ECON2123 Problem Set 4Name: LEE Yi Xuan JeremyStudent ID: 20129656

Part I1. False. Eventually the economy will converge to a constant level of output

per worker, therefore the growth rate of output will be equal to zero in the long run no matter what the saving rate is.

2. False. When the saving rate is equal to zero, capital is equal to zero and output is equal to zero. Therefore there will not be any consumption when the output is equal to zero.

3. False. A higher saving rate will lead to an increase in steady-state levels of output per effective worker and capital per effective worker. This means that there will be higher growth until the economy reaches a higher, balanced growth path.

4. False. In steady state, output per effective worker is constant.5. False. The investment has to cover both depreciated capital and

population growth.

Part II

1. a) U.S. consumption per capita = $3000b) Mexico consumption per capita = 6000 pesosc) Mexico consumption per capita = $600d) Mexico consumption per capita in US prices = $800e) Proportion of Mexico’s standard of living relative to U.S.:

Exchange rate method = 20%PPP method = 26.7%There is a difference.

2. a) Yes. This is because if the scale of operations is multiplied by a factor m, the output will also be multiplied by a factor m.

b) YN

=√K √NN

=√ KNc) No. This is because if capital per worker is multiplied by a factor m,

the output per worker will only multiply by a factor √m.

d) Y=√K=K12

gY ≈12gK

Therefore, the growth rate of output is roughly equal to half

the growth rate of capital.e) Required growth rate of capital = 4%f) The ratio of capital to output over time will keep increasing over time

as more capital will be required to sustain the same growth level of output.

g) No. Due to the decreasing returns to capital over time, the increase in capital will lead to smaller and smaller increases in output over time. Given this condition it is not possible to sustain the 2% growth rate

forever in this economy.3. a) No. This is because if capital per worker over time and population is

multiplied by a factor m, the output per worker over time will only multiply by a factor 0.5m.

b)Y tN

=0.5 √K t√NN

=0.5√ K tNc)

K t+1N

−K tN

=0.5 s √ K tN −δK tN

At steady state, K t+1N

−K tN

=0

Therefore, 0.5 s√ KN=δ KN

and KN

=0.25( sδ )2

d) YN

=0.5√ K tN =0.5√0.25 ( sδ )2=0.25 sδCN

=YN

−δ KN

=0.25 sδ−0.25δ( sδ )

2

=0.25s(1−s )δ

e)YN

=0.25 sδ=0.25 0.16

0.08=0.5

KN

=0.25( sδ )2

=0.25( 0.160.08 )2

=1CN

=YN

−δ KN

=0.5−0.08 (1 )=0.42

f)YN

=0.25 sδ=0.25 0.32

0.08=1

KN

=0.25( sδ )2

=0.25( 0.320.08 )2

=4CN

=YN

−δ KN

=1−0.08 (4 )=0.68

g)

h) The output per worker is increasing at a decreasing rate, while the growth rate of output per worker is decreasing.

3.

a) { YAN

=√ KAN

IAN

=s YAN

=(δ+gA+gN ) KAN

Therefore,

(i). KAN

=( sδ+gA+gN )

2

=1

Level of output per worker, Y/N

Time, t

1

0.5

t

s0

s1

(ii). YAN

=√ KAN

=1

(iii). 0%(iv). 4%(v). 4% + 2% = 6%

b) (i). KAN

=( sδ+gA+gN )

2

=0.64

(ii). YAN

=√ KAN

=0.8

(iii). 0%(iv). 8%(v). 8% + 2% = 10%An increase in the rate of technological progress will:Decrease the steady-state levels of capital per effective worker;Decrease the steady-state levels of output per effective worker;Increase the rate of growth of output per worker.

c) (i). KAN

=( sδ+gA+gN )

2

=0.64

(ii). YAN

=√ KAN

=0.8

(iii). 0%(iv). 4%(v). 4% + 6% = 10%People are better off in a) because the capital per effective labor and output per effective labor are higher in a).