Appendix Derivation of Euler Equation – Substitution Method A social planner chooses harvesting levels to maximize:

subject to (1)-(6) and (10a) from the main body of the paper. The need to link harvests with the preference parameter

requires a single optimality condition. As all constraints are given by

equalities, a method of substitution is followed (Azariadis, 1993). All the biological and economic constraints are substituted into (A.1), changing the choice variable from

to

Later in the appendix we show that this method is equivalent to the method of Lagrangian multipliers. First, labor endowments are normalized to one such that

. Noting that

and substituting (10a), we can rewrite (A.1) as:

To account for the ecological components, we start by substituting (4) and (5) into (3) to get:

Substituting (6) and (2) into (A.3) then yields:

.

Finally, substituting (1) into (A.4) yields:

Due to the recursive nature of the MPB dynamics and the forest stocks, one must continually substitute π, B, X and Y into (A.5) to fully account for the shadow value of B, X, and Y. The resulting expression can then be substituted into (A.2) to fully account for equations (1)(6) and (10a). This substitution procedure also changes the choice variables from harvests to the stock of adult trees. A similar solution procedure is described in Azariadis (1993) and was used in previous research to study the effect of cattle cull rates on the age structure of a cattle stock (Aadland, 2004). Following the series of substitutions outlined above and taking derivatives with respect to

yields our first-order condition for welfare maximizing levels of harvest:

where (A.7) (A.8)

Equations (A.7) and (A.8) indicate the complexity of the “seed effect” terms succinctly presented as

and

in the text.

Alternative Dynamic Optimization – Method of Lagrangian Multipliers An alternative solution method is to use the method of Lagrangian multipliers. The present value Lagrangian expression for the problem is:

where

is the rate of fire ignition (not a shadow price). The key variables introduced by the

optimization procedure are the co-state variables given by

. They can be interpreted as the

(shadow) value of an additional unit of each state variable (

) in period

. They provide a signal to the decision maker in period of the opportunity costs or gains of harvests. First-order conditions with respect to the control and state variables ( , and

) are:

,

,

,

First-order conditions with respect to the co-state variables (

) are:

Optimality condition (A.10) requires harvesting to be expanded until the net marginal benefits of harvesting in the current period just equals the marginal cost of harvesting (in terms of forgone use of labor to produce Q):

Net marginal benefits subtract the opportunity cost of harvests value of an additional adult tree in period

, which is the discounted

.

For the harvestable stock of adult trees, equation (A.11) can be rewritten as:

When adult trees are optimally harvested, the value of an additional unit of harvestable adult trees in period ,

has three components. The first term is the marginal non-market value of

an additional standing adult tree. The second term is the forgone value of harvests from leaving the tree standing. The third term is the

marginal value of an adult tree left standing in

period . The value of adult trees throughout the ecosystem is differentiated from that of the harvestable stock following (A.12):

The value of an additional adult comes from three additive sources. The first follows from its availability for harvests where this magnitude is diminished by natural mortality ( ), the probability of a successful beetle attack ( ) and the fire risk – not only in total ( marginal increase in fire risk from an additional adult (

) but the

). The second is through

the contributions of adults in period to the seed base in period

. The third is a reduction in

the value of an additional adult tree that follows from an increase in beetles. Equation (A.13) provides the value of an additional unit of the seed base under optimal harvests:

The seed base in has value through its (net) own growth plus its contribution to the young tree class. The value of an additional young tree from (A.14) is:

The first term captures the value of a young tree that transitions to being a harvestable adult and the contribution to the marginal fire risk. A young tree also has value as it contributes to the seed base (second term) and contributes to its own growth (third term) net of its contributions to adults and fire mortality. The final value the optimal program uncovers is that of the beetle stock from (A.15):

An additional beetle in period causes a loss of harvestable adult trees (first term) and a loss in value due to the increase in the

stock of beetles (second term).

Equations (A.16) through (A.20) require the ecosystem dynamics to follow the given laws of motion. The equations of optimality must be simultaneously solved over all time periods given the initial conditions on the state variables and the transversality conditions (Azariadis, 1993, p.211):

for

While little insight can be garnered analytically about optimal choices

over time, some analytical insight can be found from an inspection of the optimized steady state, an analysis of which is available from the authors by request.

Reconciling the Substitution and Lagrangian Multiplier Methods In contrast to the more traditional Lagrangian multiplier method, we chose to present the results from the substitution procedure as described in the first section of this Appendix. The substitution approach has two benefits for our application. First, the effect of all eleven firstorder conditions (A.10) through (A.20) can be conveyed in a single equation found by equating (11) and (13) in the main text. Second, while shadow values capture the net benefits of the state variables, they are not the best tool to highlight the dual role that adult trees serve. Adult trees provide timber (and nontimber) benefits while also serving as the hosts that support future MPB populations. Shadow values collapse all these future values into a single measure. We found it more straightforward to convey this tension between leaving adult trees and harvesting adult trees using the substitution method, which decomposes the shadow values into their benefit and cost components. Next, we reconcile the substitution and Lagrange procedures by showing that they produce the same solution. Start by setting (A.10) equal to (A.11) and solving for

Equation (A.21) shows that

:

from the main text is the shadow value of harvestable adult trees.

Plugging this expression into (A.12) evaluated in period

, we find

Multiplying through by , simplifying the third term, and substituting for

Substituting for

Substituting for

from (A.15a) and

and rearranging:

:

from (A.10):

Each subsequent substitution for

introduces a new shadow value for the harvestable adult

stock and MPB stock in the next period. In the text, we represent this intertemporal relationship between MPB shadow values using a time summation from

to infinity. In words, a change

in the MPB population in one period has impacts on future populations due to the recursive nature of the beetle dynamics. The relationship between

and

reflects the fact that beetles

kill adult trees, which in turn eliminates timber and nontimber values. This effect can be expressed as a sum of chained partial derivatives and incorporated into (A.22) to give:

Substituting for

Substituting for

from (A.13a):

from (A.14a):

Substituting for

and

Each subsequent substitution for

from (A.13a) and combining like terms:

and

introduces a new shadow value for the seed and

young tree stock in the next period. However, each subsequent substitution for

also

introduces a new shadow value for the harvestable adult stock. This introduction of a new in each period gives rise to a new produce two

(and thus

in each period as well. Repeated substitutions after

) in each period. One

captures the fact that adult trees provide

the seeds needed for forest regeneration (a benefit). The other

captures how new seeds

produce young trees that exacerbate fire severity and kill adult trees in the future (an opportunity cost). In the text, we represent both of these impacts using a summation (from

to infinity)

of chained partials similar to the notation employed with the MPB term. Employing this “summed partial derivative” notation and rearranging we can rewrite the above expression as:

This expression is identical to the first-order condition presented in the main paper. The left side of the expression is equation (13) while the right side is equation (11).

Sensitivity Analysis Our model incorporates climate change, changing preferences for public forests, and fire suppression to produce a MPB outbreak that peaks at 30.903 trees per hectare killed by MPB. When the impact of climate change is held constant, the peak of the MPB outbreak is lowered by 19.007 trees per hectare. This implies that MPB outbreak severity would decrease by 61.5% in the absence of climate change. When preferences for non-timber ecosystem services are held constant, the peak of the MPB outbreak is lowered by 11.663 trees per hectare. This implies that MPB outbreak severity would decrease by 37.7% if preferences for non-timber ecosystem services remain unchanged. Removing the influence of fire suppression reduces the peak of the MPB outbreak by 1.730 trees per hectare. This implies that the absence of fire suppression would lower MPB outbreak severity by 5.6%. This section investigates the sensitivity of these results to the choice of parameter values presented in Table 1. Due to the complexity of the model, an analytical comparative analysis is not possible. Instead we report how the percent increase in MPB-induced tree mortality changes for a 5% change in each parameter value (Table A1).

Table A1. Sensitivity of model results to a 5% change in parameter values % decrease in maximum MPB-induced tree mortality due to the absence of:

Results with benchmark parameter values

Climate change

Preference change

Fire suppression

61.5

37.7

5.6

δX

Benchmark parameter values 0.001

(0.00095, 0.00105)

(61.5, 61.5)

(37.7, 37.7)

(5.6, 5.6)

δY

0.004

(0.0038, 0.0042)

(63.1, 60.0)

(38.3, 37.3)

(6.0, 5.2)

bY

0.0018

(0.00171, 0.00189)

(61.9, 61.1)

(38.0, 37.6)

(5.7, 5.5)

bA

0.1

(0.095, 0.105)

(63.7, 59.3)

(38.3, 36.9)

(6.5, 5.3)

a1960

157,653

(149771, 165536)

(60.6, 62.3)

(37.3, 38.2)

(5.6, 5.6)

φ d ν ρ z

4,500 0.015 0.5 0.03255 0.00054

(4275, 4725) (0.01425, 0.01575) (0.475, 0.525) (0.0309, 0.0342) (0.00051, 0.00057)

(62.3, 60.7) (60.4, 62.8) (65.2, 54.3) (61.4, 61.6) (61.3, 61.7)

(38.2, 37.3) (37.8, 37.9) (35.9, 37.0) (36.0, 39.5) (37.6, 37.8)

(5.6, 5.6) (5.3, 6.1) (8.0, 2.3) (5.6, 5.6) (5.5, 5.7)

I1960

100

(95, 105)

(63.3, 59.8)

(38.1, 37.2)

(7.1, 4.9)

β

0.96

(0.9597, 0.9634)

(61.5, 61.5)

(37.8, 37.7)

(5.6, 5.6)

Parameter range

Results across the parameter range

Notes. The percentage decreases sum to slightly more than 100% due to the interaction effects of climate change, preferences, and fire suppression.