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: s1 S psucs w s1 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 n1 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 : n1 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