The effect of the background risk in a simple chan

Authors: Jinkwon Lee

Publish Date: 2008/01/04

Volume: 36, Issue: 1, Pages: 19-41

PDF Link

Abstract

We experimentally investigate the effect of an independent and exogenous background risk to initial wealth on subjects’ risk attitudes and explore an appropriate incentive mechanism when identical or similar tasks are repeated in an experiment Taking a simple chance improving decision model under risk where the winning probabilities are negatively related to the potential gain we find that such a background risk tends to make riskaverse subjects behave more risk aversely Furthermore we find that riskaverse subjects tend to show decreasing absolute risk aversion DARA and that a random round payoff mechanism RRPM would control the possible wealth effect This suggests that RRPM would be a better incentive mechanism for an experiment where repetition of a task is usedThis paper is a revised version of a chapter in the author’s PhD thesis and the main revision was done when the author was affiliated with the Centre for Experimental Economics University of York The author is very grateful to John Hey for his comments and supports The author also thanks Miguel CostaGomes Werner Güth Dan Levine Chris Starmer the editor and an anonymous referee for their helpful comments and suggestionsFirst we show that x N  = M/2 Define Dleft x right = uleft w 0 + M x right uleft w 0 right x cdot uprime left w 0 + M x right For a riskneutral person left uleft w 0 + M x right uleft w 0 right right mathordleft/ vphantom left uleft w 0 + M x right uleft w 0 right right left M x right right kernnulldelimiterspace left M x right = uprime left w 0 + M x right Hence D N left x N right = uleft w 0 + M x N right uleft w 0 right x N cdot left uleft w 0 + M x N right uleft w 0 right right mathordleft/ vphantom left uleft w 0 + M x N right uleft w 0 right right left M x N right right kernnulldelimiterspace left M x N right = 0 Solving this gives x N  = M/2 Now we show that x A   M/2 Since x N  = M/2 we need to check D A x N Note that for a riskaverse person left uleft w 0 + M x right uleft w 0 right right mathordleft/ vphantom left uleft w 0 + M x right uleft w 0 right right left M x right uprime left w 0 + M x right right kernnulldelimiterspace left M x right uprime left w 0 + M x right Hence D A left x N right D N left x N right = D A left M mathordleft/ vphantom M 2 right kernnulldelimiterspace 2 right = uleft w 0 + M mathordleft/ vphantom M 2 right kernnulldelimiterspace 2 right uleft w 0 right M mathordleft/ vphantom M 2 right kernnulldelimiterspace 2 cdot uprime left w 0 + M mathordleft/ vphantom M 2 right kernnulldelimiterspace 2 right 0 But dD A /dx  0 by SOC Hence To satisfy D A x A  = 0 x A should be higher than x N  = M/2 Thus x A   M/2 QEDDefine v = ku where k is a concave transformation of u ie k  0 k′  0 and k″  0 Hence ku/u  k′u ⬄ ku  u·k′u Then v is more risk averse than u at every initial wealth level If we show that v gives up more than u does we are done Solving v’s optimization problem generates FOC as following Hleft x right = kleft uleft w 0 + M x right right x cdot kprime left uleft w 0 + M x right right cdot uprime left w 0 + M x right kleft uleft w 0 right right = 0 Suppose x = x A which was the solution of D A x A  = 0 which implies x A cdot uprime left w 0 + M x A right = uleft w 0 + M x A right uleft w 0 right Denote uleft w 0 + M x A right = u A uw 0  = u 0 and Hx A  = H A where u A   u 0   0 Then by a simple calculation we can show H A = kleft u A right kleft u 0 right left u A u 0 rightkprime left u A right Note that since k′u  0 k″u  0 and u A   u 0   0 left kleft u A right kleft u 0 right right mathordleft/ vphantom left kleft u A right kleft u 0 right right left u A u 0 right right kernnulldelimiterspace left u A u 0 right kprime left u A right That is kleft u A right kleft u 0 right left u A u 0 rightkprime left u A right 0 Thus H A   0 while D A  = 0 Since it is easily seen that dH A /dx  0 we find that x A ′ which satisfies H A  = 0 should be higher than x A Thus x A ′  x A In other words a more riskaverse person v chooses a higher x than u does QEDBy totally differentiating FOC we get dx A mathordleft/ vphantom dx A dw 0 right kernnulldelimiterspace dw 0 = left uprime left w 0 + M x A right uprime left w 0 right x A cdot uprime prime left w 0 + M x A right right mathordleft/ vphantom left uprime left w 0 + M x A right uprime left w 0 right x A cdot uprime prime left w 0 + M x A right right left 2uprime left w 0 + M x A right x A uprime prime left w 0 + M x A right right right kernnulldelimiterspace left 2uprime left w 0 + M x A right x A uprime prime left w 0 + M x A right right Since u′  0 u″  0 operatornamesgn left dx A mathordleft/ vphantom dx A dw 0 right kernnulldelimiterspace dw 0 right = operatornamesgn left uprime left w 0 + M x A right uprime left w 0 right x A cdot uprime prime left w 0 + M x A right right Thus uprime left w 0 + M x A right uprime left w 0 right x A uprime prime left w 0 + M x A right 0 would imply dx A /dw 0   0 But note that DARA preference implies that v = −u′ is a concave transformation of u where v′  0 and v″  0 Eeckhoudt Gollier and Schlesinger 2005 That is DARA implies that v is more risk averse than u If we solve the maximisation problem of v then we get the FOC textIleft x A prime right = left uprime left w 0 + M x A prime right uprime left w 0 right x A prime cdot uprime prime left w 0 + M x A prime right right = 0 where x A ′ is a givingup amount satisfying FOC Ix A ′ = 0 From Proposition 2 we know x A ′  x A since v is more risk averse than u This implies that Ix A   0 since x A ′ which is higher than x A should satisfy Ix A ′ = 0 and by SOC dIx A /dx A   0 if and only if v is concaveNote that textIleft x A right = left uprime left w 0 + M x A right uprime left w 0 right x A cdot uprime prime left w 0 + M x A right right = dx A mathordleft/ vphantom dx A dw 0 right kernnulldelimiterspace dw 0 Thus Ix A   0 implies dx A /dw 0   0 QEDWe here mainly follow Gollier and Pratt 1996 Note from Proposition 2 that if a person is more risk averse at any wealth level then he or she would choose a higher x Hence if we show that a riskaverse person who faces an independent and exogenous background risk to initial wealth would be more risk averse than one who does not face the background risk to initial wealth then we are done Denote vleft w 0 + M x right = Euleft w varepsilon + M x right If we denote z = w 0 + M x then vleft z right = Euleft z + varepsilon right Since we know that vprime prime left z right mathordleft/ vphantom vprime prime left z right vprime left z right right kernnulldelimiterspace vprime left z right uprime prime left z right mathordleft/ vphantom uprime prime left z right uprime left z right right kernnulldelimiterspace uprime left z right implies x v   x u at any z we only need to show vprime prime left z right mathordleft/ vphantom vprime prime left z right vprime left z right right kernnulldelimiterspace vprime left z right uprime prime left z right mathordleft/ vphantom uprime prime left z right uprime left z right right kernnulldelimiterspace uprime left z right Rearranging vprime prime left z right mathordleft/ vphantom vprime prime left z right vprime left z right right kernnulldelimiterspace vprime left z right = Euprime prime left z + varepsilon right mathordleft/ vphantom Euprime prime left z + varepsilon right Euprime left z + varepsilon right right kernnulldelimiterspace Euprime left z + varepsilon right uprime prime left z right mathordleft/ vphantom uprime prime left z right uprime left z right right kernnulldelimiterspace uprime left z right the condition implies Egleft zvarepsilon right = Euprime prime left z + varepsilon right cdot uprime left z right Euprime left z + varepsilon right cdot uprime prime left z right 0 By noting gzEɛ = 0 since Eɛ = 0 we find the condition implies that gzɛ should be concave in ɛ at any z that is g 22 zɛ  0 This implies −u″z/u′z  −u″″z/u‴z when evaluated at ɛ = 0 By our assumptions of u‴  0 and u″″  0 this could be satisfied So this is the necessary condition In fact it is easily shown that u‴  0 and u″″  0 are sufficient for vprime prime left z right mathordleft/ vphantom vprime prime left z right vprime left z right right kernnulldelimiterspace vprime left z right uprime prime left z right mathordleft/ vphantom uprime prime left z right uprime left z right right kernnulldelimiterspace uprime left z right by using the implied relationship of Euprime prime left z + varepsilon right = Eleft Aleft z + varepsilon right cdot uprime left z + varepsilon right right EAleft z + varepsilon right cdot Euprime left z + varepsilon right Aleft z right cdot Euprime left z + varepsilon right where A denotes the absolute risk aversion coefficient QED

Keywords: