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Unformatted text preview: Review this slide at home! a,b constants X,Y random variables E(aX+bY)=aE(X)+bVar(Y) Var(aX+bY)=a 2 Var(X)+b 2 Var(Y)+2abCov(X,Y) Cov(X,Y) = covariance between X and Y = measures the extent to which X and Y vary together (“covary”) ov ,Y)>0 hen X is large (above its mean), Y Cov(X,Y)>0 b when X is large (above its mean), Y tends to be large as well. Cov(X,Y)<0 b when X is large (above its mean), Y tends to be small. Standard deviation : sd(X)= √ Var(X) Correlation coefficient : ρ = Cov (X,Y)/sd(X)sd(Y) Like covariance but unitless1 < ρ < 1 ρ >0 b positive correlation ρ <0 b negative correlation Review: The PrincipalAgent model The principalagent model Principal hires a sales agent Agent’s sales ( z ) : z = e + x e = agent’s effort = ndom factors not controlled by the agent x = random factors not controlled by the agent . E(x)=0 and Var(x)= σ 2 Cost of effort for the agent: c(e)=(k/2)e 2 Agent’s best alternative gives him utility B. Compensation contract: I(z)=F+pz Review: Optimal contract with a risk neutral agent p=1 b agent gets 100% of sales F<0 The principal “sells” the firm to the agent for a price F . Why is this optimal? Setting p=1 maximizes total value (“the size of the pie”) Choice of F ensures that the principal gets the largest possible fraction of total value (“gets the largest possible piece of the largest pie”) Optimal compensation contracts Why don’t we observe p=1 much more frequently? We may observe it more than we think b sale of assets to agent (who becomes his own principal). Agent may not have the money to pay the principal ( F<0 ). There is something we have been missing: RISK Incentive pay b agent’s pay depends on z b z is uncertain b pay is uncertain/risky The greater p the greater the risk of the agent’s pay If the agent dislikes risk (is risk averse ), this may be a problem Risk neutrality and risk aversion Suppose that an agent is given the following two alternatives (yearly salary): A) €40,000 B) €80,000 with probability 0.5 and €0 with probability 0.5 oth alternatives have the same expected pay: Both alternatives have the same expected pay: €40,000 But they are clearly not equally risky ! Being risk neutral implies that the agent is exactly indifferent between the two alternatives. Risk neutral agent b only cares about expected pay If the first alternative paid €39,999, the agent would prefer the second one even if it is much riskier. Risk neutrality and risk aversion When given the two alternatives (yearly salary): A) €40,000 B) €80,000 with probability 0.5 and €0 with probability 0.5 ost people prefer the first one. Most people prefer the first one. In fact, most people prefer, for example, €39,800 over the second alternative … even if the expected pay with the second alternative is €40,000 People who behave in this way ( dislike risk ) are called risk averse . Risk neutrality and risk aversion Most of you have studied at least some Finance Basic insight of Financial Economics:...
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This note was uploaded on 02/03/2011 for the course ECON 302 taught by Professor Pablo during the Spring '11 term at Universidad Carlos III de Madrid.
 Spring '11
 Pablo

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