3. Prove Lemma 1 of the handout Risk Aversion.

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1 IDEA Economics of Risk and Uncertainty List of Exercises Expected Utility, Risk Aversion, and Stochastic Dominance. 1. Prove that, for every pair of Bernouilli utility functions, u 1 ( ) and u 2 ( ), and for every pair of wealth levels, W 1 and W 2, there is no loss of generality in assuming that (a) u 1 (W 1 ) = u 2 (W 2 ), (b) u 1 (W 1 ) = u 2 (W 1 ) and u 1 (W 2 ) = u 2 (W 2 ), (c) u 1 (W 1 ) = u 2 (W 1 ) and u 1(W 1 ) = u 2(W 1 ). 2. Let us consider an economic agent with a Bernouilli utility function u(x) such that u ( ) > 0, u ( ) < 0. The agent (it is better to call him individual because he is not a spy) faces the possibility of suering an accident. The probability that an accident occurs is exogenous and equal to π. The individual can buy units of insurance with a unitary premium equal to p. That is, he receives the net amount m if the accident occurs, provided he pays p if the accident does not occur. Is the individual demand for insurance a decreasing function of the premium p? 3. Prove Lemma 1 of the handout Risk Aversion. 4. Prove Lemma 2 of the handout Risk Aversion. (Hint: You should use both Lemma 1 and Jensen's inequality.) 5. Consider the portfolio selection problem. Let us dene the following variables: W 0 = initial wealth, W = nal wealth, A = value of the investment in the risky asset, M = wealth held as money (the interest rate on money is zero), a = proportion of wealth invested in the risky asset, R = 1 + r = gross return on the risky asset. Suppose that the investor's Bernouilli utility function is quadratic, u(w ) = αw 2 + βw, where α, β > 0. (a) Graph this function. Does it exhibit risk aversion? (b) Find the optimal values of A and a. Prove that, other things equal, 1

2 (i) The optimal levels of A and a are lower for wealthier investors. (ii) The optimal level of A is increasing in β. (ii) The optimal level of A is decreasing in α. 6. Suppose that the investors has a Bernouilli utility function of the CARA type, u(w ) = e γw with γ > 0. (a) Graph this function. Does it exhibit risk aversion? (b) Using the rst-order conditions for optimization, prove that the optimal level of A is independent of the initial wealth and decreasing in the risk aversion parameter γ. Prove also that the optimal level of a is a decreasing function of the initial wealth. 7. Suppose that the investor has an isoelastic (or CRRA) Bernouilli utility function, W σ when 0 < σ < 1 or σ < 0, u(w ) = σ ln W when σ = 0. (a) Graph this function. Does it exhibit risk aversion? (b) Use the rst-order conditions to prove that the optimal level of a is independent of the initial wealth and that the optimal level of A is an increasing function of the initial wealth. 8. In this problem you are asked to investigate the eect which proportional taxation of interest income has on risk taking by an investor. Let us assume that there is a risky asset and a riskless asset with gross return R f = 1 + r f. Suppose that the capital income from the two assets is proportionally taxed. The marginal tax rate t is between 0 and 1. Let us assume that the tax features a loss oset. Therefore, if the interest income is negative the investor receives a subsidy equal to t times what he has lost. Assume also that the investor chooses A so as to maximize the expected utility of his nal wealth. His utility function is twice dierentiable and exhibits risk aversion and decreasing absolute risk aversion. (a) Using a comparative statics argument, nd the derivative describes the eect which an increase in wealth has on risk taking. A W 0, which (b) Using a comparative statics argument, derive an expression for the derivative A t, which describes the eect which an increase in the tax rate has on the risk taking. Show that A t = A ( ) 1 t r f W 0 A. 1 + (1 t)r f W 0 2

3 Note that this expression implies that the risk taking is increasing in t if r f = 0, i.e., if the riskless asset is money. (c) In this context, we can interpret (1 t)a as the private risk taking, since the investor only receives the interest or bears a loss on (1 t) of each dollar invested in the risky asset. Prove that, if r f is positive, private risk taking decreases when t increases. 9. (The risk aversion oriented class theory by Laont and Kihlstrom.) The labor owners face a choice. On the one hand, they can supply their labor in a competitive labor market at a xed wage s per unit. On the other hand, they can become entrepreneurs (!!) and produce with the production function ỹ = output, ỹ = f(l, x), where x = production shock, which is a random variable taking values in [0, x], where 0 < x <, L = labor used as an input, which is hired in the competitive labor market before the realization of x is known. Assume that f(0, x) = 0 for 0 x x, f(l, 0) = 0 for L 0, and f(l, x) > 0, otherwise. Furthermore, f Lx (L, x) > 0 and f LL (L, x) < 0, if L 0 and 0 < x x, f x (L, x) > 0, if L > 0 and 0 x x. Moreover, assume that every labor owner begins with one unit of labor and an initial wealth equal to W 0. Assume also that entrepreneurial activities consume all of the labor owned by individuals who choose to be entrepreneurs (!!!). Finally, assume that legal restrictions prohibit from taking decisions which imply ( a positive probability of bankruptcy. Thus L must be chosen to be below W0 ) s. Under these restrictions, the entrepreneur i chooses L between 0 and ( W0 ) s to maximize E (ui (W0 + pf (L, x) sl)), where u i is individual i's Bernouilli utility function and p equals the competitively determined output price. Assume that each individual i is risk averse and has a positive marginal utility of income. (a) Prove that if f(l, x) = h(l)x, then every entrepreneur demands a positive amount of labor if and only if ( ) s h (0)E ( x) >. p (b) Prove that if entrepreneur i's utility function is CARA then his demand for labor is a decreasing function of the wage. 3

4 (c) You are asked the same as in (b) but assuming now that the utility function exhibits an Arrow-Pratt measure of absolute risk aversion which is decreasing. (d) Let s i be the wage at which individual i is indierent between being an entrepreneur and working. Prove that if individual 1 has more absolute risk aversion than individual 2, then s 1 < s 2. (c) Now assume that s is set by the market. The marginal entrepreneur is the individual who is indierent between working at the competitive wage and being an entrepreneur. Show that all individuals who are more absolutely risk averse than the marginal entrepreneur enter the labor market, while all individuals who are less absolutely risk averse become entrepreneurs. 10. Suppose that a rm produces a single output with one input, labor. Let f(l) represent the output produced when L units of labor are employed. The function f( ) is strictly concave. Let s be the competitively determined wage. Let p be the competitively determined price of the output. Assume that p is unknown when L is chosen. The rm manager therefore chooses L to maximize the expected utility E (u i ( pf(l) sl)), where the Bernouilli utility function u i is increasing and strictly concave. Let L u be the maximizing choice for L when the utility function is u( ). Dene p u by u (p u f (L (p u )) sl (p u )) = E (u ( pf (L u ) sl u )), where L (p u ) maximizes p u f(l) sl. Thus p u is the certainty equivalent price of output for u. (a) Show that p u < E ( p). (b) Also show that p u1 < p u2 if u 1 is more absolutely risk averse than u 2 in the sense of Arrow-Pratt. 11. Show that if in the previous problem p is subjected to a mean-preserving spread, then p u will decline. 12. Suppose that there are two risky assets with random returns R1 and R 2, respectively. Assume that R 1 and R 2 are independent and have the same mean. We know further that R d 2 = R1 + ε and that R 1 and ɛ are independent. Does this imply that R 2 is more risky than R 1 in the sense of Michael Rotschild and Joseph Stiglitz? Show that if these are the only assets available to a risk averse expected utility-maximizing individual, this individual will invest more in asset 1 than asset Consider an investor who lives for two periods and who receives income from two sources in period 1 but who has no income in period 2. One source of 4

5 rst-period income is inherited wealth W 0. The other rst-period income source is wages. The wage income received is random and is denoted s. This investor can, if he chooses, save and invest part of his inherited wealth W 0. However, this savings decision must be made, unfortunately, before s is known. Thus, saving must be chosen to be between zero and W 0, and all wages are consumed in period 1. Each dollar saved earns a random return R, where R is independent of s. In making his saving choice, the investor maximizes the expected value of the utility function u (c 1, c 2 ) = V 1 (c 1 ) + V 2 (c 2 ), where c i is the consumption in period i. Assume that V i > 0, V i < 0 and that the absolute risk aversion measure of V 1 is decreasing with respect to c 1. Under these conditions, how will investor's optimal savings choice change if s becomes more risky in the sense of Mike and Joe? 14. Assume that u is a Bernouilli utility function of a single real variable. Assume that the marginal utility of income is positive and that u exhibits risk aversion. (a) For a given initial wealth W 0 and random variable z, Pratt dened the risk premium π (W 0, z) by u (W 0 + E ( z) π (W 0, z)) = E (u (W 0 + z)). Prove that if the distribution of z becomes more risky in the sense of Rotschild- Stiglitz (R-S), then the risk premium increases. Interpret this result. the risk not. (b) Now dene µ (W 0, z) by u (W 0 + E ( z)) = E (u (W 0 + z + µ (W 0, z))). (i) Interpret µ (W 0, z). (ii) Can you prove that µ (W 0, z) rises when z becomes more risky in R-S sense? If not, give an example to show why not. (iii) Can you prove that µ (W 0, z) rises if u becomes more absolutely averse in the Arrow-Pratt sense? If not, give an example to show why (c) Pratt also denes the probability premium p (W 0, h) by u (W 0 ) = 1 2 [1 + p (W 0, h)] u (W 0 + h) [1 p (W 0, h)] u (W 0 h). Show that p (W 0, h) is an increasing function of h, assuming W 0 is xed. Interpret the result. 5

6 15. Let us assume an economy with two assets, A and B. The probability distribution of asset A's return is uniform on the interval [m a, m + a], while the probability distribution of asset B's return is uniform on the interval [m b, m + b], with 0 < a < b < m. Prove that all risk averse investors prefer asset A to asset B. You must check that the conditions on integrals in the paper of Rotschild and Stiglitz (1970) are satised. 16. Prove the Proposition (p. 2) of the handout Stochastic Dominance. 6