1. Lecture 2: Savings, Capital Accumulation and Output Key Issues on Savings and Growth 1.Understanding the effects of the savings rate on capital and output per capita 2.Key findings are that an increase in the saving rate leads to: higher growth in output for some time, a higher standard of living no permanent increase in the rate of growth 3.Understanding the “Golden Rule” which indicates the savings rate (and associated level of capital) that maximises consumption in the steady state

2. Structure of the Lecture 1.Interactions between output and capital 2.Understanding the long-run equilibrium (or steady state) 3.Implications of alternative savings rates 4.Implications for consumption and the “golden rule” 5.Getting a sense of magnitudes 6.Physical Capital vs. Human Capital

3. 1. Interactions between Output and Capital At the center of the determination of output in the long run are two relations between output and capital (or a two-way causality running between output and capital): – The amount of capital determines the amount of output being produced – The amount of output determines the amount of saving and, in turn, the amount of capital accumulated over time

4. Capital, Output, and Saving/Investment Figure 11 - 1 Interactions between Output and Capital

5. First Causality - Capital’s effect on output Higher capital per worker leads to higher output per worker Since the focus is on the role of capital accumulation, we make the following assumptions: – The size of the population, the participation rate, and the unemployment rate are all constant – There is no technological progress

6. Second Causality - Output’s effect on Capital To derive the second relation, between output and capital accumulation, we proceed in two steps: – Step 1: We derive the relation between output and investment (Inv fn of Y) – Step 2: We derive the relation between investment and capital accumulation (K fn of Inv)

7. Step 1: Output and Investment Three assumptions are made to derive the relation between output and investment: – We assume the economy is closed: – Instead of I + G + X = S + T + M, we exclude the external sector (X and M), therefore: I + G = S + T, or I = S + (T – G) – We assume public saving is equal to zero i.e. T – G = 0, therefore: I = S – We assume that private saving is proportional to income, so: S = sY, where s is the savings rate between 0 and 1 – Combining the two relations gives: I = sY (that is investment is proportional to output i.e. higher output implies higher investment (and higher savings)

8. Step 2: Investment and Capital accumulation The evolution of the capital stock is given by (where ∂ denotes the rate of depreciation): Note: Investment (I) is a flow which contributes to capital (K) a stock Combining the relation I=sY and the relation from investment to capital accumulation, the expression from output to capital accumulation is given (per worker) as:

9. Investment and Capital accumulation Capital per worker at the beginning of the year t+1 is equal to capital per worker at the beginning of year t, adjusted for depreciation, plus investment per worker during year t (which is equal to the savings rate times output per worker during year t) Rearranging terms, we can express the equation as follows: That is, the change in the capital stock per worker (left side) is equal to savings per worker minus depreciation (right side)

10. 2.Understanding long-run equilibrium or steady state Our two main relations are: Combining the two relations (substituting the first into the second), we can study the behavior of output and capital over time (or the dynamics of capital and output)  First relation: Capital determines output (Y on left hand side)  Second relation: Output determines capital accumulation (K on left hand side)

11. This relation describes what happens to capital per worker. The change in capital per worker from this year to next year depends on the difference between two terms:  If investment per worker exceeds depreciation per worker, the change in capital per worker is positive: Capital per worker increases  If investment per worker is less than depreciation per worker, the change in capital per worker is negative: Capital per worker decreases  Key drivers of capital accumulation are:  The rate of savings (s) which is positively associated with the capital accumulation of capital stock  The rate of depreciation (or the proportion of capital stock that breaks down or becomes useless) which is negatively associated with the capital accumulation of capital stock Capital formation, savings and depreciation Change in capital from year t to year t + 1 _ Depreciation during year t _ = Investment during year t =

12. Capital formation, savings and depreciation In Fig 11.2 output per worker is on the vertical axis and capital per worker on the horizontal axis Blue – f(K/N) Green - sf(K/N) Purple – ∂ (K/N) Change in capital = CD (as investment > depreciation) For any particular savings rate (s) capital formation will tend toward a steady state where investment = depreciation

13. Notes on Fig 11.2  Output per worker f(K/N) increases with capital per worker, but by less and less as capital per worker increases due to decreasing returns to capital  Investment per worker sf(K/N) has the same shape as the output per worker relation but is lower by factor s ( between 0 and 1) (the savings rate)  Depreciation per worker ∂K/N increases in proportion to capital per worker (a straight line with slope ∂)  At K*/N, output per worker and capital per worker remain constant at their long-run equilibrium levels  When capital and output are high (above K*/N), investment is less than depreciation, and capital decreases

14. Notes on Fig 11.2  AB gives output per worker at the initial level of capital per worker (K 0 /N)  AC gives the level of investment per worker at the initial level of capital per worker (K 0 /N) (due to s)  AD gives the level of depreciation per worker at the initial level of capital per worker (K 0 /N) (due to ∂)  As CD = AC –AD and CD>0 therefore investment per worker exceeds depreciation per worker so capital per worker increases  Over time output per worker and capital per worker increases due to the process of capital accumulation until a point of long-run equilibrium (or steady state) is reached at K*/N where output per worker and capital per worker remain constant at Y*/N and K*/N (where savings and related investment equals the rate of depreciation)

15. Steady state capital and output The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. In steady state, the left side of the equation above equals zero, then: Given the steady state of capital per worker (K*/N), the steady-state value of output per worker (Y*/N), is given by the production function (where the * is indicative of a steady state):

16. 3. The Implications of Alternative Saving Rates Three observations about the effects of the saving rate on the growth rate of output per worker are: 1.The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero (in the absence of technological progress). 1.As the economy converges on a steady state of output per worker, the long run growth rate of output is equal to zero no matter what the savings rate. 2.Furthermore, the economy would have to save a greater and greater portion of its output to due to decreasing returns to capital. (Krugman’s “Stalinist growth” – increasing state-controlled savings rate) 2.The saving rate determines the level of output per worker in the long run. Other things equal, countries with a higher saving rate will achieve higher output per worker (and a higher standard of living) in the long run (Fig 11.3 Y 1 /N > Y 0 /N). 3.An increase in the saving rate will lead to higher growth of output per worker for some time, but not forever (Fig 11.4 (without technological progress) and 11.5 (with technological progress)

17. The Implications of Alternative Saving Rates A country with a higher saving rate achieves a higher steady state level of output per worker. The Effects of Different Saving Rates Figure 11 - 3

18. The Implications of Alternative Saving Rates An increase in the saving rate leads to a period of higher growth until output reaches its new, higher steady-state level (and growth returns to 0, indicated by a 0 slope). The Effects of an Increase in the Saving Rate on Output per Worker Figure 11 - 4

19. The Implications of Alternative Saving Rates Where there is technological progress the economy will have positive growth of output per worker even in the long run. Therefore, an increase in the saving rate leads to a period of higher growth until output reaches a new path where the rate of growth is equal to the initial growth rate as AA has the same slope as BB) The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress Figure 11 - 5

20. 4. Implications for consumption and the “golden rule” An increase in the saving rate always leads to an increase in the level of output per worker. But output is not the same as consumption as output is either consumed or saved i.e. Y = C + S. Consider what happens for two extreme values of the saving rate:  An economy in which the saving rate is (and has always been) 0 is an economy in which capital is equal to zero. In this case, output is also equal to zero, and so is consumption. A saving rate equal to zero implies zero consumption in the long run, as both capital and labour are needed for production via the production function).  Now consider an economy in which the saving rate is equal to one (100%): People save all their income. The level of capital, and thus output, in this economy will be very high. But because people save all their income, consumption is equal to zero. A saving rate equal to one also implies zero consumption in the long run.  The level of the saving rate (not 0 or 1) that yields the highest level of consumption in steady state is known as the “golden-rule” level of capital.

21. Implications for consumption (Fig 11.6) The golden rule savings rate is given as S G where consumption per worker is maximised (in Fig 11.6) Increases in the savings rate (when below the “golden rule” level) results in lower consumption initially and increased consumption in the long run (or increased steady state consumption) Increases in the savings rate (when above the “golden rule” level) decreases consumption not only initially but also in the long run (steady state) Above the golden rule level the increase in capital due to the increased savings rate leads to only a small increase in output (due to negative return to factors) which is too small to cover the increased depreciation. The economy carries too much capital.

22. Implications for consumption An increase in the saving rate leads to an increase and then to a decrease in steady-state consumption per worker. The Effects of the Saving Rate on Steady- State Consumption per Worker Figure 11 - 6

23. Policy issues In practice, most economies operate below the golden rule S G The public policy challenge is to attempt to try and get savings up and closer to S G The trade-off: an increase in the savings rate toward S G will result in lower consumption now but higher consumption later This raises inter-generational issues i.e. current generations will reduce consumption and future generations will gain (due to the larger capital accumulation) Politically, future generations do not vote! In SA the debate has been about the appropriateness of budget surplus which contributes to savings In the US about the funding of the Social Security systems and how the impacts on US savings

24. US Case: Social Security and Savings  One way to run a social security system is the fully-funded system, where workers are taxed, their contributions invested in financial assets, and when workers retire, they receive the principal plus the interest payments on their investments. This contributes to savings – even though govt savings grow in composition, but there is no negative effect on capital accumulation e.g. government pension funds invest in capital projects based on returns for future retirees).  Another way to run a social security system is the pay-as-you-go system, where the taxes that workers pay are the benefits that current retirees receive. This results in reduced saving and reduced capital accumulation. In anticipation of demographic changes, the Social Security tax rate has seen increases, and contributions are now larger than benefits, leading to the accumulation of a Social Security trust fund. Bt this is not enough as the trust fund is far smaller than the value of the benefits promised to future retirees. As each year taxes are used to pay these benefits the US system is basically a pay-as-you-go system contributing to loser savings rates.

25. Based on the workings at section 11.3, the steady state output per worker is equal to the ratio of the saving rate to the depreciation rate. A higher saving rate and a lower depreciation rate both lead to higher steady state capital per worker and higher steady state output per worker. Numerical examples: – If the savings rate and depreciation rate are both equal to 10% per year. Then both Y*/N and K*/N =1 – If the savings rate doubles to 20% (and there is no change to the depreciation rate) then Y*/N doubles to 2 and K*/N rises from 1 to 4. Conclusion: an increase in the savings rate leads to an increase in the steady state level of output (But how long does it take for output to reach its new steady state level? – See Fig 11.7) 5. Getting a Sense of Magnitudes

26. Dynamic effects It takes a long time for output to adjust to its new, higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth. The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker Figure 11 - 7

27. Dynamic effects In Fig. 11.7 (a) after an increase in saving from 10% to 20% (with no change in depreciation), the level of output per worker (Y/N) increases from 1 to 2 over a number of years, each year by a smaller amount (computed year after year using equations 11.8 and 11.7) In Fig. 11.7(b) the growth rate of output per worker (Y/N) is plotted, with a high initial growth rate returning to zero over time (hence the conclusion that a change in the saving rate will not lead to a permanent change to the growth rate of output – even though it does effect the growth rate for quite some time)

28. In steady state, consumption per worker is equal to what is left after enough is put aside to maintain constant level of capital, i.e. output per worker minus depreciation per worker. Knowing that: then: and These equations are used to derive Table 11-1 in the next slide, comparing the effect of different rates of saving on consumption given a constant rate of depreciation equal to 10%. Magnitudes and the “golden rule”

29. Table 11-1 The Saving Rate and the Steady-state Levels of Capital, Output, and Consumption per Worker Saving Rate, s Capital per Worker, (K/N) Output per Worker, (Y/N) Consumption per Worker, (C/N) 0.0 0.11.0 0.9 0.24.02.01.6 0.39.03.02.1 0.416.04.02.4 0.525.05.02.5 0.636.06.02.4 –––– 1.0100.010.00.0 Magnitudes and the “golden rule”

30. In Table 11.1 steady state consumption per worker is largest when s = 0,5 i.e. the “golden rule” is associated with saving rate of 50% Below 50% savings - increases in savings lead to increases in long run steady state consumption Above 50% savings – increases in savings leads to lower levels of long run steady state consumption In reality, US Savings are around 17% of GDP, SA around 16% of GDP, China around 30% of GDP

31. The set of skills of the workers in the economy is called human capital. An economy with many highly skilled workers is likely to be much more productive than an economy in which most workers cannot read or write. When the level of output per workers depends on both the level of physical capital per worker, K/N, and the level of human capital per worker, H/N, the production function may be written as: An increase in capital per worker or the average skill of workers leads to an increase in output per worker. 6. Physical versus Human Capital

32. A measure of human capital may be constructed as follows: Suppose an economy has 100 workers, half of them unskilled and half of them skilled. The relative wage of skilled workers is twice that of unskilled workers. Then: The Major implication is that in the long run, output per worker depends not only on how much society saves but also how much it spends on education. Extending the production function

33. The emphasis on human capital has motivated a new generation of growth models – called endogenous growth models (such as those of Robert Lucas and Paul Romer) Such endogenous growth models allow for steady growth even without technological progress, where growth depends on variables such as the saving rate and the rate of spending on education. In these models steady state growth – in the form of increases in output per worker - can be generated by increases both of physical capital per worker and human capital per worker. Therefore, both the savings rate and the fraction of output spent on education and training become key policy variables. Next week – we will focus on growth models that allow for technological progress (and there is always the possibility that improved human capital itself is a key driver of technological progress) Endogenous Growth