J Problem Set 2 - Solution

J.1 Two-period Intertemporal Optimization

Given the expression for the utility function: \[u(c) = \frac{c^{1-\sigma}-1}{1-\sigma},\] we know that marginal utility is: \[u'(c)=c^{-\sigma},\] while the derivative of marginal utility is: \[u''(c)=-\sigma c^{-\sigma-1}.\] Thus, because $u''(.)$ must be negative for the function to be concave, we have $\sigma>0$. Utility should be increasing and concave, because the more consumption people have, the more happy they are. However, they benefit less from an extra dollar of consumption, the more consumption they already have. (decreasing marginal utility, which implies concavity)
This is straight from lecture 3.⁵¹

Using the equation from question 2, we can write: \[\frac{\beta c_1^{-\sigma}}{c_0^{-\sigma}}=\frac{1}{1+r} \qquad \Rightarrow \qquad \frac{c_1}{c_0}=\beta^{1/\sigma} (1+r)^{1/\sigma}\]
The intertemporal budget constraint is: \[c_0 + \frac{c_1}{1+r} = f_0 + y_0 + \frac{y_1}{1+r},\] This intertemporal budget constraint was derived in class. It comes from the fact that: \[c_0 = y_0 + (f_1-f_0)\] and: \[c_1 = y_1 + (1+r)f_1\]
Using this intertemporal budget constraint, together with the optimality condition for saving: \[\begin{aligned} & \left(1 + \beta^{1/\sigma}(1+r)^{1/\sigma-1} \right)c_0 = f_0 + y_0 + \frac{y_1}{1+r} \\ & \Rightarrow \quad \boxed{c_0 = \frac{1}{1 + \beta^{1/\sigma}(1+r)^{1/\sigma-1}} \left( f_0 + y_0 + \frac{y_1}{1+r}\right)}. \end{aligned}\]

which implies: \[\boxed{c_1 = \frac{\beta^{1/\sigma}(1+r)^{1/\sigma}}{1 + \beta^{1/\sigma}(1+r)^{1/\sigma-1}} \left( f_0 + y_0 + \frac{y_1}{1+r}\right)}\]

This is the formula we hadin class, for $\sigma = 1$. This case is the one we saw in the class, because when $\sigma$ approaches $1$, we have: \[\lim_{\sigma \to 1} \frac{c^{1-\sigma}-1}{1-\sigma}=\log(c).\]

You can see this in many different ways. The simplest way is to write that: \[c^{1-\sigma}=e^{(1-\sigma)\log(c)}=\exp\left((1-\sigma)\log(c))\right).\]

Then, we use: \[\lim_{x \to 0} \frac{e^{ax}-1}{x}=a.\]

Indeed, the limit of $(e^{ax}-1)/x$ when $x$ goes to $0$ is by definition the derivative of $e^{ax}$ at $x=0$. Thus, since the derivative of $e^{ax}$ is $ae^{ax}$, we get that the derivative at $x=0$ of $e^{ax}$ is $a$. Using that formula for $x = 1-\sigma$ and $a=\log(c)$ allows to show: \[\lim_{(1-\sigma) \to 0} \frac{e^{\log(c)(1-\sigma)}-1}{1-\sigma}=\log(c).\]

Therefore, we get: \[\lim_{\sigma \to 1} \frac{c^{1-\sigma}-1}{1-\sigma}=\log(c).\]

Assume $\sigma = 1/2$. If $r = 1\%$, then according to this Google Spreadsheet, $c_0$ is equal to $44,776, and $c_1$ is equal to $45,676. If $r = 2\%$, then $c_0$ is equal to $44,554 and $c_1$ is equal to $46,354. Consumption $c_0$ thus falls by $222, approximately -0.5% in percentage terms.
Assume $\sigma = 1$. If $r = 1\%$, then according to this Google Spreadsheet, $c_0$ is equal to $45,000 and $c_1$ is $45,450. If $r = 2\%$, then $c_0$ is equal to $45,000 and $c_1$ is equal to $45,900. Consumption $c_0$ does not change, this is the case we have seen in class.
Assume $\sigma = 2$. If $r = 1\%$, then according to this Google Spreadsheet, $c_0$ is equal to $45,112 and $c_1$ is equal to $45,337. If $r = 2\%$, then $c_0$ is equal to $45,223 and $c_1$ is equal to $45,673. Consumption $c_0$ increases by $111, or approximately 0.25%.
Whether an increase in real interest rates leads to a fall or an increase in consumption depends on $\sigma$, which can be seen on this formula (it is crucial for this that $y_1=0$, or that second-period income is zero): \[c_0 = \frac{1}{1 +\beta^{1/\sigma}(1+r)^{1/\sigma-1}} \left( f_0 + y_0\right).\] When $1/\sigma-1>0$, or $\sigma<1$, an increase in the real interest rate leads to lower consumption today, and more saving. Conversely, when $1/\sigma-1<0$, or $\sigma>1$, an increase in the real interest rate leads to higher consumption today, and less saving. Finally, when $\sigma=1$, the interest rate has no effect on current consumption $c_0$ or saving.
Well, as often in economics, the answer is that “it depends” ! Here, it depends on what your preferences are. If you have log preferences, then you should not change your saving at all. If your preferences are “very concave”, or $\sigma>1$, then you’ll save less when interest rates go up because everything else equal, an increase in the interest rates will raise your consumption in period 1 relative to period 0, so in order to offset that you will want to raise consumption in period 0 (marginal utility is fast decreasing, so you want to have equal consumption at both dates. If your preferences are “not very concave” or $\sigma<1$, then you want to save more because the higher interest rate is rewarding you for doing so. (and you don’t mind tilting a bit your consumption towards period one, because marginal utility decreases less fast). In practice, it appears that the effect of interest rates on consumption through this channel are very limited: this could be due either to the fact that preferences are log, or (more likely !) that people are not as rational as economists, and do not think about real interest rates that much when they choose to consume more or less… During Lecture 12, we shall see that interest rates have an effect on consumption through other, more powerful channels. (this matters for the analysis of monetary policy)

Needless to say, you can’t say that during an exam!↩