Chapter 16: Models of Long-run Growth

1. 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

2. 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

3. 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 20.7.

4. 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.

5. 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.

6. 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.

7. 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: yY/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.

8. 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.

9. 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:

10. 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.

11. 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.

12. The Solow Diagram

13. 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.

14. Disequilibrium in the Solow Model

15. 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.

16. 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.

17. 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)

18. 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.

19. Steady state in the Solow Model

20. 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.

21. Increase in investment

22. Increase in population growth rate

23. 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.

24. 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.

25. 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.

26. 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

27. Solow Model with Technology • First, similar to the simple Solow model, we find that (16.7) • Substituting into the production function yields (16.8)

28. 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!

29. Solow diagram with technological progress

30. 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.

31. Permanent increase in the saving rate in the Solow model

32. 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.

33. Technological progress in the Solow model

34. 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

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

36. 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.

37. 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.

38. The Evolution of Output per Capita in Five Rich Countries Since 1950

39. Per Capita Real GDP

40. 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.

41. 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

42. 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.

43. 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.

44. Per Capita GNP and Annual Growth Rates

45. 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

46. Growth rates and initial income levels, OECD countries, 1960-1995

47. 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.

48. 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)

49. 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.

50. 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.