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Economic Growth

- Core hypothesis: Economic growth doesn’t just happen; rather, it is endogenous, and depends on the choices society makes about political and economic organization, policies, and history.
- In this module, we look at
- History of thought on growth
- Stylized facts of growth
- Early models of growth, from Malthus to Solow
- Current models of endogenous growth

History:The first revolution: Adam Smith (1723-1790)

- Theory of wealth creation, public policy, and economic growth

Saving and investment are by-products and precursors of domestic and foreign trade

size of the market

division of labour

efficiency

The first revolution:Adam Smith

- Saving and investment stimulate growth
- direct effects through accumulationof capital
- indirect effects through labour productivity
- further indirect effects through interaction with exchange and trade, through foreign investment
- domestic market can take the place of foreign markets

The first revolution:Adam Smith

- Smith’s reference to ‘private misconduct’ and the ‘publick extravagance of government’
- Problem of public corruption and what economists now call “regulatory capture”

- Distinction between quantity and quality
- Quality enhances the productivity of workers and other technological inputs to production, and permits further technical innovation to occur

- Mutual advantages of trade and growth, links to geography
- First recognition of the concept of comparative advantage

The first revolution:Adam Smith

Benefits from division of labour

- If specialization increases efficiency
- and wealth and, thereby, economic
- growth, then ...

... just about anything that increases efficiency by the same amount, other things being equal, should be expected to have the same effect on growth.

The first revolution:Adam Smith

- Implications for growth

If foreign trade enlarges themarket and thus facilitates further division of labour à la Smith, thereby increasing wealth and growth, then ...

- ... all other equivalent means of
- increasing the efficiency or quality
- of labour, capital, and land should be
- expected to affect economic growth in the same way.

The first revolution:Adam Smith

- Smith on education, efficiency, and growth
- Distinction between the quantity and quality of labour education, by increasing labour productivity, also increases efficiency and growth
- Smith feared the economic, political, and social consequences of inferior education among the masses
- He favoured public support for education
- First recognition of the external economic benefit to society of mandatory universal education

The first revolution:Adam Smith - Summing up

- Economic growth = increase in the quantity and quality of the three main factors of production: labor, capital, and land
- Growth accounting is based on this classification
- Two shortcomings:
- Fixed quantity of land – diminishing returns
- Increase in the labour force does not reallycount as a source of economic growth

Adam Smith’s followers

Thomas Malthus

Question of population growth and its effect

on economic growth

David Ricardo

Impact of the distribution of wealth and of foreign trade

Malthus: A Formal Model

Ld=labor demand

Labor

(Pop.)

Ls=labor supply

Real Wage

CBR

CBR, CDR

NRI=0

CDR

Real Wage

w*

Effects of CharityA Malthusian Perspective

Growth shifts

Ls curve up thus

reducing the

effective wage.

Labor

(Pop.)

Ls2

3

Ls

The effective

wage falls until

CBR=CDR, leaving

the level of living as

was prior to charity.

Ld

Real Wage

CBR

CBR, CDR

CBR>CDR

=> Growth

2

Worker receives

w*-c from labor

and c in charity.

CDR

Real Wage

(w*-c)

w*

(w*+c)

1

Ld

Ld2

Labor

(Pop.)

1

Ls2

4

Ls

Real Wage

CBR

CBR, CDR

3 population grows

NRI=0

CDR

Real Wage

w*

w2

5: wage returns to w*

2

Adam Smith’s followers

- John Stuart Mill
- rejected Malthus’s prediction that population would outgrow productive capacity
- more and better education would restrain population growth
- distribution a different matter than production but can be changed through policy

Adam Smith’s followers

- Karl Marx
- Economic mechanisms driving production and distribution are closely related
- Anticipates Henry Ford’s comment on the importance of income as a determinant of aggregate demand
- General equilibrium effects are important

- The limits to growth observed by Malthus are inescapable ‘technological unemployment’

- Economic mechanisms driving production and distribution are closely related

Adam Smith’s followers

- Alfred Marshall
- organization as a fourth factor of production
- made explicit the connection between education and growth
- distribution of income and wealth matters for efficiency and growth

‘Knowledge is our most powerful engine of production ... Organization aids knowledge’

Adam Smith’s followers

- Joseph Schumpeter
- technology through invention, innovation, and entrepreneurship
- rent-seekers motivated by monopoly profits
- perfectly competitive markets may not be very conducive to economic growth
- no rent to capture under perfect competition

Adam Smith’s followers

- John Maynard Keynes

Accumulation of capital

‘Science and technical inventions’

‘I draw the conclusion that, assuming no important wars and no important increase in population, the economic problem may be solved, or be at least within sight of solution, within a hundred years.’

Stylized Facts of Growth

Per capita growth rate

Stylized Facts of Growth

Return to Capital

Stylized Facts of Growth

Why is the rate of depreciation increasing?

Stylized Facts of Growth

Capital-Output Ratio

Stylized Facts of Growth

Investment rates

Enter mathematics: Harrod and Domar

- Paul Samuelson’s
- Foundations of Economic Analysis(1948)
- laid the basis for mathematical economics, including the modelling of dynamic interactions among macroeconomic variables

- Foundations of Economic Analysis(1948)

Enter mathematics: Harrod and Domar

Net investment equals the increase in the capital stock

… net of depreciation due to physical or economic wear and tear

High level of investment entails an increasing level of the capital stock

High levels of saving and investment are good for growth even if they are stationary, that is, not increasing

By continuously augmenting the capital stock ...

- … even stationary levels of saving and investment relative to output drive output higher and higher, thus generating economic growth
- Flows of investment add to the stock of capital

Enter mathematics: Harrod and Domar

Efficiency is crucial for growth

High level of efficiency stimulates growth by ...

… amplifying the effects of a given level of saving and investment on the rate of growth of output

All that is required is a steady accumulation of capital through saving and investment

A given level of efficiency, including the state of technology will, then translate the capital accumulation into economic growth

Enter mathematics: Harrod and Domar

- Harrod and Domar expressed the dynamic relationship between saving, efficiency, and growth in a simple equation

So, Samuelson’s work neatly formalized, simplified, and summarized the essence of almost 200 years’ theorizing about economic growth

The Harrod-Domar model

The Harrod-Domar model

- Economic growth depends on three factors:
- A. the saving rate
- B. the capital/output ratio
- C. the depreciation rate

The Harrod-Domar model:Mathematics

- Notation:
- Y denotes national income
- K denotes capital stock
- S denotes saving
- Y denotes national income

The Harrod-Domar model:Mathematics

- Assumptions:
- Saving is proportional to income: S=sY
- Capital-output ratio is constant: K=vY
- Investment (newly produced capital goods) must be allocated between increasing the stock of capital and replacing depreciated capital: I=K+K
- At equilibrium S=I (desired saving =desired investment)

The Harrod-Domar model:Mathematics

- Harrod-Domar equation
- From the capital-output ratio assumption, we can write K=v Y.
- Substituting into the expression for investment, we have I=v Y+vY
- Using the equilibrium condition, we then have
sY= v Y+vY or D Y/Y=s/v-d

- Example: s=0.2, v=3, d=0.04 yields a growth rate of roughly 3%.

The Harrod-Domar model

- Shortcomings:
- Neither theory nor empirical evidence seemed to provide much support for the capital/output ratio as an exogenous behavioural parameter in the model
- a more elaborate formulation of the link between capital and output was called for

- The model did not leave much room for the other crucial factor of production, labor
- population or labor-force growth is absent from the formula, which explains output growth solely by saving and efficiency

- Neither theory nor empirical evidence seemed to provide much support for the capital/output ratio as an exogenous behavioural parameter in the model

The second revolution: The neoclassical model

Since population growth is basically a demographic phenomenon and, hence, exogenous from an economic point of view, it must follow that economic growth is also exogenous

According to Solow, saving behaviour was no longer relevant for long-run growth, nor was efficiency in a broad sense, except insofar as it mattered for technology

Economic growth was considered immune to economic policy, good or bad

Even so, saving and efficiency play an important role for growth over long periods, that is, the medium term

The second revolution:The neoclassical model

Solow showed how the capital/output ratio, rather than being exogenously fixed as in the Harrod-Domar model,

- … is better viewed as an endogenous variable, which moves over time and ultimately reaches long-run equilibrium

Once attained, the long-run equilibrium is consistent with not only a constant capital/output ratio

- … but also with a constant rate of growth of output per capita, a constant rate of interest, and a constant distribution of national income between labour and capital, all of which seemed to apply to the real world

The Neoclassical ModelMathematics

- Output is produced via a production function which uses capital and labor as inputs
where the parameter a is between 0 and 1.

- Taking logs on both sides and differentiating yields
- Here, g is the rate of growth of output in percentage terms, n is the exogenously given rate of growth of the labor force (or equivalently, of population), and is the rate of growth of the capital stock.

The Neoclassical ModelMathematics

- From empirical work by Kuznets, it is plausible to assume that the long-run capital-output ratio is constant, which implies that
- Plugging into the growth equation, then, we have

The Neoclassical ModelMathematics

- Thus, in the Solow model, the long-run rate of growth is determined entirely by the exogenously given rate of population growth.
- It also follows that in the long-run, there can be no growth in per capita output

- Since we obviously have seen significant increases in standard of living since the onset of industrialization in the early 1800’s, the model must be modified if it is to explain this.

The Neoclassical ModelMathematics

- We can explain the observed growth in per capita output by assuming that technological change makes the labor input more productive over time, due to factors such as better technology or better education of the workforce. With this assumption, the production function becomes
- B represents some initial state of technology
- e is the base of the natural logarithm
- Labor productivity grows at the rate q
- We refer to as the efficiency unit equivalent of the labor input

The Neoclassical ModelMathematics

- Log differentiating the production function now gives
- As before, taking the long-run capital-output ratio as constant yields
- So, we now have that growth is exogenous, being driven by productivity improvements, but per capital growth is now positive and equal to q.

The Neoclassical ModelMathematics

- Comparing the growth equation for the Solow model with that of the Harrod-Domar model, we see that we must now have
- If all the parameters n,q,s,v, and d are exogenously given, then we would generally not expect the equality above to hold.
- Mathematically, the Harrod-Domar model is now over-identified.
- Solow resolved this over-identification by assuming that the capital-output ratio, rather than being exogenously specified, was a function of the other parameters of the model:

The Neoclassical ModelMathematics

- How do we know the capital-output ratio is the right parameter to make endogenous?
- Consider the original definition of investment:
- This can be re-written as
- Since saving must equal investment in the long-run, I/Y=s, and we may then solve the equation above for the capital-output ratio as
- Hence, changes in any of the right-hand parameters will affect the value of the capital-output ratio.

The Neoclassical ModelMathematics

- We can also use this result to solve for the rate of growth of capital in terms of other parameters of the model:
- Substituting for the rate of growth of capital in the Solow growth equation yields
- This equation tells us that if we increase saving, then the economy will grow as long as the capital-output ratio remains constant.
- We turn next to the question of whether this ratio will in fact remain constant.

The Neoclassical ModelMathematics

- Dynamics of the capital-output ratio
- Define the following ratios of capital and output per efficiency unit at time t:
- The percentage rate of change of the first ratio is
where we use the relationship between the rate of change of capital to the capital-output ratio to arrive at the right-hand side of the equation.

The Neoclassical ModelMathematics

- We can also write the production function in terms of the two ratios as
- Now, since
substituting into the expression for the rate of growth of capital, we get a key equation:

The Neoclassical ModelMathematics

- The Solow differential equation
- If k is small, so that is large, the the rate of change of k will be positive, so the capital stock will increase. On the other hand, if k is large, will be small, so that the rate of change will be negative.
- This means that if we start at a low level of capital, the economy will accumulate capital, while if we somehow started with a large amount of capital, we will decumulate it.
- Hence, independently of where the economy starts, it will evolve toward a steady-state at which

The Neoclassical ModelMathematics

- Steady-state
- Set the time derivative of k to zero and solve for k
- Using the definition of k, we can find the steady-state values of capital and output:

The Neoclassical ModelMathematics

- Note that the steady-state capital stock and flow of output are actually growing, but in a balanced way, at the same rate, so that the capital output ratio remains constant at
- Income distribution in the Solow model
- Standard results from producer theory tell us that at the competitive equilibrium, inputs are paid their marginal products. For the simple model with only capital and labor inputs, these are given by

The Neoclassical ModelMathematics

The Neoclassical ModelMathematics

- Hence, for the Cobb-Douglas specification of technology, each factor of production is paid a constant share (a for labor, 1-a for capital) of output. This is consistent with data for modern industrial economies, where labor receives 2/3 of total output, while capital receives 1/3.
- This also gives us a way to calibrate the model, since it says we should set a=2/3.
- Since the capital-output ratio is constant, it also follows that along a balanced growth path, interest rates will remain constant.
- For labor, the real wage will grow at the rate g-n=q, since labor productivity is growing over time.
- Calibration spreadsheet

The third revolution:Endogenous growth

- The neoclassical growth model seemed unable to answer some burning questions about economic growth
- Is technological change exogenous from an economic point of view?
- Do economists really have nothing to say about economic growth in the long run?

- If output per capita grows at a rate that depends solely on - in fact, is equal to - the rate of technological progress, then why is it that the growth performance of different countries differs so radically over long periods?
- What does the neoclassical model tell us about relative growth performance anyway?

The third revolution:Endogenous growth

- Key idea
- Technologyis not exogenous
- Technology depends on economic factors
- Technological improvement depends on
- Innovation – “Learning by doing”
- Education
- Basic research

- Technical innovation is external to firms’ decisions
- Basic research generates new technologies available to all
- Education and on-the-job learning spill over from one firm to the next

Endogenous Growth:Mathematics

- Human capital investment
- Focus again on Cobb-Douglas production:
- We assume that some fraction h of the workforce is engaged in innovation – basic research, fine-tuning technical processes within the firm, independent invention, or other educational pursuits. The remaining fraction (1-h) provides labor input for firms.
- The effect of human capital accumulation on production is via A, which we now assume is an increasing function of the average amount of human capital accumulation
- For specificity, we assume the production function is given by

Endogenous Growth:Mathematics

- Increasing returns property
- The inclusion of human capital accumulation effects on productivity implies that the production function now exhibits increasing returns to scale. To see this, suppose we increase the labor and capital inputs by some factor l. This will increase the average labor supply by the same factor. Hence, the effect on output will be
- Because the human capital effect is external, the increasing returns will not affect individual firms’ profit maximizations.

Endogenous Growth:Mathematics

- Growth with human capital investment
- Taking logs and differentiating the production function gives us
- On a balanced growth path where output and capital grow at the same rate g, we will have
- Hence, as in the case of exogenous technical progress, we will have positive growth per capita, but due in this framework to the productivity enhancing effects of economic activities associated with human capital accumulation.

Endogenous Growth:Mathematics

- Income distribution
- As in the Solow model, factor shares are given by
- Also as in the Solow model, wages grow over time since output grows more rapidly than population. Since capital and output grow at the same rate, interest rates do not grow.

- As in the Solow model, factor shares are given by

Endogenous Growth:Mathematics

- Steady-state
- We can also replicate the dynamic analysis from the Solow model.
- Then

Endogenous Growth:Mathematics

- Since
- Steady-state capital stock is then

Saving Behavior

- Last missing ingredient to a fully specified economic model
- Handle by positing preferences over consumption over time
- A key parameter is consumer’s degree of patience or impatience, measured by how little or much they discount future utility

- Consumers face budget constraintswhich allow them to trade off consumption today for consumption tomorrow
- Saving generates returns in excess of the actual amount saved when interest rates are positive

- Handle by positing preferences over consumption over time

Saving Behavior:Mathematics

- Two formulations of consumer model: overlapping generations and dynastic models
- In both models, we assume time is split up into discrete periods (planning time).
- In overlapping generations model, consumers live finite lives. We will make the simplifying assumption that the number of periods of life is 2.
- Overlapping generations optimization problem is then
subject to

- Here, 0<b<1 is the consumer’s discount factor, which measures the degree of her impatience
- ris the market interest rate (which, recall, will be determined by the production side of the economy), and c2/(1+r) is the present value of second-period consumption determined by the market interest rate

Saving Behavior:Mathematics

- The easiest way to solve this problem is to substitute for second-period consumption from the budget constraint into the utility function, and then take a first-order condition with respect to first-period consumption:
- The required first-order condition is

Saving Behavior:Mathematics

- In terms of this model, the per capita rate of growth in consumption is given by
- From the growth equations, the per capita growth rate for consumption from the technology side of the economy is give by g-n.Hence, we will have and we can determine the equilibrium interest rate as
- So, interaction of growth induced by technology together with degree of patience determines equilibrium interest rates.
- Cases:
- Exogenous population growth only: g=n
- Growth with technical progress: g-n=q>0

Saving Behavior:Mathematics

- Dynastic Model
- For this model, we assume consumers act to optimize the discounted utility stream of a long-lived family, and hence solve
subject to

- Analysis of this model is much harder than for the overlapping generations model, so we will simplify by looking not at the market version of the model, but at a social planning version

- For this model, we assume consumers act to optimize the discounted utility stream of a long-lived family, and hence solve

Saving Behavior:Mathematics

- Social planner’s problem:
subject to

- Substituting from constraints into the objective function, the problem simplifies to

Saving Behavior:Mathematics

- Analyzing the social planner’s problem for this model is not simply a matter of taking first-order conditions and solving. To see why, we normalize the labor supply to 1, since it doesn’t change over time. Then, taking first-order conditions gives us the so-called Euler equations:
- In the exercises for this module, you show that by letting d=(1-a)b and making a suitable change of variable, the second-order difference equation generated by the first-order conditions can be converted into the first-order difference equation

Saving Behavior:Mathematics

- Graphing the difference equation gives us the following picture

Saving Behavior:Mathematics

- Properties of the difference equation

- Two steady-states, one at z=ab
and the second at z=1

- If we start below the first steady- state, or anywhere between it and the steady-state at 1 and iterate the difference equation, we will converge to the first steady-state.
- If we start anywhere to the right of the steady-state at 1, we will diverge.

Only the lower steady-state is optimal. At the upper steady-state, with z=1, we would have which says that all production is being devoted to the accumulation of capital, with none for consumption.

Saving Behavior:Mathematics

- It is possible to show that the planner’s problem can be represented in the simpler form
provided we know the so-called value function V(K). While the math is beyond the scope of what we can do here, it can be shown that this function will exist under suitable assumptions about discounting.

- This formulation of the optimal capital accumulation program for the economy is known as the dynamic programming formulation. In the exercises for this module, you showed that for this model, the value function is

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