Numbers large and small
Imagine you’re in the market for a small car, something like a Citroën C1 maybe, a Volkswagen Up! or a Toyota Aygo. Looking at the catalogues, you find there is some variation in the price, depending on the power of the engine and the trim, but they’re all around the £10,000 mark, plus or minus a few thousand pounds. You certainly wouldn’t expect any of these manufacturers to offer a version in this segment that is 2, 3 or 4 times that price. It stands to reason that there is no demand for a £40k Fiat 500 or Ford Ka, doesn’t it? Even the most expensive Audi A1 costs less than twice the price of the cheapest variant.
Likewise, you would not expect the leftmost of two semidetached houses next to one another to be offered at 2, let alone 3 or 4 times the price of the rightmost one (unless it was a ruin). Yet for other stuff, we seem not to be in the least surprised to see very similar goods or services offered at even larger multiples. Supermarket chain Tesco sells a 2-litre bottle of its own brand of Cola at £0.50, 1/3 of what it charges for a similar bottle of a well-known brand. Moreover, it sells an ‘everyday value’ variant for £0.17 per bottle — making the £1.50 branded version nine times more expensive.
The car wash adjacent to the supermarket sells a range of washes, ranging from £2.99 to £11.99. Whether the dearest one is really four times as good as the cheapest is an open question, but even at that price ratio, it does not remotely feel as outrageously expensive as a £45,000 Renault Twingo would.
The small price bias
A recent paper by Tristan Roger, a financial economist at the Université Paris-Dauphine and colleagues may shed some light on this phenomenon. The researchers examined the target prices financial analysts issue, and found that these are considerably bolder for small price stocks than for large price stocks. Both when they are optimistic and when they are pessimistic, they forecast larger movements for the cheap shares.
This so-called small price bias is consistent with recent neurological research cited in the paper. People tend to use a linear scale for small numbers, and a logarithmic scale for large numbers. This means that when we compare small numbers, we look at the distance, or the difference, between them, while for large numbers, we look at ratios. Another way of looking at this is that we look for absolute differences with small numbers, and relative differences for large ones.
As a consequence, we tend to underrepresent the absolute distance between large numbers. We consider two cars, costing £10,999 and £11,995 respectively, as pretty similarly priced: we see them as less than 10% apart and pay little attention to the fact that the difference is nearly £1,000 (a considerable sum for most people). At the same time, we look at the two bottles of cola, and see that the difference in price is £1.00, rather than that one is three times as expensive as the other.
One particular area where this quirk is exploited by pump-and-dump scammers is that of so-called penny stocks ** or penny shares. These are shares that trade at a very low value (originally for less than a dollar). Some people see them as an effortless way to make money, because they believe “it is so much easier for a stock to go from $0.25 to $0.50 than from $50 to $100.” An increase by an insignificant amount in one case seems much more likely than a doubling in value in the other.
But we don’t need scammers to be misled — we can perfectly do this ourselves. Interest rates, both for mortgages and for savings accounts, are still near historic lows. The difference between the available options is usually of the order of 0.5–1%, and that looks pretty inconsequential. But if we are actually comparing accounts or loans with, say, interest rates of 1.5% and 2.25%, a 0.75% difference means that we will pay 50% more interest (or earn 50% more) with the second one than with the first one. If you borrow £100,000 over 20 years, the former will cost you about £16,500 in interest; the latter will cost you nearly £9,000 more. This is not immediately apparent from the difference of 0.75%.
But is it really as simple as that?
A washing machine is not quite as expensive as a small car, but it is still for most people a considerable outlay. Currys, a popular seller of white goods, has a cheap ‘Essentials’ one for £150, and eleven more options below £200. The most expensive one costs £1,500 — ten times the price of the cheapest, and there are ten more washing machines above £1,000. Arguably we may not be comparing like with like — we should not do so with a small Fiat and a large Mercedes either. But while these two cars may not be substitutes for each other, washing machines arguably are. We buy them primarily for their utility, not for their looks or the hipness of their brand, and that utility is not wildly different across the range. A factor 10 — more than you’ll see for cheap groceries — is remarkable. Even when looking at just one brand we find that the most expensive Bosch costs more than 3 times the cheapest model.
At the other end of the scale, consider the bewildering array of coffees available at Starbucks. If we exclude the espresso and look only at full-size beverages, we see that, even the most expensive large latte is still not twice as expensive as a standard Americano.
The large/small numbers effect would have predicted the price range of relatively expensive washing machines to be more like that of cars, and that of relatively cheap coffees to be more like that of cola, but we see the opposite. Does this mean the phenomenon described in the paper is not real? As is the case for so many biases and fallacies, the strength of this effect can be highly dependent on the context. It will in practice also combine with other influences which in turn may boost or attenuate it, or completely dominate it.
Leigh Caldwell, the author of The Psychology of Price, sees a theoretical opportunity for much more expensive beverage at Starbucks, a Blue Mountain organic latte or something. This would push up the anchor and make the £4.50 Frappucino look better. But it’s not happening, presumably because coffee such a habitual purchase that they feel it wouldn’t work. Consumers do indeed buy coffee more frequently than most groceries, and that might lock the common price point for a coffee very tightly in the customer’s perception.
That German washing machines tend to command higher prices than, well, other ones, is not surprising — the image of superior engineering and quality is strong and persistent. But how come Bosch gets away with a ratio of 1:3? Successful differentiation through additional features that have value in the eye of the buyer — higher capacity, faster spin, quicker quick wash programme, quieter operation — undoubtedly helps. Its reputation helps too. Hoover’s cheapest model is £260, and its most expensive one £600 (a ratio of 1:2.3); Beko’s are £180 and £400 respectively (1:2.2).
According to Leigh Caldwell, real product differences, competition, and how much permission customers give a supplier to differentiate probably make a bigger impact than this psychological effect. In the pure liquid world of investments, the psychology becomes visible because all those other effects are arbitraged away by a relatively efficient market. But in the consumer space category noise makes the psychological signal hard to discern.
Nevertheless, next time you are comparing prices of substitutable goods or services, it could be a smart move to use both perspectives. Look at both the difference in price and the ratio. Taking the ‘other’ view might reveal whether what you thought was a bargain really is.
Originally published at koenfucius.wordpress.com on May 4, 2018.
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