# The stochastic discount factor $M_t$ in a simple RBC model

In these class notes, page 3, the author defines the stochastic discount factor as $M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$.

I'm trying to find the rationale behind it.

A cash flow at period t, is usually discounted by $\prod^t_{i=0} \frac{1}{1+r_i}$ (if I'm not mistaken). I was thinking of using equation (5) - in the notes - where we have that $\frac{1}{1+r_t}=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$.

However, since $\frac{E_{t-1}(u'(C_{t}))}{u'(C_{t})}\neq 1$, I'm not exactly sure how to proceed.

Also, just below equation (10), the author states that $E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$. How is this possible? Isn't the tower property exactly the inverse: $I_0\subseteq I_t\implies E(E(M_{t+1}|I_t)|I_0)=E(M_{t+1}|I_0)$ ?

Any help would be appreciated.

• It sounds to me as though the rationale is given right below that equation? "The firm discounts by this ... equivalent value of the future utils." (p. 3). Is there some part of this that you think needs further justification? As for the 2nd equation, could you explain a bit further where you think iteration comes in? – user7935 Sep 20 '16 at 14:46
• @Timo what's given is some lines of text. I'm looking for more formal deduction. Also, I didn't understand your second question. – An old man in the sea. Sep 22 '16 at 12:10
• I'm afraid I can't help you with a formal deduction on that. I may be wrong, but it sounds like it's something new rather than something following from the previous. With regards to the second equation you invoke the tower property, relevant for iterations of expectations. I don't understand where the iteration of expectations is supposed to be happening, perhaps you could explain a bit further? Also, I don't actually see the cited equation, but I do see $E_tM_{t+1} = β_{t+1} E_tu'_0(C_{t+1})/u'_0(C_0)$. Is that the one you are referring to? – user7935 Sep 23 '16 at 13:16
• Actually, that should be $β^{t+1}$ rather than $β_{t+1}$ – user7935 Sep 23 '16 at 13:31
• @Timo the notation is the following: $E_t(X)=E(X|I_t)$, where $I_t$ is our information set at time $t$. – An old man in the sea. Sep 24 '16 at 5:43

Somewhat inconsistent/careless use of indices. In

$$M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$$

it is assumed that we stand at period $0$ for any $t$ ahead we want to look at.

In

$$E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$$

it is assumed that we stand at any $t$ and we look one period ahead.

The two expressions don't give the same formulae for any given $t$, and there lies the problem/confusion.

• Alecos, do you have an idea of which one the author meant? – An old man in the sea. Sep 23 '16 at 22:12
• And by the way could you also indicate some references for this subject. I'm interested not only in the theoretical part, but also from a more empirical perspective, with a bit of insight on calibration problems that these models may have. – An old man in the sea. Sep 24 '16 at 5:47