# Why does a higher risk aversion leads to a lower intertemporal elasticity of substitution?

Mathematically, a higher risk aversion leads to a lower intertemporal elasticity of substitution (there is an inverse relationship). But why?

If I become more risk-averse, I would like to smooth my consumption. So I will increase my participation in the financial market to avoid or reduce the risk. But how does that lower my intertemporal elasticity of substitution? Like why would I become less responsive to changes in the interest rate if I become more risk-averse? What's the intuition here?

Or because I just wanna smooth my consumption, I don't really care about changes in the interest rate?

Risk-lovers, on the other hand, although also participate in the financial market, do not care about smoothing their consumption? Like if the interest rate increases, they can drastically reduce their current consumption for more consumption in the future, but risk-aversers do not care about the higher interest rate?

Mathematically, a higher risk aversion leads to a lower intertemporal elasticity of substitution (there is an inverse relationship). But why?

The (relative) measure of risk aversion is measured as: $$r(x) = -x\frac{u''(x)}{u'(x)}$$

Consider an intertemporal utility function $$u(x_1) + \beta u(x_2)$$. Maximising this with respect to an intertemporal budget constraint gives the following first order condition: $$\frac{u'(x_2)}{u'(x_1)} = \frac{1+r}{\beta}.$$ where $$r$$ is the interest rate. This states that the marginal rate of substitution should equal the interest rate.

Now, for a given utility level, the indifference curve $$x_1(x_2)$$ can be implicitly defined as: $$u(x_1(x_2)) + u(x_2) = \bar u.$$ Using the implicit function theorem, we can compute its slope: $$\frac{\partial x_1}{\partial x_2} = - \frac{u'(x_2)}{u'(x_1)}.$$ The slope of the indifference curve is therefore equal to the negative of the marginal rate of substitution. Now, differentiate this again with respect to $$x_2$$ : \begin{align*} \frac{\partial^2 x_1}{\partial x_2 \partial x_2} &= - \frac{u''(x_2) u'(x_1) - u'(x_2) u''(x_1) \dfrac{\partial x_1}{\partial x_2}}{(u'(x_1))^2},\\ &= - \frac{u''(x_2)}{u'(x_1)} - \frac{u'(x_2)}{u'(x_1)} \frac{u''(x_1)}{u'(x_1)}\dfrac{ u'(x_2)}{u'(x_1)},\\ &= -\frac{u''(x_2)}{u'(x_2)}\frac{u'(x_2)}{u'(x_1)} - \left(\frac{u'(x_2)}{u'(x_1)}\right)^2 \frac{u''(x_1)}{u'(x_1)},\\ &= \frac{r(x_2)}{x_2} \frac{(1+r)}{\beta} + \left(\frac{(1+r)}{\beta}\right)^2 \frac{r(x_1)}{x_1}. \end{align*} This shows that the curvature of the indifference curve at the optimum is increasing in the measure of risk aversion.

Concerning it's relationship to the elasticity of intertemporal substitution, we intuitively see that the higher the relative risk aversion, the higher the curvature of the indifference curve, so the more the slope at the indifference curve, $$\frac{u'(x_2)}{u'(x_1)}$$, will change as a response to a change in the ratio $$\frac{x_2}{x_1}$$. Vice versa, the higher the relative risk aversion, the less the ratio $$\frac{x_2}{x_1}$$ will change in response to a change in the slope $$\frac{u'(x_2)}{u'(x_1)}$$ (e.g. due to a change in interest rates).

To see this mathematically, consider he intertemporal elasticity of subsitution: $$\varepsilon_s = -\frac{\partial \ln(x_2/x_1)}{\partial \ln(u'(x_2)/u'(x_1))}$$ This measures the percentage change in $$x_2/x_1$$ due to a one percentage point change in the marginal rate of substitution.

We can write everything in terms of $$x_2$$ (as we did above), then take derivatives of the numerator and denominator with respect to $$x_2$$ and finally evaluate it at the optimum: \begin{align*} \varepsilon_s &= -\frac{\dfrac{x_1}{x_2}\left(\dfrac{x_1 - x_2 \dfrac{\partial x_1}{\partial x_2}}{(x_1)^2}\right)}{-\dfrac{1}{\dfrac{u'(x_2)}{u'(x_1)}} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r)}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{x_1}\right)},\\ &= -\frac{\dfrac{1}{x_2} - \dfrac{\dfrac{\partial x_1}{\partial x_2}}{x_1} }{\dfrac{1}{\dfrac{\partial x_1}{\partial x_2}} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{x_1}\right)},\\ &= -\frac{\dfrac{1}{x_2} - \dfrac{- \dfrac{1+r}{\beta}}{x_1} }{ -\dfrac{\beta}{1+r} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{\beta}\right)},\\ &= \frac{\dfrac{x_1}{x_2} + \dfrac{1+r}{\beta}}{ \left(\dfrac{x_1}{x_2}r(x_2) + \dfrac{1+r}{\beta} r(x_1)\right)},\\ \end{align*}

The last expression gives the elasticity as a function of the relative slope $$x_2/x_1$$ and the relative risk aversions. Now, if the relative risk aversion is constant, say $$r(x_2) = r(x_1) = \sigma$$ this simplifies to $$\varepsilon_s = 1/\sigma$$.

Nevertheless, we see that if the risk aversion increases, i.e. $$r(x_2)$$ and $$r(x_1)$$ increase, then $$\varepsilon_s$$ decreases.

If I become more risk-averse, I would like to smooth my consumption. So I will increase my participation in the financial market to avoid or reduce the risk. But how does that lower my intertemporal elasticity of substitution? Like why would I become less responsive to changes in the interest rate if I become more risk-averse? What's the intuition here?

There is a difference between consumption and saving. The higher your risk aversion, the bigger the curvature of the indifference curve, i.e. the lower $$\varepsilon_s$$. And so the smaller the change of $$x_2/x_1$$ due to a change in, for example the interest rate $$r$$.

You can think of the curvature of the indifference curve as the degree of complementarity between $$x_1$$ and $$x_2$$. The higher the risk aversion the more $$x_1$$ and $$x_2$$ become complements. So I will try to consume $$x_1$$ and $$x_2$$ together which is equal to increasing consumption smoothing.

In order to smooth my consumption, I will need to lend and borrow on the financial market. So if my income over time changes a lot, I will have to save and borrow a lot. On the other hand, if my income is reasonable smooth over time, I will not need to borrow or save a lot in order to smooth out out my consumption.

Or because I just wanna smooth my consumption, I don't really care about changes in the interest rate?

This is a mis-understanding, risk averse consumers actually do care about changes in the interest rate.

• First of all, a change in interest rates affects their saving decision. It is true, however, that changes in the interest rate will not change $$x_2/x_1$$ a lot. So although my relative consumption over time will not be affected a lot by changes in the interest rate, the amount that I save or borrow can change substantially.

• A change in the interest rates also changes my utility. Assume for simplicity that I am infinitely risk aversion, so I will always always choose to set $$x_1 = x_2$$. In this case, my intertemporal budget constraint becomes:

\begin{align*} &x_1 + \frac{x_2}{(1+r)} = y_1 + \frac{y_2}{(1+r)},\\ \to &x_1\frac{2+r}{1+r} = y_1 + \frac{y_2}{1+r},\\ \to &x_1 = \frac{(1+r)y_1 + y_2}{2+r} \end{align*}

Then: $$\frac{\partial x_1}{\partial r} = \frac{y_1(2+r) - [(1+r)y_2 + y_2]}{(2+r)^2} = \frac{y_1-x}{2+r}.$$ So if my consumption $$x_1$$ is smaller than my income in period 1 $$y_1$$ (i.e. I save in period 1), I will be able to consume more if interest rates increase. On the other hand, if $$x_1 > y_1$$ (So I borrow in period 1), then I will be worse of if interest rates increase. So changing interest rates will change my utility depending on whether I am a net saver or borrower.

Risk-lovers, on the other hand, although also participate in the financial market, do not care about smoothing their consumption? Like if the interest rate increases, they can drastically reduce their current consumption for more consumption in the future, but risk-aversers do not care about the higher interest rate?

Again risk loving (or neutral) consumers also care about interest rates. Consider for simplicity a risk neutral consumer with utility function $$u(x_1, x_2) = x_1 + \beta x_2$$. Then it is clear to see that if:

1. $$\beta < \frac{1}{1+r}$$, I will consume everything in period 1, so $$x_1 = y_1 + \frac{y_2}{1+r}$$ and $$x_2 = 0$$. Utility will be equal to $$y_1 + \frac{y_2}{1+r}$$, which is decreasing in $$r$$. Higher interest rates lower my discounted future income, so I will be worse off.
2. $$\beta > \frac{1}{1+r}$$, then I will consume everything in period 2, so $$x_1 = 0$$ and $$x_2 = (1+r) y_1 + y_2$$. Then utility is equal to $$\beta (1+r)y_1 + \beta y_2$$ which is increasing in $$r$$. Higher interest rates increase my consumption in period 2.

As a conclusion, the consumer will gain by an interest rate increase if she is a net saver and will loose if she is a net borrower.

It is true that normally a risk neutral consumer will not respond her saving decisions by a lot if interest rates change. However if, for example due to the interest rate increase we shift from a situation where $$\beta < 1/(1+r)$$ to a situation where $$\beta > 1/(1+r)$$ then suddenly she will move all consumption from period 1 to period 2, which will mean a big shift from borrowing to saving.

• @@ tdm. Thanks for the answer. But with this question, "Mathematically, a higher risk aversion leads to a lower intertemporal elasticity of substitution (there is an inverse relationship). But why?", I didn't actually mean mathematically..sorry. If you can provide an intuition, that will be great. Like if my risk aversion coefficient goes up, why would I become less responsive to changes in the interest rate? That is my intertemporal elasticity of substitution goes down. May 19, 2021 at 2:05