# What behaviour do different values of Relative Risk Aversion imply?

What behaviour does it imply that Relative Risk Aversion of consumption, say R(c) is less than 1, equal to 1 or greater than 1?

I am reading an article of Diamond & Dybvig (1983), about Bank Runs, Deposit Insurance and liquidity and they assume a relative risk aversion greater than 1. Unfortunately, some of my microeconomics are a couple of years ago. I know the formulas and so on, but am unsure of the intuitive implications on behaviour for different values of Relative Risk Aversion.

## 1 Answer

In the D&D model, We deal with a 2 periods economy where the agent can invest in a short term project that yields $0$ or a long term project that yields $R$;

It is possible to prove that an optimal insurance scheme (such as the use of a mutual fund or a bank) flattens the yield curve.

It is possible to show that the optimal portfolio problem solves:

$max_{\{c_1\}}\{\lambda u (c_1) + (1-\lambda) u(\frac{c_2 R}{1-\lambda})$

where $\lambda$ is the share of impatient consumers who consume in period 1 (those who "face a liquidity shock"), and $R$ is the yield on long term projects. The solution to the problem above gives: $\frac{u'(c_1)}{u'(c_2)} = R$.

Under the assumption of $u'>0, u''<0$, the condition $-\frac{cu''(c)}{u'(c)} > 1$ is technical; it ensures that $\forall c$, the first derivative of $cu'(c)$ is negative, or that $cu'(c)$ is decreasing in c. Given $R>1$ this implies that $Ru'(R)>1u'(1)$ and so given that $c_1 = 1, c_2=R$ we have that $\frac{u'(c_1)}{u'(c_2)} = \frac{u'(1)}{u'(R)} > R$ .

Going back to the FOC of the optimal porfolio problem, in order to reach the equality we therefore need to have $c_1^*>1$ and $c_2^*<R$ so that to insure that $\frac{u'(c_1)}{u'(c_2)}$ decreases enough. This means that an insurance scheme determines a flattening of the yield curve

The bottom line is that there is not much intuition on why the condition holds but you can easily think of why it has to hold (given standard assumptions on the utility function) by looking at $lim_{c\to x} (-\frac{cu''(c)}{u'(c)})$ when $x= 0$ and $x=\infty$.

For more details on the DD model look at: Tirole Jean, "The Theory of Corporate Finance"