Experimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival


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1 Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuoustime model of expeimentation in which a iskavese entepeneu cannot fully divesify the idiosyncatic isk fom his business investment. He makes consumption/savings and business exit decisions jointly, while leaning about the unknown quality of the poject ove time. Using the closedfom solutions, we show that (i) the entepeneu may stay in business even though the poject s net pesent value (NPV) is negative; (ii) entepeneuial isk avesion eodes option value and lowes pivate poject value so that a sufficiently iskavese entepeneu may exit even when the NPV is positive; (iii) a moe iskavese o a moe pessimistic entepeneu exits ealie; and (iv) the model can geneate a positive elation between wealth and entepeneuial suvival duation fom undivesifiable idiosyncatic isk without liquidity constaints. JEL Classification: D81, D83, D91, E21, L26 Keywods: bandit poblems, eal options, leaning, pivate fim value, suvival, pecautionay savings, incomplete makets We thank Dan Benhadt, Patick Bolton, Joao Gomes, Steve Genadie, Tom Sagent, Andy Skzypacz, Steve Zeldes, and semina paticipants at BU, Columbia, 2006 Econometic Society Summe Meetings at Minnesota, NIU, NYU, Penn State Confeence on Financial Fictions o Technological Diffeences, Rocheste, and UIUC fo helpful comments. This pape was peviously ciculated unde the title: Leaning, Investment and Entepeneuial Suvival: A Real Options Appoach. Depatment of Economics, Boston Univesity, 270 Bay State Road, Boston MA 02215, USA, and Depatment of Finance, the Hong Kong Univesity of Science and Technology, Clea Wate Bay, Kowloon, Hong Kong. Tel: (+852) Columbia Business School, 3022 Boadway, Uis Hall 812, New Yok, NY 10027, and National Bueau of Economic Reseach, Cambidge, MA. Tel.:
2 1 Intoduction We analyze a twoamed bandit poblem in which one am is safe and offes a deteministic payoff, and the othe am is isky and has unknown quality. If the isky am is good, then it geneates a payoff highe than the safe am afte an exponentially distibuted andom time. If it is bad, then it geneates a payoff lowe than the safe am foeve. The decision make is isk avese and chooses between these two ams, ecognizing the dynamic natue of this decision poblem. In addition, the decision make chooses his intetempoal consumption/savings decision, and cannot fully divesify the idiosyncatic isk of the isky am. One application of this type of bandit poblems is entepeneuship. Conside an example in which a iskavese entepeneu decides whethe to stay in his isky business o to quit and accept a safe job. If the entepeneu stays in his isky business, then he eceives pofits fom an investment with unknown quality. If the investment poject is of low quality, then it epesents a business failue and thus geneates smalle pesent values than the safe poject. Howeve, if this poject is of high quality, then it geneates lage payoffs afte the aival of pofitable gowth oppotunities. The aival of these oppotunities may be modeled as a Poisson pocess. Bandit poblems have been analyzed extensively unde isk neutality in the liteatue. 1 To the best of ou knowledge, we ae the fist to analyze the impact of the decision make s isk avesion on his expeimentation stategy. The effect of isk avesion on expeimentation is potentially impotant when the entepeneu cannot fully divesify the idiosyncatic isk of the isky poject fo easons such as moal hazad and infomation asymmeties. As a esult, the entepeneu s intetempoal consumption/savings decisions become intetwined with his expeimentation stategy. Ample empiical evidence documents that an entepeneu s investment oppotunities have substantial undivesifiable idiosyncatic isks. 2 The entepeneu s wellbeing depends heavily on the outcome of his investments. Moeove, the quality of his investment oppotunities is often unknown to the entepeneu befoe stating the poject. By expeimenting with his investment oppotunities, the entepeneu leans about the quality of 1 See the ealy elated contibution by Jovanovic (1979, 1982). See Begemann and Valimaki (2006) fo a suvey and efeences cited theein. 2 See Genty and Hubbad (2004), Heaton and Lucas (2000), and Moskowitz and VissingJogensen (2002), among othes. 1
3 these oppotunities ove time. Unlike models based on isk neutality as in the liteatue, the isk fom expeimentation cannot be fully divesified and hence demands a isk pemium fom the entepeneu s pespective. Hence, entepeneus may exhibit diffeent choice behavio due to the undivesifiable idiosyncatic poject isk. That is, we expect that entepeneus isk attitudes play an impotant ole in detemining thei intedependent consumption/savings and investment/disinvestment decisions. In this pape, we use entepeneuial investment as a pimay application of ou model. While entepeneuial activities have othe impotant dimensions, such as how much to invest, and how to finance investment, fo simplicity, we focus on the exittiming aspect of entepeneuial activities. We extend the standad twoamed bandit models by using a utilitymaximization famewok to analyze the effects of the entepeneu s isk avesion and uninsuable idiosyncatic isks on his leaning and exit decisions. We show that leaning has option value. That is, exit is not a nowoneve decision. The entepeneu may stay in business fo a while and continue to expeiment with his isky investment because he waits fo new infomation to come in, hoping the quality of the isky poject is high. We also show that isk avesion eodes the option value of leaning when the entepeneu cannot fully divesify the idiosyncatic isk fom his investment. The net pesent value (NPV) ule often ecommended by pactitiones and economics and business textbooks can go wong when applied to analyzing the entepeneuial exit decisions. To elaboate on this point, we stat with the constant absolute isk avesion (CARA) utility specification. We show that when the entepeneu s degee of isk avesion is not vey high (including isk neutal entepeneu as a special case), he continues to stay in business even though the NPV of the isky poject is negative. This esult may povide a potential explanation fo Hamilton s (2000) empiical evidence that entepeneuial pojects do not seem to geneate enough NPV. We also show that a moe iskavese entepeneu exits ealie because the nontaded isky poject is less valuable to a moe iskavese entepeneu. When the entepeneu is sufficiently isk avese, he may exit at a time even when the NPV of the isky poject is positive. Finally, we study the impact of wealth effect on business exit decisions. We use a powe utility function which featues constant elative isk avesion (CRRA). Ou model geneates a 2
4 positive elation between entepeneuial wealth and suvival duation without liquidity constaints. 3 The intuition behind this esult is as follows: Fist, highe wealth makes the iskavese entepeneu moe pone to take isks. This point has been made by Cessy (2000) in a static model. Second, in ou dynamic model, the entepeneu has pefeences fo intetempoal consumption smoothing. Highe wealth gives the entepeneu moe buffe to guad against futue income fluctuation. Thus, entepeneus with highe wealth suvive longe in business. While ou analysis boows insights fom the standad eal options appoach to investment, 4 ou model diffes fom this appoach along seveal dimensions. Fist, the standad eal options appoach to investment assumes that all idiosyncatic isk is divesifiable o investos ae isk neutal. Thus, the decision make s isk avesion does not play any diect ole in conventional eal options models (afte the usual isk pemium coection). Miao and Wang (2007) is an exception to the liteatue. As they analyze optimal investment timing decision when the entepeneu cannot fully divesify the idiosyncatic isks (also see Hendeson (2005) and Hugonnie and Moellec (2005)). Unlike Miao and Wang (2007), hee we analyze an exit (disinvestment) poblem when the entepeneu leans about the quality of his investment oppotunity. Second, in the standad eal options appoach, the investment cost o the outside option is exogenous. In ou model, the outside option is endogenous in the sense that its value is given by the optimal lifetime utility obtained fom the altenative safe job and hence is wealth dependent. Finally, thee is no leaning in the standad eal options appoach, while in ou model the entepeneu leans about the quality of the investment oppotunity ove time and makes the exit decision based on the updated belief about this quality. Ou model studies a single entepeneu s decision poblem and does not involve any stategic expeimentation issues. 5 Unlike most continuoustime expeimentation models, which assume that uncetainty is geneated by a Bownian motion, ou model assumes that uncetainty about the poject quality is esolved upon the aival of a Poisson shock, as in Kelle et al. (2005) and othe elated papes cited theein. Unlike these papes, which assume isk neu 3 Evans and Jovanovic (1989) HoltzEakin et al. (1994) ague that liquidity constaints ae impotant fo entepeneuial enty and exit decisions. 4 See Bennan and Schwatz (1985), McDonald and Siegel (1986), and Dixit and Pindyck (1994). 5 See Bolton and Hais (1999), Genadie (1999), and efeences cited theein fo models of stategic expeimentation. 3
5 tality, in ou model the decision make is isk avese and also makes consumptionsavings decisions. Despite this added complexity, we ae still able to deive closedfom solutions fo two commonly used utility specifications (CARA and CRRA). The emainde of the pape is oganized as follows. Section 2 sets up the model and the decision poblem. Section 3 analyzes the model with the CARA utility specification. Section 4 studies the wealth effect on entepeneuial exit decisions by using the CRRA utility specification. Section 5 concludes. Poofs ae elegated to an appendix. 2 The Model In this section, we fist set up the model. We then descibe the entepeneu s leaning pocess. Finally, we fomulate his decision poblem. 2.1 Setup Conside an infinite hoizon continuous time model. An entepeneu deives utility fom a consumption pocess {c (t) : t 0} accoding to the following expected utility function: [ ] E e ρt u(c t ) dt, (1) 0 whee ρ > 0 is the ate of time pefeence, and u(c) is concave and stictly inceasing. Since the discount ate does not play an impotant ole in ou analysis, we assume that ρ is equal to the constant inteest ate. The entepeneu stats with initial wealth x 0. He makes consumption/saving decisions ove time by boowing o lending at the constant iskfee inteest ate. He also decides whethe to continue in business o to quit and accept a safe job. If the entepeneu accepts the safe job, he obtains a constant flow z > 0 of income pe peiod. If he stays in business, he eceives income fom a isky investment. The quality of the isky investment could be high o low. The entepeneu does not know about this quality. The entepeneu may only lean about the quality of this isky investment by taking (expeimenting with) this isky investment. Any time the entepeneu leaves this isky investment, he no longe eceives infomation about the quality of this investment. Let 0 < p 0 < 1 denote his pio belief that the isky investment is of low quality. If the isky investment is indeed of 4
6 low quality, the entepeneu will eceive a pepetual income steam at a constant flow ate y 1 > 0. In ode to have a nontivial tadeoff fo the entepeneu, assume that the payoff fom the low quality investment is stictly dominated by the income fom the safe job, in that y 1 < z. If the investment is of high quality, then the entepeneu will eceive a jump of his income fom y 1 to a pemanently highe level y 2 at the stochastic time. The aival time fo the pemanent jump fom y 1 to y 2 occus andomly with pobability λ t, ove a small time inteval t, if the isky investment poject is of high quality. The aveage time fo the entepeneu to eceive an income jump (fom the high quality isky investment poject) is 1/λ, because of the exponential distibution fo the stochastic aival time. Given the stochastic aival natue of the pemanent aise fo the highquality investment, the quality of the isky investment may neve be fully discoveed fo sue in any finite time. While the safe job pays moe than the income fom the low quality investment (y 1 < z), the entepeneu may eceive a highe income level in the futue if the quality of the isky investment tuns out to be high. Thus, the entepeneu may have incentives to continue in business and expeiment with the isky investment. Since the entepeneu does not know the quality of the isky investment, he leans about it by updating his pio using the Bayes Rule given the infomation about the past investment outcomes. The entepeneu tades off the benefit of discoveing moe infomation about the quality of the isky investment with the cost of eceiving a lowe income level y 1 < z today. When the entepeneu s belief about the quality of the isky poject is pessimistic enough, he quits fom his business and accepts the safe job. We assume that the entepeneu is not well divesified and has limited access to the capital maket. In ode to focus on the effect of undivesified investment isk, we assume that the entepeneu can only tade a iskfee asset to patially insue himself against his poject isk. Specifically, let {x (t) : t 0} denote the wealth pocess and {y (t) : t 0} denote the isky income pocess. Then, the entepeneu s wealth pocess {x (t) : t 0} satisfies dx t = (x t + y t c t ) dt. (2) The entepeneu s objective is to maximize his lifetime utility function (1), subject to his wealth accumulation dynamics (2), his business oppotunity sets (including the safe job and 5
7 the isky investments), and cetain egulaity conditions to ule out Ponzi schemes Leaning We now descibe the entepeneu s leaning poblem. Let p(t) denote the entepeneu s posteio belief that the quality of the isky investment is low at time t, given that the entepeneu has eceived no jump in income until time t. Ove a small time inteval t, the entepeneu s income jumps to y 2 pemanently with pobability λ t, conditional on the investment being of high quality. The Bayes ule implies that the posteio pobability p (t + t) is given by p (t + t) = p(t) p(t) + (1 p(t)) (1 λ t). (3) Taking the limit as t goes to 0 gives the following diffeential equation 7 dp(t) = λp(t)(1 p(t))dt, (4) with initial belief p(0) = p 0. The Bayesian updating pocess (4) implies the following popeties: (i) the posteio belief p(t) is inceasing in time t; (ii) the change of belief ṗ(t) is symmetic in p(t) and (1 p(t)); and (iii) the speed of leaning inceases with the aival ate λ. The intuition behind popety (i) is as follows. The longe the entepeneu takes the isky investment and does not eceive any income jump, the moe likely the investment is of low quality. In the limit (t ), belief p (t) will convege to 1 and the entepeneu will lean the tuth. Popety (ii) eflects the fact that the quality of the isky investment has to be eithe high o low. Thus, that ate of incease in posteio belief p(t) is equal to the absolute value of the ate of decease in the posteio belief (1 p(t)) that investment is of low quality. Popety (iii) states that the entepeneu leans the investment quality faste when the aival of income jump is faste. 6 Fo the CARA utility, we impose the tansvesality condition given in the appendix. Fo the CRRA utility, we use the boowing constaint x t y 1/. It tuns out that this constaint is neve binding at optimum. 7 Note that dp(t) dt = lim t 0 p (t + t) p(t) t = lim t 0 p(t) t λ (1 p(t)) t = λp(t) (1 p(t)). 1 λ (1 p(t)) t 6
8 Solving (4) gives the following explicit fomula fo the posteio belief p(t): 8 p(t) = [ ( ) ] e λt, (5) p 0 povided that the entepeneu s income emains at y 1 up to time t. Of couse, at any time when the entepeneu s income jumps to y 2, the investment is evealed to be of high quality with pobability 1. Figue 1 plots the dynamics of the posteio belief p(t) ove time t. Diffeentiating the belief s dynamic evolution equation (4) gives p (t) = λp (t)(1 2p(t)) 0 if p(t) 1/2. (6) Fom the peceding, we conclude that the posteio belief p(t) is convex in time t, fo p 1/2 and concave in time t, fo p 1/2. When the entepeneu believes that the isky investment is moe likely to be of high quality (p(t) < 1/2), the convexity of p( ) implies that the speed of leaning, ṗ(t), is inceasing ove time. Intuitively, leaning eveals infomation, which is against the pio that the isky investment is moe likely to be of high quality (p 1/2). On the othe hand, when the entepeneu believes that it is moe likely that the isky investment is of low quality (p > 1/2), then the incemental leaning, which povides moe signal confiming his pio, is less valuable. Thus, the speed of leaning, ṗ(t), is deceasing ove time. [Inset Figue 1 Hee] 2.3 Decision poblem We solve the entepeneu s decision poblem by dividing it into two subpoblems. Fist, we solve the entepeneu s deteministic consumption/saving poblem when he eceives a constant level of income foeve. Second, we solve the entepeneu s decision poblem in which he needs to lean about the quality of the isky investment and to choose his optimal consumptionsaving plan accodingly. We stat with the fist poblem. Let V (x; w) denote the coesponding value function given that the entepeneu s cuent wealth level is x and his income level is constant ove time and 8 See the appendix fo deivations. 7
9 equal to w. 9 Then V (x; w) satisfies subject to V (x; w) = max c 0 e t u (c t ) dt (7) dx t = (x t + w c t ) dt, x 0 = x, (8) and the no Ponzi games condition. By a standad agument, V (x; w) satisfies the following Bellman equation: V (x; w) = max c (x + w c) V x (x; w) + u(c). (9) Since ρ =, we can easily show that the entepeneu s optimal consumption is equal to x+w, the sum of inteest income and constant wage income, which ae constant ove time. Theefoe, his wealth emains constant at x at all times. Consequently, the value function is given by V (x; w) = 1 u (x + w). (10) Next, we tun to the entepeneu s decision poblem when he stays in business. Intuitively, the entepeneu will continue to expeiment with the isky investment, povided that thee is sufficiently high benefits fom potentially discoveing the isky investment to be high quality. Namely, it is optimal fo the entepeneu to expeiment with the isky investment, povided that the posteio belief about the isky investment to be low quality p t is sufficiently low. This intuition suggests that the posteio belief p t is an additional state vaiable fo the entepeneu s decision making. Let W (x, p) be the value function when the cuent wealth level and belief ae given by x and p, espectively. Let τ be the fist time when the entepeneu stops expeimenting with the isky investment and accepts the safe job. The pinciple of optimality implies that W (x, p) satisfies [ τ ] W (x, p) = max c, τ 0 e t u (c t ) dt + e τ V (x τ ; z), (11) subject to the belief updating pocess (4) and 9 Notice that hee income w is not a state vaiable. dx t = (x t + y t c t ) dt, x 0 = x, p 0 = p. (12) 8
10 We poceed heuistically to deive the optimality conditions. Fist, when the entepeneu continues to stay in business, it follows fom the pinciple of optimality that the value function W (x, p) satisfies the Bellman equation [ t ] W (x, p) = max E c 0 e t u (c t ) dt + e t W (x t, p t ), (13) fo a shot time t. We apply Ito s Lemma and let t goes to zeo to obtain W (x, p) = max c u(c) + DW (x, p), (14) whee we define DW (x, p) = (x + y 1 c) W x (x, p) + λ(1 p) (V (x; y 2 ) W (x, p)) + λp(1 p)w p (x, p). (15) The intuition behind equation (14) is as follows. The entepeneu optimally stays in business and chooses his consumption pocess c such that the flow measue of his value function W (x, p) on the left side of (14) is equal to the ight side of (14), which is given by the sum of the instantaneous utility payoff u(c) and the instantaneous expected changes of the value function given by (15). These changes consist of thee tems on the ight side of equation (15). The fist tem gives the maginal incease of the value function fom saving of (x + y 1 c) units when the entepeneu does not discove that his investment is high quality. The second tem gives the expected incease in the value function if the isky investment is evealed to be high quality. Note that λ t (1 p) measue the conditional likelihood that isky investment will be evealed to be high quality ove a small time inteval t and (V (x; y 2 ) W (x, p)) is the change in the value function associated with the discovey of the isky investment to be high quality. The last tem measues the impact of ational leaning about the quality of the isky investment on the entepeneu s value function. When the entepeneu updates his belief, he changes his expectation about the futue and hence his value function. Second, when the entepeneu stops expeimenting and accepts the safe job, the following valuematching condition is satisfied W (x, p) = V (x; z). (16) 9
11 This equation implicitly detemines a bounday p (x), which states that belief p is a function of wealth x. It is intuitive that if the entepeneu attaches moe pobability to the low quality investment, then the value function will be smalle. That is, W (x, p) is deceasing in p. Thus, fo any given wealth level x t, if the posteio belief p t p (x t ), then expeimenting with the isky investment fo some time is optimal. On the othe hand, if the posteio belief p t p (x t ), then abandoning the isky investment and accepting the safe investment is optimal. Finally, fo the bounday p (x) to be optimal, the following smoothpasting conditions must also be satisfied (see, fo example, Kylov (1980) and Dumas (1991)): W x (x, p (x)) = V x (x; z), (17) W p (x, p (x)) = 0. (18) Intuitively, the above two smooth pasting conditions captue the optimal tadeoffs fo the entepeneu when he chooses the boundaies between expeimenting with the isky business and exiting to take the safe job. Fom a mathematical point of view, the above decision poblem is a twodimensional combined contol and stopping poblem, which is geneally had to solve. This is due to the fact that the boundaies fo the patial diffeential equation (14) ae also pats of the solutions, (not inputs) of the poblem. In ode to obtain a moe detailed chaacteization of the undelying economics, we next analyze the optimization/leaning poblem by consideing two widely used utility specifications in the next two sections. 3 Suvival without wealth effect: CARA utility In this section, we assume that the entepeneu s pefeences ae epesented by CARA utility function u (c) = e γc /γ, whee γ > 0 is the constant absolute isk avesion paamete. The paamete γ also measues the entepeneu s pecautionay motive (Kimball (1990)). It is well known that the CARA utility specification is analytically convenient fo a vaiety of applications in economics due to its lack of wealth effect. 10 We adopt this utility function as a fist step 10 See Caballeo (1991) and Wang (2004) fo applications of CARA utility to incomplete makets consumptionsaving models. 10
12 towads the undestanding of the effect of leaning on the consumptionsaving and investment decisions fo a isk avese (and pecautionay) entepeneu. 3.1 Solution We fist solve the deteministic consumption/saving poblem in which the entepeneu eceives a steam of constant level w of income. Note that the entepeneu s consumption is constant ove time and given by (x + w). Using ou ealie analysis and equation (10), we may conclude that the value function is given by V (x; w) = 1 exp [ γ (x + w)]. (19) γ We now tun to the entepeneu s decision poblem when he stays in business. We take advantage of the CARA utility s lack of wealth effect and (19) to conjectue that the value function takes the following fom: W (x, p) = 1 exp [ γ (x + f(p))], (20) γ whee f(p) is a smooth function to be detemined. Given the functional foms in (19) and (20) fo the CARA utility specification, we note that the wealth state vaiable x is cancelled out in the value matching condition (16) and the smoothpasting conditions (17). Thus, the poblem is educed to a one dimensional one and the belief bounday p (x) does not depend on the wealth level x. Consequently, we simply use p to denote the constant belief theshold. We pesent the solution in the following poposition. Poposition 1 Let u (c) = e γc /γ. Suppose 0 < δ 1 < λ γ ( ) 1 e γδ 2, (21) whee δ 1 = z y 1 and δ 2 = y 2 z. Then the value function W (x, p) is given by (20) and the optimal consumption ule is given by c (x, p) = (x + f (p)), fo p < p, (22) whee f (p) satisfies the diffeential equation f(p) = y 1 + λ (1 p) 1 [ ] 1 e γ(y 2 f(p)) + λp(1 p)f (p), (23) γ 11
13 subject to f ( p) = z/. Finally, the belief theshold p is given by p = 1 γδ 1 λ (1 e γδ 2 ). (24) We fist discuss the assumption of this poposition. The assumption given in (21) ensues that the belief theshold p lies in (0, 1). If z y 1 o δ 1 0, then the isky investment dominates the safe investment fo evey possible ealization of the isky investment payoff. Theefoe, the entepeneu will always stay with the isky investment. On the othe hand, if γδ 1 λ ( 1 e γδ 2 ), then the entepeneu will always stay with the safe investment. This situation happens when any one of the following conditions is satisfied: (i) the entepeneu is sufficiently isk avese, i.e., γ is high enough; (ii) the cost of expeimenting with the isky investment, epesented by δ 1, is sufficiently high; o (iii) the benefit fom expeimenting with the isky investment, epesented by λ and δ 2, is sufficiently small. We next tun to the consumption ule. Equation (22) implies that the entepeneu consumes the annuity value of the sum of financial wealth x and cetainty equivalent nonfinancial wealth f (p). Equation (23) eveals that the cetainty equivalent nonfinancial wealth f (p) comes fom thee souces. The fist is y 1, the income if the investment quality is pemanently low, which is eflected by the fist tem on the ight side of (23). The second is the incease in cetainty equivalent wealth if the investment quality is evealed to be high ove the next instant. Since the pobability of a jump of income fom y 1 to y 2 is λ (1 p) t ove the time inteval t, the entepeneu faces income fluctuation. Consequently, he has a pecautionay motive to save pat of his wealth to buffe against income fluctuation. The second tem on the ight side of (23) captues pecautionay savings. As expected, pecautionay savings depend on the entepeneu s isk attitude γ. Finally, the thid souce of nonfinancial wealth is due to leaning, which is eflected by the last tem on the ight side of (23). Note that this component is negative because f(p) is deceasing in belief p. Intuitively, belief is deteioating ove time (afte contolling fo the upside (captued by the second tem in (23)). Hence, leaning by the ational entepeneu on aveage bings bad news, elative to myopic ones who simply ignoe the dynamics of belief updating. Because the entepeneu gadually updates his pio ove the investment quality, he changes his valuation of investment payoffs ove time. 12
14 3.2 Beliefs and pivate fim value To futhe undestand the detemination of the belief theshold, we fist analyze the valuation of the pivate fim. Since the entepeneu is subject to the uninsuable idiosyncatic investment isk and the investment poject cannot be taded in the maket, we cannot use the standad valuation method unde complete makets to pice the pivate fim. We adopt the cetainty equivalent appoach in the liteatue on the picing of nontaded assets. 11 We define pivate fim value as the lowest level of the compensating diffeential needed in ode to make entepeneu willing to give up his isky investment oppotunity and taking the safe job. Let q denote this pivate fim value. Then by definition, we have W (x, p) = V (x + q; z). (25) Given the explicit functional foms of the value functions in (19) and (20), we can show that the pivate fim value is given by q = f (p) z/. Figue 2 plots the function f (p). This figue eveals the following: Fist, f (p) is a deceasing function. This esult is intuitive. When the entepeneu becomes moe pessimistic about the quality of the isky investment, his valuation of the fim should be lowe. Second, at the belief theshold p, cetainty equivalent wealth f(p) is equal to the pesent value of income fom the safe job, z/, and theefoe the pivate fim value q is zeo. That is, at p the entepeneu is indiffeent between stay and exit. This fact follows fom the value matching condition (16). Thid, at p the cuve f (p) is tangent to the line z/. This fact follows fom the smoothpasting conditions (17)(18). Finally, fo the posteio belief p < p, the pivate fim value q = f (p) z/ is always positive. Thus, the entepeneu will stay in business fo p < p. When the entepeneu is pessimistic enough such that p eaches the theshold value p, the entepeneu s pivate valuation of the fim is zeo that he is willing to exit and accept the safe job. [Inset Figue 2 Hee] 11 See Capente (1998), Detemple and Sundaesan (1999), Hall and Muphy (2000), Kahl, Liu and Longstaff (2003), among othes, on nontaded asset valuation such as employee stock options. 13
15 We can offe anothe intepetation fo p by ewiting (24) as follows: (z y 1 ) e γ(x+z) = (1 p) λ [V (x; y 2 ) V (x; z)], (26) whee V (x; w) is the value function fo CARA utility and is given in (19). This equation eveals that the entepeneu will stop expeimenting until the cost of expeimenting is equal to the associated benefit. The cost is the income loss (z y 1 ) measued in tems of units of maginal utility e γ(x+z). The benefit occus if the entepeneu discoves that the quality of the poject is high with pobability (1 p) and if thee is an income jump with aival ate λ. When this situation happens, the entepeneu obtains a pepetual incease of income, and hence he obtains benefit measued by the diffeence in the value function: [V (x; y 2 ) V (x; z)]. 3.3 Suvival duation Fom the peceding analysis, we know that the entepeneu exits fom his business when his belief about the investment quality is pessimistic enough. Moe specifically, when the belief pocess p (t) cosses the theshold value p given in (24), the entepeneu exits. Using equation (5), we can deive the entepeneu s suvival duation, T = 1 λ ln ( 1/p0 1 1/ p 1 ), fo p 0 < p. (27) The above equation measues the time that the entepeneu will wait (povided that he does not discove that the poject is high up until time T.) Note that duation T in (27) gives the maximum time that the entepeneu will expeiment with the isky poject. This equation implies that the entepeneu stays in business longe if the initial pio p 0 is smalle o if the belief theshold p is lage. This esult is intuitive. A smalle value of the initial pio p 0 implies that the entepeneu is moe optimistic about the investment poject. He will then believe that the chance of getting a high quality investment is lage. Thus, he will expeiment with the isky investment longe. A lage value of the belief theshold p implies that the degee of pessimism tiggeing exit is highe. Thus, it will take a longe time fo the entepeneu to each this tigge value. That is, the entepeneu will stay in business longe. Since the belief theshold p is impotant fo suvival duation, we conduct futhe compaative statics analysis egading p. 14
16 Coollay 1 Given the assumptions in Poposition 1, the belief theshold p inceases with λ and δ 2, and deceases with γ and δ 1. It is natual to expect that the option value of leaning about the quality of the isky investment is highe when the aival ate λ of the high quality is highe, o the cost of leaning δ 1 is smalle, o the benefit of leaning δ 2 is lage, ceteis paibus. When the option value of leaning is highe, the belief theshold p is highe. Fo fixed λ, this esult implies that the entepeneu stays in business longe fo a highe value of δ 2 o a lowe value of δ 1. Although a highe value of λ also aises p, thee is anothe opposite effect. That is, a highe value of λ makes the entepeneu lean faste, as seen fom the duation suvival function (27). Thus, the oveall effect of an incease in λ on the suvival duation is ambiguous. Tun to the effect of isk avesion. A lage coefficient of absolute isk avesion γ induces a stonge pecautionay motive fo the entepeneu. Thus, an incease in γ aises pecautionay savings and lowes the cetainty equivalent nonfinancial wealth o the entepeneu s valuation of pivate equity f (p). As shown in Figue 3, this fact implies that a lage γ lowes the belief theshold p at which the entepeneu is willing to stop leaning. Thus, a moe isk avese entepeneu exits ealie. In paticula, when γ is lage enough, p goes to zeo. This esult implies that when the entepeneu is sufficiently isk avese, he will exit immediately and accept the safe job. This is consistent with the conventional wisdom that entepeneus ae moe likely to be isk takes, ceteis paibus. [Inset Figue 3 Hee] 3.4 NPV ule, isk avesion, and option value of leaning When thinking about whethe to stay in business o exit, people often apply the following net pesent value (NPV) ule: Fist, calculate the pesent value of the expected steam of income geneated fom the entepeneuial activities. Second, calculate the pesent value of the expected income geneated fom the safe job. Finally, detemine whethe the diffeence between the two the NPV is geate than zeo. If it is, then stay in business; othewise, exit and take the safe job. Using this method, Hamilton (2000) finds that the NPV fo many 15
17 entepeneus is significantly less than zeo. This aises the question as to why entepeneus still want to stay in business. We now use ou model to shed light on this question. We can compute the pesent value of the expected steam of income geneated fom the isky investment as follows: [ Ta ] P V (p) p e t y 1 dt + (1 p) E e t y 1 dt + e t y 2 dt, (28) 0 whee T a is the stochastic aival time of the pemanent income jump at Poisson intensity λ, and the expectation is taken with espect to the exponential distibution fo the aival time T a. Solving the above equation gives P V (p) p y 1 + (1 p) y 1 + λy 2 /, (29) + λ 0 T a whee p is the pobability that the quality of the investment is low. Note that the above pesent value calculation (28)(29) ignoes the fact that the entepeneu s belief will be updated ove time. In macoeconomics, this assumption is sometimes invoked as an appoximation fo an othewise complicated Bayesian dynamic optimization poblem. This assumption is dubbed anticipated utility following Keps (1998) and Sagent (1993, 1998). 12 While the entepeneu s belief p in (29) is coect, the entepeneu ignoes that his belief is changing ove time by holding his belief p unchanged, as if he is myopic. This is the same as in these anticipated utility models. Thus, accoding to the NPV ule (28) as in the anticipated utility analysis of Keps and Sagent, thee is a value of belief theshold p above which the entepeneu is so pessimistic that he pefes to exit. This value is defined by the equation P V (p ) = z. (30) Solving this equation gives p = 1 δ 1 (1 + ). (31) δ 1 + δ 2 λ To see how this NPV ule (as in anticipated utility models) can go wong, we fist conside the isk neutal case. 12 See Cogley and Sagent (2005) fo a ecent quantitative analysis on the appoximation of anticipated utility model fo Bayesian decision making poblems. 16
18 Coollay 2 Let the entepeneu be isk neutal. Suppose δ 1 < λδ 2. Then the belief theshold is given by and the cetainty equivalent wealth is given by p = 1 δ 1 λδ 2, (32) ( z ) f (p) = P V (p) + p p P V ( p) ( ) Ω (p) λ Ω ( p) (33) whee Ω (p) p/ (1 p). To intepet equation (32), we ewite it as δ 1 = λ (1 p) δ 2. (34) Thus, fo the iskneutal entepeneu, at the belief theshold p, the income loss fom expeimentation, δ 1, is equal to the pesent value of the income gain fom expeimentation, λ (1 p) δ 2 /. Given the assumption δ 1 < λδ 2, we can easily show that p < p given in (32). That is, the ational entepeneu exits too ealy elative to the entepeneu who applies the NPV ule as in anticipated utility analysis. Moeove, we can show that P V ( p) < z/. That is, the entepeneu may continue to stay in business even if the NPV is negative. The eason fo this behavio is that thee is an option value of leaning. The option value of leaning is given by the second tem in equation (33). Equation (33) shows that the cetainty equivalent wealth is equal to the pesent value of the investment payoffs plus the option value of leaning. Note that the option value of leaning is positive since P V ( p) < z/. We next conside the case with isk avesion. Coollay 1 shows that the belief theshold p deceases with isk avesion γ. The intuition is that an incease in the degee of isk avesion lowes cetainty equivalent wealth f(p) and eodes the option value of leaning, as illustated in Figue 3. To compae with the NPV ule, Figue 4 plots p as a function of γ. It shows that thee is a citical value γ of the isk avesion paamete such that p = p. When γ < γ, we have p > p, implying that the entepeneu may stay in business even though the NPV is negative. By contast, when γ > γ, we have p < p, implying that the entepeneu exits too 17
19 ealy even though the NPV is positive. This esult seems supising. The intuition is simple. When the degee of isk avesion is sufficiently lage, the entepeneu s pecautionay savings motive dominates and hence substantially eodes the option value of leaning. Fom the peceding discussion, we conclude that fo the isk avese entepeneu, the NPV ule can be wong fo two easons: (i) it ignoes the option value, and (ii) it does not captue isk avesion and the undivesifiable idiosyncatic isks. Theefoe, the NPV ule may imply eithe too ealy o too late exit, depending on the degee of isk avesion and the value of leaning. [Inset Figue 4 Hee] 4 Wealth effect without liquidity constaint: CRRA utility So fa, we have analyzed the effect of leaning on consumption/saving and exit decisions fo the CARA utility specification. The upside of the CARA utility specification is that we ae able to deive analytical solutions so that we can conduct an intuitive analysis. The downside of this specification is that we abstact away fom the wealth effect. To analyze this effect, we now conside the CRRA utility specification, u(c) = c 1 α /(1 α), whee α > 0 is the coefficient of constant elative isk avesion. The log utility coesponds to α = 1. Fist, conside the case with constant income w. Since ρ =,we may immediately conclude that the value function given in (10) is given by V (x; w) = 1 (x + w) 1 α, x > w/. (35) 1 α Plugging this expession into the HJB equation (14), we obtain a patial diffeential equation fo W (x, p). Unlike the CARA utility specification analyzed in the pevious section, thee is no closedfom expession fo W (x, p). Howeve, we can still chaacteize the belief bounday. Poposition 2 Let u(c) = c 1 α /(1 α), α > 0 and α = 1, and x > y 1 /. Suppose [ (x ) (x + z) + 1 α y2 0 < (z y 1 ) < λ 1]. (36) (1 α) x + z 18
20 Then the belief bounday p (x) is given by 13 (z y 1 ) (1 α) p (x) = 1 [ ( ) 1 α ]. (37) λ (x + z) x+y2 x+z 1 Moeove, p (x) > 0, and p (x) deceases in α fo any given x > y 1 /. Much intuition fo the case with the CARA utility caies ove fo case with the CRRA utility. Fist, simila to (21), assumption (36) ensues that the belief bounday p (x) lies in (0, 1). To undestand this bounday, we ewite (37) as (z y 1 ) (x + z) α = (1 p (x)) λ (V (x; y 2 ) V (x; y 1 )). (38) This equation admits a simila intepetation to that fo (26). So we do not epeat it hee. Second, Poposition 2 also shows that the belief bounday deceases with the isk avesion paamete α. That is, a moe isk avese entepeneu is moe likely to exit. A simila esult is obtained in the case of CARA utility. The intuition is also simila. Since fo the CRRA utility pecautionay motive inceases with the isk avesion paamete α, an incease in α aises pecautionay savings. equity. This incease in tun lowes the entepeneu s valuation of pivate It is impotant to emphasize that unlike the CARA utility specification, the belief bounday depends on wealth. Futhemoe, this bounday inceases with wealth. This esult has an impotant implication. That is, a wealthie entepeneu stays in business longe. This esult seems not obvious since an incease in wealth also aises the outside option value o the lifetime utility V (x; z) achieved fom the safe job. Thus, this incease also aises the entepeneu s incentive to exit. Howeve, a highe wealth level makes the entepeneu less isk avese as in the static model (Cessy (2000)). Moeove, a highe wealth level povides moe buffe fo the entepeneu to guad against the intetempoal investment isk. These static and dynamic effects povide the entepeneu with incentives to stay in business. Poposition 2 implies that the latte two effects dominate. 13 Fo log utility, we take limit as α goes to 1 to obtain p (x) = 1 (z y 1 ) λ (x + z) ln ( x+y2 x+z ). 19
21 [Inset Figue 5 Hee] Figue 5 plots the belief bounday p (x) fo α = 2 and 5. The belief bounday patitions the state space fo (x, p) into two egions. Fo state vaiables in the egion below the belief bounday, the entepeneu stays in business. When the state vaiables move into the egion above the belief bounday fo the fist time, the entepeneu exits and accepts the safe job. Figue 5 also illustates that an incease in α lowes the belief bounday. 5 Conclusion Entepeneus often do not know the qualities of thei investment oppotunities. They lean about them by expeimenting with the pojects. Moeove, in the pocess of expeimentation, entepeneus ae often exposed to the poject s idiosyncatic isks, which cannot be fully hedged. Motivated by these two impotant fictions in entepeneuial settings, we develop a model of expeimentation/leaning whee the decision make is subject to uninsuable idiosyncatic isk. We show that a naive NPV calculation ignoes the option value of leaning. Hence, an entepeneu may stay in business even though the NPV of his business investment oppotunity is negative. In addition, isk avesion and beliefs ae impotant fo entepeneuial suvival. In paticula, a moe isk avese o a moe pessimistic entepeneu exits ealie. When the entepeneu is sufficiently isk avese, he may exit even though the NPV is positive. This esult eflects the fact that isk avesion eodes the option value. We also show that wealth is positively elated to suvival duation even though thee is no liquidity constaint fo utility specification that captues wealth effect (such as CRRA). Ou model thus povide an altenative explanation (othe than liquidity constaints) fo the empiically documented positive elationship between entepeneuial liquid wealth and investment/expeimentation theshold. We have intentionally chosen ou model setup in a pasimonious way in ode to delive the economic insights into the effect of leaning on entepeneuial suvival when makets ae not complete. Pasimonious modeling comes with some limitations. Fist, we do not model enty decision in ode to focus on entepeneuial suvival. Second, ou model cannot quantitatively 20
22 addess the pivate equity pemium puzzle (Moskowitz and VissingJogensen (2002)). To undestand the size of empiically obseved pivate equity pemium, one may need a geneal equilibium model in which individuals invest in both public and pivate equity. Finally, in ode to deive intuitive and analytical esults, we assume a stylized income pocess. It would be inteesting to conside moe ealistic income pocesses and calibate the model so that one can confont the model with data. We leave these issues fo futue eseach. 21
23 Appendix A Poofs Deivation of Equation (5): A simple way to solve fo the posteio belief p(t) is as follows. Multiplying a powe function of p(t), say Ap(t) A 1, on both sides of (4) gives Ap(t) A 1 p (t) = Aλp(t) A Aλp(t) A+1. (A.1) Define q(t) = p(t) A. Then, we obtain q(t) Aλq(t) = λa [q(t)] (A+1)/A. (A.2) Choosing A = 1 yields q(t) + λq(t) = λ. (A.3) Solving gives ( q(t) = q(0)e λt + 1 e λt). (A.4) Inveting q(t) = p(t) 1 gives equation (5). Q.E.D. Poof of Poposition 1: specification, we have By the fistode condition u (c) = W x (x, p) and the CARA utility u (c) = W x (x, p) γ at the optimum. Using the conjectued functional fom given in (20) yields (A.5) W x (x, p) = exp [ γ (x + f(p))] = γw (x, p), (A.6) W p (x, p) = f (p) exp [ γ (x + f(p))] = f (p)w x (x, p) (A.7) Using equations (19) and (20) yields V (x; y 2 ) = W x (x, p) e γ(y2 f(p)). γ (A.8) Plugging the above equations (A.5)(A.8) into (14) gives W x γ = W x γ + (y 1 f(p)) W x + λ (1 p) W x γ 22 [ 1 e γ(y 2 f(p)) ] + λp(1 p)w x f (p). (A.9)
24 Dopping W x gives 0 = y 1 f(p) + Fo 0 < p < 1, we may wite λ (1 p) γ f (p) = 1 p [1 exp ( γ (y 2 f(p)))] + λp(1 p)f (p). (A.10) [ f(p) y1 λ(1 p) 1 ( )] 1 e γ(y 2 f(p)). (A.11) γ Given the conjectued fom of the value function (20), the valuematching condition (16) and the smooth pastingconditions (17)(18) become f( p) = z (A.12) f ( p) = 0. (A.13) Using δ 1 = z y 1 and δ 2 = y 2 z, and plugging (A.12)(A.13) into (A.11) give ( 0 = 1 e γδ 2 δ 1 λ (1 p) 1 γ ). (A.14) Reaanging yields (24). To ensue that the posteio belief theshold to lie within the inteesting egion (0, 1), we assume (21). 14 Q.E.D. Poof of Coollay 1: The following ae staightfowad compaative statics esults: p λ = γδ 1 λ 2 (1 e γδ = 1 p > 0, 2 ) λ (A.15) p γ = δ 1 λ (1 e γδ < 0, 2 ) (A.16) p δ 2 = γ 2 δ 1 λ (1 e γδ 2 ) 2 e γδ 2 > 0. We tun to the effect of isk avesion on the posteio theshold. We have p γ δ 1 γδ 1 δ 2 = λ (1 e γδ + 2 ) λ (1 e γδ 2 e γδ 2 2 ) = δ 1 λ (1 e γδ 2 ) 2 To show p/ γ < 0, we only need to show the function ( γδ 2 e γδ 2 + e γδ 2 1 (A.17) ). (A.18) h(x) xe x + e x 1 < 0 fo all x > 0. (A.19) This is tue since h(0) = 0 and h (x) = xe x < 0 fo x > 0. Q.E.D. 14 The tansvesality condition lim t E [ e t W (x t, p t) ] = 0 must also be satisfied. 23
25 Poof of Coollay 2: Taking limit in (23) as γ goes to zeo, we obtain the diffeential equation fo f (p) unde isk neutality, ( + λ (1 p)) f (p) = y 1 + λ (1 p) y 2 + λp (1 p) f (p). (A.20) It can be easily veified the function given in (33) is the solution to the above equation. Finally using conditions (A.12)(A.13), we obtain (32). Q.E.D. Poof of Poposition 2: Given the CRRA utility specification, the fistode condition with espect to consumption gives c = (W x (x, p)) 1/α. (A.21) Thus, at the optimum, u (c) = 1 1 α (W x(x, p)) (1 α)/α. (A.22) Substituting the above two equations into the Bellman equation (14) and evaluating at the belief bounday (x, p (x)), we obtain W (x, p (x)) = 1 1 α (W x(x, p (x))) (1 α)/α (A.23) ( + x + y 1 (W x (x, p (x))) 1/α) W x (x, p (x)) +λ(1 p (x)) (V (x; y 2 ) W (x, p (x))) + λp(1 p (x))w p (x, p (x)) Substituting the valuematching and smoothpasting conditions (16)(18) into the peceding equation, we obtain V (x; z) = = 1 ( 1 α (V x(x; z)) (1 α)/α + x + y 1 (V x (x; z)) 1/α) V x (x; z) +λ(1 p (x)) (V (x; y 2 ) V (x; z)) (A.24) α 1 α (V x(x; z)) (1 α)/α + (x + y 1 ) V x (x; z) + λ(1 p (x)) (V (x; y 2 ) V (x; z)). Plugging the explicit expession fo the value function V (x; w) given in (35) fo w = z and y 2 into the peceding equation and simplifying yield equation (37). We now show that p (x) > 0. Fist, we ewite (37) as [ ( λ(1 p (x)) x + z ) ] ( (α 1) (z y 1 ) x + z ) α ( = λ(1 p (x)) x + y ) 2 1 α. (A.25) 24
26 Apply the Implicit Function Theoem to (A.25) to deive [ ( λ p (x) x + z ) 1 α ( x + y ) ] 2 1 α (A.26) ( = λ (1 p (x)) x + z ) α [ ( α λ (1 p (x)) x + z ) ] ( (α 1) (z y 1 ) x + z ) α 1 ( λ (1 p (x)) (1 α) x + y ) 2 α. Substituting (A.25) into the second tem on the ight side yields [ ( λ p (x) x + z ) 1 α ( x + y ) ] 2 1 α [ ( = λ (1 p (x)) x + z ) α ( α x + z ) 1 ( x + y 2 ) 1 α (1 α) ( x + y 2 ) α ]. (A.27) We will show the expessions in the two squae backets have the same sign by consideing two cases. Case I. 0 < α < 1. Then since z < y 2 To show p (x) > 0, we only need to show O Let ( x + z ) 1 α ( x + y ) 2 1 α < 0. (A.28) ( x + z ) α ( α x + z ) 1 ( x + y 2 ( x + y 2 x + z ) 1 α (1 α) ( ) α 1 < α ( x + y 2 x + z k = x + y 2 x + z x + y 2 ) α < 0. (A.29) ) 1. (A.30). (A.31) Then, by assumption, k > 1. By the Mean Value Theoem and the fact that α (0, 1), whee k 0 [1, k]. The desied esult follows. k α 1 = αk α 1 0 (k 1) < α (k 1), (A.32) Case II. α > 1. Then ( x + z ) 1 α ( x + y ) 2 1 α > 0. (A.33) 25
27 To show p (x) > 0, we only need to show ( x + z ) α ( α x + z ) 1 ( x + y 2 ) 1 α (1 α) ( x + y 2 ) α > 0. (A.34) O, ( x + y 2 x + z ) α 1 > α ( x + y 2 x + z By the Mean Value Theoem and the fact that α > 1, ) 1. (A.35) whee k 1 [1, k]. The desied esult follows. k α 1 = αk α 1 1 (k 1) > α (k 1), (A.36) Finally, to pove p (x) deceases with α fo any given x, we only need to pove the expession [ (x ) α y2 1] (A.37) 1 α x + z deceases with α. Let b = (x + y 2 ) / (x + z). By assumption b > 1. We now compute ( ) b 1 α 1 1 α = (b b ln b + αb ln b bα ) α (1 α) 2. (A.38) b α Define the function g (α) = b b ln b + αb ln b b α. (A.39) It is sufficient to pove g (α) < 0. We obseve that g (α) is concave since g (α) = (ln b) 2 b α < 0. Consequently, we solve the fistode condition g (α) = b ln b (ln b) b α = 0 to conclude that g (α) is maximized at the value α = 1. Since g (1) = 0, we conclude that g (α) < 0 fo all α > 0 and α 1. Q.E.D. 26
28 Refeences Begemann, D. and Valimaki, 2006, Bandit Poblems, fothcoming in Steven Dulauf and Lay Blume, eds., The New Palgave Dictionay of Economics, 2nd ed. Macmillan Pess. Bennan, M. J., and E. S. Schwatz, 1985, Evaluating Natual Resouce Investments, Jounal of Business, 58, Bolton, P. and C. Hais, 1999, Stategic Expeimentation, Econometica 67, Caballeo, R. J., 1991, Eanings uncetainty and aggegate wealth accumulation, Ameican Economic Review 81(4), Cogley, T., and T. J. Sagent, 2005, Anticipated Utility and Rational Expectations as Appoximations of Bayesian Decision Making, woking pape, New Yok Univesity Cessy, R., 2000, Cedit Rationing o Entepeneuial Risk Avesion? An Altenative Explanation fo the Evans and Jovanovic Finding, Economics Lettes 66, Dixit, A. and R. Pindyck, 1994, Investment Unde Uncetainty, Pinceton Univesity Pess, Pinceton, NJ. Dumas, B., 1991, Supe contact and elated optimality conditions, Jounal of Economic Dynamics and Contol 4, Evans, D. and B. Jovanovic, 1989, An Estimated Model of Entepeneuial Choice Unde Liquidity Constaints, Jounal of Political Economy 97, Genty, W., Hubbad, R., Entepeneuship and household saving. Woking pape. Columbia Business School, New Yok. Genadie, S., 1999, Infomation Revelation Though Option Execise, Review of Financial Studies 12,
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