WEEK IXEconomic Growth Model

WEEK IXEconomic growth• Improvement of standard of living of society due to increase

in income therefore the society is able to consume more goods & services (measurement: output per person or GDP per capita).

• refers to growth of potential aggregate output/economic capacity i.e. natural rate of output (Yn). Thus economic growth is concerned with the long-run trend in aggregate output due to structural causes.

• In the contrary, a short-run and medium-run variation in economic growth (Yt) is called the business cycle.

•  Economic growth is a dynamic analysis:1. Change over time: A trend in the long-run (in which the

business cycle moves up and down, creating fluctuations around the trend)

 

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WEEK IX2. Compare across countries: at PPP (purchasing power

parity) prices in which GDP is measured on how much consumers must pay for the same "basket of goods" in each country.

WEEK IX• Dynamic analysis of economic growth: force of

compounding1. A large increase in GDP per capita over time2. A convergence of GDP per capita across

• Models of economic growth to explain the conclusions including:1. Adam Smith (“Wealth of Nations”, 1776); factors affect

economic growth: (i) specialization and technology advancement and (ii) market expansion

2. Joseph Schumpeter (“the Theory of Development”, 1908); Economic growth depends on innovation of entrepreunership in producing goods & services.

3. (Roy) Harrod (“An Essay in Dynamic Theory“, 1939) - (Evsey) Domar (“Capital Expansion, Rate of Growth, and Employment”, 1946); Economic growth is determined by aggregate demand side particularly investment.

4. (Robert) Solow (“Contribution of the Theory of Economic Growth”, 1956). The Solow’s model demonstrates how capital stock/saving, population, technology progress affects level of output and growth.

WEEK IXBuilding blocks of the Solow’s growth model:

1. Aggregate production function

• Production function: a specification of the relation between aggregate output and inputs in production

• Assumptions: 1. Output (Y) is a function of 2 inputs i.e. capital stock (K)

and labor (N).  

2. State of technology of the production function is constant return to scale in which the increase all the inputs (K & N) by the same percentage will increase output (Y) by exactly the same percentage.

  

WEEK IX3. Decreasing returns to factors (recall: Law of diminishing

marginal product) in which increases in a certain input (e.g. K), given other inputs (e.g. L), lead to smaller and smaller increases in output (= ∆Y/∆K).

4. The production function can also be written in relative term to labor. If x = 1/N:

Output per worker (Y/N) is the function of capital per worker (K/N). This relation:• The use of labor as a denominator in the above

equation implies that labor does not affect the relationship between economic growth and capital stock.

• Can be drawn in a 2-dimensional graph• Upward-sloping curve• Decreasing returns to capital

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Aggregate production curve: capital accumulation

WEEK IXSources of increase in GDP per capita:• Increase in K/N: capital accumulation (i.e. a movement along the

aggregate production curve)• Improvement in the state of technology: technological progress

(i.e. a shift of aggregate production curve)• Capital accumulation by itself can sustain a higher level of

output (Y/N) but cannot sustain a high growth rate of output [∆(Y/N)] due to the decreasing returns to capital. Thus economy requires larger and larger increases in the level of capital per worker (i.e. investment and saving) to sustain steady increase in output.

• Sustained growth requires technological progress.

WEEK IXEffects of improvement in state of technology

WEEK IXOutput and capital: 2 relations• Capital stock (K) output (Y): Production function• Output (Y) saving (S) investment (I) = capital

accumulation (∆K): Saving/investment function

WEEK IXModel I: Economic growth due to capital accumulation

• Thus assumption: (i) size of population and labor is constant (growth of population = 0) and (ii) no technological progress

• 1st relation: K Y

Time index (t) for Y and K ONLY: higher K/N leads to higher Y/N

• 2nd relation: Y S I • Assumption: closed economy thus Y = C + I + G

I = (Y – C – T) + (T – G)I = S + (T – G)

• Assumption: (T – G) = 0 thusI = SInvestment = private saving

WEEK IX• Private saving is proportional to income:

S = s*Y

Where s: saving rate (0 < s < 1)

It = sYt

Saving rate, s, has no relations with level of income, Y. Time index, t, for I and Y thus the higher output (Y), the higher saving (S) and the higher investment (I).

• 2nd relation: I = ∆K• Capital: stock concept at a point in time, while investment:

flow concept during a given period• Capital depreciates per year at rate: δ

K t+1 = Kt – δ*Kt + It

WEEK IXK t+1 = (1 – δ)*Kt + It

• Substitute I by sY and dividing both sides by N

N

Ys

N

K

N

K ttt )1(1

N

Ys

N

K

N

K

N

K tttt 1

N

K

N

Ys

N

K

N

K tttt 1

WEEK IXImplications of changes in saving rate• Production function = saving/investment function

• K/N Y/N therefore ∆(K/N) ∆(Y/N)• ∆(K/N) = I/N – δK/N• Interpretation: Increases in investment (I/N) will increase

capital stock (K/N) which (1) produce higher output (Y/N) as well as (2) generates higher depreciation (δK/N):• I/N > δK/N K/N • I/N < δK/N K/N

WEEK IXSteady state capital & output: capital accumulation

WEEK IX• Investment curve has the same shape as the production function but

it is lower (= sY)• Depreciation curve has a straight line that proportional to capital

• If an economy’s initial K/N < K*/N (i.e. at K0/N), then I/N > δK/N so that K/N increases over time until K/N = K*/N;

• If an economy’s initial K/N > K*/N, then I/N < δK/N so that K/N falls over time until K/N = K*/N;

• Y/N also moves around its Y*/N as K/N moves toward K*/N• Point K*/N that indicates I/N = δK/N whereby investment and

depreciation reaches its equilibrium (or ∆K/N = 0) therefore the level of output (Y/N) and capital stock (K/N) is constant over time.

WEEK IX• K*/N is called a steady-state capital (and Y*/N is a steady-state

output).

• The steady-state reflects an economic equilibrium in the long-run. • The steady-state suggests that economies converge in the long run:

• Countries with low initial levels of capital and output per worker will grow rapidly as K/N and Y/N will rise until they reach their steady state values.

• Countries with high initial levels of capital and output per worker will grow slowly as K/N and Y/N will fall until they reach their steady state values.

WEEK IXImplication of change in saving rate on economic growth 1. Saving rate has no effect on the long-run growth rate of output

per worker which is equal to 0;2. The saving rate determines the level of output per worker in the

long run

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3. An increase in the saving rate will lead to higher growth of output per worker for some time but not forever

WEEK IXGolden rule of level of capital• Does a maximum saving (i.e. s = 100%) will always be good

for economy?• Economic policies can be used to determine saving rate

thus it can reach the steady-state: (i) positive public saving; (ii) tax incentives to affect private saving

• What matters for society is not how much is produced (Y) but how much they consume (C).• Society is willing to maximize level of consumption of

goods & services at certain saving rate (0 < s < 1).•  Golden rule of level of capital is a level of capital stock

associated with the certain saving rate that maximizes level of consumption.

  

WEEK IXGolden rule of level of capital

WEEK IXModel II: Economic growth due to technological

progress

Technological progress and production function• Technological progress results in: (i) larger quantities of

output for given inputs, (ii) better products, (iv) new products, (iv) larger variety of products and (iv) more service.

• State of technology: how much output can be produced from given inputs; A.

Y = F(K, N, A)

Where AN is effective labor indicates:• Technological progress reduces labor needs to produce

given output• Technological progress increases output that can be

produced with a given labor

WEEK IX• Assumption of constant returns to scale:

xY = F(xK, xAN)

• By substitute X = 1/AN to get in relative term of labor

• Output per effective worker is a function of capital per effective worker

)1,(AN

KF

AN

Y

WEEK IXAggregate production curve: technological progress

WEEK IX• 1st relation: K Y

• 2nd relation: I = ∆K

I = S = sY

AN

Ys

AN

I

AN

Ksf

AN

I

WEEK IX• 3rd relation: δK/I ∆K

• Growth rate of effective labor = gA + gN in which gA is growth rate of technological progress, gN is growth rate of labor

KggKI NA )(

AN

Kgg

AN

INA )(

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Steady-state capital and output: technological progress

WEEK IXSteady state (= balance growth)• K/AN and Y/AN are constant• Growth of K/N and Y/N = growth of technological progress

(gA)• Or growth of K and Y = sum of population growth and rate

of technological progress (gN + gA).

Implications of increase in saving rate• Changes in saving rate; s, do not affect steady-state growth rate

∆(K/AN) but increase the steady-state level of output per effective worker (K/AN)

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Implications of an increase in saving rate

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