Proofs for Stress and Coping - An Economic Approach Klaus Wälde 56 October 2017

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1 A Appendix Proofs for Stress and Coping - An Economic Approach Klaus älde 56 October 2017 A.1 Solution of the maximization problem A.1.1 The Bellman equation e start from the general speci cation of a Bellman equation in continuous time under uncertainty (see älde, 1 or Sennewald, 2007). Given the objective function (11), the state variable (t) and the control m (t), it reads V ( (t)) max m(t) u ( (t)) v (m (t)) + 1 dt E tdv ( (t)) (A.1) The di erential of the value function reads, given the law of motion in (12) for (t) (see e.g. älde, 2012, ch ), n dv ( (t)) V 0 ( (t)) p o b (t) 0 (t) 1 m (t) dt Forming expectations E t yields dv ( (t)) V 0 ( (t)) + fv ( (t) g (t)) V ( (t))g dq g (t) + fv ( (t) (t)) V ( (t))g dq (t) n p o b (t) 0 (t) 1 m (t) dt + E h fv ( (t) g (t)) V ( (t))g g dt + E fv ( (t) (t)) V ( (t))g ( (t)) dt; (A.2) where we assume that the jump size of g (t) in (1) and the frequency of jumps of q g (t) are independent, as is the stress e ect (t) of an outburst and the corresponding Poisson process q (t). All sources of uncertainty are taken into account when the expectations operator E t is applied. As the expected numbers of jump of q g (t) over a small time interval dt is given by g dt; only the expectations with respect to g (t) need to be formed. They are denoted by E h as this expectation refers to the random size of h (t) in (1). The same reasoning holds for random outburst e ects (t) where expectations are denoted by E Using the de nition of in (13), dividing by dt and replacing this expression in the above general speci cation (A.1) yields V ( (t)) max m(t) u ( (t)) v 0 m (t) 1+ + V 0 ( (t)) [ (t) 1 m (t)] + g E h V ( (t) g (t)) V ( (t)) + ( (t)) E h V ( (t) (t)) V ( (t)) The utility function u ( (t)) is given by the expression from (10). The rst-order condition for m (t) in (A.3) requires ; (A.3) (1 + ) v 0 m (t) V 0 ( (t)) 1 (A.4) Assuming an interior solution, marginal costs from coping on the left-hand side must equal marginal gains on the right. As stress (t) is a bad and not a good, V 0 ( (t)) is negative and the right-hand side of this rst-order condition indeed re ects a positive marginal gain. 56 Klaus älde, Johannes-Gutenberg University Mainz, Gutenberg School of Management and Economics, Jakob-elder-eg 4, Mainz, waelde@uni-mainz.de, 1

2 A.1.2 Solving by guess and verify e start with a guess J ( ) for the value function V ( ) which reads J ( ) 0 1 The guess implies J 0 ( ) 1 and gives us two free parameters, 0 and 1 e need to verify that this guess satis es the rst-order condition and the Bellman equation. The rst-order condition (A.4) is satis ed if and (1 + ) v 0 m 1 1 (A.5) Concerning the Bellman equation and given the guess, the jump terms read E h V ( g) V ( ) E h ( 0 1 [ g]) ( 0 1 ) 0 1 E h g E h g E V ( ) V ( (t)) E ( 0 1 [ ]) ( 0 1 ) The Bellman equation (A.3) as a whole then reads [ 0 1 ] w [M ] e v 0 m 1+ 1 [ 1 m] + g 1 E h g + ( ) 1 wme (we + ) v 0 m m g E h g ( ) (A.6) where the utility function from (10) for s was employed. Note that for s ; the Bellman equation reads [ 0 1 ] v 0 m 1+ 1 s 1 m g E h g ( ) as wm (w + ) turns into from (10). Only the direct negative e ect of stress matters. The attention e ect is no longer there as the individual is o the job. Mechanically speaking, the Bellman equation for > s is a special case of the Bellman equation for s where w 0 and s Now collect constant terms and terms proportional to in (A.6), 0 1 wme (we + ) v 0 m m g E h g ( ), 0 wme + v 0 m m + g E h g + ( ) (( ) 1 (we + )) (A.7) As the arrival rate for outbursts ( ) is a constant from (7) for all stress levels, we treat it as a constant here. The Bellman equation (A.7) holds if two conditions are ful lled simultaneously. The rst makes sure that the right-hand side equals zero. This pins down the rst parameter of the value function, 1 we + (A.) The second makes sure that the left-hand side equals zero and pins down the second parameter 0 ; 0 wme v 0 m 1+ + we + 1 m + g E h g + ( ) wme v 0 m 1+ + we + 1 m + g h + ( ) (A.) 2

3 where we used which follows from (1) and the de nition of h and before (1). Note that optimal coping from (A.5) with (A.) is given by (1 + ) v 0 m we + 1, m E h g h (A.10) 1 we + 1 (1 + ) v 0 For the case where the individual is sick and earns no wage, w 0 as discussed after (A.6), coping m s while sick drops relative to standard coping m to A m s 1 s (1 + ) v 0 The value function in closed form hen we plug in values for 0 and 1 into our veri ed guess, we obtain the value of optimal behaviour. e obtain three versions, one for each range of described after (11), V ( ) 0 > > 1 [0; [ 0 [ ; s [ 0 [ s ;1[ 0 we+ we+ > >; for 2 0; ; s [ s ; 1[ ; where the 0 are versions of the value from (A.). In detail, > wme v 0 m 1+ + we+ 1 m + g h 0 wme v 0 m 1+ + we+ 1 m + g h + > 0; > v 0 [m s ] m + g h + >; for 2 ; s [ s ; 1[ The value function is discontinuous at with unchanged slope and (apart from a special case) discontinuous at s with a kink. The discontinuity at results from the occurrence of outbursts. The discontinuity at s results from the drop of p to p s and the kink is due to the change in coping intensity. Figure The value function as a function of stress with the tolerance level and the sickness-level s 3

4 A.2 Proof of the outburst theorem Proof of (i) The stress level rises by (15a) if (t) 1 m > 0 hen 1 m, the deterministic part of (12) is negative for any stress levels (t) and zero for Hence, if > ; _ (t) > 0 for some (t) > 0 the individual is stress prone. Proof of (ii) e understand the role of (t) by remembering that from (16) and (13), 1m 1m p b 0 For a given stress level (t) ; we can ask whether is larger or smaller than (t) Under equality, (t) 1m p b 0 Solving this for p in an attempt to draw b this into g. 2, we get p b 1m (t) + 0, 1m (t) Hence, when > 1 m (t) ; is smaller than (t) and the individual is a bad stabilizer. A.3 The instantaneous expected change of stress The di erential equation (23) for the instantaneous expected change hen we want to understand how stress changes in expectation for a given stress level (t) ; we can proceed as in the derivation of the Bellman equation in app. A.1.1. Instead of computing E t dv ( (t)) for (A.1), we compute E t df ( (t)) for some general function f () given the stochastic di erential equation for (t) from (12). Formally, we compute the in nitesimal generator as presented e.g. in Protter (15, ex. V.7), for many applications, see älde (2012, ch. 10.2). e get n E t df ( (t)) f 0 ( (t)) p o b (t) 0 (t) 1 m (t) dt + E h ff ( (t) g (t)) f ( (t))g g dt + E ff ( (t) (t)) f ( (t))g ( (t)) dt; where the analogy to (A.2) is obvious. Specifying f (x) x to be the identity function and dividing by dt we get E t d (t) dt (t) 1 m (t) g E h g (t) ( (t)) E (t) (t) 1 m (t) g h ( (t)) E (t) where the second equality used (A.10). hen we take the di erent stress regions from (12) into account with the corresponding optimal coping levels from (14) and we de ne 1 m + g h and s 1 m s + g h as in the main text after (23), we obtain E t d (t) dt (t) (t) s (t) s where is the mean of (t) as de ned after (5). 57 ; for 0 (t) (t) s s (t) ; 57 I am deeply indebted to Matthias Birkner from the Institute for Mathematics at the Johannes Gutenberg University Mainz for explaining the more subtle issues related to these applications to me. See Birkner et al. (2017) for work in progress studying an analytical description of the density of stress building on forward Kolmogorov equations. 4

5 Does stress fall? Under which conditions can we expect stress to fall? Focusing on 0 (t) to start with, the answer is given by E t d (t) dt 0, (t) To understand this condition in detail, one should distinguish four cases? and? 0 where the sign of from (24) is given by the sign of de ned after (23). e can distinguish these four cases analytically as follows, case > > a b c d >, >; > 0 0 and su ciently The four cases are illustrated in the following gure. > > small, i.e. where > 0 large, i.e. > where > 0 small, i.e. where 0 large, i.e. where 0 Figure The expected change of stress for di erent signs of and (case a and b covered in main text) It seems natural to consider > 0 to represent the normal case. As 1 m > 0; the second term in the de nition of after (23) needs to be strongly negative to make negative. Even when the individual under consideration is mildly overcon dent, i.e. subjective expectations exceed objective ones, > h, would still remain positive. This is the reason for focusing on > 0 in the main text. 5

6 The analysis of (t) > proceeds in analogy and is discussed jointly with g. 6 in the main text. References (for web appendix) Birkner, M., Scheuer, N. and älde, K., 2017, Computing means and densities for stochastic di erential equations with jumps. ork in progress, JGU Mainz Protter, P., 15, Stochastic Integration and Di erential Equations. Springer-Verlag, Berlin Sennewald, K., 2007, Controlled stochastic di erential equations under Poisson uncertainty and with unbounded utility. Journal of Economic Dynamics and Control älde, K., 1, Optimal Saving under Poisson Uncertainty. Journal of Economic Theory älde, K., 2012, Applied Intertemporal Optimization. Know Thyself - Academic Publishers. Free download at 6