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Chapter 16: Models of Long-run Growth

- Neoclassical Model of Growth
- The Solow Model
- The Solow Model with Technology
- Long-run Growth in the Solow model
- The Convergence Hypothesis
- Growth Models with Human Capital
- Endogenous Growth Theories
- The AK Model
- The Romer Model
- Evidence of Growth
- Evaluation of Endogenous Growth Theory

Learning objectives

- Understanding the drivers of long-term growth
- Using the neoclassical model of growth to assess changes in growth rates
- Introducing human capital into growth models
- Assessing the contribution of endogenous growth theories
- Gaining perspective into why countries grow at different rates

Economic growth

- Economic growth is the change in GDP on an annual basis.
- A rule of thumb is that a country growing at g percent per year will double its per capita GDP every 70/g years.
- To see this, let Y(t) be per capita GDP at time t and let Y0be the initial value of per capita GDP. Then, Therefore, per capital GDP will double when Yt=2Y0.
- The rule of thumb can be worked out as follows. First, 2Y0=Y0egt. Then, solving for t,t=log 2/g. We get the rule of thumb by noting: log 20.7.

Economic growth

- Therefore, if the U.S. grew at 3% per annum for the past 30 years while China grew at 8% per annum, then GDP per capita will double approximately every 23 years in the U.S. and every 9 years in China.
- Within a generation, the average American will be three times as rich as his grandparents, while a Chinese person would be more than 18 times richer.
- Over a fairly short period of time, small differences in compound growth rates can lead to large differences in per capita incomes.

Long-run economic growth

- The rate of growth that is consistent with an economy’s natural rate of output and the position of the long-run aggregate supply curve seen in earlier chapters.
- Therefore, when policymakers are interested in improving the long-run potential of an economy and economists are attempting to understand why countries grow at different rates, these are the models that are relevant.

Neoclassical Model of Growth: The Solow Model

- The neoclassical framework for long-run growth is the Solow model.
- The Solow model is comprised of a production function and a capital accumulation function.
- The production function refers to the inputs of capital (K) and labour (L) necessary to produce output (Y).
- It is assumed to have the Cobb-Douglas form and is given by:

(16.1)

where α is a number between 0 and 1.

Production function

- This production function exhibits constant returns to scale, so that if all inputs are doubled, then output will double.
- We are interested in finding output per worker or per capita output, so we write the production in terms of output per worker and capital per worker, which are given respectively by: yY/L, k=K/L, which gives y=kα.
- With more capital per worker, firms produce more output per worker. But, there are diminishing returns to capital worker, so each additional unit of capital increases the output per worker by less than the previous unit of capital.

Capital accumulation equation

- The second equation of the Solow model is an equation that explains how capital is accumulated in the economy.
- The capital accumulation equation is given by:

(16.2)

- According to (16.2), the change in the capital stock, is equal to the gross investment, sY, minus depreciation, dK. The change in capital stock is given by the variable differentiated with respect to time, so
- The notation simply denotes the change in capital stock per period, so it is the continuous time version of Kt+1-Kt.

Capital accumulation equation

- Assumes that workers save a constant fraction, s, of their combined wage and rental income.
- Because the economy is closed, savings equals investment and investment is used solely to accumulate capital.
- The other assumption relates to the depreciation of the capital stock that occurs during production, which is a constant fraction, d, of the capital stock. Depreciation happens every period regardless of how much output is produced.
- Now, we have the capital accumulation equation in per worker terms:

Capital accumulation equation

- This equation says that capital per worker in each period is determined by three factors.
- The first is that investment per worker, sy, increases capital, k, while depreciation per worker, dk, reduces k. There is also a reduction in k because of population growth, the nk term.
- Each period there are nL new workers since the model assumes a constant growth rate of n in the population, which is also assumed to have a 100% labour force participation rate.
- So, if there were no new investment and no depreciation, capital per worker would decline because of the increase in the labour force. Because k is known as capital per worker, it is also referred to as the capital-labour ratio.

Solving the Solow Model

- To solve the Solow model, we take the production function and capital accumulation equation together.
- The production function in terms of output per worker is given by:
- The capital accumulation function in terms of capital per worker is:
- The equilibrium is found where the two curves meet. To find this point which is the steady state of the economy.

Equilibrium in the Solow Model

- The Solow diagram consists of two curves, which are functions of k, capital per person or the capital-labour ratio.
- The first depicts the amount of investment per person:
- The curve has the same shape as the production function but is translated in terms of the function s.
- The second curve is the amount of new investment per person required to keep the amount of capital per worker constant: (n+d)k. Population growth and depreciation will both reduce the amount of capital per person, which is captured here.

Capital deepening/widening

- When the economy is below k* and say capital per worker is increasing, then capital deepening occurs. Below k*, the amount of investment per worker exceeds the amount needed to keep capital per worker constant, so k increases over time. Capital deepening will continue until the economy has reached a steady state.
- If the economy, on the other hand, began with a capital stock per worker that was larger than k*, then we would be to the right of k*. The amount of investment per worker provided by the economy is less than the amount needed to keep the capital-labour ratio constant. The amount of capital per worker will decline. This decline will cease when the amount of capital per worker falls to k*. This is known as capital widening.

Solving for the steady state

- We now add the production function to find the steady-state value of output per worker.
- At k*, the steady state output of the economy is at is at y*. If different countries have different values of y*, then some will be richer and some will be poorer.
- So, we need to do one more thing which is to solve the model and find out what determines y*. We can do this by finding the steady-state quantities of capital per worker and output per worker.
- Recall that the steady-state quantity of capital per worker is determined by the condition:
- This means that there is no change in the capital-labour ratio over time so that the economy is at its steady state.

Solving for the steady state

- We also need the two key equations of the Solow model, the production function and capital accumulation function.
- The first equation is output per worker which is given by
- The second is capital per worker which is given by
- Substitute the first equation into the second and we get
- Setting this equation to zero yields the steady-state quantity of capital per worker, as follows:

(16.3)

Solving for the steady state

- We now need to find the steady-state output per worker, which is found by substituting (16.3) into the production function (16.1). So, y* is given by:

(16.4)

- Therefore, y* is now written in terms of the parameters of the model and we have a solution.

Different steady states

- Countries which have higher savings and investment rates will tend to be richer than those who do not. These countries accumulate more capital per worker and thus more output per worker.
- In contrast, countries which have high population growth will be poorer. And, in these countries, a larger portion of savings will be required to keep the capital-labour ratio constant. The tendency for capital widening will make capital deepening more difficult and these countries will tend to accumulate less capital per worker.
- However, when the economy is in steady state, it can still experience “shocks” along these lines where key parameters like savings rates and population change.
- This explains and fits the empirical fact that countries have different levels of per capita income.

Rate of economic growth

- In the simple version of the Solow model, there is no per capita growth when the economy is at its steady state.
- In the simple Solow model, this means that economies will grow until they reach their steady state and eventually growth will stop altogether!
- Although the outcome of the Solow model fits the empirical observation that countries have different per capita incomes, it does not explain why economies grow persistently over long periods of time. To generate sustained growth in per capita income, we have to introduce technological progress.

Solow Model with Technology

- Thus, we include a technology variable, A, to the production function:

(16.5)

- The technology variable is viewed as “labour-augmenting” in this formulation. Alternatively, it could take the form of “capital-augmenting.”
- This means that a unit of labour is more productive when the level of technology is higher.
- However, technology in this model is exogenous. It is an important assumption of the model, but one that has been viewed as akin to “manna from heaven.” Somehow technology comes into the economy from the outside regardless of what is happening within the economy.

Solow Model with Technology

- Rewriting the production function in terms of

(16.6)

where

- We will now refer to as the output-technology ratio or “output per effective unit of labour.”
- We also need to rewrite the capital accumulation function:
- Combining the new versions of the production function and capital accumulation function, we have the Solow diagram with technological progress.

Solow Model with Technology

- The economy is in steady state and grows at the rate of technological progress, g.
- Capital, output, consumption and population are growing at constant rates as a result, and so the economy is said to be growing along a balanced growth path.
- Solving for for the steady state in the Solow model with technology, first recall that the steady-state output per effective unit of labour is determined by the production function and the condition that

Solow Model with Technology

- First, similar to the simple Solow model, we find that

(16.7)

- Substituting into the production function yields

(16.8)

Solow Model with Technology

- In terms of output per worker, we rewrite the above as

(16.9)

- What is notable and different here is that output y* and A both depend on time, so an economy can grow at the rate of technological progress.
- So, output per worker is determined by technology, the investment rate, and population growth rate.
- Now, if g=0, then the result would be identical to the Solow model without technology.
- And, changes in the investment rate and population growth rate affect the long-run level of output per worker but not the long-run growth rate of output per worker!

Changing the long-run level of output per worker

- Suppose that there is a permanent increase in the saving rate from s to s’.
- The result is very similar to the Solow model with no technological progress. At the initial capital-technology ratio, investment exceeds the amount needed to keep the capital-technology ratio constant. An upward shift in the s curve, therefore, speeds up the rate of growth temporarily until the new steady state is reached.
- Fundamentally, however, g has not been affected and thus the rate of economic growth remains the same as before.
- Therefore, policies that say permanently increase the rate of investment will push the economy to a higher steady state level of income, but it will not increase its long-run rate of growth.

Long-run growth in the Solow model

- The Solow model shows that sustained economic growth occurs only in the presence of technological progress.
- Without technological progress, capital accumulation will encounter diminishing returns.
- However, if there was improvement in technology, then labour productivity will grow and capital accumulation will also increase because it slows down the diminishing returns to capital.

Technological progress in the Solow model

- Technological progress has allowed the existing inputs to be used more efficiently, so the production function has shifted upward.
- At the initial capital-technology ratio there is now a higher level of steady state output.
- Part of this increase in output is due to productivity advancement or total factor productivity (TFP), and the other part is ensuing capital accumulation.
- Global Applications 16.1 The Solow residual

Annual Growth Rates of Output per Capita and of Technological Progress, 1950-87

Technological progress in the Solow model

- There is a clear association between periods of high output growth and high technological progress.
- These developed nations follow similar experiences with much better performance over 1950-73 than 1974-87, implying that all the most significant developed economies are subject to similar technology shocks.

The convergence hypothesis

- Consider the key equation from the Solow model again:
- Output per effective unit of labour will decline as capital stock rises because of diminishing returns to capital. Therefore, if an economy has low levels of capital stock, then it should realise higher returns to its capital than a country which is developed.
- If these economies have the same levels of technology, investment rate, and population growth, then the developing country will grow faster. The output per worker gap between these countries will narrow over time as both economies approach the steady state.

Empirical Evidence

- In terms of annual growth rates, it is clear that Japan, which started at a much lower level of development at the start of this period, grew faster than other more developed economies.
- Several observations can be made. Firstly, growth rates were much higher between 1950-1973 than 1974-2000. Secondly, the ratio of real per capita output between 2000 and 1950 shows that in the space of 50 years the size of these economies has increased substantially from 2.6 times for the U.S. economy to 11.4 times for Japan. It is true over a longer period of time.
- It is clear from these tables that even slight differences in growth performance, when compounded over a long time, can result in remarkably different levels of output.

Empirical Evidence

- If there is convergence, then there should be an inverse relationship between a nation’s starting level of income and subsequent growth.
- Evidence from 1880-1973

Empirical Evidence

- However, if we look at a more recent period (1960-1995) there is no clear relationship.
- DeLong (1988) argued that results found by Baumol (1986) relied too much on the countries which were industrialised and therefore had data which was available from the earlier period.

Conditional convergence

- Another inference that can be drawn from the Solow model is that countries will converge in growth rates conditioned on differences in steady states.
- This means that countries are converging to their own steady states. This is known as conditional convergence or sigma convergence.
- This concept tests for convergence by looking at the actual distribution of world income levels. If convergence is to be observed, then the variance of income levels should be falling over time.

Conditional convergence

- Some poor and middle income countries (particularly China and Korea) have a tendency to grow faster, and catch up with the richer nations, whereas there are many other poor countries that actually record growth rates that are lower than those in the OECD.
- In terms of the world income distribution this implies that instead of seeing convergence, you may actually observe polarization between rich and poor nations.
- The negative relationship in the following figure looks like there is conditional convergence among these nations, the world’s richest countries in this later as well as earlier period.
- Global Applications 16.2 Feldstein-Horioka Puzzle

Growth Models with Human Capital

- The simple assumptions surrounding labour in the Solow model can be revised to include human capital, that is, workers with skills and education, so growth could be driven by human capital instead of ‘manna from heaven.’
- In 1988, Lucas extended the neoclassical model by assuming that individuals spend time accumulating skills, such as by going to school. Mankiw, Romer and Weil (1992) thought of human capital accumulation in the same way as physical capital accumulation by forgoing consumption.
- We will now incorporate human capital into the Solow model by introducing a concept of labour that includes human capital.

Growth Models with Human Capital

- Output, Y, is produced by physical capital, K, labour, L, and h which is human capital per person, in a Cobb-Douglas production function. This can be seen as:

(16.10)

- Technology is still assumed to grow exogenously at rate, g. However, individuals in this economy accumulate human capital by spending time learning new skills instead of working.
- So, if an individual devotes 1-μ fraction of time learning (and μ amount of time working), then

(16.11)

Growth Models with Human Capital

- An increase in time spent accumulating human capital will increase the growth rate of human capital, as seen as follows:

(16.12)

- The inference to be drawn here is that h is very similar to A in the labour-augmenting version of the simple Solow model.

Growth Models with Human Capital

- The solution to the neoclassical model with human capital in steady state is virtually identical to the simple version of the Solow model.
- We start by writing the production function in terms of output per worker:

(16.13)

- The parameter 1-μ is assumed to be constant and exogenously determined, so individuals accumulate human capital at a constant rate. Since it is constant, the equation should look very similar to the Solow model from before.
- So, y and k will grow at the constant rate, g, the rate of technological progress, which is now determined by human capital accumulation, so that g=1-μ.

Growth Models with Human Capital

- Since 1-μ is constant, the output per worker equation is still given by:

(6.14)

- The capital accumulation equation can be similarly written as:

(6.15)

- To finish the solution, we first set
- And find

Growth Models with Human Capital

- Substituting this condition into the output per worker equation gives the steady-state value of output:

(16.6)

- Rewriting this in terms of output per worker:

(16.7)

- This shows that y* should grow at the same rate as 1-μ.

Growth Models with Human Capital

- The extended Solow model sheds some additional light on why some countries are rich and others are poor.
- Some countries do well because they have high investment rates in physical capital, low population growth, high levels of technology and spend a large fraction of time accumulating human capital.
- In steady state, though, this extended Solow model still predicts that per capita output grows at the rate of technological progress, g, as in the simple model.

Endogenous Growth Theories

- It is clear, therefore, that the main determinant of long-run growth is technological progress. Yet, this is the one variable that is determined outside of the neoclassical models.
- A newer strand of models, known as endogenous growth theory, attempts to incorporate the determinants of the main engine of growth, technology, into the framework.
- If we can understand what determines A, then we can explain the growth rate of y*.
- This is what endogenous growth theories try to do.

The AK Model

- The simplest version of this strand of theories is the AK Model.
- Let us start with a modified Solow-type production function where =1.
- The production function is now given by: Y=AK (16.8)
- A is a positive constant and K is broad capital. Broad capital represents physical capital, but also human capital.
- Though it is a notion of broad capital, we simplify the analysis to say that capital is accumulated as individuals save in the economy rather than consume.

The AK Model

- Capital accumulation equation:

(16.9)

where s is the investment rate and d is the rate of depreciation, both are assumed to be constant.

- We assume that no population growth and treat these as per capita variables for simplicity.
- There are two curves in the following diagram, where the dK line represents the amount of investment needed to replace the depreciation of the capital stock. The sY curve is the total investment in the economy as a function of the capital stock. Because Y is a linear function of K, this is a straight line.

The AK Model

- When we assume that total investment is larger than depreciation, the capital stock will grow. This is the key feature of the AK model. If the economy starts at point K0. At every point to the right of K0, the investment exceeds depreciation so the economy continues to grow. In fact, the capital stock is always growing and growth never stops in this model!
- The key difference between the AK model and the Solow model is that this model assumes constant returns to capital, while the Solow model assumed diminishing returns so that each unit of capital was less productive than the previous unit.

The AK Model

- The marginal product in the AK model, in contrast, is always A. This means that growth can continue perpetually. And, it means that the growth rate of output is equal to the growth rate of capital because A is constant. This implies that the growth rate of the economy is an increasing function of the investment rate.
- However, the AK model does not provide an intuition as to why constant returns to scale is achievable with broad capital. For instance, if the capital had positive spillover effects so that it made other capital productive, then we might not expect diminishing returns to set in. This intuition is provided by the next set of models.

The Romer model

- The endogenous growth model developed by Romer endogenises technological progress by introducing researchers who invent new ideas and spur technological advancement.
- In these types of models, technological progress is driven by R&D (research and development) and can explain why rich countries experience persistent growth.
- It also goes to show why technology transfer is so important for developing countries and why poor countries have lower levels of technology if they do not have much R&D.

The Romer model

- Similar to the Solow model, the Romer model has two main elements representing the production function and a set of equations explaining how the inputs into production evolve over time.
- The key point is that technology can be thought of as ideas, which are generated by researchers.
- So, similar to the neoclassical models, for a given level of technology, A, the production function exhibits constant returns to scale.
- But, because ideas are also an input into production, the production function exhibits increasing returns with respect to technology. This derives from the non-rivalrous nature of ideas!

The Romer model

- The accumulation equations for capital and labour are identical to the Solow model in the Romer model. The only difference is the treatment of technological progress, which is now not assumed to be driven by factors outside of the model.
- Instead, Romer introduces a production function for ideas. The Romer model offers A(t) as the stock of knowledge or ideas that have been invented through history up to time t.
- Then, is the number of new ideas produced at any given time. So, it is equal to the number of people inventing new ideas, LA, multiplied by the rate at which they discover these new ideas, .

The Romer model

- The production function for A is thus given by:

(16.20)

where the discovery rate, , could be constant or could be an increasing or decreasing function of A.

- There are essentially three possibilities.
- Non-changing rate of invention, so it is simply a constant stream of inventions.
- The second is a “standing on shoulders of giants” notion so is an increasing function of A.
- Conversely, a third possibility is a “crowding out” effect where is a decreasing function of A.

The Romer model

- This framework also suggests that there are two types of workers in this model, those who produce LYand those who invent LA, so LY+LA=L.
- The set up is similar to the Solow model except that there are two types of workers and there is an ideas equation.
- Except that in this model, the rate of technological progress is generated by workers, LA, and does not rely on exogenous shocks.
- So long as inventive activities continue, the economy will grow and A is now determined within the model.

Evidence of growth

- Since 1950, the fraction of the labour force that consists of scientists and engineers engaged in R&D had increased three-fold. However, average growth rates in the U.S. are no higher today than they were from 1870 to 1929.
- Investment rates in the U.S. also increased greatly since 1950 without similar evidence of an increase in growth rates.
- Finally, the U.S., Bolivia and Malawi all grew at around the same rate despite significant differences in the level of human capital, investment and also R&D personnel.

Technology

- Endogenous growth theory says that technology is determined within the model and produced by having researchers. If poorer countries do not have investment funds for R&D facilities and personnel, then they will have a lower level of technology than rich countries.
- This main driver of growth is not exogenously given but very much related to the factors within the model, such as the level of development, capital stock, human capital, etc.
- These theories also suggest that developing countries can catch up by imitating the technology in developed countries rather than re-invent the wheel. But the “catch up” process is probably both more costly and complex than the models indicate.
- Global Applications 16.3 Does the “New Economy” measure up to the inventions of the past?

Evaluation of endogenous growth theories

- Crucially, the driver of growth in this type of model is a constant rate of invention that may vary positively or negatively with the number of researchers in the economy.
- The rate of invention and innovation appear to be exogenous to the model. So, although endogenous growth models go further than neoclassical models in trying to explain how technological progress, the driver of long-run growth, occurs, it still leaves many questions unanswered.
- New growth theories, moreover, emphasise institutions, infrastructure, health, social capital, legal contracting regimes, political instability, conflict and war, among many others.

Summary

- We have covered the main models of long-run economic growth.
- We started with neoclassical growth models, namely the Solow model.
- We also considered an extended version of the Solow model that included human capital, known as the Lucas formulation.
- After discussing the limitations of the Solow model, we moved to consider endogenous growth theories or new growth theories.
- The simplest formulation of an endogenous growth model was the AK model, which could explain persistent growth by modifying the assumption of constant returns in the neoclassical framework.

Summary

- Then, we considered the Romer model which introduced a new production equation for ideas into the mix. Ideas and inventions can generate increasing returns, thus providing for persistent growth of an economy.
- We also examined the notion of technology transfers in versions of the convergence hypothesis.
- We concluded with an assessment of endogenous growth theories and how they still leave key factors, such as what drives innovation, unanswered.

Summary

- We also looked briefly at new growth theories more widely which attempt to understand why some countries seem to persistently lag behind. The evidence suggests that there are numerous factors, such as institutions, political stability, which seem to contribute to developing countries falling further behind.

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