2 marksMicroeconomicsChoice under uncertaintyInsurance, risk sharing and intertemporal choiceHardabout 4 min

GATE 2021 Question 32: Choice under uncertainty

Suppose Vijay has purchased a high-speed car worth ₹1000000. During the purchase, an Insurance company has shared the latest available road safety survey, wherein it is mentioned that, due to heavy congestion on roads, there is 40% chance of an accident within the first year of car purchase resulting in loss of the car value by 60%. Vijay's utility function for wealth ( $W$ ) is given by $U(W)=\ln(W)$ . If Vijay plans to buy an accident insurance having a premium of 30%, then he will purchase an insurance of ₹______ (round off to the nearest integer).

Answer and worked solution

Answer: 990476

Let $I$ be the insurance coverage Vijay purchases. He pays a premium of $0.30\times I$ regardless of outcome, and receives a payout of $I$ only if an accident occurs.

Wealth if no accident (probability 0.6): $W_{no} = 1{,}000{,}000 - 0.30I$

Wealth if accident (probability 0.4): starting wealth minus the 60% car-value loss ( $600{,}000$ ), minus the premium, plus the insurance payout:

$W_{yes} = 1{,}000{,}000 - 600{,}000 - 0.30I + I = 400{,}000+0.70I$

Expected utility:

$EU(I) = 0.6\ln(1{,}000{,}000-0.30I) + 0.4\ln(400{,}000+0.70I)$

Maximizing: taking the derivative with respect to $I$ and setting it to zero:

$\frac{dEU}{dI} = 0.6\cdot\frac{-0.30}{1{,}000{,}000-0.30I} + 0.4\cdot\frac{0.70}{400{,}000+0.70I} = 0$

Solving this equation (verified via symbolic and independent numerical optimization) gives:

$I^* \approx 990{,}476$

Since the premium rate (30%) is well above what would be actuarially fair for a 40% accident probability, Vijay optimally buys close to but not quite full coverage — consistent with the general principle that risk-averse agents buy less than full coverage when insurance is priced above actuarially fair rates.

Therefore Vijay will purchase insurance coverage of approximately ₹990,476.

The trap: 1000000

Students sometimes assume risk-averse agents always buy full insurance coverage (true only when premiums are actuarially FAIR), missing that the 30% premium rate here is set well above the 40% accident probability times the loss fraction in a way that makes full coverage sub-optimal — this specific problem requires setting up and solving the full expected-utility maximization rather than assuming a textbook full-coverage shortcut.

Plain-text version

Suppose Vijay has purchased a high-speed car worth ₹1000000. During the purchase, an Insurance company has shared the latest available road safety survey, wherein it is mentioned that, due to heavy congestion on roads, there is 40% chance of an accident within the first year of car purchase resulting in loss of the car value by 60%. Vijay's utility function for wealth (W) is given by U(W)=ln (W). If Vijay plans to buy an accident insurance having a premium of 30%, then he will purchase an insurance of ₹ (round off to the nearest integer).

Choice under uncertainty has come up 27 times in 7 exams since 2007.

What this question tests

This question is on Insurance, risk sharing and intertemporal choice, part of Choice under uncertainty in Microeconomics.

How insurance and saving let people smooth consumption across states and over time.

TestStop has 8 past-paper questions on this subtopic; the nearest in year are below.

IIT JAM 2022 · Q51An individual faces wealth Rs. 10 lakh with probability 0.75 and Rs. 7 lakh with probability 0.25. With…

Take this paper as a timed mock More on Insurance, risk sharing and intertemporal choice

Question from the GATE 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.

GATE 2021 Question 32: Choice under uncertainty

Answer and worked solution

Answer: 990476

Let $I$ be the insurance coverage Vijay purchases. He pays a premium of $0.30\times I$ regardless of outcome, and receives a payout of $I$ only if an accident occurs.

Wealth if no accident (probability 0.6): $W_{no} = 1{,}000{,}000 - 0.30I$

Wealth if accident (probability 0.4): starting wealth minus the 60% car-value loss ( $600{,}000$ ), minus the premium, plus the insurance payout:

$W_{yes} = 1{,}000{,}000 - 600{,}000 - 0.30I + I = 400{,}000+0.70I$

Expected utility:

$EU(I) = 0.6\ln(1{,}000{,}000-0.30I) + 0.4\ln(400{,}000+0.70I)$

Maximizing: taking the derivative with respect to $I$ and setting it to zero:

$\frac{dEU}{dI} = 0.6\cdot\frac{-0.30}{1{,}000{,}000-0.30I} + 0.4\cdot\frac{0.70}{400{,}000+0.70I} = 0$

Solving this equation (verified via symbolic and independent numerical optimization) gives:

$I^* \approx 990{,}476$

Therefore Vijay will purchase insurance coverage of approximately ₹990,476.

The trap: 1000000

Plain-text version

Suppose Vijay has purchased a high-speed car worth ₹1000000. During the purchase, an Insurance company has shared the latest available road safety survey, wherein it is mentioned that, due to heavy congestion on roads, there is 40% chance of an accident within the first year of car purchase resulting in loss of the car value by 60%. Vijay's utility function for wealth (W) is given by U(W)=ln (W). If Vijay plans to buy an accident insurance having a premium of 30%, then he will purchase an insurance of ₹ (round off to the nearest integer).

Choice under uncertainty has come up 27 times in 7 exams since 2007.

What this question tests

This question is on Insurance, risk sharing and intertemporal choice, part of Choice under uncertainty in Microeconomics.

How insurance and saving let people smooth consumption across states and over time.

TestStop has 8 past-paper questions on this subtopic; the nearest in year are below.

IIT JAM 2022 · Q51An individual faces wealth Rs. 10 lakh with probability 0.75 and Rs. 7 lakh with probability 0.25. With…

Question from the GATE 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.