1 mark, −0.333MicroeconomicsChoice under uncertaintyRisk aversion, certainty equivalent and risk premiumHardabout 3 min

GATE 2021 Question 16: Choice under uncertainty

Consider an individual who maximizes her expected utility having Bernoulli utility function $u(w)=\alpha-\beta e^{-rw}$ ; $w>0$ is wealth. The individual displays ______ relative risk aversion.

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Answer and worked solution

Answer: (B)

This utility function, $u(w)=\alpha-\beta e^{-rw}$ , is the standard CARA (Constant Absolute Risk Aversion) utility function.

Absolute risk aversion: $ARA(w) = -\dfrac{u''(w)}{u'(w)}$ .

$u'(w) = r\beta e^{-rw}$ , $u''(w) = -r^2\beta e^{-rw}$ .

$ARA(w) = -\frac{-r^2\beta e^{-rw}}{r\beta e^{-rw}} = r \quad \text{(a constant, confirming the CARA name)}$

Relative risk aversion: $RRA(w) = ARA(w)\times w = r\times w$ .

Since $RRA(w) = rw$ is directly proportional to wealth $w$ , and $r>0$ , relative risk aversion increases as wealth increases — even though absolute risk aversion stays constant.

Therefore the answer is (B) increasing.

The trap: (A) constant

This is a deliberately-set trap: CARA utility is famous for having CONSTANT absolute risk aversion (that's literally what 'CARA' stands for), so a student who recognizes the functional form and recalls 'CARA = constant risk aversion' without noticing the question specifically asks about RELATIVE (not absolute) risk aversion will confidently but incorrectly select (A).

Plain-text version

Consider an individual who maximizes her expected utility having Bernoulli utility function u(w)=α-β e^(-rw); w>0 is wealth. The individual displays relative risk aversion. (A) constant (B) increasing (C) decreasing (D) uncertain

Also written as: Consider an individual who maximizes her expected utility having Bernoulli utility function u(w)=alpha-beta e^(-rw); w>0 is wealth. The individual displays relative risk aversion.

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

What this question tests

This question is on Risk aversion, certainty equivalent and risk premium, part of Choice under uncertainty in Microeconomics.

How the shape of utility reveals attitude to risk and how much a person would pay to avoid it.

TestStop has 16 past-paper questions on this subtopic; the nearest in year are below.

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Answer and worked solution

Answer: (B)

This utility function, $u(w)=\alpha-\beta e^{-rw}$ , is the standard CARA (Constant Absolute Risk Aversion) utility function.

Absolute risk aversion: $ARA(w) = -\dfrac{u''(w)}{u'(w)}$ .

$u'(w) = r\beta e^{-rw}$ , $u''(w) = -r^2\beta e^{-rw}$ .

$ARA(w) = -\frac{-r^2\beta e^{-rw}}{r\beta e^{-rw}} = r \quad \text{(a constant, confirming the CARA name)}$

Relative risk aversion: $RRA(w) = ARA(w)\times w = r\times w$ .

Since $RRA(w) = rw$ is directly proportional to wealth $w$ , and $r>0$ , relative risk aversion increases as wealth increases — even though absolute risk aversion stays constant.

Therefore the answer is (B) increasing.

The trap: (A) constant

Plain-text version

Also written as: Consider an individual who maximizes her expected utility having Bernoulli utility function u(w)=alpha-beta e^(-rw); w>0 is wealth. The individual displays relative risk aversion.

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

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

GATE 2021 Question 16: Choice under uncertainty

Answer and worked solution

What this question tests

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GATE 2021 Question 16: Choice under uncertainty

Answer and worked solution

What this question tests

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