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2 Uncertainty Dixit: Optimization in Economic Theory (Chapter 9)

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3 1,2,3,….,m States of the world p 1, p 2,…..p m probabilities Y 1, Y 2,…..,Y m income in state i F(Y 1, Y 2,…..,Y m, p 1, p 2,…..p m ) - objective function Expected utility, U - von Neumann Morgenstern utility function

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4 Risk Aversion Y 1, Y 2, p 1, p 2 Expectation of Y: p 1 Y 1 + p 2 Y 2 Y1Y1 Y2Y2

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5 Insurance Y 1 < Y 2 Premium $1 buys $b compensation in the bad state. $x → $bx Y 1 – x + bx, Y 2 – x

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6 But: 1 = pb (Competition in the insurance industry) Full Insurance

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7 Action to reduce the risk (Care) Y 1 < Y 2 Cost z determines p 1 = p(z). p’(z) < 0. + Marginal benefit Marginal cost

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8 Care & Insurance zero expected profit:

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10 r a random variable

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11 Safe and Risky asset

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12 OR: interior solution:

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13 Managerial Incentives Owner hires a Manager for a project Project (if it succeeds) yields V Probability of success is p or q ( p > q ). The manager determines the probability Cost of the higher probability p is e. Manager’s salary is w.

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14 First Best (the owner can observe the manager’s quality) His expected profit: assume: and: Then Owner can get:

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15 The owner cannot observe the manager’s quality If the owner pays the manager according to success or failure Pays x if success, and y if failure Incentive for manager Participation constraint indifference

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16 Owner’s expected payoff: Make x, x-y small

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17 Owner’s expected payoff: same as First Best

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18 and owner’s expected payoff:

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19 owner’s expected payoff: We assumed (high quality worker is better) first best

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20 Cost-Plus Contracts Quantity produced q at cost c Government pays R >qc A firm with costs c 1 o r c 2 ( > c 1 ) Government knows prob. p 1 p 2 Government chooses R 1 R 2 c 1 c 2

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