#### Transcript Long-Run Economic Growth

Long-Run Economic Growth Prof Mike Kennedy 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 10 1988 1987 1986 1985 1984 1983 1982 1981 1980 Growth experiences: Three large groups of countries 15 World Emerging market and developing economies Advanced economies 5 0 -5 -10 -15 Growth experiences: The developing world 15 10 5 0 -5 -10 -15 Emerging market and developing economies Central and eastern Europe Commonwealth of Independent States Developing Asia Middle East, North Africa, Afghanistan, and Pakistan Sub-Saharan Africa Growth experiences: Advanced economies seem to move together 15 10 5 0 -5 Canada Japan United States Euro area -10 -15 United Kingdom Why growth matters: Small changes make a big difference over a lifetime 135 High gorwth (3.5%) $133,000 120 Moderate growth (3.0%) 105 $113,000 Low growth (2.5%) 90 75 60 $95,000 45 30 15 0 y-2015 y-2020 y-2025 y-2030 y-2035 y-2040 y-2045 y-2050 The Sources of Economic Growth • The relationship between output and inputs is described by the production function: Y = AF(K, N) • For Y to grow, either quantities of K or N must grow or productivity (A) must improve, or both. The Growth Accounting Equation • The growth accounting equation: ΔY ΔA ΔK ΔN αK αN Y A K N ∆Y/Y is the rate of output growth; ∆K/K is the rate of capital growth; ∆N/N is the rate of labour growth; ∆A/A is the rate of productivity growth. The Growth Accounting Equation (continued) aK = elasticity of output with respect to capital (about 0.3 in Canada); aN = elasticity of output with respect to labour (about 0.7 in Canada).1 • The elasticity of output with respect to capital/labour is the percentage increase in output resulting from a one per cent increase in the amount of capital stock/labour. 1 Recall the table on income shares in lecture on the national accounts (Chap 2). The Growth Accounting Equation (continued) • There is another way to derive the equation using logs. The production function can be written as: ln(Y) = ln(A) + αKln(K) + αNln(N) • The term “ln” means the natural log of the variable in question. • Since the first derivative of the natural log of a variable is approximately equal to the proportional change then: dln(Y) = dln(A) + αKdln(K) + αNdln(N) • This is approximately equal to growth accounting equation in slide 7. – Note that dln(X) is approximately ∆X/X for somewhat small changes (around 5% or less). Growth Accounting • Growth accounting measures empirically the relative importance of capital stock, labour and productivity for economic growth. • The impact of changes in capital and labour is estimated from historical data. • The impact of changes in total factor productivity is treated as a residual; that is, not otherwise explained. ΔA ΔY ΔK ΔN αK αN A Y K N Growth Accounting and the Productivity Slowdown • Output growth was rapid during 1962-1973 and then slowed in 1974-2006. • Much of the decline in output growth can be accounted for by a decline in productivity growth. • The slowdown in productivity starting in 1974 was widespread, suggesting a global phenomenon. The Post-1973 Slowdown in Productivity Growth • Explanations of the reduced growth in productivity are: – Output measurement problem: • Quality of output and inputs • Shifts to lower productivity sectors • Measurement problems have always been there – Technological depletion and slow commercial adaptation: • The easy stuff has been used up • Firms slow to take up new technologies The Post-1973 Slowdown in Productivity Growth (cont’d) • The dramatic rise in oil prices: – Old capital was energy intensive and thus inefficient – Timing and the fact that the slowdown was international in scope make this an attractive story – But price of capital did not fall and energy was not that important for several sectors – As well, productivity should have picked up when oil prices fell in the 1980s – it did somewhat but later • The beginning of a new industrial revolution: – – – – The beginning of the computer age Takes time to adopt new technologies Have seen some pick up in productivity The industrial revolution was like this The More Recent Experience: After a Pickup, Total Factor Productivity Has Slowed… 0.55 LnTFP Trend TFP 0.50 0.45 0.40 0.35 0.30 … with Implications for Growth 4.5 Trend Labour Trend Capital (adjusted) 4.0 Trend TFP 3.5 Potential output production function Actual growth 3.0 1.2 2.5 1.1 2.0 0.9 0.9 1.5 0.7 1.1 0.6 0.9 1.0 1.1 0.8 1.1 1.1 0.5 1.0 0.8 0.5 0.0 0.4 0.1 1985-90 1990-95 1995-00 2000-05 Note the calculations are based on the growth accounting framework shown in slide 9 2005-10 0.1 2010-15 Contributions to potential growth since 1985 (based on growth accounting equation) 3.50 Trend Labour 3.00 Trend Capital (adjusted) Trend TFP 2.50 2.00 1.50 1.00 0.50 0.00 Potential output production function There have been many advances over the past 40 years Growth Dynamics: The Neoclassical Growth Model • Accounting approach is just that – it is not an explanation of growth. • The neoclassical growth model: – clarifies how capital accumulation and economic growth are interrelated; – explains the factors affecting a nation’s long-run standard of living; – is suggestive of how a nation’s rate of economic growth evolves over time; and – can say something about convergence – do poor countries/regions catch up? Assumptions Underlying the Growth Model • Assume that: – population (Nt) is growing; – at any point in time the share of the population of working age is fixed; – both the population and workforce grow at a fixed rate n; – the economy is closed and there are no government purchases. Setup of the Model of Economic Growth • Part of the output produced each year is invested in new capital or in replacing worn-out capital (It). • The part of output not invested is consumed (Ct). Ct = Yt – It The per-Worker Production Function • The production function in per worker terms is: yt = Atf(kt) (6.5) yt = Yt/Nt is output per worker in year t kt = Kt/Nt is capital stock per worker in year t At = the level of total factor productivity in year t • When the production is written like (6.5) it is often called the “intensive form”. Graph of the per-Worker Production Function • The production function slopes upward. As we move rightward, K is rising faster than N so that k increases. • With more capital, each worker can produce more output. • The slope gets flatter at higher levels of capital per worker due to diminishing MPK. Steady States • In a growth model, equilibrium is defined by something called the steady state. • A steady state is a situation in which the economy’s output per worker (yt), consumption per worker (ct), and capital stock per worker (kt) are constant; these ratios do not change over time. • Remember that these variables are all ratios to Nt so that for example both Yt and Nt are growing but yt, the ratio of the two, is constant. Steady States (continued) • Holding productivity growth constant, the economy reaches a steady state in the long run. • Since yt, ct and kt are constant in a steady-state, Yt, Ct and Kt all grow at rate “n”, the rate of growth of the workforce. • As noted above, this is the definition of the steady state. Characteristics of a Steady State • Gross investment in year t is: It = (n + d)Kt • Kt grows by nKt in a steady state, which ensures that K/N is constant. • Kt depreciates by dKt where d is the capital depreciation rate. • Is this equation consistent with what we have already studied about investment? Characteristics of a Steady State (continued) • We can show that it is consistent with what we have already studied. Start by differentiating (K/N) and setting that derivative to zero (i.e., fulfilling the condition that K/N does not change). Using “Δ” to represent changes: K NK KN K N K N N2 N N N 0 K nK (note ΔN/N = n, the growth rate of the labour force, and we have multiplied the first expression by N) • Using the gross investment identity (I = K* – K + dK) and remembering that ΔK = K* – K = nK in the steady state we get: I = (n + d)K (See Addendum 2 for the discrete version) Characteristics of a Steady State (continued) • Consumption is total output less the amount used for investment. Ct = Yt – (n + d)Kt (6.7) • Put Eq. (6.7) in per-worker terms. • Replace yt with Atf(kt) (Eq. (6.5)). ct = Atf(kt) – (n + d)kt (6.8) Steady-State Consumption per Worker • An increase in the steady-state capitallabour ratio has two opposing effects on consumption per worker: 1) it raises the amount of output a worker can produce, Af(k); and 2) it increases the amount of output per worker that must be devoted to investment, (n + d)k. Steady-State Consumption per Worker (continued) • The Golden Rule level of the capital stock maximizes consumption per worker in the steady state. • At that point the slope of the production function (it’s derivative wrt to k) equals (n + d), the slope of the investment line. • From this we can show that r = n, sometimes referred to as the biological interest rate. • The key here is to use the definition of the user cost of capital assuming no taxes and a price of capital normalised to equal one: MPKG = (n + d) = (r + d) implying that r = n Steady-State Consumption per Worker (continued) • The model shows that economic policy focused solely on increasing capital per worker may do little to increase consumption possibilities of the country citizens if we are close to the Golden Rule level of k. • Empirical evidence is that, given existing starting conditions, a higher capital stock would not lead to less consumption in the long run; i.e., economies are away from kG. • We will assume that an increase in the steady-state capital-labour ratio raises steady-state consumption per worker. Reaching the Steady State • We haven’t described how an economy would reach a steady state. • Why will the described economy reach a steady state? • Which steady state will the economy reach? • The piece of information we need is saving. • Assume that saving in this economy is proportional to current income: St = sYt (6.9) where “s” is a number between 0 and 1. • It represents the faction of current income saved. Reaching the Steady State (continued) • National saving (in this case, private saving as there is no government in the model) has to equal investment. • Here we set our simple saving function equal to the amount of investment that is required to maintain the capital-labour ratio constant: sYt = (n + d)Kt (6.10) Reaching the Steady State (continued) • Put Eq. (6.10) in per-worker terms. • Replace Yt with Atf(kt) (Eq. (6.5)) sAf(k) = (n + d)k (6.11) • Subscript t is dropped because the variables are constant in the steady state. Steady-State Capital-Output Ratio • Equation 6.11 says that, in the steady state, the capital-labour ratio must ensure that saving per worker and investment per worker are equal. • k* is the value of k at which the saving curve and the steady-state investment line cross. • k* is the only possible steady-state capitaloutput ratio for this economy. Equilibrium in the Solow-Swan growth model (note that the position k* is stable) The Steady-State Consumption per Worker • Steady-state output per worker is: y* = Af(k*) • Then the steady-state consumption per worker is: c* = Af(k*) – (n + d)k* • While steady-state investment per worker is: i* = (n + d)k* • Note that the steady-state investment curve slopes upward because a higher k means more I is needed to maintain it. The Model’s Implications • The economy’s capital-labour ratio will converge to k*. • It will remain there forever, unless something changes. • In this steady state the capital-labour ratio, output per worker, investment per worker, and consumption per worker all remain constant over time. • The model determines an equilibrium but not growth – that is given by assumption. The Model’s Implications (continued) • If the level of saving were greater than the amount of investment needed to keep k constant, then that extra saving gets converted into capital and k rises. • If saving were less than the amount needed to keep k constant, the reverse would happen – k would fall. • Note that there is no reason to suppose that the steady state is at a point of maximum consumption – the “Golden Rule”. The Model Implications (continued) • You could get a situation in a poor country where, at low levels of income, saving is below (n + d)k – that is, it is not high enough for the country to reach the level of income in other countries, • Here it is possible to have a low steady state, which is unstable. – A negative shock pushes the economy towards poverty – A positive shock has the opposite effect • Stability here requires that the saving function crosses (n + d)k from above. • Note that the unstable point is bounded by two stable points. s The Determinants of Long-Run Living Standards • Long-run well-being is measured here by the steady-state level of consumption per worker. • Its determinants are: 1) the saving rate (s); 2) the population growth rate (n); 3) the rate of productivity growth (how fast A grows). Long-Run Living Standard and the Saving Rate • A higher saving rate implies a higher living standard. – The increased saving rate raises output at every level of capital per worker. • A steady-state with higher output and consumption per worker is attained in the long run. The Saving Rate (continued) • An increase in the saving rate has a cost – a fall in current period consumption. • As before, in the decision to consume, there is a trade-off between current current and future consumption. • Beyond a certain point, the cost of lost consumption today will outweigh the future benefits. The Saving Rate (continued) • It is also the case that a policy that increases saving will generate a temporary spurt in the growth rate. • Since y = Y/N and N is growing at a constant rate (n), then as we move to a new and higher k* output must grow faster than n at least temporarily. Long-Run Living Standard and Population Growth • Increased population growth tends to lower living standards. • When the workforce is growing rapidly, a larger part of current output must be devoted to just providing capital for the new workers to use. • Absent here is any effect increased population may have on output – increased immigration of highly skilled workers would improve growth by raising TFP. Population Growth (continued) • However, a reduction in population growth means: – lower population and lower total productive capacity; – lower ratio of working-age people to the population and perhaps an unsustainable pension system. • In some countries, low population growth can be raised by encouraging immigration and/or higher female participation. An increase in N, the stock of labour, with the growth rate, n, constant: The case of China • We can think of China’s growth as due, in part, to an increase in the total stock of labour, N. • Imagine a great migration from the country side to the manufacturing sector, raising the level of N but with n unchanged. • The initial effect would be a drop in the capital-labour ratio, shifting it leftward. • At the now lower k, investment is larger than what is required to maintain it constant. The K/N ratio now moves rightward and growth increases during this adjustment period. Implications of a rise in N, with n constant Long-Run Living Standard and Productivity Growth • The model accounts for the sustained growth by incorporating productivity growth. • Increased productivity will improve living standards: – it raises y at every k; – then saving per worker increases; – and a higher k* is attained. • This is important: – without productivity improvements, living standards would remain unchanged once the economy achieves a steady state. Productivity Growth (continued) • A one-time productivity improvement shifts the economy only from one steady state to a higher one. • Only continuing increases in productivity can perpetually improve living standards. • Remember again, productivity growth is exogenous in this model. Total Factor Productivity (TFP) in Canada has slowed recently … 0.55 0.53 LnTFP 0.51 0.49 0.47 0.45 0.43 0.41 0.39 0.37 0.35 Trend(TFP) … this has affected the growth of output per worker The growth of output per worker 1.80 Trend Capital (adjusted)/labour ratio Trend TFP 1.60 Potential output per unit of labour 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Do Economies Converge to Each Other? • Unconditional convergence is a situation when the poor countries eventually catch up to the rich countries so that in the long run, living standards around the world become more or less the same. Certain conditions apply: – For example, if the only difference is K/N but all else is the same (s, n, A) then the model predicts that living standards in countries will converge to that of other countries. – Note that the further away a country is from the lead country, the faster it will grow, given the above conditions. – Some evidence suggests yes – poorer countries have tended to grow faster than richer ones. – Trade and capital flows may be routes that facilitate unconditional convergence (Fischer’s results). Do Economies Converge? (continued) • Conditional convergence is a situation in which living standards will converge only within a group of countries with similar characteristics. – OECD and developing economies could be considered two such groups but even here the effect is not clear cut. – This result occurs if there are differences in s, n and A, all else the same. – The further away an economy is from the steady state, the faster it will grow. What Convergence Would Look Like in an Idealized World 7 Required growth rate to catch up within 50 years 6 Required growth rate = – 0.02 – 2.06 ln(Initial position) 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Initial position vis-à-vis US economy 0.7 0.8 0.9 1 Unconditional Convergence: Some evidence from 20 advanced economies 7 KOR Actual growth = 1.79 – 1.62ln(Initial position) (3.43) R² = 0.83 6 Growth rate from 1960 to 2011 SGP 5 HKG 4 JAP PRT 3 FIN ESP ITA AUT BEL DEU FRA NED SWE GBR 2 NOR ISL CAN AUS DEN 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Initial position vis-à-vis US in 1960 0.7 0.8 0.9 1 A different perspective on convergence Australia GDP as % of US GDP 1.6 1 Austria Belgium End of convergence 1.4 0.9 Canada Denmark 0.8 1.2 0.7 France Finland Germany 1 0.6 Hong Kong Iceland 0.8 0.5 Italy Japan 0.6 0.4 Korea Netherlands 0.3 0.4 0.2 Norway Portugal Singapore 0.2 0.1 Spain Sweden 0 0 United Kingdom United States Another look at unconditional convergence for 15 OECD Economies (1960-2013) Japan GDP as % of US GDP 1 Germany UK 0.9 France Italy 0.8 Canada Spain 0.7 Australia Belgium 0.6 Netherlands 0.5 End of convergence Sweden Denmark 0.4 Austria Finland 0.3 USA Do Economies Converge? (continued) • Most studies find support for the idea of conditional convergence. • Studies show that low saving (including human capital, often measured as spending on education) in developing countries are important in explaining growth differences. • Capital flows are again important. • Other studies highlight the importance of competition, well-functioning labour markets and macroeconomic policy. Unconditional convergence does not exist among emerging markets economies (1960-2011) Average growth rate 1960-2011 7 CHN KOR 6 SPR 5 HUN 4 MAL IDA IND KEN BLZ EGY 3 CHL PAN LES PAK MOR BRA MEX 2PAR COL CRA PHLGTA BGD ECU NIG SAF KEN 1 ALG -0.1 GNA BUR CAM NIC 0 CDI 0.1 GRC 0.3 0.5 0.7 0.9 1.1 Position vis-à-vis US in 1960 -1 LIB -2 ISR There is no real convergence among emerging market economies (1960-2011) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 USA Bangladesh Belize Bolivia Brazil Chad CAR China Costa Rica Dominican Republic Egypt Gabon Greece Guyana Hungary Indonesia Kenya Liberia Malawi Mauritania Morocco Nicaragua Pakistan Papua NG Philippines Seychelles Sri Lanka Sudan Togo Tunisia Uruguay Zambia Algeria Barbados Benin Botswana Burundi Camaroon Chile Colombia Côte d'Ivoire Ecuador Fiji Ghana Guatemala Honduras India Isreal Lesotho Madagascar Malaysia Mexico Nepal Nigeria Panama Paraguay Senegal Sierra Leone South Africa The Bahamas Trinidad-Tobago Turkey Venezuela Required Growth Rate for Emerging Market Economies to close the “Gap” over 30 Years 11 10.5% 10 9 Required growth (shown on top of bars) to close the gap in 30 years for initial position (shown on horizontal axis) Average growth rate from 1960-2012 8 7 6.5% 6 5 4.7% 4 3.6% 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1 -2 -3 Initial position vis-à-vis US in 1960 1 Implication of the Neoclassical Growth Theory • The neoclassical model highlights the role and importance of productivity. • However, it assumes, rather than explains, productivity – the crucial determinant of living standards (see first part of Chapter 6). • In other words, the model does not explain growth in output per capita which is of great interest. Endogenous Growth Theory • In part, because of the failure or difficulties of the unconditional convergence hypothesis, economist started to look for other explanations, focussing in particular on the role of productivity. • Endogenous growth theory tries to explain productivity growth within the model (endogenously). • An implication of endogenous growth theory is that a country’s growth rate depends on its rate of saving and investment, not only on exogenous productivity growth. Setup of the Endogenous Growth Model • Assume that the number of workers remains constant. • This implies that the growth rate of output per worker is simply equal to the growth rate of output. • The aggregate production function is: Y = AK (6.12) where A is a positive constant capturing productivity. • The marginal product of capital (MPK) is equal to A and does not depend on the capital stock (K). • The MPK is not diminishing, it is constant. • This is a major departure from the previous growth model. Constant MPK and Human Capital • One explanation of constant MPK is human capital – the knowledge, skills and training of individuals. • As an economy’s physical capital increases, its human capital stock tends to increase in the same proportion. • Workers get better at using capital – there is learningby-doing – an idea due to Kenneth Arrow. • An example of learning-by-doing is computers. Over time the efficiency of computing has increased 43 million times. While increased computing power has played a role, most is due to improved algorithms; i.e., humans got better at using computers. Constant MPK and Research and Development • Another explanation of constant MPK is research and development (R&D) activities. • The resulting productivity gains offset any tendency for the MPK to decrease. • As the economy grows, firms have an incentive to invest in R&D. The Model of Endogenous Growth • The implications of the model in equilibrium. – Assume that national saving, S, is a constant fraction s of aggregate output, AK, so that S = sAK. – In a closed economy I = S. – As we know, total gross investment equals net investment plus depreciation I = ∆K + dK The Model of Endogenous Growth (continued) • Therefore: (6.13) ΔK dK sAK ΔK or sA d (6.14) K ΔY and sA d (6.15) Y • Since the growth rate of output is proportional to the growth rate of capital stock. • Increases in s will raise growth since it leads to a higher capital stock. Implication of the Endogenous Growth Model • The endogenous growth model places greater emphasis on saving, human capital formation and R&D as sources of long-run growth. • Higher saving and capital formation generate investment in human capital and R&D raising A. • Remember, in the neoclassical (or Solow-Swan) growth model, over time, the saving rate affects only the level of output, not growth. Economic Growth and the Environment • So far we have assumed that there are no natural limits to growth, like declining non-renewable resources or the environment. • The empirical facts: – Levels of many pollutants rise and then fall as economy grows. – The costs of controlling pollution are rising but remain relatively constant as a fraction of GDP. – Pollution emissions per unit of GDP have been falling since the late 1940s. Economic Growth and the Environment (continued) • During the rapid initial economic growth phase, the impact of output growth overwhelms the improvements in pollution-abatement technology. • Near the steady state economic growth slows down and technological progress in pollution control overwhelms the impact of economic growth. • These results are often due to policy choices. CO2 Emissions per capita are Levelling off … 25 CO2 emissions (metric tons per capita) 20 15 EUR AUS 10 CAN GBR USA 5 0 JPN … and Falling as a Fraction of GDP 0.9 CO2 emissions (kg per 2005 PPP $ of GDP) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 CAN GBR USA EUR Time to act? Government Policies and Long-Run Living Standards • Government policies that are useful in raising a country’s long-run standard of living are: – polices to raise the saving rate; – policies to raise the rate of productivity. Policies to Affect the Saving Rate • One view is to let individuals decide for themselves. • The other view is that government actions are needed: – They can try to affect the return on saving, but the effect is small. – By taxing consumption a government can exempt from taxation income that is saved. – A government can increase the amount that it saves by reducing its deficit – this seems to have some positive benefits. National saving rates have been declining 35 30 25 20 15 10 AUS CAN FRA JPN GBR USA DEU Affecting Productivity Growth: Improving Infrastructure • Some research finds a link between productivity and the quality of nation’s infrastructure. • Other research finds that public investments cannot explain cross-country differences. • There is a political dimension here: powerful members of parliament/legislatures funnel funds into their ridings – Japan had lots of infrastructure investment but not much growth. • Higher growth in productivity may lead to more infrastructure, and not vice versa; that is, richer countries may want better infrastructure, like roads, schools, hospitals and communications networks. Affecting Productivity Growth: Building Human Capital • Recent research finds a strong connection between productivity growth and human capital. • Governments affect human capital through education policies, training programs, health programs, etc. • Productivity growth may increase if barriers to entrepreneurial activity are removed (cutting red tape) and competition increased. Spending on Education is Declining in Some Countries 8 Education expenditure (% of GNI) 7 6 AUS CAN 5 FRA DEU JPN 4 GBR USA 3 2 Affecting Productivity Growth: Research and Development • Direct government support of basic research is a good investment for raising productivity – the resulting knowledge spreads through the economy. • Some economists believe that even commercially-oriented research deserves government aid. • Here public-private partnerships may help. Affecting Productivity Growth: Industrial Policy • Industrial policy is a growth strategy in which the government attempts to influence the country’s pattern of industrial development. • The arguments for the industrial policy are borrowing constraints, spillovers, and nationalism. The danger is favouritism. • “Government’s are not very good at finding winners, but losers are good at finding governments.” Sylvia Ostry Affecting productivity Growth: Market Policy • Market policy is government restriction on free markets. • Economists favour respect for property rights and a reliance on free markets to allocate resources efficiently. • The reasons for government to interfere are: market failures and efficiency vs. equity trade-off. • A social safety net may encourage workers to be more in favour technical change, which can be disruptive Addendum 1: Solving the Model for Key Variables • We can use the neo-classical growth model to solve for various key variables. • To determine the steady-state capital-labour ratio (k*), start with the equilibrium condition that S = I in per capita terms. Thus: (n + d)k* = sAk*α • This implies that k* is: 1 (1 ) sA k (n d) * Addendum 1: Solving the Model for Key Variables (continued) • Suppose we want to know the Golden Rule level of the capital-labour ratio, kG. • When the marginal productivity of k equal n + d, we know that consumption is maximised and the capital-labour ratio = kG. Addendum 1: Solving the Model for Key Variables (continued) • Assuming that the production function, in intensive form, is Cobb-Douglas (y = Akα), then the marginal product of k is αAkα–1. Substituting kG into this relationship and setting it equal to n + d, we get: αAkGα-1 = n + d • It then follows that: 1 (1 ) A kG (n d) • Once we know kG we can now solve for y, investment and c. Addendum 1: Solving the Model for Key Variables (continued) • If we wanted to know what saving rate (s) would get us to kG, (whose value we now know) we go back to our old friend saving = investment and assume that we were at the point kG. sAkGα = (n + d)kG • Then s is given by: n d 1 s kG A – The strategy is to go to the point kG and ask the question: what must have been s to get us to this point. – Once we know kG we can solve for y and investment, which of course gives us c. Addendum 1: Solving the Model for Key Variables (continued) • Using the saving = investment identity we can solve for the effects of other changes. • Suppose we want to know what is the effect of higher labour for growth (n’). From the S-I identity we have: sAkα = (n’ + d)k • Which yields a new k equal to: 1 (1 ) sA kn' (n'd) • We can use kn’ (the capital/labour ratio resulting from n’) to calculate the new y, investment and c. Addendum 1: Solving the Model for Key Variables (continued) • A productivity improvement (call it A’) can be handled in a similar fashion; i.e., through the saving-investment identity. sA’kα = (n + d)k • Then as before, we solve for a new k, 1 (1 ) sA' kA' (n d) • Once we have kA’ we proceed as before and get the other variables of interest. • Remember, when re-calculating y, adjust it for the now larger productivity, A’. Addendum 2: Solving the Model for Key Variables in Discrete Time • Start with the following Kt+1 – Kt = It – dKt, which is the capital accumulation identity and can be written as: Kt+1 = (1 – d)Kt + It • The labour force (or population) evolves as: Nt+1 = (1 + n)Nt, where n is the growth rate of labour. • Divide both sides by Nt+1 bearing in mind the growth of labour equation and that St = It: K t 1 Kt sAKt N1 dKt t N t 1 (1 n)N t (1 n)N t (1 n)N t Addendum 2: Solving the Model for Key Variables in Discrete Time (con’t) • Remembering that k = K/N, the equation on the previous slide can be written as: s (1 d) kt 1 kt A kt (1 n) (1 n) • This expression shows what is called the law of motion of capital. Simply stated, it shows how the capital stock per worker (kt+1) evolves over time if the term in square brackets were to change. • As can be seen, kt+1 depends on the amount of capital already in place [kt, times (1-d)/(1+n)] as well as the amount of new capital being added, sAktα, divided by (1+n). • Next we show that the implied steady state capital labour ratio is the same as already derived. Addendum 2: Solving the Model for Key Variables in Discrete Time (con’t) • Multiplying both sides by 1+n we get: (1+n) kt+1 – (1–d) kt = sAktα • In the steady kt+1 = kt = k the equation becomes: (n+d)k = sAkα • The steady state level of capital per worker (k*) is then: 1 (1 ) sA k* (n d) • This is identical to the expression shown in Addendum 1. Raising Growth Can Be a Slow Process: Patience is required to see the results