Economics2010c:Lecture6 Quasi-hyperbolicdiscounting · Figure 7: Mean Illiquid Wealth of...
Transcript of Economics2010c:Lecture6 Quasi-hyperbolicdiscounting · Figure 7: Mean Illiquid Wealth of...
Economics 2010c: Lecture 6Quasi-hyperbolic discounting
David Laibson
September 18, 2014
Outline:
1. Hyperbolic Discounting
2. Hyperbolic Euler Equation
3. Lifecycle simulations
4. Method of Simulated Moments (MSM)
1 Hyperbolic Discounting
Check the item below that you would most prefer.
1. 15 minute free massage now
2. 20 minute free massage in an hour
Check the item below that you would most prefer.
1. 15 minute free massage in 168 hours
2. 20 minute free massage in 169 hours
Read and van Leeuwen (1998): Food experiment.
² Choose for Next Week: Fruit (74%) or Chocolate (26%)
² Choose for Today: Fruit (30%) or Chocolate (70%).
Read, Loewenstein & Kalyanaraman (1999): Video experiment
² Choose for Next Week: Low-brow (37%) or High-brow (63%)
² Choose for Today: Low-brow (66%) or High-brow (34%).
Evidence from gyms (Della Vigna and Malmendier 2004).
² Average cost of gym membership: $75 per month.
² Average number of visits per month: 4.
² Average cost per visit: $19.
² Cost of “pay-per-visit:” $10.
² Quasi-hyperbolic discounting (Phelps and Pollak 1968, Laibson 1997):1 2 3
= () + (+1) + 2(+2) +
3(+3) +
² For exponentials: = 1 = () + (+1) +
2(+2) + 3(+3) +
² For “quasi-hyperbolics”: 1 = () +
h(+1) +
2(+2) + 3(+3) +
i
² To build intution, assume that ' 12 and ' 1
f1 2 3 g = f1 121
21
2 g
= () +1
2[(+1) + (+2) + (+3) + ]
² Relative to the current period, all future periods are worth less (weight 12)
² Most (for this example, all) of the discounting takes place between thecurrent period and the immediate future.
² There is little (for this example, no) additional discounting between futureperiods.
= () +1
2[(+1) + (+2) + (+3) + ]
² Preferences are dynamically inconsistent.
² At date we prefer to be patient between + 1 and + 2
² At date + 1 we want immediate grati…cation at + 1
+1 = (+1) +1
2[(+2) + (+3) + (+4) + ]
Akerlof (1992) and O’Donoghue and Rabin (1999) on procrastination:
² Assume = 12 and = 1.
² Suppose exercise (cost 6) generates delayed bene…ts (value 8).
² Exercise Today? ¡6 + 12[8] = ¡2 0
² Exercise Tomorrow? 0 + 12[¡6 + 8] = 1 0
² Agent would like to exercise tomorrow.
Predictions:
² Procrastination when costs precede bene…ts (Della Vigna and Malmendier2004, 2006; Ariely and Wertenbroch 2002; Augenblick, Niederle, andSprenger 2013)
² Downward sloping consumption paths within pay-cycle (e.g., Shapiro 2005,Mastrobuoni and Weinberg 2009)
² Willingness to use commitment: savings (Ashraf, Karlan, and Yin, 2006;Beshears et al 2013), student productivity (Ariely and Wertenbroch, 2002;Houser et al., 2010, Chow 2011; Augenblick, Niederle, and Sprenger,2013), cigarette smoking (Gine, Karlan, and Zinman, 2010), workplaceproductivity (Kaur, Kremer, and Mullainathan, 2010), and exercise (Milk-man, Minson, and Volpp, 2012; Royer, Stehr, and Sydnor, 2012).
Goal account usage(Beshears et al 2013)
FreedomAccount
FreedomAccount
FreedomAccount
Goal Account10% penalty
Goal account20% penalty
Goal accountNo withdrawal
35% 65%
43% 57%
56% 44%
2 Hyperbolic Euler Equation
² Let represent consumption.
² Let represent cash-on-hand.
² Let ~ represent iid stochastic income.
² Let represent gross interest rate.
² So +1 = ( ¡ ) + ~+1
² A (Markov) strategy is a map from state to control .
² Let be the continuation-value function, be the current-value functionand be the consumption function. Then:
() = ( ()) + E[ ((¡ ()) + )]
() = ( ()) + E[ ((¡ ()) + )]
() = argmax() + E[ ((¡ ) + )]
² Note that accumulates utils exponentially.
² Note that accumulates utils quasi-hyperbolically.
² Envelope Theorem. 0() = 0( ())
² First-order-condition. 0( ()) = E
h 0((¡ ()) + )
i
² Identity linking and () = ()¡ (1¡ )( ())
2.1 Problem is recursive
² Start with .
² Find : () = argmax
() + E[ ((¡ ) + )]
² Find : () = ( ()) + E[ ((¡ ()) + )]
² In this way, generate an operator : 7!
2.2 Can also derive an Euler Equation
We have
0() = Eh 0(+1)
i= E
" 0(+1)¡ (1¡ )0(+1)
+1+1
#
= E
"0(+1)¡ (1¡ )0(+1)
+1+1
#
So,
0() = E"
Ã+1+1
!+
Ã1¡ +1+1
!#0(+1)
See Harris and Laibson (2003).
3 Lifecycle simulations (Angeletos et al 2001)
3.1 Demographic Assumptions
² Mortality
² Retirement (PSID)
² Dependents (PSID)
² Three educational groups: NHS, HS, COLL
² Stochastic labor income (PSID)
3.2 Dynamic Budget Constraints
² Credit limit: (.30)( ¹) (Calibrate from SCF.)
² Real after-tax rate of return: 375%
² Real rate of return on illiquid investment: 500%
² Real credit card interest rate: 1175%
² State variables: liquid wealth, illiquid wealth, autocorrelated income.
² Choice variables: liquid wealth investment, illiquid wealth investment.
3.3 Preferences
² Instantaneous utility function: CRRA=2.
² Quasi-hyperbolic discounting: 1 2 3
² For exponentials: = 1 For hyperbolics: = 07
² For exponentials: = Exponential For hyperbolics: = Hyperbolic
² Calibration: Pick value of Exponential (094) that matches empirical re-tirement wealth: median ‘wealth to income ratio’ ages 50-59.
² Pick Hyperbolic the same way (096)
Figure1: Discount functions
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40 45 50Year
Dis
coun
t fun
ctio
n
Quasi-hyperbolic
Exponential
Hyperbolic
Source: Authors’ calculations. Exponential: δt, with δ=0.939; hyperbolic: (1+αt)-γ/α, with α=4 and γ=1; and quasi-hyperbolic: {1,βδ,βδ2,βδ3,...}, with β=0.7 and δ=0.957.
Source: Authors' simulations.The figure plots the simulated average values of consumption and income for households with high school graduate heads.
Figure 2: Simulated Mean Income and Consumption of Exponential Households
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
20 30 40 50 60 70 80 90
Age
Inco
me,
Con
sum
ptio
n
Income
Consumption
Source: Authors' simulations.The figure plost the simulated life-cycle profiles of consumption and income for a typical household with a high school graduate head.
Figure 3: Simulated Income and Consumption of a Typical Exponential Household
0
10000
20000
30000
40000
50000
60000
70000
20 30 40 50 60 70 80 90
Age
Inco
me,
Con
sum
ptio
n
Income
Consumption
Source: Author's simulations.The figure plots average consumption over the life-cycle for simulated exponential and hyperbolic households with high-school graduate heads.
Figure 4: Mean Consumption of Exponential and Hyperbolic Households
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
20 30 40 50 60 70 80 90
Age
Con
sum
ptio
n
Hyperbolic
Exponential
Source: Authors' simulations.The figure plots the simulated mean level of liquid assets (excluding credit card debt), illiquid assets. total assets, and liquid liabilities for households with high school graduate heads.
Figure 5: Simulated Total Assets, Illiquid Assets, Liquid Assets, and Liquid Liabilities for Exponential Consumers
0
25000
50000
75000
100000
125000
150000
175000
200000
Ass
ets
and
Liab
ilitie
s
Total Assets
Illiquid Assets
Liquid Assets
-2000
-1800
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
20 30 40 50 60 70 80 90Age
Liquid Liabilities
Source: Author's simulations.The figure plots mean total assets, excluding credit card debt, over the life-cycle for simulated exponential and hyperbolic households with high school graduate heads.
Figure 6: Mean Total Assets of Exponential and Hyperbolic Households
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
20 30 40 50 60 70 80 90
Age
Tot
al A
sset
s
Hyperbolic Total Assets
Exponential Total Assets
Source: Authors' simulations.The figure plots average illiquid wealth over the life-cycle for simulated exponential and hyperbolic households with high school graduate heads.
Figure 7: Mean Illiquid Wealth of Exponential and Hyperbolic Households
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
20 30 40 50 60 70 80 90
Age
Illqu
id A
sset
s
Hyperbolic
Exponential
Source: Authors' simulations.The figure plots average liquid assets (liquid wealth excluding credit card debt) and liabilities (credit card debt) over the life-cycle for simulated exponential and hyperbolic households with high school graduate heads.
Figure 8: Mean Liquid Assets and Liabilities of Exponential and Hyperbolic Households
0
20000
40000
60000
80000
100000
120000
Ass
ets
and
Liab
ilitie
s
Exponential Assests
Hyperbolic Assests
-6000
-5000
-4000
-3000
-2000
-1000
0
20 30 40 50 60 70 80 90
Age
Exponential Liabilities
Hyperbolic liabilities
Exponential Hyperbolic Empirical Data
% with liquid 112 73% 40% 42%
mean liquid assetsliquid + illiquid assets 050 039 008
% borrowing on “Visa” 19% 51% 70%
mean borrowing $900 $3408 $5000+
C-Y comovement 003 017 023
¢ ln() = ¡1¢ln() + +
4 MSM Estimation(Laibson, Repetto, Tobacman 2008):
² Use the Method of Simulated Moments (Pakes and Pollard 1989).
² Pick parameter values to minimize the gap between simulated data andempirical data.
– Substantial retirement wealth accumulation (SCF)
– Extensive credit card borrowing (SCF, Fed, Gross and Souleles 2000,Laibson, Repetto, and Tobacman 2000)
– Consumption-income comovement (Hall and Mishkin 1982, many oth-ers)
4.1 Data
Statistic % borrowing on ‘Visa’? 0.68 0.015
(% Visa)
borrowing / mean income 0.13 0.01( Visa)
C-Y comovement 0.23 0.11(CY )
median 50-59 3.88 0.25weighted mean 50-59 2.60 0.13
(wealth)
4.2 Estimator
Estimate parameter vector and evaluate models wrt data.
² = N empirical moments
² () = analogous simulated moments
² () ´ ( ()¡)¡1 ( ()¡)0, a (scalar) loss function
² Minimize loss function: = argmin()
² is the MSM estimator.
² Pakes and Pollard (1989) prove asymptotic consistency and normality.
² Speci…cation tests: () » 2( ¡#)
4.3 Results
² Exponential ( = 1) case: = 0846 (0.025);
³ 1´= 217
² Hyperbolic case:( = 07031 (0.109) = 0958 (0.007)
³ ´= 301
(Benchmark case:h
i= [10375 005 11152])
Punchlines:
² estimated signi…cantly below 1.
² Reject = 1 null hypothesis with a t-stat of 3.
² Speci…cation tests reject only the exponential model.
BenchmarkModel
Exponential Hyperbolic Data Std err
Statistic:(1 )
= 846
( )
= 703
= 958
% 0.67 0.634 0.68 0.015
0.15 0.167 0.13 0.01
0.293 0.314 0.23 0.11
-0.05 2.69 2.60 0.13
() 217 3