Human Capital - Income Differences

Question 1

Since y=k to the a h to the b

We get logy= alogk + blogh

What is final equation for logy* pg 16

Answer 1

a/1-a-β logsK + β/1-a-β log sH -
- a+β/ 1-a-β log(n+g)

So all have same denominator! Numerators: a, β, a-β

Question 2

Logy =
a/1-a-β logsK + β/1-a-β log sH -
- a+β/ 1-a-β log(n+g)

What do we do for the solow model

B) expression

C) find elasticity of y* with respect to sK and n+g assuming a=0.35, β=0

Answer 2

No human capital so set β=0 leaves us with

B)
Logy*(solow) = a/1-a logsK - a/1-a log(n+g)

C)
Differentiate with respect to sK
= a/1-a
So elasticity with respect to sK is = 0.35/1-0.35 = 0.54

Differentiate with respect to n+g
= -a/1-a
So elasticity w respect to n+g is = -0.35/1-0.35 = -0.54

Question 3

So logy* = a/1-a-β logsK + β/1-a-β log sH -
- a+β/ 1-a-β log(n+g)

What happens when we differentiate with respect to sK

B) do the differentiation

Answer 3

Gives us elasticity of y* (output) with respect to sK (how much output changes following change in savings rate for physical capital)

dlogy*/dlogsK = a/1-a-β (basically just sK disappears)

Question 4

Do the same differentiation but with respect for sH

B) respect for n+g

C) given this, what are our elasticities of y* with respect to sK, sH and n+g if a=0.35 β=0.4

D) compare to solow model

Answer 4

dlogy*/dlogsH = β/1-a-β

B)
dllogy*/dlog(n+g) = -a+β/1-a-β

C)
1.4
1.6
-3

D) Compared to Solow results of elasticity with respect to sK=0.54 and n+g=-0.54 (so output changes way more in HC model rather than solow!)

Question 5

Look at differences in incomes across countries

Suppose a=0.35 β=0.4 again, and sK and sH in rich country are 2x sK and sH in a poor country.

So sKr = 2sKp
sHr = 2sHp

Also assume rich countries n+g (growth of economy) is 20% lower so
nr + g = 0.8(np+g)

Final expression for logyr - logyp

Answer 5

Remember rules of logs: a - = combine by dividing
E.g logsKr - logsKp becomes logsKr/sKp

So we get
Logyr - logyp =
a/1-a-β logsKr/sKp + β/1-a-β log sHr/sHp
- a+β/ 1-a-β log(nr+g/np+g)

Same as logy* expression, just did the rule of logs!

Question 6

Logyr - logyp = logyr/yp
a/1-a-β logsKr/sKp + β/1-a-β log sHr/sHp
- a+β/ 1-a-β log(nr+g/np+g)

Then how can we simplify further using values given pg18

C) then find final value of logyr/yp

Answer 6

We know sKr= 2sKp

So
Logyr - logyp =
a+β / 1-a-β log2 - a+β/ 1-a-β log0.8

(Idk why the 2nd term β/1-a-βlogsHr/sHp disappears but)

C) use ln’s on my calculator (idk why log gave diff answer) to get 3ln2 - 3ln0.8 = 2.75

Question 7

So logyr/yp = 2.75

How do we find income differences now?

Answer 7

As logyr/yp = 2.75

Take e’s
yr = e to the 2.75 x yp
yr = 15.6yp

I.e income in y*r is 15.6x more than poor!

Question 8

How would this compare to Solow model values

Answer 8

Doing same method but β=0 we get rich income only 1.6x more than poor!

Thus shows how HC model allows to show cross-country income differences!

Question 9

Recall Solow model massively overpredicts rates of return differences (e.g MP 100x larger in poor than rich)

What is the MPK* and MPH* expressions
(in BGP)

Answer 9

MPK* = a(n+g)/sK
MPK* = β(n+g)/sH

In Solow was ay to the a-1/a

Question 10

Given still sKr = 2sKp and
nr+g = 0.8(np+g)

What do we find for our values of MPKr and MPHr (in terms of the poor countries MPK and MPH, since looking at rate of return differences cross country)

Answer 10

0.8/2 = 0.4

so MPKr = 0.4MPKp
MPHr = 0.4MPHp

Marginal product in rich is only 40% of MP of poor. More realistic than Solows huge overestimation (recall we said the reason difference being not so big in reality is international capital markets)

Question 11

Does this and Solow model predict unconditional convergence (to SAME EXACT steady state levels)

b) conditional convergence?

Answer 11

No, if countries have different steady states, rich remain rich poor remain poor. so no unconditional convergence (i.e country convergence)

However condititional convergence; i.e converge to their own diffrent steady state

Question 12

If a poor country and rich country have the same steady state. Which should grow faster

Answer 12

Poor

Question 13

At what rate (λ) does y converge to y* (steady state)

b) what about rate for Solow model

Answer 13

λ = (1-a-β) (n+g+δ)

e.g if a = b = 1/3 and n+g+δ=6%
λ= 2%

b) β=0
λ=4% i solow model

i.e with human capital, slower convergence! more accurate

Question 14

Do countries with low initial income relative to their balanced growth path have lower or higher growth rates?

Answer 14

Higher - so conditional convergence (to their own steady state) faster

Question 15

Mankiw estimates how much of cross-country variation in income per capita is explained by 2 factors

what are the 2 factors and how much

Answer 15

more than 50% of variation is explained by savings rates and population growth

Question 16

HC model - how much does it account for cross-country variation in come

Answer 16

80%