nourishment

sand. We find that upward-sloping input demand cannot be sustained at high costs of nourishment sand; eventually sanddemand turns downwards.Fig. 7(b) illustrates this result for the case of high erosion and high cost of nourishment sand.

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4.84944.84944.8494

4.99074.99074.9907

5.1325.1325.132

5.27335.27335.2733

5.41465.41465.4146

5.5565.556

5.556

5.69735.6973

5.6973

5.8386

5.83865.8386

5.97995.9799

5.9799

6.12126.1212

6.1212

Cost of Nourishment Sand

BaselineErosionRate

Sand Use

0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

3.156

6

3.371

1

3.37

11

3.585

7

3.58

57

3.58

57

3.80

02

3.80

02

3.80

02

4.01

47

4.0147

4.01

47

4.01

47

4.22

93

4.22

93

4.22

93

4.44

38

4

.4438

4.4438

4.65

83

4.65

83

4.87

29

5.0874

Cost of Nourishment Sand

BaselineErosionR

ate

Sand Use

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.51.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Fig. 7. (a) Sand use with endogenous but physically constrainedx0high erosion, low sand cost. (b) Sand use with endogenous but physically constrained

x0high erosion, high sand cost.



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To summarize, when the background erosion rate is high, greater nourishment activity will increase the scarcity of

nourishment-quality sand and likely raise its price. For communities that are already building out their beaches to the

maximum feasible width, this will lead to an acceleration of nourishment (increased sand use) in the short run. If the price

of sand continues to rise, eventually communities will find it in their interest to reduce the initial beach width as they

reduce the nourishment interval. As in the case with exogenous x0, eventually sand becomes so expensive that abandoning

the shoreline defense is optimal.

6. Discussion

Resource economics fundamentally deals with connections between natural capital and financial capital. Optimal

mineral extraction hinges on whether capital in the bank grows faster than the value of capital in the ground. In fisheries

and forestry, optimal management entails balancing growth of the biological resource with growth of the financial

resource. Viewing a managed beach as a renewable natural resource, one sees again this deep capital-theoretic connection.

Optimal management from a beach communitys perspective depends on the rates of decay of sand and financial capital.

When nourishment sand decays slower than the rate of foregone interest, an increase in variable cost or increased

nourishment sand decay could accelerate beach nourishment. This result is more nuanced when initial beach width is

made endogenous, but it does not disappear entirely.

By focusing on a representative community, our model produces testable hypotheses about real coastal communities in

general. For empirical work, one would need cross-community variation in erosion rates, base property values and

values of beach width (from hedonic studies), and records of nourishment activities. Such empirical work is especiallyimportant for future research because actions of individual communities could have consequences at the spatial scale of an

entire coastline. That is, the effects of small bumps in an otherwise uniform coastline can propagate over large spatial

scales in economically meaningful time scales [1]. If our positive model describes real behavior, spatially heterogeneous

property values and erosion rates will lead to spatially heterogeneous nourishment interventions. Whether this

process would then lead to a more or less spatially uniform coastline than one that has not been altered by humans is an

open question.

A complementary approach to analyzing beach nourishment would be to use numerical dynamic programming [19].

This approach allows for a more general description of the dynamics such that the state variable evolves as a function of

absolute time, rather than time since the last nourishment event. It also permits the choice of initial beach width for each

rotation. The drawbacks include increased computational burden and the difficulty of generalizing comparative static or

dynamic results. In our model, we are able to obtain some analytical results before relying on numerical methods.

Nevertheless, with no previous literature on beach nourishment as a dynamic capital accumulation problem and with

increasing development pressure in the coastal zone, we submit that our Faustmann-style rotation and numerical dynamicprogramming are both worthwhile to pursue.

Although most of our model results conform to basic intuition, the possibility that nourishment could increase in

frequency as the variable costs increase is unsettling. As sea level rises or storm patterns change in response to climate

change, an increase in the baseline erosion rate is not unlikely. At the same time, more people are living in the coastal zone,

and property values continue to rise. Nourishment sand that can be recovered through dredging may eventually become

scarce [10,15], and one would expect variable costs to increase. If communities respond by nourishing more often, this

effect could feed back on itself and further accelerate nourishment. Beaches then become increasingly artificial, and the

ecological costs of nourishment would grow substantially. In the current policy regime, environmental impacts of

nourishment are cataloged but not counted as costs to weigh against nourishment benefits. In a future of accelerated

nourishment activity, policy-makers could begin to require these non-market costs to be counted.

Our model does not address the funding mechanism for beach nourishment. We implicitly assume that a dynamically

optimal nourishment projectfrom the communitys perspectivecan be funded in some manner. Real projects receive

funding from federal, state, and local governments as well as private sources. Both the William J. Clinton and George W.Bush administrations have supported reducing the federal share of nourishment, though Congress passed the 2002 Water

Appropriations Bill with $47.1 million more for beach nourishment than the $87.6 million that President Bush requested

[23]. An important question for future research is how a change in the availability of federal funds for nourishment will

affect the prevalence and frequency of nourishment. Conventional wisdom suggests that there would simply be less

nourishment, but our capital-theoretic model suggests that outcomes could be more complicated. Whether a reduction in

the federal share will increase or decrease nourishment activity could depend on the allocation of federal funds across fixed

and variable costs.

Acknowledgments

This research was funded by the NSF Biocomplexity Program (Grant #DEB0507987). We thank Sathya Gopalakrishnan

for valuable research assistance and Tom Crowley, Ling Huang, Mike Orbach, Joe Ramus, Jim Wilen, Junjie Zhang, and twoanonymous reviewers for helpful comments.

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Appendix A. Supplementary data

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jeem.2008.07.011.

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Beach Nourishment as a Dynamic Capital Accumulation Problem

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