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![Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Example
◮ Suppose you cast a dice. If your get x ∈ {1,2,..,6}, you
receives $x.
◮ The expected utility is
EU(x) =
1
6
[u(1)+u(2)+u(3)+u(4)+u(5)+u(6)]
◮ For example, if u(x) =
√
x, then EU(x) = 1.805303682](https://image.slidesharecdn.com/lecture2-130420210559-phpapp01/75/Lecture-2-10-2048.jpg)
![Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Independence
Definition
A preference over lotteries satisfies independence
assumption if, for any lotteries L′,L′′,L′′′ such that L′ L′′,
αL′ +(1−α)L′′′ αL′′ +(1−α)L′′′ for any α ∈ (0,1].
◮ Roughly put, if you expand two lotteries with a common
lottery, it should not affect preference over the two
lotteries.](https://image.slidesharecdn.com/lecture2-130420210559-phpapp01/75/Lecture-2-13-2048.jpg)
Uploaded byLoc Ha
Lecture 2
This document summarizes key points from Lecture 2 of an Intermediate Microeconomics course. It introduces risk and expected utility, defines risk attitudes like risk aversion and risk neutrality, and discusses how expected utility theory can represent preferences over risky outcomes. It also provides an example of how insurance markets work based on individuals' risk attitudes and the incentives of insurance companies.
Lecture 2
- 1. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Intermediate microeconomics: Lecture 2 March 7, 2013
- 2. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk ◮ Many economic choice is risky in the sense that the outcome is not certain. ◮ Project choice ◮ Investment decision ◮ Occupational choice ◮ Marriage decision
- 3. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risky choice as lottery ◮ A decision problem under risk can be interpreted as a choice of “lotteries”. ◮ Project choice ◮ If the future economic condition is good, your project yields $1000 ◮ If it is bad, your project yields $0. ◮ If the probability of having a good economy is 20%, it is a lottery such that $1000 with 20% and $0 with 80%
- 4. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Notation ◮ If the probability of having a good economy is 20%, it is a lottery such that $1000 with 20% and $0 with 80% ◮ Formally, the above lottery can be written as L = (x1,x2;p1,p2) = (1000,0;0.2,0.8)
- 5. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Example ◮ Which is your favorite lottery? ◮ L1 = (10,10;0.5,0.5) ◮ L2 = (0,20;0.5,0.5) ◮ L3 = (5,10;0.2,08)
- 6. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Preference over lotteries ◮ As Lecture 1, define a preference over lotteries, e.g., L1 ≻ L2 ≻ L3. ◮ If your taste is transitive and complete, it is rational preference. ◮ Then, you have a utility function U(L) that represents your taste for lotteries. ◮ However, you might want more intuitive and useful representation.
- 7. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Expected value ◮ If it is a lottery, how about the average or expected value? ◮ That is, U(L) = p(x1)x1 +p(x2)x2 ◮ More generally, if there are N possible outcomes, U(L) = ∑ n p(xn)xn
- 8. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Problem ◮ Note that if L1 = (10,10;0.5,0.5) and L2 = (0,20;0.5,0.5), then U(L1) = U(L2) if U is the expected value. ◮ Thus, unless you find L1 and L2 indifferent, this is not right way to represent your preference. ◮ In other words, the expected value cannot reflect your attitude toward risk.
- 9. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Expected utility ◮ There is an alternative way. ◮ Expected utility theory represents your preference with the following form U(L) = p(x1)u(x1)+p(x2)u(x2) ◮ More generally, if there are many possible outcomes U(L) = ∑ n p(xn)u(xn)
- 10. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Example ◮ Suppose you cast a dice. If your get x ∈ {1,2,..,6}, you receives $x. ◮ The expected utility is EU(x) = 1 6 [u(1)+u(2)+u(3)+u(4)+u(5)+u(6)] ◮ For example, if u(x) = √ x, then EU(x) = 1.805303682
- 11. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market When does EU exist? ◮ We learn that a preference can be represented by a utility if and only if the preference is rational. ◮ When can a preference over lotteries be represented by an expected utility? ◮ In other words, what is the condition that guarantees existence of u(x) such that L′ L′′ if and only if ∑ n pL′ (xn)u(xn) ≥ ∑ n pL′′ (xn)u(xn)
- 12. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Continuity Definition A preference over lotteries satisfies continuity assumption if, for any lotteries L′,L′′,L′′′ such that L′ L′′ L′′′, there exists α ∈ (0,1) such that αL′ +(1−α)L′′′ ∼ L′′ ◮ Roughly put, a small change of a probability can make a small change in your preference.
- 13. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Independence Definition A preference over lotteries satisfies independence assumption if, for any lotteries L′,L′′,L′′′ such that L′ L′′, αL′ +(1−α)L′′′ αL′′ +(1−α)L′′′ for any α ∈ (0,1]. ◮ Roughly put, if you expand two lotteries with a common lottery, it should not affect preference over the two lotteries.
- 14. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market The condition ◮ The following result is known. Proposition There exists an expected utility that represents a preference over lotteries if and only if the preference is rational and satisfies the continuity and independence assumptions.
- 15. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk attitude ◮ The expected value cannot capture a risk attitude. ◮ If L1 = (10,10;0.5,0.5) and L2 = (0,20;0.5,0.5), then what can we say about U(L1) vs. U(L2) if U is the expected utility? ◮ Can the expected utility capture a risk attitude?
- 16. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk aversion ◮ Suppose u(x) = xα (other functional form can be also possible) ◮ Suppose you prefer L1 to L2, that is, you are risk averse. ◮ Then choose α ∈ (0,1), say 1/2 ◮ Then, 1 2101/2 + 1 2101/2 = 101/2 = 3.162 > 1 2201/2 = 2.236. ◮ So α < 1 represents that you are risk averse.
- 17. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk aversion and concavity ◮ If u(x) is strictly concave, then the agent is always risk averse. ◮ u(x) = ln(x) ◮ u(x) = √ x
- 18. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk loving ◮ Suppose you prefer L2 to L1, that is, you are a risk lover. ◮ Then choose α > 1 in your xα. ◮ Then, 1 2102 + 1 2102 = 102 = 100 < 1 2202 = 200. ◮ So α > 1 represents that you are a risk lover.
- 19. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk loving and convexity ◮ If u(x) is strictly convex, then the agent is always risk averse. ◮ u(x) = x3 ◮ u(x) = ex
- 20. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk neutrality ◮ If α = 1, then the expected utility becomes the expected value. ◮ 1 210+ 1 210 = 10 = 1 220. ◮ That is, L1 and L2 are indifferent. ◮ So α = 1 means that you are risk neutral.
- 21. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk neutrality and linearity ◮ if u(x) is linear, then the agent is always risk averse. ◮ u(x) = 4x ◮ u(x) = 0.3x
- 22. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Certainty equivalent ◮ L = (x,y;p,1−p) ◮ Consider z such that u(z) = pu(x)+(1−p)u(y) ◮ That is, you are indifferent between certain z dollar and lottery L. ◮ Then, z is called certainty equivalent of L ◮ If you are risk averse, z < px +(1−p)y. ◮ If you are risk neutral, z = px +(1−p)y.
- 23. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk premium ◮ Risk premium of L is an extra money that can be paid to compensate the risk of L. ◮ Formally, we can define risk premium as px +(1−p)y −z. ◮ Note that if the agent is risk neutral z = px +(1−p)y and thus, the risk premium is always 0.
- 24. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Example ◮ L = (10,60;0.5,0.5) and u(x) = 100x − x2 2 ◮ pu(x)+(1−p)u(y) = 1 2(1000−50)+ 1 2 (6000−1800) = 2625
- 25. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Example ◮ Certainty equivalence z solves u(z) = 2625 or 0 = z2 −200z +5250 ◮ The certainty equivalent is z = 200− √ 40000−21000 2 ≈ 31 ◮ Risk premium is 1 2 10+ 1 2 60−31 = 4
- 26. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Insurance ◮ Consider the following simple insurance market. ◮ Suppose that the probability of having an accident is known as p. ◮ The accident could make your income from 100 to 0. ◮ Insurance guarantees you to have 100 all the time as long as you pay $q.
- 27. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Expected utility ◮ So with the insurance, your utility is U(100−q) . ◮ On the other hand, without the insurance, your expected utility is (1−p)U(100)+pU(0) .
- 28. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Buyer’s problem ◮ You buy the insurance if your utility from buying is higher than your expected utility from facing uncertainty. ◮ That is, (1−p)U(100)+pU(0) ≤ U(100−q)
- 29. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Seller’s problem ◮ The insurance company has to make sure that ◮ she can sell the product (Sale’s condition) ◮ the profit is not negative (Profitability condition)
- 30. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Sale’s condition ◮ Recall the certainty equivalent of L = (0,100;p,1−p) is z such that U(z) = (1−p)U(100)+pU(0) ◮ Then, the consumer prefers to buy the insurance as long as 100−q ≥ z or q ≤ 100−z ◮ Thus, the insurance company has to choose q so that it satisfies the above condition to sell the insurance.
- 31. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Profitability condition ◮ The insurance company gets q but they have to pay 100 with probability p. Thus, their expected profit given q is q −100p. ◮ Thus, q has to be q ≥ 100p to keep her profit non-negative.
- 32. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Trading condition ◮ The trade occurs if and only if the sale’s condition and the profitability condition are both satisfied. ◮ In other words, q has to be 100−z ≥ q ≥ 100p
- 33. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk neutral buyer ◮ If you are risk neutral, z = (1−p)100. Then, 100−z = 100p ◮ Thus, q = 100p is the only value that satisfies the trading condition 100−z ≥ q ≥ 100p ◮ Then, the expected profit is always 0.
- 34. Intermediate microeconomics: Lecture 2 Risk andExpected utility Risk attitude Application to insurance market Risk aversion ◮ Recall that if you are risk averse, z < (1−p)100. Then, 100−z > 100p ◮ Thus, by choosing q such that 100−z > q > 100p, the company earns a positive profit and the consumer buys the insurance.