Models of Economics Growth

Models of Economics Growth: Capital Accumulation- Growth models go beyond last section's "growth accounting” framework.- Models of the growth process: - simplification: focus on essentials - identify key parameters and outcomes- model interrelationships between key variables.- The two models below stress the importance of "capital accumulation". - Physical capital is the focus: machines, tools, buildings, infrastructure.- Some characteristics of capital (K): - Capital is a produced input. - Capital is a productive asset: - an asset pays a stream of returns over it’s lifetime. - capital generates a stream of valuable services i.e. value of the extra output produced with the capital.- Creating/ buying K involves investment: pay now reap future returns. - Capital is a durable asset but wears out over time (depreciation)- Durable: lasts for many periods – there is an important time element in decisions to invest in K. - Depreciation means that a country’s stock of K will shrink unless there is replacement investment. 1 - Economic history and the importance of capital:- Developed countries: underwent industrializations.- Capital accumulation is necessary to build modern industry (factories, machines, infrastructure)- Early development economics (1950s-1970s): strong emphasis on boosting capital accumulation. e.g. W.W. Rostow (1960) Stages of Economic Growth: boost the savings (and investment rate) to achieve -- “takeoff”.W. A. Lewis (1955) Theory of Economic Growth: “central problem of economic development is to understand the process by which a community … investing 4-5 per cent of its national income … converts…to saving 12-13 per cent …” - Allen Global Economic History – plots of K per L vs. GDP per L (see next page)- first plot: each data point is a country; - second and third plots: also includes evolution of the relationship over time for Italy and Germany. - Graphs suggest a positive relationship: higher GDP per worker associated with higher K/L.- Growth accounting and development accounting studies (last set of notes): most suggest that differences in K/L are important.2 3 Harrod-Domar Model- See: Easterly The Elusive Quest for Growth Ch. 2 (a practictioner’s criticial view)- Developed in the 1940s.- Focus: capital accumulation as the key to growth.- Model highlights two key parameters (k and s): (1) k = incremental capital-output ratio (assumed a constant) (ICOR)= extra capital needed to produce an extra unit of output= K/Y (this is inverse of K productivity: Y/K)- ICOR is often treated as constant determined by technology and industrial structure of the economy.(2) s = savings rate (share of output or income that is saved)4 - Aggregate (economy-wide) production function in Harrod-Domar:"Fixed-coefficients" (Leontief) technology:Yt = min(aLt , bKt)- this says that Yt is the minimum of aLt or bKt - it implies that producing one unit of Y requires "1/a" units of L and "1/b" units of K (so: Y = min(1,1)=1 ) - notice that having L=2/a and K=1/b still gives only one unit of output: Y=min(2,1)=1 i.e. the extra L is unproductive unless it has K to work with.- Let’s represent the production function in a graph.- Isoquants: combinations of K and L that produce the same quantity of output.- Leontief production: isoquants are right-angles (through the ray from the origin a/b).5 - Poor economies? - Surplus labour seems plausible (like pt. 0 at LBig above)- lots of L, relatively little K (so K/L < a/b).- much of the L is ‘unproductive’ or ‘surplus’ due to lack of K. - there is a bottleneck on growth from lack of capital.- raise K to raise output (raising L has no effect).- the growth equation is (assuming b is constant):Y = b K- ‘b’ measures the productivity of extra K.- The incremental capital-output ratio (ICOR) k= 1/b:1/b = K/Y - this is the amount of extra K needed to ease the capital shortage bottleneck enough to raise output by 1. 6 - Where does K come from? Capital is created by investment spending (I):Investment (I) = K (this version ignores depreciation’ with constant depreciation: K =I-K where  is depreciation rate: see Solow model)- Investment spending is financed via savings (S) i.e. some of the economy’s income is not being spent on current consumption.Savings (S) = Investment (I)- Assume a simple economy-wide savings equation:S = s Y s= savings rate (constant)- savings is assumed to be domestic (from own country).- savings is a constant share of output (national income).7 - So output growth in the labour surplus economy is:Y = bK = K/k by definition of k=1/bY = I/k = S/k = sY/k (assumes: S=sY )Y/Y = s/k growth rate of GDP equals s/k- Capital accumulation is key to growth in a labour surplus economy.- industrialize ! - a lack of K is keeping workers unproductive and Y low. - “Financing gap” approach to development is rooted in this kind of thinking: - a target growth rate implies a target level of savings (given k).- development policies should aim to provide the needed finance.- Policies that raise "s" will raise output growth.8 - Aggregate savings and National accounting (from intro economics):Y = GDP G = government spending T = taxes (less transfers) C = consumption X = exports NCI = net capital inflow I = investment spending M = importsAn identity: GDP equals aggregate spending across sectors.Y = C + I + G + X – MSolve for I and then add and subtract T :I = (Y-T-C) + (T-G) - (X-M)Y-T-C = private domestic savingsT-G = government savings (budget surplus if >0)X-M = net exports : <0 financed by borrowing abroad (foreign savings) >0 financed by lending abroad. (outflows of domestic saving)= -(net foreign savings) (or Net Capital Inflow)9 - So three sources of savings: Source PoliciesDomestic private financial development, taxes and savingsGovernment savings budget surplusesSavings from abroad foreign aid, ownership rules, exchange rate policy, stability, rule of law.- Recall Allen’s discussion of the ‘Standard Model’ followed by countries who industrialized after Britain: development of a financial system was one of the four parts of this strategy. - Financial system: offers a variety of secure ways to save; provides ways in which businesses can borrow; matches savings to borrowers. 10 - Model suggests growth rates would also be affected by altering "k" - lower k: more growth- Interpretation of a lower k: more productive investments.- Labour? - implicitly it is in excess supply here (surplus).- unemployment or underemployment of labour depends on the capital growth.- Is much of labour in poor country agriculture or in the informal sector underemployed?11 - Easterly: Harrod-Domar “most widely used growth model”- Early development economics, planners and the World Bank used (and some still use) versions of this model for their policy prescriptions.- often multi-sector versions (a Leontief production function and estimates of ICOR (k) for each major industry): Input-Output Models. - Given “k” the growth rate equation tells what the target value of “s” must be to achieve a growth rate target in a given sector.- Construct a plan to allocate savings (investment) between industries to achieve a growth target. Allows via input-output structure for interdependence between industries (outputs in one industry may be an input in another industry).- Bhagwhati (in India in Transition) experience of India and the Soviet Union in the post-WWII period.- savings rates were reasonably high. - BUT investment was unproductive (high k) - lack of competition - poor incentives. i.e., economic institutions proved to be a barrier to growth- High savings and investment only produce high growth if capital is productive.- Was this true elsewhere in 1950s and 1960s? North Africa, parts of Latin America.12 - Problems with the Harrod-Domar Model and the "Financing Gap Approach" to Development: (see Easterly):- Empirical support: - the link between investment and growth is not as simple as the model suggests (other factors matter);- predictive power: Easterly’s Figure 2.1 for Zambia; model’s prediction of which countries would grow fast (wrong ones).- Technological assumptions: - Leontief production function does not allow for different input intensities in production: low K/L need not mean that capital shortages constrain growth.- common for LDCs to use very different input mixes than MDCs in the same industry.- Model ignores incentives: - low investment may partly reflect incentives e.g. returns to investment are low for some reason.- then additional finance (savings) may produce little growth.- Growth diagnostics approach: can methods be developed to identify constraints on growth in individual countries?- Poor countries: too little finance or no incentive to invest? 13 (D. Rodrik: no single recipe, answer is case specific – growth diagnostics)Solow Growth Model- See: any second-year macroeconomics text; see website for online sources. - Developed by R. Solow in the late-1950s (Solow model, Solow-Swan model).- The dominant model of growth in mainstream, "neoclassical economics".- It makes neoclassical technical assumptions: - substitution between types of inputs is assumed possible; - diminishing returns.- An alternative to the Harrod-Domar model which was built upon a non- neoclassical production function.Model Components:(1) Aggregate Production functionYt = F(Kt, Lt)Yt = GDP, output at time t.Lt = amount of labour at time t.Kt = quantity of capital at time t.14 - Production function assumptions:- K and L are substitutes in production.- K intensive and L intensive ways of producing a given Y.- Constant returns to scale:- double K and L and output (Y) doubles.- Due to constant returns to scale:Yt = F(Kt, Lt) divide by Lt: Yt /Lt = F(Kt, Lt)/ LtYt /Lt = F(Kt/Lt, 1) GDP per worker depends on capital per worker.15 - Diminishing returns:- Marginal product: extra output from an extra unit of input.- If K rises, with L constant, the marginal product of K will fall.- If L rises, with K constant, the marginal product of L will fall.- A given rise in K/L gives a smaller rise in Y/L the higher is K/L.- technically: same first derivative as for diminishing returns to K.- intuitively: K/L rising, so K is growing faster than L, extra K has less L to work with so its productivity falls.- Picture: Production function becomes flatter at higher K/L.- Support for diminishing returns: see figures from Allen (above and see the previous set of notes)16 (2) Capital accumulation:- Capital accumulation part of the model is like Harrod-Domar model.- K grows due to investment;- but let’s allow for depreciation of existing capital as well. i.e. assume that a share () of the K stock wears out each period.- in the model  is treated as a constant. Kt = (Investment at time t) - (Depreciation of existing stock)= It -  KtInvestment (It) = Savings (St)Savings function (as in Harrod-Domar):St = sYt s= savings rate (a constant)So: Kt = sYt -  Kt- is the capital accumulation relationship.17 (3) Labour force growth:Lt = gLt labour grows at a constant rate g (some current treatments set g=0).- Put the three parts of the model together!- Since:Yt /Lt = F(Kt/Lt, 1)How Y/L changes over time depends on whether K/L is rising, falling or constant: i.e., on how the growth rates of K and L compare: K/L rising if: Kt/Kt - Lt/Lt > 0 K/L falling if: Kt/Kt - Lt/Lt < 0K/L constant if: Kt/Kt - Lt/Lt = 0We have: Lt/Lt = gKt/Kt = (sYt -  Kt)/ Kt= sYt/Kt - So answer to the question of how K/L changes depends on whether: Kt/Kt - Lt/Lt = sYt/Kt -  - g > 0 , < 0 , = 018 - The key equation: sYt/Kt -  - g > 0 , < 0 , = 0Is usually multiplied through by K/L to help present the model in graphs.- After multiplying through by K/L the key equation of the model is: K/L and Y/L rising if: s(Yt/Lt) – (g+)(Kt/Lt) > 0 K/L and Y/L falling if: s(Yt/Lt) – (g+)(Kt/Lt) < 0K/L and Y/L constant if: s(Yt/Lt) – (g+)(Kt/Lt) = 019 - Graph: s(Yt/Lt) – (g+)(Kt/Lt) > 0, =0 or <0 s(Yt/Lt) = savings per unit of L= capital growth per unit L generated by savings.= falling slope reflects diminishing returns i.e., s Yt/Lt = s F(Kt/Lt, 1)(g+)(Kt/Lt) = amount of capital growth per unit L required to keep K/L constant. i.e. to maintain K/L: g K is needed to equip each new worker with the same level of K as old workers. the Kt capital that wears out must be replaced.20 - The model has an equilibrium: (Y/L)ss, (K/L)ssStart instead with: Low K/L: - capital growth per L exceeds K growth required to keep K/L constant.- K/L rises, Y/L rises.- As K/L rises, capital required to keep K/L constant grows steadily at rate of (g+.- Savings per L grow at a diminishing rate due to diminishing returns in producing output. - K/L eventually stops growing.- when required capital growth equals actual capital growth.(Constant returns numerical example?Say that: g= .05 (5%) K-growth: 12%Output must be growing between 5%-12% (e.g. 9%)Next period - Investment in K grows at 9% (S = sY = I) - Say depreciation is 1% then K has grown 8%. - So: - L now grows at 5% - Y grows 5-8% (say 7%)Next period: - K has grown by 7%-1%=6% - L grew 5% - Y grows 5-6% (6%) i.e., K growth eventually falls to equal L growth: here 5% )21 Start with: High K/L: - capital growth per L is less than K growth required to keep K/L constant.- K/L falls, Y/L falls.- As K/L falls, the capital required to keep K/L constant falls at a steady rate of (g+.- Savings per L falls more slowly due to diminishing returns in producing output. (i.e., drop in Y/L is relatively small for a fall in K/L when K/L is high)- K/L eventually stops falling.- when required capital growth equals actual capital growth.- At the "steady state" equilibrium (K/L)ss: s(Yt/Lt) – (g+)(Kt/Lt) = 0- equilibrium in that Y/L and K/L are constant over time.- But: Y, K and L are all growing over time at a rate of g. i.e., equal to the labour growth rate. (‘balanced growth’)- Consumption per worker? Consumption (C) = Income – Savings- In per worker terms: (Y/L) – (sY/L) = (1-s)Y/L- Notice that higher K/L raises Y/L but at a cost (lower C). - Is there a best value of ‘s’ that balances these effects and maximizes C/L?22 Comparative statics: How does the steady state equilibrium change if key parameters change?(1) Change in the savings rate (s):- Rise in s:- at the old equilibrium K/L, savings per L are higher than what is required to keep K/L constant.- K/L rises. - Capital required to keep K/L constant at its higher level grows faster than savings: eventually get to a new equilibrium: K/L and Y/L higher than before.23 (2) Rise in g (labour force growth):- At the old equilibrium, capital growth required to keep K/L constant is now higher than before.- K/L falls.- capital required to keep K/L constant shrinks at a faster rate then savings, eventually a new equilibrium is attained with a lower K/L and lower Y/L.24 (3) Technology improves (or institutions or organization improves):- Form of F(K, L) changes so that more output is produced by current K, L (total factor productivity has risen). 1- e.g. in Cobb-Douglas “A” rises. ( Yt = A Kt Lt )- Savings per L rise. - Capital growth is higher than is required to keep K/L constant. K/L rises.- Capital growth required to keep K/L at its increasingly higher level grows faster than savings. Eventually, K/L stops rising.- K/L and Y/L higher after the technological improvement.(a depreciation rate comparative static could also be done but is not so interesting –  is treated as if it is not an economic variable)25 Predictions: Why might Y/L Differ between Rich and Poor Countries?- Imagine Poor and Rich economies operate along the lines of the Solow model.- Poor have lower levels of GDP per capita (Y/L) than Rich countries. - Why might this occur? - Two types of situation are suggested by the model:(A) If these are equilibrium differences, the Solow growth model suggests these differences could arise because:(1) Poor countries have lower savings rates than Rich countries;(2) Poor countries have higher labour force growth than Rich countries; (3) Poor countries have inferior technology, organization or institutions compared to Rich countries (so lower F).(any of these or some mix of them could explain lower Y/L)- To encourage growth, Poor countries should:- Adopt policies that will raise their savings rates.- Adopt policies that will lower labour force growth (e.g., slow population growth).- Improve their technology, organization, improve institutions so they get more output per worker for a given K/L.26 (B) Some Poor vs. Rich country Differences may be Transitional:- Say Poor country has the same equilibrium K/L and Y/L as a Rich country.- Then the Poor Country is poorer since it is still moving toward its equilibrium. - Differences will be eliminated once the equilibrium is reached. (convergence)27 Predictions Regarding Differences in Growth Prospects: - Comparative statics asked how the equilibrium was affected by differences in g, s and technology.- For economies with the same “s”, “g” and the same production function:- countries with low Y/L should grow faster (shrink less) than countries with high Y/L (low Y/L country is further from their common steady state)- For economies starting with the same initial Y/L, same production function:- high "s" and/or low "g" countries will see most growth (least shrinkage) in Y/L.- low “s” and/or high “g” countries will see least growth (most shrinkage) in Y/L. i.e., as each economy moves to its steady state (steady state is higher for high s, low g country) - Same initial Y/L, same s, same g: - country with a higher production function (F) will have more growth (less shrinkage).28 Evidence in Support of the Solow Growth Model:- Scatterplots from Jones and Vollrath or Mankiw and Scarth:- positive association between current GDP per person and investment as a share of GDP (averaged over the previous few decades).- negative association between current GDP per person and average annual population growth over the previous few decades.- note: if the labour force participation rate is constant population growth will equal labour force growth.- scatterplots are consistent with Solow model. 29 30 Mankiw, Romer and Weil: Quarterly Journal of Economics 1992- Can the neoclassical (Solow) growth model explain observed cross-country variation in GDP per worker?- Started with a Cobb-Douglas production function. - extra wrinkle: assumes production function is shifting upward at a constant rate (see below) e.g. reflecting technological progress.- Linear regression approach: ln(Yt/Lt) = a + b ln(I/Y) + c ln(g+)- Fits this relation to data for a large sample of countries 1965-85, i.e. choose values of the coefficients (a, b and c) that best fit the data. - Value of Y/L is for 1985- Investment rate (I/Y) plays the role of “s”: average 1960-85- g is the working population growth rate: average 1960-85.-  : suggests it is around .03-.04 for most countries.- Actual version also assumes constant growth in technological progress (dropped in the regression equations below).- Regression results:98 country sample: (excludes 8 major oil producers) ln(Yt/Lt) = 5.48 + 1.42 ln(I/Y) -1.97 ln(g+)75 country sample: (excludes countries with poor data quality) ln(Yt/Lt) = 5.36 + 1.31 ln(I/Y) -2.01 ln(g+)OECD (industrialized) countries only (22 countries): ln(Yt/Lt) = 7.97 + 0.50 ln(I/Y) -0.76 ln(g+)31 - Results and theory:- Signs of the coefficients are consistent with the Solow model.- higher investment (savings) rate higher Y/L.- higher population growth smaller Y/L.- Estimates are statistically significant for the two larger samples.- The estimated equation explains 60% of variation in Y/L in 1985 between countries for the two larger samples.- But estimated coefficients are larger than theory suggests. - Extended version adds human capital accumulation to the model: - coefficients still predicted signs.- coefficient sizes more reasonable.- 80% of variation in Y/L explained. 32 Some Problems and Extensions of the Neoclassical (Solow) Model(1) Steady-state equilibrium has a constant value of Y/L:- There appears to be an upward trend in Y/L. - maybe there is a steady state in Y/L rather than Y. i.e. Y/L grows at a constant rate in equilibrium. - Technology tends to improve over time. - Could have allow a constant shift upward in the production function over time e.g. Y=F(K,L) = A KL1- let A in the Cobb-Douglas grow at a constant rate. - Common alternative (labour-augmenting technical progress):- Main effect of technology is to increase efficiency of L. - Say that N is the measure of labour input:Yt = F(Kt, Nt)Nt = et Lt where et = measure of efficiency of a unit of L. Lt = units of labour (workers) – as before. - let technological improvements raise the productivity of labour (boost “et”)- say that “e” grows at a rate of - then N grows by (g+ ) each year (L grows at rate g).- Now assuming constant returns (as before):Yt/Nt = F(Kt/Nt, 1) 33 - A steady state requires a constant K/N : s(Yt/Nt) - (g+ +) (Kt/Nt) = 0- like before except N replaces L and (g+ +) replaces g+.- this maintains a constant: K/N and Y/N- K/L and Y/L will actually be growing in the steady state: - K, N and Y grow at rate (g+) - L grows at rate g - so Y/L grows at rate - Implication: differences in  will determine relative growth in Y/L between countries. 34 (2) Determination of population growth or savings rates:(a) Population growth- It can be argued that savings rates and population (labour force) growth rates depend on the current level of Y/L. i.e., not constants as in basic model. - Malthusian population growth: sub g = ∙(Yt/Lt – y ) where: ysub = subsistence level Y/L.  >0 a parameter showing the effect of Y/L on population (labour force) growth. - Diagrams? - imagine g growing as K/L and Y/L grow. - K growth required to maintain K/L grows since L growth rises. - Malthusian population growth means less Y/L growth for a given rise in s, F.- But don’t have the Classical model result of no growth. (can get closer to this by adding a fixed input too)- Is Malthusian population growth more appropriate for an LDC?- Does the Malthusian relationship break down at higher Y/L? (demographic transition)- if so: g may rise with Y/L at low Y/L g may start to decline after some threshold level of Y/L.- imagine what this could do to the steady-state.35 (b) Savings rate (s): constant in the basic model. Savings rates and Y/L:- s likely low in poor countries (little income left after meeting basic needs)- s seems likely to rise initially with Y/L. (see for example Weil's Figure 3.8)- More generally: maybe "s" rises gradually over some range of Y/L.- this can give low-, medium- and high-level equilibria (draw this!) - note: medium-level equilibrium is unstable.- Interest?- low-level steady state equilibrium is a “poverty trap” type outcome: low income  low savings, low investment   low productivity  low K/L ( of L)- vicious circle: poor because you are poor!- is this the situation of LDCs today?- high-level steady state could be identified with MDCs.36 37- A “Big Push” strategy? - a one-time shock to the economy that raised K/L sufficiently could switch a LDC into a MDC. i.e. it would grow from the low-level to the high-level steady state (show this).- Jeffrey Sachs (2005) End of Poverty suggests that foreign aid might be able to do something like this.(others e.g. William Easterly dispute this)- Other determinants of investment and savings:- Are they determined by “fundamentals” – culture, governance, history, religion, geography, etc. - if so what if anything can be done?- Above: more savings leads to more investment.- but low investment may partly reflect incentives e.g. returns to investment are low for some reason.- then additional finance (savings) may produce little growth.- Growth diagnostics approach: can methods be developed to identify constraints on growth in individual countries?- too little finance (low savings) or no incentive to invest? e.g Dani Rodrik.38 - Interest Rates and the Solow model. - focus is on capital accumulation, investment and savings but where is the interest rate?- interest rates affect cost of borrowing to finance investment and the return to saving: likely to be important to capital accumulation.- slope of the aggregate production function (F): indicates the amount by which extra K (extra K/L) raises value of output, Y (Y/L). i.e. related to the return on investment in extra K. - diminishing returns and this? Return on K investment should fall as K/L rises.- interest rates are linked to this return (slope of production function) - return on K must be as good as on financial investments: savings are directed to the higher return investment (tends to equalize returns) - return on K > interest rate: borrow more to finance more K, extra borrowing raises the interest rate. - Interesting implication: other things equal returns on K investment should be higher in LDCs where K/L is low.- with globalized financial markets will savings flow to LDCs?(will look at this in more detail later)- Fuller versions of the model: savings a result of household maximization problem (end result is not too different from basic Solow).39 (3) Human capital.- Too simple: focus on one type of labour and an aggregate measure of capital.- Versions of the model with education variables typically perform better.(more of observed growth is explained: Mankiw, Romer and Weil)(4) Technology:- Technological change is not explained in the model. - Implicitly it is independent of Y/L, the amount of investment, labour force growth etc.- Investment and technological change may be closely linked in practice.- Research and Development expenditures as investment.- in the previous model  and therefore growth in Y/L would depend upon “s”.- Virtuous circles?- More saving, more investment, more R&D, higher return to K investment (diminishing returns pushed back), more investment, more advances in knowledge, etc.- higher Y/L, bigger markets, higher potential returns to innovation, more R&D, more innovation, higher productivity, higher Y/L, etc. - Population growth: does it feed technological change? - New (Endogenous) Growth theory: interest in links like these.40 (5) What about economic institutions?- Aren’t institutions important?- Eastern Europe / Soviet Union vs. the West? - Could be introduced into the model through form of F (like technology they may affect productivity of inputs).- Do institutions help determine the savings and investment rates? Are they higher in countries with good institutions e.g. rule of law, high levels of trust, etc. - Do institutions change in predictable ways with income per person? - Some work on this (historical, empirical): little consensus. 41

Models of Economics Growth

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