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.

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- 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.

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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)

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- 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).

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- 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.

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- 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).

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- 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.

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- Aggregate savings and National accounting (from intro economics):

Y = GDP G = government spending T = taxes (less transfers)C = consumption X = exports NCI = net capital inflowI = 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)

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- 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.

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- 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?

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- 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.

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- 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?

(D. Rodrik: no single recipe, answer is case specific – growth diagnostics)

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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.

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- 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.

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- 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)

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(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.

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(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

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- 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

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- 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.

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- 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% )

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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?

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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.

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(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.

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(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).

e.g. in Cobb-Douglas “A” rises. ( Yt = A KtLt1- )

- 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)

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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.

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(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)

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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).

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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.

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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+)

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- 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.

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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)

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- 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.

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(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:

g = ∙(Yt/Lt – ysub)

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.

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(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.

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- 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.

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- 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).

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(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.

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(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.

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