Cobb Douglas Question?

Question

Assume that a country's production function is Y = K0.3L0.7 and L increases with a constant rate n=0.01

a. What is the per-worker production function y = f(k)?

b. If the saving rate (s) is 0.2 and the depreciation rate (δ) is 0.19, what are capital per worker, production per worker, and consumption per worker in the steady state?

c. What is the steady-state growth rate of output per worker and of capital per worker?

d. What is the steady-state growth rate of total output and of total capital?

e What is the Golden Rule level of capital per worker and output per worker?

f If s increases to 0.4 calculate again b and c. Explain

g

Assume now that the production function is Y = K0.3(E*L)0.7 and L increases with a constant rate n = 0.01 and E increase with a constant rate g = 0.2. The saving rate (s) is 0.2 and the depreciation rate (δ) is 0.19. Calculate again c and d.

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If you could type out the steps so I can follow along, that would be great. Thanks!

Answer 1

Solow Growth Model. such a long question...

Y = K^0.3L^0.7

Y = f(K, L)

Y = f(K/L, L/L)

y  = f(k)

Therefore,

y = k^0.3

Where y = output per worker (think GDP Per Capita) = Y/L, k = K/L = capital per worker.

b.

Δy = f(Δk)

In Maths,

Δk/k = ΔK/K - ΔL/L

Given,

ΔL/L = n = 0.01

ΔK = I = S - D = sY - δK

Hence,

Δk/k = ΔK/K - ΔL/L

         = sY/K - δK/K - 0.01

         = sY/K - δ - 0.01

         = sY/K(L/L) - δ - 0.01

         = sy/k - δ - 0.01

  ÃƒÂŽÃ‚”k  = sy - k(δ + 0.01)   ----------------------- (1)

From (1)

Δk  = sy - k(δ + 0.01)

At steady state,

Δk = 0

sy - k(δ + 0.01) = 0

When δ = 0.19, s = 0.2

0.2y - 0.2k = 0

0.2y = 0.2k

since y = k^0.3

0.2k^0.3 = 0.2k

1 = k/(k^0.3)

k = 1

y = k^0.3 = 1

c = 1-s = 1-0.2 = 0.8

hence, cY = 0.8*1 = 0.8

c.

output per worker growth = capital grow per worker growth = 0

d.

Growth rate of total output = capital growth rate = labour growth rate = 0.01

e. Golden rule = Max (C/L)

                         => (d+n) = MPk

                              0.01 + 0.19 = 0.3k^(-0.7)

                           (3/2)^10 = k^7

                                k = 1.847

                                y = 1.202

f: you should be able to solve using the above method...

g. I'm skipping the derivation, use the below equation to calculate instead.

Δk  = sy - k(δ + n + g)

Answer 2

a:

Y=K^0.3 * L^0.7

Y/L=(K^0.3 * L^0.7)/L

y=(K/L)^0.3=k^0.3

y=k^0.3

b:

n=0.01

δ=0.19

s=0.2

Δk=0 (given by definition of steady state)

s*k^0.3=(δ+n)k

0.2*k^0.3=(0.19+0.01)k

0.2/0.2=k^0.7

k=1

y=k^0.3=1^0.3=1

c=y-s*k^0.3=1-0.2=0.8

c:

Steady state growth rate Δy=0 (given by definition of steady state) Δk=0 (given by definition of steady state)

d:

Steady state growth rate:

Y/L=(K^0.3 * L^0.7)/L

Δy=0

Y/L=1

(K^0.3 * L^0.7)/L=1

(K/L)^0.3=1

L^0.3=K^0.3

ΔL=0.01=+1%

ΔK=0.01=+1%

ΔY=K^0.3 * L^0.7 =

= (1.01^0.3)*(1.01^0.7)=1.01^1 =1.01=+1%

e:

0=(k^0.3-(0.19+0.01)k)'

0.2=0.3/k^0.7

k^0.7=1.5

k=1.784674184

s*k^0.3-(0.19+0.01)k=0

s*1.784674184^0.3 =0.2*1.784674184

s=1.5*0.2=0.3

y=1.189782789

f:

b (f):

n=0.01

δ=0.19

s=0.4

Δk=0 (given by definition of steady state)

s*k^0.3=(δ+n)k

0.4*k^0.3=(0.19+0.01)k

0.4/0.2=k^0.7

k^0.7=2

k=2.691800385

y=k^0.3= 2.691800385^0.3= 1.345900193

c=y-s*y=y(1-s)=y(1-0.4)=

=0.6y=0.6*1.345900193 = 0.807540116

c (f):

Steady state growth rate Δy=0 (given by definition of steady state) Δk=0 (given by definition of steady state)

Explanation: With different saving rates economy will have different steady states.

g:

b (g):

Production function per worker will reamin the same:

y=k^0.3

n=0.01

δ=0.19

g=0.2

s=0.2

Δk=0 (given by definition of steady state)

s*k^0.3=(δ+n+g)k

0.2*k^0.3= (0.19+0.01+0.2)k

0.2/0.4=k^0.7

0.5=k^0.7

(1/2)^(1/0.7)= (1/2)^(10/7) =k

k=0.371498572

y=k^0.3= 0.371498572^0.3= 0.742997145

c=y-sy=y(1-s)=y(1-0.2)= 0.8y=

=0.742997145*0.8=0.594397716

c (g):

Steady state growth rate Δy=0 (given by definition of steady state) Δk=0 (given by definition of steady state)

d(g):

Steady state growth rate:

Y/EL=(K^0.3 * EL^0.7)/EL

Δy=0

Y/EL=1

(K^0.3 * EL^0.7)/EL=1

(K/EL)^0.3=1

(EL)^0.3=K^0.3

ΔL=0.01=+1%

ΔE=0.20=+20%

ΔK=1.01*1.2=1.212=+21.2%

ΔY=K^0.3 * (EL)^0.7 =

= (1.212^0.3)*((1.01*1.2)^0.7)=

=1.0593776 * 1.144067988=1.212=+21.2%