Lecture 9.ppt

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Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall • Macroeconomics, 5/e • Olivier Blanchard

Saving, Capital Accumulation, and Output

Chapter 11

Q:Does the saving rate affect growth?

A: If the production function exhibits decreasing returns to capital, an increase in the saving rate can only affect the growth rate temporarily. In the long run, saving does not affect growth, but does affect the level of output per worker.

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At the center of the determination of output in the long run are two relations between output and capital:

The amount of capital determines the amount of output being produced.

The amount of output determines the amount of saving and, in turn, the amount of capital accumulated over time.

11-1 Interactions between Output and Capital

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Capital, Output, and Saving/Investment

Figure 11 - 1


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Under constant returns to scale, we can write the relation between output and capital per worker as follows:

, 1Y K

FN N

The Effects of Capital on Output


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Since the focus here is on the role of capital accumulation, we make the following assumptions:

The size of the population, the participation rate, and the unemployment rate are all constant.

There is no technological progress.



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With these two assumptions, our first relation between output and capital per worker, from the production side, can be written as

Y

Nf

K

Nt t

In words, higher capital per worker leads to higher output per worker.



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To derive the second relation, between output and capital accumulation, we proceed in two steps:

1. We derive the relation between output and investment.

2. We derive the relation between investment and capital accumulation.

The Effects of Output on Capital Accumulation


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We make three assumptions to derive the relation between output and investment:

We assume the economy is closed.

We assume public saving, T – G, is equal to zero.

We assume that private saving is proportional to income, so

Combining these two relations gives:

I S T G ( )

I sYt t

S sY

I S


Output and Investment


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The evolution of the capital stock is given by:

denotes the rate of depreciation

I sYt t Combining the relation from output to investment, , and the relation from investment to capital accumulation, we obtain the second important relation we want to express, from output to capital accumulation:

K K It t t 1 1( )

K

N

K

Ns

Y

Nt t t 1 1( )


Investment and Capital Accumulation


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Investment and Capital Accumulation

Rearranging terms in the equation above, we can get the change in capital per worker over time:

In words, the change in the capital stock per worker (left side) is equal to saving per worker minus depreciation (right side).

K

N

K

Ns

Y

Nt t t 1 1( )

K

N

K

Ns

Y

N

K

Nt t t t 1

Output and Capital per Worker:


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Our two main relations are:

Combining the two relations, we can study the behavior of output and capital over time.

First relation:Capital determinesoutput.

Y

Nf

K

Nt t

K

N

K

Ns

Y

N

K

Nt t t t 1

Second relation:Output determines

capital accumulation

11-2 The Implications of AlternativeSaving Rates

N

K

N

K tt 1

Change in capitalfrom year t to year t + 1

N

K t_

Depreciationduring year t

_

N

Ksf t

=

Investmentduring year t

=

Dynamics of Capital and Output

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This relation describes what happens to capital per worker. The change in capital per worker from this year to next year depends on the difference between two terms:

If investment per worker exceeds depreciation per worker, the change in capital per worker is positive: Capital per worker increases.

If investment per worker is less than depreciation per worker, the change in capital per worker is negative: Capital per worker decreases.



N

K

N

K tt 1

Change in capitalfrom year t to year t + 1

N

K t_

Depreciationduring year t

_

N

Ksf t

=

Investmentduring year t

=

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When capital and output are low, investment exceeds depreciation, and capital increases. When capital and output are high, investment is less than depreciation, and capital decreases.

Capital and Output Dynamics

Figure 11 - 2

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The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. In steady state, the left side of the equation above equals zero, then:

* *K Ksf

N N

* *Y Kf

N N

Given the steady state of capital per worker (K*/N), the steady-state value of output per worker (Y*/N), is given by the production function:

K

N

K

N

K

Nt t t

1 = sf

K

Nt


Steady-State Capital and Output

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Capital Accumulation and Growth in France in the Aftermath of World War II

Table 1 Proportion of the French Capital Stock Destroyed by the End of World War II

Railways Tracks 6% Rivers Waterways 86%

Stations 38% Canal Locks 11%

Engines 21% Barges 80%

Hardware 60% Buildings (numbers)

Roads Cars 31% Dwellings 1,229,000

Trucks 40% Industrial 246,000

When WWII ended in 1945, France had suffered some of the heaviest losses of all European countries. A vivid picture of the destruction of capital is provided by the numbers in Table 1.

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Three observations about the effects of the saving rate on the growth rate of output per worker are:

1. The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero.

2. Nonetheless, the saving rate determines the level of output per worker in the long run. Other things equal, countries with a higher saving rate will achieve higher output per worker in the long run.

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


The Saving Rate and Output

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A country with a higher saving rate achieves a higher steady- state level of output per worker.

The Effects of Different Saving Rates

Figure 11 - 3

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An increase in the saving rate leads to a period of higher growth until output reaches its new, higher steady-state level.

The Effects of an Increase in the Saving Rate on Output per Worker

Figure 11 - 4

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An increase in the saving rate leads to a period of higher growth until output reaches a new, higher path.

The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress

Figure 11 - 5

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The Saving Rate and Consumption

An increase in the saving rate always leads to an increase in the level of output per worker. But output is not the same as consumption. To see why, consider what happens for two extreme values of the saving rate:

An economy in which the saving rate is (and has always been) zero is an economy in which capital is equal to zero. In this case, output is also equal to zero, and so is consumption. A saving rate equal to zero implies zero consumption in the long run.

Now consider an economy in which the saving rate is equal to one: People save all their income. The level of capital, and thus output, in this economy will be very high. But because people save all their income, consumption is equal to zero. A saving rate equal to one also implies zero consumption in the long run.

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The level of capital associated with the value of the saving rate that yields the highest level of consumption in steady state is known as the golden-rule level of capital.



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An increase in the saving rate leads to an increase and then to a decrease in steady-state consumption per worker.

The Effects of the Saving Rate on Steady- State Consumption per Worker

Figure 11 - 6

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Social Security, Saving, and Capital Accumulation in the United States

One way to run a social security system is the fully-funded system, where workers are taxed, their contributions invested in financial assets, and when workers retire, they receive the principal plus the interest payments on their investments.

Another way to run a social security system is the pay-as-you-go system, where the taxes that workers pay are the benefits that current retirees receive.

In anticipation of demographic changes, the Social Security tax rate has seen increases, and contributions are now larger than benefits, leading to the accumulation of a Social Security trust fund.

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Assume the production function is:

Y K N

Y

N

K N

N

K

N

K

N

fK

N

K

Nt t

Output per worker is:

Output per worker, as it relates to capital per worker is:

Given our second relation, K

N

K

N

K

Nt t t

1 = sf

K

Nt

K

N

K

Ns

K

N

K

Nt t t t 1 Then,

11-3 Getting a Sense of Magnitudes

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Steady-state output per worker is equal to the ratio of the saving rate to the depreciation rate.

A higher saving rate and a lower depreciation rate both lead to higher steady-state capital per worker and higher steady-state output per worker.

2

* *Y K s s

N N


The Effects of the Saving Rate on Steady-State Output

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The Dynamic Effects of an Increase in the Saving Rate

It takes a long time for output to adjust to its new, higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth.

The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker

Figure 11 - 7

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In steady state, consumption per worker is equal to output per worker minus depreciation per worker.

Knowing that:

C

N

Y

N

K

N

C

N

s s s s

2 1then:

2

* *Y K s s

N N

and

2

*K s

N

These equations are used to derive Table 11-1 in the next slide.


The U.S. Saving Rate and the Golden Rule

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The U.S. Saving Rate and the Golden Rule

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Table 11-1 The Saving Rate and the Steady-State Levels of Capital,Output, and Consumption per Worker

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The set of skills of the workers in the economy is called human capital.

An economy with many highly skilled workers is likely to be much more productive than an economy in which most workers cannot read or write.

The conclusions drawn about physical capital accumulation remain valid after the introduction of human capital in the analysis.

11-4 Physical versus Human Capital

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When the level of output per workers depends on both the level of physical capital per worker, K/N, and the level of human capital per worker, H/N, the production function may be written as:

Y

Nf

K

N

H

N

,

( , )

An increase in capital per worker or the average skill of workers leads to an increase in output per worker.


Extending the Production Function

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A measure of human capital may be constructed as follows:

Suppose an economy has 100 workers, half of them unskilled and half of them skilled. The relative wage of skilled workers is twice that of unskilled workers.

Then:

HH

N [( ) ( )] .5 0 1 5 0 2 1 5 0

1 5 0

1 0 01 5


Extending the Production Function

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An increase in how much society “saves” in the form of human capital—through education and on-the-job-training—increases steady-state human capital per worker, which leads to an increase in output per worker.

In the long run, output per worker depends not only on how much society saves but also how much it spends on education.


Human Capital, Physical Capital, and Output

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In the United States, spending on formal education is about 6.5% of GDP, compared to about 16% investment in physical capital. This comparison:

Accounts for the fact that education is partly consumption.

Does not account for the opportunity cost of education.

Does not account for the opportunity cost of on-the-job-training.

Depreciation of human capital is slower than that of physical capital.


Human Capital, Physical Capital, and Output

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A recent study has concluded that output per worker depends roughly equally on the amount of physical capital and the amount of human capital in the economy.

Models that generate steady growth even without technological progress are called models of endogenous growth, where growth depends on variables such as the saving rate and the rate of spending on education.

Output per worker depends on the level of both physical capital per worker and human capital per worker.

Is technological progress unrelated to the level of human capital in the economy? Can’t a better-educated labor force lead to a higher rate of technological progress? These questions take us to the topic of the next chapter: the sources and the effects of technological progress.


Endogenous Growth

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