Economics 2010c: Lecture 4 Precautionary 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 +1 Why? Marginal utility is convex when is in the CRRA class. An increase in uncertainty, raises the expected value of marginal utility. This increases the motive to 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 the fact that future labor income is uncertain instead of being fixed at its mean value.
Greater income uncertainty increases motive to save (even if expected value of future income is unchanged). Prediction tested using variation in income uncertainty across occupations. Dynan (1993) finds that income uncertainty does not predict consumption growth. Carroll (1994) finds a robust relationship.
2 Liquidity constraints. Since the 1990 s consumption models have emphasized the role of liquidity constraints (Zeldes, Carroll, Deaton). Two key assumptions of these buffer stock models. 1. Consumers face a borrowing limit e.g. This matters whether or not it actually binds in equilibrium (e.g., atom at zero income). 2. Consumers are impatient
Predictions: Consumers accumulate a small stock of assets to buffer transitory income shocks Consumption weakly tracks income at high frequencies (even predictable income) Consumption strongly tracks income at low frequencies (even predictable income) 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 1 1 with = =2 Discount factor, =0 9 Gross rate of return, =1 0375 Infinite horizon
Solution method is numerical. Let ( )( ) sup { ( )+ ( ( )+ +1 )} [0 ] +1 = ( )+ +1 Solution given by: lim ( )( ) Iteration of Bellman operator is done on a computer (using a discretized state 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 is no 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: abuffer stock consumer with cash-on-hand (and a non-tradeable claim to all future labor income) an eat-the-pie consumer with cash-on-hand (and a tradeable claim to all future labor income); so the eat-the-pie consumer has current tradeable wealth = + X =1 Note that eat-the-pie consumption function lies above optimal consumption function
Optimal policy function is concave and bounded above by the lower envelope of the 45 degree line and eat-the-pie consumption function
0.04 Density of Income Process 0.035 0.03 0.025 Density 0.02 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Income (partitioned into 100 cells)
-3 Converging value functions -4-5 -6 v(x) -7-8 -9-10 0 0.5 1 1.5 2 2.5 3 3.5 4 cash-on-hand
1.4 1.2 Consumption Functions Consumption Function for Eat-the-Pie Problem 1 Consumption 0.8 0.6 Consumption Function for Liquidity Constraint Problem 0.4 0.2 45 degree line (liquidity constraint) 0 0 1 2 3 4 5 6 Cash-on-hand
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 variability ofconsumptiongrowthnextperiod? If cash-on-hand is high this period, what can we say about the variability ofconsumptiongrowthnextperiod?
When is close to 0, " +1 # as 0 When is large, is well approximated by an affine function, = + implying that and " # +1 +1 ' [ ( + )] + +1 " # ( + +1 ) ( + ) ' + " { [ ( + )] + ' +1 } + " # [1 ] 1 =0 as 1 #
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 set up 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 discrete space, and then iterating until convergence)
5.1 Example of discretization of buffer stock model (optional) 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 =0and equal to a value that is sufficiently large that you never expect an optimizing agent to reach However, should not be so large that you lose too much computational speed. Finding a sensible is an art and may take a few trial runs. 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 must be chosen judiciously. Define Γ( ) as the set of feasible consumption values that satisfy constraints (1) and (2). So the Bellman Equation for the discretized problem becomes: ( ) = sup { ( )+ ( ( )+ +1 )} Γ( )
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 I assume 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 lowest valued cells of the Γ( ) correspondence. Specifically, it must be the case that for every value of for all This implies that ( min {Γ( )})+ 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 hazard rate for retirement) work with tractable densities (always try the uniform density in discrete time; try brownian motion and/or poisson jump processes in continuous time) 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 horizon 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 functions (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: Functional Equation: ( city) =max city 0 ( )( city) =max city 0 n (city city 0 ) + ( 0 city 0 ) o n (city city 0 ) + ( 0,city 0 ) o How many different states are there? X ³ =0 This is a large number when is large.
For example: ³ =! ( )!! Let =100 =50 so ³ = 100! 50!50! =1029 And that s just one value of To put this in perspective, a modern supercomputer can do a trillion calculations per second. So a supercomputer could go through one round of Bellman operator iteration in 10 10 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 space into blocks of $1000. Imagine that you bound each of your four assets between $0 and $1,000,000. Your state space has 1000 4 elements. So each round of Bellman iteration requires the computer to do 1000 4 computations.