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Chapter 18: Problem 4

Suppose there is a \(50-50\) chance that an individual with logarithmic utility from wealth and with a current wealth of \(\$ 20,000\) will suffer a loss of \(\$ 10,000\) from a car accident. Insurance is competitively provided at actuarially fair rates. a. Compute the outcome if the individual buys full insurance. b. Compute the outcome if the individual buys only partial insurance covering half the loss. Show that the outcome in part (a) is preferred. c. Now suppose that individuals who buy the partial rather than the full insurance policy take more carc when driving, reducing the damage from loss from \(\$ 10,000\) to \(\$ 7,000\). What would be the actuarially fair price of the partial policy? Does the individual now prefer the full or the partial policy?

Short Answer

Expert verified
The individual prefers to buy partial insurance and take more care when driving, as it results in a higher expected utility compared to full or partial insurance without taking extra care.

Step by step solution

01

Derive the utility function

The individual's utility function is written in logarithmic form, which means: \[u(W) = \ln(W)\] where \(W\) represents the wealth of the individual.
02

Calculate expected utilities for each scenario

For each scenario, the expected utility can be calculated as follows: \[E[u(W)] = p \cdot u(W_{1}) + (1-p) \cdot u(W_{2})\] where \(W_{1}\) and \(W_{2}\) are the individual's wealth after facing loss or no loss, respectively, and \(p\) is the probability of facing the loss. Let's calculate the expected utilities for each scenario:
03

a. Full insurance

In this case, the individual pays the actuarially fair premium to cover the entire loss. The premium is equal to the probability of the loss multiplied by the amount of the loss: \[Premium = p \cdot L = 0.5 \cdot \$ 10,000 = \$ 5,000\] When an accident occurs (probability 0.5): \[W_{1} = \$ 20,000 - \$ 10,000 + \$ 10,000 - \$ 5,000 = \$ 15,000\] When there is no accident (probability 0.5): \[W_{2} = \$ 20,000 - \$ 5,000 = \$ 15,000\] Expected utility: \[E[u(W)] = 0.5 \cdot \ln(\$ 15,000) + 0.5 \cdot \ln(\$ 15,000) = \ln(\$ 15,000)\]
04

b. Partial insurance

The individual pays a premium to cover half the loss: \[Premium = 0.5 \cdot L/2 = 0.5 \cdot \$ 5,000 = \$ 2,500\] When an accident occurs (probability 0.5): \[W_{1} = \$ 20,000 - \$ 10,000 + \$ 5,000 - \$ 2,500 = \$ 12,500\] When there is no accident (probability 0.5): \[W_{2} = \$ 20,000 - \$ 2,500 = \$ 17,500\] Expected utility: \[E[u(W)] = 0.5 \cdot \ln(\$ 12,500) + 0.5 \cdot \ln(\$ 17,500)\]
05

c. Partial insurance with more careful driving

The individual pays a premium to cover the reduced loss of \(L = \$ 7,000\): \[Premium = 0.5 \cdot L/2 = 0.5 \cdot \$ 3,500 = \$ 1,750\] When an accident occurs (probability 0.5): \[W_{1} = \$ 20,000 - \$ 7,000 + \$ 3,500 - \$ 1,750 = \$ 14,750\] When there is no accident (probability 0.5): \[W_{2} = \$ 20,000 - \$ 1,750 = \$ 18,250\] Expected utility: \[E[u(W)] = 0.5 \cdot \ln(\$ 14,750) + 0.5 \cdot \ln(\$ 18,250)\]
06

Compare expected utilities

Now, we will compare the expected utilities for all three scenarios: a. Full insurance: \(E[u(W)] = \ln(\$ 15,000)\) b. Partial insurance: \(E[u(W)] = 0.5 \cdot \ln(\$ 12,500) + 0.5 \cdot \ln(\$ 17,500)\) c. Partial insurance with more care: \(E[u(W)] = 0.5 \cdot \ln(\$ 14,750) + 0.5 \cdot \ln(\$ 18,250)\) We can observe that (by either calculating numerically or just observing the values of W): \(E[u(W)]_{a} > E[u(W)]_{b}\), implying the individual prefers full insurance over partial insurance. Next, we compare (a) and (c): \(E[u(W)]_{a} < E[u(W)]_{c}\), implying the individual prefers partial insurance with more careful driving over full insurance.
07

Conclusion

The individual prefers to buy partial insurance and take more care when driving, as it results in a higher expected utility compared to full or partial insurance without taking extra care.

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