MACRO-1. The simplified Solow growth model. Suppos

Solution

a

To determine if the aggregate production function

Y=K^{\frac{1}{2}} N^{\frac{1}{2}}

represents constant returns to scale, we can scale both inputs by a factor of

t

:

Y(tK, tN) = (tK)^{\frac{1}{2}} (tN)^{\frac{1}{2}} = t^{\frac{1}{2}} K^{\frac{1}{2}} t^{\frac{1}{2}} N^{\frac{1}{2}} = t Y(K, N)

. Since the output scales by

t

when both inputs are scaled by

t

, this indicates constant returns to scale

b

If both

N

and

K

increase by

5\%

, we can express this as

N' = 1.05N

and

K' = 1.05K

. The new output will be

Y' = (1.05K)^{\frac{1}{2}} (1.05N)^{\frac{1}{2}} = 1.05^{\frac{1}{2}} Y = 1.05^{0.5} Y

. The percentage increase in

Y

is approximately

2.5\%

since

1.05^{0.5} \approx 1.025

c

To express

\frac{Y}{N}

as a function of

\frac{K}{N}

, we have

\frac{Y}{N} = \frac{K^{\frac{1}{2}} N^{\frac{1}{2}}}{N} = K^{\frac{1}{2}} N^{-\frac{1}{2}}

. Letting

x = \frac{K}{N}

, we can rewrite this as

\frac{Y}{N} = x^{\frac{1}{2}}

. If

N

and

K

both increase by

5\%

, then

\frac{Y}{N}

will also increase by approximately

2.5\%

d

To find the steady-state levels of

\frac{K}{N}

and

\frac{Y}{N}

for different saving rates, we use the formula for steady-state capital per worker:

\frac{K}{N} = \frac{s}{\delta}

, where

\delta = 0.05

. For

s = 0.1

,

\frac{K}{N} = \frac{0.1}{0.05} = 2

. For

s = 0.5

,

\frac{K}{N} = \frac{0.5}{0.05} = 10

. For

s = 0.9

,

\frac{K}{N} = \frac{0.9}{0.05} = 18

. The corresponding

\frac{Y}{N}

can be calculated as

\frac{Y}{N} = \left(\frac{K}{N}\right)^{\frac{1}{2}} = \sqrt{\frac{K}{N}}

. Thus,

\frac{Y}{N}

for

s=0.1

is

\sqrt{2}

, for

s=0.5

is

\sqrt{10}

, and for

s=0.9

is

\sqrt{18}

Answer

a: Yes, it represents constant returns to scale.

b: The percentage increase in

Y

is approximately

2.5\%

.

c:

\frac{Y}{N} = \left(\frac{K}{N}\right)^{\frac{1}{2}}

; percentage increase is approximately

2.5\%

.

d: Steady-state

\frac{K}{N}

for

s=0.1

is

2

,

s=0.5

is

10

,

s=0.9

is

18

; corresponding

\frac{Y}{N}

values are

\sqrt{2}

,

\sqrt{10}

,

\sqrt{18}

.

Key Concept

The Solow growth model illustrates the relationship between capital, labor, and output in an economy.

Explanation

The questions explore the implications of production functions, returns to scale, and steady-state conditions in the context of economic growth.

Solution

a

To show that the aggregate production function

Y=K^{\alpha}(A L)^{1-\alpha}

exhibits constant returns to scale, we scale both inputs by a factor of

t

:

Y(tK, tAL) = (tK)^{\alpha}(A(tL))^{1-\alpha} = t^{\alpha}K^{\alpha}(A L)^{1-\alpha} = tY(K, AL)

. Since the output scales by

t

when both inputs are scaled, it confirms constant returns to scale

b

If

AL

and

K

both increase by

5\%

, we can express this as

AL' = 1.05(AL)

and

K' = 1.05(K)

. The new output is

Y' = (1.05K)^{\alpha}(1.05AL)^{1-\alpha} = 1.05^{\alpha + (1-\alpha)} Y = 1.05Y

. Thus, the percentage increase in

Y

is

5\%

. For

y

, since

y = \frac{Y}{AL}

, the percentage increase in

y

is also

5\%

c

Expressing

y

as a function of

k

, we have

y = f(k) = k^{\alpha}

. To show decreasing returns to factors, we take the derivative:

\frac{dy}{dk} = \alpha k^{\alpha - 1}

. Since 0 < \alpha < 1,

\frac{dy}{dk}

decreases as

k

increases, indicating decreasing returns to factors

d

In steady state, we have

sY = \delta K

. Substituting

Y = K^{\alpha}(AL)^{1-\alpha}

gives

sK^{\alpha}(AL)^{1-\alpha} = \delta K

. Rearranging leads to the steady-state level

k^* = \left(\frac{s}{\delta}\right)^{\frac{1}{1-\alpha}}

. The steady-state levels of

y

and

c

can be calculated as

y^* = f(k^*)

and

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

e

To maximize steady-state consumption

c

, we differentiate

c = (1-s)f(k)

with respect to

s

and set the derivative to zero. The optimal saving rate can be found by solving the first-order condition

f

The elasticity of steady-state output with respect to saving is given by

\frac{\partial y^{*}}{\partial s} \cdot \frac{s}{y^{*}}

. This measures how responsive the steady-state output is to changes in the saving rate

Answer

The aggregate production function exhibits constant returns to scale, and a 5% increase in

AL

and

K

results in a 5% increase in both

Y

and

y

. The function

y = f(k)

shows decreasing returns to factors, and the steady-state levels can be calculated based on the saving and depreciation rates.

Key Concept

The Solow growth model illustrates how savings, population growth, and technological progress affect economic output and growth.

Explanation

The analysis shows the relationships between inputs and outputs in the production function, demonstrating key concepts like returns to scale and steady-state conditions.

AskSia

Plus