Week 7 & 8: Growth Flashcards
Is GDP per capita a good measure of economic wellbeing?
- Many short-comings, including:
(1) no account taken of inequality;
(2) doesn’t include ‘unpriced’ economic activities that impact on well-being, such as
pollution, global warming, child-care / housework;
(3) some components of GDP may reflect ‘bads’.
• However, evidence suggests an imperfect but positive correlation between GDP
per-capita and happiness (see BAG Ch 10 (3rd edition) for further discussion).
• So we will focus on modelling growth in GDP per-capita and assume it is a reasonable, if flawed, measure of economic wellbeing.
Summary of growth facts:
- A lot of variation through time in BOTH GDP per-capita and growth rates.
- A lot of variation across countries in BOTH GDP per-capita and growth rates.
- Some countries have grown very fast and moved from relatively low income to
very high income - some countries have gone in the opposite direction. - No consistent pattern in relationship between the level of GDP per-capita and
growth rate. In the case of high-income countries, economies that start off with
lower GDP per-capita (in 1950) tend to see higher growth rates. But this is not
apparent for other country groupings.
How to compute growth rates:
If we want to calculate the average annual growth rate over n years beginning
in year t, then
𝑦𝑡+𝑛 = (1 + 𝑔)^𝑛 𝑦𝑡
which can be re-written as
𝑔 = (𝑦𝑡+𝑛/𝑦𝑡)^1/n - 1
Or take the natural logarithms of growth
G = (ln[yt+n]-ln[yt])/n
The Solow Growth Model without technological change
❑ Assume an aggregate production function of the following form:
𝑌 𝑡 = 𝐹(𝐾𝑡, 𝑁𝑡, 𝐴𝑡)
where
• Y is real aggregate output (GDP) as defined in previous lectures.
• K is the capital stock.
• N is the number of workers in the economy.
• A is the level of technology.
Output is a function of capital, workers, and technology.
• K, the (physical) capital stock, is the real value of physical capital in the
economy at a particular point in time. It includes factories, production lines,
office buildings, computer equipment, machinery, tools etc. It is used by
workers to produce output.
• N is total employment in the economy, the number of people working in
the economy.
• A is technology. Think of technology as the stock of ideas (knowledge) that
allows the combination of K and N to produce level of output Y.
• Assume A is fixed at 1. That is, we will initially look at the Solow Model in
a world without technological change.
This allows us to focus on the role of how changes in the capital stock and
in population influence growth.
- assume decreasing returns to labour and to capital
- constant returns to scale: 𝑥𝑌 𝑡 = 𝐹(𝑥𝐾𝑡, 𝑥𝑁𝑡)
Rewrite in per-worker terms: x=1/Nt
Output per-worker is a function of the capital stock per-worker.
The relationship between 𝑦𝑡 and 𝑘𝑡 is summarised in the following figure.
SGM w/o tech. Change: Output per workers depends on capital per worker
∆𝑘𝑡+1 =∆𝐾𝑡+1/𝑁𝑡− 𝑔𝑁𝑘t
- Capital stock increases with investment (𝐼𝑡), but falls with depreciation.
• Investment is the formation of new capital stock.
• ∆𝑘𝑡+1depends on the balance between actual investment (determined by (i))
and required investment (determined by (ii)).
• As long as actual investment is higher than required investment, the total stock of capital p.w. will increase, and vice versa.
• There is a point where actual investment equals required investment, that is how much we invest in capital p.w. is exactly how much is needed to keep
capital p.w. constant.
The SGM w/o tech. Change, graph:
- At k’, actual investment is greater than required investment; this is shown by
the distance AB.
This implies that capital p.w. is increasing and so too is output p.w.; the economy moves to the right as indicated by the arrow.
This process will continue as long as actual investment is greater than required investment, i.e. as long as the actual investment schedule is above the required investment line. - The increase in capital p.w. will eventually stop.
As we move to the right along the horizontal axis, higher levels of capital per worker increase actual investment but by less and less, while required investment keeps increasing in proportion to capital.
For some level of capital p.w. , k, actual investment equals required
investment. Once the economy reaches k, it stays there.
• There is a long-run equilibrium or steady-state at G, with the steady state capital p.w. of k* and the steady state output p.w. of y* .
• At k’’, actual investment is smaller than required investment; this is shown by the distance DC.
This implies that capital p.w. is decreasing and so too is output p.w.; the economy moves to the left as indicated by the arrow.
This process will continue as long as actual investment is less than required investment, i.e. as long as the actual investment schedule is below the required investment line.
• This fall in capital p.w. will eventually stop when actual investment equals required investment.
• Steady-state consumption p.w. is the vertical distance between the production
function p.w. and the actual investment schedule (the vertical distance FG in our
diagram).
• In steady-state, consumption per-worker is also constant [c* = (1 − s) y
The growth of GDP per-worker is zero in steady-state, as is the growth of capital
per-worker. This is a strong conclusion in this simple form of the model.
This result is due to
(a) our decreasing returns to capital assumption: as k increases y increases but at
a decreasing rate, and
(b) required investment increases linearly
“Golden rule” steady-state
A higher saving rate, s, leads to higher output p.w. as we saw earlier.
If a government wanted to maximise output p.w. in the long-run it COULD
encourage as high a saving rate as possible.
But would this be a sensible course of action?
→ Welfare depends on consumption per-capita, not output per-capita. Govt is
likely to be interested in the saving rate that maximises the steady-state
consumption p.w..
Assume the economy is initially in steady-state at point A, where
• the population growth rate is 𝑠𝑜𝑙𝑑.
• capital p.w. is 𝑘𝑜𝑙𝑑
• output p.w. is 𝑦𝑜𝑙𝑑
Then there is an increase in the population growth rate to 𝑠𝑛𝑒𝑤at time t’ so
that 𝑠𝑜𝑙𝑑 < 𝑠𝑛𝑒𝑤.
• An increase in the saving rate implies that the savings curve shifts up.
• At the initial point, actual investment is greater than required investment, so
capital p.w. will increase and the economy will move as indicated by the black arrow. This process will continue as long as actual investment is greater than required investment.
Capital p.w. will increase (as will output p.w) until we reach the new steady-state at B.
→ Hence, a higher saving rate leads to higher capital p.w. (as well as output p.w.) in the long run.
• There will be a short-term increase in the growth rate of output, but the growth rate of output will gradually return to its previous population growth rate 𝑔𝑁, as the economy approaches the new steady state.
• Likewise, capital grows at a rate above 𝑔𝑁 between steady-states. But the growth rate of capital gradually goes back to 𝑔𝑁.
• Consumption p.w. is maximised where the vertical distance between the
required investment line and the production function p.w. is greatest.
This is the point on the required investment line where the slope of the production function and the slope of the required investment line are the same. This implies
MPK = δ + 𝑔𝑁
This steady state is known as the “Golden Rule” steady state.
The golden rule saving rate, 𝑠𝐺 , is indicated in the previous diagram
How can we analyze the effect of corruption on the economy with the Solow model?
Intuitively, an increase in the level of corruption should decrease the expected rate of return to investments.
• The empirical literature also finds that corruption affects economic performance through a negative impact on investment.
➢Model an increase in corruption as a decrease in the saving rate. According to the simple Solow model, an increase in corruption modelled by a decrease in the saving rate should have
• strong effects on the level of GDP per worker
• but no effect on the long-run growth rate of GDP per worker.
Variation in capital-output across countries
Capital-output ratio in steady state:
Kt/Yt = s/(𝛿 + 𝑔𝑁)
■ At the steady state, output per capita and capital per capita are constant, so the model
cannot explain why GDP per capita varies for economies which have reached steady
state.
But GDP per-capita changes when there is a gap between current GDP per-capita and
steady state GDP per-capita. For example, from a starting point below steady state, there
is an increase in output per capita and capital per capita as an economy approaches the
steady state.
Summary of growth facts
1) (a) A lot of variation in GDP per-capita through time.
(b) A lot of variation in growth rates of GDP per-capita through time.
2) (a) A lot of variation in GDP per-capita across countries.
(b) A lot of variation in growth rates of GDP per-capita across countries.
3) Some countries have grown very fast and moved from relatively low income to very high income - some countries have gone in the opposite direction.
4) No consistent pattern in relationship between the level of GDP per-capita and its growth rate. In the case of high-income countries, economies that started off with lower GDP per-
capita (in 1950) tend to see higher growth rates. But this is not apparent for other country groupings
Variation in growth rates across time
■ Per capita growth is zero in steady state in the model, so the model cannot explain
growth for economies that are in steady state.
But, for given values of the saving rate, depreciation rate and growth rate of population, the model predicts that a country’s growth rate in output per capita will decline as it moves towards the steady state, so there is a change through time, but in
a very predictable way.
■ The model predicts that economies with a higher saving rate have a higher output
per capita in steady state, everything else equal.
A higher saving rate leads to more capital per capita being accumulated, and
countries with more capital per capita have more output per capita.
■ The model predicts that economies with a LOWER population growth rate
should enjoy higher output per capita in steady-state, everything else equal.
A lower population growth results in capital being spread across fewer workers so
this leads to higher capital per capita. Countries with more capital per capita have
more output per capita
■ There is no growth in steady state in this model, so growth rates across countries
that are in steady state cannot be explained by the model.
But, for given values of the saving rate, depreciation rate and growth rate of
population, the model predicts that an economy with a lower level of capital per
capita (and hence a lower output per capita) will grow faster than an economy with a
higher level of capital per capita (and so a higher output per capita), both moving
towards the same steady state.
Solow Growth Model with (exogenous) technical change
• Technology = the way inputs to the production process are transformed to
make output.
→ In the Solow model, it is how capital and labour combine to produce
output.
• Technological progress occurs when the application of new ideas leads to the
same value of inputs producing more or improved or new output. It can also
lead to a larger variety of output.
Technological progress leads to increases in output for given amounts of
capital and labour.
Solow Growth Model with (exogenous) technological change
❑Consider the role of technological progress in the aggregate production
function of the following form:
𝑌 𝑡 = 𝐹(𝐾𝑡, 𝐴𝑡𝑁𝑡)
where
• 𝐴𝑡 measures the level of technology.
Technological progress increases 𝐴𝑡, which all else equal, leads to an increase
in 𝑌 𝑡.
𝐴𝑡𝑁𝑡 is called effective labour, or sometimes the efficiency of labour.
Solow Growth Model with (exogenous) technological change: savings curve and investment
❑ The required investment line requires a little more thought.
• Recall the required investment line in the previous model was given by:
𝛿 + 𝑔𝑁 𝑘𝑡, where 𝑘𝑡 in this case was capital per-worker.
All else equal, 𝑘𝑡 will tend to fall as the capital stock depreciates at rate 𝛿
and as the population grows at rate 𝑔𝑁.
So 𝛿 + 𝑔𝑁 𝑘𝑡 is the investment p.w. that would be required to keep 𝑘𝑡
constant.
Solow Growth Model with (exogenous) technological change
• We extend our analysis to include technological growth.
Note here that
𝑘~𝑡 =𝐾𝑡/𝐴𝑡𝑁𝑡
Capital p.u.e.l. will fall through [a] depreciation, [b] population growth and now [c] growth in technology.
o the required investment line is (𝛿 + 𝑔𝑁 + 𝑔𝐴)𝑘~𝑡.
→ This means (𝛿 + 𝑔𝑁 + 𝑔𝐴 )𝑘~𝑡 is the investment p.u.e.l. that is required to keep 𝑘𝑡 constant.
Here 𝛿 is the depreciation rate and 𝑔𝑁 is the population growth rate (as before).
Now we also have 𝑔𝐴 which is the technology growth rate, which we assume is
exogenous to the model. That is ∆𝐴𝑡+1/𝐴𝑡 = 𝑔𝐴.
Endogenous technological progress
- Endogenous growth theory aims to develop models which directly explain growth in technology, and thereby offer a richer explanation for growth in output per head.
• First strip the Solow model back to absolute basics - exclude physical capital and
population growth.
• A very simple Cobb-Douglas production function in this case becomes 𝑌 𝑡 = 𝐴𝑡𝑁𝑌,𝑡 where 𝐴𝑡 is the level of technology.
𝑁𝑌,𝑡 is labour input into the production of goods and services.
This production function retains the basic feature of the Solow model - growth of output per worker depends on growth of technology.
• Add an equation which explains growth in A, i.e. the creation of new ideas.
The creation of ideas can be thought of as a form of production which itself
requires inputs. Model the production of new ideas with a simple
production function of the form
∆𝐴𝑡+1 = 𝐴𝑡𝑁 𝐴,𝑡
where 𝑁 𝐴,𝑡 is labour input into the production of ideas
The model is completed by assuming a resource constraint of the form 𝑁𝑌,𝑡 + 𝑁 𝐴,𝑡 = 𝑁 where N is the total labour force, which is assumed to be fixed in this very simple model.
The number of workers in the technology sector is a fixed proportion of the labour force so that 𝑁 𝐴,𝑡 = 𝑞𝑁
▪ Notice an important difference between the two inputs in our model:
Labour is a rival input: labour which is used in one sector can’t also be used in the other sector so, for given N, it is not possible to increase 𝑁 𝐴,𝑡 without reducing 𝑁𝑌,𝑡 . (Capital is also a rival input.)
But technology is a non-rival input: the stock of ideas (𝐴𝑡) used in the
production of goods and services can also be used in the production of new idea
Growth rate of technology: simplified to gA=qN
▪ Consider again the production function for new ideas
∆𝐴𝑡+1 = 𝐴𝑡𝑞𝑁
Consider a modified production function
∆𝐴𝑡+1 = 𝐴𝑡𝛽𝑞𝑁
where 0 < β < 1. There are diminishing returns to knowledge in the production of new knowledge.
The growth rate of new knowledge is given by
𝑔𝐴 = ∆𝐴𝑡+1/𝐴𝑡= 𝐴𝑡 𝛽−1𝑞𝑁
