Question

Intermediate Macroeconomics! Thank you!!

Suppose that the economy is summarized by the Solow economy with technological progress:

Production Function: Y=10K^{.3}(LE)^{.7}

Savings rate: s= .2

Depreciation rate: δ= .1

Population Growth rate: n= .02

Technological growth rate: g= .01

a) Derive the per effective worker production function for this economy.

b) Based on your answer in part (a), derive the formula for marginal product of capital (MPK) and show that the per effective worker production function exhibits diminishing marginal product of capital.

c) Solve for steady state values of capital per effective worker, k; output per effective worker, y: investment per effective worker, i; and consumption per effective worker, c.

d) The condition for golden rule is given as: MPK=n+δ+g, where MPK is marginal product of capital derived in part (b) above. Use this condition for golden rule to find the golden rule level capital per effective worker & output per effective worker.

Answer #1

Y = 10K^{0.3}LE^{0.7}

(a)

Dividing both sides by LE,

Y/LE = 10 x (K/LE)^{0.3}

y = 10 x k^{0.3} [y = Y/LE and k = K/LE]

(b)

MPk = dy/dk = (10 x 0.3) / k^{0.7} = 3 /
k^{0.7}

As k increases, MPk decreases, so there is diminishing marginal product of capital.

(c)

In steady state,

s / (n+δ+g) = k / y

0.2 / (0.02 + 0.1 + 0.01) = k / (10 x k^{0.3})

0.2 / 0.13 = k^{0.7} / 10

k^{0.7} = 2 / 0.13 = 15.38

k = (15.38)^{(1/0.7)} = 49.64

y = 10 x (49.64)^{0.3} = 10 x 3.23 = 32.3

i = s x y = 0.2 x 32.3 = 6.46

c = y x (1 - s) = y x (1 - 0.2) = 32.3 x 0.8 = 25.84

(d)

At Golden Rule level,

MPK = n+δ+g

3 / k^{0.7} = 0.13

k^{0.7} = 3 / 0.13 = 23.08

k = (23.08)^{(1/0.7)} = 88.59

y = 10 x (88.59)^{0.3} = 10 x 3.839 = 38.39

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