Lesson 5 of 8
Furthest from the shop
The same rate cut raises the present value of remote stages far more than near ones. Computed on the five-stage chain, by hand, with the factors shown.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- One rate cut does not raise every stage's value by the same amount. The further a payoff sits from consumption, the more its present value rises.
- On a five-stage chain, a cut from 6 percent to 3 percent raises the one-period factor by 2.9 percent and the five-period factor by 15.4 percent.
- This is the arithmetic behind the claim that a credit boom appears first and hardest in the stages furthest from the shopper.
This lesson is a calculation. Take the chain from the structure course, apply a discount rate to each stage's distance from consumption, then change the rate and look at what moved.
The calculation, in five steps
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Set the distances. Stage 1 output reaches a consumer in one period, stage 2 in two, and so on to stage 5 in five. The chain is the one from the previous course.
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Pick two rates. Six percent, then three percent. Round numbers, chosen so the factors can be checked with any calculator.
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Compute the factors at 6 percent. One divided by 1.06 raised to the number of periods: 0.9434, 0.8900, 0.8396, 0.7921, 0.7473.
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Compute them at 3 percent. One divided by 1.03 raised to the same powers: 0.9709, 0.9426, 0.9151, 0.8885, 0.8626.
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Take the ratio at each distance. 2.9 percent, 5.9 percent, 9.0 percent, 12.2 percent, 15.4 percent. The same cut, five different effects.
How much a cut from 6 percent to 3 percent lifts each stage
- One period out, the shop2.9 percent
- Two periods out5.9 percent
- Three periods out9.0 percent
- Four periods out12.2 percent
- Five periods out15.4 percent
What the numbers say
A rate cut is not a uniform stimulus. It is a change in the relative price of distant output against near output, and the gradient is steep. The remote stage gains more than five times as much as the stage next to the shopper.
That is the whole of the claim that the boom shows first at the top of the chain. It does not require anybody to be irrational, or to have been told anything. It requires only that firms discount.
Why the gradient steepens with length
Because discounting compounds. Each additional period multiplies by the same factor again, so a change in the factor is applied more times the further out you go. The ratio between the two factor series is itself a compound of the ratio of the two rates.
That is why lengthening the chain does not merely add another bar. It adds a taller one than the last.
What this does and does not establish
It establishes that a rate change has different effects at different distances from consumption, which is arithmetic and is not controversial.
It does not establish that a boom therefore happens. That requires the rest of the mechanism: that the rate fell for a reason that released no resources, that firms act on it, and that consumers push back. The arithmetic here is a necessary step in the argument and not the argument.
Check yourself
By how much does a cut from 6 percent to 3 percent raise the five-period discount factor?
- By 2.9 percent
- By 15.4 percent, from 0.7473 to 0.8626
- By exactly half
Show the answer
Answer: B.
The one-period factor rises only 2.9 percent over the same cut, which is the gradient the whole lesson is about.Why does the effect get larger with distance?
- Because distant projects are riskier
- Because discounting compounds, so a change in the per-period factor is applied more times the further out the payoff sits
- Because remote stages are more capital intensive
Show the answer
Answer: B.
The ratio between the two factor series is itself a compound of the ratio of the rates, which is why each further stage gains more than the last.What does this arithmetic establish on its own?
- That credit expansion causes depressions
- Only that a rate change has different effects at different distances from consumption
- That firms are irrational during booms
Show the answer
Answer: B.
The rest of the mechanism is needed for anything more: that the rate fell without releasing resources, that firms act, and that consumers push back.Sources
- Hayek, Prices and Production, first published 1931, second revised edition 1935, lecture III, sections 4 and 5, for the argument that a fall in the rate changes the discounted marginal product differently across stages and so shifts factors towards the earlier ones.
- Mises, The Theory of Money and Credit, German original 1912, English translation by Batson 1934, part three, chapter 5, section 4, for the claim that a reduced loan rate necessarily lengthens the average period of production.
- Bohm-Bawerk, The Positive Theory of Capital, German original 1889, English translation by Smart, book II, chapter II, for why a longer method is worth considering at all.
What this leaves out
- This is a heterodox tradition. The mainstream account of the same phenomenon differs, and nothing here is a forecast or a view on any security.
- Every number here is invented arithmetic on an invented chain, chosen so a reader can recompute it. No industry is described, and the two projects in the example are illustrative.
- The calculation assumes one discount rate applies across all distances. Real projects face a term structure and their own credit spread, and the previous course showed that a policy rate reaches the long end only loosely.
- It also assumes stages map cleanly onto periods, which the structure course showed is a device rather than a measurement.