Thursday, October 16, 2014

Excel Homework Due Wednesday 10/22 at 11 PM

This one is on insurance under private information.  A person could be a low risk (low probability of loss) or a high risk (high probability of loss) but there are no distinguishing features that lets the insurance company know which type an individual is.  The individuals know which type they are, but if low risk types get better rates than the both types have incentive to claim they are low risk.  The market must somehow address this incentive problem.

I didn't have a chance to talk about this today, but we will make some simplifying assumptions on the supply side to make the equilibrium analysis easier.  In actual insurance, there are costs - writing policies, determining whether claims are legitimate or not, etc.  And sometimes insurance companies will not pay on a claim.  For example, in a car accident where both parties have their own insurance and where the accident was clearly the fault of one driver only, sometimes the goal of the insurance company for the driver who was not at fault is to collect from the other insurance company and there can be disputes about that.  We abstract entirely from all this reality in our models.  Our focus is to understand how the market addresses the incentive problem, not to understand all other aspects of insurance.


  1. Do I use -(1-p)/p (p is the low risk loss probability) to calculate the slope of low risk type?

  2. I wonder where you got that from. It looks like the right answer to an entirely different representation of the fair odds line than what we're doing. For starters, you have a negative sign in front, but in our representation the line slopes up.

    Here's the relevant algebra. In this, Pi is the premium, I is the coverage, F is the fixed load, v is the variable load and p is the probability of loss. The pricing formula for insurance that we did in class on Thursday has:

    Pi = F + vI

    When the insurance is actuarially fair, as we want for the lines you are talking about, v = p. So

    Pi = F + pI

    However, in the graphs we have I on the vertical axis and Pi on the horizontal axis. So, solving for I

    pI = Pi - F or I = -F/p +Pi/p.

    So the intercept is -F/p and the slope is 1/p.

  3. I understand the above, but when calculating the utility for a fixed load, do we factor in the probability that a person is defined as "low risk?" Or is that probability irrelevant to the situation?

    1. First I should note that in the Excel worksheets I used the letter q, where above I use the letter p. So in the Excel, qL is the probability of loss for the low risk type and qH is the probability of loss for the high risk type.

      Now as to your question, there are two different perspectives two take here. One is from the point of view of the individual who purchases insurance. This person knows the type. It is not random from this perspective. A low risk type of individual knows she is low risk. A high risk type of individual knows he is high risk. In computing the expected utility for the individual, there is no randomness about type, only randomness on whether loss occurs.

      The other perspective is from the point of view of the insurance company that sells the policy. Does the insurance company that sells to a particular individual know the individuals type? Ahead of time the answer is no, the insurance company doesn't know, so the type is random.

      The next question is whether the insurance company learns the type of the individual based on the policy that the person selects. In a separating equilibrium, yes the insurance company does learn the type this way. So after observing the individual's choice, the insurance company knows the type.

      In a pooling equilibrium, however, there is only one policy offered. It is the same policy regardless of the type of individual. So the individual's choice conveys no information to the insurance company. In this case, the type remains random to the insurance company and the probability of low risk matters in the expected profit calculation for the insurance company.

  4. Im having a hard time understanding the certain equivalent of the original lottery questions, specifically what the lottery and money for certain mean. Could I perhaps get some help on how to start this question?

  5. The certainty equivalent is the amount of money that for certain gives the same utility as the as the expected utility of the original lottery. So if CE is the certainty equivalent it satisfies the equation.
    u(CE) = pu(W-L) +(1-p)u(W).
    It is also the case that you did this sort of calculation in the previous homework, so you should be familiar with the reasoning. All that is different here are the specific values of the parameters.