Frank Cowell: Microeconomics Exercise 8.12 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Frank Cowell: Microeconomics Ex 8.12(1): Question purpose: to develop an analysis of insurance where terms are less than actuarially fair purpose: to develop an analysis of insurance where terms are less than actuarially fair method: model payoffs in each state-of-the-world under different degrees of coverage. Find optimal insurance coverage. Show how this responds to changes in wealth method: model payoffs in each state-of-the-world under different degrees of coverage. Find optimal insurance coverage. Show how this responds to changes in wealth

Frank Cowell: Microeconomics Ex 8.12(1): model Use the two-state model (no-loss, loss) Use the two-state model (no-loss, loss) Consider the person’s wealth in extremes Consider the person’s wealth in extremes  if uninsured: (y 0, y 0  L)  if fully insured: (y 0  κ, y 0  κ) Suppose partial insurance is possible Suppose partial insurance is possible  if person insures a proportion t of loss L…  …pro-rata premium is tκ So if a proportion t is insured wealth is So if a proportion t is insured wealth is  ([1  t]y 0 + t [y 0  κ], [1  t][y 0  L] + t [y 0  κ])  which becomes(y 0  tκ, y 0  tκ + [1  t]L)

Frank Cowell: Microeconomics Ex 8.12(1): utility Put payoffs (y 0  tκ, y 0  tκ + [1  t]L) into the utility function Put payoffs (y 0  tκ, y 0  tκ + [1  t]L) into the utility function Expected utility is Expected utility is Therefore effect on utility of changing coverage is Therefore effect on utility of changing coverage is Could there be an optimum at t =1? Could there be an optimum at t =1?

Frank Cowell: Microeconomics Ex 8.12(1): full insurance? What happens in the neighbourhood of t = 1? What happens in the neighbourhood of t = 1? We get We get Simplifying, this becomes [Lπ  κ] u y (y 0  κ) Simplifying, this becomes [Lπ  κ] u y (y 0  κ)  positive MU of wealth implies u y (y 0  κ) > 0  by assumption Lπ <κ  so [Lπ  κ] u y (y 0  κ) < 0 In the neighbourhood of t =1 the individual could increase expected utility by decreasing t In the neighbourhood of t =1 the individual could increase expected utility by decreasing t Therefore will not buy full insurance Therefore will not buy full insurance

Frank Cowell: Microeconomics Ex 8.12(2): Question Method Standard optimisation Standard optimisation Differentiate expected utility with respect to t Differentiate expected utility with respect to t

Frank Cowell: Microeconomics Ex 8.12(2): optimum For an interior maximum we have For an interior maximum we have Evaluating this we get Evaluating this we get So the optimal t ∗ is the solution to this equation So the optimal t ∗ is the solution to this equation

Frank Cowell: Microeconomics Ex 8.12(3): Question Method Take t* as a function of the parameter y 0 Take t* as a function of the parameter y 0 This function satisfies the FOC This function satisfies the FOC So to get impact of y 0 : So to get impact of y 0 :  Differentiate the FOC w.r.t. y 0  Rearrange to get  t* /  y 0

Frank Cowell: Microeconomics Ex 8.12(3): response of t * to y 0 Differentiate the following with respect to y 0 : Differentiate the following with respect to y 0 : This yields: This yields: On rearranging we get: On rearranging we get:

Frank Cowell: Microeconomics Ex 8.12(3): implications for coverage Response of t * to y 0 is given by Response of t * to y 0 is given by The denominator of this must be negative: The denominator of this must be negative:  u yy ( ⋅ ) is negative  all the other terms are positive The numerator is positive if DARA holds The numerator is positive if DARA holds Therefore ∂t * /∂y 0 < 0 Therefore ∂t * /∂y 0 < 0 So, given DARA, an increase in wealth reduces the demand for insurance So, given DARA, an increase in wealth reduces the demand for insurance

Frank Cowell: Microeconomics Ex 8.12: Points to remember Identify the payoffs in each state of the world Identify the payoffs in each state of the world  ex-post wealth under…  …alternative assumptions about insurance coverage Set up the maximand Set up the maximand  expected utility Derive FOC Derive FOC Check for interior solution Check for interior solution Get comparative static effects from FOCs Get comparative static effects from FOCs