Microeconomics 3200/4200:

Transcription

1 Microeconomics 3200/4200: Part 1 P. Piacquadio p.g.piacquadio@econ.uio.no September 25, 2017 P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

2 Example (1) Suppose I take an urn with 2 blue and 2 red balls. Would you prefer 100 NOK for sure......or a gamble that pays 0 if a blue ball is extracted and 400 NOK if the red one is extracted? What if there were 2 blue balls and 4 red ones? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

3 Example (2) Individual A has an income of 1000$ per month. He bikes to office on sunny days and takes public transport on rainy days. He needs to choose: better to buy a monthly public transport subscription at 100$ or a daily ticket at 2$ if it rains? What do we learn from his choice? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

4 Example (3) Individual B has a wealth evaluated at 50000$. She anticipates that with a small (but positive) probability, a home accident may occur (fire, water damage, thieves, etc...). What coverage should she choose, if an insurance contract that pays k 100$ in case of (a larger) damage would cost k$? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

5 The basic concepts (1) The assumption is that there is a moment in which an event occurs. Before the event occurs, the agent does not know what will happen, i.e. which state of the world will realize. After the event, the true state of the world is revealed and payoffs realize. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

6 The basic concepts (2) The set of possible states of the world is S. A specific state of the world is s S. Each state of the world s is associated to a (monetary) payoff c s. A payoff can be money or a vector of consumption goods (or any consequence). The list of payoffs for each state of the world is a vector (c 1,...,c s,...,c S ), named contingent consumption plan (or prospect). P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

7 The basic concepts (3) Consider a simple case with two states of the world (blue or red ball extracted) and a monetary payoff in each of these (c 1 and c 2 ). Can we draw indifference curves in that space? What do they tell? what assumptions should we make on preferences? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

8 A simple utility function Let π 1 and π 2 be the probabilities over the two states of the world. Then, a common utility function is: u (c 1,c 2,π 1,π 2 ) = π 1 v (c 1 ) + π 2 v (c 2 ) This is the von Neumann-Morgerstern (vn-m) utility function or the expected utility function. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

9 The vn-m utility function The function v is the cardinal utility function or felicity function. If we replace v by v = a + bu (with b > 0), the resulting utility function still represents the same preferences! Assume: ū (c 1,c 2,π 1,π 2 ) ū ( c 1,c 2,π 1,π 2 ). By definition: Now substitute v = a + bu: π 1 v (c 1 ) + π 2 v (c 2 ) π 1 v ( c 1) + π2 v ( c 2). π 1 (a + bu (c 1 )) + π 2 (a + bu (c 2 )) π 1 ( a + bu ( c 1 )) + π2 ( a + bu ( c 2 )). Simplifying, π 1 u (c 1 ) + π 2 u (c 2 ) π 1 u ( c 1) + π2 u ( c 2), that is, u (c 1,c 2,π 1,π 2 ) u (c 1,c 2,π 1,π 2 ). P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

10 The properties of the vn-m utility function The vn-m is an expected utility function of the form u (c 1,c 2,π 1,π 2 ) = E s [u (c s )] where expectations E over the states S = {1,2} are taken wrt the subjective probabilities π 1 and π 2. When c 1 = c 2, the MRS equals the ratio of the subjective probabilities. Independence. The choice that individuals plan to make in one state of nature is independent of what they plan to make in different states of nature. With three states, u (c 1,c 2,c 3 ), the MRS 12 is independent of c 3. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

11 Risk aversion The agent is risk averse if preferences are convex or if the utility function u is quasi-concave. If the individual has vn-m utility, u (c 1,c 2,π 1,π 2 ) = π 1 v (c 1 )+π 2 v (c 2 ), then she is risk averse if and only if v is concave. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

12 Risk loving The agent is risk loving (or risk prone) if preferences are concave or if the utility function u is quasi-convex. If the individual has vn-m utility, u (c 1,c 2,π 1,π 2 ) = π 1 v (c 1 )+π 2 v (c 2 ), then she is risk loving if and only if v is convex. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

13 Risk neutral The agent is risk neutral if preferences are both convex and concave (indifference curves are linear). If the individual has vn-m utility, u (c 1,c 2,π 1,π 2 ) = π 1 v (c 1 )+π 2 v (c 2 ), then she is risk loving if and only if v is linear. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

14 Risk premium (1) How much would an agent be willing to pay to avoid (or get) the risk of a contingent consumption plan? First, define the mean (or expected value) of a contingent consumption plan as the c such that: c = π s c = π s c s = E s [c s ]. }{{} =1 P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

15 Risk premium (2) Second, compute the certainty equivalent. It is the consumption ξ such that getting ξ for sure is as desirable as the contingent consumption plan: or, with vn-m utilities, u (ξ,ξ,π 1,π 2 ) = u (c 1,c 2,π 1,π 2 ) v (ξ ) = E s [v (c s )]. The risk premium is: risk premium = c ξ, i.e. the difference between the mean and the certainty equivalent. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

16 Indices of risk aversion (1) Assume the utility is vn-m and that the function v is twice differentiable: the first order derivative is v c (c) and the second order derivative is v cc (c). Absolute risk aversion. The index of absolute risk aversion is a function: a(c) = v cc (c) v c (c) P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

17 Indices of risk aversion (2) Relative risk aversion. The index of relative risk aversion is a function: ρ (c) = c v cc (c) v c (c) P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

18 Concavity of v and risk aversion Theorem. Concavity and risk aversion. If v is more concave than v, then the indices of risk aversion for v are higher than for v, i.e. a v (c) a v (c) and ρ v (c) ρ v (c). P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

19 Examples of functions v Constant absolute risk aversion (CARA): v (c) = 1 α e αc Constant relative risk aversion (CRRA): v (c) = c1 ρ 1 ρ P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

20 Optimal demand for insurance (1) A person has a wealth of 35000$ and might incur a loss of 10000$ with probability 1%. She can purchase an insurance paying K$ if the loss occurs, for a premium of γk$. Her choice is to set K to optimaze the contingent consumption plan that pays: C b = 25000$ + K γk with probability π=1% C g = 35000$ γk otherwise P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

21 Optimal demand for insurance (2) The budget constraint. Consider C b = 25000$ + K γk, and C g = 35000$ γk with K = 0. Both contingent consumption plans are affordable and on the budget constraint. Thus: C g = C g C g C b C b C b = = (35000$ γk) (35000$) (25000$ + K γk) (25000$) = γk (1 γ)k = γ 1 γ Remark: γ and 1 γ play the role of prices. Note, it is independent of probabilities. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

22 Optimal demand for insurance (3) From the point of view of the insurer, the expected profit is: P = γk πk (1 π) 0 = γk πk Suppose that on average the insurer breaks even, i.e. makes a zero profit. Then, γ = π is the fair insurance premium. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23

23 Optimal demand for insurance (4) For the consumer, the optimum is to equalize marginal rate of substitution with relative prices: MRS = π u(c g ) C g (1 π) u(c = γ b) 1 γ = π 1 π C b But we saw that, when C g = C b, the MRS equals the relative probabilities. Thus, the optimal insurance is K = 10000$. What if the insurance price is not fair? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, / 23