Intergenerational Transmission of Risk Preferences and ...

Intergenerational Transmission of Risk Preferences and Entrepreneurship∗ Matthias Doepke

Fabrizio Zilibotti

Northwestern

Zurich

January 2011

1

Motivation

Goal: study endogenous transmission of risk aversion in a model where agents are subject to two types of risk. On the one hand, agents face some risk when young, which has a strong downside component.On the other hand, they face (with some probability) ”entrepreneurial opportunities” with a stronger upside component (but also some downside risk) in mature age. Key assumptions: Young agents are by ”nature” more risk tolerant than old agents. Parents behave paternalistically with respect to their children’s choice – they use their own risk tolerance in order to evaluate the lotteries facing the young. Parents can influence their children risk tolerance, possibly at some cost. Trade-off: transmitting risk tolerance is good both for its direct effect on expected utility (under our assumption below) and because it foster entrepreneurship which may be good for welfare. However, a risk-tolerant child may be insufficiently careful in young age and take risks that parents do not like. ∗

Preliminary and incomplete. Financial support by the National Science Foundation (grant SES-0519265) and the Alfred P. Sloan Foundation is gratefully acknowledged. Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208 (e-mail: [email protected]).

Main predictions: depending on the extent of the young-age risk and the probability of future entrepreneurial opportunities, the society may end up with more or less risk aversion and entrepreneurs. More entrepreneurs means more output and possibly higher growth.

2 Model Environment Consider a model where altruistic parents invest in children’s risk tolerance. We parameterize risk tolerance by a ∈ [a, a ¯] where we assume risk tolerance to be increasing in a. We assume that parents can affect their children’s risk tolerance with probability 1 − π. With probability π, preference transmission fails and children get the ”default” risk tolerance a ˆ.1 The total mass of the adult population is constant and normalized to unity. Parents are endowed with time-separable Von Neumann-Morgenstern utility, with function u (c; a) , where u is increasing and concave in c and increasing in a. Children evaluate lotteries with felicity uY (c; a) . We assume that u induces more risk averse than uY . The timeline is the following: 1. Parents endowed with utility a choose the children’s utility, a0 . 2. Young children face some low-risk lottery (cSY ). With probability λB they have the option of replace the low-risk default lottery with a high-risk (bad ) lottery (cRY ). 3. Grown-up children face some low-risk lottery (cOY ). With probability λG they have the option of replace the low-risk default lottery with a high-risk (good ) lottery (cRY ) that we interpret as an entrepreneurial opportunity. 1

This avoids the possibility of multiple steady states where, for instance, all risk-tolerant (riskaverse) parents choose to have risk-tolerant (risk-averse) kids and risk-tolerant parents choose entrepreneurship while risk-averse parents choose to be workers. When this happens, many initial conditions can be stationary equilibria.

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More precisely, we construct an OLG model, with the following timing. In the first period (youth) agents receive the preferences and face the juvenile lottery. In the second period (adulthood) agents face the entrepreneurial lottery and transmit preferences to their kids. All choices depend on two numbers: the individual state (inherited preferences) and the aggregate state (distribution of preferences). Definition 1 Let a ˜ to be such that EuY (cRY ; a ˜) = EuY (cSY ; a ˜) . To simplify the notation, we assume that the parent who cannot influence his child’s preferences (probability π) is aware of his inability. The value function of such a parent is simply v N O (a; Ω, Γ) = max {(1 − IG ) × Eu (cSO (Ω) ; a) + IG × Eu (cRO (Ω) ; a)} IG

Define the value function of a parent who can influence his offspring choice to be       v (a; Ω, Γ) =

max

a0 ∈[a,¯ a],IG

(1 − IG ) × Eu (cSO (Ω) ; a) + IG × Eu (cRO (Ω) ; a)

+z × ((1 − IB (a0 )) × Eu (cSY ; a) + IB (a0 ) × (Eu (cRY ; a)))    ¡ ¢   +βz × (1 − π) v (a0 ; Γ (Ω) , Γ) + πv N O (a0 ; Γ (Ω) , Γ)

where notation is as follows • β is the discount factor • IB ∈ {0, 1} indicates whether bad opportunities are accepted or declined • IG ∈ {0, 1} indicates whether entrepreneurial opportunities are accepted or declined • cSY and cSO denote the low-risk lotteries in youth and mature age • cRY and cRO denote the high-risk lotteries in youth and mature age • Ω is the aggregate state vector of the economy • Γ (Ω) is a law of motion of the state vector, Ω0 = Γ (Ω) 3

          

• Note that the occupational choice yields the following simple solution

IG (a; Ω) =

   1 if Eu (cRO (Ω) ; a) > Eu (cSO (Ω) ; a)   0 if

else

Next, define the child value function to be  ¡ ¢   (1 − IB ) × EuY (cSY ; a0 ) + IB × EuY (cRY ; a0 ) vchild (a0 ; Γ (Ω) , Γ) = max ¡ ¢ IB   +β × (1 − π) v (a0 ; Γ (Ω) , Γ) + πv N O (a0 ; Γ (Ω) , Γ)

    

Hence, ¡ ¢ª © IB (a0 ) = arg max (1 − IB ) × EuY (cSY ; a0 ) + IB × EuY (cRY ; a0 ) IB

which has the following simple solution:

IB (a0 ) =

   1 if EuY (cRY ; a0 ) > EuY (cSY ; a0 )   0 if

else

We introduce the following assumptions Assumption 1 For all feasible Ω, Eu (cRO (Ω) ; a ˜) < Eu (cSO (Ω) ; a ˜) . Assumption 2 Eu (cRY ; a ¯) < Eu (cSY ; a ¯) . The first assumption guarantees that, for any distribution of preferences, if IG (a, Ω) = 1, then IB (a) = 1 whereas if IB (a) = 0 then IG (a, Ω) = 0. In words, it is not possible to induce the child to take entrepreneurship opportunities and to reject bad opportunities. The second assumption guarantees that parents dislike their children taking bad opportunities, irrespective of their own risk aversion. Note that the first assumption may fail if the entrepreneurial return is too high. However, 4

there are ways to controll this through technological assumptions. See discussion below. We will also introduce an assumption that avoids strange results arising from reasons related to the cardinality of the utility function. Assumption 3 Let φmax be the maximum sustainable equilibrium wage, i.e., such that Eu (cRO (φmax ) ; a ¯) = u (φmax ; a ¯) . Then, (Eu (cSO ; a ¯) − Eu (cSO ; a ˜)) is decreasing in w for all w ≤ φmax .

3 Preliminary Results Lemma 1 Parents choose either a0 = a ˜ or a0 = a ¯, where EuY (cRY ; a ˜) = EuY (cSY ; a ˜) Intuitively, parents want to give their children the largest a since this increases their utility. However, they fear that children will take bad opportunity. So, they may choose either the largest possible a or the highest level of risk tolerance such that children will not run into troubles. Define: maxIG {(1 − IG ) × Eu (cSO (Ω) ; a) + IG × Eu (cRO (Ω) ; a)} U (a0 , a; Ω, Γ) = +z × ((1 − IB (a0 )) × Eu (cSY ; a) + IB (a0 ) × (Eu (cRY ; a))) ¡ ¢ +βz × (1 − π) v (a0 ; Γ (Ω) , Γ) + πv N O (a0 ; Γ (Ω) , Γ) . Then, U is a piece-wise increasing function of a0 with a downward discontinuity at a0 = a ˜, since then the child starts taking the bad lottery. Thus, U is maximized either at a0 = a ˜ or az a0 = a ¯. Moreover, the value function v (a, Ω) is strictly increasing in a with two kinks, respectively, at the level of risk tolerance that induce the parents to take the entrepreneurial lottery and at the level that induces a cultural transmission choice that makes the child take/not take the risky lotteries 5

(recall that the child only take the entrepreneurial lottery if he takes the juvenile lottery). Hence, without loss of generality, we can restrict attention to initial distribution with a positive mass at a ˜, a ˆ and a ¯. Any other initial distribution would lead to one such distribution after one period. Moreover, U (¯ a, a; Ω) − U (˜ a, a; Ω) is increasing in a, since U (¯ a; a, Ω, Γ) − U (˜ a; a, Ω, Γ) = z × ((Eu (cRY ; a) − Eu (cSY ; a))) +βz (1 − π) × (v (¯ a; Γ (Ω) , Γ) − v (˜ a; Γ (Ω) , Γ)) ¡ NO ¢ +βzπ × v (¯ a; Γ (Ω) , Γ) − v N O (˜ a; Γ (Ω) , Γ) and the parent’s dislike of juvenile risk increases with his risk aversion (i.e., it decreases with a). The following Lemma can then be proven. Lemma 2 Let a0 = A (a; Ω, Γ) denote the policy function for the choice of the child’s preferences. Given Ω and Γ: 1. If a ˜ ∈ A (¯ a; Ω, Γ), then a ˜ = A (˜ a; Ω, Γ) = A (ˆ a; Ω, Γ) (i.e., if some parents with a=a ¯ choose a0 = a ˜, then all parents with a = a ˜ and all parents with a = a ˆ choose 0 a =a ˜) 2. If a ¯ ∈ A (min{ˆ a, a ˜}; Ω, Γ) , then a ¯ = A (max{ˆ a, a ˜}; Ω, Γ) = A (¯ a; Ω, Γ) (i.e., if some parents with a = min{ˆ a, a ˜} choose a0 = a ¯ then all other parents choose 0 a =a ¯ 3. If both a ¯ ∈ A (max{ˆ a, a ˜}; Ω, Γ) and a ˜ ∈ A (max{ˆ a, a ˜}; Ω, Γ) , then a ¯ = A (¯ a; Ω, Γ) and a ˜ = A (min{ˆ a, a ˜}; Ω, Γ) (i.e., if parents with a = max{ˆ a, a ˜} are indifferent between a0 = a ¯ and a0 = a ˜, then all parents with a = a ¯ choose a0 = a ¯, whereas all 0 parents with a = min{ˆ a, a ˜} choose a = a ˜.

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4 Interpretation of the Lotteries We interpret the juvenile lottery as follows Eu (cSY ; a) = pSY × u (cSY,L ; a) + (1 − pSY ) × u (cSY,H ; a) Eu (cRY ; a) = (1 − λB ) Eu (cSY ; a) + λB Eu (J; a) , where λB is the arrival rate of ”dangerous” juvenile opportunities, i.e., (sub)lotteries that parents dislike while children may like to take if they are not sufficiently risk averse. The parameter λB is important since it measures the exposure to ”juvenile” risk. In contrast the variance of the lottery cSY can be interpreted as the unavoidable risk. Note that these lotteries is assumed to be independent of Ω. We give the adult lotteries an occupational choice interpretation. In particular, we let Eu (cSO (Ω) ; a) = pW × u (wW L (Ω) ; a) + (1 − pW ) × u (wW H (Ω) ; a) Eu (cRO (Ω) ; a) = (1 − λG ) E (cSO (Ω) ; a) + λG (pE × u (wEL (Ω) ; a) + (1 − pE ) × u (wEH (Ω) ; a)) , where λG is the arrival rate of ”entrepreneurial” opportunities, i.e., (sub-)lotteries that adult agents like as long as they are sufficiently risk tolerant. The parameter λG is also important since it measures the arrival rate of entrepreneurial opportunities. We denote by wW = pW × wW L + (1 − pW ) wW H , wE = pE × wEL + (1 − pE ) wEH , where wW and wE are the (average) ”wages” determined in general equilibrium.

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5 General Equilibrium We now move to a general equilibrium setting. We assume (similar to DZ QJE 08) that the aggregate production function consists of two activities, entrepreneurs and workers Y = F (XW , XE ) , where F is assumed to feature constant returns to scale (relaxed later) and decreasing returns to each input. Each worker supplies one efficiency unit of W type labor earning a safe wage wW . The assumption that workers face no wage risk is for simplicity. Instead, entrepreneurs are subject to idiosyncratic uninsurable shocks. After deciding to be an entrepreneur, an agent supplies (1 + σ) efficiency units of E-type labor with probability pE . In this case, he earns ³ a return ´ pE (1 + σ) wE . With probability 1 − pE he will only be able to supply 1 − σ 1−p E ´ ³ pE units of labor, and earns 1 − σ 1−pE wE . Note that under these assumptions XE = NE where NE is the number of agents choosing to be entrepreneurs. Likewise, XW = 1 − NE . Profit maximization implies that wE = FXE (1 − NE , NE ) and wW = FXW (1 − NE , NE ) . The demand for entrepreneurs from firms will decrease in the entrepreneurial premium. More precisely, the inverse demand function is given by wE XE (1 − NE , NE ) ≡η= , wW XW (1 − NE , NE )

(1)

where the right hand-side is decreasing in NE since the numerator is decreasing and the denominator is icreasing in NE .

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State vector

In a general setting, the aggregate state contains two elements, i.e., the entrepreneurial 2 and worker average wage, {wE , wW } ∈ (R+ ) . However, the two wages are not independent as they are linked through the technology F and the feasibility condition in the labor market. This means that, in a world of constant technology, 8

the entrepreneurial premium η is a sufficient statistic for all consumers’ choices (preference transmission and occupational choice). Thus, Ω = η and η 0 = Γ (η) where Γ is an arbitrary law of motion. We will therefore henceforth set Ω = η.2

7

Steady state

In a steady state, the skill premium η is constant, Γ (η) = η. This implies that the proportion of entrepreneur, NE , is also constant. Preferences will be a key determinant of the occupational choice. Recall that the support of the steady-state distribution of risk aversion can be collapsed to three points, a ∈ {ˆ a, a ˜, a ¯}, which are independent of the state vector. Two cases are possible. Case 1 a ˜a ˆ, then ηaˆ > ηa¯ , namely, there exists a range of entrepreneurial premia such that the a ¯ type become entrepreneurs while the a ˆ type become workers. The ranking of ηP REF depends instead on parameters, and three situations are possible: (i) ηP REF < ηa¯ < ηaˆ , (ii) ηa¯ < ηP REF < ηaˆ , (iii) ηa¯ < ηaˆ < ηP REF . The following Lemmas can be established. Lemma 3 If η < max{ηP REF, ηa¯ }, then the steady-state supply of entrepreneurs is NE = 0. If η < ηP REF , then, the a ˆ-type agents strictly prefer to endow their children with a ˜ preferences. Thus, there is no positive measure of a ˆ-type agents in the steady state. If η < ηa¯ , then, risk-tolerant (¯ a-type) agents strictly prefer to decline entrepreneurial opportunities. So, will do, a fortiori, a ˆ-type and a ˜-type agents. Thus, there are no entrepreneurs in the steady state. Lemma 4 If η > max{ηP REF, ηaˆ }, then the steady-state supply of entrepreneurs is NE = 1. 10

If η > ηP REF , then, the a ˆ-type agents strictly prefer to endow their children with risk tolerance (i.e., they choose a ¯). Thus, there is no positive measure of riskaverse (˜ a-type) agents in the steady state. If, in addition, η > ηaˆ , then, the a ˆtype agents (who are the more risk averse in the society) strictly prefer to take entrepreneurial opportunities. So, will do, a fortiori, a ¯-type. Thus, there are no workers in the steady state. Clearly, this possibility cannot arise in general equilibrium, since the Inada condition in the production fucntion imply that the entrepreneurial premium would become for a positive fraction of workers. Lemma 5 (i) If η ∈ (ηaˆ , ηP REF ) , then the steady-state supply of entrepreneurs is NE = π. (ii) If η ∈ (ηP REF, ηa¯ ) , then the steady-state supply of entrepreneurs is NE = 1 − π. Clearly, the two cases are mutually exclusive, since (i) only arises if ηa¯ < ηaˆ < ηP REF while (ii) only arises if ηP REF < ηa¯ < ηaˆ . Consider (i), first. In this case, there are no a ¯-type agents in the steady state. Yet, the a ˆ-type agents (who are in measure π in any steady state) strictly prefer to take entrepreneurial opportunities. On the contrary, the a ˜-type agents (who are in measure 1 − π in this steady state) decline as usual entrepreneurial opportunities. Consider (ii), next. In this case, there are no a ˜-type agents in the steady state. The a ˆ-type agents (who are in measure π) decline entrepreneurial opportunities. On the contrary, the a ¯-type agents (who are in measure 1 − π in this steady state) take entrepreneurial opportunities. In summary, we have the following steady-state supply functions. If ηP REF < ηaˆ , then

If ηP REF

   =0        ∈ [0, 1 − π]    < ηaˆ , then NE 1−π       ∈ [1 − π, 1]       1

11

if

η < max {ηP REF , ηa¯ }

if

η = max {ηP REF , ηa¯ }

if η ∈ [max {ηP REF , ηa¯ } , ηaˆ ] if

η = ηaˆ

if

η ∈ [ηaˆ , ηmax ].

(2)

If ηP REF

   =0        ∈ [0, π]    > ηaˆ , then NE π       ∈ [π, 1]       1

if

η < ηaˆ

if

η = ηaˆ

if

η ∈ [ηaˆ , ηP REF ]

if

η = ηP REF

(3)

if η ∈ [ηP REF , ηmax ]

In both cases it is a monotonic entrepreneurial ”supply” function. Thus, given the downward sloping demand function (1) the steady state is unique. See Figure XXX. INSERT FIGURE Note that an exogenous increase in λB affects only ηP REF . In particular, a higher ηP REF implies less risk-tolerant population, fewer entrepreneurs and a higher entrepreneurial risk. In contrast, an increase in unavoidable risk (e.g., parameterized by an increase in the risk associated to cSY ) has the opposite effect: parents want to turn their kids more risk tolerant. Thus, ”unavoidable” risk tends to increase entrepreneurship in society. WE STILL NEED TO IMPOSE AN EXPLICIT CONDITION ON THE MINIMUM WAGE SUCH THAT AGENTS WHO TAKE THE ENTREPRENEURIAL RETURN MUST ALSO TAKE THE JUVENILE RISK.

8 Equilibrium Law of Motion The previous section restricts attention to constant sequences of entrepreneurial premia. However, outside of the steady state the equilibrium law of motion consistent with the individual choices need not imply a constant sequence of entrepreneurial premia and fraction of entrepreneurs. When technology is constant, the deep state variable is the distribution of preferences in the population.

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Let m (a) denote the p.d.f. of agents with the preference level a ∈ [a, a ¯]. As discussed above, A (a; Ω, Γ) ∈ {ˆ a, a ¯}. Thus, after one period the p.d.f. collapses to a two- or three-mass point distribution with positive mass at most at three points: a ˆ, a ˜, a ¯. Restricting attention to recurrent states, we claim that m (ˆ a) = π and m (˜ a) + m (¯ a) = 1 − π. Thus, µ ≡ m (¯ a) is a sufficient statistics. Given the assumption of constant technology and population, µ pins down uniquely the entrepreneurial premium, η = η (µ) . Thus, we can write the state variable as Ω = η (µ) , and the equilibrium law of motion as η (µ0 ) = Γ (η (µ)) , where µ ∈ [0, 1 − π]. TO BE COMPLETED...

9 GROWTH

Yt+1

1 = α

µZ

Z

Nt

α

Nt+1

x¯ (i) di + 0

¶ x (i) di Q1−α α

Nt

where Q is a fixed factor (land), normalized to unity. To make innovation we need entrepreneurs. Entrepreneurs are necessary to produce new varieties. Worker are needed to produce goods. The old varieties are sold in competitive markets. We assume that the productivity of both workers and entrepreneurs is indexed by Nt Nt x¯ + (Nt+1 − Nt ) x = Nt XtW ⇔ x¯ + (1 + g) x = X W and

µ

Nt+1 − Nt ξNt

¶ = XtE ⇔

1+g = XE ξ

The competitive final producer max 1 α

µZ

Nt

Z α

Nt+1

x¯ (i) di + 0

¶ Z x (i) di −

Nt

Nt+1

p¯ (i) x¯ (i) di + 0

13

Z

Nt

α

p (i) x (i) di Nt

FOC yields p¯ (i) = x¯ (i)α−1 p (i) = x (i)α−1 Next, the problem of the intermediate producer. The old goods are produced competitively, thus 1 wW α−1 p¯ (i) = ≡ ωW , x¯ (i) = ωW N The new goods are monopolies. The profit ”per variety” yields is Π (i) = p (i) x (i)− ωW x (i) . The maximization of Π (i) subject to the demand constraint yields: 1 ³ ω ´ α−1 ωW W p (i) = p = , x (i) = x = , Π (i) = Π = (p − ωW ) x = (1 − α) α α

µ

α ωW

α ¶ 1−α

We can now solve for ωW : X

W

= x¯ + (1 + g) x = ωW

à ωW =

1 α−1

1 + (1 + g) α XW

1 1−α

+ (1 + g)

!1−α

à =

1 ³ ω ´ α−1

W

α

1 1−α

1 + α ξX XW

E

!1−α .

Note that ωW ≥ 1. Even in the absence of growth and entrepreneurship, workers earn a wage equal to N. We can also solve for the entrepreneurial return. Denote by wE the average entrepreneurial wage, and let ωE ≡ wE /N. Then: µ ωE = ξΠ = ξ (1 − α)

α ωW

α ¶ 1−α

à = ξ 1−α (1 − α)

1

α 1−α ξX W 1

1 + α 1−α ξX E

!α .

α

α

Note that ωE ∈ [0, ξ (1 − α) α 1−α ]. We must therefore assume that ξ > (1 − α)−1 α− 1−α . Otherwise, the equilibrium will entail no entrepreneurship. We assume the entrepreneurial return to be risky. In particular, the entrepreneur ignores the portion of projects that he will be able to pursue. With probability κ he will be able to run (1 + σ) N project (and, hence, earn (1 + σ) ωE ), with probability 14

¢ ¡ ¢ ¡ κ κ N (and, hence, earn 1 − σ 1−κ ωE ) 1 − κ he will only be able to run 1 − σ 1−κ projects. We can then define the entrepreneurial premium α wE ξX W 1−α . η≡ = (1 − α) α 1 wW 1 + α 1−α ξX E

Because of risk aversion, this premium must be positive. Hence, we have a upper bound to the number of entrepreneurs in equilibrium: α

(1 − α) α 1−α ξ ³ ´ X < α 1 (1 − α) α 1−α ξ + 1 + α 1−α ξ E

Utility functions must be rescaled in order to avoid that Assumption 3 fails. We must make sure that u (cSO (wW ) ; a ¯) − u (wW ; a ˜) is decreasing in wW for all wE ≤ φmax even with a growing wage (recall that in the growth model above wE grows at the rate g). For example, with CRRA utility what works is ³ u (ct ) =

ct

´1+a

γwE,t−1

−1

1+a

where γ is a constant (which must be taken to be sufficiently large) and wE,t−1 is the wage level in the previous period. Note that c and w will grow at the same rate in steady state.

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