In a risky world, a pessimist assumes the worst will happen. Someone who ignores risk altogether is an optimist. Consumption decisions are mathematically simple for both the pessimist and the optimist because both behave as if they live in a riskless world. A consumer who is a realist (that is, who wants to respond optimally to risk) faces a much more difficult problem, but (under standard conditions) will choose a level of spending somewhere between that of the pessimist and the optimist. We use this fact to redefine the space in which the realist searches for optimal consumption rules. The resulting solution accurately represents the numerical consumption rule over the entire interval of feasible wealth values with remarkably few computations.
Dynamic Stochastic Optimization
1Carroll: Department of Economics, Johns Hopkins University, Baltimore, MD, http://www.econ2.jhu.edu/people/ccarroll/, email@example.com 2Slacalek: European Central Bank, Frankfurt am Main, Germany, http://www.slacalek.com/, firstname.lastname@example.org 3Tokuoka: International Monetary Fund, Washington, DC, email@example.com.
Solving a stochastic consumption, investment, portfolio choice, or similar continuous intertemporal optimization problem using numerical methods generally requires the modeler to choose how to represent a policy or value function. A common approach is to use low-order polynominal splines that exactly match the function (and maybe some derivatives) at each of a finite set of gridpoints, and then to assume that the matching polynomial is a good representation elsewhere.
This paper argues that, at least in the context of a standard consumption problem, there is a better approach, which relies upon the fact in the absence of uncertainty the optimal consumption function has a simple analytical solution. The key insight is that, under standard assumptions, the consumer who faces an uninsurable labor income risk will consume less (the consumer will engage in ‘precautionary saving’) than a consumer with the same path for expected income but who does not perceive any uncertainty as being attached to that future income. That is, the perfect foresight riskless solution provides an upper bound to the solution that will actually be optimal. A lower bound is provided by the behavior of a consumer who has the subjective belief that the future level of income will be the worst that it can possibly be. This consumer, too, behaves according to the analytical perfect foresight solution, but his certainty is that of an extremely overconfident pessimist.
Using results from Carroll (2011b), we show how to use these upper and lower bounds to tightly constrain the shape and characteristics of the solution to the ‘realist’s problem (that is, the solution to the problem of a consumer who correctly perceives the risks to future income and behaves rationally in response.
After showing how to use the method in the baseline case, we show how refine the method to encompass an even tighter theoretical bound, and how to extend it to solve a problem in which the consumer faces both labor income risk and rate-of-return risk.
It turns out (see Carroll (2011a) for a proof) that this problem can be rewritten in a more convenient form in which choice and state variables are normalized by the level of permanent income, e.g., using nonbold font for normalized variables, . When that is done, the transformed version of the consumer’s problem is
For the remainder of the paper we will assume that permanent income grows by a constant factor and is not subject to stochastic shocks. (The generalization of the analysis below to the case of permanent shocks is relatively straightforward.)
For comparison to our new solution method, we use the endogenous gridpoints solution to the microeconomic problem presented in Carroll (2006). That method computes the level of consumption at a set of gridpoints for market resources that are determined endogenously using the Euler equation. The consumption function is then constructed by linear interpolation among the gridpoints thus found.
Carroll (2011a) describes a specific calibration of the model and constructs a solution using five gridpoints chosen to capture the structure of the consumption function reasonably well at values of near the infinite-horizon target value. (See those notes for details).
Unfortunately, the endogenous gridpoints solution is not very well-behaved outside the original range of gridpoints targeted by the solution method. (Though other common solution methods are no better outside their own predefined ranges). Figure 1 demonstrates the point by plotting the amount of precautionary saving implied by a linear extrapolation of our approximated consumption rule (the consumption of the perfect foresight consumer minus our approximation to optimal consumption under uncertainty, ). Although theory proves that precautionary saving is always positive, the linearly extrapolated numerical approximation eventually predicts negative precautionary saving (at the point in the figure where the extrapolated locus crosses the horizontal axis).
This problem cannot be solved by extending the upper gridpoint; in the presence of serious uncertainty, the consumption rule will need to be evaluated outside of any prespecified grid (because starting from the top gridpoint, a large enough realization of the uncertain variable will push next period’s realization of assets above that top). While a judicious extrapolation technique can prevent this problem from being fatal (for example by carefully excluding negative precautionary saving), the problem is often dealt with using inelegant methods whose implications for the accuracy of the solution are difficult to gauge.
As a preliminary to our solution, define as end-of-period human wealth (the present discounted value of future labor income) for a perfect foresight version of the problem of a ‘risk optimist:’ a consumer who believes with perfect confidence that the shocks will always take the value 1, . The solution to a perfect foresight problem of this kind takes the form2for a constant minimal marginal propensity to consume given below. We similarly define as ‘minimal human wealth,’ the present discounted value of labor income if the shocks were to take on their worst possible value in every future period (which we define as corresponding to the beliefs of a ‘pessimist’).
A first useful point is that, for the realist, a lower bound for the level of market resources is , because if equalled this value then there would be a positive finite chance (however small) of receiving in every future period, which would require the consumer to set to zero in order to guarantee that the intertemporal budget constraint holds. Since consumption of zero yields negative infinite utility, the solution to realist consumer’s problem is not well defined for values of , and the limiting value of the realist’s is zero as .
Given this result, it will be convenient to define ‘excess’ market resources as the amount by which actual resources exceed the lower bound, and ‘excess’ human wealth as the amount by which mean expected human wealth exceeds guaranteed minimum human wealth:
We can now transparently define the optimal consumption rules for the two perfect foresight problems, those of the ‘optimist’ and the ‘pessimist.’ The ‘pessimist’ perceives human wealth to be equal to its minimum feasible value with certainty, so consumption is given by the perfect foresight solution
The ‘optimist,’ on the other hand, pretends that there is no uncertainty about future income, and therefore consumes
It seems obvious that the spending of the realist will be strictly greater than that of the pessimist and strictly less than that of the optimist. Figure 2 illustrates the proposition for the consumption rule in period .
Proof is more difficult than might be imagined, but the necessary work is done in Carroll (2011b) so we will take the proposition a fact and proceed by manipulating the inequality:
where the fraction in the middle of the last inequality is the ratio of actual precautionary saving (the numerator is the difference between perfect-foresight consumption and optimal consumption in the presence of uncertainty) to the maximum conceivable amount of precautionary saving (the amount that would be undertaken by the pessimist who consumes nothing out of any future income beyond the perfectly certain component). Defining (which can range from to ), the object in the middle of the last inequality isand we now define which has the virtue that it is linear in the limit as approaches .
Given , the consumption function can be recovered from
Thus, the procedure is to calculate at the points corresponding to the log of the points defined above, and then using these to construct an interpolating approximation from which we indirectly obtain our approximated consumption rule by substituting for in equation (13).
Because this method relies upon the fact that the problem is easy to solve if the decision maker has unreasonable views (either in the optimistic or the pessimistic direction), and because the correct solution is always between these immoderate extremes, we call our solution procedure the ‘method of moderation.’
Results are shown in Figure 3; a reader with very good eyesight might be able to detect the barest hint of a discrepancy between the Truth and the Approximation at the far righthand edge of the figure.
Often it is useful to know the value function as well as the consumption rule associated with a problem. Fortunately, many of the tricks used when solving the consumption problem have a direct analogue in approximation of the value function.
Consider the perfect foresight (or ‘optimist’) case in period :
We apply the same transformation to the value function for the problem with uncertainty (the realist’s problem):
However, as with the consumption approximation, we can do even better if we realize that the function for the optimist’s problem is an upper bound for the function in the presence of uncertainty, and the value function for the pessimist is a lower bound. Analogously to (10), define an upper-caseand an upper-case version of the equation in (12): and if we approximate these objects then invert them (as above with the and functions) we obtain a very high-quality approximation to our inverted value function at the same points for which we have our approximated value function:
Although a linear interpolation that matches the level of at the gridpoints is simple, a Hermite interpolation that matches both the level and the derivative of the function at the gridpoints has the considerable virtue that the derived from it numerically satisfies the envelope theorem at each of the gridpoints for which the problem has been solved.
Carroll (2011b) derives an upper limit for the MPC as approaches its lower bound. Using this fact plus the strict concavity of the consumption function yields the proposition that
The solution method described above does not guarantee that approximated consumption will respect this constraint between gridpoints, and a failure to respect the constraint can occasionally cause computational problems in solving or simulating the model. Here, we describe a method for constructing an approximation that always satisfies the constraint.
Defining as the ‘cusp’ point where the two upper bounds intersect:
Again defining , the object in the middle of the inequality isAs approaches , converges to zero, while as approaches , approaches .
As before, we can derive an approximated consumption function; call it . This function will clearly do a better job approximating the consumption function for low values of while the previous approximation will perform better for high values of .
For middling values of it is not clear which of these functions will perform better. However, an alternative is available which performs well. Define the highest gridpoint below as and the lowest gridpoint above as . Then there will be a unique interpolating polynomial that matches the level and slope of the consumption function at these two points. Call this function .
Using indicator functions that are zero everywhere except for specified intervals,
To construct the corresponding refined representation of the value function we must first clarify one point: The upper-bound value function that we are constructing will be the one implied by a consumer whose spending behavior is consistent with the refined upper-bound consumption rule.
For , this consumption rule is the same as before, so the constructed upper-bound value function is also the same. However, for values matters are slightly more complicated.
Start with the fact that at the cusp point,
But for all ,
Thus far we have assumed that the interest factor is constant at . Extending the previous derivations to allow for a perfectly forecastable time-varying interest factor would be trivial. Allowing for a stochastic interest factor is less trivial.and in this case the previous analysis applies once we substitute this MPC for the one that characterizes the perfect foresight problem without rate-of-return risk. The more realistic case where the interest factor has some serial correlation is more complex. We consider the simplest case that captures the main features of empirical interest rate dynamics: An AR(1) process. Thus the specification is
The consumer’s problem in this case now has two state variables, and , and is described by
We approximate the AR(1) process by a Markov transition matrix using standard techniques. The stochastic interest factor is allowed to take on 11 values centered around the steady-state value and chosen [how?]. Given this Markov transition matrix, conditional on the Markov AR(1) state the consumption functions for the ‘optimist’ and the ‘pessimist’ will still be linear, with identical MPC’s that are computed numerically. Given these MPC’s, the (conditional) realist’s consumption function can be computed for each Markov state, and the converged consumption rules constitute the solution contingent on the dynamics of the stochastic interest rate process.
In principle, this refinement should be combined with the previous one; further exposition of this combination is omitted here because no new insights spring from the combination of the two techniques.
The method proposed here is not universally applicable. For example, the method cannot be used for problems for which upper and lower bounds to the ‘true’ solution are not known. But many problems do have obvious upper and lower bounds, and in those cases (as in the consumption example used in the paper), the method may result in substantial improvements in accuracy and stability of solutions.
CARROLL, CHRISTOPHER D. (2006): “The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems,” Economics Letters, pp. 312–320, http://www.econ2.jhu.edu/people/ccarroll/EndogenousGridpoints.pdf.
(2011b): “Theoretical Foundations of Buffer Stock Saving,” Manuscript, Department of Economics, Johns Hopkins University, http://www.econ2.jhu.edu/people/ccarroll/papers/BufferStockTheory.