Macroeconomics
Macroeconomics
10th Edition
ISBN: 9780134896441
Author: ABEL, Andrew B., BERNANKE, Ben, CROUSHORE, Dean Darrell
Publisher: PEARSON
Question
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Chapter 6, Problem 6NP

a

To determine

To calculate:Steady state values of Output per worker, Capital per worker, Consumption per worker, Investment per worker

a

Expert Solution
Check Mark

Answer to Problem 6NP

the steady-state value of the capital per worker is k=16 .

the steady-state value of output per worker is y=24 .

the steady-state value of consumption per worker is c=21.6 .

Explanation of Solution

Given Information:

Households save 10% of income, so savings, s=0.1 . The population growth rate is given as n=0.01 and the depreciation rate is given as d=0.14 .

At steady stae:

Output per worker

Capital per worker

Consumption per worker are equal.

To determine the steady-state value of the capital per worker, use the equation sf(k)=(n+d)k .

Households save 10% of income, so savings, s=0.1 . The population growth rate is given as n=0.01 and the depreciation rate is given as d=0.14 .

Substitute the given values s=0.1 , yt=f(kt)=6k into the equation and solve for k.

  sf(k)=(n+d)kt

  0.1×6k=(0.01+0.14)k

  4k=k

Continue solving for k by dividing both sides by k .

  4kk=kk

  4=k

  42=(k)2

  16=k

Therefore, the steady-state value of the capital per worker is k=16 .

To find the steady-state value of output per worker, substitute the calculated value for capital-labor ratio from the previous step into the given per-worker production function, y=6k .

  y=6k

  =616

  =24

Hence, the steady-state value of output per worker is y=24 .

To find the steady-state value of consumption per worker, use the equation c=y(n+d)k , substituting the values of steady-state output per worker from the previous step, y , steady-state value of the capitaI-labor ratio k and the given values of population growth rate and depreciation rate,

  n=0.01 and d=0.14

  c=y(n+d)k

  c=24(0.01+0.14)(16)

  c=21.6

Therefore, the steady-state value of consumption per worker is c=21.6 .

To fund the steady-state value of investment per worker, use the equation I=(n+d)k , substituting the values of steady-state output per worker from above, y , steady-state value of the capital-labor ratio k and the given values of population growth rate and depreciation rate, n=0.01 and d=0.14 .

  I=(n+d)k

  =(0.01+0.14)(16)

  =2.4

Hence, the steady-state value of investment per worker is I=2.4 .

Economics Concept Introduction

Introduction:

Steady state is a situation at which investment is equal to depreciation. It implies all the investment done is used to replace the depreciating capital.

b)

To determine

Value of capital-labor ratio required to double steady state value of output per capita, Amount of savings to achieve output per worker.

b)

Expert Solution
Check Mark

Answer to Problem 6NP

The steady-state value of capital-labor ratio needed is k=64 .

Households would need to save 20% of income.

Explanation of Solution

Given Information:

Households save 10% of income, so savings, s=0.1 . The population growth rate is given as n=0.01 and the depreciation rate is given as d=0.14 .

To determine the steady-state value of the capitallabor ratio needed to double the steady-state value of output per capita, double the value for y in part a, y=24 and set it equal to the production function, y=6k , so 48=6k . Solve for k.

  48=6k

  486=6k6

  64=k

So to double the steady-state value of output per capita, the steady-state value of capital-labor ratio needed is k=64 .

To determine the fraction of income households would need to save to have a double steady-state value of output per capita than in part a, use the steady-state value of capitaI-Iabor ratio from the previous step, k=64 . Substitute this value, double the value of output, y=48 and the values of population growth rate and depreciation rate from part a., n=0.01 and d=0.14 in the equation sf(k)=(n+d)k .

Solve for s.

  sf(k)=(n+d)k

  s(48)=(0.01+0.14)(64)

  48s=(0.15)(64)

  48s=9.6

Continue solving for s ,

  48s=9.6

  48s48=9.648

  s=0.2

So households would need to save 20% of income.

Economics Concept Introduction

Introduction:

Steady state is a situation at which investment is equal to depreciation. It implies all the investment done is used to replace the depreciating capital.

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