Lecture 11 - Solow Model

Question 1

Capital per worker (k) formula

, and what assumption do we make?

Answer 1

k = K/N

N is labour which we assume grows overtime at rate n

Question 2

How do we find how k changes as capital stock and labour force change?

(With labour force growth n)

Answer 2

We take logs of the previous equation to get
lnk=lnK - lnN

Then differentiate with respect to time dk/dt

Question 3

What is the final equation for capital per worker with population growth? (Solow model with population growth)

2. Using this, what increases k and what decreases it?

Answer 3

𝑖 − (δ+𝑛)𝑘 =0

2. Investment per worker (𝑖) increases k , while depreciation and labour force growth (δ+𝑛) decrease it.

Question 4

Why do we equate it equal to 0?

Answer 4

To find the steady state equilibrium.

Steady state is k* (natural dynamics always end up here)

Question 5

Solow model with population growth diagram (pg 5)

2. How does the steady state model shows a balanced growth path (all real variables grow at same rate)

Answer 5

(δ+𝑛)k known as break even investment (instead of depreciation)

2. Because output per worker (y) = Y/N and k=K/N and N grows at rate n, output and capital grow at the same rate.

Thus Proves steady state shows a balanced growth path!!!

Question 6

So Y and K (output and capital) grow at the rate n.

What is this called?

Answer 6

Balanced growth path.

Question 7

What happens if n increases

Answer 7

n is part of breakeven investment, so a shift upwards

reduces steady state k* to a lower one.

Question 8

Why does this happen? (k falling)

Answer 8

As we see the shift upwards in breakeven investment.

We can see breakeven investment>saving and

Not enough saving to keep k constant to match the increases population growth so k falls.

Question 9

Note: theory suggest living standards are lower in countries with higher birth rates (increased n)

Question 10

Now we can incorporate unemployment to this Solow model…

What assumption do we make

Answer 10

There is a natural rate of unemployment (u)

It is a proportion

Question 11

What is the expression for labour employed then?

Answer 11

(1-u)N

N is labour (which grows at rate n)

E.g if u=0.05 and N=100
Labour employed is 95

Question 12

Sub our labour employed equation into the cobb Douglas function, assume A=1

Answer 12

𝑌=((1−𝑢)𝑁) to the power of V 𝐾 to the 1-V

Question 13

Then find output per worker by dividing by n

Answer 13

y = 1-u to the v (K/N) to the 1-v

Then ends up as
y= (1-u) to the v [k] to the 1-v

k=K/N

Question 14

Assume a policy is introduced that reduces the natural rate of unemployment (u) at time T.

What happens, and show graphically (output time graph)

Answer 14

Output rises immediately as more workers employed (sharp rise in diagram)

It continues to rise as we have the new K/N ratio. (More workers per capital so output rises)

Question 15

What does this look like in Solow model?

Answer 15

Rise/Shift in investment (recall investment expressed as function of output: i=sf(k), output f(k) rises as a result of the fall in unemployment)

So an increase in the steady state k*

Question 16

Relationship between income/output per worker (y) and effective investment rate (another way to write the steady state sf(k)=(δ+n)k

Answer 16

After rearranging…. (cover later)

y= s/(δ+n) to the a/1-a

Where s/(δ+n) is the effective investment rate

Relationship: if s increases y increases.
More saving, more investment (i=sf(k)) , increase k. With more k increased y (we have shown K and Y grow at same rate n)
Also more n, lower y

Question 17

How is the effective investment rate link with the Solow model? (Use an increase of n)

Answer 17

Effective investment rate is given by

s / (δ+n)

Using this, if n (growth of labour force) increases, steady state k falls (shown by Solow model shift up in break even investment).

This fall in capital per worker means income per person (y) falls too. (See this through y = (s/δ+n) to the a/1-a

Question 18

Two perspectives on the impact of labour force/population growth

Answer 18

Malthus
Kremer

Question 19

Malthus main idea

2. When does population growth stop, and what assumption does this depend on?

Answer 19

Population growth halts income growth. (Shown by effective investment rate formula: increased n reduces y.

2. Population growth n stops when we reach subsistence level of income (the income needed to survive). This depends on the assumption that adults’ only role is having children.

Question 20

How is income needed to survive expressed? (Malthus’ idea)
(Use the effective investment rate equation)

Answer 20

yPov = (s/δ+nPov) to the α(1−α)

nPov is the high population growth
yPov is the subsistence income level (needed to keep to survive) if not child dies as pop growth>food supply

Question 21

Why is Malthus’ theory dismissed

Answer 21

It ignores that other factors generate growth (technical progress).

Question 22

Kremer’s argument

Answer 22

Population growth drives economic growth.

Question 23

2 ways Kremer said population growth increases growth

2. Why does population growth faciliate the 2 ways.

Answer 23

Population growth drives economic growth since it encourages innovation and technical progress

2. Population growth adds to human diversity for innovation. (Humans are not homogenous i.e we’re all different!!)

Question 24

How to model such technical progress that Kremer iterates drives growth, put it into the Solow model

Just RMB inital function and steady state final equation for now!

(Hint: to start adjust production function)

Answer 24

Let y=Y/EN (E is efficiency of workers, so it becomes output per efficient worker)

y = F(K/EN , 1) = f(k)

Where k = K/EN (capital per efficient worker)

Then take logs to get
lnK - lnN = lnE and differentiate.

We get i − δ𝑘 = 𝑑𝑘/dt + 𝑛𝑘 + 𝑔𝑘

Set dk/dt=0 for steady state and rearranging gets us

Final equation!!!
𝑖 − (δ + 𝑛 + 𝑔)𝑘 =0
g is rate of efficiency growth

Question 25

What does output per worker increase by?

Answer 25

Because y= Y/EN, we see output per worker increase by the rate g (rate of efficiency growth)

So we get output and income gains

Question 26

So with technical progress, at what rate does…

1. Output per worker grow at
2. Total output grow at
3. Capital per effective worker
4. Output per effective worker

Answer 26

1. g
2. n + g
3. 0
4. 0

So only total output and output per worker increase from technical progress (NOT CAPITAL PER EFFECTIVE WORKER, OR OUTPUT PER EFFECTIVE WORKER)

Question 27

If an economy moves from a steady state with positive population growth to a zero population growth rate, then in the new steady state, total output growth will be ______, and growth of output per person will be ______.

Answer 27

Lower; (since total output growth grows at n+g, so without n now so smaller)

Same as it was before. (Output per worker grows at g, and so with no n it doesn’t make a difference