哈佛大学：《中级微观经济学》（英文版）Lecture 6 Choice Under Uncertainty

Economics 2010a Fa|2002 Edward L. Glaeser Lecture 6

Economics 2010a Fall 2002 Edward L. Glaeser Lecture 6

6. Choice Under Uncertainty Representing Uncertainty: Lotteries and Compound Lotteries b. Axioms of Expected Utility C. The Expected Utility Theory d. Empirical Challenges to Expected Utility Theory-the Paradox Business e. Application: Crime and punishment

6. Choice Under Uncertainty a. Representing Uncertainty: Lotteries and Compound Lotteries b. Axioms of Expected Utility c. The Expected Utility Theory d. Empirical Challenges to Expected Utility Theory—the Paradox Business e. Application: Crime and punishment

n lecture 4, we discussed utility over time, and thought about a utility function of the form u(cl, c2, ..cr) that was defined over future periods of consumption We were particularly interested in the time-separable form of this utility function (c12,.cr)=∑(c and we even assumed that Br=B Uncertainty has many similarities. Think of consumption is differentstates of nature rather than time periods

In lecture 4, we discussed utility over time, and thought about a utility function of the form uc1, c2,.. cT that was defined over future periods of consumption. We were particularly interested in the time-separable form of this utility function uc1, c2,.. cT   t1 T tuct and we even assumed that t  t Uncertainty has many similarities. Think of consumption is different "states of nature" rather than time periods

You can imagine a two states world (it rains, it shines), or an 11 states world(the sum of two dices come up anything from 2-12)or an almost unimaginably large state space in space which actually live Then the utility function can be thought of as being defined over states of the world u(c1, c2,.cs) Where cs is consumption in state s In fact- this is the most important point-if you define all forms of consumption as state contingent consumption-good x in state s, then the mwg discussion in chapters 1-3 can handle uncertainty with no changes whatsoever

You can imagine a two states world (it rains, it shines), or an 11 states world (the sum of two dices come up anything from 2-12) or an almost unimaginably large state space in space which actually live. Then the utility function can be thought of as being defined over states of the world: uc1, c2,.. cS where cs is consumption in state s. In fact– this is the most important point– if you define all forms of consumption as "state contingent consumption" – good x in state s, then the MWG discussion in chapters 1-3 can handle uncertainty with no changes whatsoever

The state-space analogue to time separability is state separability of (c2,cs)=∑ ui(cs In some cases, we prefer the even more extreme assumption that (c.2,cs)=∑p(c Where ps is a constant. We can assume that >Ps=1, so that these constant S- terms can be interpreted as"subjective probabilities"(i.e weights that add to one)

The state-space analogue to time separability is state separability of uc1, c2,.. cS   s1 S uics In some cases, we prefer the even more extreme assumption that: uc1, c2,.. cS   s1 S psucs where ps is a constant. We can assume that  s1 S ps  1, so that these constant terms can be interpreted as "subjective probabilities" (i.e. weights that add to one)

In the case that the value of ps equals the objective probability of state s occurring (for each state s), we call this an expected utility model

In the case that the value of ps equals the objective probability of state s occurring (for each state s) , we call this an expected utility model

We will derive this in a second more formally- but if we believe that assets span-ie there are enough assets so that you can actually by and sell goods in each state of the world then consumers maxImize ∑p(e)+(-∑kc Where ks indicates the cost of consumption in state s, which leads to first order conditions: Tsu(cs)=nps or u(c h-p k The ratio of the marginal utility of consumption in the different states equals the ratio of the prices divided by the ratio

We will derive this in a second more formally– but if we believe that assets span– i.e. there are enough assets so that you can actually by and sell goods in each state of the world, then consumers maximize:  s1 S psucs   w   s1 S kscs where ks indicates the cost of consumption in state s, which leads to first order conditions: su cs  ps or u cz u cs  kzps kspz The ratio of the marginal utility of consumption in the different states equals the ratio of the prices divided by the ratio

of the probabilities Note that for these first order conditions to make sense u()must be concave n particular, if the ratio of the probabilities equals the ratio of the prices-this would be true if all bets were fair- consumption levels are equal across states But more generally- economics tells you to equalize marginal utilities of consumption not total utilities

of the probabilities. Note that for these first order conditions to make sense u. must be concave. In particular, if the ratio of the probabilities equals the ratio of the prices– this would be true if all bets were fair– consumption levels are equal across states. But more generally– economics tells you to equalize marginal utilities of consumption not total utilities

Back to mwG and the more formal treatment MWG Definition 6.B. 1 A simple lottery L is a list L=(pl,pN) with Pn>0 for all n and>pn=l where pn n=1 is interpreted as the probability of an outcome n occurring A simple lottery is a point in the n-1 dimensional simplex, i.e. the set △ P∈界.x4 ∑Pn

Back to MWG and the more formal treatment MWG Definition 6.B.1: A simple lottery L is a list L  p1,...pN  with pn  0 for all n and  n1 N pn  1 where pn is interpreted as the probability of an outcome n occurring. A simple lottery is a point in the N  1 dimensional simplex, i.e. the set   p   N :  n1 N pn  1

MWG Definition 6.B.2 Given K simple lotteries Lk=(pI..PN) k= 1, 2, ...,K, and probabilities ak >0 with ∑αk=1, the compound/ ottery LI,... LK;a1,.ax)is the risky alternative that yields the simple lottery Lk with probability ak for k=1,.K For any compound lottery, LI,... LK;al..ax), the corresponding reduced lottery is the simple lottery k=∑a∑ap2,∑

MWG Definition 6.B.2: Given K simple lotteries Lk  p1 k ,...pN k , k  1, 2, . . . ,K, and probabilities k  0 with k k  1, the compound lottery L1,...LK;1,...K  is the risky alternative that yields the simple lottery Lk with probability k for k  1, . . .K. For any compound lottery, L1,...LK;1,...K  , the corresponding reduced lottery is the simple lottery: Lk   k kp1 k , k kp2 k ,.. k kpN k