Human Capital

Question 1

So revisit: shortcoming of Solow model

B) Shortcoming of endogenous (new) growth theory

Answer 1

Does not explain cross-country income differences

B) explains worldwide rises in living standards overtime, but still not cross-country income differences

So now add human capital accumulation to Solow model to account for this omission from both!

Question 2

Human capital: is it non-excludable and non-rival

Answer 2

It is rival (one’s skills can only be employed for one firm/task at a time) and excludable (firms own human capital, thus prevent use from other firms)

Question 3

So now we include human capital to Solow model.

Production function

B) what does a ‘skilled’ worker supply?

Answer 3

Y = K to the a H to the β (AL) to the 1-a-β

H: stock of human capital

B)
A skilled worker supplies 1 unit of L, and some amount of H

Question 4

Assuming 0 depreciation δ, what is our

net increase in capital (K.)

Answer 4

A) K. = sKY

Only slight change is sK (little K) instead of s
sK is fraction of output (saved) devoted to physical capital accumulation

Question 5

What about human capital accumulation H.

Answer 5

H. = sHY

sH (little H) is fraction of output saved devoted for human capital accumulation

(Accumulated the same way as K i.e giving up consumption and instead saving, measured in output)

Question 6

Prove y=k to the a h to the β is the intensive production function (pg5)

Answer 6

Y/AL = K to the a H to the β (AL) to the 1-a-β / AL
= K to the a H to the β/ (AL) to the a+β
= (K/AL) to the a (H/AL) to the β

Then put in small caps
y = k to the a h to the β

Question 7

Now look at law of motion of physical capital:
Recall is key equation for Solow model,

B) what do we do with it

Answer 7

k = K/AL

B) take logs
Logk = logK - logA -logL

Question 8

What does differentiating with respect to time do? Expression

(Which can be simplified)

Answer 8

Give us growth rates

k./k = K./K - L./L - A./A
= sKY/K - n - g

As recall K. = sKY

Question 9

Important: obtain Locus of k.
By x both sides by k to get k. on its own! Then solve for final answer by subbing in values Pg7

Answer 9

k. = sKY/K (k) - (n+g)k
As k=K/AL
= sKY/K (K/AL) - (n+g)k
= sKY/AL - (n+g)k
Then as y = Y/AL
= sKy - (n+g)k
And then recall y=k to the a h to the β
= sKk to the a h to the β - (n+g)k

Looks hard but isn’t if rmb steps

Question 10

So k. = sKk to the a h to the β - (n+g)k

Locus of k. = 0

What does it mean

Answer 10

sKk to the a h to the β - (n+g)k = 0

It is steady state, combination of human and physical capital that keeps capital per effective worker fixed/constant

Question 11

So what is the expression for the level of k at which it is fixed?

Hint; expression for k, and C! pg 7

Answer 11

Fixed where k.=0 i.e
sKk to the a h to the β - (n+g)k = 0

Or
k = Ch to the β/1-a

Where C = [sK / n+g] to the 1/1-a

Question 12

Now law of motion of human capital

What is our log form expression

B) same process, differentiate with respect to time and x both sides by h (not k this time since human capital)

We get same answer except replace k with h! State final answer

Answer 12

Logh = logH - logA - logL

(Got from taking logs of h=H/AL)

B)
sHk to the a h to the β - (n+g)h

(Physical capital one was
sKk to the a h to the β - (n+g)k

Question 13

Locus of h. = 0 what is it

Answer 13

sHk to the a h to the β - (n+g)h = 0

The combination of human and physical capital that keeps human capital per effective worker fixed/constant

Question 14

So what is the expression for the level of h at which it is fixed?

Hint; expression for k, and B!
Pg 9

Answer 14

k = Bh to the 1-β/a

Where B = [n+g/sH] to the 1/a

Question 15

What do we get when loci of k.=0 and h.=0?

Answer 15

Steady state where k h and y are constant in balanced growth path

Question 16

What do K H and Y grow at?

Answer 16

n+g !
K = kAL H = hAL Y=yAL

Since k h y constant in balance growth path as mentioned, only influencer is AL. Which grows at rate n+g!!!

Question 17

What about K/L H/L Y/L

Answer 17

As we found K H Y grow at n+g, just divide by L now,
means n is no longer matters, so grow at rate g!

E.g K/L is kAL/L
= kA
Therefore just grows at rate g!!!

Question 18

Draw locus of k. = 0 diagram pg11 (own drawing)

Explain what k.=0 curve looks like

And what below and above line mean

Answer 18

K.=0 line is upward sloping and concave as S.O.C finds DMR.

along the line - combinations of k and h that keep k constant !
above the line - k. Is falling.
Below the line k. Is increasing

Question 19

Draw locus of h. = 0 diagram pg12 (own drawing)

Explain what h.=0 curve looks like

And what below and above line mean

Answer 19

Upward sloping but with increasing returns!

Left of curve - h is increasing towards the line
Right of curve - h is decreasing towards the line

Question 20

Then combine 2 diagrams to get the steady state where k.=0 and h.=0

I.e k h and y constant, K H Y grow at n+g, K/L H/L Y/L grow at rate g
(I didnt draw diagram but is just the simple intersection)

Question 21

Effect of an increase in sK

B) does growth rate change?

C)) diagram pg 13

Answer 21

Steady state levels of k h rise

B) but no permanent change in growth rate, only temporary growth rate during transition to SS.

C) shift upwards in k.=0 curve

Question 22

Difference between technology diffusion across countries vs human capital

Answer 22

Human capital is within individuals - diffusion less rapid opposed to tech

So human capital slower diffusion can explain cross country differences, which endogenous and original solow can’t!

Question 23

Move (n+g)k and (n+g)h for both locus expressions:

sKk* to the a hto the β = (n+g)k
sHk* to the a hto the β = (n+g)h

k* h* are steady state values of k,h.