watchfully waiting: medical intervention as an optimal investment decision
TRANSCRIPT
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Journal of Health Economics 31 (2012) 349– 358
Contents lists available at SciVerse ScienceDirect
Journal of Health Economics
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atchfully waiting: Medical intervention as an optimal investment decision
lisabeth Meyera,∗, Ray Reesb,1
Helmholtz Center Munich, Ingolstaedter Landstr. 1, 85764 Neuherberg, GermanyCenter for Economic Studies (CES), University of Munich, Schackstr. 4, 80539 Munich, Germany
r t i c l e i n f o
rticle history:eceived 23 April 2009eceived in revised form 29 January 2012ccepted 2 February 2012vailable online 10 February 2012
EL classification:18
a b s t r a c t
Watchfully waiting involves monitoring a patient’s health state over time and deciding whether to under-take a medical intervention, or to postpone it and continue observing the patient. In this paper, we considerthe timing of medical intervention as an optimal stopping problem. The development of the patient’shealth state in the absence of intervention follows a stochastic process (geometric Brownian motion).Spontaneous recovery occurs in case the absorbing state of “good health” is reached. We determine opti-mal threshold values for initiating the intervention, and derive comparative statics results with respectto the model parameters. In particular, an increase in the degree of uncertainty over the patient’s devel-
8161
eywords:atchfully waiting
eal optionsedical decision making
opment in most cases makes waiting more attractive. However, this may not hold if the patient’s healthstate has a tendency to improve. The model can be extended to allow for risk aversion and for sudden,Poisson-type shocks to the patient’s health state.
© 2012 Elsevier B.V. All rights reserved.
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. Introduction
“Watchfully waiting” involves monitoring a patient’s healthtate over time and deciding whether or not to undertake a med-cal intervention – an operation, course of treatment or somether medical procedure. This process is characterized by thehree essential properties of an investment decision: the exis-ence of uncertainty, in this case concerning the development ofhe patient’s health state and, possibly, the outcome of medicalntervention; irreversibility, in the sense that the results of thentervention may be impossible or extremely costly to reverse; andexibility of timing, implying that a key aspect of the decision isxactly when to make the intervention. At each point in time, theres the possibility of making the intervention or of postponing it andbserving how the patient’s health state evolves. Recent develop-ents in the theory of investment under uncertainty, also referred
o as real options approach,2 have greatly increased our under-tanding of how such decisions should be taken, and this paper is
∗ Corresponding author. Tel.: +49 89 3187 3747; fax: +49 89 3187 3375.E-mail addresses: [email protected] (E. Meyer),
[email protected] (R. Rees).1 Tel.: +49 89 2180 3914; fax: +49 89 2180 3915.2 See Dixit and Pindyck (1994) for an introduction.
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oncerned with the application of this theory to the watching andaiting process.
A key insight of the theory concerns the option value of waiting. standard proposition of traditional investment decision theory
s that an investment should be undertaken if (when) the presentalue of its benefits exceeds the present value of its costs. This how-ver ignores the possibility that information about the inherentlyncertain costs and benefits may be revealed over time, and so itay pay to postpone the investment beyond this point in order to
ollect more information. That is, the option to wait has a positivealue, while making the investment, in the present case carryingut the medical intervention, extinguishes this option and sacri-ces this value. Thus, typically, the net present value will have toe – possibly substantially – greater than zero to compensate forhe loss of the option value.
In this paper we provide a formal model of the watching andaiting process and derive threshold values for initiating medi-
al intervention. This amounts to an optimal stopping problem:he patient’s health state follows a stochastic process, and medi-al intervention terminates this process and restores the patient’sealth. We also allows for the possibility of spontaneous recovery.his occurs if the patient’s health state reaches the absorbing state
f “good health”.In our baseline model, the patient’s development is describedy a geometric Brownian motion process. Here, increases in thereatment cost and in the average rate of change of the health state
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50 E. Meyer, R. Rees / Journal of H
aise the optimal threshold value, so that waiting becomes optimalor a wider range of health states. According to the standard theoryf investment under uncertainty, an increase in the variance of thetochastic process also makes waiting more attractive, because itaises the option value. This does not always apply in our model: ifhe patient’s health state has a tendency to improve, an increase inhe degree of uncertainty may work in favour of medical interven-ion. Intuitively, spontaneous recovery becomes less likely in thisase, because uncertainty raises the probability that the patient’sealth state evolves against its central tendency.
The model can also be extended to allow for risk aversion onhe patient’s side. We also introduce a second type of uncertainty,epresented by a Poisson jump process. This captures the possibil-ty of sudden, discontinuous deterioration in the patient’s healthtate. Depending on the degree of deterioration, a health shock mayake immediate intervention optimal. Even if this is not the case,
irect treatment becomes relatively more attractive compared tohe baseline model, to an extent determined by the parameters ofhe Poisson process.
The purpose of the present paper is to explore the applicationf the main ideas of modern investment theory to the watchfullyaiting decision at a general level. We want to clarify the kinds of
nsights that may be gained and the ways that intuitions may behallenged, especially by examining the comparative statics resultsf the models. Clearly, concrete applications to specific types ofedical conditions will require far more detailed models and cali-
ration with real data.The paper closest to ours is Driffield and Smith (2007), who also
iscuss the applicability of the real options approach to watchfullyaiting. Their paper is less general than ours in that they only con-
ider a hypothetical example, and do not derive comparative staticsesults. The authors emphasize that more data needs to be col-ected to corroborate the simulation results. However, they arguehat conventional approaches to medical decision making, suchs decision tree models, suffer from similar or even more severeroblems.
Whynes (1995) establishes a framework for identifying optimalimes of transfer from a watchfully waiting program to direct inter-ention. Spontaneous resolution of the disease is possible, and itsrobability is assumed to increase over time according to specificunctional forms. As a main result, increases in the probability ofelf-resolution and in the costs of direct treatment raise the opti-al time of transfer, making watchfully waiting more attractive,hereas increases in the monitoring costs and in the probability ofisease recurrence have an opposite effect.
Lasserre et al. (2006) are also concerned with the decision wheno initiate medical treatment. They consider the case of HIV therapyn a two-period model. While early intervention has the downsidehat it causes resistance effects, waiting may result in a deteriora-ion of the patient’s health state, so that treatment may no longer beffective. An interesting result is that the magnitude of the resis-ance effect becomes irrelevant for the treatment decision if it isufficiently high.3
In practice, watchfully waiting is often recommended for dis-ases which are progressing only slowly, and where the benefitsf existing therapies are possibly limited. Typical examples are
lowly growing cancers (e.g., lymphoma, prostate cancer), which inany cases would not cause severe problems during the patient’sifetime. Furthermore, watchfully waiting is often the preferred
3 Further removed from the concerns of the present paper, Palmer and Smith2000) and Pertile (2008) apply the real options approach to study the decision todopt new health care technologies. Attema et al. (2010) consider the decision of aountry to stockpile antiviral drugs to prepare for a pandemic.
sa
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conomics 31 (2012) 349– 358
pproach for dealing with common diseases such as children’s mid-le ear infections, where many cases resolve spontaneously. Otherxamples involve inguinal hernia, abdominal aortic aneurysm andild chronic hepatitis C. Generally, recommendations vary accord-
ng to the type and severity of illness, and according to the patient’sge and readiness to comply with a treatment regime that requiresctive monitoring.4
The remainder of the paper is organized as follows. The modelssumptions are presented in the next section. In Section 3, theodel is solved under the assumption that the patient’s health state
volves according to a geometric Brownian motion. In Section 4,wo extensions are discussed: the case of risk aversion, and theossibility of sudden, Poisson-type shocks to the patient’s healthtate. Section 5 concludes.
. The model
We consider a patient who suffers from a particular disease. Herealth state at time t is denoted by ht ∈ [0, 1], where ht = 1 representsgood health”, ht = 0 represents death, and time is continuous. Theatient has a utility function u(ht) with the standard properties:
′(ht) > 0; u′′(ht) ≤ 0
Thus, she is (weakly) risk averse with respect to lotteries overealth states. We normalize u(0) = 0 and u(1) = 1. In the absencef medical intervention, the development of the patient’s healthtate is uncertain, and a central aspect of the model is the choicef a stochastic process with which to characterize this uncertainty.ere, we consider two cases:
.1. Geometric Brownian motion (GBM)
In this case, the development of the health state is describedy a geometric Brownian motion with drift rate and variancearameter � > 0:
dht
ht= ˛dt + �dz
The variable z represents a standardized Wiener process, whosencrement is normally distributed with mean zero and variance dt.hus starting from any value h0 ∈ (0, 1), the patient’s condition mayave a central tendency of improvement ( > 0), stability ( = 0), oreterioration ( < 0), but its development will also be affected by aandom element for the better or worse, to an extent determined byhe value of �. These random changes are uncorrelated over time,nd can cumulatively cause the patient’s health state to diverge sig-ificantly from its central tendency. Moreover, ht = 0 and ht = 1 arebsorbing states. If reached by the stochastic process, the patientemains there for the rest of the time. Thus, reaching ht = 1 woulde “spontaneous recovery” and ht = 0 “premature death”.
.2. GBM with Poisson jump process
In the second case, the patient’s health state still evolves accord-ng to the GBM process defined above, but there may also beudden, discontinuous health shocks whose arrival times follow
Poisson distribution. This process is described by the equation
dht
ht= ˛dt + �dz + dq
4 References for these examples are: Ardeshna et al. (2003), Bill-Axelson et al.2005), American Academy of Pediatrics (2004), Fitzgibbons et al. (2006), Valentinet al. (1999), and Wong and Koff (2000).
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here dq denotes the Poisson component. Defining � as the arrivalate of a Poisson event and � as the rate of deterioration, we assumehat:
q ={
0 with probability 1 − �dt−� with probability �dt
At every moment, there is a probability �dt that the patient’sealth state will deteriorate to a value (1 − �)ht during the next
nfinitesimal time interval (dt). We assume that � is a known, posi-ive number, although the model can be generalized to allow it alsoo be a random variable. Moreover, the cases of non-fatal emergen-ies, where � ∈ (0, 1), and sudden death (� = 1) must be consideredeparately.
The perspective of our model is that of a regulator who max-mizes consumer welfare. The decision-maker can observe theatient’s health state at any time t, and knows the stochasticrocess governing her future development. Whatever the currentealth state ht ∈ (0, 1), medical intervention will restore the patiento good health, but involves a cost c which may be composed of aesource cost and a utility cost (pain, discomfort). Future costs andenefits are discounted at a rate �. The decision-maker internal-
zes the patient’s utility and the total cost. If the resource cost iset out of government funds, it might be augmented by a shadow
rice of public expenditure. Alternatively, the model could be inter-reted to include only the patient’s cost share. These are of courseimplifying assumptions. More realistically, one would expect aositive relationship between the patient’s current health statend the likely treatment outcome. As discussed below, the qual-tative results of the model would still hold under more generalssumptions.5
In the following, the model will be solved for an infinite timeorizon. This makes the problem more easily tractable as timeoes not enter as a state variable. Likewise, we ignore caseshere watchfully waiting is recommended based on the patient’s
dvanced age only. For convenience, we will omit the time indicesrom now on whenever there is no risk of confusion.
. Continuous disease progression
In this section, the problem is analyzed under the assumptionhat the patient’s health state follows a pure GBM process. We alsossume risk neutrality on the patient’s side, thus setting u(h) = h.
.1. Decision under certainty
It is helpful to consider the decision problem under certainty� = 0) first. Here, the patient’s health state evolves according tohe function ht = h0e˛t. Of course, this is not a realistic assumption,ut it serves as a baseline against which the results of the stochasticodel can be compared. If the patient’s condition has a tendency
o get worse, intervention is optimal as soon as the health stateeaches or falls below the level h ≡ 1 − �c. At the critical healthtate, the discounted benefit of remaining in the current healthtate forever (h/�) just equals the discounted benefit of being in fullealth for the rest of time (1/�) minus costs. This type of decisionule was first described by Jorgenson (1963). It implies that waiting
ay be beneficial even if there is no uncertainty. The gain of waitingerives from the fact that the treatment costs are discounted overime, which makes early intervention less attractive.
5 We also abstract from any monitoring costs related to watchfully waiting. Theseould be introduced as a flow cost per time unit.
a
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conomics 31 (2012) 349– 358 351
If > 0, in contrast, there are no gains to waiting. Instead, it isptimal to intervene immediately or not at all. As the patient willecover even without treatment, the question is whether the ben-fit from immediate recovery exceeds the costs. The critical valueelow which medical intervention is optimal is implicitly definedy the function
(h; ˛, �, c) = h
� − ˛− ˛
�(� − ˛)h�/˛ −
(1�
− c)
(1)
here � (h*) = 0. We can state the following:
roposition 1. Under certainty, the threshold value h* is character-zed by
(i) ≤ 0 : h∗ = 1 − �c = hii) > 0: h* < 1 − �c
In the second case, the threshold value is strictly decreasing in theodel parameters ˛, �, and c.6
Intuitively, an increase in leads to accelerated recovery, sohat intervention becomes optimal for a smaller range of healthtates. Notice that the function h*(˛) has a kink at = 0 (see Fig. 4).oreover, higher treatment costs or steeper discounting of future
osts reduce the benefits of immediate intervention. These effectsill be discussed in more detail below.
.2. Characterization of the decision problem
The decision problem under uncertainty can be solved usingechniques of stochastic dynamic programming. We consider anptimal stopping problem: As long as there has been no interven-ion, the decision-maker may either continue waiting, in whichase the patient gets a flow utility from her current health state,r intervene to obtain the discounted benefit of treatment7:
=∫ ∞
0
e−��d� − c = 1�
− c
The optimal time of intervention is chosen according to:
axt
E
[∫ t
0
e−��u(h�)d� + e−�t˝
]
Since the optimal stopping time t* is stochastic, we are effec-ively looking for a threshold value h*, such that interventionecomes optimal for h ≤ h*:
roposition 2. The optimal stopping time is given by:
∗ = inf{t|ht ≤ h∗}Moreover, the value function F(h) is strictly increasing on the inter-
al [h*, 1].
Proposition 2 guarantees that the interval (0, 1) can be uniquelyivided into a continuation (waiting) and a stopping (treatment)egion. A precondition is that the relative benefit of waiting versusirect intervention must be higher for better health states. Thisay still apply under more general assumptions than those made
bove.
According to the Bellman principle of optimality, the valueunction F(h) is defined as the expected discounted sum of pay-ffs (patient utility minus costs) when optimal decisions are made
6 All technical proofs are relegated to Appendix A.7 Recall intervention restores the patient to good health.
3 ealth Economics 31 (2012) 349– 358
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52 E. Meyer, R. Rees / Journal of H
rom the present time period onward. In the waiting region, where ∈ (h*, 1), the following differential equation must be satisfied:
(h) − �F(h) + ˛hF ′(h) + 12
�2h2F ′′(h) = 0 (2)
The derivation of Eq. (2) will not be discussed in detail here,s it has become standard in problems of investment underncertainty.8 In the following, we will briefly outline the methodf solution.
The homogenous part of the differential equation (omitting(h)) has solutions of the form Ahˇ, where
1/2 = 12
− ˛
�2±
√(12
− ˛
�2
)2+ 2�
�2
re the roots of the associated characteristic equation. We define1 as the positive root and ˇ2 as the negative one. Moreover, it cane verified that
0(h) =
⎧⎨⎩
h
� − ˛if /= �
h − h ln h
� + (1/2)�2if = �
s a particular solution of Eq. (2).9 Thus, the general solution is giveny
(h) = F0(h) + A1hˇ1 + A2hˇ2 (3)
here A1 and A2 are constants to be determined.Some remarks are in order here. Firstly, we will not explicitly
onsider the case = �. It is not interesting in itself and also, itay be shown that h* is a continuous function of at that point.
econdly, our approach differs from other models in that we dollow for the case > �. This case is usually ruled out to preventnbounded growth of the state variable. Here, since the healthtate is bounded above, this constraint is not necessary. It shoulde emphasized that the case with positive drift rate applies to lessevere conditions, so from a medical perspective, it may not be aselevant as the opposite case, where health is deteriorating.
To determine the optimal threshold value h* together with A1nd A2, the following boundary conditions are applied:
(1) = 1� − ˛
+ A1 + A2 = 1�
(4)
(h∗) = h∗
� − ˛+ A1(h∗)ˇ1 + A2(h∗)ˇ2 = 1
�− c (5)
′(h∗) = 1� − ˛
+ A1ˇ1(h∗)ˇ1−1 + A2ˇ2(h∗)ˇ2−1 = 0 (6)
Eq. (4) says that the patient will remain in good health once sheas reached h = 1. Eq. (5) implies that at the critical health state, theecision-maker is indifferent between waiting and intervention.ccording to Eq. (6), the two branches of the value function – for
he waiting and stopping region – must meet tangentially at thisoint. This is often called the “smooth-pasting” condition.10 As theenefit of treatment is assumed to be independent of h, we have
′(h*) = 0.While in the standard model of investment under uncertainty,he constant A1 drops out, it generally takes a value different from
8 See Dixit and Pindyck (1994) and Øksendal (2003) provides a more formal treat-ent.9 Note that the particular solution for the case = � is based on the indefinite
ntegral of the logarithmic function, h ln h − h, which causes the terms of Eq. (2) toancel out.10 See Dixit (1993). Intuitively, a small change in the health state only has second-rder effects around the optimal threshold value, see Shackleton and Sødal (2005).
ig. 1. Value function for = − 0.2, � = 0.2, � = 0.05, c = 1; represented by the hori-ontal line for values h < h*.
ero in our model. As a consequence, the value function is notlways convex: rather, convexity is guaranteed only if < 0 and1 > 0 (Fig. 1). In the opposite case, the value function is convex foralues close to h*, but concave for higher values of the health stateFig. 2). As will be discussed below, this property may lead to inter-sting comparative statics results. In Figs. 1 and 2, notice that thealue function corresponds to the horizontal line for health statesn the range (0, h*], and to the solution function of Eq. (2) for healthtates in the interval (h*, 1].
Unfortunately, the optimal threshold value cannot be obtainednalytically. However, the existence of a solution can be directlyroven.
roposition 3. A unique solution for the optimal threshold value h*xists in the interval (0, h).
Since h corresponds to the Jorgenson criterion for the treat-ent decision under certainty, we can infer that the introduction
0.4 0.6 0.8 1 1.2 1.418.5
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19.5
h
h*=0.49
F
Fig. 2. Value function for = 0.2, � = 0.2, � = 0.05, c = 1.
E. Meyer, R. Rees / Journal of Health Economics 31 (2012) 349– 358 353
0.02 0.04 0.06 0.08 0.10.7
0.72
0.74
0.76
0.78
0.8
ρ
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α = − 0.2
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0.4
0.5
0.6
0.7
0.8
0.9
1
α
h*
σ = 0
σ = 0.2
σ = 0.06
Fig. 4. Effect of on h*.
0.1 0.2 0.3 0.4 0.5 0.6
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1h*
α = − 0.2
α = 0.2
α = 0.04
Rta
FosibfiCfit
As mentioned, if > �, an increase in the variance term may havea positive effect on h* (as shown for the case = 0.2), even thoughits size tends to be very small. A key factor here is the probability
Fig. 3. Effect of � on h*.
.3. Comparative statics
How does the solution vary with the model parameters? To get clearer picture of the direction and magnitude of the effects, weonducted a number of numerical simulations. For our central case,e chose the parameter values = ± 0.2, � = 0.2, � = 5% and c = 1. We
onsidered a wide range of variations around these values, and thexamples shown here are representative of the general effects.
The effects of the parameters c, and � are similar as in theeterministic case:
esult 1. The optimal threshold value for the treatment decisionnder uncertainty is decreasing in the treatment cost c, in the driftate and in the discount rate �.
Intuitively, an increase in the cost c only affects the payofff medical intervention, which becomes less attractive comparedith watchfully waiting. Thus in equilibrium, the threshold valueust be reduced. This can be shown by totally differentiating the
quation system (4)–(6). In contrast, an increase in raises thexpected value of future health states in the absence of inter-ention. In this case, the comparative static expressions becomeifficult to interpret. However, the negative effect of can behown to follow from the monotonicity of the decision problem,stablished in Proposition 2. That is, if increases, the optimalhreshold value must be lowered to keep the decision-maker indif-erent between waiting and immediate intervention. Finally, anncrease in � also has a negative effect: as the discount rate repre-ents the opportunity cost of present versus future investment inealth, an increase in this variable makes waiting more attractive.his is shown in Fig. 3.
Fig. 4 illustrates the effect of the drift rate under certainty andncertainty. If only a small amount of uncertainty is introduced� = 0.06), the main effect is that the kink in the function h*(˛) ismoothed out. The figure also shows that an increase in the varianceerm � has ambiguous effects: for negative values of ˛, a decreasen the optimal threshold value is observed as the level of uncer-ainty increases. This reflects the positive option value of waiting.
n contrast, if ≥ �, the opposite may be true: for higher values of, the solid line representing the case � = 0.2 lies (slightly) abovehe graph of the deterministic model, and that of the case � = 0.06.
σFig. 5. Effect of � on h*.
esult 2. If < �, an increase in the variance term � has a nega-ive effect on the optimal threshold value h*. If > �, the effect ismbiguous.
ig. 5 shows the optimal threshold value as a function of the levelf uncertainty. In cases where ˛ < �, the intuition known fromtandard real options models is at work. Waiting allows a morenformed decision to be made, in the sense that intervention cane avoided in cases where it turns out to be unnecessary or inef-cient, and this advantage increases with the level of uncertainty.onvexity of the value function (as in the example of Fig. 1) is a suf-cient, but not necessary condition for the optimal threshold valueo depend negatively on �.11
11 See Meyer (2009).
3 ealth Economics 31 (2012) 349– 358
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54 E. Meyer, R. Rees / Journal of H
ith which the patient recovers in the absence of treatment. Heren increase in uncertainty reduces the likelihood of spontaneousecovery, thus making medical intervention relatively more attrac-ive. To see why this is true, consider the case without uncertainty,
= 0. Here, the patient recovers for sure whenever is positive. Inontrast, introducing uncertainty reduces the probability of recov-ry below one. While this effect is present for all > 0, however, ithould be mentioned that the positive effect on the optimal thresh-ld value will occur only for higher values of the drift rate.12
To summarize our results so far, we found that uncertaintyakes watchfully waiting more favourable if the patient’s health
ends to get worse, or if the rate of improvement lies below theiscount rate. This applies even to medically severe conditions. Ithould be emphasized that this result is based on the assumptionhat medical intervention will still be effective at a later point inime if delayed. Also, the option value of waiting is higher in caseshere the patient’s health state is relatively stable than in cases
f rapidly progressing disease. Fig. 4 illustrates this point: the dif-erence between the graphs for the deterministic and stochasticases is largest if is close to zero. This finding is consistent withhe fact that watchfully waiting is often recommended for slowlyrogressing diseases.
.3.1. Probability of treatmentSo far, we have considered the threshold value h* as a measure
or the level of treatment provided to the patient. Alternatively, oneight be interested in the probability that medical interventionill be necessary, i.e., that the threshold value h* is reached by the
tochastic process. Starting from an initial health state h0 ∈ (h*, 1),his is given by13:
=
⎧⎪⎨⎪⎩
h1−2˛/�2
0 − 1
(h∗)1−2˛/�2 − 1if /= �2
2ln h0
ln h∗ if = �2
2
The probability of reaching a given boundary h* is increasing inhe degree of uncertainty if the patient’s health state is improving
> 0), and vice versa.14 Notice that there is zero probability thathe health state evolves in the interval (h*, 1) without ever reach-ng one of the boundaries. Thus the probability of spontaneousecovery amounts to 1 − �.
Now, consider the effects of and � on the probability of medicalntervention. An increase in the drift rate reduces the probabilityhat the lower boundary is reached, and also causes a reductionn h*. Both effects work in the same direction, making medicalntervention less likely. If is negative, an increase in the levelf uncertainty also has unambiguous effects: in this case, the prob-bility of medical intervention is reduced. In contrast, if ˛ > 0, anncrease in the degree of uncertainty may raise the probability of
ntervention because a worsening of the health state becomes moreikely. This may be true even in cases where the threshold value forhe treatment decision is reduced.12 A similar effect is discussed by Brock et al. (1989), who also consider a prob-em with absorbing barrier. The authors showed that the option value may indeedecome negative in this case. Nevertheless, this case has not been further explored
n the applied literature. We believe that the our example of spontaneous recoveryrom a disease provides an intriguing application of the theory.13 See Dixit (1993).14 See the discussion above. The proof is available on demand.
4
caotrnh
s
λFig. 6. Effect of � on h*, where = 0.2, � = 0, � = 0.4.
. Extensions
In this section, we discuss two extensions of the baseline model:rstly, the case of risk aversion, and secondly, the possibility ofudden deterioration of the patient’s condition. Since the effectsf risk aversion cannot be satisfactorily examined without makingore specific assumptions on the disease, the focus will be on the
econd issue.
.1. Risk aversion
To allow for the more realistic case of risk aversion on theatient’s side, it is convenient to assume constant relative riskversion. Using the utility function u(h) = h , for ∈ (0, 1), the cal-ulations of the last section can be generalized, and comparativetatics results with respect to the parameter can be derived.s is known from the literature on decision-making under uncer-
ainty, equals one minus the coefficient of relative risk aversion.he question of how risk aversion affects the decision for orgainst watchfully waiting should be considered at a more generalevel: Above, we assumed that the outcome of medical inter-ention is deterministic, whereas waiting involves uncertainty. Inhis context, risk aversion works against watchfully waiting. Moreenerally, however, the effect may depend on the level of risk asso-iated with medical intervention. In particular, surgery carries aisk of death in the short run, which makes it less attractive forisk averse patients. This point deserves further attention and willopefully be examined in future studies.
.2. Sudden deterioration of the health state
So far, it has been assumed that the patient’s health state evolvesontinuously, without any “dramatic” changes. In practice, leavingn illness untreated may carry a risk of sudden and severe deteri-ration. For example, patients suffering from inguinal hernia facehe risk that the hernia may become incarcerated, a problem whichequires immediate medical attention. Also, patients with abdomi-
al aortic aneurysms face a risk of rupture, which is associated withigh mortality rates.In general, the model with Poisson jumps can capture theseituations quite well. Notice that the stochastic process we
E. Meyer, R. Rees / Journal of Health E
0 0.2 0.4 0.6 0.80.5
0.55
0.6
0.65h*
ctDmtpiise
mbtdm
Ri
ir
rrocb
�Gpvfsavf
5
tuingtiddttt
it
φ
Fig. 7. Effect of � on h*, where = 0.2, � = 0, � = 0.4.
onsider is not limited to describing one singular event, but rather,he patient’s condition deteriorates in steps as health shocks arrive.epending on the extent of deterioration, medical interventionay or may not be optimal when a health shock occurs. Formally,
he continuation region can be split into two parts: in the lowerart, where h(1 − �) < h*, medical intervention becomes optimal
mmediately after a Poisson jump. In contrast, if h(1 − �) > h*, wait-ng remains superior. Depending on the model parameters, theecond case may not exist. That is, if h* < 1 − �, a jump will alwaysnd in the stopping region.
An interesting finding is that in the first case, the optimal treat-ent decision will not depend on the actual rate of deterioration,
ut only on the arrival rate of a Poisson event �. Thus for “catas-rophic” changes in the patient’s condition, the exact degree ofeterioration is no longer relevant. Proposition 3 summarizes the
ain results of this section.esult 3. In the model with Poisson jumps, the threshold value h*s a strictly increasing function of the arrival rate �. Also, it is strictly
0 0.2 0.4 0.6 0.8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
h*
α = − 0.2
α = 0.2
Fig. 8. Effect of � on h*, where � = 0.2, � = 1.
tcfst
twiastwuch
sidetva
conomics 31 (2012) 349– 358 355
ncreasing in � ∈ (0, 1) up to a certain value of the deteriorationate.
The proof and calculations for the problem with Poisson jumpsely on the following idea: First, define a cutoff value for the cur-ent health state h, such that medical intervention is just (weakly)ptimal when a health shock occurs: that is, h(1 − �) = h∗. Then,onditions are imposed to make sure that the upper and lowerranches of the value function meet tangentially at h.
Figs. 6 and 7 illustrate the comparative statics effects of � and, respectively. For simplicity, we assumed that the variance of theBM process is zero, so that all uncertainty results from the Poissonrocess. Finally, it should be mentioned that the optimal thresholdalue increases discontinuously when the health shocks becomeatal (� = 1). In this case, a medical intervention may come too late,o the net present benefit of treatment can no longer be guar-nteed as a minimum utility level. As a consequence, the thresholdalue may even exceed the Jorgenson level h. Fig. 8 illustrates thisor varying values of �.
. Conclusion
We have shown that watchfully waiting represents an impor-ant treatment alternative in cases where the course of illness isncertain. The existence of uncertainty raises the benefit of wait-
ng whenever the patient’s condition has a tendency to get worse,et of the discount rate, and particularly when the disease is pro-ressing slowly. When there is a strong tendency to recovery, onhe other hand, an increase in the level of uncertainty may workn favour of immediate intervention. Moreover, increases in therift rate of the GBM process, in the treatment costs and in theiscount rate have a negative effect on the threshold value for thereatment decision. In contrast, the possibility of sudden shocks tohe patient’s health state works in favour of immediate interven-ion.
The model is based on the assumption that the benefit of med-cal intervention, relative to watchfully waiting, is decreasing inhe patient’s health state. For some diseases, the opposite may berue: that is, medical treatment may be effective only in less severeases, whereas waiting (or palliative treatment) would be indicatedor severely ill patients. In that context, watchfully waiting corre-ponds to a passive treatment strategy, and the chances of curativereatment are small.
We believe that the more interesting areas of application arehose where curative treatment is still possible after a period ofaiting. For instance, several types of cancer are slowly progress-
ng, but can be diagnosed through modern screening methods at very early stage. In many cases, the disease would not becomeymptomatic during the patient’s lifetime, and there are concernshat screening may lead to over-treatment. Therefore, watchfullyaiting plays an important role, even though reliable threshold val-es may not yet be available.15 Arguably, both medical research andonsequent application of the ideas discussed in this paper couldelp improve decision-making.
In deriving the results of this paper, we have made severalimplifying assumptions. For example, we have used rather styl-zed stochastic processes to characterize the progression of theisease. Furthermore, we have assumed that the costs and ben-fits of medical intervention are constant over time, and over
he patient’s health state. While these assumptions may not beery realistic, they are useful to analyze the treatment decisiont a general level. The next step is to apply this approach to15 See Bill-Axelson et al. (2005).
3 ealth E
swtop
A
oLoc
A
A
A
m
∫
h
h
t
A
−i
∫
h
�
dbp
A
p
a
h
�
h
n
A
1
F
dtT
G
at
u
sictoot
A
A
56 E. Meyer, R. Rees / Journal of H
pecific treatment decisions. Then, future studies may find thatatchfully waiting is beneficial also for patients who, according
o current practice guidelines, would receive immediate treatmentr, possibly, for illnesses where active monitoring is not currentlyracticed.
cknowledgements
We would like to thank Hugh Gravelle for a helpful discussionf the ideas underlying this paper, and Mathias Kifmann, Reinereidl, Markus Reisinger and Marco Sahm for insightful commentsn later drafts. Two referees provided very thorough and helpfulomments. Responsibility for the final version is entirely our own.
ppendix A. Proofs
.1. Proof of Proposition 1
.1.1. First part ( < 0)The decision-maker chooses the time of intervention t so as to
aximize the discounted net benefit of the patient:
t
0
e−��(h0e˛�)d� − e−�tc +∫ ∞
t
e−��d�
= h0
− �(e(˛−�)t − 1) − e−�tc + 1
�e−�t
The first-order condition reads:
0e(˛−�)t + �e−�tc − e−�t = 0
This can be solved for the optimal time of intervention and theealth state at that point:
t∗ = − 1˛
ln(
h0
1 − �c
)h∗ = 1 − �c
It can be easily shown that the second-order condition holds at*.
.1.2. Second part ( > 0)Without intervention, the patient recovers at time t =
(ln h0)/˛. At the threshold value, the decision-maker must bendifferent between waiting and intervening, so that:
t
0
e−��h0e˛�d� +∫ ∞
t
e−��d� = 1�
− c
Solving the integrals and inserting t yields Eq. (1).Next, we show that the function � (·) is strictly increasing in the
ealth state. The first derivative can be written as:
h = 1� − ˛
(1 − h�/˛−1) > 0
Clearly, for h ∈ (0, 1), both factors take the same sign. The positive
erivative implies that the optimal threshold value for this case lieselow the Jorgenson value, as it can be shown that � (h) takes aositive sign..1.3. Comparative staticsBy the implicit function theorem, the effects of the model
arameters take opposite signs as the partial derivatives � ˛, � �
(
A
conomics 31 (2012) 349– 358
nd � c. The partial derivative with respect to can be written as16:
�˛ = h
(� − ˛)2− h�/˛
(� − ˛)2+ h�/˛
˛(� − ˛)ln h
Dividing by h/(� − ˛)2 and rearranging yields:
(�/˛−1) ln h(�/˛−1) − h(�/˛−1) + 1 > 0
This inequality holds for all exponents and h ∈ (0, 1).The effect of � is described by:
� = − h
(� − ˛)2+ ˛(2� − ˛)
�2(� − ˛)2h�/˛ − h�/˛
�(� − ˛)ln h + 1
�2
This expression is decreasing in h ∈ (0, 1) and becomes zero for = 1. Thus it must be positive at the optimal threshold value.
Finally, we have � c = 1. In conclusion, all three variables have aegative effect on the threshold value h*.
.2. Proof of Proposition 2
The following proof is based on Dixit and Pindyck (1994, pp.28–130).
The Bellman value function is recursively defined as:
(ht) = max{u(ht) + e−�dtE[F(ht+dt)|ht], ˝(ht)}This reflects the idea that as long as waiting remains optimal, the
ecision-maker will face a similar problem in the next infinitesimalime period (t + dt). For simplicity, we denote F(ht) − ˝(ht) by G(ht).hen:
(ht) = max{u(ht)dt + e−�dtE[˝(ht+dt)|ht] − ˝(ht)
+ e−�dtE[G(ht+dt)|ht], 0} (7)
Now, consider only the first three terms in the maximum oper-tor (related to the waiting region). Dividing by dt and letting dt goo zero yields:
(ht) + limdt→0
e−� dtE[˝(ht+dt)|ht] − e0˝(ht)dt
= u(ht) − �˝(ht) + limdt→ 0
E [d˝(ht)]dt
Clearly, this is increasing in ht since we assume to be con-tant. The last part of the continuation value is also increasing in ht
f the value function is. This is because the stochastic processesonsidered in this paper exhibit positive persistence of uncer-ainty. That is, an increase in ht shifts the probability distributionf future health state towards higher values, in the sense of first-rder stochastic dominance. A fixed-point argument implies thathe solution of Eq. (7) is again an increasing function.
.3. Proof of Proposition 3
.3.1. ExistenceTo prove the existence of a threshold value, we first solve Eqs.
4) and (6) for the constant A1:
1 = 1� − ˛
h∗ − (˛/�)ˇ2(h∗)ˇ2
ˇ2(h∗)ˇ2 − ˇ1(h∗)ˇ1
16 Notice that: dd˛
h�/˛ = − �
˛2 h�/˛ ln h.
ealth E
Af
˚
tb
˚
t“˚(
a
A
isa
˚
A
s
˚
A
1ta
F
tt
b
A
yc
0
St
A
hiifmki
daoFp
A
A
at
u
Ff
F
˛
ˇ
G(h) = h
� + �� − ˛+ B1h�1 + B2h�2 (10)
E. Meyer, R. Rees / Journal of H
An analogous expression results for A2. After inserting A1 and2 into Eq. (5) and rearranging, we obtain the following implicit
unction:
(h) = (1 − ˇ1)h1+ˇ1−ˇ2 − (1 − ˇ2)h + ˛
�(ˇ1 − ˇ2)hˇ1
+ (1 − �c)(
1 − ˛
�
)(ˇ1hˇ1−ˇ2 − ˇ2); ˚(h∗) = 0
We show that ˚(0) and ˚(h) have opposite signs. Continuityhen implies that there exists a root within the interval (0, h). Toegin with, the expression
(0) = (1 − �c)(
1 − ˛
�
)(−ˇ2)
akes the same sign as � − ˛. For the second part, we consider thesymmetric” function ˜ (h) = ˚(h)hˇ2 . This takes the same sign as(h) for h > 0. At h = 1 − �c, the function value reduces to:
1 − ˛
�ˇ1
)h
1+ˇ1 −(
1 − ˛
�ˇ2
)h
1+ˇ2 + ˛
�(ˇ1 − ˇ2)h
ˇ1+ˇ2
Note that sign(1 − (˛/�)ˇ1) = sign(� − ˛), and that 1 − (˛/�)ˇ2 islways positive.
.3.2. Case 1 > � ⇒ ˚(h) > 0
We have ˇ1 < 1 and 1 − (˛/�)ˇ1 < 0. As h is smaller than one,t is clear that h
1+ˇ1< h
2ˇ1 and h1+ˇ2
< hˇ1+ˇ2 . Replace the two
maller terms in ˜ (h) by the greater ones. The following inequalitypplies:
˜ (h) >(
1 − ˛
�ˇ1
)(h
2ˇ1 − hˇ1+ˇ2 ) > 0
.3.3. Case 20 ≤ < � ⇒ ˚(h) < 0Here, we have ˇ1 > 1 and 1 − (˛/�)ˇ1 > 0. Replace h
1+ˇ2 by themaller term h
ˇ1+ˇ2 to obtain the following comparison:
˜ (h) <(
1 − ˛
�ˇ1
)(h
1+ˇ1 − hˇ1+ˇ2 ) < 0
This completes the proof.
.3.4. UniquenessThe following proof relies on ideas of Dixit (1989, Appendix A).The threshold value h* can be written as a function h(A1, A2) ∈ (0,
), implicitly defined by the boundary conditions (4)–(6). Supposehere are two solutions, (A1, A2) and (A′
1, A′2). Eq. (5) can be written
s:
(A1, A2) = h(A1, A2)� − ˛
+ A1 h(A1, A2)ˇ1 + A2 h(A1, A2)ˇ2
Because of the smooth-pasting condition (6), the derivatives ofhe value function with respect to A1/2 reduce to the partial deriva-ives:
dF
dAi= ∂F
∂Ai
= h(Ai, Aj)ˇi for i, j = 1, 2
Due to the upper boundary condition (4), the sum A1 + A2 muste constant. Thus there must be a number a such that:
′1 = A1 + a; A′
2 = A2 − at
conomics 31 (2012) 349– 358 357
The two points can be connected by a line segment (x = A1 + ta, = A2 − ta), for t ∈ [0, 1]. Applying the fundamental theorem of cal-ulus, we obtain:
= F(A′1, A′
2) − F(A1, A2) =∫ 1
0
(∂F
∂A1
dx
dt+ ∂F
∂A2
dy
dt
)dt
= a
∫ 1
0
[h(A1 + ta, A2 − ta)ˇ1 − h(A1 + ta, A2 − ta)ˇ2 ]dt
ince h(·) ∈ (0, 1), we have h(·)ˇ1 < 1 < h(·)ˇ2 . This implies that a = 0,hat is, the solution must be unique.
.4. Comparative statics of and c
Since affects the value function only via the GBM process, itas an unambiguous effect on the optimal threshold value: if ˛
s increased, the expected value of future health state is raised,n the sense of first-order stochastic dominance, and thus theunction F(ht) − ˝(ht) is shifted upwards (see Section 2). By the
onotonicity of the decision rule, h* must be reduced in order toeep the decision-maker indifferent between waiting and directntervention.17
The effect of the treatment cost can be derived by totallyifferentiating the equation system (4)–(6). From the upper bound-ry condition, we obtain dA2 = − dA1. Then, the total differentialf equation the value-matching condition, also considering that′(h*) = 0, simplifies to dA1(hˇ1 − hˇ2 ) = − dc. From the smooth-asting condition, we obtain:
dh∗
dc= ˇ1hˇ1−1 − ˇ2hˇ2−1
F ′′(h∗)(hˇ1 − hˇ2 )< 0
This must be negative because ˇ1 > 0 > ˇ2 and F ′′ (h*) > 0.
.5. Model with Poisson jumps
.5.1. Method of solutionWe define the boundary between the two subregions (discussed
bove) as h ≡ h∗/(1 − �) and assume h < 1. The differential equa-ion (2) generalizes to18:
(h) − �F(h) + ˛hF ′(h) + 12
�2h2F ′′(h) + �{F[(1 − �)h] − F(h)} = 0
(8)
In the lower part of the continuation region, h ∈ (h∗, h), we have[(1 − �)h] = ˝. Under risk neutrality, the value function takes theorm:
(h) = �
� + �
(1�
− c)
+ h
� + � − ˛+ A1hˇ1 + A2hˇ2 (9)
The first two terms constitute the particular solution for the case /= � + �, and the roots ˇ1 and ˇ2 are given by:
1/2 = 12
− ˛
�2±
√(12
− ˛
�2
)2+ 2(� + �)
�2
In contrast, in the interval (h, 1), the value function becomes:
17 See also Friedman and Johnson (1997): the authors show that the tools of mono-one comparative statics can be applied to real options problems.18 See Dixit and Pindyck (1994).
3 ealth E
t
�
mU�
t
G
Afgttmt
A
spo(aT
�
tCstt
Tp
a
R
A
A
A
B
B
D
D
D
D
F
F
J
L
M
M
Ø
P
P
S
V
58 E. Meyer, R. Rees / Journal of H
The special case /= � + �� is omitted, and �1 and �2 are solu-ions of the following equation:
(�) ≡ −(� + �) + ˛� + �2
2�(� − 1) + �(1 − �)� = 0 (11)
Since Eq. (11) has no analytical solution, the roots must be deter-ined numerically. However, it is clear that the function �(�) is-shaped with a positive root �1 and a negative root �2 (because(0) < 0).
To “connect” the two subregions, we apply the following condi-ions:
(h) = F(h), G′(h) = F ′(h)
The first condition requiring continuity is straightforward.ccording to the second condition, the two branches of the value
unction must meet tangentially at h. Intuitively, this is because theeometric Brownian motion crosses h smoothly.19 Together withhe above boundary conditions, we can establish an equation sys-em that can be solved for the four constants and for h*. Finally, it
ust be verified whether the solution fulfills h* < 1 − �, such thathe upper subregion exists.
.5.2. Comparative staticsAn intuitive proof can be derived by approximating the Pois-
on jump process to a process of continuous deterioration in theatient’s health state.20 Assume that the jumps arrive infinitelyften (�→ ∞) and that the jump size becomes infinitesimally small� → 0). Furthermore, let the limiting processes be such that theverage rate of deterioration is fixed at ≡ ��. Now, we apply aaylor expansion to the term �F[(1 − �)h]:
F[(1 − �)h] = �(
F(h) − �hF ′(h) + 12
(�h)2F ′′(h) + · · ·)
= �F(h) − hF ′(h)
The second-order term can be ignored (because the factor �ends to zero), and the same holds for all terms of higher order.omparing the differential Eq. (8) to that of the case without Pois-on jumps, this implies nothing else than a decrease in the drift rateo − . Therefore, an increase in raises the threshold value for
he treatment decision.Clearly, for h ∈ (h∗, h), the value function is independent of �.hus the solution only depends on the arrival rate of the Poissonrocess, but not on the extent of deterioration.
19 We do not show this formally here. Dixit and Pindyck (1994, pp. 130–132)., give proof of the smooth-pasting condition (6), which can be applied here as well.20 See also Murto (2007).
W
W
conomics 31 (2012) 349– 358
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