[4] Long-Run Economic Growth: Solow Model

Question 1

The importance of economic growth for poor countries
[4]

1- poor
2- rich mainly
1-both

Answer 1

A change in the long-run rate of economic growth will have huge effects on living standards in the long run.

To help poor countries rise out of poverty, we will be making a huge difference in the lives of billions of people,

as well as creating new markets for our exports.

political influence on the world stage is correlated to economic power. Richer economies are more likely to have a say on world politics than poorer ones.

Question 2

What does the Solow Growth Model look at? [2]

Answer 2

looks at the determinants of economic growth and the standard of living in the long run

Question 3

How Solow model is generally different from Chapter 3 (national income) of Mankiw and Taylor’s textbook? [4]

Answer 3

K is no longer fixed: investment causes it to grow, depreciation causes it to shrink.

L is no longer fixed:population growth causes it to grow.

The consumption function is simpler.

No G or T (only to simplify presentation; we can still do fiscal policy experiments).

Question 4

Define output per worker ? [in terms of letters for both]

Define capital per worker?

Answer 4

y = Y/L

k = K/L

Question 5

What was the Production function?

How do we transform the production function

Answer 5

Y = F (K, L )

Assume constant returns to scale:
zY = F (zK, zL ) for any z > 0

Pick z = 1/L. Then
Y/L = F (K/L , 1)
y = F (k, 1)
y = f(k) where f(k) = F (k, 1)

f(k) is the “per worker production function,” it shows how much output one worker could produce using k units of capital.
This is the very same production function as in chapter 3. It is just expressed it differently.

Question 6

What does the production function look like graphically and what causes this shape?

What are the axis labels?

Answer 6

: this production function exhibits diminishing MPK.

X AXIS: Capital per worker, k

Y AXIS: Output per worker, y (Tip; Y for Y axis)

Question 7

What is the national income identity?

and in per worker terms?

Answer 7

Y = C + I (remember, no G )

In “per worker” terms:
y = c + i

where c = C/L and i = I/L

Question 8

What is the consumption function?

What does each variable represent?

Answer 8

Consumption function: c = (1–s)y
(per worker)

s = the saving rate, the fraction of income that is saved
(s is an exogenous parameter)

Note: s is the only lower case variable that is not equal to its upper case version divided by L

Question 9

What is saving per worker? [3 steps of simplifying]

hint; remember consumption func

Answer 9

saving (per worker) = y – c
= y – (1–s)y
= sy

Question 10

Show investment equals savings?

Answer 10

National income identity is y = c + i
Rearrange to get: i = y – c = sy
(investment = savings, like in chap. 3!)
[saving (per worker) = y – c ]

Using the results above, i = sy = sf(k)

Question 11

What doesn’t appear explicitly in any of the solow growth model equations?

What can we assume?

Answer 11

The real interest rate r does not appear explicitly in any of the Solow model’s equations. We can assume that investment still depends on r, which adjusts behind the scenes to keep investment = savings at all times (e.g., I=S).

Question 12

Def of depreciation?

what is used to denote it?

Answer 12

= the fraction of the capital stock that wears out each period

delta = d
therefore ‘dk’ is used

Question 13

What is the equation [2] for capital accumulation?

Answer 13

Change in capital stock = investment – depreciation

Δk                        =        I         –        dk
Since  i  =  sf(k) , this becomes:

      Δk              =        sf(k)         –        dk

Question 14

Why is the capital accumulation equation the Solow model’s central equation?

Answer 14

Determines behaviour of capital over time…
…which, in turn, determines behaviour of all of the other endogenous variables because they all depend on k. Δk = sf(k) – dk

E.g.,
income per person: y = f(k)
consump. per person: c = (1–s) f(k)

Question 15

What is the steady state of capital?

Answer 15

Δk = sf(k) – dk

If investment is just enough to cover depreciation [sf(k) = dk ],

then capital per worker will remain constant: dk = 0.

This constant value, denoted k*, is called the steady state capital stock.

Question 16

What is the steady state of capital graphically?
What point is it ?
And how do we know if we are approaching it?

Whats on the axis?

Answer 16

the point where the straight line of depreciation and the sf(k) investment line intersect [k*]

Remember, the change in capital Δk, is measured by the difference in investment and depreciation, so the larger the gap between curve investment line and straight depreciation line, the greater capital accumilation. As long as k < k, investment will exceed depreciation, and k will continue to grow toward k.

X axis: capital per worker
Y axis: Investment and depreciation

Question 17

EX:
Continue to assume s = 0.3, d = 0.1, and y = k ^1/2

Use the “equation of motion”
Δk = sf(k) – dk
to solve for the steady-state values of k, y, and c.

Answer 17

Δk = 0 in steady state

…..

slide 33 : k* =9 , y* = k*^0.5 = 3 ,

               c* = (1 - S)y* = 0.7x3 = 2.1

Question 18

An increase in the savings rate raises ___ causing _to grow toward a ___ ___ ___:

What happens to the sf(k) investment line in this case?

changes in _ and/or _ affect national saving. In the Solow model , we can simply change the ____ saving rate to analyze the impact of ___ policy changes

Answer 18

investment…
K
new steady state

Shifts up the sf(k) investment line, creating a new steady state

G and/ or T
exogenous
fiscal

Question 19

Higher s -> ?

And since y = f(k) , higher k* -> ?

Thus, the Solow model predicts that countries with higher rates of ___ and investment will have higher levels of ___ and __ ___ ___ in the long run.

Answer 19

Higher s -> higher k*.

And since y = f(k) , higher k* -> higher y* .

Saving investment

capital , and income per worker

Question 20

If we start with k

Answer 20

s

tax cuts
government spending
standard of living

Question 21

In the static model of Chapter 3, a fiscal expansion __ ___ investment.

The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the ___ ___, reduces ___.

If we were initially in a steady state (in which investment just covers ____), then the fall in investment will cause ___ ___ __, labour productivity and __ __ __ to fall toward a new, lower steady state.

(If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to ___ more slowly, and ___ their steady-state values.)

Answer 21

In the static model of Chapter 3, a fiscal expansion crowds out investment.

The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment.

If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labour productivity and income per capita to fall toward a new, lower steady state.

(If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

Question 22

International Empirical Evidence from the World Bank:

A scatterplot showing data for 96 countries: High investment is associated with __ ___ __ __, as the Solow model predicts.

Answer 22

high income per person

Question 23

Different values of s lead to different __ ___.

How do we know which is the “best” steady state?
Economic well-being depends on ____, so the “best” steady state has the highest possible value of ___ per person: c* = ?

An increase in s:
- leads to higher k* and y, which may raise c

• reduces consumption’s share of income (1–s), which may lower c*

Answer 23

Different values of s lead to different steady states.

How do we know which is the “best” steady state?
Economic well-being depends on consumption, so the “best” steady state has the highest possible value of consumption per person: c* = (1–s) f(k*)

An increase in s:
- leads to higher k* and y, which may raise c

• reduces consumption’s share of income (1–s), which may lower c*

Question 24

What is the Golden Rule level of capital?

Answer 24

the Golden Rule level of capital, the steady state value of k that maximizes consumption.

Question 25

How do we find the golden rule level of capital ?

Answer 25

To find it, first express c* in terms of k:
c = y* - i*
= f (k) - i
= f (k) - dk

    [ i* = Δk + dk ] because  Δk = 0 ,   [ i* = dk ]     

Then, graph f(k) and dk, and look for the point on x-axis where the gap between them is biggest.

This is the point where c* = f(k) - dk is equals the slope of the production func. equals the slope of the depreciation line: MPK = d (delta) slide 41

The problem is to find the value of k* that maximizes c* = f(k) - dk.
Just take the first derivative of that expression and set equal to zero: f’(k) - d = 0
where f’(k) = MPK = slope of production function and d = slope of steady-state investment line.

Question 26

Does the economy have a tendency to move towards a steady state?

Achieving the Golden Rule requires that policymakers adjust __

This adjustment leads to a new steady state with ___ ___

But what happens to consumption during the transition to the Golden Rule?

Answer 26

The economy does NOT have a tendency to move toward the Golden Rule steady state.

s

This adjustment leads to a new steady state with higher consumption.

Question 27

How can policy makers affect the national savings rate?

x2

Answer 27

policymakers can affect the national saving rate:
- changing G or T affects national saving

- holding T constant overall, but changing the structure of the tax system to provide more incentives for private saving (i.e., shifting from income tax to consumption tax in such a way that leaves total revenue unchanged)

Question 28

Starting with too much capital: if k* > k*Gold

then increasing c* requires a fall in __
In the transition to the Golden Rule, consumption is ____ at all points in time.

The change in the saving rate doesn’t immediately change k, so y doesn’t change immediately.

But the fall in s causes a fall in investment [because saving equals ____] and a rise in ____ [because c = (1-s)y, s has fallen but y has not yet changed.].

Answer 28

then increasing c* requires a fall in s.

In the transition to the Golden Rule, consumption is higher at all points in time.

But the fall in s causes a fall in investment [because saving equals investment]

consumption

Question 29

What is Break-Even Investment?
What is the equation for it?

What does each variable mean? [2]

Answer 29

the amount of investment necessary to keep k constant. (d + n)k

Break-even investment includes:
d k to replace capital as it wears out
n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)
Assume that the population – and labour force – grow at rate n (n is exogenous) ΔL / L = n

Question 30

With Population growth, what does the law of motion for k become?

Answer 30

Δk = sf (k) - (d + n) k

Change in Capital = actual inv. - break-even inv.
[both investments are in ‘per worker’ terms]

Question 31

What is the impact of increase in population growth?

[^ n]

Answer 31

An increase in n causes an increase in break-even investment, leading to a lower steady-state level of k.

Question 32

Thus, the Solow model predicts that in the long run countries with ____ population growth rates will have ___ levels of capital and___ ___ ___.

Answer 32

Thus, the Solow model predicts that in the long run countries with higher population growth rates will have lower levels of capital and income per worker.

Question 33

What is the golden rule of capital stock?

Answer 33

In the Golden Rule Steady State, the marginal product of capital net of depreciation equals the population growth rate.
MPK - d =n

Question 34

What is Mathulus model of 1798 neglect that meant his prediction that population growth will outstrip the Earth’s ability to produce food, leading to the impoverishment of humanity was never realised?

Answer 34

Malthus neglected the effects of technological progress
Predicts population growth will outstrip the Earth’s ability to produce food, leading to the impoverishment of humanity.
Since Malthus, world population has increased sixfold, yet living standards are higher than ever.

Question 35

What evidence is there for the Solow Growth model? x2

Answer 35

Evidence, from very long historical periods:

As world pop. growth rate increased, so did rate of growth in living standards

Historically, regions with larger populations have enjoyed faster growth.

Question 36

SUMMARY:

The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its __ ___ and negatively on its __ ___ ___.

An increase in the saving rate leads to:

• higher ____ in the long run
• faster growth ____
• but not faster ___ ___ growth.

Answer 36

The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate and negatively on its population growth rate.

An increase in the saving rate leads to:

• higher output in the long run
• faster growth temporarily
• but not faster steady state growth.

Question 37

If the economy has MORE capital than the Golden Rule level, then reducing saving will ___ consumption at all points in time, making all generations __ __.

If the economy has LESS capital than the Golden Rule level, then increasing saving will increase ____ for ___ generations, but reduce consumption for the ___ generation.

Answer 37

If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off.
If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation.