Macroeconomics Homework: Production, Growth, and Steady State Analysis

This document presents a comprehensive solution to a macroeconomics homework assignment. The assignment explores key concepts such as output per worker, capital per worker, and productivity, analyzing how these factors interact within production functions. The solution delves into comparative analysis, examining scenarios where countries have differing levels of output and capital. It further investigates the law of motion for aggregate capital, per-worker capital, and the determination of steady-state capital and output. The assignment also considers the impact of population growth on economic growth and output per worker, comparing the steady states of countries with different population growth rates. The solution references relevant economic literature to support its analysis.

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MACROECONOMICS
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Macroeconomics 2
Part A
1. Output per worker as a function of capital per worker
Y = AKαN1-α
Y/N = (AKαN1-α)/N
y = AKαN1-ΑN-1
y = AKαN-α
y = A( K
N )α
y = Akα
Output per worker as a function of capital per worker and productivity
y/A = Akα/A
y = kα
2. When the capital per worker is the same yet country A output per worker is twice the level of
country B, then country A is twice productive than country B. Thus, productivity of country A is
2kα while the productivity of country B is kα.
3. Output per worker of A = yt
Output per worker of B = (1/2)*yt
Doubling output per worker of B = (2*(1/2)*yt) = yt

Macroeconomics 3
Thus, if the prime minister of country B doubles the level of capital per worker, his country will
become as rich as country A.
4.
Output per worker of B is 0.5yt
Output per worker of A is 1yt
It = sYt
It/N = sYt/N
It/N = syt
0.05*0.5yt = It/1
It = 0.025yt
Since It is the same, then:
0.025yt = syt
Therefore s of A is 0.025
6.
Consumption of B:
Ct/N = (1-s)Yt/N
ct = (1-s)yt

Macroeconomics 4
Assuming the output level per worker is 0.5;
ct = (1-0.05)0.5
ct = 0.475
Consumption of A:
ct = (1-0.025)1
ct = 0.975
Thus, the average person in country A is twice as rich as in B and also consumes twice as much.
Part B
1. Law of motion for aggregate capital
kt+1≈kt(1-δ-n)+it
2. Law of motion for capital in per worker terms:
kt+1= 1
1+ n(sAkαt + (1- δ) kt)
3. Steady state of capital per worker:
Set kt+a=kt =k
k= 1
1+ n(sAkα +(1- δ)k)
k(1+n)= (sAkα +(1-δ)k)
k+nk=sAkα +k-δk

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k+nk-k-δk=sAkα
(n + δ)k=sAkα
k
kα = sA
n+δ
k(1-α)=A( s
n+δ )
Therefore, k* = [A^( 1
1−α )] [( s
n+δ )^( 1
1−α )]
Steady state of output per worker
y* = Ak*α
y* = A[A^( 1
1−α )][( s
n+δ )^( 1
1−α )]
y* = A^( 1
1−α )][( s
n+δ )^( α
1−α )]
4. At the steady state, the country is at: Dk = s*f(k)- n*k-d*k (Kerschner, 2010). When the
population growth rate increases, n*k increases. Thus, n*k represents the decrease in capital
stock per labor unit due to increasing labor (Mankiw, 2014).
5. At country EU, the steady state at GN,EU = 0. Thus, since the labor force is not increasing,
output per worker and capital is not changing. At country UK, the steady state at GN,UK > 0. Since
the labor force is increasing, the output per worker increases while capital per worker decreases.
Therefore, EU has richer people while EU has a higher GDP growth.

Macroeconomics 6
References:
Kerschner, C. (2010). Economic de-growth vs. steady-state economy. Journal of cleaner
production, 18(6), 544-551.
Mankiw, N. G. (2014). Principles of macroeconomics. Cengage Learning.