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Long-Run Economic Growth

Understand the effect of capital accumulation on labor productivity.

Understand the effect of labor force growth on labor productivity.

Understand the effect of technological change on labor productivity and the standard of living.

Explain balanced growth, convergence, and long-run equilibrium.

Explain the determinants of technological change.

Appendix: Discuss the contributions of capital, labor, and efficiency to the growth rate of real GDP.

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Learning Objectives
After studying this chapter, you should be able to:
6.1
6.2
6.3

6.4
6.5
6.A

The surprising economic rise of India

India’s 1950 real GDP per capita was less than $1,000 in 2010 U.S. dollars, less than 7% of 1950 U.S. real GDP per capita. By 1975, India’s GDP per capita fell to 5.5% of the U.S.

Between 1993 and 2011, real GDP per capita in India grew at an average annual rate of 5.5% versus U.S. growth of 1.5%.

India remains very poor, with a population of 1.2 billion. One-half of the population or more is employed in agriculture, and nearly half of women and one-quarter of men are unable to read or write.

Countries with similar histories and geographies, like Bangladesh and Pakistan, had about the same levels of per capita GDP in 1950, but India has grown rapidly while Bangladesh and Pakistan have stagnated.

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Real GDP has increased substantially over time in the United States and other developed countries.

What are the main factors that determine the growth rate of real GDP per capita?

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Key Issue and Question

Issue:

Question:

Economic growth rates

In Chapter 5, we saw that the level of labor productivity was determined by:

Capital per worker

Efficiency (total factor productivity, TFP)

In this chapter, we will look at the long-run growth rate of the economy, which is strongly affected by:

The rate of technological change

The growth rate of the labor force

We will see how to determine an equilibrium growth rate, or steady state growth rate.

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Understand the effect of capital accumulation on labor productivity.

6.1

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Learning Objective

The Solow growth model

The Solow growth model has become the foundation for how economists think about economic growth.

Solow growth model A model that explains how the long-run growth rate of the economy depends on saving, population growth, and technological change.

We begin with the aggregate production function for real GDP per worker; y is real GDP per worker, k is capital per worker (or the capital-labor ratio), and A measured the overall level of economic efficiency: total factor productivity.

For now, we will assume total factor productivity is fixed at 1:

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Capital accumulation

We need to consider how the capital stock changes over time: capital accumulation.

Investment will increase the amount of capital

Depreciation will decrease the amount of capital

Depreciation rate The rate at which the capital stock declines due to either capital goods becoming worn out by use or becoming obsolete.

In our model, we will have:

Closed economy: no exports/imports

No government sector: no taxes, government expenditures

Then real GDP per worker (y) can be divided into consumption per worker (c) and investment per worker (i):

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Investment

Let s be the fraction of total output per worker that is saved: the national saving rate. Since savings must equal investment, we have:

Substituting this into the equation y = c + i, we obtain:

Also, substituting the production function into the expression for investment:

Because of diminishing marginal returns, increases in the capital-labor ratio must cause smaller and smaller increases in investment per worker.

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Investment per worker increases at a decreasing rate

Notice that investment is proportional to real GDP.

Because of diminishing marginal returns, increases in the capital-labor ratio must cause smaller and smaller increases in investment per worker.

Investment per worker, real GDP per worker, and the capital-labor ratio

Figure 6.1

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Depreciation

Let d be the (constant) depreciation rate.

Then the amount of depreciation is proportional to the capital-labor ratio :

Depreciation and the capital-labor ratio

Figure 6.2

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The steady state

Equilibrium in the Solow growth model occurs when

The capital-labor ratio is constant

This implies the real GDP per worker is also constant

We call this equilibrium a steady state.

Steady state An equilibrium in the Solow growth model in which the capital-labor ratio and real GDP per worker are constant but capital, labor, and output are growing.

Capital-labor ratio is a stock variable: measured at a point in time.

Investment and depreciation are flow variables: measured per period of time.

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The steady state and the bathtub analogy

Change in the level of water = Water flowing in – Water flowing out.

Steady state occurs when Water flowing in = Water flowing out.

Change in the capital-labor ratio = Investment – Depreciation

If investment = depreciation, the capital-labor ratio is constant: the steady state.

The steady state and the bathtub analogy

Figure 6.3

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Equilibrium in the Solow growth model

Change in the capital-labor ratio:

In the steady state, Δk = 0; so k* is the equilibrium capital-labor ratio if:

Equilibrium in the Solow growth model

Figure 6.4

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Transition to the steady state

Suppose , and the initial capital-labor ratio is .

An example of transition to the steady state

Table 6.1

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Calculating steady-state values of k and y

In the steady state, we have . Rewrite this as:

Since , we can write:

So the steady-state capital-labor ratio is $27 per worker.

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Saving rates and growth rates

An increase in the saving rate would

Shift the investment function up, and

Result in a higher steady-state level of the capital-labor ratio.

This will increase the level of real GDP per worker.

But it will not necessarily increase the growth rate of real GDP per worker.

So we cannot explain differences in growth rates with differences in saving rates.

An increase in the saving rate

Figure 6.5

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Distinguish levels and growth rates here; the distinction between a high standard of living and an improving standard of living. Perhaps allow the class to discuss which is more desirable.

The connection between investment and income

The Solow growth model predicts that higher saving rates will result in higher real GDP per worker. Do we see this in real-world data?

The graph shows the Solow model’s prediction is confirmed: higher savings/investment seems to result in higher income.

The basic Solow model explains some, but not all, facts about growth.

While most economists believe higher savings/ investment causes higher, income, there is a possible reverse causality argument.

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Macro Data

Understand the effect of labor force growth on labor productivity.

6.2

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Learning Objective

Labor force growth and the steady state

We will continue to assume total factor productivity is constant, but now we will allow the labor force to grow.

If the labor force grows more (less) quickly than the capital stock, then the capital-labor ratio falls (rises).

If the labor force is growing at rate n, the amount of dilution of the capital-labor ratio is:

Now, to maintain a steady state, investment must equal the sum of depreciation and dilution. We call this the break-even investment:

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To understand why dilution equals nk, keep in mind that k = K/L and n is the growth rate of L. So if L

grows by 3% (that is, n = 3%), then K also has to grow by 3% in order to keep k constant.

Equilibrium in the Solow growth model

Change in the capital-labor ratio:

In the steady state, Δk = 0; so k* is the equilibrium capital-labor ratio if:

In the graph, notice that the depreciation line has been relabeled to break-even investment.

Labor force growth in the Solow model

Figure 6.6

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An increase in the labor force growth rate

What if the labor force grows more quickly?

The capital stock gets diluted faster—current investment cannot keep up with the growing labor force.

The break-even investment line becomes steeper.

The steady-state value of k* falls.

An increase in the labor force growth rate

Figure 6.7

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An increase in the labor force growth rate

Note that while the level of real GDP per worker will fall, the steady-state growth rate of real GDP per worker will not fall.

(The same would be true for an increase in the depreciation rate.)

So again, differences in labor-force growth rates cannot explain differences in steady-state growth rates across countries.

An increase in the labor force growth rate

Figure 6.7

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A decrease in the labor force growth rate

According to the United Nations’ Population Division, the world’s population growth rate averaged 1.7% per year between 1950 and 2010. The following table shows the Population Division’s forecasts for the population growth rates for different regions in the world:

In every region, the forecast is for slower population growth. The slower population growth should reduce the growth rate of the labor force.

What effect does the Solow growth model predict this reduction will have on the standard of living in the world?

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Solved Problem

A decrease in the labor force growth rate

What effect would a decrease in the labor force growth rate have on the standard of living?

Step 1 Review the chapter material.

Step 2 Use a graph to determine how a decrease in labor force growth rate influences the Solow growth model. The break-even investment line should become less steep.

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Solved Problem

A decrease in the labor force growth rate

What effect would a decrease in the labor force growth rate have on the standard of living?

Step 3 Determine the effect on the capital-labor ratio. At the old equilibrium k1*, investment exceeds break-even investment; so k will gradually rise to the new equilibrium, k2*.

Step 4 Determine the effect of the capital-labor ratio on the standard of living. At the greater capital-labor ratio, labor is more productive, so the standard of living rises. The regions with the greatest decline in population growth should see the largest improvement in standard of living, ceteris paribus.

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Solved Problem

Understand the effect of technological change on labor productivity and the standard of living.

6.3

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Learning Objective

Technological change and the Solow growth model

The following will not cause different steady-state growth rates of real GDP per worker (or real GDP per capita, which is very similar):

Changes in the saving rate

Changes in the labor force growth rate

Changes in the depreciation rate

Up till now, we have assumed total factor productivity (A in the production function ) was constant.

Can technological change explain the different growth rates?

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Technological change

Improvements in technology or efficiency need not affect capital and labor equally.

Labor-augmenting technological change Improvements in economic efficiency that increase the productivity of labor but that do not directly make capital goods more efficient.

To analyze labor-augmenting technological change, we will return to the levels (i.e. not per-worker) version of production: .

We will replace L with effective units of labor: , where E is the efficiency of labor.

Think of as the number of effective workers in the economy.

Let g be the growth rate of labor-augmenting technological change

n is the growth rate of the number of workers

So n+g is the growth rate of the effective labor force

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Technological change and the steady state

Output per effective worker:

Capital per effective worker:

Now, to keep capital per effective worker constant (the steady state), we need enough investment to offset:

Depreciation: dk

Dilution from labor force growth: nk

Dilution from effective labor growth: gk

So the change in capital becomes:

The steady state () requires k* such that:

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Equilibrium with technological change

The steady state () requires k* such that:

In the steady state, y is constant.

But y is real GDP per effective worker:

Real GDP must grow at the same rate as effective worker: n+g

But population grows at rate n, so real GDP per worker (or per capita) grows at rate g.

Equilibrium with technological change

Figure 6.8

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Steady state growth rates in the Solow growth model

We finally have an explanation for real GDP per capita varying across countries:

Changes in the underlying rate of labor-augmenting technological change will affect the steady-state growth rate of the standard of living.

The table summarizes the steady-state growth rates in our model.

Steady state growth rates of key variables in the Solow growth model

Table 6.2

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Explain balanced growth, convergence, and long-run equilibrum.

6.4

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Learning Objective

Balanced growth

In the steady-state equilibrium, some key quantities are growing. Which ones?

The capital-labor ratio

Real GDP per worker

In addition, they grow at the same rate: g, the growth rate of the efficiency of labor.

Balanced growth A situation in which the capital–labor ratio and real GDP per worker grow at the same constant rate.

Before a country reaches the steady state, growth of these variables may not be constant.

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Convergence to the balanced growth path

The balanced growth path is the equilibrium for the economy over time.

Germany and Japan experienced large shocks to their capital stocks during WWII.

After WWII, each experienced fast growth, then returned to balanced growth (at a higher level, for Japan).

Post-World War II convergence in Germany and Japan

Figure 6.9

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Convergence to the steady state

Assume that in 1939, the German economy was in steady state, on its balanced growth path.

WWII destroyed much of German capital stock

After WWII, labor-augmenting technology change continued, causing growth in real GDP per worker (or per capita)

But capital-labor ratio was artificially low—below equilibrium—so it started to grow: another source of growth for real GDP per worker

This is (positive) convergence to the steady state.

Summary of adjustments to the steady state

Table 6.3

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Will China’s standard of living exceed the U.S.?

2011 real GDP per capita in the U.S. was nearly six times that of China.

U.S. growth averaged 1.9% per year since the late 1980s.

China averaged 8.9% per year over the same time frame.

If rates continued, China could surpass the U.S in 2038.

China must maintain TFP growth rates, which seems unlikely.

U.S. has higher investment in R&D, leading to TFP advances.

Chinese growth is partially due to transitional economy.

Higher balanced growth path plus convergence growth.

Demographic problems with low birthrate in China.

Working-age population expected to decline by 30%.

Evidence that Chinese growth rate is slowing:

Fewer firms relocating production to China (diminishing cost advantage).

Japan also grew very fast in 1970s, but its growth rate slowed.

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Making the Connection

Do all countries converge to the same balanced growth?

2010 real income per capita for the average country was $9,400.

U.S. income per capita was $47,400.

Lowest income per capita was $300 in the Congo.

Convergence describes the action of poor countries catching up to rich countries in terms of income per capita.

Some countries (i.e., Japan) converge, while others do not (i.e., Zaire).

Different savings rates and labor force growth rates.

Differing levels of TFP growth.

Conditional convergence means countries all converge to their own balanced growth path.

After controlling for factors leading to different balanced growth paths, convergence occurs at about 2% per year.

Would take about 35 years for poor country to close half the gap.

Gaps in income per capita may never disappear.

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Explain the determinants of technological change using the endogenous growth model.

6.5

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Learning Objective

Capital accumulation and endogenous growth

Growth from capital accumulation eventually dies out, so technological change is the ultimate determinant of the growth rate of labor productivity and the standard of living.

The Solow growth model assumes TFP growth occurs, but doesn’t explain why.

Endogenous growth theory A theory of economic growth that tries to explain the growth rate of technological change.

There are many difference endogenous growth models. We focus on two approaches:

Assume capital is not subject to diminishing marginal returns.

Assume research firms produce new technology and ideas.

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AK growth models: reconsidering diminishing returns

Could capital not be subject to diminishing returns?

For conventional capital (machinery, buildings), seems unlikely

For human capital and knowledge, possibly

Human capital The accumulated knowledge and skill that workers acquire from education and training or from life experiences.

Some economists even argue that knowledge is subject to increasing returns.

If human capital and knowledge are relatively important components of overall capital, then capital may not be subject to diminishing returns.

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An AK growth model

Assume labor is constant (and equal to 1); then we have a production function:

Here Y is real GDP, A is (constant) total factor productivity, and K is capital. The exponent on K is 1, so capital has constant returns: marginal product is A.

Since labor is constant at 1, real GDP = real GDP per worker (or per capita).

So we can just consider growth in real GDP.

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An AK growth model

Investment is still a constant fraction of output.

Water flowing in = sY = sAK

With a constant labor force (growth rate n = 0), depreciation removes capital.

Water flowing out = dK

The change in capital is the difference between the water flowing into the tub of capital stock and the water flowing out of the tub of capital stock.

Dividing each side by K gives an expression for the growth of capital.

So the growth rate of real GDP per worker depends on the national saving rate.

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Two-sector growth model: the production of knowledge

Alternative model: introduce research firms or research universities that produce new ideas and new technology (E).

Labor supply L split between manufacturing (fraction 1-p) and research (fraction p) sectors.

Effective labor in manufacturing:

Production function for manufacturing:

Assume constant returns to scale for production function (double labor and capital implies double real GDP).

Production for research: where pEL is effective labor in research sector, z is researcher productivity.

Capital accumulation: (same as for Solow growth model from earlier)

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Growth rate of labor-augmenting technological change

In the Solow growth model, technological change was exogenous.

Here, we explicitly model it. We have the production of research ; dividing both sides by E gives:

But /E is the growth rate of technological change.

So policies that increase z (the productivity of researchers) or p (the proportion of the labor force devoted to research) will raise the growth rate of technological change.

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What explains recent economic growth in India?

Barry Bosworth of the Brookings Institution and Susan Collins of the University of Michigan used growth accounting techniques to explain growth in India.

Labor productivity growth increased from 2.4% per year between 1978 and 1993 to 4.6% per year between 1993 and 2004.

Why did labor productivity growth rates rise so rapidly?

TFP growth increased from 1.4% per year to 2.7% per year after 1993, leaving over half the increase in labor productivity due to faster TFP growth.

TFP growth has slowed since 2004. According to The Conference Board, “India’s transition to a higher growth path had been … resource-consuming and … constrained by a continuing need for reforms.”

Nicolas Eberstadt of the American Enterprise Institute expressed concerns about varying population growth rates in the North and South of India.

Future growth prospects are in the south and northern cities.

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Making the Connection

Policies to promote economic growth

Increase the national saving rate

Budget deficits decrease saving rate, while budget surpluses increase saving rate

Favorable tax treatment of savings increases saving rate

Promote research and development

Tax credits for spending on R&D

Offset public good nature of R&D

Enforce patents, copyrights

Directly fund research (universities and research institutions)

Increase human capital

Public education

Tax credits for education

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Policies to promote economic growth

Increase population growth

Larger countries can devote more people to research

Over very long run, countries with higher population experience greater technological change

But higher population growth decreases steady-state real GDP per capita

Reduce income taxes

High income tax rates decrease work incentives; so lower tax rates may increase available labor supply L

But if taxes are too low, cannot fund infrastructure for growth

Reform the political process and establish rule of law

In some countries, property rights and the rule of law are poorly enforced

This diminishes entrepreneurs’ incentives to innovate, start businesses

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Should the government invest in green energy?

In late 1990s, investors were irrationally exuberant about e-commerce, leading to the “dot-com bubble”.

E-commerce firms were able to raise funds without evidence of viability.

Is the same thing happening with green energy firms—except worse, because the government is choosing to invest our money?

If solar energy is a good investment, why wouldn’t private investors make the investment?

Supporters argue green energy has positive externalities, so private markets will invest too little.

Long history of similar government investments: railroads, petroleum industry, nuclear energy, aviation.

However policymakers still need to make wise choices.

Can we trust federal government to make good decisions?

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Making the Connection

Answering the key question

“What are the main factors that determine the growth rate of real GDP per capita?”

Long-run growth rate is determined by technological change.

If we use a broader definition of capital, higher saving rates could lead to faster long-term growth.

Devoting more resources to the production of new ideas and technology may also increase the long-run growth rate.

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Discuss the contributions of capital, labor, and efficiency to the growth rate of real GDP.

6.A

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Learning Objective

Growth accounting

How much of real GDP growth can we attribute to:

Technology?

Labor?

Capital?

The procedure of growth accounting allows us to determine this for a given country.

We will find that total factor productivity (technology) growth has been the most important determinant of economic growth.

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The growth accounting equation for real GDP

We begin with the Cobb-Douglas production function:

The extra output from an additional unit of capital is:

So if the capital stock grows, and all other inputs remain constant, the growth rate of real GDP equals capital’s share of income () multiplied by the growth rate of the capital stock.

By symmetry, if the labor force grows, and all other inputs remain constant:

The growth rate of real GDP equals labor’s share of income, multiplied by the growth rate of the labor force.

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Real GDP growth

Real GDP growth can also come from increases in total factor productivity: if labor and capital do not change, then

Putting these three elements together, we obtain an overall expression for overall real GDP growth:

(Real GDP Growth) = (Contribution from capital) + (Contribution from labor) + (Contribution from total factor productivity)

We can observe everything in this equation apart from the total factor productivity growth. This will be the residual in our estimates, known as the Solow residual.

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Growth accounting for the United States

We can determine the shares of capital, labor, and total factor productivity to growth by noting that for 1950-2010,

Real GDP grew at 3.3% per year

Capital’s share of income has averaged about 1/3 in the United States.

The capital stock grew at 3.3% per year.

The labor force grew at 1.7% per year.

Sources of growth for the United States, 1950-2010

Table 6A.1

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The first point explains the real GDP growth line.

The second and third points combine to explain the contribution of capital: one-third of 3.3%.

The second and fourth points combine to explain the contribution of labor: two-thirds of 1.7%

The contribution of TFP is the residual: 3.3% - 1.1% - 1.2%.

Total factor productivity as the ultimate source of growth

Can governments spur economic growth by encouraging accumulation of capital goods?

Yes, but only for a period of time, due to diminishing marginal returns.

Nicholas Crafts performed a growth-accounting exercise for the Soviet Union; his equation for real GDP per worker (y) was:

Accounting for labor productivity growth in the Soviet Union, 1920-1985

Table 6A.2

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Total factor productivity as the ultimate source of growth

But to keep labor productivity high, the Soviets had to increase the capital-labor ratio (k) beyond balanced growth levels, neglecting consumption goods, and eventually efficiency suffered (TFP decreased after 1970).

In market economies, efficiency continues to improve; entrepreneurs’ livelihoods depend on it.

After the collapse of the Soviet Union in 1991, Russia now has a more market-oriented system.

Accounting for labor productivity growth in the Soviet Union, 1920-1985

Table 6A.2

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