# Finding the Optimal Infinite-Horizon Propensity to Consume

In some of the slides I called this R, but on the chalkboard I called it a better name: B_infinity.

**Notation:**

B_infinity = optimal infinite-horizon propensity to consume in the Merton model

S = elasticity of intertemporal substitution

rho = impatience

r = safe real interest rate

mu^2 / (2 * A * sigma^2) = investor’s surplus

(See the earlier post about finding investor’s surplus)

**Terminological Note. **What I call the “elasticity of intertemporal substitution” or “EIS” is often called the “intertemporal elasticity of substitution” or “IES.” My view is that intertemporal substitution is a phenomenon, of which the elasticity is a measure. So I hate to separate the words “intertemporal substitution.” Other economists think that “elasticity of substitution” is a well-known measure, and this is applying it in an intertemporal context. So my terminology emphasizes measuring a particular economic phenomenon, while the other way of talking about it emphasis applying a particular measure in a new context.

**The Formula: **

Using some very cool, but quite advanced math that extends Robert Merton’s model to carefully distinguish risk tolerance 1/A from the elasticity of intertemporal substitution S, the formula for the optimal infinite horizon propensity to consume is

B_infinity = S rho + (1-S) [ r + investor’s surplus]

*This formula is actually about average values of r and investor’s surplus over the rest of your life**, using appropriate weights for the average. Thus B_infinity is affected very little by business cycle fluctuations. *

However, if people’s view of the long-run future changed, then B_infinity would change. For example, I think if China lets regular citizens get a decent return for their savings, this would change the Chinese people’s view of the long-run, lifetime average values of r and investor’s surplus and would cause B_infinity (and therefore consumption) to go up dramatically. (A preview of a future post on my main blog!)

**Reasonable Values:**

**The Elasticity of Intertemporal Substitution S. **What I just said about China only works if the elasticity of intertemporal substitution S is less than 1. And I truly believe that S is less than 1. To me, any value between .1 and .6 or so is a thinkable value. But I need to tell you that some people truly believe that the elasticity of intertemporal substitution S is as high as 2.

You can see, however, that a value of 1 for S is extremely convenient. And many, many, economic models use a value S=1 for that reason.

**Impatience. **Many people are probably quite impatient–maybe up to 8% per year. Others are not impatient at all, which would be zero. It is even quite possible for someone to have rho be negative, which simply means that even at a zero interest rate and zero investor’s surplus, they would prefer to consume more in the future than now. *However* though there is no problem with negative impatience in a finite-horizon where death looms up ahead, negative impatience does very weird things in infinite horizon models, technically. These technical problems make many economists think that negative impatience is unreasonable, but it isn’t. Most people don't *really *expect to live forever.

**The Real Interest Rate. **I think of 3% per year as a reasonable value of the real interest rate to use for retirement planning. One part of the reasoning behind this is that the government has borrowed a lot of money. That tends to push real interest rates up when the economy is at the natural level of output (which it isn’t right now).

**Investor’s Surplus.** Let me tweak things a bit to get a round number. Suppose the risk premium is 3% per year, the standard deviation of the risky index mutual fund (or ETF) is 15% over a year (I am assuming the return of the Great Moderation over most of your life because of getting monetary policy better figured out, one way or another), and a risk aversion of 2. Then on an annual basis, investor’s surplus is .0009/(2 * 2 * .0225) = .01 = 1%. So that is 1% per year.

**Interpretation Note: **

Below, please calculate B_infinity according to the formula, even if the value turns out 0 or negative. If B_infinity is ever 0 *or *negative, it means that in an actual infinite horizon model, very weird things would happen. *But* a negative B_infinity can still be plugged into the finite-horizon formula in the next post. If B_infinity is zero, as the next post will argue, it means that the finite horizon propensity to consume is just 1/time left to live. So if you have 40 years left to live, consume 2.5% of what you have this year.

**Problems:**

**Answers:**