Using the FOCs, show that W - s∗ is larger than s∗ for any β < 1. 2. Show that s∗ is a weakly increasing function of W (this can be done without FOCs). 3. Show that s∗ is a weakly increasing function of β (this too can be done without FOCs). 4. If u(x) = log(x), give the optimal savings as a function of β and W.
A consumer who
will consume c1 in period 1 and c2 in period 2 exhibits pure time discounting if they evaluate their consumption using the utility function
U(c1; c2) = u(c1) + βu(c2)
where 0 < β < 1, and β < 1 captures the idea that consumption in
the future is not worth as much as current consumption. Note that
U(·; ·) depends on two arguments, while u(·) depends on only one
argument. We assume that u0(·) > 0, u00(·) < 0, and suppose that
the consumer can split their current wealth, W, between the two
periods as they please. In this problem, you are supposed to find
properties of the the optimal savings, s, as the solution to
max0≤s≤W U(W - s; s) = u(W - s) + βu(s):
1. Using the FOCs, show that W - s∗ is larger than s∗ for any
β < 1.
2. Show that s∗ is a weakly increasing function of W (this can be
done without FOCs).
3. Show that s∗ is a weakly increasing function of β (this too can
be done without FOCs).
4. If u(x) = log(x), give the optimal savings as a function of β and
W.
Step by step
Solved in 2 steps