Exercises · 8. Growth: Solow and Beyond

Exercises

Problems to check your understanding of this chapter. The solutions are folded away — try them yourself first.

Medium

1MediumCounterexample

In a Solow economy with depreciation $\delta = 0.1$ , find steady-state consumption $c^* = y^* - \delta k^*$ for saving rates $s = 0.2$ , $s = 0.4$ , and $s = 0.8$ .

Does eating more in the steady state always follow from saving more?

Verdict

The claim is false — at a saving rate of $0.8$ steady-state consumption drops to $1.6$ , less than the $2.4$ eaten at $0.4$

A Country Cannot Grow on Machines Alone

Saving rate

0.4

— steady state

k^* = 16

y^* = 4

c^* = 2.4

Show solution

Counterexample and verification

For $s = 0.2$ : $y^* = 2$ and $k^* = 4$ , so replacing worn machines costs $\delta k^* = 0.4$ and $c^* = 2 - 0.4 = 1.6$ .
For $s = 0.4$ : $y^* = 4$ , $k^* = 16$ , replacement costs $1.6$ , and $c^* = 4 - 1.6 = 2.4$ .
Up to here, saving more does mean eating more.
But at $s = 0.8$ , $y^* = 8$ and $k^* = 64$ , replacement alone eats $6.4$ , and $c^* = 8 - 6.4 = 1.6$ — back down again.
Pile up too many machines and most of the output leaks into maintaining them.
Steady-state consumption peaks at $s = 0.5$ , where $c^* = 2.5$ .

Hard

2Hard

A household with interest rate $r = 0.04$ and income $y = 4$ holds assets $a = 10$ .

Find its optimal next-period assets for patience $\beta = 0.80$ , $0.90$ , and $0.95$ , then find the smallest $\beta$ (on a two-decimal grid) at which this household stops running its assets down.

Where the Saving Rate Comes From

Rate

0.04

, patience

0.90

— at assets

10

: consumption

7.4

, saving

7

Show solution

Reading the value-iteration policy at assets $10$ : $\beta = 0.80$ gives $a' = 4$ (consumption $10.4$ ), $\beta = 0.90$ gives $a' = 7$ (consumption $7.4$ ), and $\beta = 0.95$ gives $a' = 9$ (consumption $5.4$ ). Resources are $14.4$ in all three cases — each step of patience moves food from today's plate to tomorrow's storehouse.
Not running assets down requires $a' \ge 10$ .
At $\beta = 0.95$ , $a' = 9$ still falls one step short; at $\beta = 0.96$ , $a' = 10$ (consumption $4.4$ ) holds the line for the first time.
The border sits near where $\beta (1 + r) = 1$ , that is $\beta = 1 / 1.04 \approx 0.9615$ : the point where patience, grossed up by the interest rate, exactly balances.