程序代写 Choice Under Uncertainty

Choice Under Uncertainty
() Expected Utility 1 / 32

Consider driving from A to B for an appointment.

Copyright By PowCoder代写 加微信 powcoder

If you get there in time with probability π1 = 95%, payo§ is x.
If there is a tra¢ c jam (with probability π2 = 4.8%) you are late and payo§ is 0.
If an accident (with probability π3 = 0.2%) you will be late AND will have to repair your car, so the payo§ is y.
Represent this as a lottery [+x,π1;0,π2;y,π3]
More generally, a lottery is
[x1,π1;…;xS,πS] withπs 0and ∑πs =1
The prizes (x1 , …, xS ) are real numbers (amounts of money). () Expected Utility

Preferences over lotteries
Let L denote the set of all lotteries.
We assume that agents have preferences over this set; that is, just as in ordinal utility theory an agent has a preference relation  on L that satisÖes the usual assumptions of ordinal utility theory (asymmetric, transitive, and continuous).
By these assumptions we can represent such preferences over lotteries with a continuous utility function V: L ! R so that
LL0 ()V(L) 0,a > 0 =) V ([π1,x1;π2,x2]) < V ([π1,x1 +a;π2,x2]) Assume that agents dislike risk or are risk-averse. () Expected Utility Risk aversion Let E [L] denote the expected value of the prize of lottery L S E[L]=∑πsxs =π
x. Consider the degenerate lottery [E [L] , 1] that pays E [L] with certainty. An agent is risk neutral if V (L) = V ([E[L],1]), i.e., the risk in L (the variation of payo§s between states) is irrelevant to the agentó he cares only about the expectation of the prize. The agent is risk averse if V (L) < V ([E[L],1]). Such agent is willing to give up some wealth on average in order to avoid the randomness of the prize in L. () Expected Utility Certainty equivalent and risk premium Let V be some utility function on L, and let L be some lottery with E[L]. The certainty equivalent of L under V satisiÖes V (L) = V ([CE(L),1]) CE (L) is the level of (non-random) wealth that yields the same utility as the lottery L. If preferences over lotteries are monotonic and satisfy risk aversion, each lottery has precisely one certainty equivalent. The risk premium is the di§erence between the expected prize of the lottery, and its certainty equivalent, RP(L) = E[L]CE(L). Risk aversion is equivalent to the assumption that the risk premium is positive, E [L] > CE (L).
() Expected Utility 5 / 32

Geometric interpretation
Suppose S = 2 and Öx (π1,π2) = (0.4,0.6).
A lottery [x1, 0.4; x2, 0.6] is then just a point in (x1, x2) space.
The lotteries on the 45 degree line (x1,x2) = (z,z) are risk-free because their prize is non-random .
The lotteries with expected prize z are located on a straight line that is orthogonal to the vector of probabilities π.
For such lotteries
E[L]=π
x =π1×1 +π2×2 =z =π1z+π2z
so that π1 (x1 z) + π2 (x2 z) = π
(x z) = 0, hence vectors π and (x z ) are orthogonal.
() Expected Utility 6 / 32

Geometric interpretation
() Expected Utility 7 / 32

Geometric interpretation
By monotonicity of preferences the indi§erence curves must be strictly decreasing from left to right.
Moreover, risk aversion implies that
E[L]>z foranyLsuchthatV(L)=V([z,1]).
The indi§erence curve that goes through (z,z) lies to the right and above the line where E[L] = z.
The indi§erence curve is tangent to the E [L] = z line at (z , z ). Then the gradient
rV ([z,π1;z,π2]) = λπ.
is collinear to the vector of probabilities at this point.
() Expected Utility

Geometric interpretation
() Expected Utility 9 / 32

Risk-aversion vs. Convexity
Take some lottery L (not on the x2 = x1 line) and consider the indi§erence curve to which it belongs.
Given monotonicity and continuity, this indi§erence curve cuts through the x2 = x1 line at exactly one point, CE (L) .
Risk aversion per se does not imply convexity of the indi§erence curves.
It implies only that the indi§erence curve is above E [L] = z line, so that RP (L) > 0.
Likewise, convexity of indi§erence curves does not imply risk aversion.
The von NeumannñMorgenstern expected utility representation requires assumptions which imply convexity of the indi§erence curves over L.
() Expected Utility 10 / 32

Certainty equivalent and risk premium
() Expected Utility 11 / 32

The von NeumannñMorgenstern representation
Exploits the additional structure given by the probabilities Function v is the Expected Utility representation of V if
S V([π1,×1;…;πS,xS])= ∑πsv(xs).
The above requires restrictions on V, i.e., the axioms (assumptions)
that V needs to satisfy.
1 State independence. All that matters is the statistical distribution of
outcomes. The state itself is just a label with no signiÖcance per se.
2 Consequentialism. Only the consequences (the distribution over the prizes) matter, and not the way we get there.
3 Irrelevance of common alternatives
L1  L2 () [L1,1π3;L3,π3]  [L2,1π3;L3,π3].
() Expected Utility 12 / 32

Allais Paradox
Two choices of two lotteries
Lottery1:2mwith Pr=1 versus
Lottery2:2mwith Pr=0.89, 10mwith Pr=0.1, 0with Pr=0.01 and
Lottery3:2mwith Pr=0.11, 0with Pr=0.89 versus
Lottery4:10mwith Pr=0.1, 0with Pr=0.9
() Expected Utility 13 / 32

Allais Paradox
The usual choices are
1 preferred to 2 and 4 preferred to 3
For expected utility representation there should exist v (2) , v (10) , v (0) so that
v (2) > 0.89v (2)+0.1v (10)+0.01v (0) () 1  2 and
0.11v (2) + 0.89v (0) < 0.1v (10) + 0.9v (0) () 4  3 but this requires 0.11v (2) 0.11v (2) 0.1v (10)+0.01v (0) () 1  2 0.1v (10)+0.01v (0) () 4  3 Expected Utility Allais Paradox Since Allais Paradox obviously violates the Expected Utility representation, one of the above axioms must be violated. Problem 2 in the homework digs into this deeper and asks which one. We are going to IGNORE this and proceed with the Expected Utility anyway. Convexity of the indi§erence curves in (x1,x2) space is more important. Under Expected Utility representation the indi§erence curves of a risk-averse agent are convex. () Expected Utility Illustration of the proof Consider a binary lottery [xlow , π; xhigh , 1 π]. Calculate v (xlow ) and v (xhigh ). The points (xlow,v(xlow)), (E[x],E[v(x)]), and (xhigh,v(xhigh)) lie on one straight line. Expected utility of this lottery E[v (x)] = πv(xlow)+(1π)v(xhigh). The certainty equivalent v (CE(x)) = E[v (x)] hence CE(x) = v1 (E[v (x)]). Since for a risk-averse agent CE(x) < E[x] this requires E[v (x)] < v (E[x]) agent is risk averse if and only if v is a strictly concave function. () Expected Utility Jensenís inequality: the strict convex combination of two values of a function is strictly below the graph of the function if and only if the function is concave. () Expected Utility 17 / 32 Risk aversion and concavity The risk premium is positive (and thus the agent is risk averse) if and only if v is strictly concave. If v has no curvature (v00 (x) = 0, i.e. v is an a¢ ne function). Then CE (x) = E [x] and the risk premium is zero. This is the utility function of a risk-neutral agent If the agent is risk-averse, v (x) is a concave function Their weighted sum S V([π1,x1;...;πS,xS])= ∑πsv(xs) is a concave function of (x1 , ..., xS ) . Hence the sets above the indi§erence curves of V are convex sets. () Expected Utility Measures of risk aversion Arrow-Pratt coe¢ cient of absolute risk aversion ARA is a local measure of the degree that an agent dislikes risk A (w) := v00 (w). v0 (w) A (w ) is invariant to a¢ ne transformations of the utility function, i.e., if v and v ̃ are equivalent von NM utility functions then Av (w ) = Av ̃ (w ) for all w . Suppose wX = wY but X ís utility function v is more concave than Y ís, there exists a concave function g such that v(w) = g(v ̃(w)). X will always demand a larger risk premium than Y for exposing himself to the same risk. X is globally more risk averse than Y . But also Av (w ) > Av ̃ (w ) for all w .
() Expected Utility 19 / 32
A does indeed measure the degree of risk aversion

CARAó DARAó IARA
The utility function v has constant absolute risk aversion, or CARA, if A does not depend on wealth,
A0 (w) = 0.
v exhibits decreasing (increasing) absolute risk aversion, or DARA, if
richer are less (more) absolutely risk averse than poorer ones A0 (w) < 0,(>)0.
Consider a lottery in which you may lose $10 with 50% chance. CARA means that a millionaire requires the same payment to enter this lottery as a beggar.
IARA means that the millionaire requires a larger payment than the beggar. (seems not very probable).
Most likely the millionaire will enter this lottery for a signiÖcantly smaller payment than the beggar, which is DARA.
() Expected Utility 20 / 32

CRRAó DRRAó IRRA
The coe¢ cient of relative risk aversion,
R (w ) := wA (w ) .
If R is independent of wealth, the utility function is of constant
relative risk aversion, CRRA.
Similarly, deÖne increasing (IRRA) or decreasing (DRRA) relative risk aversion.
It is less clear what we should expect. Experimental evidence suggests 0 < R(w) < 4 for all w, but is inconclusive on CRRA vs. DRRA vs. IRRA. () Expected Utility 21 / 32 Precautionary saving and prudence An agent is prudent if his optimal saving increases with the amount of uncertainty of his future wealth. The coe¢ cient of absolute prudence P (w) := v000 (w). v00 (w) An agent is prudent if and only if P (w) > 0.
Suppose there is only a risk-free bond with price β, but S = 2, each state equally likely.
The state-contingent endowment of the agent is ω0 today and
(ω1 x,ω1 +x) tomorrow.
x is endowment risk, and because there is only a risk-free bond, this risk cannot be hedged; the agent must bear some of it.
The agent maximizes intertemporal utility by choosing an optimal amount of bonds z,
maxv(ω0 βz)+δ12v(ω1 x+z)+δ12v(ω1 +x+z).
() Expected Utility 22 / 32

Precautionary saving and prudence
βv0(ω0 βz)=δ21v0(ω1 x+z)+δ12v0(ω1 +x+z)
The LHS is the marginal utility today of reducing bond holdings, and the RHS is the discounted expected marginal utility tomorrow of increasing bond holdings.
Recall that v0 > 0 and decreasing (v00 < 0 by risk-aversion) hence the LHS increases with z while the RHS decreases with z. If an increase in x (more uncertainty) increases tomorrowís expected marginal utility, then z should increase to restore the FOC. An increase of x has such e§ect on the RHS if and only if v0 is convex (by Jensenís inequality), i.e. v000 > 0.
Since v00 < 0 the resuting P (w) > 0.
() Expected Utility 23 / 32

Precautionary saving and prudence
w1 x + z w1 + x + z w Figure: Precautionary savings
Expected Utility

What to assume?
Experimental evidence suggest the utility function v (w) that is
1 strictly increasing and strictly concave;
2 DARA (A0 (w) < 0) 3 with 0 < R(w) < 4 for all w. All standard speciÖcations of v belong to HARA, hyperbolic absolute risk aversion class DeÖne absolute risk tolerance T (w) := 1/A(w). A utility function v is of HARA class if T (w ) = a + bw . () Expected Utility 25 / 32 T0(w), is called cautiousness. HARA utility functions are constant cautiousness utility functions, since T 0 (w ) = b. With the above deÖnition for T (w) A(w) = 1 = 1 ARA coe¢ cient T (w ) a + bw R(w) = wA(w)=wa+b1 RRAcoe¢cient HARA utility is DARA if and only if b > 0.
HARAv isCARAifb=0,withA(w)=1/a.
HARAv isDRRAifandonlyifa<0. it is CRRA if a = 0, with R (w) = 1/b. Asaspecialcase,v(w)=ln(w)hasA(w)=w1 =)T(w)=w. This is HARA with a = 0 and b = 1. () Expected Utility 26 / 32 A general HARA utility function ln (w + a) v (w) = aexp(w/a) (a + bw )11/b / (b 1) To check that these functions work as expected, note v0(w)= 1 , v00(w)= 1 , T(w)=a+w topline w + a (w + a)2 v0 (w) = exp(w/a), v00 (w) = 1a exp(w/a), T (w) = a middle v0 (w) = (a+bw)1/b , v00 (w) = (a+bw)11/b , T(w)=a+bw bottomline if b = 0 otherwise. () Expected Utility 27 / 32 Portfolio choice problem Suppose an investor with initial wealth w0 can buy a risky asset a or a riskless asset b. The price of a is q and the price of b is 1. The payo§ from a in state s is rs; the payo§ from b is R in every s. The payo§ from the portfolio (a, b) in state s ys (a, b) = rs a + Rb. The rs are distributed according to distribution π and the investor has vNM utility function ∑πsv(ys (a,b)). () Expected Utility 28 / 32 Portfolio choice problem The investorís decision problem as a constrained optimization problem. ∑ πsv (ys (a,b)) ! max subject to qa+b = w0 sothatb=w0qa ys(a,b) = rsa+Rbforanys. SS Restate this as ∑πsv(Rw0 +(rs Rq)a) = ∑πsv(βs)!max a Rw0 +(rs Rq)a. () Expected Utility Let a (w0 ) denote the investorís holding of the risky asset and let A (w ) be his coe¢ cient of the absolute risk-aversion. We will see that a0(w0)>0ifandonlyifA0(w)<0, foreveryw that is DARA investor holds more of the risky asset when she is wealthier. For IARA investor exactly the opposite is true. The Örst order condition for every s ∑πsv0(βs)(rs Rq)=0 s=1 di§erentiate this with respect to w0 ∑πsv00(βs)(rs Rq)R+∑πsv00(βs)(rs Rq)2
a0(w0)=0 () Expected Utility Portfolio choice problem Solve this for a0 (w0) a0 (w0) = ∑Ss=1 πsv00 (βs)(rs Rq)R ∑Ss=1 πsv00 (βs)(rs Rq)2 Recall that for a risk averse investor v00 < 0, hence sign of a0 (w0) coincides with the sign of ∑Ss=1 πsv00 (βs)(rs Rq). Recall also A(w) = v00 (w) so that v0 (w) ∑πsv00 (βs)(rs Rq) = ∑πsv0 (βs)(rs Rq)
A(βs). () Expected Utility 31 / 32 Portfolio choice problem (rs Rq)
A(Rw0) > (rs Rq)
A(Rw0 +(rs Rq)a) = (rs Rq)
A(βs)
ifandonlyifA0(w) < 0. Hence ∑πsv0(βs)(rs Rq)
A(βs)>A(Rw0)
∑πsv0(βs)(rs Rq)=0
The equality follows from the Örst order condition (1). Thus
a0(w0)>0ifandonlyifA0(w)<0i.e. forDARAagent () Expected Utility 32 / 32 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com