Risk choices in OLG models∗

Elena Rancoita†

February 15, 2013

Abstract

In this paper, I investigate the efficiency of the savings allocation among differenttypes of investment opportunities in an overlapping generations (OLG) economy. Asimple OLG model is presented where the current cohort can decide how much to investin a risky and in a safe technology and, with this choice, it determines the incomes offuture generations. If each generation is selfish, it does not internalize the pecuniaryexternalities that its investment choices place on the incomes of future generations.For this reason, the competitive equilibrium investment allocation is inefficient in that,if wealth redistribution was possible across generations, there would exist a Paretosuperior allocation. It is shown that this allocation is characterized by a higher shareof risky investments and can be achieved by a simple redistributive policy of incomesacross generations. This result has relevant policy implications because it justifies theexistence of PAYGO social security systems under broader assumptions than in the restof the literature. The simplicity of the model allows also to show that a redistributivepolicy across generation and higher share of risky investments characterize the firstbest.

∗Preliminary and incomplete: comments are welcome!†Center for Doctoral Studies in Economics (CDSE), University of Mannheim. elena.rancoita@gess.uni-

mannheim.de. I would like to thank Klaus Adam, Daniel Haremberg, Dirk Krüger, Philip Jung, MicheleTertilt for their useful comments and suggestions.

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IntroductionHow much of the economy’s resources should we invest in technologies with uncertain re-turns? A natural tension arises between the objectives of consumption smoothing and thoseof growth, though with complete markets decentralized decisions should lead to efficient in-vestment allocations. Yet, it is well known that overlapping generations (OLG) cause marketincompleteness because agents cannot trade before their birth with agents of older genera-tions. Thus, investment risks cannot be shared and traded across generations and choicesare potentially inefficient (Acemoglu (2007)).

The objective of this paper is to spell out an inefficiency arising in OLG models whichhas not yet been considered in the literature. This inefficiency arises due to the agentsnot internalizing the externalities that their risky decisions place on future generations. Toillustrate this inefficiency, I set up a model in which agents can transfer their wealth intothe future through investments in two types of production: a safe one and one subject toa technology shock. The analysis is built around the following questions: Which share ofinvestments is allocated to the risky technology in the competitive equilibrium? Can thisallocation be Pareto improved if at least two cohorts can trade risk? Which policies canachieve a Pareto improvement? Are those policies also a first best?

I present a simple OLG economy where each generation lives for two periods. In the firstperiod agents work and split their income either in the risky or in the safe technology. In thesecond period agents consume their investment returns. Thereby, the aggregate investmentchoices of the current period determine the capital stock and, thus, the incomes of futureperiods. If each generation maximizes its utility (excluding bequests and dynasty utilityfunctions), then it does not internalize two pecuniary externalities on the utility of futuregenerations, which occur even with perfectly correlated returns and wages, as discussed inthe literature. Both these externalities are due to the labor income from work of each cohortbeing determined by the investments of the previous cohort.

The first externality concerns agents of future generations having the possibility to reinvestthe outcome of past investments in the same “lottery”. Namely, the investment decisions ofthe generation born at time t determine the capital stock in t + 1 and, thus, the wages ofthe generation born and working in t + 1. This income is invested at the end of t + 1 inthe same production technologies. Intuitively, this investment cycle is similar to playing alottery and then gambling once again with the outcome of this lottery: the variance of theoutcome at the end of the second period is higher than at the end of the first period (onecould always win or always lose, so the outcomes more extreme like in the St. Petersburgparadox). In the same way, for each generation, the consumption risk due to its investmentsin the risky technology is lower than for any following generation, which can reinvest theincome produced by those investments. Then, even if all generations are equally risk averse,the current young generation prefers the previous young generation to invest less in the riskytechnology.

The second externality arises because a one percent increase in the share of risky in-vestments chosen by the generation born at time t has a higher marginal utility for that

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generation than for the generation born at time t+ 1. This effect depends on the concavityof the production technology because the investments at time t determine the wages of thegeneration working in period t + 1. These wages will be invested by the generation born attime t+1 in the same concave production function. In particular, due to the concavity of theCobb Douglas production technology, a marginal change in the share of risky investments attime t has a lower impact on the utility of the generation born in t+ 1 than on the utility ofthe generation born in t. This means that the generation born in t+ 1 would prefer a higherlevel of investment in the risky technology than the one chosen by the previous generation.

Following these arguments, I show that the second externality prevails on the first oneand that the competitive equilibrium allocation can be improved upon by an allocation witha higher share of risky investments. Due to the previous arguments, this higher share of riskyinvestments increases the utility of future cohorts, but decreases the one of the current cohortwhich suffers from the increased consumption volatility. Thus, for a Pareto improvement,the current cohort needs to be compensated. I show that such a Pareto improvement can beachieved with a state contingent sharing rules for output across generations.

It is important not to confuse the questions tackled in this paper with what is known inthe literature as dynamic inefficiency problem.

OLG economies are said to be dynamically inefficient if individual decisions over savingsdetermine an inefficient capital accumulation in that does not optimize the steady stateconsumption. Namely, if people have to save for their retirement, they might be inducedby the fear of future poverty to save too much, decreasing the expected return of capitalwhich then further encourages over-accumulation. It should be remembered that problemsof dynamic inefficiency generally abstract from portfolio decisions and only look at the trade-off between savings and consumption. This latter being empirically very large as found inKrüger and Kubler (2006). In other words, the main issue of this literature is to assess theinefficiency of private consumption-savings decisions and how a Pareto superior allocationcan be achieved (Phelps(1961), Diamond (1965), Acemoglu (2007)).

The inefficiency that I am going to describe could be collocated in the literature dealingwith the so called intergenerational risk sharing problems: if incomes of different generationsare not perfectly correlated due to a financial market incompleteness, a redistributive policyacross generations diversifies the sources of consumption risk for each generations improvingupon the competitive equilibrium allocaiton. Usually the literature regarding intergenera-tional risk sharing problems aims at showing that PAYGO social security systems are welfareimproving (Gordon and Varian (1988), Shiller (1999), Smetters (2002), Krüger and Kubler(2006), Ball and Mankiw (2007), Bohn (2009) ).

The remainder of this paper is structured as follows. In the first section, I revise theliterature on dynamic inefficiency and inter-generational risk sharing in order to clarify thedifferences and similarities with my findings. In Sections 2 and 3, the model is presented andthe inefficiency is discussed in a context of incomplete financial markets due to missing Arrowsecurities and constraint participation. In Section 4, I generalize the findings of Section 3to the case of complete Arrow Debreu securities and I discusse the policy implications. In

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Section 5, I show that the policy here presented characterizes the first best.

1 Related literatureTo my knowledge, there is no literature on the inefficiency presented in this paper. Whilemodels on dynamic inefficiency are characterized by endogenously determined wages, butignore investment portfolio allocation, models on optimal inter-generational risk sharing as-sume exogenous wages (except Krüger and Kubler (2006)) and consumption risk diversifi-cation through different income sources (for instance returns to capital investments and anunfunded pension system).

Inter-generational risk sharing models generally aims at explaining the existence of PAYGO1

social security systems or other redistributive policies across generations as a cure to somemarket incompleteness. Although the most natural incompleteness appears in OLG modelswhere “markets do not appear able to pool lotteries faced by different non overlapping gener-ations” (Gordon and Varian (1988)), often the incompleteness due only to the constrainedparticipation to future and previous lotteries of non overlapping generations and the one dueto missing Arrow securities are not distinguished. I believe that a distinction is useful becausethose two types of market incompleteness need different policies for a Pareto improvement:In an OLG economy with an incomplete set of Arrow securities, a sufficient condition for aPareto improvement is the non perfect correlations of incomes (the return to capital for theold and the wage for the young) of two consecutive generations. Instead, the policy that Iwill present doesn’t require that condition.

Gordon and Varian (1988) and Ball and Mankiw (2007) analyze how the redistribution ofincomes across generations can achieve a Pareto improvement upon the competitive allocationin a setting where cohorts are endowed with an iid exogenous wage and can invest only in onerisky production. In this framework, each generation has an idiosyncratic shock if return’sdistribution is iid, therefore there is room for welfare improvement when allowing for anunfounded pension system. While Gordon and Varian (1988) focus on the conditions allowingfor an ex-ante and ex-post incentive compatible transfer policies, Ball and Mankiw (2007)look at the equilibrium allocation chosen if agents could to trade contingent consumptionclaims at time 0, when the first generation is born.

Shiller (1999) discusses different forms of social security (intragenerational, intergenera-tiona dn international) and derives policy advices on which direction future reforms of the USsystem should go. In particular, he argues that, in the debate on replacing unfounded socialsecurity system with a founded one, it is often argued that investing in stocks would be muchmore profitable than financing a unfounded social security system. As an example, Shillerconsiders a simple framework where agents live for two periods, earn exogenous stochasticincomes in both periods and can decide whether to store their saving or invest them in arisky technology. He shows that private agents invest in risk less than a social planner, if the

1For PAYGO social security system is meant a social security system where pensions are financed by taxeson labor income of younger generations

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investment returns and income of the young generation are uncorrelated. Instead, higher isthe correlation between investment returns and income of the young generation less is thesavings fraction that the social planner invests in the risky technology. With high positivecorrelation between returns and wages the social planner could invest even less than privateagents in the risky production.

Krüger and Kubler (2006) examine risk investment decisions in OLG economies wherewages of future generations are endogenously determined by past investments and marketsare incomplete due to missing assets and to constraint participation to the securities market.Krüger and Kubler (2006) aims at justifying the existence of a PAYGO social security systemwith the possibility of pooling the consumption risk across generations. Krüger and Kublerargue that the positive welfare effects due to risk pooling might be larger than the crowdingout effect of capital due to the financing of an unfounded social security system.

My objective is substantially different from Krüger and Kubler (2006), because I want toanalyze whether the share of investments in risky technology is inefficient in OLG modelswhere future income is endogenously determined by past investments. However, the findingspresented here can extend the results of Krüger and Kubler (2006): not only I will provein Proposition 2 that the PAYGO social security system is welfare improving under muchbroader assumptions than Krüger and Kubler (2006), but also that it is a first best.

Namely, it will be illustrated that in the same set up of Krüger and Kubler (2006) myinefficiency can justify the existence of a social security system without any assumption onthe correlation between incomes of different generations. Thus, the assumption of incompletemarkets due to missing Arrow Debreu securities is not necessary to show Pareto improvementof the competitive equilibrium of a PAYGO system in OLG models with agents living for adeterministic and finite time.

From a technical point of view, the present model is close to De Marzo, Kaniel, Kremer(2004), (2006) and (2007) who show that inefficient investments can occur due to endogenousrelative wealth concerns arising because of constrained participation to some market.

2 Model: simple set up with CARA utility functionIn this section, I present the basic version of the model with CARA utility function. Thisutility function allows to write the expression for the competitive equilibrium in an intuitiveway, but it needs a continuous (normal) distribution of consumption, thus, of the technologyshock in order to simplify computations. Given that agents have only two investment oppor-tunities, this implies an incomplete financial market not only because of constrained agents’participation, but also because of missing assets2. In Section 4, the model will be solved forthe case of CRRA utility function and complete financial markets showing that results do

2I distingush between incompleteness due to constrained participation to the financial markets and dueto missing assets. The first occurs when different agents cannot cannot participate to the same risk market(e.g. agents live in different eras or agents lives in different geographical areas and goods are local). Thelatter if the number of assets with linearly independent returns is lower that the number of possible shocksrealizations (states of the world)

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not depend on the non perfect correlation between the incomes of consecutive generations.3

Consider a world with infinite overlapping generations. Each generation consists of acontinuum of risk averse agents living for two periods and there is no population growth.When agents are young, they work and invest all their earnings. There exist both a riskyand a non risky investment opportunity: For example, all firms produce energy either withFossil Fuel Power Stations (FFPS) or with Nuclear Power Plants (NPP), the latter beingsubject to technology shocks. Young agents decide over the fraction of savings (wages)devoted to each investment technology. During their oldness, agents earn the returns fromtheir investments and consume.

Firm sector

There exists two types of firms: firms of type I (Innovative, the NPP) and C (Conserva-tive, the FFPS). For each production technology there exists a continuum of atomless firmsin perfect competition maximizing their profits. The aggregate production function of bothtypes of firms is a Cobb Douglas:4

Y It+1 = At+1

(KIt

)α (LIt+1

)1−α, (1)

At+1 ∼iid N(A, σ2

), (2)

Y Ct+1 =

(KCt

)α (LCt+1

)1−α(3)

where α ∈ (0, 1), and Ljt , Kjt for j = I, C represent the total capital and working hours

employed in each type of firm respectively. Notice that the only difference between the twoproduction technologies is that firms of type I are subject to a technology shock, implyingthat the risk is the only difference between the two investments. For the discussion it is easierto work with the intensive forms of the production functions

yIt+1 = At+1(kIt)α, (4)

yCt+1 =(kCt)α, (5)

where kjt , yjt for j = I, C are the capital and output per working unit respectively. Forsimplicity, at the end of each period, all capital depreciates.

Optimization problem of a representative young agent

At time t, a representative young agent , born at time t, supplies inelastically lIt workinghours to a firm of type I and lCt working hours to a firm of type C, such that the totalamount of hours worked is smaller than the maximum time available, l lIt + lCt ≤ l <∞. As

3Note that my proofs require only the strict concavity of the utility function and no further properties.Therefore, the use of a CARA or a CRRA utility function should not affect the results, but only simplifycomputations according to the shock distribution.

4For the production function, I indicate capital with the time index t in order to underline that theinvestment decision is taken at time t.

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remuneration of his work, a young agent receives a total wage wt which is the sum of thewages he receives from each firm

wt = wIt lIt + wCt l

Ct . (6)

As it is shown in Appendix D, the assumption of inelastic labor supply does not influencethe results, but it is a practical simplification of computations. Agents are risk averse andhave CARA utility function

u(ct) = 1− e−γct (7)

where γ > 0 quantifies the risk aversion. In order to focus on the determination of the optimallevel of risky investments, assume that a young agent saves all his income and chooses theamount of income, kIt ∈ [0, wt], that he invests in the risky technology and kCt ∈ [0, wt] theinvestment in the non risky technology. A young agent maximizes the expected utility of oldage consumption, ct+1 with respect to kIt and kCt . Therefore, the optimization problem of ayoung agent is the following:

maxkIt , k

Ct

Et [u (ct+1)] (8)

such that the budget constraints hold:

ct+1 ≤ rIt+1kIt + rCt+1k

Ct (9)

wt = kIt + kCt (10)

where rjt+1 is the gross return from investment in firm of type j = I, C and Et[.] indicatesexpectations at time t+ 1 conditional on the information until time t, At ≡ {A0, .., At}. 3.

Optimization problem of a representative old agent

At time t+1, agents are old and decide how much to consume. Formally, the optimizationproblem of an old agent is the following:

maxct+1

u (ct+1) (11)

under the budget constraint

ct+1 ≤ rIt+1kIt + rCt+1k

Ct (12)

wt = kIt + kCt (13)

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2.1 The competitive equilibriumThe competitive equilibrium can be computed for each cohort by backward induction: firstfinding the equilibrium on the consumption good market, then on the investment market.

The equilibrium on the good market at each period is found solving the optimizationproblem of the old agents under the resources constraint of eq. (10) and (??) and the firms’optimization problem. Given that firms are in perfect competition, the factors of production,labor and capital are remunerated at their marginal productivities:

rI∗t+1 = αAt+1(kIt)α−1

, wI∗t+1 = (1− α)At+1(kIt)α, (14)

rC∗t+1 = α(kCt)α−1

, wC∗t+1 = (1− α)(kCt)α. (15)

Notice that the equilibrium values of the returns and wages depend on the average capitalinvested in the correspondingly type of firm, kIt and kCt . The utility function of old agentsis strictly increasing in the consumption good, so that the budget constraint is binding andthe total consumption is given by

c∗t+1 = rI∗t+1kIt + rC∗t+1k

Ct . (16)

Let us compute kI∗t and kC∗t , the optimal individual investment in the risky and non riskytechnology, respectively. The maximization problem at the beginning of time t for each youngagent is the following:

maxkIt , k

Ct

E[u(ct+1)|At] (17)

st

c∗t+1 = rI∗t+1kIt + rC∗t+1k

Ct , (18)

wt = kIt + kCt . (19)

where At = [A0, ..., At] indicates the vector of all shocks from time zero to t. In AppendixA.1, it is shown that for kIt → 0 the utility, as function of kIt , degenerates to a vertical lineand for kIt = wt, the optimal individual level of risky investment is zero. Therefore, it isreasonable to look for internal solutions of kIt . As it is shown in Appendix A.1, the FOC ofthe problem is given by

−rC∗t+1 + α(kIt)α−1

A− σ2

2 α22(kIt)2α−1

= 0 (20)

This leads to the following result

Lemma 1. Under the CARA utility function specification, there exists one and only oneequilibrium level of investments which corresponds to the fixed point kI∗t of

kI∗t =Et[rIt(kI∗t)]− rCt

(kI∗t)

γV ar [rIt (kI∗t )] (21)

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and such that kI∗t ∈ (0, wt).

Proof. See Appendix A.1.

Eq. (21) is a Sharpe ratio where the denominator is the variance of the risky returnmultiplied by the risk aversion parameter instead of the standard deviation of the riskyreturn, as in its common formulation. The interpretation is quite simple: the equilibriumlevel of risky investments is increasing in the difference between the expected return of therisky and the risk free investments weighted by the inverse of the risky return’s variancetimes the inverse of agents’ risk aversion.

This result together with the equilibrium consumption, wages and interest rate allows usto define the competitive equilibrium as follows

Definition 1. A competitive equilibrium is a set of quantities and prices{{c∗t} ,

{kI∗t},{kC∗t

}, rI∗t , r

C∗t , wI∗t , w

C∗t

}∞t=0

such that

• the quantities{{c∗t}j ,

{kI∗t}j,{kC∗t

}j

}∞t=0

are solutions to the agents’ optimization prob-lems in eq. (8-10) and (11-13) and

• the prices{{rI∗t},{rC∗t

},{wI∗t

},{wC∗t

}}∞t=0

clear the markets.

3 Inefficiency of the competitive equilibrium and Paretoimprovement

OLG economies are, by construction, incomplete markets because future cohorts cannottrade risk before their births. This may generate inefficiencies because each generation, ifmaximizing its own utility, does not internalize the effects of its choices on the welfare offuture generations.

In order to discuss the inefficiency of the competitive equilibrium computed in the previoussection, I define a Pareto superior equilibrium to the competitive equilibrium as follows

Definition 2. An allocation is Pareto superior to the competitive equilibrium if it increasesthe expected utility of at least one generation (say the generation born at t+1) and

• keeps the previous generation’s expected utility at least at the level of the competitiveequilibrium

• keeps the following generation’s expected utility at least at the level of the competitiveequilibrium

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Due to the inefficiency previously discussed, one might expect that the competitive equilib-rium is not Pareto efficient and can be improved upon. Each cohort places two externalitieson future cohorts and thus it might over/under-invest in risk. This is due to the increasedrisk in the income of future generations for which the current generation does not care whenoptimizing. Namely, the wage of the cohort born at time t+ 1 is the sum of their wages fromworking in firms of type I and of type C and competition on the factor markets implies thatin equilibrium wages in both sectors equal the marginal productivities of work, which aredetermined by the investment choices at time t. At first sight, a shift in investments fromfirms of type C to firms of type I implies an increase in the wages of future generations,because it corresponds to an increase in the marginal productivity of work in firms of typeC, but it might imply a decrease as well as an increase in the income from working in firmsof type I.

If, for example, the generation born at time t invested all its wealth in the firm producingenergy with NPP and a negative shock occurred, there would be no way for the cohort bornat time t+ 1 to hedge this bad shock realization. Therefore, one might expect that if agentsdon’t care about future generations, they over-invest in the risky production.

On the contrary, an increase of investments in firms of type I (risky investments) canhave two possible positive externalities on future generations. The first externality can isgenerated by the compounding of thelottery over generations: One generation invests in thea lottery and the following generation invests the outcome of the lottery in the same lottery.In the long run the variance of the return of the lottery will increase, but the mean of thelottery will not change. Then, if the expected return of the lottery is equal to the return ofa safe investment, a risk averse individual should prefer investing in the safe investment toplaying repeatedly the lottery. If the expected return is higher, he might prefer the lottery.In this model, the capital invested at time t in the technology I determines the wage of thefuture generation and can be reinvested by this generation. Therefore, especially when theexpected return of the risky investment is equal to the return of the safe one, an increase inkIt can harm the generation born at time t + 1. Consider a lottery with returns σ and −σwith probability 0.5. Suppose that the generation born in t decides to invest, on average, kItin the lottery, the distribution of the lottery’s return would be given byk

It (1 + σ) pr = 0.5kIt (1− σ) pr = 0.5

(22)

Assume that the generation born at time t + 1 could reinvest the outcome of the lottery inthe same lottery. Then the distribution of the lottery’s returns for the generation born int+ 1 would be given by

kIt (1 + σ)2 pr = 0.25kIt (1− σ) (1 + σ) pr = 0.5kIt (1− σ)2 pr = 0.25

(23)

Both lotteries have mean 1, but the variance of the lottery returns is higher for the generation

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born at t+ 15. Consequently, if both generations are equally risk averse, the generation bornat time t+ 1 would prefer a lower kIt than the one born at time t.

The second externality concerns a marginal change of investments in technology I at timet determining a lower marginal change in the utility of the generation born at time t+1 thanof the one born at time t. This occurs because kIt determines the wages of the generation bornat time t+1, but wages enter the utility function of that generation only after being invested.So, while kIt enters the utility function of the generation born at time t via the interest rate,it affects the utility of the generation born at time t+1 via the interest rates realized at timet + 2 on the investment of the wage. If the parameter of the production functions α is lessthan one, then this implies that consumption of the generation born at time t + 1 is a lessconcave function of kIt than the one of the generation born at time t. This means also thatthe generation born at t+ 1 is less risk averse toward the volatility in consumption inducedby the investment decisions taken at time t than the generation born at time t. Suppose, forexample that agents born at time t earn wages w and decide the share x ∈ [0, 1] to investin a risky production function of the type yRt = g (xw) = A (xw)α, where A = 1 + σ withprobability 0.5 and A = 1 − σ with the same probability and α < 1. If agents do no investin the risky production, they can invest in a non risky production yNRt = f (xw) = (xw)α.Consider a first scenario, where agents consume all the production in t + 1 and earn utilityfrom consumption: U = u (f (xw) + g (xw)). Then, under the assumption of strictly concaveutility function u (.), in equilibrium x ∈ (0, 1). Now assume that in t+1 agents for simplicitycannot consume and can only invest in the non risky production function. In this way, anyeffect due to the principle of playing repeatedly the same lottery is shut down. If agents earnutility in t+2 from the consumption of all production, then their utility is a concave functionof a concave transformation of the sum of productions at time t: U = u (f (f (xw) + g (xw))).In other words, this means that the preferences over the risky decision are different from thefirst scenario. Namely, one could rewrite the utility and U = h (f (xw) + g (xw)), where h isless concave than u if α < 1. The same reasoning is schematized below:

scenario 1 : wt = 1

Axα

↗↘

(1− x)α+

= ct+1 ⇒ u(ct+1) = [Axα + (1− x)α]1−γ

1− γ , γ > 1

scenario 2 : wt = 1

Axα

↗↘

(1− x)α+

= wt+1

wt+1 = Axα + (1− x)α → wαt+1 = ct+2 ⇒ u (ct+2) = [Axα + (1− x)α]α(1−γ)

1− γ , γ > 1

Both these externalities are due to the fact that generations cannot trade before theirbirth, thus, cannot insure against the risky investment decisions of the previous generations.

5For a full specification of the distribution of the outcomes s a function of the times that the lottery isplayed, see Appendix E (to be done)

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Allowing for redistributive policies across generations it is possible to show

Proposition 1. There exists an allocation(kI,Pt , {ρ (At+1)}

)Pareto superior to the compet-

itive equilibrium. This allocation is such that kI,Pt > kI∗t and the generation born at time treceives a share ρ (At+1)of production decreasing in At+1.

Proof. See Appendix B.

Proposition 1 states that the competitive equilibrium allocation can be Pareto improvedupon by a higher level of investments in the risky technology accompanied by an appropriateredistributive policy such that when a bad shock occurs, the generation born at time t

receives a higher share of output then when a positive shock occurs. Proposition 1 showsalso that the effect of reinvestment in a Cobb Douglas production function prevails on theincrease in risk due to repeatedly playing the same lottery with the previous outcome of thelottery. However, increasing the investments in the risky production reduces the welfare ofthe generation born at time t, because, as shown for Lemma ??, the competitive equilibriumis unique and maximizes their utility. But, it is possible to compensate the generation bornat time t with a state contingent redistribution of output policy. Under the assumption thatall generations are risk averse, a reduction in the volatility of their incomes, consequentlyconsumption, determines an increase in the expected utility. Intuitively, the generation bornat time t can be compensated if, for example, in correspondence to a negative (positive)shock it receives a higher (lower) share of output than in the competitive equilibrium.

In terms of policy implications I am justifying the existence of a PAYGO social securitysystem even under the assumption of perfect correlation between the return to capital claimedby the old cohort and the wage earned by the young ones. This is a more general result thanthe findings of Krüger and Kubler (2006). The authors justify intergenerational risky sharingsystem in the form of PAYGO because it allows the diversification of old age incomes.

4 Generalization of the results to CRRA utility func-tion

The previous result can be generalized to CRRA utility functions. In this case, the interpre-tation of Proposition 1 becomes clearer and one can write a simple example with a completeset of Arrow Debreu securitieswhere the results do not change.

Let us consider the economy described in Section 2, but CRRA utility function

u(ct) = c1−γt

1− γ , (24)

whereγ > 0 is the risk aversion parameter. Let the stochastic process be a binary process

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which consists only of a bad and positive shock:

At+1 =

A+ σ

A− σ(25)

for simplicity assume that both cases occur with equal probability. Wages and interest ratesin equilibrium are determined in the same way as with the CARA utility function. Let usdefine as xt ∈ [0, 1] the share of income invested by a representative young agent in therisky investment technology: xt ≡ kIt

wt. Optimizing over xt is equivalent to optimizing over

kIt , but it makes computations easier in this case. As shown in Appendix C.1 and C.2, thecompetitive equilibrium share of risky investment, x∗t , is given by the fixed point of

x∗t = rCt+1 (x∗t ) [1−B (x∗t )][rI∗t+1(x∗t ,−σ)− rCt+1 (x∗t )]B (x∗t )− [rI∗t+1(x∗t , σ)− rCt+1 (x∗t )]

. (26)

It is possible to show that eq (26) does not depend on the previous level of capital becausewα−1t can be simplified on the left hand-side of the equation. This is a useful property for

the proof of

Proposition 2. There exists an allocation(xPt , ρ, ρ

)Pareto superior to the competitive equi-

librium. This allocation is such that xPt > x∗t and the generation born at time t receives ashare ρ (ρ) if a negative (positive) shock occurs, where ρ > α and ρ > ρ. Where ρ and ρare, respectively, the share of production which goes to the old generation if a negative or apositive technology shock occurs at time t+ 1.

Proof. Proof, see Appendix C.3.

Intuitively, proposition 2 tells us that, as in proposition 1, the generation born at time t+1prefers that the previous cohort invested more in the risky technology. However, increasingthe share of risky investment xt is detrimental for the cohort born at time t. A possiblecompensation is to share the output contingently on the realization of the shock: if a negativeshock occurs the cohort born at time t receives a higher share of the total production thanin the competitive equilibrium. Instead, if an positive shock realizes, that cohort will receivea lower share of income than in the case of positive shock. This redistribution of outputdecreases the volatility of the old cohort’s income and, consequently, of its consumption.

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5 The first best solutionLet us consider the optimization problem of a social planner who maximizes the discountedsum of utilities of all generations6

maxct, kIt , kCt , it E0 {∑∞t=0 β

tu(ct)} , β ∈ (0, 1) (27)

under the constraints

ct = At(kIt)α

+(kCt)α− it+1 (28)

it = kIt + kCt (29)

where u(c) = c1−γ

1−γ and At is distributed as in Section 4. That is, in each period, the socialplanner decides how much production to transfer under the form of investment to the futurecohort, it+1, and how to split the investments among the two technologies, kIt and kCt . Themaximization problem implies the following two FOCs

kIt : Et−1

{u′(ct)

[At(kIt)α−1

−(it−1 − kIt

)α−1]}

= 0 (30)

it : −u′(ct−1) + βEt−1

{u′(ct+1)

(it − kIt

)α−1}

= 0. (31)

The model cannot be analytically solved, although one could characterize some propertiesof the equilibrium in a ways similar to Proposition 2. I solved the model numerically usingtime iteration and parameter values α = 0.3, β = 0.9, γ = 3, A = 1. The outcomes of thenumerical solution are reported in Figure 1. As we can see on the left side, the olde generationreceives a higher share of output than in the competitive equilibrium, as in Proposition 2.In particular, the volatility of the old generation consumption is reduced because when anegative shock occurs, the old generation receives a larger fraction of output than when apositive shock occurs. On the right, the percentage of total investment in the risky investmentis shown. The picture shows that not only a higher share of risky investments is Paretoimproving, but also first best.

6The choice of the social planner problem is arbitrary and one might criticize the unequal weighting ofthe utilities of different cohorts. However, this formulation allows to apply the Bellman theorem and to solvethe infinite horizon maximization problem.

14

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

wt−1

Sh

are

of p

rod

uct

ion

co

nsu

me

d b

y th

e o

ld g

en

era

tion

, ρ

SP positive shockSP negative shockCE positive shockCE negative shock

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.467

0.468

0.469

0.47

0.471

0.472

0.473

0.474

0.475

0.476

wt−1

Sh

are

of risk

y in

vest

me

nts

, x t

CESP

6 ConclusionsThis paper contributes to the existing literature on inefficiencies in OLG models, most no-tably on dynamic inefficiencies and risk sharing among generations, by pointing out a newinefficiency regarding the investment portfolio. Selfish agents maximizing their utility do notconsider the pecuniary externalities that the investment decisions exert on future generations.Insurances against the risky choices of previous cohorts are not possible because future gen-erations are not yet born when these decisions are taken. This yields inefficient investmentchoices. In particular, when agents can choose between a risky investment opportunity and asafe one, the share of savings allocated to the risky investments can be Pareto improved witha simple redistributive policy. More specifically, agents overall benefit more from a largershare of risky investments than from the one chosen in the competitive equilibrium. A Paretoimprovement can be achieved by redistributing more (less) resources than in the competitiveequilibrium to old agents when negative (positive) shocks occur. The basic model is highlyabstract and uses several simplifications such as keeping the savings-consumption decisionexogenous, to emphasize the main issues. However, the results are robust to different typesof utility functions (CRRA and CARA) and to market incompleteness due to missing assets,providing a rational for risk sharing across generations also when future wages and returnsare perfectly correlated. In the final part I analyzed also the firs best solution. This latteris characterized by the same properties of the Pareto improving allocation. In the first bestsolution one cannot abstract from the consumption.savings decision. My findings are robustalso to this type of decision and, thus, possibly exist also in a dynamic inefficient economy.Future research should be devoted to a more quantitative analysis of the different inefficiency.

15

Appendix A

A.1: Competitive equilibrium with CARA utility functionmaxkIt kCt

E[u(ct+1)|At] (32)

st

c∗t+1 = rI∗t+1kIt + rC∗t+1k

Ct (33)

wt = kIt + kCt (34)

Substituting the constrain in the utility function and taking the expectation we obtain

Et

[1− e−γ[rI∗

t+1kIt+rC∗

t+1(1−kIt )]]

= 1− e−γrC∗t+1(1−kIt )−γEt[rI∗

t+1]kIt+0.5γ2V ar(rI∗t+1)(kIt )2

(35)

which implies the following FOC:

−rC∗t+1 + E[rI∗t+1

]− γkIt V ar

(rI∗t+1

)= 0 (36)

⇒ kIt =Et[rI∗t+1

]− rCt+1

γV ar (rI∗t+1)(37)

where rI∗t+1 = αAt+1(kIt)α−1

and rC∗t+1 = α(kCt)α-1

.

A.2: Existence and uniqueness of the competitive equilibrium withCARA utility function

Lemma 1.: Under CARA utility function specification, there exists one and only one equilib-rium level of investments which corresponds to the fixed point, kI∗t of the following expression

kI∗t(kI∗t)

=Et[rIt(kI∗t)]− rCt

(kI∗t)

γV ar (rIt (kI∗t )) (38)

and such that kI∗t ∈ (0, wt).

Proof to Lemma ??.

First let us consider the boundaries. For kIt → 0, the FOC becomes

limkIt→0−rC∗t+1 + E

[rI∗t+1

]− γkIt V ar

[rI∗t+1

](39)

= −αwα−1t + lim

kIt→0αA

(kIt)α−1 [

1− αγσ2(kIt)α]

= +∞ (40)

this means that the overall utility function is increasing for low levels of investments in therisky technology (its interest rate explodes).

16

For kIt → wt, the FOC becomes:

limkIt→wt

−rC∗t+1 + E[rI∗t+1

]− γkIt V ar

[rI∗t+1

](41)

= limkIt→wt

−α(wt − kIt

)α−1+ αA

(kIt)α−1

− γσ2α2(kIt)2α−1

= −∞. (42)

That is, the utility function is strictly decreasing for large investments in the riskytechnology. Being the utility, as a function of kIt , two times continuously differentiable onthe interval for kIt ∈ (0, wt),the marginal utility equal to ∞ for kIt → 0 and equal to −∞ forkIt → wt , then there exists at least one kIt ∈ (0, wt) satisfying the FOC, by the continuity ofthe sing of the first derivative. By the strict concavity of the utility function the maximumis unique.

17

Appendix B

B.1: Proof Proposition 1

Proposition 1: There exists an equilibrium Pareto superior to the competitive equilibrium.This equilibrium is such that kI,Pt > kI∗t and the generation born at time t receives a shareρ (At) increasing in At.

Let us consider the generations born at time t and t + 1 and suppose that it is possi-ble to write a contract contingent on the states of the world, all the possible outcomes ofthe technology shock At, prescribing the investment in risky capital, kIt , and the share ofproduction distributed to the generation born at time t, it (At). The contract defines anallocation Pareto superior to the competitive equilibrium outcome if it satisfies the followingoptimization problem:

maxkt,it

Et [u(ct+1)] (43)

st

ct+1 = yt+1 − it+1

= At+1(kIt)α

+(it − kIt

)α− it+1 ∀t (44)

u(ct) ≥ u (45)

where the first constrain is the budget constraint of the generation born at time t and thesecond constraint is the participation constraint of the generation born at time t+1, where urepresents the utility achieved with the competitive equilibrium. Notice that the competitiveequilibrium corresponds to an allocation (it (At) , kt) = (α, kt). There exists a real numberβ ∈ < such that the previous optimization problem is equivalent to

maxkIt , it

Et [u(ct) + βEt+1 [u(ct+1)]] (46)

st

ct+1 = yt+1 − it+1

= At+1(kIt)α

+(it − kIt

)α− it+1 ∀t (47)

The FOCs of the problem are given by

it : γ exp {−γyt + γit}+Et[−γβ exp {−γyt+1 + γit+1}

(α (it − kt)α−1

)]= 0 (48)

kt : Et[−γ exp {−γyt+1 + γit+1}

(rIt+1 − rCt+1

)]= 0 (49)

18

From eq() we obtain:

exp {−γyt + γit} = βEt[exp

{−γAt+1

(kIt)α

+ γit+1}]

exp{−γ

(it − kIt

)α} (α (it − kt)α−1

)(50)

Taking log on both sides of the equation

−γyt + γit = log(β) + log(Et[exp

{−γAt+1

(kIt)α

+ γit+1}])

−γ(it − kIt

)α+ (α− 1)log

(α(it − kIt

))(51)

which implies

it +(it − kIt+1

)α+ (1− α)

γlog

(α(it − kIt

))=

yt + 1γlog(β) + 1

γlog

(Et[exp

{−γAt+1

(kIt)α

+ γit+1}])

(52)

Then the left hand side is increasing in it+1 and the right hand side is increasing in yt, thusin At+1. We could rewrite the expression in the following simpler way

it = ρ(At)yt (At) (53)

where ρ(.) is an increasing function of At+1 and is the share of output given to the old (thegeneration born at time t). By definition, it must be less than the overall production, thatis ρ(At) ≤ 1, and, in order to keep eq.() in equilibrium, one it is optimal to increase it whenytincreases. Roughly speaking, by the continuity of both hand sides of eq. (), this means thatfor large positive (negative) shocks, the generation born at time t earns a larger (smaller)share of output than in the competitive equilibrium. As all the other generations, the cohortborn at time t+ 1 is risk averse. The policy of eq() implies a larger volatility in the incomeof this generation, thus also in its consumption, which itself decreases the expected utility.This loss in utility, intuitively, will be compensated by the optimal investment policy kI, Pt .

Let us consider the second FOC (eq. ()):

Et[−γ exp {−γyt+1 + γit+1}

(rIt+1 − rCt+1

)]= 0 (54)

Et[−γ exp {−γ (1− ρ(At+1)) yt+1}

(rIt+1 − rCt+1

)]= 0 (55)

⇒ Et[γ exp {−γ (1− ρ(At+1)) yt+1} rIt+1

]= (56)

Et[γ exp {−γ (1− ρ(At+1)) yt+1} rCt+1

](57)

19

The right hand side of eq () can be solved as follows:

exp {−γ(1− ρ) (it − kt)α}ˆ ∞−∞

12√σ2π

(58)

exp{−γ (1− ρ)At+1k

αt −

12σ2 [At+1 − A]2

}At+1k

α−1t γdAt+1 = (59)

exp {−γ(1− ρ) (it − kt)α} kα−1t γ

ˆ ∞−∞

12√σ2π

(60)

exp{− 1

2σ2

[A2t+1 − 2AAt+1 + A2 + 2σ2γ(1− ρ)At+1k

αt

]2}At+1dAt+1 = (61)

kα−1t γ

ˆ ∞−∞

12√σ2π

exp{− 1

2σ2

[At+1 −

(A− σ2γ(1− ρ)kαt

)]2(62)

− 12σ2

[2Aσ2γ(1− ρ)kαt − σ4γ2(1− ρ)2k2α

t

]}At+1dAt+1 = (63)(

A− σ2γ(1− ρ)kαt)kα−1t γ exp

{−γ(1− ρ)Et [yt+1] + γ2 1

2(1− ρ)2V ar (yt+1)}

(64)

The left hand side of eq() equals

γ exp{−γ (1− ρ(At+1)) yt+1 + γ2(1− ρ)2V ar (yt+1)

}rCt+1 (65)

which implies that eq() can be rewritten as

Et[rIt+1

]− (1− ρ)γV ar

(rIt+1

)kt = rCt+1 (66)

⇒ kt =Et[rIt+1

]− rCt+1

(1− ρ)γV ar (rIt+1)(67)

Now, let us define

g(kt) ≡Et[rIt+1

]− rCt+1

(1− ρ)γV ar (rIt+1)(68)

then we have thatg(kt) = 1

1− ρf(kt) (69)

As it was shown, f(kt) is a strictly concave function, which means it intersects at maximumtwice the x-axis. It was shown in Appendix A that f(kt) = 0 in kt = 0 and in kt ∈ (0, wt).Therefore, also g(kt) intersect the x-axis in the same points. Moreover, for all kt ∈ (0, wt),the first derivative of g(kt) is in absolute value higher than the one of f(kt) because ρ ∈ (0, 1).This means that g(kt) intersects the bisector in a point kI,Pt > kI∗t .

20

Appendix C

C.1: competitive equilibrium with CRRA utility functionThe maximization problem of young agent j is the following:

maxxjt

Et[u(cjt+1

)](70)

under the budget constraint:

cjt+1 ≤ rIt+1kIjt + rCt+1k

Cjt (71)

kIjt = xjtwt (72)kCjt = ((1− xjt)wt) (73)

where rkt+1 is the rent of capital for investment in firms k = I, C.

Optimization problem of old agents

At time t + 1, agents are old and decide their optimal consumption. The optimizationproblem of old agents is the following:

maxcjt+1

u(cjt+1

)(74)

under the budget constraint

cjt+1 ≤ rIt+1kIjt + rCt+1k

Cjt

kIjt = xitwt (75)kCjt = ((1− xit)wt) (76)

And the market clearing conditionˆj

cjt+1dj =ˆj

(rIt+1k

Ijt + rCt+1k

Cjt

)dj

Wages and interest rates are determined in equilibrium as with CARA utility function.By backward induction one can solve the optimization problem of the young agents. Let uscall

maxkjt

E[u(cjt+1)|At] (77)

21

st

cj∗t+1 = rI∗t+1kIjt + rC∗t+1k

Cjt (78)

kCjt = wt − kIjt (79)

we can rewrite the budget constraint as

cj∗t+1 = rI∗t+1kIjt + rC∗t+1

(w∗t − k

Ijt

)(80)

where xt =´jxjtdj. Then the FOC of the optimization problem is given by:

cjt+1(σ)−γ[rI∗t+1(σ)− rC∗t+1

]+ cjt+1(−σ)−γ

[rI∗t+1(−σ)− rC∗t+1

]= 0 (81)

Contrary to the CARA case, the CRRA utility is not degenerate for kIt → 0. Namely, forkIt → 0, then, the interest rate of the risky investment tends to infinity. Given that cjt+1 ispositive, this means that the utility function is strictly increasing in kIjt , that is, it is optimalto set kIjt = 1. The same reasoning hold for all kIt such that

(rI∗t+1(σ)− rC∗t+1 > 0

)∧(rI∗t+1(−σ)− rC∗t+1 > 0

)(82)

⇒ rI∗t+1(−σ)− rC∗t+1 > 0 (83)

⇒ (A− σ)(kIt)α−1

−(wt − kIt

)α−1> 0 (84)

⇒ k < wt

(A− σ)1

1−α

1 + (A− σ)1

1−α

< wt (85)

In the same way, the utility function is strictly decreasing in kIjt if(rI∗t+1(σ)− rC∗t+1 < 0

)∧(rI∗t+1(−σ)− rC∗t+1 < 0

)(86)

⇒ rI∗t+1(σ)− rC∗t+1 < 0 (87)

⇒ k ∈

wt (A+ σ)

11−α

1 + (A+ σ)1

1−α

, wt (88)

Let us compute now the internal solution for(rI∗t+1(σ)− rC∗t+1 > 0

)∧(rI∗t+1(−σ)− rC∗t+1 < 0

).

We can call

B(kIt)

=[

rI∗t+1(σ)− rC∗t+1−rI∗t+1(−σ) + rC∗t+1

] 1γ

(89)

22

we can rewrite the FOC as:

cjt+1(−σ)B(kIt)− cjt+1(σ) = 0 (90)[

rI∗t+1(−σ)kIjt + rCt+1

(wt − kIjt

)]B(kIt)−[

rI∗t+1(σ)kIjt + rCt+1

(wt − kIjt

)]= 0 (91)[

rI∗t+1(−σ)− rCt+1

]kIjt B

(kIt)

+ rCt+1wtB(kIt)−[

rI∗t+1(σ)− rCt+1

]kIjt − rCt+1wt = 0 (92)

which implies

kIjt =rCt+1

(1−B

(kIt))wt

[rI∗t+1(−σ)− rCt+1]B (kIt )− [rI∗t+1(σ)− rCt+1](93)

The competitive equilibrium is then given by the fixed point of kI∗t ≡ kIjt(kIt)

= kIt . Noticethat the competitive equilibrium does not depend on the wealth in the economy at thebeginning of t, because one can divide by wt eq (?) on both sides and obtain an expressiondepending only by the share of income invested in the risky production.

C.2: existence and uniqueness of the competitive equilibrium withCRRA utility function

AS shown in Appendix C.1, for kIt < wt

((A−σ)

11−α

1+(A−σ)1

1−α

), then the optimal individual in-

vestment in the risky production function is kIjt = 1. Therefore, none of the point in[0, wt

((A−σ)

11−α

1+(A−σ)1

1−α

))is an equilibrium. For kIt ∈

(wt

((A+σ)

11−α

1+(A+σ)1

1−α

), wt

]the optimal in-

dividual investment is kIjt = 0, that is, there is no equilibrium also in this interval.

kIjt = rCt+1(1−B(kIt ))wt[rI∗t+1(−σ)−rCt+1]B(kIt )−[rI∗

t+1(σ)−rCt+1] is continuous in[wt

((A−σ)

11−α

1+(A−σ)1

1−α

), wt

((A+σ)

11−α

1+(A+σ)1

1−α

)]and it is also strictly decreasing in kIt .Moreover, for value before the interval the functionassumes value 1 and afterward value 0. Therefore, rCt+1(1−B(kIt ))wt

[rI∗t+1(−σ)−rCt+1]B(kIt )−[rI∗

t+1(σ)−rCt+1] must in-tersect the bisector once. This implies that there exists a unique fixed point of eq (?) in the

interval[wt

((A−σ)

11−α

1+(A−σ)1

1−α

), wt

((A+σ)

11−α

1+(A+σ)1

1−α

)],thus, a unique equilibrium.

C.3: Proof proposition 2 (to be reviewed)The generation born at time t is indifferent with the competitive equilibrium if

0.5ct+1 (α, σ)1− γ

1−γ

+ 0.5ct+1 (α,−σ)1− γ

1−γ

= 0.5ct+1 (ρ, σ)1− γ

1−γ

+ 0.5ct+1

(ρ, σ

)1− γ

1−γ

(94)

23

which implies

(αyt+1 (σ))1−γ + (αyt+1 (−σ))1−γ = (ρyt+1 (σ))1−γ +(ρyt+1 (−σ)

)1−γ(95)

ρ(ρ)

=

[(αyt+1 (σ))1−γ + (αyt+1 (−σ))1−γ −

(ρyt+1 (−σ)

)1−γ] 1

1−γ

yt+1 (σ) (96)

Now let us consider the optimization problem of the generation born at time t+ 1

maxρ,ρ,xt

Et [u (ct+2)] (97)

st

ct+2 = α (yt+2 (ρ,At+2)) , ρ ={ρ, ρ

}(98)

= α[At+2x

αt+1 + (1− xt+1)α

]wαt+1 (99)

= α[At+2x

αt+1 + (1− xt+1)α

](1− ρ)α (At+1x

αt + (1− xt)α)αwαt (100)

ρ(ρ)

=

[(αyt+1 (σ))1−γ + (αyt+1 (−σ))1−γ −

(ρyt+1 (−σ)

)1−γ] 1

1−γ

yt+1 (σ) (101)

Let us call Zt+2 ≡ α[At+2x

αt+1 + (1− xt+1)α

]wαt and substitute ρ

(ρ), then the maximization

problem can be written as

maxρ,xt

Et+1

0.5

(Zt+2

(1− ρ

(ρ))α

((A+ σ)xαt + (1− xt)α)α)1−γ

1− γ (102)

+0.5

(Zt+2

(1− ρ

)α((A− σ)xαt + (1− xt)α)α

)1−γ

1− γ

(103)

Let us consider the FOC wrt ρ

ρ : Et+1[Z1−γt+2

] {(1− ρ

(ρ))α(1−γ)−1

((A+ σ)xαt + (1− xt)α)α(1−γ) (−1)∂ρ∂ρ

(104)

−(1− ρ

)α(1−γ)−1((A− σ)xαt + (1− xt)α)α(1−γ)

}= 0(105)

In correspondence to the competitive equilibrium value of ρ and ρ the first derivative becomes:

24

Et+1[Z1−γt+2

] {− (1− α)α(1−γ)−1

[((A+ σ)xαt + (1− xt)α)α(1−γ) ∂ρ

∂ρ(106)

+ ((A− σ)xαt + (1− xt)α)α(1−γ)]} (107)

Let us consider first

∂ρ

∂ρ= −

[(αyt+1 (σ))1−γ + (αyt+1 (−σ))1−γ −

(ρyt+1 (−σ)

)1−γ] 1

1−γ−1(108)

yt+1 (−σ)1−γ

yt+1 (σ) ρ−γ (109)

which for ρ = α becomes∂ρ

∂ρ

∣∣∣∣∣ρ=α

= −yt+1 (−σ)1−γ

yt+1 (σ)1−γ (110)

Then eq (?) becomes

Et+1[Z1−γt+2

] {− (1− α)α(1−γ)−1

[−yt+1(−σ)1−γ

yt+1(σ)1−γ ((A+ σ)xαt + (1− xt)α)α(1−γ) (111)

+ ((A− σ)xαt + (1− xt)α)α(1−γ)]} (112)

Et+1[Z1−γt+2

] {− (1− α)α(1−γ)−1 ((A− σ)xαt + (1− xt)α)α(1−γ) (113)1−

(((A− σ)xαt + (1− xt)α)1−α

((A+ σ)xαt + (1− xt)α)1−α

)1−γ (114)

which is positive. Being u(ρ, xt) strictly concave in ρ, then the optimal value of ρ is largerthat α.

The rest of the proof is still on paper, but it goes in the following order:

1. one can show that ρ is increasing in xt

2. one can rewrite ct+2 ans a function of ct+1 and compute the relative risk aversionof u(ct+2) toward ct+1(xt). Measuring the relative risk aversion toward ct+1 is thenequivalent to measure the relative risk aversion toward the investment choice xt. Itturns out that the relative risk aversion toward xt is γα− α+ 1 which is lower than γ.

ct+2 = Zt+2

wαtcαt+1

(1− ρρ

)α(115)

The relative risk aversion of u(ct+2(ct+1)) with respect to ct+1 is given by

RRA = −ct+1

∂2u(ct+2)∂2ct+1∂u(ct+2)∂ct+1

(116)

25

where∂u(ct+2)∂ct+1

= c−γt+2Zt+2

wαtαcα−1

t+1

(1− ρρ

)α(117)

∂2u(ct+2)∂2ct+1

= −γc−γ−1t+2

[Zt+2

wαtαcα−1

t+1

(1− ρρ

)α]2

(118)

+c−γt+2Zt+2

wαtα(α− 1)cα−2

t+1

(1− ρρ

)α(119)

Then the RRA becomes

RRA = ct+1γc−γ−1

t+2

[Zt+2wαt

αcα−1t+1

(1−ρρ

)α]2c−γt+2

Zt+2wαt

αcα−1t+1

(1−ρρ

)α −c−γt+2

Zt+2wαt

α(α− 1)cα−1t+1

(1−ρρ

)αc−γt+2

Zt+2wαt

αcα−1t+1

(1−ρρ

)α (120)

= γc−1t+2

[Zt+2

wαtαcαt+1

(1− ρρ

)α]− α + 1 (121)

= γα + 1− α < γ (122)

That is the generation born at time t + 1 is less risk averse toward the risky decisionsinfluencing ct+1 (that is xt) than the generation born at time t. Therefore, the optimalequilibrium level of xt maximizing the utility of the generation born at time t + 1 isxPt > x∗t .

26

Appendix D

D.1: Sketch of the proof that an endogenous labor decision doesnot change the results of propositions 1, 2Agents decide in the firs period of life about the optimal investments in the risky and in thenon risky technology. This investment is measure in capital per hours worked. Given thatagents derive utility uniquely from the consumption of both goods and there is no disutilityfrom working, their utility depends only on the capital-labor ratio. If one optimized overboth capital and labor, the optimization would give not unique solutions. Therefore, tofix the labor levels does not change the equilibrium allocation which is unique only in thecpaital-labor ratio.

27

Appendix E

E.1: Social planner problemThe social planner maximizes the infinite discounted sum of utilities of all generations withrespect to:

• the optimal share of investment in risk xt, t = 0, 1, ...

• the optimal share of output going to the young generation it

maxρ,x

∞∑t=0

βtu (ct) (123)

st

ct = (At (xt−1it−1)α + ((1− xt−1) it−1)α)− it (124)

take two following periods

(1− γ)−1{[(At (xt−1it−1)α + ((1− xt−1) it−1)α)− it]1−γ

+βEt [(At+1 (xtit)α + ((1− xt) it)α)− it+1]1−γ}

(125)

FOCS

xt : 0.5 [ct+1 (σ)]−γ[rIt+1 (σ)− rCt+1

]it (At) + (126)

0.5 [ct+1 (−σ)]−γ[rIt+1 (−σ)− rCt+1

]it (At) = 0 (127)

it (At) : − [ct (At)]−γ + αiα−1t βEt

[c−γt+1 (At+1x

αt − (1− xt)α)

]= 0 (128)

28

References[1] Laurence Ball and N. Gregory Mankiw. Intergenerational risk sharing in the spirit of

arrow, debreu, and rawls, with applications to social security design. Journal of PoliticalEconomy, 115(4):523–547, 08 2007.

[2] H. Bohn. Intergenerational risk sharing and fiscal policy. Journal of Monetary Eco-nomics, 56(6):805–816, 2009.

[3] G. Demange. On optimality in intergenerational risk sharing. Economic Theory, 20(1):1–27, 2002.

[4] P. DeMarzo, R. Kaniel, and I. Kremer. Technological innovation and real investmentbooms and busts. Journal of Financial Economics, 85(3):735–754, 2007.

[5] P.M. DeMarzo, R. Kaniel, and I. Kremer. Diversification as a public good: Communityeffects in portfolio choice. The Journal of Finance, 59(4):1677–1716, 2004.

[6] P.M. DeMarzo, R. Kaniel, and I. Kremer. Relative wealth concerns and financial bubbles.Review of Financial Studies, 21(1):19–50, 2008.

[7] R.H. Gordon and H.R. Varian. Intergenerational risk sharing. Journal of Public eco-nomics, 37(2):185–202, 1988.

[8] Dirk Krueger and Felix Kubler. Pareto-improving social security reform when financialmarkets are incomplete!? American Economic Review, 96(3):737–755, June 2006.

[9] P.A. Samuelson. An exact consumption-loan model of interest with or without the socialcontrivance of money. The journal of political economy, 66(6):467–482, 1958.

[10] R.J. Shiller. Social security and institutions for intergenerational, intragenerational, andinternational risk-sharing. In Carnegie-Rochester Conference Series on Public Policy,volume 50, pages 165–204. Elsevier, 1999.

[11] Kent A. Smetters. Trading with the unborn: A new perspective on capital incometaxation. Working Papers wp066, University of Michigan, Michigan Retirement ResearchCenter, May 2004.

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