Assignment for Mathematics for Economists Fall 2016

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Dinah Alexander
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1 Due date: Mon. Nov. 1. Reading: CSZ, Ch. 5, Ch. 8.1 Aignment for Mathematic for Economit Fall 016 We now turn to finihing our coverage of concavity/convexity. There are two part: Jenen inequality for concave/convex function and it implication in expected utility theory and the value of information; contrained optimization. Below you will find a hort et of note on the value of information (in the finite etting where thing are eaiet). A to contrained maximization: CSZ 5.8 give more practice with Lagrangean multiplier; CSZ 5.9 fixe firmly in your mind that the multiplier are the derivative of the value function. For twice continuouly differentiable function, CSZ 5.10 take you through a proof of the alternating ign of principal minor of the Jacobean implie concavity reult. We will cover it in lecture with a quick review of the neceary linear algebra. In the Spring emeter, you will be taking Econometric I, remembering, or learning, or re-learning linear algebra hould happen before then. A Very Short Introduction to the Value of Information There i a joint ditribution pdf, q(, ), for a random ignal, S and a random tate X, that i, q(x, ) = P rob(x = x, S = ), everything taking value in finite pace, X for X and S for S. The time line i: S = i oberved for ome S, but the value of X i not oberved, perhap becaue it han t happened yet; the deciion maker pick an action a A, ince it can be, and uually i, a function of, thi i written a(); the value of X = x i oberved; von Neumann-Morgentern utility of u(a, x) i received. The expected utility maximization problem i to pick a function from obervation to action, a(), o a to maximize (1) ( ) E u(a(s), X) = x, u(a(), x)q(x, ). One can olve thi problem by formulating a complete contingent plan, a(), or one can cro that bridge when one come to it, that i, wait until S = ha been oberved and figure out at that point what a() hould be. Let = x q(x, ), rewrite E u = x, u(a(), x)q(x, ) a () q(x,) x u(a(), x). β(x ) := q(x,) i the poterior probability that X = x after S = ha been oberved. A i well-known and eay to verify, if a () olve max a x u(a, x) β(x ) at each with > 0, then a ( ) olve the original maximization problem in ( ). In other word, the dynamically conitent optimality of bet reponding to belief formed by Bayeian updating arie from the linearity of expected utility in probabilitie. The piece of the baic tatitical deciion model are: 1

2 β 0 (x) := q(x, ), the marginal of q on X, known a the prior ditribution a in the probability ditribution that the deciion maker ha prior to oberving and incorporating any information contained in the ignal; := x q(x, ), the marginal of q on S, the ditribution of the ignal; and β( ) = q(,), the poterior ditribution, i.e. the conditional ditribution of X given that S =. The martingale property of belief i the obervation that the prior i the π-weighted convex combination of the poterior, for all x, (3) E (β(x S)) = β(x ) = q(x, ) = q(x, ) = β 0 (x). To put thi more directly, the average of the poterior i the prior, or, thinking of B a a random vector of belief and S a the random ignal, thi i repeating the law of iterated expectation, E (E (B S)) = E B. Belief at, β( ) belong to (X ) and have ditribution π. Re-writing, Information i a ditribution, π ( (X )) having mean equal to the prior. Blackwell (1950, 1951) howed that all ignal tructure are equivalent to uch ditribution. Now, the value of the information, π, i the increae in expected utility that come from receiving ignal and acting optimally in repone. To get at thi, for β (X ), define (4) V (β) = max a A x u(a, x) β(x). Being the maximimum of affine function, V ( ) i convex, thi becaue it epigraph, {(β, r) : V (β) r} i, by the definition, equal to the interection of a et of cloed half-plane. Returning to the increae in expected utility that come from receiving ignal and acting optimally in repone, if there i no information, the expected utility i V (β 0 ), o the value of π i (5) V (β) dπ(β) V (β (X ) 0). An eay obervation: if information never lead to a change in the action picked at β 0, then the value of π i 0. Problem A. Prove the eay obervation jut given. [If you tart working very hard on thi problem, then you are doing it the wrong way.] B. From Chapter 8.1: and C. [Infratructure invetment] X = G or X = B correpond to the future weather pattern, the action are to Leave the infratructure alone or to put in New infratructure, and the ignal,, i the reult of invetigation and reearch into the ditribution of future value of X. The utilitie u(a, x) are given in the following table where c i the cot of the new infratructure.

3 Good Bad L 10 6 N (10 c) (9 c) 1. After you ee a ignal S =, you form your poterior belief β(g ) and β(b ). For what value of β(g ) do you optimally chooe to leave the infratructure a it i? Expre thi inequality a L if c > M, give the value of M in term of β(g ), and interpret. [You hould have omething like leave the old infratructure a it i if cot are larger than expected gain. ]. If c = 0.60, that i, if the new infratructure cot 0% of the damage it prevent, give the et of β(g ) for which it optimal to leave the old infratructure in place. For the ret of thi problem, aume that c = Suppoe that the original or prior ditribution ha β(g) = 0.75 o that, without any extra information, one would put in the N ew infratructure. We now introduce ome ignal tructure. Suppoe that we can run tet/experiment that yield S = G or S = B with P (S = G G) = α 1 and P (S = B B) = γ 1. The joint ditribution, q(, ), i Good Bad G α 0.75 (1 γ) 0.5 B (1 α) 0.75 γ 0.5 Give β( S = G ) and β( S = B ). Verify that the average of the poterior belief i the prior, that i, verify that β( x) = β( ). 4. Show that if α = γ = 1, then the ignal tructure i worthle. 5. Give the et of (α, γ) ( 1, 1 ) for which the information tructure trictly increae the expected utility of the deciion maker. [You hould find that what matter for increaing utility i having a poitive probability of changing the deciion.] D. [The value of repeated independent obervation] Thi problem continue where the previou problem left off. Now uppoe that the tet/experiment can be run twice and that the reult are independent acro the trial. Thu, P (S = ( G, G ) G) = α, P (S = ( G, B ) G) = P (S = ( B, G ) G) = α(1 α), and P (S = ( B, B ) G) = (1 α) with the parallel pattern for B. 1. Fill in the probabilitie in the following joint ditribution q(, ). Verify that the average of poterior belief i the prior belief. ( G, G ) ( G, B ) ( B, G ) ( B, B ) Good. Give β(g ( G, G )), β(g ( G, B )), β(g ( B, G )), and β(g ( B, B )). 3. Show that if α = β = 1, then the ignal tructure i worthle. 4. Give the et of (α, β) ( 1, 1 ) for which the information tructure trictly increae the expected utility of the deciion maker. Bad 3

4 5. Explain why the et i larger here than it wa in the previou problem. E. [A more theoretical problem] We ay that ignal tructure A i unambiguouly better than ignal tructure B if every deciion maker, no matter what their action et and what their utility function are, would at leat weakly prefer A to B. Thi problem ak you to examine ome part of thi definition. Let u uppoe that the random variable take on two poible value, x and x, with probabilitie p > 0 and p = (1 p) > 0, that there are two poible action, a and b, that the ignal take on two value and, that the ignal tructure i given by and that the utilitie are given by x x αp (1 γ)p (1 α)p γp x x a u(a, x) u(a, x ) b u(b, x) u(b, x ) We aume that α > 1 and γ > 1 and we let V (α, γ) denote the maximal expected utility when the ignal are ditributed a above and the utilitie are above. 1. Show that V (α, γ) V ( 1, 1), that i, the information tructure with α > 1 and γ > 1 i unambiguouly better than the tructure with α = γ = 1.. Give utilitie uch that V (α, γ) > V ( 1, 1 ), that i, how that ome deciion maker trictly prefer the information tructure with α > 1 and γ > 1 to the tructure with α = γ = Show that for any utilitie, if α > α 1 and γ > γ 1, then V (α, γ ) V (α, γ). 4. Give utilitie uch that V (α, γ ) V (α, γ) for α > α 1 and γ > γ 1. F. [Another theoretical problem] A way to make information le valuable i to cramble it. Let T be a random ignal that ha ditribution P ( S = ) and uppoe that, conditional on S =, T i independent of X. (Note that thi rule out the more valuable cae of repeated obervation above.) Show that the ignal tructure T i (weakly) le valuable than the ignal tructure S. [Do not tart thi problem until you have found a good definition of conditional independence.] G. The Joint Stock Act of 1844 (in Britain) allowed for the formation of joint-tock companie that were, for the firt time, not government granted monopolie. The Limited Liability Act of 1855 meant that one could loe up to and including one entire invetment in a firm, but no more. It i not urpriing that thi led to more invetment. Let X be a random variable taking both negative and poitive value and having E X = (1 + r) for r > 0 being the expected rate of return, and let Y = max{0, X}. 1. Show that ay firt order tochatically dominate ax for all a 0. 4

5 . Show that it i poible to have a random variable X be rikier than X and have Y := max{0, X } firt order tochatically dominate Y = max{0, X}. 3. Conider the two problem max a 0 [u(w 0 a) + ρe u(w 1 + ax)] and max a 0 [u(w 0 a) + ρe u(w 1 + ay )] where 0 < ρ < 1. Give general condition under which the olution to the econd problem i larger than the olution to the firt problem, and prove that your condition deliver thi concluion. H. The deciion maker ha wealth w 0 at preent and next period wealth i a random variable W 1. Saving can be put into a rik-free aet with rate of return r. Conider the problem max a 0 [u(w 0 a) + ρe u(w 1 + a(1 + r))] where 0 < ρ < Give ufficient condition for a to be larger when W 1 become rikier and prove that your condition deliver thi concluion.. Now uppoe that one can alo invet b 0 in a riky aet with return X 0 per each unit inveted and that E X > (1 + r). Conider the problem max a,b 0 [u(w 0 (a + b)) + ρe u(w 1 + a(1 + r) + bx)] where X i tochatically independent of W 1. Under the condition you found above, can you tell what happen to a and b if X become rikier? What about if W 1 become rikier? I. From Chapter 5.8: , , and J. From Chapter 5.8: 5.8.0, and K. From Chapter 5.8: 5.8., 5.8.3, and L. From Chapter 5.9: 5.9., 5.9.5, 5.9.6, and

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