Math help

[Cheezdoodles]

If you have a first order condition:

E([1+R]p^t[U'[Ct+1]])=U'[Ct]

Note: p^t is some constant discount factor, U' denotes Utility partially derivated on [Ct] or [Ct+1] ) (marginal utility in time period t and time period t+1.

My professor claims that you can do this:
Divide by U'[Ct] on both sides, and rewrite to:

E[1+R]S[t+1]=1 where S[t+1]= p^t[U'[Ct+1]]/U'[Ct]

However, my intuition says that this is illegal, as U'[Ct] was not in expectations form to begin with, and he cannot just write it in, nor is it possible to remove the expectations of S[t+1] unless S[t+1]is constant, which it is not , as he calls it the stochastic discount factor.

Expectations of two variables that are not constant: E(x)E+cov(x,y)

Is it "legal" to do what he does? If so why?

 

[V3]

However, my intuition says that this is illegal, as U'[Ct] was not in expectations form to begin with,

The first order condition stated that it is equal to the expectation of R and t.

and he cannot just write it in, nor is it possible to remove the expectations of S[t+1] unless S[t+1]is constant, which it is not , as he calls it the stochastic discount factor.

If I remember right, as long as R and t are independent the expected value operator is multiplicative.

 

[Cheezdoodles]

Erm, i should have made the note that T+1 is just time subscript

 

[V3]

C is the random variable ?

If (1+R) is the only random variable that the expected value operator worked on, you can treat the rest as constant. Because the operator only work on random variable. If not than the random variables need to be independent.

This is what ? finance ? economy ?

 

[Cheezdoodles]

Its some bs finance theory called consumtion capm.

Problem is that in the next step, from E[1+R]S[t+1]=1

He uses the expected value rule for random variables E[XY]=E[X]E[Y] + cov.

E[1+R]S[t+1]=E[St]+ E[St]E[rSt] + cov [R,St]


Which would imply that the U'[c] inside S are random.

This is what confuses me, as if the utility functions are not constant, he cannot divide by the utility on the left hand side and create S, as the left hand side does not have an expectation, wheras the right hand side does.

 

[V3]

In E([1+R]p^t[U'[Ct+1]]) = U'[Ct]

U'[Ct] is the result of expected value of E([1+R]p^t[U'[Ct+1]]), so you can treat it as constant. It is all the bits inside the expected value operator that you need to consider which is the random variable.

So for

E([1+R]p^t[U'[Ct+1]]) = U'[Ct]

Let X = 1+R and Y = p^t[U'[Ct+1]] and k = U'[Ct] where X,Y are random variables, etc and k is a constant.

=> E[XY] = k
=> E[X]E[Y] + cov[X, Y] = k (in your case, it seems cov[X, Y] = 0 ie X and Y are independent)
=> E[X]E[Y] = k
=> E[X]E[Y]/k = 1 as long as k is not zero.
Let E = E[Y]/k
=> E[X]E = 1

Expected value cannot equate to a random variable. Because expected value doesn't return a random variable. It's an expected value, it'll be pretty useless if it is random.