<p>A practical or mathematical bound? For a practical bound, people have argued that very large values of $\rho$ are impossible to reconcile with non-financial risk taking. Indeed, some values of $\rho$ may not be compatible with crossing the street. </p>

<p>Perhaps that's a bit glib, but consider <a href="http://web.utk.edu/~wneilson/EcLett-Calibration.pdf" rel="nofollow">Neilson and Winter (2001)</a> which find that CRRA utility and observed wage premia for dangerous jobs are compatible with values of $\rho$ such that  $0&lt;\rho&lt;&lt;1$. They then point out that if $\rho$ were very very large and the model properly specified then the dangerous job wage premium would also be large or alternatively (and implausibly) that people place a very, very low value on their own lives (like $3,000). </p>

<p>Generally speaking, data and models implying very high values of $\rho$ like a basic equity premium setup and the relative volatility of equity returns and consumption) are often taken as signs of model miss-specification instead of alternative measures of risk preferences. </p>

<p>While we can't talk about globally infinite risk aversion, we can consider what happens as If you take the limit $\rho\rightarrow\infty$ around a particular value of wealth $W_0$. As it goes to infinity you get a step-function utility function with $U(W) = -\infty, W &lt; W_0$ and $U(W) = C, W \ge W_0$. </p>

CC BY-SA 3.0
2014-12-01T23:03:47.560Z
Bounds derived from economic reasoning would be interesting. At ρ→∞ marginal utility is decreasing to fast to make sense.
Vayu
CC BY-SA 3.0
2014-12-02T03:42:49.237Z
I've been mulling over how one might quantify "some values of $\rho$ are not compatible with crossing the street." That is, write down a model, solve it for some value of $\rho$, and show that it is highly implausible with respect to empirical predictions (crossing the street is an empirical prediction). Or is that simply the far bounds on something like the grid search in Carroll's DP notes? (See Figure 24 [here](http://www.econ2.jhu.edu/people/ccarroll/solvingmicrodsops/), for example.)
CompEcon
CC BY-SA 3.0
2014-12-02T11:49:27.990Z
I tried to address CompEcon's question this in more detail in the body of my answer. Vayu, I'm not sure that intuition holding. In CRRA utility, a 1% change in consumption always gives the same percent increase in utility for all values of parameter $\rho$.
BKay
CC BY-SA 3.0
2014-12-01T21:51:11.717Z
2014-12-02T17:49:45.703Z
2014-12-02T17:49:45.703Z
BKay
<p>In the benchmark model of representative household intertemporal utility maximization problem (the "Ramsey model", see for example, <a href="http://down.cenet.org.cn/upfile/8/200751171644184.pdf" rel="nofollow">Barro &amp; Sala-I-Martin (2004), Economic Growth (2n ed)</a>, <strong>ch. 2</strong>, using a Constant Relative Risk Aversion utility function</p>

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

<p>results in the optimal rule for per capita consumption growth
$$\frac {\dot c}{c} = \frac 1{\theta} (r-\rho)$$</p>

<p>where $r$ is the net rate of return on assets and $\rho$ the rate of pure time preference.</p>

<p>$\theta$ is the coefficient of relative risk aversion (for which the OP uses $\rho$ in the question).</p>

<p>Benchmark values for post-war western economies are $r=0.06$ and $\rho=0.02$. So we would have</p>

<p>$$\frac {\dot c}{c} = \frac {0.04}{\theta}$$</p>

<p>Therefore for $\theta =2$ we get $\frac {\dot c}{c}=0.02$ which is consistent with historical data, while for $\theta = 10$ we would get $\frac {\dot c}{c}=0.004$.  </p>

<p>In another aspect in this model, the coefficient of relative risk aversion is inversely related to the gross savings rate, so a very-high value of $\theta$ would imply a counter-factually low steady-sate savings rate (where "gross savings rate" should be interpreted broadly, as any form of deferred consumption, including investments in human capital, something that also implies that the concept of "capital" and its share should be extended to include "human capital" also).</p>

CC BY-SA 3.0
2014-12-03T05:00:22.267Z
2016-10-01T21:37:53.247Z
2016-10-01T21:37:53.247Z
Alecos Papadopoulos
<p>For instance, Cagetti (2003) estimates ρ > 2 by targeting the median wealth of households,
Gourinchas and Parker (2002) find ρ &lt; 2 by targeting the mean consumption of households,
Chetty (2006), using the effects of wage changes on labor supply, finds that ρ &lt; 2.</p>

<p>Marginal utility of consumption is decreasing very fast when ρ > 2, still you can find in macro or finance even ρ = 10. In experimental or in labor ρ is close to 0.</p>

<p>Is there any bound on this coefficient?</p>

<utility>
Is risk aversion bounded in CRRA utility?
Vayu
# Is risk aversion bounded in CRRA utility?

(Asked by: username_0 on 2014-12-01)

For instance, Cagetti (2003) estimates ρ > 2 by targeting the median wealth of households,
Gourinchas and Parker (2002) find ρ < 2 by targeting the mean consumption of households,
Chetty (2006), using the effects of wage changes on labor supply, finds that ρ < 2.

Marginal utility of consumption is decreasing very fast when ρ > 2, still you can find in macro or finance even ρ = 10. In experimental or in labor ρ is close to 0.

Is there any bound on this coefficient?

**Tags:** utility


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## 2 answers

### Answer by username_1 on 2014-12-01. Score: 4

A practical or mathematical bound? For a practical bound, people have argued that very large values of $\rho$ are impossible to reconcile with non-financial risk taking. Indeed, some values of $\rho$ may not be compatible with crossing the street.

Perhaps that's a bit glib, but consider [Neilson and Winter (2001)](http://web.utk.edu/~wneilson/EcLett-Calibration.pdf) which find that CRRA utility and observed wage premia for dangerous jobs are compatible with values of $\rho$ such that $0<\rho<<1$. They then point out that if $\rho$ were very very large and the model properly specified then the dangerous job wage premium would also be large or alternatively (and implausibly) that people place a very, very low value on their own lives (like $3,000).

Generally speaking, data and models implying very high values of $\rho$ like a basic equity premium setup and the relative volatility of equity returns and consumption) are often taken as signs of model miss-specification instead of alternative measures of risk preferences.

While we can't talk about globally infinite risk aversion, we can consider what happens as If you take the limit $\rho\rightarrow\infty$ around a particular value of wealth $W\_0$. As it goes to infinity you get a step-function utility function with $U(W) = -\infty, W < W\_0$ and $U(W) = C, W \ge W\_0$.

> **Comments:**
> **username_0**: Bounds derived from economic reasoning would be interesting. At ρ→∞ marginal utility is decreasing to fast to make sense.  
> **username_2**: I've been mulling over how one might quantify "some values of $\rho$ are not compatible with crossing the street." That is, write down a model, solve it for some value of $\rho$, and show that it is highly implausible with respect to empirical predictions (crossing the street is an empirical prediction). Or is that simply the far bounds on something like the grid search in Carroll's DP notes? (See Figure 24 [here](http://www.econ2.jhu.edu/people/ccarroll/solvingmicrodsops/), for example.)  
> **username_1**: I tried to address username_2's question this in more detail in the body of my answer. username_0, I'm not sure that intuition holding. In CRRA utility, a 1% change in consumption always gives the same percent increase in utility for all values of parameter $\rho$.  

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### Answer by username_3 on 2014-12-03. Score: 2

In the benchmark model of representative household intertemporal utility maximization problem (the "Ramsey model", see for example, [Barro & Sala-I-Martin (2004), Economic Growth (2n ed)](http://down.cenet.org.cn/upfile/8/200751171644184.pdf), **ch. 2**, using a Constant Relative Risk Aversion utility function

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

results in the optimal rule for per capita consumption growth
$$\frac {\dot c}{c} = \frac 1{\theta} (r-\rho)$$

where $r$ is the net rate of return on assets and $\rho$ the rate of pure time preference.

$\theta$ is the coefficient of relative risk aversion (for which the OP uses $\rho$ in the question).

Benchmark values for post-war western economies are $r=0.06$ and $\rho=0.02$. So we would have

$$\frac {\dot c}{c} = \frac {0.04}{\theta}$$

Therefore for $\theta =2$ we get $\frac {\dot c}{c}=0.02$ which is consistent with historical data, while for $\theta = 10$ we would get $\frac {\dot c}{c}=0.004$.

In another aspect in this model, the coefficient of relative risk aversion is inversely related to the gross savings rate, so a very-high value of $\theta$ would imply a counter-factually low steady-sate savings rate (where "gross savings rate" should be interpreted broadly, as any form of deferred consumption, including investments in human capital, something that also implies that the concept of "capital" and its share should be extended to include "human capital" also).


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