Economics2010c:Lecture6 Quasi-hyperbolicdiscounting · Figure 7: Mean Illiquid Wealth of...
Transcript of Economics2010c:Lecture6 Quasi-hyperbolicdiscounting · Figure 7: Mean Illiquid Wealth of...
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Economics 2010c: Lecture 6Quasi-hyperbolic discounting
David Laibson
September 18, 2014
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Outline:
1. Hyperbolic Discounting
2. Hyperbolic Euler Equation
3. Lifecycle simulations
4. Method of Simulated Moments (MSM)
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1 Hyperbolic Discounting
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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%).
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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.
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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
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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.
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= () +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) + ]
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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.
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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).
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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%
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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 .
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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.
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² Envelope Theorem. 0() = 0( ())
² First-order-condition. 0( ()) = E
h 0((¡ ()) + )
i
² Identity linking and () = ()¡ (1¡ )( ())
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2.1 Problem is recursive
² Start with .
² Find : () = argmax
() + E[ ((¡ ) + )]
² Find : () = ( ()) + E[ ((¡ ()) + )]
² In this way, generate an operator : 7!
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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).
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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)
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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.
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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)
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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.
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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
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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
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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
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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
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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
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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
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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
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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() + +
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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)
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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)
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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()
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² is the MSM estimator.
² Pakes and Pollard (1989) prove asymptotic consistency and normality.
² Speci…cation tests: () » 2( ¡#)
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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])
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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.
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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