<p>Recall that in the RCK model households choose their consumption and savings so as to maximise current and future utility (i.e. future generations). Then the objective function is given by</p>
<p><span class="math-container">$$ U = \int_{0}^{\infty} u[c(t)]e^{nt}e^{-\rho t}dt$$</span></p>
<p>The RCK model assumes the functional form of utility</p>
<p><span class="math-container">$$ u[c(t)] = \frac{c(t)^{1-\theta}-1}{1 - \theta} $$</span></p>
<p>Note that this function is increasing and concave, and that assuming that <span class="math-container">$\rho &gt; n$</span>, it ensures that lifetime utility does not diverge. This is a very practical assumption as it ensures that households would not have infinite lifetime utility and therefore allows for a well defined solution.</p>
<p>This is is called <strong>constant intertemporal elasticity of substitution (CIES) or constant relative risk aversion (CRRA) utility function</strong> (note that this isoelastic utility function is the only class of utility functions with constant relative risk aversion).</p>
<p>You can check for this by computing the risk aversion coefficient, <span class="math-container">$\frac{1}{\sigma}$</span>, where <span class="math-container">$\sigma$</span> is the intertemporal elasticity of substitution (measures the willingness on the part of the consumer to substitute future consumption for present consumption) and is defined as</p>
<p><span class="math-container">$$ \sigma =  \frac{1}{u''(c)} \frac{u'(c)}{c}$$</span></p>
<p>From the functional form of utility you obtain</p>
<p><span class="math-container">$$ \sigma = \frac{1}{\theta} $$</span></p>
<p>And therefore the risk aversion coefficient equals <span class="math-container">$\frac{1}{\sigma} = \theta$</span>, which is constant and independent of consumption.</p>
<p>Note that the intertemporal elasticity of substitution is analytically derived as the inverse of risk aversion. This follows from considering fluctuation in the marginal utility of consumption over time as a form of risk. For example, a risk-averse individual would choose to smooth their consumption path to avoid such risk (fluctuations).</p>

CC BY-SA 4.0
2020-11-04T12:13:19.000Z
no problem, if you're satisfied with the answer can you mark your question as solved?
Ali
CC BY-SA 4.0
2020-11-03T12:15:16.857Z
2020-11-19T20:36:25.753Z
2020-11-19T20:36:25.753Z
Ali
<p>We are studying Ramsey-Cass-Koopmans model (RCK) in my macro course, I am already stuck in the starting section on assumptions of the model. In particular, I do not quite understand why the ulitity function in RCK is called CRRA, constant-relative-risk aversion function and why it is the case that the coefficient of relative risk aversion is theta θ = -Cu&quot;(C)/u'(C) and is therefore independent of C.</p>
<p>Would appreciate if someone could clarify this.</p>

<macroeconomics><ramsey>
Utility function in Ramsey-Cass-Koopmans model
user30845
# Utility function in Ramsey-Cass-Koopmans model

(Asked by: username_0 on 2020-11-03)

We are studying Ramsey-Cass-Koopmans model (RCK) in my macro course, I am already stuck in the starting section on assumptions of the model. In particular, I do not quite understand why the ulitity function in RCK is called CRRA, constant-relative-risk aversion function and why it is the case that the coefficient of relative risk aversion is theta θ = -Cu"(C)/u'(C) and is therefore independent of C.

Would appreciate if someone could clarify this.

**Tags:** macroeconomics, ramsey

---

## 1 answers

### Answer by username_1 on 2020-11-03. Score: 4 | Accepted answer

Recall that in the RCK model households choose their consumption and savings so as to maximise current and future utility (i.e. future generations). Then the objective function is given by

$$ U = \int\_{0}^{\infty} u[c(t)]e^{nt}e^{-\rho t}dt$$

The RCK model assumes the functional form of utility

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

Note that this function is increasing and concave, and that assuming that $\rho > n$, it ensures that lifetime utility does not diverge. This is a very practical assumption as it ensures that households would not have infinite lifetime utility and therefore allows for a well defined solution.

This is is called **constant intertemporal elasticity of substitution (CIES) or constant relative risk aversion (CRRA) utility function** (note that this isoelastic utility function is the only class of utility functions with constant relative risk aversion).

You can check for this by computing the risk aversion coefficient, $\frac{1}{\sigma}$, where $\sigma$ is the intertemporal elasticity of substitution (measures the willingness on the part of the consumer to substitute future consumption for present consumption) and is defined as

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

From the functional form of utility you obtain

$$ \sigma = \frac{1}{\theta} $$

And therefore the risk aversion coefficient equals $\frac{1}{\sigma} = \theta$, which is constant and independent of consumption.

Note that the intertemporal elasticity of substitution is analytically derived as the inverse of risk aversion. This follows from considering fluctuation in the marginal utility of consumption over time as a form of risk. For example, a risk-averse individual would choose to smooth their consumption path to avoid such risk (fluctuations).

> **Comments:**
> **username_1**: no problem, if you're satisfied with the answer can you mark your question as solved?  

---

