The consumption-investment decision of a prospect theory household: A two-period model with an endogenous second period reference level
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Hlouskova, Jaroslava; Fortin, Ines; Tsigaris, Panagiotis Working Paper The consumption-investment decision of a prospect theory household: A two-period model with an endogenous second period reference level IHS Economics Series, No. 344 Provided in Cooperation with: Institute for Advanced Studies (IHS), Vienna Suggested Citation: Hlouskova, Jaroslava; Fortin, Ines; Tsigaris, Panagiotis (2018) : The consumptioninvestment decision of a prospect theory household: A two-period model with an endogenous second period reference level, IHS Economics Series, No. 344, Institute for Advanced Studies (IHS), Vienna This Version is available at: https://hdl.handle.net/10419/195953 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
IHS Economics Series Working Paper 344 November 2018 The consumption-investment decision of a prospect theory household: A two-period model with an endogenous second period reference level Jaroslava Hlouskova Ines Fortin Panagiotis Tsigaris
Impressum Author(s): Jaroslava Hlouskova, Ines Fortin, Panagiotis Tsigaris Title: The consumption-investment decision of a prospect theory household: A two-period model with an endogenous second period reference level ISSN: 1605-7996 2018 Institut für Höhere Studien - Institute for Advanced Studies (IHS) Josefstädter Straße 39, A-1080 Wien E-Mail: o [email protected]ffi Web: ww w .ihs.ac. a t All IHS Working Papers are available online: http://irihs. ihs. ac.at/view/ihs_series/ This paper is available for download without charge at: http://irihs.ihs.ac.at/4837/
The consumption-investment decision of a prospect theory household: A two-period model with an endogenous second period reference level ∗ Jaroslava Hlouskova Macroeconomics and Economic Policy, Institute for Advanced Studies, Vienna, Austria Department of Economics, Thompson Rivers University, Kamloops, BC, Canada Ines Fortin Macroeconomics and Economic Policy, Institute for Advanced Studies, Vienna, Austria Panagiotis Tsigaris Department of Economics, Thompson Rivers University, Kamloops, BC, Canada ∗Jaroslava Hlouskova gratefully acknowledges financial support from the Austrian Science Fund FWF (project number V 438-N32).
Abstract In this paper we analyze the two-period consumption-investment decision of a household with prospect theory preferences and an endogenous second period reference level which captures habit persistence in consumption and in the current consumption reference level. In particular, we examine three types of household depending on how the household’s current consumption reference level relates to a given threshold which is equal to the average discounted endowment income. The first type of household has a relatively low reference level (less ambitious household) and can avoid relative consumption losses in both periods. The second type of household (balanced household) always consumes exactly its reference levels. The third type of household has a relatively high reference level (more ambitious household) and cannot avoid to incur relative consumption losses, either now or in the future. Note that these households may act very differently from one another and thus there will often be a diversity of behavior. For all three types we examine how the household reacts to changes in: income (e.g., income fall caused by recession or taxation of endowment income), persistence to consumption, the first period reference level and the degree of loss aversion. Among others we find that the household increases its exposure to risky assets in good economic times if it is less ambitious and in bad economic times if it is more ambitious. We also find that in some cases more income can lead to less happiness. In addition, the less ambitious household and the more ambitious household with a higher time preference will be less happy with a rising persistence in consumption while the more ambitious household with a lower time preference will be happier if it sticks more to its consumption habits. Finally, the household will be happiest for the lowest possible current consumption reference level, i.e., not comparing at all will lead to the highest level of happiness. Keywords: prospect theory, loss aversion, consumption-savings decision, portfolio allocation, happiness, income effects JEL classification: G02, G11, E20
1 Introduction One of the most important decisions households face is consumption today versus consumption in the future. Households transfer current consumption into the future by allocating their savings into different types of assets some of which are riskier than others. These decisions are done with the knowledge that the future is risky. The expected utility theory (EUT) has been the cornerstone model for exploring these household decisions. This research deviates from the EUT model and explores, in a two-period model, the behavior of households which are characterized by reference dependent preferences (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) and by habit persistence (Abel, 1990; Campbell and Cochrane, 1999; Constantinides, 1990; Flavin and Nakagawa, 2008; Pagel, 2017) when deciding on consumption, savings, and the portfolio allocation of savings. We explore the factors that influence a household’s consumption, savings and portfolio decisions when the second period reference level is assumed to depend on first period consumption and the first period consumption reference level. Households have been observed to show a habit for consumption that persists into the future, and hence a habit persistence model combined with prospect theory preferences will provide new insights on such important life cycle decisions. By incorporating prospect theory type of preferences and habit persistence we will be able to address a number of issues on consumption and risk taking behavior that have not been explored in the literature previously. How does a household make intertemporal decisions under these two behavioral traits? Does the optimal solution depend on avoiding relative losses or not? Does the optimal choice depend on whether the household is sufficiently loss averse? Is the choice dependent on the household being less or more ambitious on targets? How do the second period reference level, consumption, risk taking, and happiness change when the first period reference level changes? Do the responses depend on the household’s level of ambition? What impact does the habit persistence in consumption have on consumption and portfolio choice? How will a household react to sudden income changes? Do happiness, current consumption and risk taking always increase when income increases? This paper will attempt to shed some light on the above questions. The first reference levels ever used in economic research were developed by Stone (1954) and Geary (1950). The Stone-Geary utility preferences involve reference dependent utility on subsistence levels of consumption and thus subsistence levels can be considered as a special type of reference points. Under such preferences households derive utility from consumption in excess of a subsistence level. Savings and portfolio choices with subsistence consumption have been explored by Achury et al. (2012).1They use a Stone-Geary expected utility model to explain the empirical observations that rich people show a higher savings rate, higher holdings 1Merton (1969, 1971) used HARA preferences to examine savings and portfolio allocations in an infinite horizon expected utility model. Achury et al. (2012) added subsistence and also habit persistence to Merton’s CRRA utility function (a subset of HARA preferences). 3
of risky assets as a fraction of personal wealth, and face a higher volatility in consumption. Another model that has been used is habit persistence. This model assumes that households derive satisfaction from consumption relative to a reference level which in turn depends on past consumption levels. Thus current consumption affects not only a household’s current marginal utility but also its marginal utility in the next period, which may explain why the more a household consumes today the more it will want to consume tomorrow. The macroeconomics and finance literature uses habit persistence models to explain many puzzles, e.g., the equity premium puzzle (Abel, 1990; Constantinides, 1990; Campbell and Cochrane, 1999), excess consumption smoothing (Lettau and Uhlig, 2000) and many business cycle patterns (Boldrin et al., 2001; Christiano et al., 2005). Reference levels are also used to compare one’s own consumption levels to others (Falk and Knell, 2004; Hlouskova, Fortin and Tsigaris, 2017). Many households are influenced by the self-enhancement motive while others are determined by the self-improvement motive. The self-enhancement motive applies when people want to feel they are better than their peers and set their references at low levels possibly reflecting the wealth of poorer people. Others with a high reference level place importance to the self-improvement motive and compare themselves with the ones who are more successful. Hlouskova, Fortin and Tsigaris (2017) use a two-period life-cycle model with a sufficiently loss averse household to investigate the impact of these psychological traits on consumption, savings, portfolio decisions, as well as on welfare. They find that the optimal solution depends on whether the household’s present value of the consumption reference levels is below, equal to, or above the present value of its endowment income. When reference levels are below the endowment income the authors associate this with the self-enhancement motive. Under this motive the household wants to avoid relative losses in consumption in any present or future state of nature (good or bad). Hence the degree of loss aversion does not affect optimal first period consumption and risky asset holdings. When reference levels are equal to the endowment income this is linked to the belonging motive (i.e., the sufficiently loss averse household belonging to a similar social class). They find that the sufficiently loss averse household’s first period consumption is the exogenous reference consumption level and such households avoid playing the stock market. Finally, reference levels above the endowment income are connected with the self-improvement motive. Households with such high reference levels cannot avoid to consume below the reference level, either now or in the future. In this case loss aversion affects consumption and risky investment negatively. The current study differs from Hlouskova, Fortin and Tsigaris (2017) in that it incorporates habit persistence into the household’s behavior. Close to our work is also a recent paper by van Bilsen et al. (2017) who investigate optimal consumption and portfolio choice paths of a loss averse household with an endogenous reference level. The uncertainty arises from risky assets and it is assumed that the time is continuous. Mainly due to loss aversion, the household’s behavior is geared towards protecting 4
itself against bad states of nature to avoid or to reduce losses. Consumption choices are found to adjust slowly to financial shocks. In addition, welfare losses are found to be substantial given consumption and portfolio selections are suboptimal. Curatola (2015) also analyzes optimal consumption-savings decisions of a loss averse household with a time varying reference level in a continuous-time framework and finds that a loss averse household can consume below the reference level in bad economic times. This is done in order to invest in risky assets and increase the likelihood that in the future consumption exceeds its reference level. This behavioral approach can explain why investors increase their exposure to risky assets during financial crises. In contrast, standard habit persistence models do not allow consumption to be below the reference level. Our research complements the work by van Bilsen et al. (2017) and Curatola (2015) in that it provides additional insights: as our model is a two-period lifecycle model we can derive closed-form solutions which allow us to conduct comparative static analysis to detect why certain adjustments happen and also to conduct a welfare analysis. In this paper, we find closed-form solutions for consumption and risk taking of a loss averse household whose endogenous second period reference level depends on current consumption (habit persistence) and on reference consumption. Households who have a relatively low first period reference level are more conservative (less ambitious), which allows them to achieve relative gains in both periods in both states of nature. Households who have a relatively high first period reference level and a low discount factor are more adventurous (more ambitious) and will thus face relative losses in the bad state of nature in the second period while they will achieve relative gains in the first period and in the good state of nature in the second period. On the other hand more ambitious households who value future consumption relatively more will have first period consumption below the reference level but will maintain future consumption in both states of nature above the endogenous second period reference level. We then conduct comparative statics and examine how these different types of households react to income changes, to changes in the first period reference level, to changes in loss aversion, and to changes in habit persistence. The main difference with respect to Hlouskova, Fortin and Tsigaris (2017), henceforth called HFT, is that this study considers also habit persistence. An increase in the consumption habit persistence will reduce current consumption but stimulate risk taking for less ambitious households, reduce both current consumption and risk taking for more ambitious households with a high time preference, and stimulate both current consumption and risk taking for more ambitious households with a low time preference. In addition, we analyze income effects, which are closely related to the effects of income taxes. Another difference between this study and HFT is that the response of first and second period consumption of less ambitious households to a change of the first period reference level is ambiguous. Finally, unlike in HFT we also consider here a scarcity constraint on consumption, i.e., the consumption in both periods can not fall below a certain value. 5
Note that the household’s first period reference level may be interpreted to equal the first period consumption of a reference household, the Joneses. Then following the Joneses2means that an increase of first period consumption of the Joneses will also trigger an increase of this household’s first period consumption.3In HFT the less ambitious household and the more ambitious household with a high time preference (low discount factor) do follow the Joneses, while the more ambitious household with a low time preference (high discount factor) does not. In this study the behavior of the more ambitious household is similar, while that of the less ambitious household may be similar or different, depending on the household’s time preference: for a lower time preference (larger discount factor) the household does follow the Joneses (like in HFT), while for a higher time preference it does not. The rest of the results are somewhat similar to HFT in terms of the impact of the exogenous parameters on the choice variables but differ in terms of magnitude. Another interesting result that was not elaborated in HFT is the reaction of the choice variables of the household to income changes. When focusing, for instance, on risk taking then less ambitious households reduce risk taking when their income falls while more ambitious households increase risk taking when their income shrinks, which is consistent with the observation that investors increase their exposure to risky assets during financial crises (see Curatola, 2015). Finally, the same finding as in HFT is that the highest utility is achieved for the lowest current consumption reference level (while keeping everything else unchanged). Thus, not comparing at all (e.g., to others) leads to the highest level of happiness. In the next section we present the model and lay out the methodology used to find the solutions. Section 3 presents the main results with a discussion and investigates the impact of income taxation. Finally, we offer some concluding remarks. 2 The two-period consumption-investment model 2.1 Model set-up Consider a household who decides on current and future consumption within a two-period model. In the first period it decides how to allocate a non-stochastic exogenous income, Y1>0, to current consumption, C1, risk-free investment, m, and risky investment, α≥0: Y1=C1+m+α=C1+S(1) Savings are composed of the risk-free investment and the risky investment, i.e., S=m+α. The net of the dollar return rf>0 represents the yield from the safe asset. The risky asset yields a stochastic net of the dollar return r. We assume two states of nature, good and bad. 2See Clark et al. (2008) and Falk and Knell (2004), among others. 3This will work through the household’s first period reference level which is equal to the Joneses’ first period consumption. 6
we introduce the following notation Ω = (1 + rf)Y1+Y2−2 (1 + rf)¯ C1(9) Kγ=(1 −p)(rf−rb)1−γ p(rg−rf)1−γ(10) ¯ CU,P 1 1=1 2Y1+Y2 1 + rf(11) λP1−P2=rf−rb (1+rf)(rg−rb+w(rf−rb)) 1−γ+δphΩ+(rg−rf)αC1=¯ C1 C2b=C2Li1−γ δ(1 −p)(1 + rf)( ¯ C1−CL)1−γ −Ω1−γ[(1 + rf)(1 + w) + M]γ δ(1 −p)(1 + rf)(1 + w)(1 + rf)( ¯ C1−CL)1−γfor ¯ C1≤¯ CP1 1(12) λP1−P5= k21 + K 1 γ γ (1 + rf)(1 + w) γ =M (1 + rf)(1 + w)γ (13) k2="δ(1 + rf)(1 + w)prg−rb rf−rb1−γ#1 γ (14) M=δ(1 + rf)(1 + w)prg−rb rf−rb1 γrf−rb+K 1 γ 0(rg−rf) rg−rb (15) αC1=¯ C1 C2b=C2L=(1 + rf)(Y1−¯ C1−CL) + Y2 rf−rb (16) Note that ¯ C1<¯ CU,P 1 1is equivalent to Ω >0.10 We present the optimal solution for first period consumption and risk taking of the less ambitious household in the following proposition. Proposition 1 Let ¯ C1<¯ CU,P 1 1and λ > max λP1−P2, λP1−P5. Then problem (6) obtains 10Note that HFT characterize the different types of household through Ω (being positive, equal to zero, or negative), while in this study we define the different types of household through their first period consumption reference levels (being smaller than, equal to, or larger than a threshold value), which we think makes more sense. However, we could equivalently describe our households through Ω. 13
a unique maximum at (C∗ 1, α∗) = CP1 1, αP1, where CP1 1=¯ C1+Ω (1 + rf)(1 + w) + M =(1 + rf)Y1+Y2+ [M−(1 + rf)(1 −w)] ¯ C1 (1 + rf)(1 + w) + M>¯ C1(17) αP1=1−K 1 γ 0M rf−rb+K 1 γ 0(rg−rf)CP1 1−¯ C1>0 (18) Proof. See Appendix B. The future relative gains, or excess consumption, are given by: CP1 2g−¯ C2=k2rg−rb rf−rbCP1 1−¯ C1>0 CP1 2b−¯ C2=k2K 1 γ 0 rg−rb rf−rbCP1 1−¯ C1>0 (19) Current relative gains, CP1 1−¯ C1, are driving both the investment in the financial market as well as future excess consumption, see (18) and (19). The higher the relative gains in the first period the higher the investment in the financial market and the higher the relative gains (excess consumption) in the future. Note that the household invests positively in the risky asset. Total savings, however, which include both risky and risk-free assets, may be either positive or negative. The household’s consumption and risk taking does not directly depend on the degree of loss aversion; however, the household needs to be sufficiently loss averse.11 Thus the optimal consumption in both periods as well as the relative consumption in both periods, risk taking and happiness are insensitive to changes in the degree of loss aversion. The effect of an increase in the first period consumption reference level on current and future consumption cannot be determined a priori, see dCP1 1 d¯ C1 = M 1+rf−1 + w M 1+rf+ 1 + w >0,if δ > ¯ δ = 0,if δ=¯ δ <0,if δ < ¯ δ (20) where ¯ δ= 1−w 1 + K1/γ γ!γrf−rb rg−rb1−γ1 p(1 + w)(1 + rf)1−γ(21) 11As shown in Proposition 1, the loss aversion parameter needs to be sufficiently large, namely λ > max λP1−P2, λP1−P5, to guarantee that the utility of (P1) at its maximum exceeds the potential maximum of (P2) at its border, λ > λP1−P2, as well as the potential maximum of (P5) at its border, λ > λP1−P5. Note that problem (P1) is a concave programming problem and its unique maximum does not depend on λ. 14
It depends on the household’s time preference, i.e., on its discount factor, as follows: a relatively high discount factor (large weight placed to the future) will cause current consumption to increase with increasing ¯ C1, while a relatively low discount factor (small weight place to the future) will cause current consumption to decrease.12 However, the effect on optimal consumption in the second period is opposite: future consumption increases with a lower discount factor and shrinks with a higher discount factor. In addition, the sensitivity of second period consumption in the bad state to the first period reference consumption depends on the probability of the good state. Relative current and future consumption decreases with an increasing first period reference level and also risk taking decreases when the current consumption reference level increases. The latter happens because the increase in the current consumption reference level decreases the relative gains in the first period discouraging investment in the risky asset. Finally, an increase in the first period reference level will reduce the household’s happiness and thus the highest possible level of happiness is achieved for the lowest possible current consumption reference level. This suggests that comparison does not make oneself happy, and indeed not comparing at all would be the best. Note that the sensitivity results with respect to the first period reference level are similar (in terms of sign) to the ones when the second period consumption reference level is exogenous (see Hlouskova, Fortin and Tsigaris, 2017), except for the sensitivity of first and second period consumption: if the second period reference level is exogenous then first period consumption always increases, and second period consumption in both states of nature always decreases, with a rising first period reference level. As stated earlier habit persistence in consumption is determined by the parameter w. An increase in wreduces optimal first period consumption (and thus also the first period relative consumption) and the level of happiness, while it increases the investment in the risky asset. The effect of an increase in won the second period reference level, however, is not unambiguous. It depends on the curvature, γ, the discount factor, δ, and on the level of habit persistence in consumption, w, itself. If the household is rather risk averse (γ > 0.5), however, then the effect of habit persistence on the second period reference level is always positive. Also the effect of won the second period consumption in the bad state can be either positive or negative. Namely the second period consumption in the bad state increases with increasing habit persistence in the first period consumption when wis below a certain threshold and it decreases with increasing habit persistence in the first period consumption when wexceeds the threshold.13 On the other hand, the impact of won the second period consumption in the good state is always positive. Note that as habit persistence in consumption, w, relates negatively to habit persistence in the current consumption reference level, 1−w, the reported dependencies hold with the opposite sign for habit persistence in the first period reference 12Note, however, that a larger persistence in consumption reduces the threshold of the discount factor, see (21), which makes it more plausible that first period reference consumption encourages current consumption. 13This threshold is a function of the parameters describing the financial market and on the curvature. 15
level. Current consumption depends positively on income, i.e., it depends positively on both first period and second period income.14 An increase in the first period income, as in good economic times, will increase current consumption by (1 + rf)/[(1 + rf)(1 + w) + M)], while an increase in the second period income (i.e., good future economic conditions) will increase current consumption by 1/[(1+rf)(1+w)+M)]. Note that the presence of habit persistence in consumption has reduced the impact of income upon current consumption relative to models without such a behavioral trait. Furthermore, an increase in income will increase second period consumption as well as the relative gains (excess consumption) in both periods, the second period reference level, the investment in the risky asset and the level of happiness. Note that a sudden reduction in income, caused by a recession or a loss of job (bad economic conditions) or by the introduction of an income tax, will cause the opposite effect and the household will thus reduce current consumption and risk taking. Note in addition that if the first period reference level is equal to a fraction of the present value of the total wealth, i.e., ¯ C1=cY1+Y2 1+rfwhere c∈0,1 2,15 then the sensitivity results will not change. This suggests that the direct income effect is stronger than the indirect effect of income through the first period consumption reference level. Table 1 summarizes the sensitivity results related to Proposition 1, which have been discussed above. Finally, it can be shown that the expected utility evaluated at the optimal choices is determined by the relative gains in the first period: (1 −γ)EUCP1 1, αP1=[(1 + rf)(1 + w) + M]γ (1 + rf)(1 + w)CP1 1−¯ C11−γ(22) The household will be more happy with a rising income, while it will be less happy with a larger first period reference level (as the first period relative consumption decreases) and a higher persistence in current consumption, see Table 1. C∗ 1=CP1 1and α∗=αP1 dC∗ 1dC∗ 2gdC∗ 2bdα∗d¯ C2d(C∗ 1−¯ C1)d(C∗ 2g−¯ C2)d(C∗ 2b−¯ C2)d(E(U(C∗ 1, α∗))) dλ = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 d¯ C1≷0≶0≶0<0>0<0<0<0<0 dw < 0>0≶0>0≶0<0>0>0<0 dYi>0>0>0>0>0>0>0>0>0 Table 1: Sensitivity results for the less ambitious household with respect to λ,¯ C1,wand Yi, i= 1,2. 14We say that some quantity depends positively (negatively) on income, if it depends positively (negatively) on both first period income and second period income. 15The fraction needs to be less than one half such that the household is less ambitious. 16
3.2 Neutral first period reference consumption: balanced households This special case applies when the household is neither less ambitious (see the previous section) nor more ambitious (see the following section). The household is balanced in the sense that it consumes exactly its reference levels, in both the first and the second period. This requires that the household’s first period reference level is equal to the threshold separating less ambitious from more ambitious households. The reference consumption is thus equal to the average of the discounted income, i.e., ¯ C1=¯ CU,P 1 1=1 2Y1+Y2 1+rf. We call this reference level the neutral first period reference consumption. Note that the neutral reference level depends explicitly on the household’s exogenous income. Note, in addition, that if the household’s total income coincides with the total income of some reference household then this current reference consumption can be viewed as an external reference consumption, as the household compares itself to someone like itself. The following corollary describes the solution of the balanced household. Corollary 1 Let ¯ C1=¯ CU,P 1 1and λ > max λP1−P2, λP1−P5. Then problem (6) obtains its unique maximum at (C∗ 1, α∗), where C∗ 1=1 2Y1+Y2 1 + rf=¯ CU,P 1 1 α∗= 0 Proof. See Appendix B. The sufficiently loss averse balanced household will consume exactly its consumption reference level in the first period, which is equal to half the current value of total income. In addition, it will not invest in the financial market even though the expected return from the risky asset is greater than the return from the safe asset. This phenomenon can help to explain the equity premium puzzle as it indicates that the risk premium is not sufficient to induce the household to invest in the risky asset. The savings will thus consist only of the risk-free investment, which can be positive, zero or negative, based on how the first period income and the discounted second period income relate to each other: S=m=1 2Y1−Y2 1 + rf >0 if Y1>Y2 1+rf = 0 if Y1=Y2 1+rf <0 if Y1<Y2 1+rf (23) Note, in addition, that also in the second period in both states of nature the household consumes exactly its consumption reference level, i.e., ¯ C2=C2g=C2b=1 2[(1 + rf)Y1+Y2] = (1 + rf)¯ CU,P 1 1= (1 + rf)¯ C1, which implies that this solution is feasible for all sub-problems (P1)–(P8) and thus can be considered a threshold solution, where the household achieves no relative gains and no relative losses in either period. 17
If the household’s income increases either in the first period and/or in the second period, while other parameters remain unchanged, including ¯ C1, then the household’s upper bound ¯ CU,P 1 1will also increase and as a result the household will become relatively less ambitious since now ¯ C1<¯ CU,P 1 1. Thus, the household will be able to avoid relative losses in both periods. If on the other hand, the household’s income falls unexpectedly, while other parameters remain unchanged, then this will reduce the household’s threshold level ¯ CU,P 1 1and thus the first period reference level will be above this new upper bound ¯ CU,P 1 1. As a result the household will become more ambitious in order to make up for the lost income. In this case its optimal consumption will be below the reference level either in the second period in the bad state of nature, problem (P2), or in the first period, problem (P5). We will discuss these cases in the next section. Suppose the household has initially a current consumption reference level below the threshold level and hence is less ambitious. Then it is hit by a sudden reduction in income, e.g., due to a loss of job in bad economic times, which triggers a decrease of the threshold level such that the household’s (constant) reference level is above the new threshold, and hence the household is more ambitious. This switch from the less ambitious (across the balanced) to the more ambitious type will change, for example, its sensitivity of risk taking with respect to income: while before the drop in income the household (which is less ambitious) takes on less risk with decreasing income, it will be eager to take on more risk with a decreasing income – with the hope to make up for the lost income – after the drop in income (when it will be more ambitious).16 Note that consumption in both periods (as well as the relative consumption in both periods), risk taking and happiness are unaffected by changes in the level of loss aversion, as well as by changes in the persistence level in current consumption. 3.3 High first period reference consumption: more ambitious households If the first period reference level exceeds the threshold level which is equal to the average of the discounted income, i.e., if ¯ C1>¯ CU,P 1 1=1 2Y1+Y2 1+rf, then the household cannot consume above its reference levels in both periods. In either the first or the second period the household will have to consume below its reference consumption, and thus will incur relative losses. A household with such a high first period reference level is called more ambitious. The optimal consumption of the more ambitious household will be either below its consumption reference level in the second period in the bad state of nature, problem (P2), or in the first period, problem (P5). Which case occurs, problem (P2) or (P5), depends on the household’s time preference, i.e., on its discount factor, and on the probability of the good state to occur. If the sufficiently loss averse household is relatively time impatient and assigns a low weight to future consumption (i.e., it has a small discount factor, or a high time preference) then the 16See the sensitivity results in Tables 1 and 2. 18
optimal solution of (6) for optimal consumption and risk taking coincides with the optimal solution of problem (P2). In this problem the optimal consumption in the first period is above its reference level, as in problem (P1). However, in the second period the household cannot avoid relative losses in the bad state of nature. Proposition 2 provides the optimal solution for this case. This case also applies if the probability of the good state of nature is sufficiently large (irrespective of the household’s time preference). On the other hand, if the discount factor is relatively large (i.e, future consumption is valued high), and the probability of the good state is not too high, then the sufficiently loss averse household will find a solution where first period consumption is below the first period reference level (suffering relative losses in the first period) but will keep future consumption above the endogenous reference level in both states of nature. The solution for this case is presented in Proposition 3. The first period reference level cannot be arbitrarily large, however. It needs to be smaller than a certain threshold, ¯ CU,P 2 1. To summarize, if the more ambitious household values first period consumption relatively high (lower discount factor), then it focuses on avoiding relative losses in the first period and thus first period consumption is above its reference level. If, however, the more ambitious household values second period consumption relatively high (larger discount factor), then it wants to prevent relative losses in the second period and consequently second period consumption exceeds its reference level. This is only true, however, if the probability of the good state is not too large. If it is larger than a certain threshold then only the first case applies, where relative losses occur in the second period in the bad state, irrespective of the household’s time preference.17 17Note that for better readability we will often omit the information on the large (small) enough probability of the good state of nature in identifying the type of household, and simply call a household that finds it optimal solution in problem (P2) “more ambitious with a high time preference”, and a household that finds its optimal solution in problem (P5) “more ambitious with a low time preference”. 19
Before proceeding further, we introduce the following notation ¯ CU,P 2 1= rg−rb rg−rfY1+Y2 1+rf−CL 1 + 2 rf−rb rg−rf (24) ¯ CU,P 5 1=Y1+Y2 1+rf−(1 + w)CL 1−w(25) k="δ(1 + rf)(1 + w)(1 −p)rg−rb rg−rf1−γ#1 γ (26) M(λ) = "δ(1 + rf)(1 + w)prg−rb rf−rb1−γ#1 γh(λKγ)1/γ −1i(27) CU L=rg−rb rg−rfY1+Y2 1 + rf−1 + 2 rf−rb rg−rf ¯ C1(28) λP2="(1 + rf)(1 + w) k+1 Kγ1/γ#γ 1 + rf−rb rg−rf+1+rf+k2 (1+rf)(1+w)+k2 CU 2L−(1 + rf)CL (−Ω) γ (29) for CL<CU 2L 1 + rf and ¯ CU,P 1 1<¯ C1<¯ CU,P 2 1 λP2−P2="rf−rb (1+rf)(rg−rb+w(rf−rb))1−γ +δp#hΩ+(rg−rf)αC1=¯ C1 C2b=C2Li1−γ δ(1 −p)(1 + rf)( ¯ C1−CL)1−γ−rg−rb rg−rf1−γ(−Ω)1−γ(30) λP4=1 (1 + rf)1−γw δ wY1+Y2 1+rf+ (1 −w)¯ C1−(1 + w)CL αC1=¯ C1 C2b=C2L−αC2g=¯ C2 C2b=C2L 1 + rf rf−rb γ (31) for ¯ C1<¯ CU,P 2 1 λP5=λP1−P5 (1 + w)( ¯ C1−CL) (1 −w)¯ CU,P 5 1−¯ C1 γ (32) λP2−P6= δp(1 + rf)2rg−rb rf−rb+w2(¯ C1−CL)1+γ [1 + δ(1 −p)(1 + rf)1−γw2]hΩ+(rg−rf)αC1=¯ C1 C2b=C2Li1+γ(33) δP2−P5=1 1−prg−rf (1 + rf)(1 + w)(rg−rb)1−γ (34) αC2g=¯ C2 C2b=C2L=(1 + rf)( ¯ C1−CL) + w(1 + rf)(Y1−¯ C1−CL) + Y2 rg−rb+w(rf−rb)(35) 20
The optimal solution for first period consumption and risk taking is given in the next proposition. Proposition 2 Let ¯ CP1 1<¯ C1<¯ CU,P 2 1,λ > max λP2, λP2−P2, λP4, λP5, λP2−P6, δ≤δP2−P5and CL< CU L. Then problem (6) obtains a unique maximum at (C∗ 1, α∗) = CP2 1, αP2, where CP2 1=¯ C1−Ω M(λ)−(1 + rf)(1 + w)>¯ C1(36) αP2=1 K01/γ +λ1/γk rg−rfCP2 1−¯ C1>0 (37) Proof. See Appendix B. Note that for a sufficiently large probability of the good state18 the threshold value of the discount factor is larger than one (δP2−P5>1) and is thus not binding. In that case Proposition 2 applies, irrespective of the household’s time preference. The reason is that the household is rather willing to accept a relative loss in the bad state of nature, which occurs with a small enough probability, than to face a relative loss in the first period, which occurs with certainty. Future relative gains (in the good state of nature) and losses (in the bad state of nature) are given by CP2 2g−¯ C2=krg−rb rg−rf1 K01 γCP2 1−¯ C1>0 ¯ C2−CP2 2b=krg−rb rg−rfλ1 γCP2 1−¯ C1>0 (38) In problem (P1) the loss aversion parameter does not affect the optimal choices but here loss aversion plays a significant role. An increase in the degree of loss aversion will result in a decline in the first period consumption, a decline in the future consumption in the good state of nature, and a decline in the endogenous second period consumption reference level, but will increase future consumption in the bad state of nature. An increase in loss aversion will also reduce relative gains in the good state of nature in the second period because of the decline in relative gains in the first period. In addition, an increase in loss aversion will reduce relative losses in the bad state of nature in the second period. Even though there are two opposite effects on relative losses in the second period arising from an increase in loss aversion it can be shown that the indirect effect from the decline in CP2 1−¯ C1overpowers the direct impact from increasing the loss aversion parameter. Finally, an increase in loss aversion will reduce the exposure to the stock market and reduce the happiness level. 18Namely for 1 > p > 1−hrg−rf (1+rf)(1+w)(rg−rb)i1−γ. Note that pmust also be larger than rf−rb rg−rb, which is implied by the assumption E(r)> rf. 21
Contrary to problem (P1), an increase in the first period reference level will increase first period consumption, see (36), which is in line with the assumption on preferring the presence to the future. Also it will increase second period consumption in the good state of nature, the second period reference level, and the investment in the financial market because the increase in ¯ C1increases relative gains CP2 1−¯ C1. However, an increase in ¯ C1will reduce future consumption in the bad state of nature. Relative gains of consumption in the first period will increase, and so will future relative gains in the good state of nature by having higher future relative losses in the bad state of nature. Similarly as in problem (P1), an increase in the first period reference level will decrease the level of happiness, i.e., not comparing at all makes one the happiest. Note that the sensitivities of the solutions (in terms of signs) with respect to loss aversion and the first period consumption reference level are the same as in the case of an exogenous second period reference level, as reported in Hlouskova, Fortin and Tsigaris (2017). An increase in the habit persistence in consumption reduces the current consumption, the relative consumption in both periods, risk taking, as well as the happiness level. Finally, the increase in the habit persistence in current consumption reduces also the second period endogenous reference level, ¯ C2, and future consumption in the good state of nature, CP2 2g, for a sufficiently large habit persistence level (where the threshold depends on the curvature parameter which is binding only for γ≤0.5), while it increases both ¯ C2and CP2 2gwhen the household is sufficiently loss averse and at the same time exhibits a lower level of habit persistence in consumption. Note that the opposite dynamics hold when we consider the effect of the habit persistence in the consumption reference level. Finally, note that the dynamics of the current consumption, current relative consumption, second period endogenous reference level and the happiness level with respect to the habit persistence are in line with the dynamics of the less ambitious households. A change in income here has profoundly different effects from those related to the less ambitious household. An unexpected decrease in income, due to, e.g., a loss of job in bad economic times, will increase first period consumption, second period consumption in the good state of nature, investment in the financial market and also the endogenous second period consumption reference level. In addition, a decrease of income increases the relative consumption in both periods. On the other hand, the second period consumption in the bad state of nature will decrease when income decreases, and so will the happiness level. These effects are opposite (in terms of sign) with respect to those reported for the less ambitious household, with the exception of the future consumption in the bad state of nature and the happiness level, which both decrease with a falling income. The reason is probably related to the fact that the more ambitious household cannot consume above its consumption reference levels at all times while the less ambitious household can always do that. Total savings actually decrease with a falling income. Note finally that if the first period reference consumption 22
Note finally that an increase of current income increases savings for both less and more ambitious households.30 The effect of second period income on savings is opposite to the effect of second period income on current consumption for both types of households. Thus, savings are discouraged for less ambitious households and for more ambitious households with a low time preference (that achieve relative gains in the second period), while they are encouraged for more ambitious households with a high time preference (that face relative losses in the bad state of nature in the second period). Effects of the first period consumption reference level Ceteris paribus, a higher first period consumption reference level is bad: both the less ambitious and the more ambitious households will be less happy with a larger first period reference level, i.e., a higher comparison level decreases happiness. Thus, the household seems to be happiest when it does not compare itself to anybody at all. For the less ambitious household being less happy with a rising first period reference level works through lower relative gains in both periods as the relative gains shrink with an increasing first period reference level. However, for the more ambitious household, this works only through larger relative losses, enhanced by the penalty on losses, as both relative gains and losses increase with an increasing first period consumption reference level. The reaction of the less ambitious household’s consumption to an increase in the first period reference level is ambiguous: A household with a smaller weight placed to the future will decrease its current consumption and increase its future consumption while the opposite happens for a household with a larger weight placed to the future. In the case of a more ambitious household with a higher time preference, the current consumption as well as the second period consumption in the good state of nature increase with increasing ¯ C1while the second period consumption in the bad state shrinks. On the other hand, in the case of a more ambitious household with a lower time preference, the increase of the first period reference level will cause a reduction in the current consumption but an increase in the future consumption.31 Thus, for instance, if ¯ C1is equal to the consumption of a reference household (the Joneses) then our household is following the Joneses32 when it is either less ambitious with a low time preference or when it is more ambitious with a high time preference. Note that in the case of an exogenous second period consumption reference level, see Hlouskova, 30Only more ambitious households with a low time preference need to be sufficiently loss averse. If the degree of loss aversion of these households, (P5), is not too large then savings will decrease with an increasing current income. This may happen for households with a small persistence to current consumption. 31This holds also for the second period consumption in the bad state of nature, when the household is sufficiently loss averse. 32The household is following the Joneses when the increase, or decrease, of the first period consumption of a reference household (the Joneses) impacts this household such that its current consumption will change in the same way as the one of the Joneses, i.e., it will increase if the current consumption of the Joneses increases and vice versa. Note that in this context the household’s first period reference consumption is equal to the current consumption of the Joneses. 29
Fortin and Tsigaris (2017), the less ambitious household follows the Joneses irrespective of its time preference. A larger first period reference level also implies a larger second period reference level, except for the more ambitious household with a low rate of time preference (high discount factor), where the effect can be positive or negative. Finally, the effect of a rising first period reference level upon the household’s investment in the risky asset is negative for the less ambitious household, and positive for the more ambitious household. Thus, the risk taking decreases for less ambitious households with an increasing current consumption reference level while it increases for more ambitious households when the current reference level increases. See Tables 1 and 2 for the sensitivity results with respect to the first period consumption reference level. Effects of loss aversion All other things equal, the degree of loss aversion does not have any effects on the less ambitious household’s happiness, nor on its consumption or its investment in the risky asset.33 On the other hand, the more ambitious household is less happy with an increasing level of loss aversion, which is triggered solely by shrinking relative gains (whenever there are gains). An increasing level of loss aversion shows opposite effects (in terms of signs) on the second period reference level, for different time preferences. The effect is negative for a high time preference, and it is positive for a low time preference. Technically speaking, this works through the impact of loss aversion on first period consumption (which is negative for a high time preference, and positive for a low time preference). Finally, a higher degree of loss aversion implies a lower investment in the risky asset for the more ambitious household, which is what one would probably expect. See Tables 1 and 2 for the sensitivity results with respect to loss aversion. Effects of the habit persistence in consumption The effect of persistence in consumption on happiness depends on whether the household’s first period optimal consumption is above or below the reference level. Whatever is smaller – either consumption or the reference level – should be followed more intensely (in the formation of the second period reference level) in order to increase happiness. If first period consumption is above the reference level, then increasing habit persistence in consumption makes the household less happy while increasing persistence in the consumption reference level makes it happier. Thus the household should intensify its persistence on the consumption target. This situation applies to the less ambitious household and the more ambitious household with a higher time preference. On the other hand, if first period consumption is below the reference 33The assumption on the degree of loss aversion to be sufficiently large is to guarantee that the maximum of (P1) exceeds the potential maxima of (P2) and (P5). 30
level, then growing habit persistence on consumption leads to more happiness while increasing persistence on the consumption reference level results in less happiness. Hence the household should stick more to its consumption habits. This applies to the more ambitious household with a lower time preference. For the less ambitious household the decrease in happiness materializes only through a decline of the first period consumption (or, equivalently, through a decline of the first period relative gains), as the second period relative gains actually increase with an increasing persistence in consumption, see Table 1. On the other hand, for the more ambitious household with a sufficiently small discount factor the decrease in happiness is triggered by a decrease of both the first period relative gains and the second period relative gains when the good state of nature occurs. Finally, for the more ambitious household with a sufficiently large discount factor the increase in happiness is caused by a decrease of the relative losses in the first period (for a sufficiently loss averse household) as well as by an increase of the relative gains in the second period. 3.5 Implications for income taxation Our analysis has important implications in terms of how a household responds to income reductions due to the impact of taxation of endowment income.34 In fact an increase in the tax rate is equivalent to a decrease in income in the model without taxes. The effects of taxation will depend on whether the household has a low or a high first period consumption reference level, hence on the type of household. Suppose suddenly income is taxed and the household is less ambitious such that it only experiences relative gains. Then increased taxation will reduce current consumption, future consumption in both states of nature, risk taking, second period reference level, relative gains and happiness. Suppose, then, the household is more ambitious with a high time preference (i.e., it values more current consumption) and thus experiences relative losses in the bad state of nature in the second period. Increased taxation of income in this case will increase current consumption, risk taking, consumption in the good state of nature, the second period reference level, current relative gains, second period relative gains in the good state of nature and second period relative losses in the bad state of nature, which is opposite to the response of a less ambitious household towards taxation of income. On the other hand, increased taxation will reduce consumption in the bad state of nature as well as happiness. Suppose, further, the household is more ambitious with a low time preference (i.e., it values more future consumption) and is thus willing to experience relative losses in the first period, then increased taxation will discourage current consumption but stimulate risk taking while the direction of future consumption is ambiguous and happiness will decrease. In terms 34The corresponding model set-up is the same as presented by (6), only income is replaced by after-tax income in all formulations, propositions and corollaries. I.e., Y1is replaced by (1 −τ)Y1and Y2is replaced by (1 −τ)Y2, where τ∈(0,1) is the tax rate of income. 31
of relative gains and losses increased taxation of income will increase relative losses in the first period as well as relative gains in the future. Finally, if the household is at the threshold level, i.e., if it is balanced, then a sudden increase in taxation (while keeping all other parameters unchanged) will reduce the present value of after-tax income and thus the threshold level of the current reference consumption ¯ CU,P 1 1will shrink, which in turn makes the household more ambitious. For the discussion of the effects of income taxes on savings it is reasonable to assume that the tax rates on current and future income are not independent. For simplicity we assume that they are the same. Then a higher income tax (which induces lower disposable income) discourages savings for the more ambitious household with a high time preference, and it also discourages savings for the less ambitious household and the more ambitious household with a low time preference provided second period income is small enough.35 On the other hand, if second period income is larger than the threshold then a higher tax rate stimulates savings for the less ambitious household and the more ambitious household with a low time preference. Note that if the household has a sufficiently large persistence in current consumption and the second period is the retirement period then more plausible is the case when second period income does not exceed its threshold.36 A particularly interesting result is the impact of taxation on risk taking. Taxation of income will discourage risk taking for less ambitious households, while for more ambitious households, irrespective of their rate of time preference, taxation will increase risk taking. Finally, taxation makes a household less happy irrespective of its first period reference level. dτ dC∗ 1dC∗ 2gdC∗ 2bdα∗d¯ C2d|C∗ 1−¯ C1|d|C∗ 2g−¯ C2|d|C∗ 2b−¯ C2|dS∗dU∗ ¯ C1<¯ CU,P 1 1<0<0<0<0<0<0<0<0≶0<0 ¯ C1>¯ CU,P 1 1,δ≤δP2−P5>0>0<0>0>0>0>0>0<0<0 ¯ C1>¯ CU,P 1 1,δ > δP2−P5<0≶0≶0>0<0>0>0>0≶0<0 Table 3: Sensitivity results for less ambitious and more ambitious households with respect to the income tax. 4 Concluding remarks In this paper we analyze the two-period consumption-investment decision of a household with prospect theory preferences and an endogenous second period reference level which captures habit persistence in consumption and in the current consumption reference level. We find that the optimal solution of a sufficiently loss averse household depends on how its first period consumption reference level relates to a given threshold which is equal to the average 35In the latter case the household, in addition, needs to be sufficiently loss averse. 36The threshold is given by (1 + rf)wY1+˜ k, where ˜ k≥0. 32
discounted endowment income. The reference level may be below, equal to, or above this threshold and hence households can be of three types. These three types are characterized by how their optimal consumption relates to their reference consumption. First there are households with a relatively low reference level (less ambitious households), which can avoid relative consumption losses in both periods. This means that they always consume above their reference levels. Second there are balanced households with a neutral reference level, which always consume exactly their reference levels. This type of household, however, is very special and can only occur when its first period reference level is equal to the average of the discounted income. Third there are households with a relatively high reference level (more ambitious households), which cannot avoid to incur relative consumption losses, either now or in the future. More precisely, a more ambitious household with a lower discount factor (high time preference) will face relative losses in the second period in the bad state of nature while a more ambitious household with a higher discount factor (low time preference) incurs relative losses in the first period. Note that the three types of household sometimes act very differently from one another and thus there is a diversity of behavior resulting from the different levels of comparison. We observe the following effects of habit persistence in consumption. A less ambitious household will be less happy with a rising persistence in consumption, but at the same time it will be happier with a rising persistence in the reference consumption. Hence it is better to stick to one’s exogenously given consumption target than to one’s consumption habits. The same applies to the more ambitious household with a high time preference. However, the situation is reverse for the more ambitious household with a low time preference: it will be happier if it sticks more to its consumption habits than to its target, i.e., if it intensifies its consumption habits. In addition clinging to one’s consumption habits decreases current consumption for the less ambitious household and the more ambitious household with a relatively high time preference, while evidence is mixed for the more ambitious household with a relatively low time preference. It is always true that more income is better, i.e., the larger the income, the happier the household – provided the first period reference level does not depend on income. However, if the reference level depends on income in the sense that it is equal to a fraction of the present value of total wealth and the household is more ambitious, then a higher income reduces happiness. This is due to the fact that in this case the indirect effect of income (through the first period reference level) outweighs the direct effect of income. We also observe that less ambitious households increase their exposure to risky assets during good economic times (i.e., when their income increases) while the more ambitious households increase their exposure to risky assets during bad economic times (i.e., when their income decreases). Finally, we obtain the same findings as in Hlouskova, Fortin and Tsigaris (2017) related to the dependence of happiness upon the current consumption reference level: the highest utility 33
is achieved for the lowest current consumption reference level (while keeping everything else unchanged). Thus, not comparing at all (e.g., to others) leads to the highest level of happiness. We also discuss the effects of taxation of endowment income: increasing the tax rate in a model with income taxes is actually equivalent to decreasing income in a model without taxes. An interesting extension would certainly be to examine the household’s optimal consumptioninvestment behavior if also capital income is taxed. 34
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Appendix A: Optimization problems Before proceeding further, we introduce (or re-introduce) the following notation Ω = (1 + rf)(Y1−¯ C1) + Y2−w0+ (w1+w2)¯ C1 = (1 + rf)Y1+Y2−w0−(1 + rf+w1+w2)¯ C1 Kγ=(1 −p)(rf−rb)1−γ p(rg−rf)1−γ k="δ(1 + rf+w1) (1 −p)rg−rb rg−rf1−γ#1 γ ,dk drb <0,dk drg <0 (47) k2="δ(1 + rf+w1)prg−rb rf−rb1−γ#1 γ =k K1/γ γ ,dk2 drb >0,dk2 drg >0 M=δ(1 + rf+w1)prg−rb rf−rb1 γrf−rb+K 1 γ 0(rg−rf) rg−rb =k"1 + 1 Kγ1/γ#=k21 + K1/γ γ = (1 + rf+w1)λP1−P51/γ ,dM drb >0,dM drg >0 M(λ) = k"λ1/γ −1 Kγ1/γ#=k2h(λKγ)1/γ −1i,dM(λ) drb <0,dM(λ) drg <0 ¯ CU,P 1 1=(1 + rf)Y1+Y2−w0 1 + rf+w1+w2 ¯ CU,P 2 1=(rg−rb) [(1 + rf)Y1+Y2]−(rf−rb)w0−(rg−rf)C2L (rg−rb)(1 + rf) + (rf−rb)(w1+w2) ¯ CL,P 4 1=(1 + rf+w1)C2L−w1[(1 + rf)Y1+Y2]−(1 + rf)w0 (1 + rf)w2 (48) ¯ CU,P 4 1=Y1+Y2−C2L 1 + rf (49) ¯ CU,P 5 1=(1 + rf)Y1+Y2−w0−(1 + rf+w1)C1L w2 (50) ¯ CU,P 6 1=1 w2rg−rb rf−rb ((1 + rf)(Y1−C1L) + Y2−C2L)−w0−w1C1L+C2L(51) ¯ CU,P 6,(iii) 1=1 w2rg−rb rf−rb ((1 + rf)(Y1−C1L) + Y2−C2L)−w0−(1 + rf+w1)C1L (52) 37
CU 2L=rg−rb rg−rf(1 + rf)(Y1−¯ C1) + Y2−rf−rb rg−rfw0+ (w1+w2)¯ C1 λP2= 1+rf+w1 k+1 Kγ1/γ −(−Ω) w0+(w1+w2)¯ C1−C2L w1 k 1−(−Ω) w0+(w1+w2)¯ C1−C2L rg−rb rg−rf γ ="1 + rf+w1 k+1 Kγ1/γ#γ ×"1 1+r+k2+w1((1 + rf)(Y1−¯ C1) + Y2)w1−(w0+ (w1+w2)¯ C1)(1 + rf+k2)−C2L CU 2L−C2L#γ ="1 + rf+w1 k+1 Kγ1/γ#γ 1 + rf−rb rg−rf+1+rf+k2 1+rf+k2+w1 CU 2L−C2L (−Ω) γ for C2L< CU 2Land ¯ CU,P 1 1<¯ C1<¯ CU,P 2 1 λP1−P2=rf−rb (rg−rb)(1+rf)+w1(rf−rb)1−γ+δphΩ+(rg−rf)αC1=¯ C1 C2b=C2Li1−γ δ(1 −p)w0+ (w1+w2)¯ C1−C2L1−γ −Ω1−γ(1 + rf+w1+M)γ δ(1 −p)(1 + rf+w1)w0+ (w1+w2)¯ C1−C2L1−γfor ¯ C1≤¯ CP1 1 λP2−P2=rf−rb (rg−rb)(1+rf)+w1(rf−rb)1−γ+δphΩ + (rg−rf)αC1=¯ C1 C2b=C2Li1−γ δ(1 −p)w0+ (w1+w2)¯ C1−C2L1−γ−rg−rb rg−rf1−γ(−Ω)1−γ λP2−P4=1 δp rf−rb (1 + rf)(rg−rb) + w1(rf−rb)1−γ λP4=1 w1δ w0+w1Y1−¯ C1+Y2−C2L 1+rf+ (w1+w2)¯ C1−C2L αC1=¯ C1 C2b=C2L−αC2g=¯ C2 C2b=C2L 1 + rf rf−rb γ for ¯ C1<¯ CU,P 2 1 38
Appendix B Problem (P1). Let ¯ C1≤(1+rf)Y1+Y2−w0 1+rf+w1+w2=¯ CU,P 1 1, i.e., (Ω ≥0) and C1≤¯ C1+Ω 1+rf+w1.38 At first, we solve the concave programming problem (P1) as an unconstrained problem, i.e., we solve two equations in two unknown variables C1and α, namely dE(U) dC1= 0 and dE(U) dα = 0 (∇E(U) = 0), obtain the optimum solution (CP1 1, αP1) and finally verify that CP1 2b≥¯ C2and CP1 2g≥¯ C2,CP1 1≥¯ C1and 0 ≤αP1≤Ω rf−rb, i.e., that the solution is also feasible. The first order conditions are 1 1+rf+w1 dE(U) dC1=(C1−¯ C1)−γ 1+rf+w1−δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ −δ(1 −p)Ω−(1 + rf+w1)(C1−¯ C1)−(rf−rb)α−γ= 0 dE(U) dα =δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ(rg−rf) −δ(1 −p)Ω−(1 + rf+w1)(C1−¯ C1)−(rf−rb)α−γ(rf−rb) = 0 (65) dE(U) dα = 0 from (65) implies the following pΩ−(1 + rf+w1)(C1−¯ C1)−(rf−rb)αγ(rg−rf) = (1 −p)Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)αγ(rf−rb) which after using the definition of Kγ, as given by (10), gives K−1 γ 0Ω−(1 + rf+w1)(C1−¯ C1)−(rf−rb)α= Ω −(1 + rf+w1)(C1−¯ C1) + (rg−rf)α This implies that α=1−K 1 γ 0 rf−rb+K 1 γ 0(rg−rf)Ω−(1 + rf+w1)(C1−¯ C1)(66) 38There is no feasible solution for (P1) if ¯ C1>¯ CU,P 1 1or if C1>¯ C1+Ω 1+rf+w1. If the latter inequality holds then C2b<¯ C2which is infeasible in (P1). 45
If we plug the last expression for αinto the C1part of the FOC in (65) we obtain (C1−¯ C1)−γ δ(1 + rf+w1) =p Ω−(1 + rf+w1)(C1−¯ C1) + 1−K 1 γ 0(rg−rf) rf−rb+K 1 γ 0(rg−rf)Ω−(1 + rf+w1)(C1−¯ C1) −γ + (1 −p) Ω−(1 + rf+w1)(C1−¯ C1)−1−K 1 γ 0(rf−rb) rf−rb+K 1 γ 0(rg−rf)Ω−(1 + rf+w1)(C1−¯ C1) −γ = rf−rb+K 1 γ 0(rg−rf) Ω−(1 + rf+w1)(C1−¯ C1)(rg−rb) γ prg−rb rf−rb under the assumption that Ω −(1 + rf+w1)(C1−¯ C1)>0 which is equivalent to C1< ¯ C1+Ω 1+rf+w1.39 After some simplifications we obtain Ω−(1 + rf+w1)(C1−¯ C1) = (C1−¯ C1)rf−rb+K 1 γ 0(rg−rf) rg−rbδ(1 + rf+w1)prg−rb rf−rb1 γ = (C1−¯ C1)M(67) which gives C1=¯ C1+Ω 1 + rf+w1+M=CP1 1 as given by (17). Note that (17) and the assumption Ω ≥0 imply that CP1 1≥¯ C1. In addition, after plugging CP1 1into (66) we obtain αP1as given in (18). It is also easy to verify that upper bounds on CP1 1and αP1, as given in (P1), are satisfied. Note finally that αP1≥0 as K0<1 which follows from E(r)> rf. Using (65), it is easy to verify that d2E(U) dC2 1<0,d2E(U) dα2<0,and ∇2E(U(C1, C2)) = d2E(U) dC2 1 d2E(U) dα2−d2E(U) dC1dα 2>0 and thus problem (P1) is a concave programming problem and (CP1 1, αP1) is its unique global maximum. Finally, CP1 2gand CP1 2bcan be written as CP1 2g=w0+ (w1+w2)¯ C1+w1+k2 rg−rb rf−rbΩ 1 + rf+w1+M(68) CP1 2b=w0+ (w1+w2)¯ C1+w1+k2K 1 γ 0 rg−rb rf−rbΩ 1 + rf+w1+M(69) 39Note that the optimal solution satisfies this inequality. 46
and thus CP1 2g−¯ C2=k2 rg−rb rf−rb ×Ω 1 + rf+w1+M≥0 CP1 2b−¯ C2=k2K 1 γ 0 rg−rb rf−rb ×Ω 1 + rf+w1+M≥0 It can be shown that (1 −γ)E(U(CP1 1, αP1)) = Ω1−γ 1 + rf+w1 (1 + rf+w1+M)γ =Ω 1 + rf+w11−γh1 + λP1−P51/γiγ(70) where λP1−P5is given by (13). As we have already mentioned the only feasible solution for C1=¯ C1+Ω 1+rf+w1is C1=¯ C1+Ω 1+rf+w1, α = 0with (1 −γ)EU¯ C1+Ω 1 + rf+w1 ,0=Ω 1 + rf+w11−γ for ¯ C1≤¯ CU,P 1 1(71) which is below the value of the expected utility function at (CP1 1, αP1 1) as M > 0 and w1>0. Thus, the maximum of (P1) is reached at (CP1 1, αP1). Note that as C2g=C2b=¯ C2for C1=¯ C1+Ω 1+rf+w1, α = 0then this point is feasible also for problems (P2)–(P4) if ¯ C1≤ ¯ CU,P 1 1(i.e., Ω ≥0) and is feasible also for problems (P5)–(P8) if ¯ C1>¯ CU,P 1 1(i.e., Ω <0) where (1 −γ)EUC1=¯ C1+Ω 1 + rf+w1 , α = 0=−Ω 1 + rf+w11−γ for ¯ C1>¯ CU,P 1 1(72) for C1L≤(1+rf)Y1+Y2−w0−w2¯ C1 1+rf+w1and ¯ C1≤(1+rf)Y1+Y2−w0 w2.40 Note in addition that for ¯ C1= ¯ CU,P 1 1(i.e., Ω = 0) is CP1 1=¯ C1, αP1= 0which is feasible for all problems (P1)–(P8). Sensitivity analysis. Equations (68) and (69) imply that dCP1 2s d¯ C1 =k2Ds+ (1 + rf)w2(73) 40To guarantee that the upper bound on C1Lis nonnegative. 47
where s∈ {b, g}and Ds= (w1+w2)1 + K1/γ γ−rg−rb rf−rb(1 + rf+w1+w2),for s=g =−(w1+w2)(rg−rf)1−K1/γ 0+(1+rf)(rg−rb) rf−rb<0 (w1+w2)1 + K1/γ γ−rg−rb rf−rb(1 + rf+w1+w2)K1/γ 0,for s=b = (w1+w2)1−K1/γ 0−(1 + rf)K1/γ 0+K1/γ γ (74) Equations (73) and (74) imply that for s=g dCP1 2g d¯ C1 <0,for k2>(1+rf)w2 −Dg⇔δ > ¯ δg = 0,for k2=(1+rf)w2 −Dg⇔δ=¯ δg >0,for k2<(1+rf)w2 −Dg⇔δ < ¯ δg (75) where ¯ δg=(1 + rf)w2 −Dgγrf−rb rg−rb1−γ1 (1 + rf+w1)p= w2 (w1+w2)(rg−rf) (1+rf)(rg−rb)+ 1 γ rf−rb rg−rb1 (1 + rf+w1)p Equations (73) and (74) imply that for i=b dCP1 2b d¯ C1 <0,for k2>(1+rf)w2 −Dband Db<0⇔δ > ¯ δband Db<0 = 0,for k2=(1+rf)w2 −Dband Db<0⇔δ=¯ δband Db<0 >0,for Db>0 or for k2<(1+rf)w2 −Dband Db<0⇔δ < ¯ δband Db<0 (76) where ¯ δb=(1 + rf)w2 −Dbγrf−rb rg−rb1−γ1 (1 + rf+w1)p for Db<0 which is equivalent to w1+w2<(1 + rf)K1/γ 0+K1/γ γ 1−K1/γ 0 . Note that the following holds when w1+w2= 1 + rf Db <0,for p < ¯p = 0,for p= ¯p >0 for p > ¯p (77) 48
where ¯p=(rf−rb)1−γ[rg−rf+ 2(rf−rb)]γ (rf−rb)1−γ[rg−rf+ 2(rf−rb)]γ+rg−rf Under this condition, i.e., when w1= (1 + rf)w,w2= (1 + rf)(1 −w) and w∈[0,1], it can also be shown that for sufficiently large ware the threshold values for δ, namely ¯ δband ¯ δg smaller than one, i.e., both CP1 2band CP1 2gcan be both increasing or decreasing with respect to ¯ C1, see (76) and (75). However, based on (77) this applies to CP1 2bonly for p < ¯p. For a sufficiently large p(p > ¯p) is CP1 2bincreasing with respect to ¯ C1. Problem (P2). Let ¯ C1≤¯ CU,P 2 1. We proceed in the following way: At first we solve problem (P2) as an unconstrained problem i.e., we solve ∇E(U) = 0, so that the FOC are satisfied, obtain the unique solution (CP2 1, αP2), verify that the objective function of (P2) is concave at (CP2 1, αP2) and that the solution is also feasible. As the utility function is differentiable at the domain under consideration, (CP2 1, αP2) is the only local extremum (namely a local maximum) and if the objective function at the border of (P2) does not exceed its value at (CP2 1, αP2), then this point is also a global maximum of (P2) when λ > λP2and ¯ CU,P 1 1≤¯ C1≤¯ CU,P 2 1. For ¯ C1≤¯ CU,P 1 1is the maximum reached at the border. The first order conditions are dE(U) dC1= (C1−¯ C1)−γ −δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ(1 + rf+w1) −λδ(1 −p)(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)α−γ(1 + rf+w1) = 0 dE(U) dα =δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ(rg−rf) −λδ(1 −p)(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)α−γ(rf−rb) = 0 (78) dE(U) dα = 0 from (78) implies the following p(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)γ(rg−rf) =λ(1 −p)Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)αγ(rf−rb) which gives 1 K01 γ(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)α=λ1 γΩ−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α 49
This implies that α= λ1 γ+1 K01 γ λ1 γ(rg−rf)−1 K01 γ(rf−rb)(1 + rf+w1)(C1−¯ C1)−Ω = λ1 γ+1 K01 γ λ1 γ−1 Kγ1 γ(rg−rf)(1 + rf+w1)(C1−¯ C1)−Ω(79) for λ > 1 Kγ. If we plug the last expression for αinto the C1part of the FOC in (78) we obtain (C1−¯ C1)−γ δ(1 + rf+w1)=w0+w2¯ C1−(1 + rf)Y1−Y2+ (1 + rf+w1)C1−γ × p 1 K01 γ+1 Kγ1 γ λ1 γ−1 Kγ1 γ −γ +λ(1 −p) 1 + λ1 γ+1 K01 γ λ1 γ−1 Kγ1 γ rf−rb rg−rf −γ After some simplifications we obtain (1 + rf+w1)(C1−¯ C1)−Ω = (C1−¯ C1)"δ(1 + rf+w1)(1 −p)rg−rb rg−rf1−γ#1 γ"λ1 γ−1 Kγ1 γ# = (C1−¯ C1)M(λ) (80) which gives C1=CP2 1≡¯ C1+(−Ω) M(λ)−1−rf−w1 In addition, after plugging CP2 1into (79) we obtain α=αP2≡1 K01/γ +λ1/γk rg−rfCP2 1−¯ C1 Note that CP2 1>¯ C1if Ω <0 and λ > 1+rf+w1 k+1 Kγ1/γγ (i.e., M(λ)>1 + rf+w1) or if Ω >0 and λ < 1+rf+w1 k+1 Kγ1/γγ (i.e., M(λ)<1 + rf+w1). What remains to be shown is when the expected utility function is strictly concave at (CP2 1, αP2). For this to hold it is sufficient to show that the following holds at (CP2 1, αP2): 50
d2E(U) dα2<0 and D≡ ∇2E(U(C1, C2)) = d2E(U) dC2 1 d2E(U) dα2−d2E(U) dC1dα 2>0. Note that CP2 2g−¯ C2=k(−Ω) M(λ)−1−rf−w1 rg−rb rg−rf1 K01 γ(81) ¯ C2−CP2 2b=k(−Ω) M(λ)−1−rf−w1 rg−rb rg−rf λ1 γ(82) which are positive for either ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1(Ω <0) and λ > 1+rf+w1 k+1 Kγ1/γγ or for ¯ C1<¯ CU,P 1 1(Ω >0) and λ < 1+rf+w1 k+1 Kγ1/γγ .41 Thus, ¯ C2−CP2 2b= (K0λ)1 γ(CP2 2g− ¯ C2).Using (78), (81) and (82) we obtain the following 1 γ d2E(U) dC2 1 |(C1,α)=(CP2 1,αP2)=−Ω M(λ)−1−rf−w1−1−γ ×"−1 + 1 + rf+w1 krg−rf rg−rb2λ−1 γ−K 1 γ 0 rf−rb rg−rf#(83) 1 γ d2E(U) dα2|(C1,α)=(CP2 1,αP2)=(rf−rb)2 k(1 + rf)rg−rf rg−rb2−Ω M(λ)−1−rf−w1−1−γλ−1 γ−K 1 γ γ (84) 1 γ d2E(U) dC1dα |(C1,α)=(CP2 1,αP2)=rf−rb krg−rf rg−rb2−Ω M(λ)−1−rf−w1−1−γλ−1 γ+K 1 γ 0 Note that (84) and λ > 1 Kγimply that d2E(U) dα2|(C1,α)=(CP2 1,αP2)<0.In addition, 1 γ2−Ω M(λ)−1−rf−w12(1+γ) D="−1 + 1 + rf+w1 krg−rf rg−rb2λ−1 γ−rf−rb rg−rf K 1 γ 0# ×rf−rb k(1 + rf+w1)rg−rf rg−rb2(rf−rb)λ−1 γ−(rg−rf)K 1 γ 0 −rf−rb k2rg−rf rg−rb4λ−1 γ+K 1 γ 02 and thus 1 γ2−Ω M(λ)−1−rf−w12(1+γ)rg−rb rg−rf2k rf−rb D=1 1 + rf+w1(rg−rf)K 1 γ 0−(rf−rb)λ−1 γ 41For ¯ C1=¯ CU,P 1 1(Ω = 0) is CP2 1=¯ C1,CP2 2g=CP2 2b=¯ C2=w0+ (w1+w2)¯ C1and αP2= 0. 51
+1 krg−rf rg−rb2λ−1 γ−rf−rb rg−rf K 1 γ 0(rf−rb)λ−1 γ−(rg−rf)K 1 γ 0 −rf−rb krg−rf rg−rb2λ−1 γ+K 1 γ 02 After some derivations we obtain 1 γ2−Ω M(λ)−1−rf−w12(1+γ)rg−rb rg−rf2k rf−rb D=1 1 + rf+w1(rg−rf)K 1 γ 0−(rf−rb)λ−1 γ −rg−rf kλ−1 γK 1 γ 0 Now it can be easily shown that if ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1and λ > 1+rf+w1 k+1 Kγ1 γγ then D > 0 and thus the local maximum is reached at CP2 1, αP2. There is no local maximum for ¯ C1<¯ CU,P 1 1and thus in this case the maximum occurs at the border. Thus, for analyzing the feasibility of CP2 1, αP2we assume that ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1 and C2L≤[(w1+w2)(1+rf)Y1+Y2]+(1+rf)w0 1+rf+w1+w2(see Lemma 1-(viii)). Note that (81) and (82) imply that CP2 2g>¯ C2and CP2 2b<¯ C2only when M(λ)−1−rf−w1>0 which holds for λ > 1+rf+w1 k+1 Kγ1 γγ . In addition, C2L≤CU 2Land λ≥λP2imply that CP2 2b≥C2L. In more detail, CP2 2b= (1 + rf)(Y1−CP2 1) + Y2−(rf−rb)αP2≥C2Lif λ1/γ w0+ (w1+w2)¯ C1−C2L −Ω−rg−rb rg−rf≥w0+ (w1+w2)¯ C1−C2L −Ω"1 + rf+w1 k+1 Kγ1/γ# −w1 k(85) For C2L< CU 2Lis the left-hand-side in (85) positive and thus (85) holds for λ≥λP2. If, on the other hand, C2L> CU 2Lthen the left-hand-side in (85) is negative and there are two possibilities for the right-hand-side: if it is non-negative then λ1/γ is non-positive and thus infeasible; if, on the other hand, the right-hand-side in (85) is non-positive then λ1/γ ≤ −Ωw1 k+C2L−w0−(w1+w2)¯ C11+rf+w1 k+1 Kγ1/γ rg−rb rg−rf(−Ω) + C2L−w0−(w1+w2)¯ C1 =λP21/γ and it can be shown that the right-hand-side of the last inequality is below 1+rf+w1 k+1 Kγ1/γ which then contradicts the other feasibility assumption, namely CP2 2g≥¯ C2and CP2 2b<¯ C2 that imply λ > 1+rf+w1 k+1 Kγ1 γγ . If these conditions are not satisfied then the maximum will be reached at the border. Note 52
in addition that (1 −γ)EUCP2 1, αP2 =−Ω M(λ)−1−rf−w11−γ"1 + k 1 + rf+w1 1 Kγ1 γ−λ1 γ!# =−(−Ω)1−γ 1 + rf+w1 [M(λ)−1−rf−w1]γ(86) What remains to show is that feasible solutions at the border do not exceed the expected utility at (CP2 1, αP2), where (P2) obtains its local maximum for ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1as well as to analyze feasible solutions at the border for ¯ C1≤¯ CU,P 1 1. The feasible solutions at the border that come into consideration are: (i) C2g=¯ C2, (ii) C2b=¯ C2, (iii) C2b=C2Land (iv) C1=¯ C1. Note that for ¯ C1=¯ CU,P 1 1, i.e., Ω = 0, is CP2 1=¯ CU,P 1 1and αP2= 0. As this is the inflation point then the maximum in this case will be reached at the border. Case (i). C2g=¯ C2when C1=1 1 + rf+w1(1 + rf)Y1+Y2−w0−w2¯ C1+ (rg−rf)α=¯ C1+Ω+(rg−rf)α 1 + rf+w1 and max n0,−Ω rg−rfo≤α≤αC2g=¯ C2 C2b=C2Lwhere αC2g=¯ C2 C2b=C2Lis given by (54). Note that condition α≥−Ω rg−rffollows from C1≥¯ C1and the upper bound on αfollows from C2b≥C2L. It can be seen that (1 −γ)EU¯ C1+Ω + (rg−rf)α 1 + rf+w1 , α=Ω+(rg−rf)α 1 + rf+w11−γ −λδ(1 −p)(rg−rb)1−γα1−γ(87) Let Ω ≥0, i.e., ¯ C1≤¯ CU,P 1 1. Then the potential maximum occurs either at α= 0 or α=αC2g=¯ C2 C2b=C2Lor at the stationary point of the function given by (87) which can be easily derived and has the value α= ¯α≡kλ 1 γ 1+rf+w1−kλ 1 γ Ω rg−rfwhen λ < 1+rf+w1 kγand is infeasible for λ > 1+rf+w1 kγ. Thus, for λ < 1+rf+w1 kγ (1 −γ)EUC1=¯ C1+Ω+(rg−rf)¯α 1 + rf+w1 ,¯α =Ω1−γ 1 + rf+w11 + rf+w1−kλ1 γγ ≤Ω 1 + rf+w11−γ = (1 −γ)EU¯ C1+Ω 1 + rf+w1 ,0 therefore point C1=¯ C1+Ω+(rg−rf)¯α 1+rf+w1,¯αcan not be a maximum of the main problem (6) as its utility function is below the utility function of ¯ C1+Ω 1+rf+w1,0which is feasible for (P1) and also corresponds to the case when α= 0. 53
The end-point α=αC2g=¯ C2 C2b=C2L, which gives C2b=C2L, is dealt with in case (iii) where it was shown that it can not be the point of the maxima. Let Ω <0, i.e., ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1. Then −Ω rg−rf≤αC2g=¯ C2 C2b=C2Lonly if C2L≤CU 2L. The potential maximum occurs either at α=−Ω rg−rfor at α=αC2g=¯ C2 C2b=C2Lor at the stationary point α= ¯α=kλ 1 γ kλ 1 γ−1−rf−w1 (−Ω) rg−rf. As lim α→+−Ω rg−rf dEU¯ C1+Ω+(rg−rf)α 1+rf+w1α, α dα = +∞ then the maximum can not occur at α=−Ω rg−rf. Regarding the stationary point ¯α, note that the value of the utility for λ > 1+rf+w1 kγis (1 −γ)EU¯ C1+Ω+(rg−rf)¯α 1 + rf+w1 ,¯α =−(−Ω)1−γ 1 + rf+w1kλ 1 γ−1−rf−w1γ(88) It is easy to see that the utility at this stationary point is below the utility at (CP2 1, αP2), as given by (86). It can be also verified that the objective function as given by (87) obtains its local maximum at α= ¯α. Finally, ¯αis feasible if ¯α≤αC2g=¯ C2 C2b=C2Lwhich holds if λ≥ 1 + rf+w1 k× CU 2L−C2L+1+rf 1+rf+w1+rf−rb rg−rf(−Ω) CU 2L−C2L γ ≡λP2 (i) As λP2 (i)< λP2then for λ≥λP2> λP2 (i)is ¯αfeasible and is the point of the maximum for (87) which does not exceed the value of the objective function at CP2 1, αP2. Case (ii). Any feasible solution of this case is also a feasible solution of (P1). Case (iii). C2b=C2Lwhen C1=Y1+Y2−C2L 1+rf−rf−rb 1+rfαand αC2g=¯ C2 C2b=C2L≤α≤(1 + rf)(Y1−¯ C1) + Y2−C2L rf−rb =αC1=¯ C1 C2b=C2L where αC2g=¯ C2 C2b=C2Lis given by (54). Note that the lower bound on αis below its upper bound only if ¯ C1≤¯ CU,P 2 1. 54
Note that λP5> λP1−P5. After plugging the stationary point CP5 1into (104) we obtain αP5=1−K1/γ 0 rf−rb+K1/γ 0(rg−rf)×(λP1−P5)1/γ λ1/γ −(λP1−P5)1/γ ×(−Ω) (105) Thus the maximum of (P5) for λ > λP5is reached at CP5 1, αP5with (1 −γ)EUCP5 1, αP5=−−Ω 1 + rf+w11−γhλ1/γ −(λP1−P5)1/γiγ(106) It can be shown that for ¯ CU,P 1 1<¯ C1<min n¯ CU,P 2 1,¯ CU,P 5 1ois the maximum of (P5) below the maximum of (P2), i.e., (1 −γ)EUCP5 1, αP5 =−−Ω 1 + rf+w11−γhλ1/γ −(λP1−P5)1/γiγ <−(−Ω)1−γ 1 + rf+w1 [M(λ)−1−rf−w1]γ = (1 −γ)EUCP2 1, αP2 (107) see (86), when k 1+rf+w1<1 as inequality (107) boils down to λ1/γ + 1 >k 1 + rf+w1λ1/γ + 1 Thus, for k 1+rf+w1= 1 both objective functions have the same value and for k 1+rf+w1>1 (P5) at its maximum exceeds (P2) at its maximum. Note finally that k 1+rf+w1<1 when δ < 1 1−prg−rf (1 + rf+w1)(rg−rb)1−γ =δP2−P5 k 1+rf+w1= 1 when δ=δP2−P5and k 1+rf+w1>1 when δ > δP2−P5. Note that for ¯ C1=¯ CU,P 2 1is the maximum of (P5) as given by (106) for λ > λP5below the maximum of (P2), which is the value of its objective function at its only feasible solution when (1 −γ)EUCP5 1, αP5 =−−Ω 1 + rf+w11−γhλ1/γ −(λP1−P5)1/γiγ ≤ −λ δ(1 −p)hw0+ (w1+w2)¯ CU,P 2 1−C2Li1−γ = (1 −γ)EU¯ CU,P 2 1, αC1=¯ C1 C2b=C2L (108) This holds for any λ > λP5and δ≤δP2−P51−λP1−P5 λ1/γγ =δ(λ)P2−P5 ¯ C1=¯ CU,P 2 1 . On the 61
other hand, the maximum of (P5) exceeds the maximum of (P2) when δ > δ(λ)P2−P5 ¯ C1=¯ CU,P 2 1 . Summary for (P5): •For C1L<¯ C1≤¯ CU,P 1 1(P1) exceeds (P5) for λ≥λP1−P5. •The following holds for for ¯ CU,P 1 1<¯ C1<min n¯ CU,P 2 1,¯ CU,P 5 1oand λ > max λP2, λP2−P2, λP5: –(P2) at its maximum exceeds (P5) at its maximum when δ < δP2−P5, –(P5) at its maximum coincides with (P2) at its maximum when δ=δP2−P5and –(P5) at its maximum exceeds (P2) at its maximum when δ > δP2−P5. •The following holds for ¯ C1=¯ CU,P 2 1≤¯ CU,P 5 1and λ > λP5: –(P2) at its maximum exceeds (P5) at its maximum when δ < δ(λ)P2−P5 ¯ C1=¯ CU,P 2 1 , –(P2) at its maximum coincides with (P5) at its maximum when δ=δ(λ)P2−P5 ¯ C1=¯ CU,P 2 1 and –(P2) at its maximum is below (P5) at its maximum when δ > δ(λ)P2−P5 ¯ C1=¯ CU,P 2 1 . Problem (P6). Let ¯ C1≤¯ CU,P 2 1. We show at first that there is no interior local maximum or minimum for (P6) which implies that the maximum will occur at the border of the set of feasible solutions for (P6). Then we check all potential feasible solutions at the border. The first order conditions are dE(U) dC1=λ(¯ C1−C1)−γ−δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ(1 + rf+w1) −λδ(1 −p)(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)α−γ(1 + rf+w1+w1) = 0 dE(U) dα =δp Ω−(1 + rf+w1)(C1−¯ C1) + (rg−rf)α−γ(rg−rf) −λδ(1 −p)(1 + rf+w1)(C1−¯ C1)−Ω+(rf−rb)α−γ(rf−rb) = 0 (109) dE(U) dα = 0 from (109) implies the expression for αgiven by (79). Note that αis feasible only for Ω ≤0 where for Ω = 0 there is only one feasible solution, namely ( ¯ C1,0) which is feasible also for (P1). Thus, assume that Ω <0 and let’s plug (79) into the C1part of the FOC in (109). After some simplifications we obtain λ1 γ(1 + rf+w1)(C1−¯ C1)−Ω= ( ¯ C1−C1)M(λ) which gives C+ 1=¯ C1+Ω M(λ) λ1 γ+1 + rf+w1 (110) 62
In addition, after plugging C+ 1from (110) into (79) we obtain α+=k rg−rf"1 K01 γ+λ1 γ#−Ω M(λ) + λ1 γ(1 + rf+w1) Next we show that the expected utility function is indifferent at (C+ 1, α+), namely, we show that at (C+ 1, α+) are d2E(U) dα2<0,and D3≡ ∇2E(U(C1, C2)) = d2E(U) dC2 1 d2E(U) dα2−d2E(U) dC1dα 2<0. Note that C+ 2g−¯ C2=k−Ω M(λ) + λ1 γ(1 + rf+w1) rg−rb rg−rf1 K01 γ(111) ¯ C2−C+ 2b=k−Ω M(λ) + λ1 γ(1 + rf+w1) rg−rb rg−rf λ1 γ(112) and thus ¯ C2−(C+ 2b) = (K0λ)1 γ((C+ 2g)−¯ C2).Using (109), (111) and (112) we obtain the following 1 γ d2E(U) dC2 1 |(C+ 1,α+)="−Ω M(λ) + λ1 γ(1 + rf+w1)#−1−γ ×"λ−1 γ+1 + rf+w1 krg−rf rg−rb2λ−1 γ−K 1 γ 0 rf−rb rg−rf#(113) 1 γ d2E(U) dα2|(C+ 1,α+)=(rf−rb)2 k(1 + rf+w1)rg−rf rg−rb2"−Ω M(λ) + λ1 γ(1 + rf+w1)#−1−γλ−1 γ−K 1 γ γ (114) 1 γ d2E(U) dC1dα |(C+ 1,α+)=rf−rb krg−rf rg−rb2"−Ω M(λ) + λ1 γ(1 + rf+w1)#−1−γλ−1 γ+K 1 γ 0 Note that (114) and λ > 1 Kγimplies that d2E(U) dα2|(C+ 1,α+)<0.In addition, 1 γ2"−Ω M(λ) + λ1 γ(1 + rf+w1)#2(1+γ) D="λ−1 γ+1 + rf krg−rf rg−rb2λ−1 γ−rf−rb rg−rf K 1 γ 0# ×(rf−rb)2 k(1 + rf+w1)rg−rf rg−rb2λ−1 γ−K 1 γ γ −rf−rb k2rg−rf rg−rb4λ−1 γ+K 1 γ 02 63
where D=∇2E(U(C1, C2))(C+ 1,α+)=d2E(U) dC2 1 d2E(U) dα2−d2E(U) dC1dα 2(C+ 1,α+). Thus, 1 γ2"−Ω M(λ) + λ1 γ(1 + rf+w1)#2(1+γ)rg−rb rg−rf2k rf−rb2 D=λ−1 γ 1 + rf+w1λ−1 γ−K 1 γ γ −λ−1 γK 1 γ γ<0 for λ > 1 Kγwhich gives that D=∇2E(U(C1, C2)) = d2E(U) dC2 1 d2E(U) dα2−d2E(U) dC1dα 2<0. From (114) it follows that d2E(U) dα2|(C+ 1,α+)≥0 for λ≤1 Kγand thus (C+ 1, α+ 1) can not be a point of local maxima for any λ > 1 and thus the maximum will occur at the border. The feasible solutions at the border for (P6) that come into consideration are given by: (i) C2g=¯ C2, (ii) C2b=¯ C2, (iii) C2b=C2L, (iv) C1=¯ C1and (v) C1=C1L. Case (i): C2g=¯ C2when C1=1 1 + rf+w1(1 + rf)Y1+Y2−w0−w2¯ C1+ (rg−rf)α=¯ C1+Ω+(rg−rf)α 1 + rf+w1 and max n0, αC1=C1L C2g=¯ C2o≤α≤min αC1=¯ C1 C2g=¯ C2=−Ω rg−rf , αC2g=¯ C2 C2b=C2L Note that this case can occur only for Ω ≤0, i.e., for ¯ C1≥¯ CU,P 1 1,43 where in the case of Ω = 0 the only feasible solutions is ( ¯ C1,0). Thus, in the following we assume that ¯ CU,P 1 1< ¯ C1≤¯ CU,P 2 1. It can be seen that (1 −γ)EU¯ C1+Ω+(rg−rf)α 1 + rf+w1 , α =−λ−Ω−(rg−rf)α 1 + rf+w11−γ −λδ(1 −p)(rg−rb)1−γα1−γ is a convex function in αand thus its maximum is reached either at α= 0, when C1L≤ ¯ C1+Ω 1+rf+w1, or at α=αC1=C1L C2g=¯ C2, when ¯ C1+Ω 1+rf+w1≤C1L≤¯ C1, or at α=αC1=¯ C1 C2g=¯ C2or at αC2g=¯ C2 C2b=C2L. Thus, the potential candidates for the maximum in this case are (a) C1=¯ C1+Ω 1+rf+w1, α = 0when C1L≤¯ C1+Ω 1+rf+w1or (b) C1=C1L, α =αC1=C1L C2g=¯ C2when ¯ C1+Ω 1+rf+w1≤C1L≤¯ C1and ¯ CU,P 5 1≤¯ C1≤¯ CU,P 6 1or (c) C1=¯ C1, α =αC1=¯ C1 C2g=¯ C2when ¯ CU,P 1 1<¯ C1≤¯ CU,P 2 1or (d) C1=¯ C1+Ω+(rg−rf)αC2g=¯ C2 C2b=C2L 1+rf+w1, α =αC2g=¯ C2 C2b=C2L!when ¯ CU,P 2 1≤¯ C1≤¯ CU,P 6 1 43Note that αC2g=¯ C2 C2b=C2L≥0 when ¯ C1≥¯ CL,P 4 1and condition (63) guarantees that C1L≥¯ CL,P 4 1. 64
Note that the point in case (a) is feasible also for (P5), when ¯ C1≤min n¯ CU,P 2 1,¯ CU,P 5 1oand is also the only feasible solution for case (ii). Based on Lemma 1-(vi), case (b) is relevant, i.e., αC1=C1L C2g=¯ C2≥0, only for ¯ CU,P 5 1≤¯ C1≤ ¯ CU,P 6 1which we do not consider here. The point in case (c) is a feasible solution for (P2). Case (d) holds only for ¯ CU,P 2 1≤¯ C1≤¯ CU,P 6 1which is not considered here. Case (iii). C2b=C2Lwhen C1=Y1+Y2−C2L 1+rf−rf−rb 1+rfαfor αC1=¯ C1 C2b=C2L≤α≤αC1=C1L C2b=C2L(115) where αC1=¯ C1 C2b=C2L=(1+rf)(Y1−¯ C1)+Y2−C2L rf−rband αC1=C1L C2b=C2L=(1+rf)(Y1−C1L)+Y2−C2L rf−rb. Then (1 −γ)EUY1+Y2−C2L 1 + rf −rf−rb 1 + rf α, α=−λ¯ C1−Y1−Y2−C2L 1 + rf +rf−rb 1 + rf α1−γ +δp 1 + rf+w1 1 + rf C2L−w0−w1Y1+Y2 1 + rf−w2¯ C1+rg−rb+rf−rb 1 + rf w1α1−γ −λδ(1 −p)w0+w1Y1+Y2 1 + rf+w2¯ C1−1 + rf+w1 1 + rf C2L−w1(rf−rb) 1 + rf α1−γ (116) Note that 1 γ d2EUY1+Y2−C2L 1+rf−rf−rb 1+rfα, α dα2=λ¯ C1−Y1−Y2−C2L 1 + rf +rf−rb 1 + rf α−1−γrf−rb 1 + rf2 −δp 1 + rf+w1 1 + rf C2L−w0−w1Y1+Y2 1 + rf−w2¯ C1+rg−rb+rf−rb 1 + rf w1α−1−γ ×rg−rb+rf−rb 1 + rf w12 +λδ(1 −p)w0+w1Y1+Y2 1 + rf+w2¯ C1−1 + rf+w1 1 + rf C2L−w1(rf−rb) 1 + rf α−1−γrf−rb 1 + rf w12 65
≥λ¯ C1−Y1−Y2−C2L 1 + rf +rf−rb 1 + rf αC1=C1L C2b=C2L−1−γrf−rb 1 + rf2 −δp 1 + rf+w1 1 + rf C2L−w0−w1Y1+Y2 1 + rf−w2¯ C1+rg−rb+rf−rb 1 + rf w1αC1=¯ C1 C2b=C2L−1−γ ×rg−rb+rf−rb 1 + rf w12 +λδ(1 −p)w0+w1Y1+Y2 1 + rf+w2¯ C1−1 + rf+w1 1 + rf C2L−w1(rf−rb) 1 + rf αC1=¯ C1 C2b=C2L−1−γ ×rf−rb 1 + rf w12 =λ¯ C1−C1L−1−γrf−rb 1 + rf2 −δp hΩ + (rg−rf)αC1=¯ C1 C2b=C2Li−1−γrg−rb+rf−rb 1 + rf w12 +λδ(1 −p)w0+ (w1+w2)¯ C1−C2L−1−γrf−rb 1 + rf w12 (117) It is sufficient for the objective function (116) to be convex if the right-hand-side of the last expression is positive which is guaranteed for λ > δp(1+rf)(rg−rb) rf−rb+w12 hΩ+(rg−rf)αC1=¯ C1 C2b=C2Li1+γ 1 (¯ C1−C1L)1+γ+δ(1−p)w2 1 [w0+(w1+w2)¯ C1−C2L]1+γ = δp h(rg−rb)(1+rf) rf−rb+w1i2 hΩ + (rg−rf)αC1=¯ C1 C2b=C2Li1+γׯ C1−C1L1+γw0+ (w1+w2)¯ C1−C2L1+γ δ(1 −p)w2 1¯ C1−C1L1+γ+w0+ (w1+w2)¯ C1−C2L1+γ =λP2−P6(118) Thus the maximum occurs at one of the end points α=αC1=¯ C1 C2b=C2Lor α=αC1=C1L C2b=C2L. Note that α=αC1=¯ C1 C2b=C2Lis feasible for (P2) and α=αC1=C1L C2b=C2Lis tackled in case (v) where it is shown that it is exceeded by another point which is feasible for (P5). For ¯ C1=¯ CU,P 2 1is the first derivative of the objective function (116) given by dEUY1+Y2−C2L 1+rf−rf−rb 1+rf, α dα =−λrf−rb 1 + rf1−γα−αC1=¯ C1 C2b=C2L−γ +δp rg−rb+rf−rb 1 + rf w11−γα−αC1=¯ C1 C2b=C2L−γ +λ δ(1 −p)w1rf−rb 1 + rf1−γ"(rg−rb)(1 + rf) rf−rb αC1=¯ C1 C2b=C2L α−αC1=¯ C1 C2b=C2L −w1#−γ α−αC1=¯ C1 C2b=C2L−γ 66
while using Lemma 1 (iv). Sufficient conditions for dEUY1+Y2−C2L 1+rf − rf−rb 1+rf, α dα <0 and thus for the objective function being decreasing are λ > p 1 δ−1 δP2−P6(1 + rf)(rg−rb) rf−rb +w11−γ =λP2−P6 ¯ C1=¯ CU,P 2 1 and δ < δP2−P6=1 (1 −p)w1 rg−rb ¯ CU,P 2 1−C1L αC1=¯ C1 C2b=C2L−w1!γ (119) Thus, for ¯ C1=¯ CU,P 2 1,δ < δP2−P6and λ > λP2−P6 ¯ C1=¯ CU,P 2 1 the maximum of the objective function (116) is reached for α=αC1=¯ C1 C2b=C2Lwhich is of the same value as the objective function of (P2) at its maximum, see (96). In case (iv) any feasible solution is also feasible for (P2) and thus this case can occur only for ¯ C1≤¯ CU,P 2 1. Case (v). C1=C1Lfor αC1=C1L C2b=¯ C2≤α≤αC1=C1L C2b=C2L, see (56), (60) and Lemma 1-(vi). Then (1 −γ)E(U(C1L, α)) = −λ(¯ C1−C1L)1−γ +δp (1 + rf)(Y1−C1L) + Y2−w0−w1C1L−w2¯ C1+ (rg−rf)α1−γ −λδ(1 −p)w0+w1C1L+w2¯ C1−(1 + rf)(Y1−C1L)−Y2+ (rf−rb)α1−γ It can be shown that for ¯ C1≤¯ CU,P 2 1and λ≥1 Kγis the objective function of case (v) decreasing and thus the maximum is reached at α=αC1=C1L C2b=¯ C2which is feasible also for (P5). The value of the expected utility function at this point is as follows (1 −γ)EUC1L, αC1=C1L C2b=¯ C2=−λ(¯ C1−C1L)1−γ+δp rg−rb rf−rb w2¯ CU,P 5 1−¯ C11−γ (120) Summary for (P6): Let λ > 1 Kγ. Then the following holds. •For ¯ C1L<¯ C1<¯ CU,P 2 1and λ > λP2−P6(P2) exceeds (P6). •For ¯ C1=¯ CU,P 2 1,δ < δP2−P6and λ > λP2−P6 ¯ C1=¯ CU,P 2 1 (P2) exceeds (P6). Note that the above mentioned conditions are sufficient, not necessary. Problem (P8). Let ¯ CU,P 1 1≤¯ C1≤¯ CU,P 2 1. As the utility function of (P8) is convex a maximum will occur at the border of the set of feasible solutions. The feasible solutions at the border that come into consideration are: (i) C2g=¯ C2, (ii) C2b=¯ C2=C2g, (iii) C2g=C2L=C2b, (iv) C2b=C2L, (v) C1=¯ C1and (vi) C1=C1L. 67
Case (i) is feasible for (P6) and was already dealt with in the proof of (P6) in case (i) and thus ¯ C1≤¯ CU,P 6 1. The only feasible solution in case (ii) is C1=CP5 1=¯ C1+Ω 1+rf+w1, α =αP5= 0 which is also feasible (and thus dealt with) for (P5) and (P6). The only feasible solution in case (iii) when C2g=C2L=C2bis C1=Y1+Y2−C2L 1+rf, α = 0 which is feasible only for ¯ C1≥Y1+Y2−C2L 1+rf. Case (iv): C2b=C2Lwhen C1=Y1+Y2−C2L 1+rf−rf−rb 1+rfαand max n0, αC1=¯ C1 C2b=C2Lo≤α≤min nαC2g=¯ C2 C2b=C2L, αC1=C1L C2b=C2Lo where αC2g=¯ C2 C2b=C2Lis given by (54), αC1=¯ C1 C2b=C2L=(1+rf)(Y1−¯ C1)+Y2−C2L rf−rband αC1=C1L C2b=C2L=(1+rf)(Y1−C1L)+Y2−C2L rf−rb. Note that for ¯ C1<¯ CU,P 2 1is αC1=¯ C1 C2b=C2L> αC2g=¯ C2 C2b=C2Land thus case (iv) has no feasible solution. Note in addition that the only feasible solution for ¯ C1=¯ CU,P 2 1is α=αC1=¯ C1 C2b=C2Las αC1=¯ C1 C2b=C2L=αC2g=¯ C2 C2b=C2Lwhich coincides with the value of (P2) at its maximum, see (96). If C1=¯ C, case (v), then any feasible solution will be feasible also for (P4). For more details in the proof see case (iv) of (P4). If C1=C1L, case (vi), then (1 −γ)E(U(C1L, α)) = −λ(¯ C1−C1L)1−γ−λδp −Ω−(1 + rf+w1)( ¯ C1−C1L)−(rg−rf)α1−γ −λδ(1 −p)−Ω−(1 + rf+w1)( ¯ C1−C1L) + (rf−rb)α1−γ(121) for 0≤α≤min nαC1=C1L C2g=¯ C2, αC1=C1L C2b=C2Lo where αC1=C1L C2g=¯ C2is given by (59) and αC1=C1L C2b=C2Lis given by (56). Based on Lemma 1-(vi) is αC1=C1L C2g=¯ C2<0 and thus (121) is infeasible. 68