Consider two individuals who face an uncertain income, w, that can have two values, 0 and 16, with equal probabilities. The individuals' utility functions are given by: u¹ = w¹/2 and u² = (w¹/2)¹/2. Use THESE utility functions to show that the individual who is more risk averse is willing to pay a higher risk premium.
Consider two individuals who face an uncertain income, w, that can have two values, 0 and 16, with equal probabilities. The individuals' utility functions are given by: u¹ = w¹/2 and u² = (w¹/2)¹/2. Use THESE utility functions to show that the individual who is more risk averse is willing to pay a higher risk premium.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Question:
Consider two individuals who face an uncertain income, w, that can have two
values, 0 and 16, with equal probabilities. The individuals' utility functions are
given by: u¹=w¹/2 and u² = (w¹/2)1/2. Use THESE utility functions to show that
the individual who is more risk averse is willing to pay a higher risk premium.
Use below method to answer above question:
3 Attitudes Toward Risk
Consider an individual whose income is given by the random variable w.
Suppose w is distributed according to the distribution function g(w), whose mean and variance are given by
E(w) = w and Var(w) = o².
• Assuming that the individual's preferences satisfy the above assumptions, we can consider her expected utility
function: E[u(w)].
• Compare the two values: E[u(w)] and u[E(w)] = u(w).
• Will she prefer the lottery w (distributed according to the distribution g(w)) or the certain value of w?
● Consider a function of one variable u(w), where w € W.
• Remember the definition of concavity (of course, in the general definition of concavity, we did not restrict
ourselves to functions of one variable):
u(w) is concave if, for any (w₁, W2) E W, we have,
pu(w₁) + (1 − p)u(w₂) ≤ u[pw₁ + (1 − p)w₂] = u[w]
for all 0 < p ≤1
This definition says: "The average of the function is smaller than the function evaluated at the average.
● In our context (of choice under uncertainty), if we think of w as a discrete random variable with two possible
outcomes (n = 2), whose probabilities are p, (1 − p), we have:
pu(w₁) + (1 − p)u(w₂) = E[u(w)]
pw₁ + (1 − p)w2 = E(w)
● Thus, we can write the inequality above as:
E[u(w)] ≤ u[E(w)]
Jensen's Inequality.
• This inequality is a version of what is known as
● It can be generalized to the discrete case with any number of outcomes n > 2 and the continuous case.
(18)
(19)
• Here is the proof for the continuous case:
• Let w be distributed according to the distribution function g(w), whose mean is E(w) = w.
Since u(w) is concave, we know that its graph lies everywhere below all its tangents (equivalently, it lies
everywhere below its supporting lines).
● In other words, for any w and z, we have:
u(w) ≤ u(z) + u'(z)(w – z)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09238b20-1e79-43dc-b0a6-3d40e032e374%2F57e9b7a4-553b-47e7-be21-abfdad47d8a8%2Fgs6azre_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question:
Consider two individuals who face an uncertain income, w, that can have two
values, 0 and 16, with equal probabilities. The individuals' utility functions are
given by: u¹=w¹/2 and u² = (w¹/2)1/2. Use THESE utility functions to show that
the individual who is more risk averse is willing to pay a higher risk premium.
Use below method to answer above question:
3 Attitudes Toward Risk
Consider an individual whose income is given by the random variable w.
Suppose w is distributed according to the distribution function g(w), whose mean and variance are given by
E(w) = w and Var(w) = o².
• Assuming that the individual's preferences satisfy the above assumptions, we can consider her expected utility
function: E[u(w)].
• Compare the two values: E[u(w)] and u[E(w)] = u(w).
• Will she prefer the lottery w (distributed according to the distribution g(w)) or the certain value of w?
● Consider a function of one variable u(w), where w € W.
• Remember the definition of concavity (of course, in the general definition of concavity, we did not restrict
ourselves to functions of one variable):
u(w) is concave if, for any (w₁, W2) E W, we have,
pu(w₁) + (1 − p)u(w₂) ≤ u[pw₁ + (1 − p)w₂] = u[w]
for all 0 < p ≤1
This definition says: "The average of the function is smaller than the function evaluated at the average.
● In our context (of choice under uncertainty), if we think of w as a discrete random variable with two possible
outcomes (n = 2), whose probabilities are p, (1 − p), we have:
pu(w₁) + (1 − p)u(w₂) = E[u(w)]
pw₁ + (1 − p)w2 = E(w)
● Thus, we can write the inequality above as:
E[u(w)] ≤ u[E(w)]
Jensen's Inequality.
• This inequality is a version of what is known as
● It can be generalized to the discrete case with any number of outcomes n > 2 and the continuous case.
(18)
(19)
• Here is the proof for the continuous case:
• Let w be distributed according to the distribution function g(w), whose mean is E(w) = w.
Since u(w) is concave, we know that its graph lies everywhere below all its tangents (equivalently, it lies
everywhere below its supporting lines).
● In other words, for any w and z, we have:
u(w) ≤ u(z) + u'(z)(w – z)
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