advanced microeconomics, uncertainty Question:Consider two individuals who face an uncertain income, w, that can have twovalues, 0 and 16, with equal probabilities. The individuals' utility functions aregiven by: u¹=w¹/2 and u² = (w¹/2)1/2. Use THESE utility functions to show thatthe individual who is more risk averse is willing to pay a higher risk premium.Use below method to answer above question:3 Attitudes Toward RiskConsider 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 byE(w) = w and Var(w) = o².• Assuming that the individual's preferences satisfy the above assumptions, we can consider her expected utilityfunction: 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 restrictourselves 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 ≤1This 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 possibleoutcomes (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 lieseverywhere below its supporting lines).● In other words, for any w and z, we have:u(w) ≤ u(z) + u'(z)(w – z)

Given: Utility function: u1=w12 u2=w1212 The wealth can be: w = 0 with the probability of 12 w = 16…

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.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.7P

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