Solow–Swan Growth Model

Solutions

Exercise 1

(a)

k^* = (0.25/0.08)^{3/2} = (3.125)^{1.5} \approx 5.525

y^* = (k^*)^{1/3} \approx 1.768

c^* = 0.75 \times 1.768 \approx 1.326

. (b)

k^*/y^* = s/(n+g+\delta) = 0.25/0.08 = 3.125

. (c) In steady state

Y/(AL)

is constant, so

Y/L = A \cdot y^*

grows at rate

g = 0.02

, i.e. 2% per year.

Exercise 2

(a) Golden Rule:

\frac{1}{3}(k^*_{GR})^{-2/3} = 0.08

gives

(k^*_{GR})^{2/3} = 4.167

, so

k^*_{GR} \approx 8.508

. (b)

s_{GR} = 0.08 \times (k^*_{GR})^{2/3} = 0.08 \times 4.167 \approx 0.333 = \alpha

. (c)

s = 0.25 < s_{GR} = 1/3

: below the Golden Rule (dynamically efficient). Raising

s

would eventually raise

c^*

, but the current generation bears a cost.

Exercise 3

(a)

k^{**} = (0.36/0.08)^{3/2} = (4.5)^{1.5} \approx 9.545

y^{**} = (9.545)^{1/3} \approx 2.121

. (b)

(2.121 - 1.768)/1.768 \approx 20\%

. (c) It remains

g = 2\%

. The savings rate increase raises the level of output per worker but not the long-run growth rate.

Exercise 4

(a)

\text{TFP} = 4.0 - (0.35)(5.0) - (0.65)(2.0) = 4.0 - 1.75 - 1.30 = 0.95\%

. (b)

0.95/4.0 = 23.75\%

. (c) Capital deepening contribution:

\alpha \times (\hat{K} - \hat{L}) = 0.35 \times 3\% = 1.05\%

. Fraction:

1.05/4.0 = 26.25\%

Exercise 5

(a)

|\lambda| = (1-\alpha)(n+g+\delta) = (2/3)(0.08) \approx 5.33\%

per year. (b) From

0.10 = e^{-|\lambda|t}

t = \ln(10)/0.0533 \approx 43

years.