B Problem Set 2

B.1 Two-period Intertemporal Optimization

Consider again the 2-period consumption problem of Lecture 2. That is, assume that there are two periods $t = 0,1$. Denote the interest rate by $r$, the discount factor by $\beta$, initial financial wealth by $f_0$, income in period $0$ by $y_{0}$, income in period $1$ by $y_1$, consumption in period $0$ by $c_0$, and consumption in period $1$ by $c_1$. Instead of logarithmic preferences $u(c)=\log c$, assume that preferences are given by: \[u(c) = \frac{c^{1-\sigma}-1}{1-\sigma},\]

Under what condition on $\sigma$ is this an increasing and concave utility function? Why should utility be increasing and concave?
Show using 4 different methods that: \[\frac{\beta u'(c_{1})}{u'(c_{0})}=\frac{1}{1+r}\]
Replacing out $u'(c)$ by its new expression, compute $c_1/c_0$.
What is the intertemporal budget constraint?
Use the intertemporal budget constraint to solve for $c_1$ and $c_0$.
Under what condition on $\sigma$ do you obtain the results we had in class when $u(c)=\log c$? Why is that?
Assume that $\sigma = 1/2$, and $f_0=0$, $y_0=\$90,000$, $y_1=0$, $\beta = 1$. What are $c_0$ and $c_1$ if $r=1\%$? What about if $r=2\%$? How much does $c_0$ change then? How much in percentage terms? (Advice: You may use a Google Spreadsheet to do all of these calculations much faster)
Same questions if $\sigma = 1$. (again, use a Spreadsheet)
Same questions if $\sigma = 2$. (again, use a Spreadsheet)
Compare the changes in $c_0$ following an increase in the real interest rate $r$ in questions 7, 8, 9. Comment.
Application: real interest rates have gone down substantially recently. Imagine you were starting your first job today. Assuming you are maximizing utility (!), should you save more, or less, for retirement purposes, than if interest rates were higher?