<p>Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable <em>growth rate</em> (see also <a href="https://economics.stackexchange.com/questions/460/why-is-economic-growth-measured-exponentially-rather-than-linearly/464#464">this post</a>), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time.</p>

<p>So one would want a model that admits a constant growth rate at its steady-state. In the benchmark deterministic/continuous time "representative household" model, the Euler equation  takes the form</p>

<p>$$r = \rho - \left(\frac {u''(c)\cdot c}{u'(c)}\right)\cdot \frac {\dot c}{c}$$</p>

<p>This is the optimal rule for the growth rate of consumption. The rate of pure time preference $\rho$ is assumed constant. The interest rate $r$ has its own way to become constant at the steady state. So in order to obtain a constant consumption growth rate at the steady state, we want the term</p>

<p>$$\left(\frac {u''(c)\cdot c}{u'(c)}\right)$$
to be constant too. The <strong>Constant Relative Risk Aversion (CRRA)</strong> utility function satisfies exactly this requirement:</p>

<p>$$u(c) = \frac {c^{1-\sigma}}{1-\sigma} \Rightarrow u'(c) = c^{-\sigma} \Rightarrow u''(c) = -\sigma c^{-\sigma-1}$$</p>

<p>So</p>

<p>$$\frac {u''(c)\cdot c}{u'(c)} = \frac {-\sigma c^{-\sigma-1} \cdot c}{c^{-\sigma}} = -\sigma $$
and the Euler equation becomes</p>

<p>$$\frac {\dot c}{c} = (1/\sigma)\cdot (r-\rho)$$</p>

<p>Barro &amp; Sala-i-Martin (2004, 2n ed.), extend the required form of the utility function when there is also leisure-labor choice (ch. 9 pp 427-428).<br>
These fundamental property extends to the case of stochastic/discrete time.</p>

<p><strong>XXXX</strong></p>

<p>To compare, if we have specified a <strong>Constant Absolute Risk Aversion (CARA)</strong> form, we would have </p>

<p>$$u(c) = -\alpha^{-1}e^{-\alpha c} \Rightarrow u'(c) = e^{-\alpha c}\Rightarrow u''(c) = -\alpha e^{-\alpha c}$$ and the Euler equation would become</p>

<p>$$\dot c = (1/\alpha)\cdot (r-\rho)$$</p>

<p>i.e.here we would obtain a constant steady-state growth in the <em>level</em> of consumption (and so a diminishing growth <em>rate</em>). </p>

CC BY-SA 3.0
2014-12-30T03:52:24.583Z
2016-10-01T21:43:32.323Z
2017-04-13T12:51:38.253Z
Alecos Papadopoulos
<p>As title says, why is CRRA utility often used in macroeconomics DSGE model? That is, the form of</p>

<p>$$u(c_t) = \frac{c_t^{1-\sigma}}{1-\sigma}$$</p>

<p>I cannot find any theoretical background around this..... After all, there is often no risk involved in DSGE models..</p>

CC BY-SA 3.0
2014-12-30T02:01:42.767Z
There might be DSGE models without risk, but I doubt very much that there are DSGE models without intertemporal choice --- that's what the D is for.
Steven Landsburg
CC BY-SA 3.0
2014-12-30T02:26:54.077Z
oh right. Thanks. I forgot that $x$ is related to intertemporal choice. haha.
Kamster
CC BY-SA 3.0
2015-04-13T20:37:05.930Z
@StevenLandsburg You have a point, but so does he: CRRA comes from matching inter temporal elasticities (i.e. growth facts), not from matching risk preferences. Hence, even without the D, we'd still see this preference choice.
FooBar
<macroeconomics><utility>
Why is CRRA utility often used in macroeconomics DSGE model?
Kamster
# Why is CRRA utility often used in macroeconomics DSGE model?

(Asked by: username_0 on 2014-12-30)

As title says, why is CRRA utility often used in macroeconomics DSGE model? That is, the form of

$$u(c\_t) = \frac{c\_t^{1-\sigma}}{1-\sigma}$$

I cannot find any theoretical background around this..... After all, there is often no risk involved in DSGE models..

**Tags:** macroeconomics, utility

> **Comments:**
> **username_1**: There might be DSGE models without risk, but I doubt very much that there are DSGE models without intertemporal choice --- that's what the D is for.  
> **username_0**: oh right. Thanks. I forgot that $x$ is related to intertemporal choice. haha.  
> **username_2**: @username_1 You have a point, but so does he: CRRA comes from matching inter temporal elasticities (i.e. growth facts), not from matching risk preferences. Hence, even without the D, we'd still see this preference choice.  

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## 1 answers

### Answer by username_3 on 2014-12-30. Score: 9 | Accepted answer

Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable *growth rate* (see also [this post](https://economics.stackexchange.com/questions/460/why-is-economic-growth-measured-exponentially-rather-than-linearly/464#464)), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time.

So one would want a model that admits a constant growth rate at its steady-state. In the benchmark deterministic/continuous time "representative household" model, the Euler equation takes the form

$$r = \rho - \left(\frac {u''(c)\cdot c}{u'(c)}\right)\cdot \frac {\dot c}{c}$$

This is the optimal rule for the growth rate of consumption. The rate of pure time preference $\rho$ is assumed constant. The interest rate $r$ has its own way to become constant at the steady state. So in order to obtain a constant consumption growth rate at the steady state, we want the term

$$\left(\frac {u''(c)\cdot c}{u'(c)}\right)$$
to be constant too. The **Constant Relative Risk Aversion (CRRA)** utility function satisfies exactly this requirement:

$$u(c) = \frac {c^{1-\sigma}}{1-\sigma} \Rightarrow u'(c) = c^{-\sigma} \Rightarrow u''(c) = -\sigma c^{-\sigma-1}$$

So

$$\frac {u''(c)\cdot c}{u'(c)} = \frac {-\sigma c^{-\sigma-1} \cdot c}{c^{-\sigma}} = -\sigma $$
and the Euler equation becomes

$$\frac {\dot c}{c} = (1/\sigma)\cdot (r-\rho)$$

Barro & Sala-i-Martin (2004, 2n ed.), extend the required form of the utility function when there is also leisure-labor choice (ch. 9 pp 427-428).  
These fundamental property extends to the case of stochastic/discrete time.

**XXXX**

To compare, if we have specified a **Constant Absolute Risk Aversion (CARA)** form, we would have

$$u(c) = -\alpha^{-1}e^{-\alpha c} \Rightarrow u'(c) = e^{-\alpha c}\Rightarrow u''(c) = -\alpha e^{-\alpha c}$$ and the Euler equation would become

$$\dot c = (1/\alpha)\cdot (r-\rho)$$

i.e.here we would obtain a constant steady-state growth in the *level* of consumption (and so a diminishing growth *rate*).


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