What the time preference maths actually shows

The claim, and the sign error inside it

There is a version of this argument that circulates constantly. It runs: Bitcoin is expected to gain purchasing power, so people prefer the future to the present, so time preference falls, so interest rates fall, so capital gets allocated properly.

I wrote a version of it myself. It is wrong, and it is wrong in a way you can check with a calculator, because the arithmetic it invokes says the opposite of what it concludes. Working out why is more useful than the conclusion was.

What the ratio is

Time preference is the rate at which somebody discounts a future good against the same good now. Call it $\rho$. Over $t$ periods, the discount factor is

$d = \frac{1}{(1+\rho)^t}$

and the present value of a future amount is

$PV = \frac{FV}{(1+\rho)^t} = FV \cdot d$

Which means the ratio of present to future value is the discount factor itself:

$\frac{PV}{FV} = \frac{1}{(1+\rho)^t}$

Here is the first thing the popular version gets backwards. It defines time preference as the ratio $PV/FV$ and then says a high ratio means a high time preference. It does not. Somebody who barely values the future at all assigns a future good a present value near zero, so their ratio is near zero. Someone indifferent between now and later has a ratio near one.

High time preference means a low $PV/FV$. The ratio is the discount factor, and it moves against $\rho$, not with it. A post that gets this backwards will get everything downstream of it backwards too, and the popular argument does.

The rate, and the second sign error

Interest is what somebody has to be paid to postpone consumption. Rearranging the same identity:

$r = \left(\frac{FV}{PV}\right)^{\frac{1}{t}} - 1$

Now apply the popular argument's own premise. It says that under a money with a fixed supply, expected future value $FV$ rises relative to present value $PV$. Put that into the formula. The ratio $FV/PV$ gets bigger, so $r$ gets bigger. The argument's premise implies a higher rate, and the argument concludes a lower one. Those cannot both be true.

The economics behind the algebra is straightforward once you look at it. A saver has two options: hold the money, or lend it. Let $g$ be the real return on simply holding, meaning the change in what one unit buys. Nobody lends for less than they would earn by doing nothing, so in equilibrium the real lending rate cannot fall below $g$:

$r_{\text{real}} \geq g$

A money that gains purchasing power puts a floor under the real interest rate. It does not push it down. This is the ordinary Fisher relation read in the direction people usually forget: if the unit itself appreciates, the compensation demanded for parting with it rises with it.

The result nobody quotes

Follow that through and you land somewhere more interesting than the slogan.

Under a money whose purchasing power is expected to rise at $g$, every project competing for capital has to clear a hurdle rate of at least $g$ before anyone will fund it rather than sit still. If $g$ is two percent, a project returning one percent in real terms does not get built, and it should not: it destroys value relative to the alternative of doing nothing.

That is the strong form of the hard-money case, and it is a genuine result. It is also the strong form of the case against, and this is where the argument has to be honest with itself.

The objection, in its best form

Raising the hurdle rate does not only screen out bad projects. It screens out every project between zero and $g$, including ones that were worth doing. A road, a hospital, a factory with a long slow payback and a low real return can be worth building and still fail the test, because the money's own return is competing with it.

This is the oldest objection to hard money and it does not go away by being ignored. Under an appreciating unit, doing nothing is a paying strategy, and an economy where doing nothing pays will do less. The counter-argument, that projects failing to beat a two percent hurdle were never worth the capital, is an assertion about the real return on public infrastructure, not a mathematical result, and the maths above cannot settle it.

What the maths does settle is that you cannot have it both ways. You cannot claim a money that appreciates and also claim it lowers the cost of capital. Those are the same variable with opposite signs.

What is actually left

Three things survive, and they are narrower than the slogan.

The first is that under a fixed supply schedule, $g$ is set by demand for the money against a supply nobody can change, rather than by a committee. That is a claim about who sets the number, not about the number being low.

The second is about dispersion rather than level. The interesting failure of an administered interest rate is not that it is too low, it is that it can sit away from where savers and borrowers would have met, for years, with no mechanism forcing it back. Whatever $g$ turns out to be, it is not a policy decision.

The third is that none of this predicts a price. The whole argument above is conditional on $g$, and I have deliberately never told you what $g$ is, because nobody knows. If it is negative, every conclusion here reverses. An argument that only works when the asset appreciates is not an argument about monetary theory, it is a position with a proof attached, which is the trap the behavioural version falls into too.

The concrete test is not a formula anyway. It is whether people who live under a failing unit of account behave the way the model says, and when you look, they mostly do not.