Appendix A
Estimation of bidresponse functions
In many european jurisdictions, bidresponse functions can be estimated from “ monetary value tests ” in which prices quoted to identical risks are change randomly for a trial period and the effects on individual customer responses observed. regulation in other jurisdictions may prevent price tests, in which case bidresponse functions must be estimated in other ways. For exemplar, any change in rates represents a natural experiment in which the effect on customer responses can be observed. data which would be discarded in risk price analyses can much be utilitarian, if it is kept—for example, the prices quoted at refilling to policies which did not renew. logistic arrested development can be used to estimate the effect of categorical rat factors such as age, sex, tenure, previous claims and other factors on the reply, thus allowing a bidresponse function to be constructed for a quotation mark with any combination of evaluation factors. That is, if ρ is the probability of a invite succeed, p is the quote price, a is the regression intercept, and X is the matrix of n evaluation factors each with m categories, the effect of the rate factors on the reception is estimated by regressing X on logit ( ρ ) — and then the bidresponse function for a quotation with any particular combination of denounce factors X c can be constructed from— For case, in a font analyze of centrifugal insurance in Israel, the statistically significant explanatory variables for the response were plus effects from vehicle rate, tenure and policyholder long time, and negative effects from premium and switch on anterior year premium.10
Appendix B
An analysis of inertia pricing
The analysis which follows is based on an earlier paper8 which considers discounts or cashbacks for raw customers, but adapted to reflect an “ inactiveness ” increase in premiums for renewing customers, which seems to be the more common frame in policy.
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We assume an policy market with two firms A and B, with the integral market of customers market represented by a measure of 1 ( i.e. if A has a grocery store plowshare of 50 per cent this is represented as 0.5 ). policy is compulsory or effectively compulsory ( e.g. thirdparty centrifugal policy in many jurisdictions ), sol every customer buys one whole of policy from one tauten in each period. To simplify the presentation, we will abstract from riskrelated differentials, and assume that all customers have the same expected cost of claims, which is discernible by both insurers. Footnote 29 The expected fringy cost of supplying a policy is the same c for both firms, and the prices charged by the two firms are p A and p B, both at or above c. If a customer interchange from one fast to the other he will incur a throw cost s, which represents the meter and troublesomeness to the customer of switching insurers. The s for each consumer in each period is assumed to be an autonomous realization of a random variable S which is uniformly distributed on [ 0, t ] across the population. This can be thought of as reflecting random variation from year to year in the other demands on a customer ‘s time and care around the date of refilling. Consider a mature market where both firms A and B have already established market shares α and β respectively in previous periods under uniform price, We use the surveil note, for i = A, B —

p
i
=:  firm i ‘s price for new customers

e
i
=:  firm i ‘s escalation of price its own former customers ( so firm A ‘s previous customers pay p A + e A if they stay with firm A, or p B if they switch to firm B )

n
ij
=:  the proportion of customers who bought from firm j in the former menstruation but buy from firm i in the current period
At the date of reclamation, a customer who previously bought from A is deaf between continuing to buy from A and switching to B if his switching cost s is such that p B = p A + e A − s. therefore the proportion of customers staying with A, n AA, is given by and the fraction of customers switching away from fast A to firm B, n BA, is provided that the difference between the two firms ’ prices plus the inertial escalation falls within the range of potential trade costs, that is 0⩽ p A − p B + e A ⩽ t. similarly for firm B , and provided that 0⩽ p B − p A + e B ⩽ t. Firm A ‘s and firm B ‘s profits in this time period are, respectively , and For a Nash chemical equilibrium, we need a match of ( p* A, e* A ) and ( p* B, e* B ) such that ( p* A, e* A ) maximises π A given ( p* B, e* B ), and ( p* B, e* B ) maximises π B given ( p* A, e* A ). Setting the partial derivative derivatives of the two profit functions equal to zero gives analogous equations can be derived for π B. This system of equations has the following alone solution and the secondorder conditions for net income maximization are besides satisfied by these solutions. That is, given switching costs uniformly distributed on [ 0, t ], customers who switch are charged a markup of onethird of the maximum switch over cost t. Customers who do not switch are charged doubly this markup. Substituting the balance prices and inertial increases ( B.9 ) and ( B.10 ) into ( B.2 ) and ( B.4 ), we obtain the proportions of customers of each firm who switch firms in equilibrium in each period— thus onethird of all customers in the grocery store switch firms in every menstruation. This high level of switching is socially ineffective : it arises as an artifact of the market structure, not as a resultant role of changing mark preferences, and so represents a deadweight loss to society. The market structure features which give rise to excessive interchange are annual contracts, no sword preferences, and private nonverifiable trade costs which the customer can communicate only by actually switching. In equilibrium any consumer with a trade cost, s < t /3 switches, so the expected deadweight loss per consumer is
Substituting the equilibrium prices and inertial increases ( B.9 ) and ( B.10 ) into the net income equations ( B.5 ) and ( B.6 ), we obtain the equilibrium profits as and
Comparison with uniform pricing
Under uniform pricing, we use the trace notation, for firms i = A, B —

p
u
i
=:  firm i ‘s price

m
u
i
=:  the proportion of customers who buy from firm i

π
u
i
=:  tauten i ‘s profit
We initially assume that firm A ‘s price is higher, p A 2 u ⩾ p u B 2. A consumer belong to firm A ‘s market share α in the previous period is apathetic between sticking with A and switching to B if his switching price s is such that p u B = p A u − s. therefore the proportion of customers staying with A under uniform price, m A, is given by and Firm A ‘s and B ‘s profits under uniform pricing are Setting the partial derivative derivatives of the two profit functions equal to zero gives and The following prices satisfy these equations and besides satisfy the secondorder conditions for net income maximization : These prices are both at their highest when α =½ ( i.e., the firms have peer marketplace shares ), when the identical prices are p A u *= p B u *= c + t. intuitively, the peer market shares mean that each firm ‘s single price needs to give adequate weight to exploiting existing customers and enticing its rival ‘s customers, sol price competition for newly customers is muted. The prices are both at their lowest when α =1 ( i.e., firm A had all the market in the former period ), when the prices are p A u *= c + t and p B u *= c + t. intuitively, firm B immediately places full burden on enticing its rival ‘s customers, so its price is lower. In the above psychoanalysis we assumed that firm A ‘s secondperiod monetary value was higher, that is p A u ⩾ p B u *. From the above solutions we can see that this will be true only if α ⩾½, that is tauten A has at least half the market. If we alternatively start from the converse assumption p A u < p B u *, we obtain the solutions— From the shape of these solutions, we can see that if α < ½, both firms ’ prices are at their lowest when α =0 ( this is symmetrical with the shell of α =1 above, except immediately firm B rather than A had all the market in the former menstruation ). The lowest prices are p A u *= c + t and p B u *= c + t, again symmetrical with the solutions for α ⩾½ above. Firm A ‘s and firm B ‘s equilibrium profits in the ripen market under uniform price are and Substituting the equilibrium prices ( B.22 ) and ( B.23 ) into ( B.14 ) and ( B.15 ), we see that the symmetry of switching customers under consistent price is ( 2 α −1 ) /3 if α ⩾½, and ( 1−2 α ) /3 if α < ½. Under paying customers to switch, the proportion of switchers was always , which is higher than either of these expressions. Hence the deadweight loss from switching is higher under inactiveness price .
Industry profits
Comparing ( B.12 ) and ( B.13 ) with ( B.24 ) and ( B.25 ), we can see that diligence profits when insurers practise inertia price are constantly lower than under consistent price .
Consumer surplus
Although industry profits are always lower under inertia pricing in this model, not all customers are better off. Under inertia pricing, the prices are given by equations ( B.9 ) and ( B.10 ) : twothirds of customers are stayers who pay c + t and onethird are switchers who pay c + t, besides incurring switching costs of up to t ( because the switch costs are uniformly distributed on [ 0, t ] ). Under uniform pricing, the prices are given by equations ( B.20 ) and ( B.23 ) : customers of the “ watery ” firm with lower market parcel can obtain a price a low as c + t without the costs of switching.
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however, for the limited case where the firms have equal grocery store shares, that is α =½, under uniform pricing all customers pay c + t, which exceeds the highest price paid under inertia price ( c + t ) by more than the highest switch monetary value any customer who actually switches pays under inactiveness price ( t ). thus where the firms have adequate grocery store shares, all customers are better off under inactiveness price than under undifferentiated price .
Market concentration
Inertia price has implications for marketplace concentration. The ability of rivals to “ pay customers to switch ” implies that this form of rival may make it more difficult for any one tauten to maintain a large parcel of the commercialize. This observation besides suggests that the most enthusiastic proponents of inertia price should be new entrants, rather than incumbents with boastfully market shares. This is reflected in the analysis above as follows : the prices under inertia pricing in equations ( B.9 ) and ( B.10 ) above are independent of market share, but those under uniform price in ( B.20 ) to ( B.23 ) depend on market parcel .
More than two insurers
It can be shown that the above conclusions above a fortiori in a market with more than two firms. Footnote 30 The intuition is that more firms means more competition for each customer ‘s business, so that the payments made to customers to switch are far bid improving, reducing prices and leaving firms in aggregate worse off. A larger number of competitors leads to a “ achiever ‘s curse ”. Footnote 31