<p>Somewhat inconsistent/careless use of indices. In </p>

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

<p>it is assumed that we stand at period $0$ <em>for any $t$ ahead we want to look at</em>.</p>

<p>In </p>

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

<p>it is assumed that we stand at <em>any</em> $t$ and we look one period ahead.</p>

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

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2016-09-23T22:12:50.440Z
Alecos, do you have an idea of which one the author meant?
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2016-09-24T05:47:09.343Z
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.
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2016-09-23T17:45:22.463Z
2016-09-23T17:45:22.463Z
Alecos Papadopoulos
<p><a href="http://www3.nd.edu/~esims1/rbc_notes_2016.pdf" rel="nofollow">In these class notes</a>, page 3, the author defines the stochastic discount factor as $M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$. </p>

<p>I'm trying to find the rationale behind it.</p>

<p>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 $\frac{1}{1+r_t}=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$. </p>

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

<p>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: $I_0\subseteq I_t\implies E(E(M_{t+1}|I_t)|I_0)=E(M_{t+1}|I_0)$ ?</p>

<p>Any help would be appreciated.</p>

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2016-09-22T12:10:18.627Z
@Timo what's given is some lines of text. I'm looking for more formal deduction. Also, I didn't understand your second question.
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2016-09-24T05:43:29.467Z
@Timo the notation is the following: $E_t(X)=E(X|I_t)$, where $I_t$ is our information set at time $t$.
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<macroeconomics><rbc>
The stochastic discount factor $M_t$ in a simple RBC model
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# The stochastic discount factor $M_t$ in a simple RBC model

(Asked by: username_0 on 2016-09-16)

[In these class notes](http://www3.nd.edu/~esims1/rbc_notes_2016.pdf), 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.

**Tags:** macroeconomics, rbc

> **Comments:**
> **username_0**: @Timo what's given is some lines of text. I'm looking for more formal deduction. Also, I didn't understand your second question.  
> **username_0**: @Timo the notation is the following: $E_t(X)=E(X|I_t)$, where $I_t$ is our information set at time $t$.  

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## 1 answers

### Answer by username_1 on 2016-09-23. Score: 0 | Accepted answer

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.

> **Comments:**
> **username_0**: username_1, do you have an idea of which one the author meant?  
> **username_0**: 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.  

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