economics 2010c: lecture 4 precautionary savings and ... · 2 liquidity constraints. since the...
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Economics 2010c: Lecture 4Precautionary Savings and Liquidity
Constraints
David Laibson
9/11/2014
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Outline:
1. Precautionary savings motives
2. Liquidity constraints
3. Application: Numerical solution of a problem with liquidity constraints
4. Comparison to “eat-the-pie” problem
5. Discrete numerical analysis (optional)
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1 Precautionary motives
How does uncertainty affect the Euler Equation?
∆ ln +1 =1
(+1 − ) +
2∆ ln +1
where ∆ ln +1 = [∆ ln +1 −∆ ln +1]2
Increase in economic uncertainty, raises ∆ ln +1 raising∆ ln +1Why?
Marginal utility is convex when is in the CRRA class. An increase in uncer-tainty, raises the expected value of marginal utility. This increases the motiveto save. Sometimes this is referred to as the “precautionary savings effect.”
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2 period Example:
0 = 0 = 1
1 has distribution function () with non-negative support.
(1) = 1
Definition: Precautionary saving is the reduction in consumption due to thefact that future labor income is uncertain instead of being fixed at its meanvalue.
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• Greater income uncertainty increases motive to save (even if expected valueof future income is unchanged).
• Prediction tested using variation in income uncertainty across occupations.
• Dynan (1993) finds that income uncertainty does not predict consumptiongrowth.
• Carroll (1994) finds a robust relationship.
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2 Liquidity constraints.
Since the 1990’s consumption models have emphasized the role of liquidityconstraints (Zeldes, Carroll, Deaton). Two key assumptions of these ‘bufferstock’ models.
1. Consumers face a borrowing limit — e.g. ≤
• This matters whether or not it actually binds in equilibrium (e.g., atomat zero income).
2. Consumers are impatient
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Predictions:
• Consumers accumulate a small stock of assets to buffer transitory incomeshocks
• Consumption weakly tracks income at high frequencies (even predictableincome)
• Consumption strongly tracks income at low frequencies (even predictableincome)
We will revisit these predictions in coming lectures.
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3 Application: Numerical solution
• Labor income iid, symmetric beta-density on [0,1]
• ≤ = cash-on-hand
• () = 1−−11− with = = 2
• Discount factor, = 09
• Gross rate of return, = 10375
• Infinite horizon
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• Solution method is numerical.
• Let
()() ≡ sup∈[0]
{() + ((− ) + ̃+1)} ∀
+1 = (− ) + ̃+1
• Solution given by: lim→∞()()
• Iteration of Bellman operator is done on a computer (using a discretizedstate and action space).
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4 Eat the pie problem
Compare to a model in which the consumer can securitize her income stream.In this model, labor income can be transformed into a bond.
• If consumers have exogenous idiosyncratic labor income risk, then there isno risk premium and consumers can sell their labor income for
0 = 0
∞X=0
−
• The dynamic budget constraint is
+1 = ( − )
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• Bellman equation for “eat-the-pie” problem:
( ) = sup∈[0 ]
{() + (( − ))} ∀
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• Guess the form of the solution.
( ) =
⎧⎨⎩ 1−1− if ∈ [0∞] 6= 1
+ ln if = 1
⎫⎬⎭• Confirm that solution works (problem set).
• Derive optimal policy rule (problem set).
= −1
−1 = 1− (1−)
1
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• Let’s compare two similarly situated consumers:
— a buffer stock consumer with cash-on-hand (and a non-tradeableclaim to all future labor income)
— an eat-the-pie consumer with cash-on-hand (and a tradeable claimto all future labor income); so the eat-the-pie consumer has currenttradeable wealth
= +
∞X=1
−
• Note that eat-the-pie consumption function lies above optimal consump-tion function
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• Optimal policy function is concave and bounded above by the lower enve-lope of the 45 degree line and eat-the-pie consumption function
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04Density of Income Process
Income (partitioned into 100 cells)
Den
sity
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0 0.5 1 1.5 2 2.5 3 3.5 4-10
-9
-8
-7
-6
-5
-4
-3
cash-on-hand
v(x)
Converging value functions
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0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
Cash-on-hand
Con
sum
ptio
n
Consumption Functions
Consumption Function for Eat-the-Pie Problem
Consumption Function for Liquidity Constraint Problem
45 degree line (liquidity constraint)
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• Finally, think about linearized Euler Equation
∆ ln +1 =1
(+1 − ) +
1
2∆ ln +1
• Is the conditional variance of consumption growth constant?
• If cash-on-hand is low this period, what can we say about the variabilityof consumption growth next period?
• If cash-on-hand is high this period, what can we say about the variabilityof consumption growth next period?
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When is close to 0,
"+1 −
#→∞ as → 0
When is large, is well approximated by an affine function, = + implying that
+1 ' [ − (+ )] + +1
and
"+1 −
#'
"(+ +1)− (+ )
+
#
'
" { [ − (+ )] + +1}−
+
#
→
" [1− ]− 1
1
#= 0 as →∞
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5 Discrete numerical analysis
• basic idea is to partition continuous spaces into discrete spaces
• e.g., instead of having wealth in the interval [0, $5 million], we could setup a discrete space
0, $1000, $2000, $3000, ... , $5,000,000
• we could then let the agent optimize at every point in the discrete space(using some arbitrary continuation value function defined on the discretespace, and then iterating until convergence)
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5.1 Example of discretization of buffer stock model (op-
tional)
Continuous State-Space Bellman Equation:
() = sup∈[0]
{() + ((− ) + +1)}
We’ll now discretize this problem.
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• Consider a discrete grid of points = {0 1 2 }
• It’s natural to set 0 = 0 and equal to a value that is sufficientlylarge that you never expect an optimizing agent to reach
• However, should not be so large that you lose too much computa-tional speed. Finding a sensible is an art and may take a few trialruns.
• Consider another discrete grid of points = {0 1 2 }
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• Now, given ∈ is chosen such that
0 ≤ ≤ (1)(− ) + ∈ for all ∈ (2)
• To make this last restriction possible, the discretized grids, and mustbe chosen judiciously.
• Define Γ() as the set of feasible consumption values that satisfy con-straints (1) and (2).
• So the Bellman Equation for the discretized problem becomes:
() = sup∈Γ()
{() + ((− ) + +1)}
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An example of discretized grids and
Choose to be divisible by ∆ Let
≡ {0∆ 2∆ 3∆ }Let the elements of by multiples of ∆ (e.g., = {13∆ 47∆}) Here Iassume that the largest element in is smaller than
Fix a cell ∈ Let represent the REMainder generated by dividing by ∆
Then, ∈ Γ() ≡
{ +∆
+
2∆
− 2∆
− ∆
}
If ∈ Γ() then − will be a multiple of ∆ so
(− ) + ∈
for all ∈
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Remark: For some (large) values of you will need to truncate the lowestvalued cells of the Γ() correspondence. Specifically, it must be the case thatfor every value of
(−min {Γ()}) + ≤
for all ∈ This implies that
min {Γ()} = max{ (−
) +
)}
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5.2 Practical advice for Dynamic Programming (optional)
When using analytical methods...
• work with ∞-horizon problems (if possible)
• exploit other tricks to make your problem stationary (e.g., constant hazardrate for retirement)
• work with tractable densities (always try the uniform density in discretetime; try brownian motion and/or poisson jump processes in continuoustime)
• minimize the number of state variables
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When using numerical methods...
• approximate continuous random variables with discretized Markov processes
• coarsely discretize exogenous random variables
• densely discretize endogenous random variables
• use Monte Carlo methods to calculate multi-dimensional integrals
• use analytics to partially simplify problem (e.g., retirement as infinite hori-zon eat the pie problem)
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• translate Matlab code into optimized code (C++)
• consider using polynomial approximations of value functions (Judd)
• consider using spline (piecewise polynomial) approximations of value func-tions (Judd)
• minimize the number of state variables (cf Carroll 1997)
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5.3 Curse of dimensionality: travelling salesman (optional)
• must map route including visits to cities
• job is to minimize total distance travelled
• state variable: a -dimensional vector representing the cities
• if the salesman has already visited a city, we put a one in that cell
• set of states is all -dimensional vectors with 0’s and 1’s as elements
=Y=1
{0 1}
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Bellman Equation:
( city) = maxcity0
n−(city city0)+ (0 city0)
oFunctional Equation:
()( city) = maxcity0
n−(city city0)+ (0,city0)
o
How many different states ∈ are there?
X=0
³
´This is a large number when is large.
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For example:
•³
´= !(−)!!
• Let = 100 = 50so³
´= 100!50!50! = 10
29
• And that’s just one value of
• To put this in perspective, a modern supercomputer can do a trillion cal-culations per second.
• So a supercomputer could go through one round of Bellman operator iter-ation in 1010 years.
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Lesson:
• Even for seemingly simple problems the state space can get quite large.
• Work hard to limit the size of your state space.
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• You will typically have state spaces that are in <
• Suppose you had = 4
• Suppose you were modelling assets and you partitioned your state spaceinto blocks of $1000.
• Imagine that you bound each of your four assets between $0 and $1,000,000.
• Your state space has 10004 elements.
• So each round of Bellman iteration requires the computer to do 10004computations.