Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Production for use wikipedia , lookup
Fei–Ranis model of economic growth wikipedia , lookup
Uneven and combined development wikipedia , lookup
Productivity improving technologies wikipedia , lookup
Okishio's theorem wikipedia , lookup
Ragnar Nurkse's balanced growth theory wikipedia , lookup
Productivity wikipedia , lookup
Growth ECON 30020: Intermediate Macroeconomics Prof. Eric Sims University of Notre Dame Fall 2016 1 / 51 Economic Growth I When economists say “growth,” typically mean average rate of growth in real GDP per capita over long horizons I Long run: frequencies of time measured in decades I Not period-to-period fluctuations in the growth rate I “Once one begins to think about growth, it is difficult to think about anything else” – Robert Lucas, 1995 Nobel Prize winner 2 / 51 US Real GDP per capita 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 50 55 60 65 70 75 80 85 Real GDP Per Capita 90 95 00 05 10 15 Linear Trend 3 / 51 Summary Stats I Average (annualized) growth rate of per capita real GDP: 1.8% I Implies that the level of GDP doubles roughly once every 40 years I Growing just 0.2 percentage points faster (2% growth rate): level doubles every 35 years I Rule of 70: number of years it takes a variable to double is approximately 70 divided by the growth rate I Consider two countries that start with same GDP, but country A grows 2% per year and country B grows 1% per year. After 100 years, A will be 165% richer! I Small differences in growth rates really matter over long horizons 4 / 51 Key Question I I What accounts for this growth? In a mechanical sense, can only be two things: I I Growth in productivity: we produce more output given the same inputs Factor accumulation: more factors of production help us produce more stuff I I I Two key factors of production on which we focus are capital and labor Labor input per capita is roughly trendless – empirically not a source of growth in per capita output The key factor of production over the long run is capital: physical stuff that must itself be produced that in turn helps us produce more stuff (e.g. machines). I Which is it? Productivity or capital accumulation? What are policy implications? I What accounts for differences in standards of living across countries? Productivity or factor accumulation? 5 / 51 Stylized Facts: Time Series 1. Output per worker grows at an approximately constant rate over long periods of time picture 2. Capital per worker grows at an approximately constant rate over long periods of time picture 3. The capital to output ratio is roughly constant over long periods of time picture 4. Labor’s share of income is roughly constant over long periods of time picture 5. The return to capital is roughly constant over long periods of time picture 6. The real wage grows at approximately the same rate as output per worker over long periods of time picture 6 / 51 Stylized Facts: Cross-Section 1. There are large differences in income per capita across countries table 2. There are some examples where poor countries catch up (growth miracles), otherwise where they do not (growth disasters) table 3. Human capital (e.g. education) strongly correlated with income per capita table 7 / 51 Solow Model I Model we use to study long run growth and cross-country income differences is the Solow model, after Solow (1956) I Model capable of capturing stylized facts Main implication of model: productivity is key I I I Productivity key to sustained growth (not factor accumulation) Productivity key to understanding cross-country income differences (not level of capital) I Important implications for policy I Downside of model: takes productivity to be exogenous. What is it? How to increase it? 8 / 51 Model Basics I Time runs from t (the present) onwards into infinite future I Representative household and representative firm I Everything real, one kind of good 9 / 51 Production Function I Production function: Yt = At F (Kt , Nt ) I Kt : capital. Must be itself produced, used to produce other stuff, does not get completely used up in production process I Nt : labor I Yt : output I At : productivity (exogenous) I Think about output as units of fruit. Capital is stock of fruit trees. Labor is time spent picking from the trees 10 / 51 Properties of Production Function I Both inputs necessary: F (0, Nt ) = F (Kt , 0) = 0 I Increasing in both inputs: FK (Kt , Nt ) > 0 and F N ( K t , Nt ) > 0 I Concave in both inputs: FKK (Kt , Nt ) < 0 and FNN (Kt , Nt ) < 0 I Constant returns to scale: F (qKt , qNt ) = qF (Kt , Nt ) I Capital and labor are paid marginal products: wt = At FN (Kt , Nt ) Rt = At FK (Kt , Nt ) I Example production function: Cobb-Douglas: F (Kt , Nt ) = Ktα Nt1−α , 0<α<1 11 / 51 Consumption, Investment, Labor Supply I I I I Fruit can either be eaten (consumption) or re-planted in the ground (investment), the latter of which yields another tree (capital) with a one period delay Assume that a constant fraction of output is invested, 0 ≤ s ≤ 1. “Saving rate” or “investment rate” Means 1 − s of output is consumed Resource constraint: Yt = Ct + It I I I Abstract from endogenous labor supply – labor supplied inelastically and constant Current capital stock is exogenous – depends on past decisions Capital accumulation, 0 < δ < 1 depreciation rate: Kt +1 = It + (1 − δ)Kt 12 / 51 Equations of Model Yt = At F (Kt , Nt ) Yt = Ct + It Kt +1 = It + (1 − δ)Kt Ct = (1 − s )Yt It = sYt wt = At FN (Kt , Nt ) Rt = At FK (Kt , Nt ) I Six endogenous variables (Yt , Ct , Kt +1 , It , wt , Rt ) and three exogenous variables (At , Kt , Nt ) 13 / 51 Central Equation I First four equations can be combined into one: Kt +1 = sAt F (Kt , Nt ) + (1 − δ)Kt I Define lowercase variables as “per worker.” kt = per-worker terms: Kt Nt . In kt +1 = sAt f (kt ) + (1 − δ)kt I One equation describing dynamics of kt . Once you know dynamic path of capital, you can recover everything else 14 / 51 Plot of the Central Equation of the Solow Model 𝑘𝑡+1 = 𝑘𝑡 𝑘𝑡+1 𝑘𝑡+1 = 𝑠𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡 𝑘∗ 𝑘∗ 𝑘𝑡 15 / 51 The Steady State I The steady state capital stock is the value of capital at which kt +1 = kt I We call this k ∗ I Graphically, this is where the curve (the plot of kt +1 against kt ) crosses the 45 degree line (a plot of kt +1 = kt ) I Via assumptions of the production function along with auxiliary assumptions (the Inada conditions), there exists one non-zero steady state capital stock I The steady state is “stable” in the sense that for any initial kt 6= 0, the capital stock will converge to this point I “Once you get there, you sit there” I Since capital governs everything else, all other variables go to a steady state determined by k ∗ I Work through dynamics 16 / 51 Algebraic Example I Suppose f (kt ) = ktα . Suppose At is constant at A∗ . Then: sA∗ k = δ ∗ ∗ ∗α y =A k ∗ 1−1 α c ∗ = (1 − s )A∗ k ∗ α i ∗ = sA∗ k ∗α R ∗ = αA∗ k ∗α−1 w ∗ = (1 − α )A∗ k ∗ α 17 / 51 Dynamic Effects of Changes in Exogenous Variables I Want to consider the following exercises: I I I What happens to endogenous variables in a dynamic sense after a permanent change in A∗ (the constant value of At )? What happens to endogenous variables in a dynamic sense after a permanent change in s (the saving rate)? For these exercises: 1. Assume we start in a steady state 2. Graphically see how the steady state changes after the change in productivity or the saving rate 3. Current capital stock cannot change (it is predetermined/exogenous). But kt 6= k ∗ . Use dynamic analysis of the graph to figure out how kt reacts dynamically 4. Once you have that, you can figure out what everything else is doing 18 / 51 Permanent Increase in A∗ 𝑘𝑡+1 = 𝑘𝑡 𝑘𝑡+1 𝑘𝑡+1 = 𝑠𝐴1,𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡 𝑘1∗ 𝑘𝑡+2 𝑘𝑡+1 𝑘𝑡+1 = 𝑠𝐴0,𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡 𝑘𝑡 = 𝑘0∗ 𝑘𝑡+1 𝑘1∗ 𝑘𝑡 19 / 51 Impulse Response Functions: Permanent Increase in A∗ 𝑦𝑡 𝑘𝑡 𝑦1∗ 𝑘1∗ 𝑘0∗ 𝑐𝑡 𝑦0∗ 𝑡𝑖𝑚𝑒 𝑡 𝑖𝑡 𝑐1∗ 𝑐0∗ 𝑤𝑡 𝑡 𝑡𝑖𝑚𝑒 𝑖1∗ 𝑖0∗ 𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑡𝑖𝑚𝑒 𝑅𝑡 𝑤1∗ 𝑤0∗ 𝑅0∗ 𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑡𝑖𝑚𝑒 20 / 51 Permanent Increase in s 𝑘𝑡+1 = 𝑘𝑡 𝑘𝑡+1 𝑘𝑡+1 = 𝑠1 𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡 𝑘1∗ 𝑘𝑡+2 𝑘𝑡+1 𝑘𝑡+1 = 𝑠0 𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡 𝑘𝑡 = 𝑘0∗ 𝑘𝑡+1 𝑘1∗ 𝑘𝑡 21 / 51 Impulse Response Functions: Permanent Increase in s 𝑘𝑡 𝑦𝑡 𝑘1∗ 𝑦1∗ 𝑘0∗ 𝑦0∗ 𝑡𝑖𝑚𝑒 𝑡 𝑖𝑡 𝑐𝑡 𝑡 𝑡𝑖𝑚𝑒 𝑖1∗ 𝑐1∗ 𝑐0∗ 𝑤𝑡 𝑖0∗ 𝑡 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡 𝑅𝑡 𝑤1∗ 𝑤0∗ 𝑅0∗ 𝑅1∗ 𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑡𝑖𝑚𝑒 22 / 51 Discussion I Neither changes in A∗ nor s trigger sustained increases in growth I Each triggers faster growth for a while while the economy accumulates more capital and transitions to a new steady state I Aren’t we supposed to be studying growth? In the long run, there is no growth in this model – it goes to a steady state! We’ll fix that. You can kind of see, however, that sustained growth must come from increases in productivity. Why? I I I No limit on how high A can get – it can just keep increasing. Upper bound on s Repeated increases in s would trigger continual decline in Rt , inconsistent with stylized facts I Bottom line: sustained growth must be due to productivity growth, not factor accumulation. You can’t save your way to more growth I Key assumption: diminishing returns to capital 23 / 51 Golden Rule I What is the “optimal” saving rate, s? I Utility from consumption, not output Higher s has two effects – the “size of the pie” and the “fraction of the pie”: I More capital → more output → more consumption (bigger I I size of the pie) Consume a smaller fraction of output → less consumption (eat a smaller fraction of the pie) I Golden Rule: value of s which maximizes steady state consumption, c ∗ I s = 0: c ∗ = 0 I s = 1: c ∗ = 0 I Implicity characterized by A∗ f 0 (k ∗ ) = δ. Graphical intuition. 24 / 51 Growth I Wrote down a model to study growth I But model converges to a steady state with no growth I Isn’t that a silly model? I It turns out, no I Can modify it 25 / 51 Augmented Solow Model I Production function is: Yt = At F (Kt , Zt Nt ) I I I Zt : labor-augmenting productivity Zt Nt : efficiency units of labor Assume Zt and Nt both grow over time (initial values in period 0 normalized to 1): Zt = (1 + z )t Nt = ( 1 + n ) t I I z = n = 0: case we just did Zt not fundamentally different from At . Mathematically convenient to use Zt to control growth while At controls level of productivity 26 / 51 Per Efficiency Unit Variables I Define kbt = ZKt Nt t and similarly for other variables. Lower case variables: per-capita. Lower case variables with “hats”: per efficiency unit variables I Can show that modified central equation of model is: kbt +1 = I h i 1 sAt f (kbt ) + (1 − δ)kbt . (1 + z )(1 + n) Practically the same as before 27 / 51 Plot of Modified Central Equation 𝑘𝑘�𝑡𝑡+1 = 𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1 𝑘𝑘� ∗ 𝑘𝑘�𝑡𝑡+1 = 𝑘𝑘� ∗ 1 �𝑠𝑠𝐴𝐴𝑡𝑡 𝑓𝑓�𝑘𝑘�𝑡𝑡 � + (1 − 𝛿𝛿)𝑘𝑘�𝑡𝑡 � (1 + 𝑧𝑧)(1 + 𝑛𝑛) 𝑘𝑘�𝑡𝑡 28 / 51 Steady State Growth I I I I Via similar arguments to earlier, there exists a steady state kb∗ at which kbt +1 = kbt . Economy converges to this point from any non-zero initial value of kbt Economy converges to a steady state in which per efficiency unit variables do not grow. What about actual and per capita variables? If kbt +1 = kbt , then: Kt Kt + 1 = Z t + 1 Nt + 1 Z t Nt Kt + 1 Z t + 1 Nt + 1 = = (1 + z )(1 + n) Kt Zt Nt Zt + 1 kt + 1 = = 1+z kt Zt Level of capital stock grows at approximately sum of growth rates of Zt and Nt . Per capita capital stock grows at rate of growth in Zt This growth is manifested in output and the real wage, but not the return on capital 29 / 51 Steady State Growth and Stylized Facts I Once in steady state, we have: yt +1 = 1+z yt kt +1 = 1+z kt Kt + 1 Kt = Yt +1 Yt w t + 1 Nt + 1 wt Nt = Yt +1 Yt Rt +1 = Rt wt +1 = 1+z wt I These are the six time series stylized facts! 30 / 51 Understanding Cross-Country Income Differences I Solow model can reproduce time series stylized facts if it is assumed that productivity grows over time I Let’s now use the model to think about cross-country income differences What explains these differences? Three hypotheses for why cross-country income differences exist: I 1. Countries initially endowed with different levels of capital 2. Countries have different saving rates 3. Countries have different productivity levels I Like sustained growth, most plausible explanation for cross-country income differences is productivity I Consider standard Solow model and two countries, 1 and 2. Suppose that 2 is poor relative to 1 31 / 51 Convergence I I Suppose two countries are otherwise identical, and hence have the same steady state. But suppose that country 2 is initially endowed with less capital – k2,t < k1,t = k ∗ 𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑘𝑘𝑗𝑗,𝑡𝑡 𝑘𝑘𝑗𝑗,𝑡𝑡+1 𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑠𝑠𝐴𝐴𝑓𝑓�𝑘𝑘𝑗𝑗,𝑡𝑡 � + (1 − 𝛿𝛿)𝑘𝑘𝑗𝑗,𝑡𝑡 𝑘𝑘 ∗ 𝑘𝑘2,𝑡𝑡 𝑘𝑘1,𝑡𝑡 = 𝑘𝑘 ∗ 𝑘𝑘𝑗𝑗,𝑡𝑡 32 / 51 Catch Up I If country 2 is initially endowed with less capital, it should grow faster than country 1, eventually catching up with country 1 Country 1 Country 2 𝑦𝑦 𝑔𝑔𝑗𝑗,𝑡𝑡+𝑠𝑠 𝑘𝑘𝑗𝑗,𝑡𝑡+𝑠𝑠 𝑦𝑦 𝑔𝑔2,𝑡𝑡 𝑘𝑘1,𝑡𝑡 = 𝑘𝑘 ∗ 𝑦𝑦 𝑔𝑔1,𝑡𝑡 = 0 𝑘𝑘2,𝑡𝑡 0 𝑠𝑠 0 𝑠𝑠 33 / 51 Is There Convergence in the Data? 16 14 Correlation between cumulative growth and initial GDP = -0.18 Y2010/Y1950 12 10 8 6 4 2 0 0 2,000 6,000 10,000 14,000 Y1950 I Correlation between growth and initial GDP is weakly negative when focusing on all countries 34 / 51 Focusing on a More Select Group of Countries 14 12 Correlation between cumulative growth and initial GDP = -0.71 Y2010/Y1950 10 8 6 4 2 2,000 4,000 6,000 8,000 10,000 14,000 Y1950 I I Focusing only on OECD countries (more similar) story looks more promising for convergence Still, catch up seems too slow for initial low levels of capital to be the main story 35 / 51 Pseudo Natural Experiment: WWII 1.2 1 0.8 US Germany 0.6 UK Japan 0.4 0.2 0 1950 I I 1960 1970 1980 1990 2000 2010 WWII losers (Germany and Japan) grew faster for 20-30 years than the winners (US and UK) But don’t seem to be catching up all the way to the US: conditional convergence. Countries have different steady states 36 / 51 Differences in s and A∗ I Most countries seem to have different steady states I For simple model with Cobb-Douglas production function, relative outputs: y1∗ = y2∗ A1 A2 1−1 α s1 s2 1−α α . I Question: can differences in s plausibly account for large income differences? I Answer: no 37 / 51 Differences in s I Suppose A∗ the same in both countries. Suppose country 1 is y∗ US, and country 2 is Mexico: y1∗ = 4. We have: 2 s2 = 4 α −1 α s1 I Based on data, a plausible value of α = 1/3. Means α −1 α = −2 I Mexican saving rate would have to be 0.0625 times US saving rate I This would be something like a saving rate of one percent (or less) I Not plausible I Becomes more plausible if α is much bigger 38 / 51 What Could It Be? I If countries have different steady states and differences in s cannot plausibly account for this, must be differences in productivity I Seems to be backed up in data: rich countries are highly productive 160,000 GDP per worker (2011 US Dollars) Correlation between TFP and GDP = 0.82 140,000 120,000 100,000 80,000 60,000 40,000 20,000 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 TFP (relative to US = 1) 39 / 51 Productivity is King I Productivity is what drives everything in the Solow model I Sustained growth must come from productivity I Large income differences must come from productivity I But what is productivity? Solow model doesn’t say 40 / 51 Factors Influencing Productivity I Including but not limited to: 1. 2. 3. 4. 5. 6. 7. Knowledge and education Climate Geography Institutions Finance Degree of openness Infrastructure 41 / 51 Policy Implications I If a country wants to become richer, need to focus on policies which promote productivity I Example: would giving computers (capital) to people in sub-Saharan Africa help them get rich? Not without the infrastructure to connect to the internet, the knowledge of how to use the computer, and the institutions to protect property rights I Also has implications when thinking about poverty within a country 42 / 51 Output Per Worker over Time 11.6 Actual Series Trend Series 11.4 log(RGDP per Worker) 11.2 11 10.8 10.6 10.4 10.2 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 43 / 51 Capital Per Worker over Time 12.6 Actual Series Trend Series 12.4 log(Capital per Worker) 12.2 12 11.8 11.6 11.4 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 44 / 51 Capital to Output Ratio over Time 3.5 3.4 K/Y 3.3 3.2 3.1 3 2.9 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 45 / 51 Labor Share over Time 0.68 0.67 Labors Share 0.66 0.65 0.64 0.63 0.62 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 46 / 51 Return on Capital over Time 0.125 0.12 Return on Capital 0.115 0.11 0.105 0.1 0.095 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 47 / 51 Real Wage over Time 3.6 Actual Series Trend Series 3.4 log(Wages) 3.2 3 2.8 2.6 2.4 2.2 1950 1960 1970 1980 1990 2000 2010 2020 Year go back 48 / 51 Income Differences GDP per Person High income countries Canada Germany Japan Singapore United Kingdom United States $35,180 $34,383 $30,232 $59,149 $32,116 $42,426 China Dominican Republic Mexico South Africa Thailand Uruguay $8,640 $8,694 $12,648 $10,831 $9,567 $13,388 Cambodia Chad India Kenya Mali Nepal $2,607 $2,350 $3,719 $1636 $1,157 $1,281 Middle income countries Low income countries go back 49 / 51 Growth Miracles and Disasters South Korea Taiwan China Botswana Madagascar Niger Burundi Central African Republic Growth Miracles 1970 Income 2011 Income $1918 $27,870 $4,484 $33,187 $1,107 $8,851 $721 $14,787 Growth Disasters $1,321 $937 $1,304 $651 $712 $612 $1,148 $762 % change 1353 640 700 1951 -29 -50 -14 -34 go back 50 / 51 6 7 log(RGDP per Person) 8 9 10 11 Education and Income Per Capita 1 1.5 2 2.5 Index of Human Capital 3 3.5 go back 51 / 51