Time-inconsistent preferences, investment and asset pricing

Economics Letters 148 (2016) 48–52

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Time-inconsistent preferences, investment and asset pricing✩ Bo Liu a , Lei Lu b,∗ , Congming Mu c , Jinqiang Yang d a

School of Management and Economics, University of Electronic Science and Technology of China, China

b

Asper School of Business, University of Manitoba, Canada

c

School of Finance, Shanghai University of Finance and Economics, China

d

Shanghai Key Laboratory of Financial Information Technology, School of Finance, Shanghai University of Finance and Economics, China

highlights • • • •

We extend the production-based asset pricing model by incorporating time-inconsistent preferences. Time-inconsistent preferences induce under-investment, over-consumption and higher risk-free rate. The naïve agents consume more and invest less than the sophisticated agents. The interest rate in the economy with naïve agents is higher than that in the economy with sophisticated agents.

article

info

Article history: Received 27 June 2016 Received in revised form 10 September 2016 Accepted 17 September 2016 Available online 22 September 2016 JEL classification: C73 D92 G11

abstract In this paper, we present a production-based asset pricing model in which agents have time-inconsistent preferences. We find that the time-inconsistent preferences lead to under-investment, over-consumption, and higher interest rate. These variables are distorted more in the economy with naive agent than the economy with sophisticated agent. In particular, the sophisticated agent invests more and consumes less than the naive agent, but invests less and consumes more than the time-consistent agent. The interest rate in the sophisticated agent economy is lower than that in the naive agent economy, but higher than that in the time-consistent agent economy. © 2016 Elsevier B.V. All rights reserved.

Keywords: Time-inconsistent preferences Investment Consumption Interest rate

1. Introduction The existing production-based asset pricing models assume that agents have constant time preferences. However, the evidence

✩ Congming Mu acknowledges the support from the Postgraduate Students Innovation Foundation of Shanghai University of Finance and Economics (CXJJ2015-318). Bo Liu acknowledges the support from the National Natural Science Foundation of China (#71373036 and #71573033) and the Science and Technology Support Program of Sichuan Province (#2013GZ0116). Jinqiang Yang acknowledges the support from the National Natural Science Foundation of China (#71202007 and #71522008), New Century Excellent Talents in University (#NCET-13-0895), Innovative Research Team of Shanghai University of Finance and Economics (#2016110241), and Fok Ying-Tong Education Foundation of China (#151086). ∗ Corresponding author. E-mail addresses: [email protected] (B. Liu), [email protected] (L. Lu), [email protected] (C. Mu), [email protected] (J. Yang).

http://dx.doi.org/10.1016/j.econlet.2016.09.015 0165-1765/© 2016 Elsevier B.V. All rights reserved.

suggests that agents’ time preferences vary over time (e.g., Thaler, 1981; Ainslie, 1992; and Loewenstein and Prelec, 1992). Timeinconsistent preferences imply that agents act relatively patiently when two payoffs are far away in time, but more impatiently when they are brought forward in time. Laibson (1997) models such time-varying impatience with a quasi-hyperbolic discount function, in which the discount rate declines with the horizon.1 Harris and Laibson (2013) capture this effect with a continuoustime model. Grenadier and Wang (2007) extend the real op-

1 Laibson (1997) uses a discrete-time discount function to model quasihyperbolic preferences, in which time is divided into two periods: the present period and all future periods. Utility in the current period is discounted exponentially with a discount rate ρ , while utility in all future periods is first discounted exponentially with discount rate ρ and then discounted by an additional factor α ∈ (0, 1].

B. Liu et al. / Economics Letters 148 (2016) 48–52

tion approach to model the investment-timing decisions by entrepreneurs who have time-inconsistent preferences. Unlike these papers, we present a production-based asset pricing model in which capital adjustment costs are continuous-time quadratic (e.g., Abel and Eberly, 1994; Eberly and Wang, 2009; and Pindyck and Wang, 2013) and agents have time-inconsistent preferences, and then examine agents’ consumption, asset prices and investment. We analyze two economies with either sophisticated agent or naive agent. The sophisticated agent correctly realizes that his future selves act according to their own preferences, while the naive agent has wrong belief that the future selves act in the interest of the current self. We have three findings. First, naive agent consumes more than sophisticated agent, and both of them consume more than time-consistent agent. Second, naive agent invests less than sophisticated agent, and both of them invest less than time-consistent agent. Finally, the time-inconsistent preferences lead to a higher interest rate. Therefore, by considering time-inconsistent preferences, this paper enriches the existing asset pricing models. 2. Model setup

Dn (t , s) =

e

−ρ(s−t )

α e−ρ(s−t )

if s ∈ [tn , tn+1 ) if s ∈ [tn+1 , ∞),

(1)

for s ≥ t. Intuitively, Eq. (1) implies that, in addition to the constant discounting rate ρ , self n values the utility after the arrival of self n + 1 by an extra discounting factor α ≤ 1. After the death of self n and the birth of self n + 1, the agent uses the discount function Dn+1 (t , s) to value his utility. Assuming agent is self n at time t and his lifetime utility over consumptions C is ∞





Dn (t , s)U (C (s))ds ,

E

state variable in this economy, and thus the value function can be denoted by V ∗ (K ). The following proposition provides the equilibrium results for the case of time-consistent preferences. Proposition 1. If agent has time-consistent preferences, the value function is 1

V ∗ (K ) =

1−γ

(b∗ K )1−γ , γ

(5) 1

where b∗ = (A − i∗ ) γ −1 φ ′ (i∗ ) γ −1 . The optimal investment–capital ratio, i∗ , solves the following equation A − i∗ =

1

φ ′ (i∗ )





   γσ2 ρ + (γ − 1) φ(i∗ ) − .

(6)

2

The optimal consumption–capital ratio is c ∗ = A − i∗ , and Tobin’s q is q∗ =

1

φ ′ (i∗ )

.

(7)

The interest rate r ∗ and the equity risk premium rp∗ are

We consider a continuous-time, production economy in which agents have quasi-hyperbolic preferences, capturing the fact that the discount rate declines over time. Following Grenadier and Wang (2007) and Harris and Laibson (2013), we assume that each period has a random lifespan, which is modeled as a Poisson process with intensity of λ. Solving such a problem with timeinconsistent preferences can be thought as the outcome of an intrapersonal game, in which the agent is represented by different selves at future periods. Each self makes consumption–investment decisions during his lifetime but also concerns with the utility received by his future selves. Let Dn (t , s) denote self n’s discount function. At time t ∈ [tn , tn+1 ), self n evaluates the utility received at future time s as Dn (t , s) given by



49

(2)

t 1−γ

where U (C ) = C1−γ . The output has an AK production technology Y (t ) = AK (t ),

(3)

where A is constant representing the productivity and capital stock K is the source of production. The capital stock K evolves with the following process dK (t ) = Φ (I (t ), K (t ))dt + σ K (t )dZ (t ),

(4)

where σ is the volatility of capital depreciation, Z is a standard Brownian motion process, I denotes the investment, and adjust2 ment cost Φ (I , K ) = φ(i)K , where i = I /K and φ(i) = i − θ2i − δ .

r ∗ = ρ + γ φ(i∗ ) −

γ (γ + 1)σ 2 2

,

(8)

rp∗ = γ σ 2 .

(9)

It is easy to verify that the growth rates of consumption, investment, capital, and output are all equal. Therefore, after scaling by capital stock K , the consumption–capital ratio c = C /K , the investment–capital ratio i = I /K , and Tobin’s q are all constant. 4. The naive agent with time-inconsistent preferences In this section, we consider a case where agent is naive—the naive agent makes consumption–investment decisions under the wrong beliefs that the future selves act in the interest of the current self.2 For example, starting with the self 0, the naive agent has the discount function D0 (t , s), the future self 1 has the discount function D1 (t , s), the future self 2 has the discount function D2 (t , s), and so on. Since naive agent incorrectly believes that all future selves use the same discount function D0 (t , s), he acts as if he can commit his future selves to behave as his current preference, and then the self 0 values the utility received after his death as α V ∗ (K ), which is usually defined as the continuation value function. We assume that the central planner maximizes the naive agent’s utility with an infinite time horizon. Let VN (K ) denote naive agent’s value function, and IN and CN the optimal investment and consumption, respectively. By dynamic programming method, VN (K ) solves the following Hamilton–Jacobi–Bellman (HJB) equation

ρ VN (K ) = max U (CN ) + Φ (IN , K )VN′ (K ) + IN

+ λ(α V ∗ (K ) − VN (K )),

σ 2K 2 2

VN′′ (K ) (10)

subject to the budget constraint CN + IN = AK . Using the first-order condition for investment IN , we have U ′ (CN ) = ΦIN (IN , K )VN′ (K ).

(11)

3. The benchmark with time-consistent preferences In this section, we consider a benchmark economy in which agent has constant discount rates. The capital stock K is the only

2 This assumption on naivety is first proposed by Strotz (1956) and then used by Akerlof (1991) and O’Donoghue and Rabin (1999a,b), among others.

50

B. Liu et al. / Economics Letters 148 (2016) 48–52

Eq. (11) characterizes the naive agent’s optimal consumption CN and investment IN . It shows that naive agent’s marginal benefit of consumption, U ′ (CN ), is equal to the marginal cost of consumption—the product of the marginal effective value of a unit of investment, ΦIN (IN , K ) units of capital, and the marginal value of a unit of capital, VN′ (K ). The equilibrium results for the economy with naive agent are summarized in the following proposition. Proposition 2. If agent is naive and has time-inconsistent preferences, the value function is VN (K ) =

1 1−γ

(bN K )1−γ ,

(12)

γ





   γ ′  φ (iN ) A − iN ρ+λ 1−α A − iN = ′ φ (iN ) A − i∗ φ ′ (i∗ )   γσ2 + (γ − 1) φ(iN ) − . 2

φ ′ (iN )

(13)

A − iN rN + γ σ 2 − φ(iN )

.

(14)



= ρ+λ 1−α −



γ (γ + 1)σ 2

rpN = γ σ 2 .

(17)





γσ2

2

A − iN A − i∗

,

γ

φ (iN ) + γ φ(iN ) φ ′ (i∗ ) ′

 γ −1 1

2

.

By using the dynamic programming method, VS (K ) solves the following HJB equation

ρ VS (K ) = max U (CS ) + Φ (IS , K )VS′ (K ) +

σ 2K 2 2

VS′′ (K )

+ λ(α V (K ; IS ) − VS (K )),

(18)

subject to the budget constraint CS + IS = AK . By taking the firstorder derivative with respect to IS , we have (19)

Eq. (19) has the same economic implication as Eq. (11). The following proposition presents the intra-personal equilibrium for the economy with sophisticated agent. Proposition 4. If agent is sophisticated and has time-inconsistent preferences, the value function is VS (K ) =

The interest rate rN and the equity premium rpN are rN

( bK )1−γ ,

U ′ (CS ) = ΦIS (IS , K )VS′ (K ).

The optimal consumption–capital ratio is cN = A − iN , and Tobin’s q is

=

1−γ

where  b = (A − iS ) ρ + (γ − 1) φ(iS ) −

1

1

1

1

 V (K ; IS ) =

IS

where bN = (A−iN ) γ −1 φ ′ (iN ) γ −1 . The optimal investment–capital ratio, iN , solves the following equation

qN =

conditioning on the intra-personal equilibrium where the current self and all future selves of sophisticated agent choose investment IS .  V (K ; IS ) is given by

1 1−γ

(bS K )1−γ ,



γ

(20) 1

where bS = (A − iS ) γ −1 φ ′ (iS ) γ −1 . The optimal investment–capital ratio, iS , solves the following equation (15)





  γσ2 (A − iS )φ ′ (iS ) = ρ + (γ − 1) φ(iS ) − 2

(16)

By comparing the optimal consumption–capital ratio, the optimal investment–capital ratio, and the interest rate between the naive and benchmark economies, we have following proposition. Proposition 3. The naive agent invests less and consumes more than the time-consistent agent, i.e., iN < i∗ and cN > c ∗ . This leads to a higher interest rate for economy with naive agent, rN > r∗ . Intuitively, with the quasi-hyperbolic preferences and wrong beliefs by naive agent, the current self values the utility received by future selves less than had he received himself. Therefore, the current self of naive agent has incentive to consume more and invest less when making consumption–investment decisions during his lifetime, and then the interest rate is increased. 5. The sophisticated agent with time-inconsistent preferences Unlike naive agent, the sophisticated agent correctly perceives that his future selves act according to their own preferences. In other words, self n of sophisticated agent makes consumption–investment decisions based on his own preferences, but fully anticipates that all future selves do likewise. Thus, self n and self n + 1 do not agree on the optimal consumption and investment. As pointed out by existing literature, this scenario might have multiple solutions due to the existence of multiple intra-personal equilibria. We focus on the most natural one—all future selves of sophisticated agent have the same investment strategies. Let VS (K ; IS ) and α V (X ; IS ) denote sophisticated agent’s value function and continuation value function respectively,



(A − iS )φ ′ (iS )  + λ 1 − α ρ + (γ − 1) φ(iS ) −

 γσ2

  . (21)

2

The optimal consumption–capital ratio is cS = A − iS , and Tobin’s q is qS =

1

φ ′ (iS )

=

A − iS rS + γ σ 2 − φ(iS )

.

(22)

The interest rate rS and the equity premium rpS are





rS

(A − iS )φ ′ (iS )  = ρ + λ 1 − α ρ + (γ − 1) φ(iS ) − + γ φ(iS ) −

rpS = γ σ 2 .

γ (γ + 1)σ 2

2

,

γσ2



2

(23) (24)

Through comparing the optimal consumption–capital ratio, the optimal investment–capital ratio, and the interest rate for benchmark, naive, and sophisticated economies, we have following proposition. Proposition 5. The sophisticated agent invests more and consumes less than the naive agent, but invests less and consumes more than the time-consistent agent, i.e., iN < iS < i∗ and cN > cS > c ∗ . The interest rate in the sophisticated agent economy is lower than that in the naive agent economy, but higher than that in the time-consistent agent economy, i.e. rN > rS > r∗ .

B. Liu et al. / Economics Letters 148 (2016) 48–52

The intuition is straightforward. The sophisticated agent correctly anticipates that his future selves act according to their own preferences. Therefore, when making consumption and investment decisions, the current self of sophisticated agent can reduce the potential loss due to the decisions made by his future selves relative to the current self of naive agent, and thus consumes less and invests more. However, the extra discounting rate α provides the incentive for the current self of sophisticated agent to consume more compared to the current self of time-consistent agent. Our result for investment is consistent with the partial equilibrium result of Grenadier and Wang (2007). They find that, under the case of flow payoff, the sophisticated agent invests more than the naive agent but less than the time-consistent agent.

By using Ito’s lemma, we have

  γ (γ + 1) 2 = −ρ − γ φ(iN ) + σ dt M (t− ) 2     dM (t )

− γ σ dZ (t ) + α

− 1 dJ (t )

bN

b∗

+λ α

1−γ

−1

bN

dt

bN

Appendix. Proofs Proof of Proposition 2. Substituting value function (Eq. (12)) into the first-order condition for investment IN (Eq. (11)) yields naive − γ1

agent’s optimal consumption–capital ratio, cN = φ ′ (iN ) γ γ −1 cN

γ γ −1

1

γ −1 γ

bN

,

1

φ ′ (iN ) γ −1 = (A − iN ) φ ′ (iN ) γ −1 . and thus we have bN = Inserting bN , cN and value function under the economy with timeconsistent preferences V ∗ into Eq. (10) generates the optimal investment–capital ratio iN , which can be solved from the following equation φ ′ (iN )

1−γ

    1−γ b∗ − γ σ dZ (t ) + α − 1 [dJ (t ) − λdt] .

In this paper, we incorporate time-inconsistent preferences into a general equilibrium production-based asset pricing model to investigate agents’ consumption and investment strategies and the interest rate. We find that the time-inconsistent agent invests less and consumes more than the time-consistent agent, and that the time-inconsistent preferences induce a higher interest rate. Moreover, the naive agent invests less but consumes more than the sophisticated agent, which leads to a higher interest rate in the naive agent economy than in the sophisticated agent economy.

A − iN =

b∗

 γ (γ + 1) 2 σ = −ρ − γ φ(iN ) + 2    

6. Conclusion

1

51





ρ+λ 1−α 

+ (γ − 1) φ(iN ) −



A − iN A − i∗

σ 2γ 2



γ

φ (iN ) φ ′ (i∗ )

.

′

No-arbitrage condition implies that the interest rate, rN , equals the negative expected rate of the change in M, given by rN = ρ + γ φ(iN ) −

γ (γ + 1)σ 2 2

 +λ 1−α



b∗

1−γ 

bN

γ (γ + 1)σ 2 = ρ + γ φ(iN ) − 2   γ ′  A − iN φ (iN ) +λ 1 − α . A − i∗ φ ′ (i∗ )

(28)

By homogeneity, we have QN (K ) = qN K where QN (K ) is the value of capital stock and qN is Tobin’s q. In equilibrium, the consumption equals the dividend for all time t, CN (t ) = AK (t ) − IN (t ). The standard valuation methodology implies that the instantaneous drift of M (t )(AK (t ) − IN (t ))dt + d(M (t )QN (t )) is zero. That is, 0 = E[M (t )CN (t )dt + d(M (t )QN (t ))]

 γ (γ + 1) 2 = cN M (t )Kt dt + −ρ − γ φ(iN ) + σ 2      b∗

+λ α



1−γ

− 1 + φ(iN ) − γ σ 2 qN M (t )Kt dt .

bN

Through simplifying the above equation we have (25)

cN qN

We next solve stochastic discount factor (SDF). For self n of naive agent who is alive at time t, the marginal rate of substitution

U ′ (CN (s)) M (s) = Dn (t , s) ′ U (CN (t ))

  γ ′  A − iN φ (iN ) = ρ+λ 1−α A − i∗ φ ′ (i∗ )   σ 2γ + (γ − 1) φ(iN ) − 2

U ′ (C (s))

between t and s is Dn (t , s) U ′ (CN (t )) . Recall that self n discounts the N continuation payoff received by his future selves with an additional factor α . Therefore, the SDF, denoted by {M (s) : s ≥ t }, is

 ΦI (IN (s), K (s))VN′ (K (s))   e−ρ(s−t ) N   ΦIN (IN (t ), K (t ))VN′ (K (t ))   if tn ≤ t ≤ s ≤ tn+1 = ∗′  −ρ(s−t ) ΦIN (IN (s), K (s))V (K (s))  αe    ΦIN (IN (t ), K (t ))VN′ (K (t ))  if tn ≤ t < tn+1 < s < ∞ −γ   −ρ(s−t ) K (s)   e   K (t )   if tn ≤ t ≤ s ≤ tn+1  ∗ 1−γ  −γ =  b K (s)  −ρ(s−t )  α e   bN K (t )  if tn ≤ t < tn+1 < s < ∞.

(27)

= rN + γ σ − φ(iN ). 2

rN

Together with Eqs. (25) and (28), we have qN = φ ′ (1i ) = N A−i N . Thus, the risk premium of equity is +γ σ 2 −φ(i ) N

rpN =



1 dt

Et

dQN + CN dt QN



− rN = φ(iN ) +

= γ σ 2.

(26)

cN qN

− rN (29)

Proof of Proposition 4. This proof is similar to the proof of Proposition 2. Substituting value function (Eq. (20)) into the firstorder condition for investment IS (Eq. (19)) yields the optimal consumption–capital ratio cS , and then we obtain bS . Substituting bS , cS , and the continuation value function for sophisticated agent,  V (K ; IS ), into Eq. (18), we obtain an equation for optimal investment–capital ratio iS . Similarly, the SDF equals the marginal U ′ (C (s))

rate of substitution, M (s) = Dn (t , s) U ′ (CS (t )) . Unlike naive agent, S the sophisticated agent’s continuation value function is Eq. (17).

52

B. Liu et al. / Economics Letters 148 (2016) 48–52 dM (t )

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