# Why is CRRA utility often used in macroeconomics DSGE model?

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..

• 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. Commented Dec 30, 2014 at 2:01
• oh right. Thanks. I forgot that $x$ is related to intertemporal choice. haha. Commented Dec 30, 2014 at 2:26
• @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. Commented Apr 13, 2015 at 20:37

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.

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).