Homework 1- Micro III - Spring 2007

1. Consider the problem (u, A) and the random variables X, X' given by

where, e.g. 50 = u(a = 1, Y = 1) and 0.3 = P(X = 2|Y = 1).

a. Let (β, 1 - β) ∈ ?(R(Y)) be a distribution over the range of Y . Give the set of β for which argmax_a∈A∫_u(a, y) dβ(y) = {1}, argmax_a∈A∫_u(a, y) dβ(y) = {2}, and argmax_a∈A∫_u(a, y) dβ(y) = {3}.

b. Give the β^X_x and the β^X'_x'. Using the previous problem, give the solutions f^∗_X and f^∗_X' to the problems max_f∈A^S E u(f(X), Y) and max_f∈A^S E u(f(X'), Y ). From these calculate V_(u,A)(X) and V_(u,A)(X').

c. Let M = (β^X_x, P(X = x))_x=1,2 ∈ ?(?(R(Y)) and M' = (β^X'_x', P(X' = x'))_x'=1,2 ∈ ?(?(R(Y)). Show directly that M is not risker than M' and that M' is not riskier than M.

d. Graph, in R², the sets of achievable, Y -dependent utility vectors for the random variables X and X'. That is, graph

F_(u,A)(X) = {(E (u(f(X), Y)|Y = 1), E (u(f(X), Y)|Y = 2)) ∈ R²: f ∈ A^S}

and

F_(u,A)(X') = {(E (u(f(X'), Y)|Y = 1), E (u(f(X'), Y)|Y = 2)) ∈ R²: f ∈ A^S}.

e. If we allow random strategies, that is, pick f according to some q ∈ ?(A^S), then the sets of achievable Y -dependent utility vectors become con (F_(u,A)(X)) and con (F_(u,A)(X')). Show that the same is true if we allow "behavioral strategies," that is, f ∈ ?(A)^S.

f. Show that con (F_(u,A)(X)) ⊄ con (F_(u,A)(X')) and con (F_(u,A)(X')) ⊄ con (F(_u,A)(X)).

g. Give a different problem, (u?, A?) for which X _­(u?,A?) X'.

2. Let X, X' be two signals, and define X'' = (X, X') to be both signals. Let S, S' and S'' be the ranges of the three signals.

a. Show directly that for all (u, A), X''_(u,A) X.

b. Show directly that {(β^X''_x'', P(X'' = x''))_x''∈S'' is riskier than {(β^X_x, P(X = x))_x∈S.

c. Interpret X as the result of a doctor's diagnostic test and X' is the result of a possible additional test. Show that if f^∗_(X,X')(x, x') = f^∗_X(x) for a problem (u, A), then there is no point in doing the extra test.

3. Most of the following results are in the Muller [1] article, which covers and extends the famous Rothschild and Stiglitz [2], [3] articles on increases in risk.

a. If q is a mean preserving spread of p, then q is riskier than p.

b. If X ∼ p (i.e. P(X ∈ A) = p(A)), Y ∼ q, and there is a random variable Z such that E (Z|X) = 0 and X + Z ∼ q, then q is riskier than p.

c. Let p₀ = δ₀, p₁ = ½δ_-1 + ½δ₁. Let p₂ = ½δ_-1 + ¼δ₀ + ¼δ₂, and p₃ = ¼δ_-2 + ¼δ₀ + ¼δ₀ + ¼δ₂ = ¼δ_-2 + ½δ₀ + ¼δ₂. Continuing in this fashion, p₄ = 1/8δ_-4 + 6/8δ₀ + 1/8δ₄, and so on.

i. Show that p₁ is a mean preserving spread of p₀.

ii. Show that p_k+1 is a mean preserving spread of p_k.

iii. Show that p_k →w p₀.

d. The previous problem showed that a sequence can become riskier and riskier and still converge to something that is strictly less risky. Show that this cannot happen if p_k([a, b]) ≡ 1 for some compact interval [a, b]. Specifically, show that if p_k([a, b]) ≡ 1, for all k, p_k+1 is riskier than p_k, and p_k → q, then q is riskier than all of the p_k.

4. In each time period, t = 1, . . ., a random wage offer, X_t ≥ 0, arrives. The X_t are iid with cdf F. The problem is which offer to accept. If offer X_t = x is accepted, utility is β^tu(x) where 0 < β < 1, and u: R₊ → R is strictly monotonic, concave, and ∫u(x) dF(x) < ∞. A "reservation wage" policy is one that accepts all offers of x or above for some x.

a. Show that the optimal policy is a reservation wage policy, and give the distribution of the random time until an offer is expected.

b. In terms of u and F, give the expected utility of following a reservation wage policy with reservation wage x.

c. If the offers are, instead, iid Y_t with cdf G and G is riskier than F, then the optimal reservation wage is higher, and the expected utility is also higher.

5. X_a is your random income depending on your action a ≥ 0, understood as money that you spend on stochastically increasing X_a. The distribution of X_a is R_a,c := cQ_a + (1 - c)µ, 0 ≤ c ≤ 1. Here, µ does not depend on a, but, if a > a', then Q_a first order stochastically dominates Qa'. The parameter c is the amount of "control" that you have, c = 0 means you have no control, c = 1, means you have the most possible.

This question asks you how a^∗ depends on c. Intuitively, increasing c ought to increase the optimal action, more control means that your actions have more effect.

Let f(a, c) = Eu(X_a - a) where u is an increasing, concave function. Increases in a pull down X_a - a, hence u(X_a - a), by increasing the direct cost, but increase X_a - a by stochastically increasing X_a.

a. Show that f(·, ·) is not, in generaly, supermodular.

b. Suppose that f(a, c) is smooth, that we can interchange integration and differentiation, and that the optimum, a^∗(c) is differentiable. The f_a := ∂f/∂a is equal to

f_a = - ∫u'(x - a)dµ(x) + cd/da[messy term with Q_a and µ].

We let m = [messy term with Q_a and µ].

i. Show that if f_a(a, c) = 0, then ∂_m(a, c)/∂_a > 0.

ii. Show that f_a,c := ∂²f/∂a∂c = ∂m/∂a > 0.

iii. Show that da^∗(c)/dc > 0.

References-

1. Alfred Muller, Comparing risks with unbounded distributions, J. Math. Econom. 30 (1998), no. 2, 229-239. MR MR1652641 (99m:90049)

2. Michael Rothschild and Joseph E. Stiglitz, Increasing risk. I. a definition, J. Econom. Theory 2 (1970), 225-243. MR MR0503565 (58 #20284a)

3. Increasing risk. II. Its economic consequences, J. Econom. Theory 3 (1971), 66-84. MR MR0503567 (58 #20284c).