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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..

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    $\begingroup$ 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. $\endgroup$ – Steven Landsburg Dec 30 '14 at 2:01
  • $\begingroup$ oh right. Thanks. I forgot that $x$ is related to intertemporal choice. haha. $\endgroup$ – Kamster Dec 30 '14 at 2:26
  • $\begingroup$ @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. $\endgroup$ – FooBar Apr 13 '15 at 20:37
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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), $\dot x/x=\gamma$, 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.

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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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