# Utility function in Ramsey-Cass-Koopmans model

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.

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).

• thanks a lot for your answer! – Joker312 Nov 4 '20 at 8:13
• no problem, if you're satisfied with the answer can you mark your question as solved? – user20105 Nov 4 '20 at 12:13