Real Default-Free Interest Rates and Intertemporal Rates of Substitution

If you were offered a sum of money now or later, which would you prefer?

Sure. Nearly everyone would agree.

You're definitely in the minority.

Having money now gives you the choice of consuming things whenever you want, so being forced to wait is almost certainly a disadvantage. There is an opportunity cost of delaying consumption. That's really the idea of a default-free real rate of return: it compensates an investor just for giving up consumption today.

In economics terms, investors get utility (which is happiness, or satisfaction) from consuming things. The marginal utility of consumption is the change in total utility from another unit of consumption. As you eat more food at a buffet, what happens to your total utility?

No. Only if you're already getting sick from eating too much.

Yes.

As you eat, your total satisfaction rises. Each bite tastes good and adds to your happiness. If it didn't, you wouldn't eat any more. So total utility rises. But the marginal utility is different. As you continue to eat more and more, what happens to your marginal utility from each bite?

Not likely. That would mean that you never get full!

Probably not. If that were the case, you wouldn't ever stop eating.

Absolutely. The more you have in your stomach, the less you value the next bite. It's the same with whatever money can buy from your investments as well. You have a marginal utility of consuming today and a marginal utility of consuming at some point in the future. More money today, a lower marginal utility of consumption today. More money in the future, a lower marginal utility of consumption in the future. Same idea.

The __intertemporal rate of substitution__ is a ratio of these two things, and it's denoted as _m_ with a tilde. $$\displaystyle \widetilde{m}_{t,s} = \frac{MU_s}{MU_t}$$ If you have some extra money today, then your marginal utility (MU) of consumption today falls. What happens to your intertemporal rate of substitution between these two time periods?

No. It will rise.

Exactly.

A lower marginal utility of consumption today means that the denominator is getting smaller, making your inter-temporal rate of substitution higher. So you decide to buy a zero-coupon bond to defer some consumption from now into the future. What happens to your intertemporal rate of substitution?

No. Now it falls.

Excellent!

You lower consumption today, increasing the MU of consumption today, and increase future consumption, lowering MU of consumption in time _s_. Both of these reduce the intertemporal rate of substitution.

Now a quick check for consistency of these ideas. If investors collectively purchase a bond, that makes the bond price rise. Okay, so remember that this is all because you wanted less money now and more in the future. That means that your discount rate is a little lower. As investors decide that they require less compensation for deferring consumption, the default-free real rate falls, and as a portion of the discount rate, that causes the whole discount rate to fall. What's the effect on the price of the zero-coupon bond?

No. Higher price.

Of course.

A lower discount rate means that future sum of money is reduced less. Of course you don't really know what your intertemporal rate of substitution will be in the future, but you can expect a certain value. That tradeoff of "today or tomorrow" that defines the intertemporal rate of substitution is really the same tradeoff of taking the price of a bond today or getting its face value one period in the future. So the default-free yield component can be stated as $$\displaystyle l_{t,1} = \frac{1 - P_{t,1}}{P_{t,1}} = \frac{1}{P_{t,1}} - 1 = \frac{1}{E_t \left[ \widetilde{m}_{t,1} \right] } - 1 $$. You can see in this formulation that the bond price and the intertemporal rate of substitution are both used in the exact same way, since they are equal. For example, a default-free, inflation-indexed, one-year bond priced at 0.9912 suggests an intertemporal rate of substitution of 0.9912.

To summarize: [[summary]]

Now

Later

It goes up

It goes down

It falls

It also rises

It stays about the same

It falls

It rises

It falls

It rises

Lower price

Higher price

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