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Exercise 8.12

Exercise 8.12. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 8.12(1): Question. purpose : to develop an analysis of insurance where terms are less than actuarially fair

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Exercise 8.12

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  1. Exercise 8.12 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

  2. Ex 8.12(1): Question • 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

  3. Ex 8.12(1): model • Use the two-state model (no-loss, loss) • Consider the person’s wealth in extremes • if uninsured: (y0, y0 L) • if fully insured: (y0 κ, y0 κ) • 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 • ([1  t]y0 + t [y0 κ], [1  t][y0 L] + t [y0κ]) • which becomes (y0 tκ, y0 tκ+ [1  t]L)

  4. Ex 8.12(1): utility • Put payoffs (y0 tκ, y0 tκ+ [1  t]L) into the utility function • Expected utility is • Therefore effect on utility of changing coverage is • Could there be an optimum at t =1?

  5. Ex 8.12(1): full insurance? • What happens in the neighbourhood of t = 1? • We get • Simplifying, this becomes [Lπ  κ] uy(y0 κ) • positive MU of wealth implies uy(y0 κ) > 0 • by assumption Lπ <κ • so [Lπ  κ] uy(y0 κ) < 0 • In the neighbourhood of t =1 the individual could increase expected utility by decreasing t • Therefore will not buy full insurance

  6. Ex 8.12(2): Question Method • Standard optimisation • Differentiate expected utility with respect to t

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

  8. Ex 8.12(3): Question Method • Take t* as a function of the parameter y0 • This function satisfies the FOC • So to get impact of y0: • Differentiate the FOC w.r.t. y0 • Rearrange to get t* / y0

  9. Ex 8.12(3): response of t* to y0 • Differentiate the following with respect to y0: • This yields: • On rearranging we get:

  10. Ex 8.12(3): implications for coverage • Response of t* to y0 is given by • The denominator of this must be negative: • uyy(⋅) is negative • all the other terms are positive • The numerator is positive if DARA holds • Therefore ∂t*/∂y0 < 0 • So, given DARA, an increase in wealth reduces the demand for insurance

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

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