Has the Profitability of Selling Car Insurance Increased in Early 2020 Due to COVID-19?

Some time this summer, I will revise my  Price Theory Problems e-book.   Here is a new problem I will add.

Every driver is required to buy car insurance.   The historical probability of experiencing an accident can be expressed by the function:  prob(accident) = f(Miles_i)  where Miles_i is the count of miles person i drives per year.

ASSUME: that the price of the premium is set as a function of this historical accident function and the insurance firm is risk neutral.

The profits of the insurance firm will rise in the short run due to COVID-19.  True, False, Uncertain explain.

My tentative answer is : True

In the typical insurance model, the aggregate economic activity where the person is driving is taken as given.  So, in the prob(accident) = f(Miles_i)  there is a KEY omitted variable.  The true accident production function can be written as;  prob(accident) =f(Miles_i, Aggregate Miles)  where Aggregate Miles are all of the other vehicle miles being driven by all of the other vehicles in the area where you drive.

Since COVID-19 is a macro shock that effectively shrinks Aggregate Miles to zero, your probability of having an accident per mile of driving falls sharply and this means that holding your premium price constant, that short run insurance profits rise.

Now, where is perfect competition here?   Are the insurance companies updating their models and offering discounts?  If there is any “cartel” methods to avoid such competition, they will try to do so.

One extension of this problem would be to introduce pay per mile insurance such that people who expect to drive less can substitute to a low mileage plan as they social distance at home. In this case, the algebra will be a bit more involved. The insurance seller experiences a revenue decline and a cost decline but based on the historical probability function, costs will decline more than revenues decline.