L13 - Long-Run Economic Growth II

Question 1

What is the golden rule?

Answer 1

• •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*
• So, how do we find the s and k* that maximizes c* ?

This is called the Golden Rule

Question 2

What is the golden rule level of capital?

Answer 2

Question 3

What does the Golden Rule of Capital Stock look like on a graph?

Answer 3

It is possible to confuse this graph with the other Solow model diagram, as the curves look similar.

Be aware the differences:

1. On this graph, the horizontal axis measures k*, not k. Thus, once we have found k* using the other graph, we plot that k* on this graph to see where the economy’s steady state is in relation to the golden rule capital stock.
2. On this graph, the curve measures f(k*), and not sf(k).
3. .On the other diagram, the intersection of the two curves determines k*. On this graph, the only thing determined by the intersection of the two curves is the level of capital where c*=0, and we certainly wouldn’t want to be there.
4. There are no dynamics in this graph, as we are in a steady state. In the other graph, the gap between the two curves determines the change in capital.

Question 4

when does the golden rule of consumption occur?

Answer 4

Using calculus, deriving the condition MPK = d is straight-forward:

The problem is to find the value of k* that maximizes:

c* = f(k*) - 𝛿k*.

Just take the first derivative of that expression and set equal to zero:

f𝛿(k*) - 𝛿 = 0

where f𝛿(k*) = MPK = slope of production function
and 𝛿 = slope of steady-state investment line.

Question 5

How does an economy move toward the Golden Rule steady state?

Answer 5

• •The economy does NOT have a tendency to move toward the Golden Rule steady state.
• Achieving the Golden Rule requires that policymakers adjust
• This adjustment leads to a new steady state with higher consumption

Remember: 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 6

What happens to consumption when we start with too much capital?

Answer 6

t₀ is the time period in which the saving rate is reduced. There is a difference in the behaviour of each variable (i) before t₀, (ii) at t₀ , and (iii) in the transition period (after t₀ ).

Before t₀:

• in a steady-state, where k, y, c, and i are all constant.

At t₀:

• 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 investment] and a rise in consumption [because c = (1-s)y, s has fallen but y has not yet changed.].
• Note that Δc = -Δi, because y = c + i and y has not changed.

After t₀:

• In the previous steady-state, saving and investment were just enough to cover depreciation. Then saving and investment were reduced, so depreciation is greater than investment, which causes k to fall toward a new, lower steady state value.
• As k falls and settles on its new, lower steady state value, so will y, c, and i (because each of them is a function of k).
• Even though c is falling, it doesn’t fall all the way back to its initial value.
• Policymakers would be happy to make this change, as it produces higher consumption at all points in time (relative to what consumption would have been if the saving rate had not been reduced).

Question 7

What happens to consumption when we start with too little capital?

Answer 7

Before t₀: in a steady-state, where k, y, c, and i are all constant.

At t₀:

• The increase in s doesn’t immediately change k, so y doesn’t change immediately.
• But the increase in s causes investment to rise [because higher saving means higher investment] and consumption to fall [because we are saving more of our income, and consuming less of it].

After t₀:

• Now, saving and investment exceed depreciation, so k starts rising toward a new, higher steady-state value.
• The behaviour of k causes the same behaviour in y, c, and i (qualitatively the same, that is).
• Ultimately, consumption ends up at a higher steady state level. But initially consumption falls.
• Therefore, if policymakers value the current generation’s well-being more than that of future generations, they might be reluctant to adjust the saving rate to achieve the Golden Rule.
• Notice, though, that if they did increase s, an infinite number of future generations would benefit, which makes the sacrifice of the current generation seem more acceptable.

Question 8

What is the formula for Population growth rate?

Answer 8

• the formula also has a continuous-time analogue as well where L(dot) can be used to represent the derivative of L with respect to time and replace ΔL in the formula

Question 9

What is the Break-even level of investment?

Answer 9

(𝛿 + n)k = break-even investment, the amount of investment necessary to keep k constant.

Break-even investment includes:

•𝛿 k to replace capital as it wears out

•nk 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)

Question 10

What is the law of motion of k under the including population growth?

Answer 10

actual investment and break-even investment are in per worker magnitudes

Question 11

What will be the new steady-state taking into account population growth?

Answer 11

Question 12

How does an increase in the population growth rate effect the steady state?

Answer 12

• The policy of the state may affect the population growth rate e.g. china’s one-child policy
• Too expensive to have children - Japan now has a declining birth rate

•Higher n –> lower k*.

•And since y = f(k) ,
lower k* –> lower y* .

•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 13

What happens when s falls?

Answer 13

a fall in s (caused, for example, by tax cuts or government spending increases) leads ultimately to a lower standard of living.

• 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 14

What is the Golden Rule with Population growth?

Answer 14

Question 15

What is the Malthusian Model (1798)?

Answer 15

Alternative perspective on population growth

The Malthusian Model (1798)

• 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.
• Malthus neglected the effects of technological progress.

Question 16

What is the Kremerian Model (1993)?

Answer 16

An alternative perspective on population growth

• Posits that population growth contributes to economic growth.
• More people = more geniuses, scientists & engineers, so faster technological progress.
• Evidence, from very long historical periods:
• As world pop. the growth rate increased, so did the rate of growth in living standards
• Historically, regions with larger populations have enjoyed faster growth

Question 17

What are some examples of technological progress?

Answer 17

• U.S. farm sector productivity nearly tripled from 1950 to 2009.
• The real price of computer power has fallen an average of 30% per year over the past three decades.

•2000: 361 million Internet users, 740 million cell phone users
2011: 2.4 billion Internet users, 5.9 billion cell phone users

•2001: iPod capacity = 5gb, 1000 songs. Not capable of playing episodes of popular TV shows.

2012: iPod touch capacity = 64gb, 16,000 songs. Can play episodes of popular TV shows

Question 18

How can technological progress be implemented into the Solow model?

Answer 18

•A new variable: E = labour efficiency

•Assume:
Technological progress is labour-augmenting: it increases labour efficiency at the exogenous rate g:

g = ΔE/E

Question 19

What is the production function of the Solow model including technological progress?

Answer 19

We now write the production function as:

Y = F(K, L x E)

where L x E = the number of effective workers.

Hence, increases in labour efficiency have the same effect on output as increases in the labour force.

• you should now be able to distinguish between the number of workers and the number of effective workers

Question 20

How has Technological progress effect the notation in the Solow model?

Answer 20

Question 21

What is break-even investment with the inclusion of technological progress?

Answer 21

(𝛿 + n + g)k = break-even investment: the amount of investment necessary to keep k constant.

Consists of:

• 𝛿k to replace depreciating capital
• nk to provide capital for new workers
• gk to provide capital for the new “effective” workers created by technological progress

Remember: k = K/LE, capital per effective worker. Technological progress increases the number of effective workers at rate g, which would cause capital per effective worker to fall at rate g (other things equal). Investment equal to gk would prevent this.

Question 22

What does the Solow model look like including Technological progress?

Answer 22

Horizontal axis should be Capital per Effective Worker

There are minor differences between this and the Solow model graph from the previous lecture :

• •Here, k and y are in “per effective worker” units rather than “per worker” units.
• •The break-even investment line is a little bit steeper: at any given value of k, more investment is needed to keep k from falling - in particular, gk is needed.
• Otherwise, technological progress will cause k = K/LE to fall at rate g (because E in the denominator is growing at rate g).
• With this graph, we can do the same policy experiments as in the previous lecture. We can examine the effects of a change in the savings or population growth rates, and the analysis would be much the same.
• The main difference is that in the steady-state, income per worker/capita is growing at rate g instead of being constant.

Question 23

What are the different Steady State growth rate in the Solow Model with technological progress?

Answer 23

Explanations:

• k is constant (has zero growth rate) by definition of the steady-state
• y is constant because y = f(k) and k is constant
• To see why Y/L grows at rate g,
• note that the definition of y implies (Y/L) = yE.
• The growth rate of (Y/L) equals the growth rate of y plus that of E.
• In the steady-state, y is constant while E grows at rate g.
• Y grows at rate g + n. To see this, note that Y = yEL = (yE) xL. The growth rate of Y equals the growth rate of (yE) plus that of L.
• We just saw that, in the steady-state, the growth rate of (yE) equals g. And we assume that L grows at rate n.

Question 24

What is the golden rule with technological progress?

Answer 24

Remember investment in the steady-state, i*, equals break-even investment.

Question 25

How does an increase in capital affect output?

Answer 25

Question 26

How does an increase in labour affect output?

Answer 26

Question 27

How can the total change in output be shown from a change in labour and capital?

Answer 27

• MPK x K –> change in output due to a change in K
• ΔK/K –> growth rate of K

when divided by Y - the share in the growth rate of output due to the change in K

Question 28

How can the formula of the Growth rate of output, capital and labour be interpreted?

Answer 28

Question 29

What happens to the output growth formula when we assume constant returns to scale?

Answer 29

Question 30

How is total output affected by technological progress?

Answer 30

• The framework up to now has ignored the role of technology.
• In practice, we know that technological progress improves the production function.
• For any given amount of input today, more output can be produced now than in the past.
• This means that it is perhaps more appropriate to write that Y = AF (K, L)
• Here, we define A as total factor productivity (TFP)
  
  • (A and E are interchangeable for technological progress)

Question 31

What is the formula for output growth including technological progress?

Answer 31

Question 32

How do you calculate TFP growth?

Answer 32

The Solow residual actually captures everything that the model does not account for

Question 33

How do you compute TFP growth using continuous-time equations?

Answer 33

Question 34

What is TFP growth highly correlated with?

Answer 34

Output growth

this would suggest that when there is a recession, it is due to a negative technology shock

although technology shock is hard to explain - it’s not like we have forgotten how to make things from the past 5 years

• does the Solow residual, therefore, capture something else other than technology?

In the 1970’s the ‘negative technology shock’ was due to OPEC increasing the price of oil massively, making a lot of technology at the time obsolete

Question 35

How else can the Solow residual be interpreted?

Answer 35

• Robert Solow intended that the Solow residual could be used to analyse growth in the long run. Edward
• Prescott suggests that the Solow residual is useful in analysing technology change over shorter durations of time
• Data in Figure 2 shows that the Solow residual fluctuates quite markedly over the sample period 1960-2010
• The fact that the Solow residual is highly correlated with movements in output suggests that a major cause of recessions is adverse shocks to technology: this view, combined with the neutrality of money, is at the heart of real business cycle (RBC) theory.
• This view is, however, highly controversial.

Question 36

What was the study performed by Mankiw, Romer and Weil (1992) used for?

Answer 36

known as the human capital model, in which adds human capital to a model of economic growth

• psi denoted the effectiveness of the education

Question 37

how do you derive the effect if psi?

Answer 37

Question 38

How does physical capital accumulate in the Human Capital Model?

Answer 38

• Similar to the equation of motion used in the basic Solow model - K is a continuous-time analogue
• the expression is in output per worker term!

Question 39

How do you solve the for the steady-state of the human capital model?

Answer 39

Question 40

What is important about the steady-state solution of the human capital model?

Answer 40

helps to explain why some countries are rich, and some countries are poor. Rich countries:

• spend a high proportion of their income on investment (s_k)
• spend more time accumulating skills (he^Ψu );
• enjoy lower population growth rates (n)
• have higher levels of technology (A)

Question is: does this model perform well empirically in explaining growth differences between rich and poor countries?

u is assumed to be the average number of years of schooling for the population - therefore, u will necessarily di§er across countries.

Question 41

How can we define per capita income relative to another country?

Answer 41

Question 42

What can we learn from the graph of relative technology against observed relative per capita income?

Answer 42

• rich countries have high levels of technology;
• strong correlation between relative A and relative y suggests that rich countries not only have high levels of physical and human capital, but use such inputs very productively;
• correlation is far from perfect - Bangladesh and Singapore have levels of A much higher than might be reasonably expected.
  
  • This suggests that we have not controlled adequately for differences in the education systems, on the job training, the nature of the labour force, and so on across countries;
• These di§erences will therefore be included in A, meaning that our estimates of A should be considered as estimates of Total Factor Productivity (TFP), and not technology.

If we interpret A as TFP, the results still indicate that the variations between the richest and poorest countries are sizable.

Question 43

What empirical data can we learn from the relative productivity graph?

Answer 43

Returning to expression y(hat)* = (sk(hat)/(x(hat))^(α/1-α)*h(hat)A(hat), we can arrive at some other rather interesting conclusions:

• richest countries in the world have GDP per capita around 32 times higher than the poorest countries
• these differences can be decomposed into differences associated with investment rates in k, h and productivity.
• richest countries in the world have investment rates around 25% of GDP versus 5% for the poorest countries
• workers in rich countries have between 10-11 years of education on average; for workers in the poorest countries, the figure is around 3 years.
• by construction, all remaining differences can be put down to differences in TFP.