Economics 2010c: Lecture 4Precautionary Savings and Liquidity

Constraints

David Laibson

9/11/2014

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)

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.”

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.

• 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.

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

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.

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

• 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).

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 = ( − )

• Bellman equation for “eat-the-pie” problem:

( ) = sup∈[0 ]

{() + (( − ))} ∀

• 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

• 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

• Optimal policy function is concave and bounded above by the lower enve-lope of the 45 degree line and eat-the-pie consumption function

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

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

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)

• 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?

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 →∞

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)

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.

• 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 }

• 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)}

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 ∈

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{ (−

) +

)}

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

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)

• 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)

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}

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.

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.

Lesson:

• Even for seemingly simple problems the state space can get quite large.

• Work hard to limit the size of your state space.

• 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.