Q1.
Emma’s utility function is given by 𝑈(𝐼) = √10 𝐼, where 𝐼 represents her annual income in thousands of Euros. To understand Emma’s attitude toward risk without knowing her income we shall input an income of €10,000 into the given formula and will progressively increase it by €10,000 till there is enough data to plot a utility function chart that we can visually deduct her level of risk based off.
10√10 = 31.6
10√20 = 44.7
10√30 = 54.8
10√40 = 63.2
10√50 = 70.7
10√60 = 77.4
10√70 = 83.6
In figure 1 the slope of total utility function decreases as the money income of Emma increases; therefore, this concave utility function represents the case of a risk-adverse individual.
(b) Suppose Emma is currently earning an income of €40,000 (𝐼 = 40). This is the only source of income. She can earn the same amount next year with certainty. She has the chance to take a new job that offers a 0.6 probability of earning €44,000 (𝐼 = 44) and a 0.4 probability of earning €33,000 (𝐼 = 33). Based on the expected value and expected utility, should she take the new job? (10 marks)
Choice A: €40 guaranteed
Choice B: 60% of €44 and 40% of €33
The expected value of B is 0.6*€44 + 0.4*€33 = €39.6
𝑈(𝐼) = √10 𝐼
€44,000 (𝐼 = 44) U= 66.32
€33,000 (𝐼 = 33) U= 57.44
E (U) B = 0.6*U(44) + 0.4*U(33)
= 0.6*66.32 + 0.4*57.44
= 39.79 + 22.98
=62.77
Choice A Utility = 10√40 = 63.24
Choice B E(U) = 62.77
Based on the expected value and expected utility, Emma should not take the new job as choice A offers a slightly higher (0.47) utility and EV than choice B. Choice A is also a guaranteed income which suits her risk profile
Given the new job in (b), would Emma be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance?
Choice B: 60% of €44 and 40% of €33
The expected value of B is 0.6*€44 + 0.4*€33 = €39.6
c) Given the new job in (b), would Emma be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance?
