ECONOMIC GROWTH

Lviv, September 2012

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Growth in Finland

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The Sources of Economic Growth

Production functionY = AF(K, N)

Decompose into growth rate form: the growth accounting equation

DY/Y = DA/A + aK DK/K + aN DN/N The a terms are the elasticities of output with

respect to the inputs (capital and labor)

The Sources of Economic Growth

Interpretation A rise of 10% in A raises output by 10% A rise of 10% in K raises output by aK times 10%

A rise of 10% in N raises output by aN times 10%

Both aK and aN are less than 1 due to diminishing marginal

productivity

The Sources of Economic Growth

Growth accounting Four steps in breaking output growth into its causes (productivity

growth, capital input growth, labor input growth)1. Get data on DY/Y, DK/K, and DN/N, adjusting for quality changes

2. Estimate aK and aN from historical data

3. Calculate the contributions of K and N as aK DK/K and aN DN/N,

respectively

4. Calculate productivity growth as the residual: DA/A = DY/Y – aK

DK/K – aN DN/N

The Sources of Economic Growth

Growth accounting and the productivity slowdown Denison’s results for 1929–1982

Entire period output growth 2.92%; due to labor 1.34%; due to capital 0.56%; due to productivity 1.02%

Pre-1948 capital growth was much slower than post-1948

Post-1973 labor growth slightly slower than pre-1973

Sources of Economic Growth in the United States (Denison) (Percent per

Year)

Productivity Levels, 1947-2005

Productivity

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Labor TFP Growth rateProductivity of K/N

Labor productivity growth may exceed TFP growth because of faster growth of capital relative to growth of labor

Technological Progress in the Solow Model

A new variable: E = labor efficiency Assume: Technological progress is labor-

augmenting: it increases labor efficiency at the exogenous rate g = ∆E/E:

We now write the production function as: Y = F(K,L*E)

where L*E is the number of effective workers. Increases in efficiency will have the same effect on

output as increases in the labor force.

The Solow Model

Basic assumptions and variables Population and work force grow at same rate n Economy is closed and for now there is no govt. (G = 0)

Ct = Yt – It

Rewrite everything in per-worker terms:

(kt is also called the capital-labor ratio)

Therefore, saving and investment per effective worker is:

sy=sf(k)

y t Yt /L tE t = output per effective worker

kt K t /L tE t = capital per effective worker

c t Ct /Lt consumption per worker

The Solow Model

The per-worker production functionyt = f(kt)

Assume no productivity growth for now (we will add it later) Plot of per-worker production function Same shape as aggregate production function

The per-worker production function

The Solow Model

Steady states Steady state: yt, ct, and kt are constant over time

Gross investment must Replace worn out capital, δKt

Expand so the capital stock grows as the economy grows, nKt

Provide capital for the new “effective” workers created by technological progress,

It = (δ+n+g)Kt

If so, then ( + n + g)k = break-even investment: the amount of investment necessary to keep k constant.

gK t

The Solow Model

Ct = Yt – It = Yt – (δ+n+g)Kt

In per-worker terms, in steady state

c = f(k)-(δ+n+g)kt

Plot of c, f(k), and (δ+n+g)kt

The relationship of consumption per worker to the capital–labor ratio in the steady state

The Solow Model

Increasing k will increase c up to a point This is kG in the figure, the Golden Rule capital-labor ratio

For k beyond this point, c will decline But we assume henceforth that k is less than kG, so c always rises

as k rises

The Solow Model

Reaching the steady state Suppose saving is proportional to current income:

St = sYt,

where s is the saving rate, which is between 0 and 1 Equating saving to investment gives

sYt = (n + d + g)Kt

The Solow Model

Putting this in per-worker terms gives

sf(k) = (n + δ + g)k Plot of sf(k) and (n + δ + g)k

Determining the capital–labor ratio in the steady state

The steady state

The only possible steady-state capital-labor ratio is k* Output at that point is y* = f(k*);

consumption is c* = f(k*) – (n + d + g)k* If k begins at some level other than k*, it will move toward k*

For k below k*, saving > the amount of investment needed to keep k constant, so k rises

For k above k*, saving < the amount of investment needed to keep k constant, so k falls

Long-run growth

To summarize: With no productivity growth, the economy reaches a steady state, with constant capital-labor ratio, output per worker, and consumption per worker.

Therefore, the fundamental determinants of long-run living standards are:

1. The saving rate2. Population growth3. Productivity growth

The fundamental determinants of long-run living standards: the saving rate Higher saving rate means higher capital-labor

ratio, higher output per worker, and higher consumption per worker

The fundamental determinants of long-run living standards: population growth

Higher population growth means a lower capital-labor ratio, lower output per worker, and lower consumption per worker

Fundamental Determinants of Long-run Living Standards: Productivity Growth

In equilibrium, productivity improvement increases the capital-labor ratio, output per worker, and consumption per worker: Productivity improvement directly improves the amount that can be produced at any capital-labor ratio and the increase in output per worker increases the supply of saving, causing the long-run capital-labor ratio to rise

Repetito

Growth Empirics: Balanced growth

Solow model’s steady state exhibits balanced growth - many variables grow at the same rate. Solow model predicts Y/L and K/L grow at the same

rate (g), so K/Y should be constant. This is true in the real world.

Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant. Also true in the real world.

Growth Empirics: Convergence Solow model predicts that, other things equal,

“poor” countries (with lower Y/L and K/L) should grow faster than “rich” ones.

If true, then the income gap between rich & poor countries would shrink over time, causing living standards to “converge.”

This does not always occur because “other things” aren’t equal. In samples of countries with similar savings &

pop. growth rates, income gaps shrink about 2% per year.

In larger samples, after controlling for differences in saving, pop. growth, and human capital, incomes converge about 2% per year.

Factor Accumulation vs. Production Efficiency

Differences in income per capita among countries can be due to differences in:1. capital – physical or human – per worker

2. the efficiency of production (the height of the production function)

Empirical studies find that: Both factors are important. The two factors are correlated: countries

with higher physical or human capital per worker also tend to have higher production efficiency.

Endogenous Growth Theory

Endogenous growth theory—explaining the sources of productivity growth Aggregate production function

Y = AK Constant Marginlal Product of Kapital

Human capital Knowledge, skills, and training of individuals Human capital tends to increase in the same

proportion as physical capital Research and development programs Increases in capital and output generate increased

technical knowledge, which offsets decline in MPK from having more capital

Endogenous Growth Theory

Implications of endogenous growth Suppose saving is a constant fraction of output: S =

sAK Since investment = net investment + depreciation, I = DK + dK

Setting investment equal to saving implies:DK + dK = sAK

Rearrange:DK/K = sA – d

Since output is proportional to capital, DY/Y = DK/K, so

DY/Y = sA – d Thus the saving rate affects the long-run growth rate

(not true in Solow model)

Endogenous (New) Growth Theory

Summary Endogenous growth theory attempts to explain,

rather than assume, the economy’s growth rate The growth rate depends on many things, such as

the saving rate, prerequisites for innovative activity, working of the financing industry, regulations (patent laws)

Most of those, can be affected by government policies

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Share of high-tech industries, %2007

Actual vs forecast output

Growth may accelerate also because substitution between labor and capital increases (the efficiency in production increases!)

Economic Growth in China, the U.S., and Japan

The mathematics of the growth problem: Y=Y(K,L) The net investment is: K’ = I – δK = Y – C – δKLet the objective function to be: max ∫ U(C)e– βt dt

Here we first move to labor intensive forms so that y = Y/L and k = K/L, and c = C/L.

Thus, the net investment equation can be written in the form:

K´= knL + Lk’, where n is the growth rate of labor force. Hencek´= y – c – (n+δ)k = φ(k) – c – (n+δ)k

Now the current value Hamiltonian can be expressed as: Ĥ = U(c) + μ(φ(k) – c – (n+δ)k)

Here u is the control and k the state. Now the necessary conditions are:

∂H/∂c = U´(c) – μ = 0μ´ = – ∂Ĥ/∂k + βμ = – μ(φ’(k) – (n+δ + β))and k´= φ(k) – c – (n+δ)k

and k(0) = k0 and lim(T→∞)λ(T) ≥ 0,

lim(T→∞) k(T) ≥ 0 and lim(T→∞) λ(T)k(T) = 0

Now, using the first– order condition μ = U(c) we get by differentiating both sides μ´ = U”(c)c’so we can rewrite c’ = (U’/U”)( φ’(k) – (n+δ) – β)

U”/U’ is the so– called elasticity of marginal utility, to be denoted by η(c). Its inverse, η(c)– 1, in turn is called the instantaneous elasticity of substitution. Thus, c’/c = – (1/η(c))(φ’(k) – (n+δ) – r) which says that consumption growth is proportional to the difference between the net marginal produce of capital accounting for population growth and depreciation, and the rate of time preference.

k´= φ(k) – c – (n+δ)k is the so– called Keynes– Ramsey rule. In the steady state, k’ = 0, optimum consumption is given by: c* = φ’(k*) – (n+δ)k*

If we maximize c* with respect to k* we get the optimum consumption as:

φ’(k*) = (n+δ)

which is the golden rule. Now, we have the steady state, the difference equations give us the dynamic paths, then we have the draw the phase diagram and analyze the dynamics of k and c.  As usual, in this kind of models, the solution has the saddle point property. Thus, there is only one (stable) trajectory which leads to the steady state.

dc/dt=0 dk/dt=0

k

steady state c

Convergence

Do income differences decrease or increase (convergence or divergence)?

Solow model predicts that income levels decrease (poor countries catch up rich countries): marginal product of capital is larger

On top of that, poor countries may also benefit from adaptation of rich countries’ technology

Convergence may occur but depend (also) on other things (conditional convergence)

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1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Suomi Ruotsi Iso-Britannia USA

Geary-Khamis $, vuoden 1990 hintoihin (vuoteen 1997)

Lähteet: Suomen Pankki ja Tilastokeskus.

Bruttokansantuote asukasta kohtiGDP per capita in several countries

Walt Rostow’sTake-offperiods

Vernon’s product cycle