Models of Economics Growth
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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 = bK = 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 = imports
An identity: GDP equals aggregate spending across sectors.
Y = C + I + G + X – M
Solve for I and then add and subtract T :
I = (Y-T-C) + (T-G) - (X-M)
Y-T-C = private domestic savings
T-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 Policies
Domestic private financial development, taxes and savings
Government savings budget surpluses
Savings 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 function
Yt = 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)/ Lt
Yt /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 - Kt
Investment (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 < 0
K/L constant if: Kt/Kt - Lt/Lt = 0
We 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 , = 0
18 - The key equation:
sYt/Kt - - g > 0 , < 0 , = 0
Is 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) < 0
K/L and Y/L constant if: s(Yt/Lt) – (g+)(Kt/Lt) = 0
19 - 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)ss
Start 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 KL1- 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