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# 1 Sample Questions: Uncertainty and Imperfect Information 1. Show that expected utility is invariant with respect to linear transformations, but not...

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1 Sample Questions: Uncertainty and Imperfect In- formation 1. Show that expected utility is invariant with respect to linear transformations, but not with respect to monotonic transformations in general. 2. Discuss the possibility of a pooling equilibrium with a perfectly competitive indus- try. 3. Show that a risk averse f rm that faces an uncertain output price will produce less than a similar f rm facing the mean price with certainty. 4. Show that an individual who is more risk averse is willing to pay a higher risk premium. 5. Show that an individual whose utility function is more concave is more risk averse. 6. Show that if the utility is quadratic then expected utility depends only on the f st two moments. 7. Show that the Pareto e cient allocation of risk between a risk neutral and risk averse individual involves full insurance. 8. Show that an individual who can buy insurance at fair premia, in a perfectly competitive insurance market, will buy full coverage. 9. Set up the portfolio choice problem of an expected utility maximizing individual who can in two assets: one risky and one risk free. Explain the optimality condi- tions and show how you can obtain the e f ects of an increase in the variance of the risky asset (without using math). 10. De f ne the e ciency frontier in the standard portfolio problem and show that if there exists a risk free asset, then the (e f ective) e ciency frontier is linear. 11. Show that if two individuals have similar information then they will have the same e ciency frontier. 12. Show that an individual will prefer a (cumulative) distribution F ( x )ove ra(cu - mulative) distribution G ( x ) (with the same range) if G ( x ) >F ( x )fora l l x . 13. Explain (no need to prove) what is meant by second order stochastic dominance and show that f rst order stochastic dominance implies second order stochastic dominance, but not the other way around. 14. Consider an individual facing two alternative uncertain payo f s: X 1 and X 2 , where X 1 and X 2 are random variables with equal expected values. Let X 2 = X 1 + e, where e is a random variable that is independent of X 1 , with E ( e ) = 0. Show that a risk averse individual will prefer X 1 . 1
15. Use a two-period model to compare the consumption/savings choice of an indi- vidual whose future income is uncertain, with an in individual facing the (same) expected future income with certainty. 16. Suppose a risk neutral competitive f rm is facing an uncertain output price, p .I t s production function is given by: y = F ( k,l )where y,k, and l are output, capital and labour respectively. Consider the two options: i. the f rm has to choose both capital and labour inputs, before p is known ii. the f rm has to choose its capital input, before p is known, but can choose its labour input after p is revealed. a. What is the ex ante ranking of the two options (i.e., the ranking, before the price is known)? b. Assuming that the f rm has to pay a fee to be able to exercise option ii, how much is it willing to pay for this option? 17. Consider a risk averse competitive f rm, facing an uncertain output price. The f rm has to make its input and output choices before it observes the output price. Show that the f rm: (i) will produce its output e ciently (ii) may produce where its marginal costs are decreasing (iii) will produce less than a similar f rm that faces the expected price with cer- tainty. 18. Suppose a risk neutral competitive f rm has to make its output decision before it ob- serves the uncertain output price, p. Let its cost function be given by C = βY + Y 2 . (i) De f ne the expected value of perfect information about p . (ii) assuming that the price distribution is given by p = ( p 1 with probability . 5 p 2 with probability . 5 how much is the f rm willing to pay for information about p ? 19. Use the backward induction principle to demonstrate the solution to a three period consumption/savings model with uncertain second and third period incomes. 20. Discuss the “principal agent” problem when actions are not veri f able (but there is no hidden information). 21. Discuss the “principal agent” problem when actions are veri f able, but the agent has private information. 2
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