# Complex prices – confused consumers?

20/06/2013 Leave a comment

This morning (20-06-13) Which? published its report on the UK mortgage market finding that 99.5% of homeowners and home-buyers struggle to identify the total cost of a mortgage deal and that only 0.5% of the surveyed homeowners ranked mortgage deals in the correct order.

This is important because with such complex prices consumers are more likely to make purchasing errors if they are unable to tell the real price difference between two products. As a result, this jeopardises the functioning of markets and the informative role prices play in them. There have been many theoretical works that showed the adverse effect of complex pricing. For example the model by Chiovenanu and Zhou (2011) show that these behavioural biases can lead to rather perverse results, for example with complex prices the increase in the number of firms will increase industry profits and harm consumers (while standard theory would assume that new entrants reduce industry profits and benefit consumers). The experimental studies conducted by London Economics for the OFT on the other hand found that complex pricing does not lead to consumers making significantly more purchasing errors. I have never been very happy with this finding and the Which? study seems to confirm my suspicion, which is based purely on some logical reasoning and play with various probabilities of errors. The bottomline is: it is rather likely that consumers make a purchasing error when faced with complex prices. This is even true when consumers are assumed to be unrealistically savvy.

Take the overly simplified situation where the consumer has to decide which of two products has the lower prices. If prices are not obfuscated then the decision is simple as long as consumers can order two numbers based on their magnitude. If prices are ‘complex’ then with a given probability consumers get the ordering wrong. With every shopping decision consumers set up a hypothesis (which we call null hypothesis in statistics) and then decide whether it is true or false. For example a consumer may go ahead with a hypothesis that price A is smaller than price B (pA < pB). For all consumer decisions between a pair of prices the ratio of true hypothesis to false hypotheses is given by T/F=R, therefore T=FR. The probability of a hypothesis being true is then R/(R+1) and false 1/(R+1).

If the consumers’ null hypothesis is pA < pB and the true relationship is also pA < pB but the consumer mistakenly refuses this and concludes that pB > pA (let us assume prices of two products are never equal) then she makes a Type 2 error (not finding a true relationship true), and we can denote its probability by β. If the null hypothesis is still pA < pB but the true relationship is now pA > pB and the consumer accepts the null and concludes that pA < pB then she makes Type 1 error (finding a false relationship true) – with a probability α.

The probability of making decision errors can therefore be summarised in the following table:

Actual relationship |
||||

True | False | Total | ||

Consumer’s finding |
True | ((1-β)R)/(R+1) | α/(R+1) | ((1-β)R+α)/(R+1) |

False | (βR)/(R+1) | (1-α)/(R+1) | (1- α+ βR)/(R+1) | |

Total | R/(R+1) | 1/(R+1) | 1 |

From the table we can get that the probability of the consumer rightly finding the true relationship true is given by ((1-β)R)/((1-β)R+α)), and the probability of finding a false relationship true is (α/((1-β)R+α)).

Therefore the consumer is more likely to choose the cheaper product if (1-β)R > α, which is unlikely to be the case if the consumer is very prone to Type 1 or Type 2 errors. Moreover I do not think that society would be happy with an outcome where the consumer makes the right decision in around 51% of the cases. If we wanted the consumers to be right in 90% of the cases we would need (1-β)R > 9α, which would mean that even very small Type 1 or Type 2 errors will mean consumers making errors in more than 10% of the cases. For example if we have very savvy – and more importantly time-millionaire – consumers than the probability of finding the true relationship between two prices may be very high at 90% (this is called the power of the study in statistics). However, depending on R, this may still not be enough to make the right decision in 90% of the cases if at the same time there is a 0.1 probability of making a Type 1 error (finding a false relationship true).