<p>In this case, the stochastic discount factor appears in equation (7), the intertemporal optimization equation. You can derive (7) by taking the first order conditions of the Lagrangean function with respect to consumption, and then investment (or capital, <code>K_t+1</code>, if you want to substitute out the capital accumulation equation). You then combine these two optimality conditions to yield (7).</p>

<p>The stochastic discount factor, then, is just the ratio of marginal utilities of consumption in period <code>t+1</code> and <code>t</code>. The ratio of prices appears in the stochastic discount factor because this is a nominal model, and you need to discount future nominal payoffs by inflation between time <code>t</code> and <code>t+1</code>.</p>

CC BY-SA 3.0
2017-08-31T06:00:15.220Z
2017-08-31T06:00:15.220Z
James Graham
<p>I can't figure out how the calculation of first order conditions was carried out.</p>

<p><img src="https://i.sstatic.net/ULipR.png" alt="The model and constraints"></p>

<p><img src="https://i.sstatic.net/JCXeh.png" alt="First order conditions"></p>

<p>I can't figure out where the stochastic discount factor came from.</p>

<utility><optimization>
Optimization of Households' utility in " Rule-of-Thumb Consumers and the Design of Interest Rate Rules " (Gali et al., 2004)
Paul
# Optimization of Households' utility in " Rule-of-Thumb Consumers and the Design of Interest Rate Rules " (Gali et al., 2004)

(Asked by: username_0 on 2015-02-01)

I can't figure out how the calculation of first order conditions was carried out.

![The model and constraints](https://i.sstatic.net/ULipR.png)

![First order conditions](https://i.sstatic.net/JCXeh.png)

I can't figure out where the stochastic discount factor came from.

**Tags:** utility, optimization


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## 1 answers

### Answer by username_1 on 2017-08-31. Score: 1

In this case, the stochastic discount factor appears in equation (7), the intertemporal optimization equation. You can derive (7) by taking the first order conditions of the Lagrangean function with respect to consumption, and then investment (or capital, `K_t+1`, if you want to substitute out the capital accumulation equation). You then combine these two optimality conditions to yield (7).

The stochastic discount factor, then, is just the ratio of marginal utilities of consumption in period `t+1` and `t`. The ratio of prices appears in the stochastic discount factor because this is a nominal model, and you need to discount future nominal payoffs by inflation between time `t` and `t+1`.


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