Question

Portugal has the following per-worker production function:

y=3k^0.05

Depreciation rate is 0.08, population growth rate is 0.02. Saving is S=0.2Y, where S is national saving and Y is national output.

(a) what are the steady state value of capital-labour ratio, output per worker and consumption per worker?

(b) Suppose that national saving increases to 0.4, what are the steady state value of capital-labour ratio, output per worker and consumption per worker?

(c) Suppose depreciation rate increases to 0.20, what are the steady state value of capital-labour ratio, output per worker and consumption per worker?

Answer 1

(a)

In steady state,

s / ( + n) = k / y

0.2 / (0.08 + 0.02) = k / (3k^0.05)

0.2 / 0.1 = k^0.95 / 3

k^0.95 / 3 = 2

k^0.95 = 6

k = (6)^{(1 / 0.95)} = 6.59 (capital labor ratio)

y = 3 x (6.59)^0.05 = 3.30 (outut per worker)

c = (1 - s) x y = (1 - 0.2) x 3.3 = 0.8 x 3.3 = 2.64 (consumption per worker)

(b)

0.4 / 0.1 = k^0.95 / 3

k^0.95 / 3 = 4

k^0.95 = 12

k = (12)^{(1 / 0.95)} = 13.68

y = 3 x (13.68)^0.05 = 3.42

c = (1 - 0.4) x 3.42 = 0.6 x 3.42 = 2.05

(c)

0.2 / (0.2 + 0.02) = k / (3k^0.05)

0.2 / 0.22 = k^0.95 / 3

k^0.95 / 3 = 0.91

k^0.95 = 2.73

k = (2.73)^{(1 / 0.95)} = 2.88

y = 3 x (2.88)^0.05 = 3.16

c = (1 - 0.2) x 3.16 = 0.8 x 3.16 = 2.53