Question

A is a good driver with a probability of 0.1 of having an accident and B is less careful and has an accident probability of 0.2. Each has a car initially valued at $10,000, and with a value of zero in the event of an accident. They each have no other wealth. They can each buy fair insurance of any positive amount up to $10,000. If  c_b is consumption (in $'s) in the "state" where an accident occurs, and  c_g is consumption in the "state" where no accident occurs, then in a diagram with c_g on the horizontal axis and  c_b on the vertical axis.

Which of the following statements is correct

(a) A's budget constraint has a slope of -0.9/0.1 and B's has slope -0.8/0.2 and both go through the point c_b = 0, c_g = 10,000  

  

(b)   B's budget constraint has a slope of -0.2/0.8 and A's has slope -0.1/0.9 and both go through the point  c_b = 0, c_g = 10,000.

  

(c)   A's budget constraint has a slope of -0.9/0.2 and B's has slope -0.8/0.1 and both go through the point  c_b = 10,000, c_g = 0.

  

(d)   A's budget constraint has a slope of -0.8/0.1 and B's has slope is -0.9/0.2 and both go through the point  c_b = 10,000, c_g = 0

Answer 1

We can solve this into 2 parts. First, lets see the scenario when no insurance is taken. In this case

C_g=10000
C_b=0

This is applicable to both. If we plot this on the chart, it will at value 10000 on horizontal axis (as the question says,C_g is on the horizontal axis). Since we already know this, it rules out options C and D, since they are with C_b=10000 and C_g=0, which is wrong.

Now lets see when there is insurance. Lets say the price of insurance is p and the amount for which insurance is taken is K (as given in question, K can be till 10000). In this case,

C_g=10000-pK
C_b=10000-pK-10000 (since value of car is zero after accident)+K (this is the value of insurance, which you will get back if accident happens).
C_b=K-pK. Rearranging, we get K=C_b/1-p

Putting this back in C_g, we get C_g=10000-p*C_b/1-p

Now, given the probability of accident is .1 for the A and .2 for the B, the price would be in the same ratio. Putting that in,

C_g=10000-.1*C_b/.9. This is a linear equation with slope of -.1/.9. This is for A. For B, the equation would come out to be C_g=10000-.2*C_b/.8, a linear equation with slope of -.2/.8.

Option B is the correct answer as the slope of A is -.1/.9 and slope for B is -.2/.8, as shown above.