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ECO 3311 Ch 5: Long Run Models The Solow Growth Model

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ECO 3311 Ch 5: Long Run Models The Solow Growth Model. Introduction. In this chapter, we learn: how capital accumulates over time, helping us understand economic growth. the role of the diminishing marginal product of capital in explaining differences in growth rates across countries.

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slide1

ECO 3311

Ch 5: Long Run Models

The Solow Growth Model

http://www.webpages.ttu.edu/vvalcarc

introduction
Introduction
  • In this chapter, we learn:
  • how capital accumulates over time, helping us understand economic growth.
  • the role of the diminishing marginal product of capital in explaining differences in growth rates across countries.
  • the principle of transition dynamics: the farther below its steady state a country is, the faster the country will grow.
  • the limitations of capital accumulation, and how it leaves a significant part of economic growth unexplained.

http://www.webpages.ttu.edu/vvalcarc

douglass north s clock
Douglass North’s “Clock”
  • 24-hour clock representing human experience
  • Starts in Africa 4-5 million years ago
  • “Civilization” starts 8,000 B.C. (agric. & permanent settlement)

in the last 3 or 4 minutes on the clock!

  • Other 23 hrs, 56 mins – humans were hunters/gatherers with very slow pop. growth

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world population and major advances in knowledge
World Population and Major Advances in Knowledge

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some facts about economic growth
Some Facts about Economic Growth
  • Worldwide economic growth is not constant. Average growth rates in industrialized countries were higher in the 20th than the 19th century and higher in the 19th than the 18th.
  • Average real incomes today in the US and western Europe are between 10 and 30 times larger than a century ago, and between 50 and 300 times larger than two centuries ago.
  • Productivity growth slowdown. Average annual growth in per capita output in the US and other industrialized countries since the early 1970s has been about a % point below its earlier level.
  • Growth miracles. Episodes where growth in a country far exceeds world averages over an extended period. E.g. Japan, South Korea, Taiwan, Singapore and Hong Kong grew at an average annual rate of over 5% from the 1960s to the 1990s.
  • Growth disasters. Episodes where growth in a country is below world averages over an extended period. E.g. in 1900 Argentina’s average income was only slightly behind world’s leaders and looked poised to become a major industrialized country. But its growth performance over most of the 20th century was dismal and as a result, it is now in the middle of the world’s income distribution. Sub-Saharan African countries such as Chad, Ghana and Mozambique have been extremely poor and have been unable to attain any sustained growth in average incomes. Consequently, their average incomes have remained close to subsistence levels while average world income has been steadily rising.

http://www.webpages.ttu.edu/vvalcarc

slide6
it builds on the production model by adding a theory of capital accumulation

developed in the mid-1950s by Robert Solow of MIT

the basis for the Nobel Prize he received in 1987

TheSolow growth modelis the starting point to determine why growth differs across similar countries

http://www.webpages.ttu.edu/vvalcarc

the solow growth model
The Solow growth model
  • capital stock is no longer exogenous
  • capital stock is “endogenized”: converted from an exogenous to an endogenous variable.
  • the accumulation of capital as a possible engine of long-run economic growth

http://www.webpages.ttu.edu/vvalcarc

the solow model dynamics of growth
The Solow Model (Dynamics of Growth)

Addresses 3 fundamental questions:

1. What is the relationship between a nation’s saving rate, population growth, technological advancement and its L-R living standards?

2. How does economic growth evolve over time? Will it accelerate, stabilize or stop?

3. Can poorer countries catch up with the richest in terms of living standards?

Robert Solow

http://www.webpages.ttu.edu/vvalcarc

setting up the model
Setting Up the Model

Start with the production model from the last chapter and add an equation describing the accumulation of capital over time.

Production

  • The production function:
    • is Cobb-Douglas
    • has constant returns to scale in capital and labor
    • has an exponent of one-third on capital
  • Variables are time subscripted as they may potentially change over time

http://www.webpages.ttu.edu/vvalcarc

slide10
Output can be used for either consumption (Ct) or investment (It)

A resource constraint describes how an economy can use its resources

Capital Accumulation

capital accumulation equation: the capital stock next year equals the sum of the capital started with this year plus the amount of investment undertaken this year minus depreciation

http://www.webpages.ttu.edu/vvalcarc

slide11
Depreciation is the amount of capital that wears out each period

the depreciation rate is viewed as approximately 10 percent

Thus the change in the capital stock is investment less depreciation

represents the change in the capital stock between today, period t, and next year, period t+1

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slide12
Labor

the amount of labor in the economy is given exogenously at a constant level

Investment

the amount of investment in the economy is equal to a constant investment rate times total output

remember that total output is used for either consumption or investment

therefore, investment equals output times the quantity one minus the investment rate

http://www.webpages.ttu.edu/vvalcarc

prices and the real interest rate
Prices and the Real Interest Rate
  • If we added equations for the wage and rental price, the MPL and the MPK would pin them down, respectively -- omitting them changes nothing.
    • the real interest rate is the amount a person can earn by saving one unit of output for a year
    • or equivalently, the amount a person must pay to borrow one unit of output for a year
    • measured in constant dollars, not in nominal dollars

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slide14
saving is the difference between income and consumption

Saving equals investment:

a unit of saving is a unit of investment, which becomes a unit of capital: therefore the return on saving must equal the rental price of capital

the real interest rate in an economy is equal to the rental price of capital, which is equal to the marginal product of capital

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solving the solow model
Solving the Solow Model
  • To solve the model, write the endogenous variables as functions of the parameters of the model and graphically show what the solution looks like and solve the model in the long run.
    • combine the investment allocation equation with the capital accumulation equation
    • netinvestment is investment minus depreciation
    • substitute the supply of labor into the production function:

(change in capital)

(net investment)

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slide16
We now have reduced our system of five equations and five unknowns to two equations and two unknowns:

The key equations of the Solow Model are these:

The production function

And the capital accumulation equation

How do we solve this model?

We graph it, separating the two parts of the capital accumulation equation into two graph elements: saving = investment and depreciation

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the solow diagram graphs these two pieces together with k t on the x axis

Investment, Depreciation

At this point,

dKt = sYt, so

Capital, Kt

The Solow Diagram graphs these two pieces together, with Kt on the x-axis:

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the solow diagram

Depreciation: d K

Investment: s Y

Investment, depreciation

Net investment

K0

K*

Capital, K

The Solow Diagram

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using the solow diagram
Using the Solow Diagram
  • the amount of investment is greater than the amount of depreciation, the capital stock will increase
  • the capital stock will rise until investment equals depreciation: this point, the change in capital is equal to 0, and absent any shocks, the capital stock will stay at this value of capital forever
  • the point where investment equals depreciation is called the steady state

http://www.webpages.ttu.edu/vvalcarc

suppose the economy starts at this k 0

Investment, Depreciation

K0

K1

Capital, Kt

Suppose the economy starts at this K0:
  • We see that the red line is above the green at K0:
  • Saving = investment is greater than depreciation
  • So ∆Kt > 0 because
  • Then since ∆Kt >0, Kt increases from K0 to K1 > K0

http://www.webpages.ttu.edu/vvalcarc

now imagine if we start at a k 0 here

Investment, Depreciation

K0

K1

Capital, Kt

Now imagine if we start at a K0 here:
  • At K0, the green line is above the red line
  • Saving = investment is now less thandepreciation
  • So ∆Kt < 0 because
  • Then since ∆Kt<0,Ktdecreases from K0 to K1 < K0

http://www.webpages.ttu.edu/vvalcarc

slide22

Investment, Depreciation

No matter where we start, we’ll transition to K*!

At this value of K, dKt = sYt, so

K*

Capital, Kt

We call this the process of transition dynamics: Transitioning from any Kt toward the economy’s steady-state K*, where ∆Kt = 0

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slide23
when not in steady state, the economy obeys transition dynamics or in other words, the movement of capital toward a steady state

notice that when depreciation is greater than investment, the economy converges to the same steady state as above

at the rest point of the economy, all endogenous variables are steady

transition dynamics take the economy from its initial level of capital to the steady state

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output and consumption in the solow diagram
Output and Consumption in the Solow Diagram
  • using the production function, it is evident that as K moves to its steady state by transition dynamics, output will also move to its corresponding steady state by transition dynamics
  • note that consumption is the difference between output and investment

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we can see what happens to output y and thus to growth if we rescale the vertical axis

Y*

K*

We can see what happens to output, Y, and thus to growth if we rescale the vertical axis:

Investment, Depreciation, Income

  • Saving = investment and depreciation now appear here
  • Now output can be graphed in the space above in the graph
  • We still have transition dynamics toward K*
  • So we also have dynamics toward a steady-state level of income, Y*

Capital, Kt

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the solow diagram with output

Investment, depreciation,

and output

Y0

Y*

Investment: s Y

Depreciation: d K

Consumption

K0

K*

Capital, K

The Solow Diagram with Output

Output: Y

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the solow model graphical analysis
The Solow Model (Graphical Analysis)

Steady-state investment

kt(d)

Output, Y

Capital, K

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slide28

The Solow Model (Graphical Analysis)

Consumption, C

Capital, K

Level of K that maximizes C

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solving mathematically for the steady state
Solving Mathematically for the Steady State
  • in the steady state, investment equals depreciation. If we evaluate this equation at the steady-state level of capital, we can solve mathematically for it
    • the steady-state level of capital is positively related with the investment rate, the size of the workforce and the productivity of the economy
    • the steady-state level of capital is negatively correlated with the depreciation rate

http://www.webpages.ttu.edu/vvalcarc

what determines the steady state
What determines the steady state?
  • We can solve mathematically for K* and Y* in the steady state, and doing so will help us understand the model better
  • In the steady state:

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slide31
If we know K*, then we can find Y* using the production function:

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slide32
This equation also tells us about income per capita, y, in the steady state:

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slide33
notice that the exponent on the productivity parameter is greater than in the chapter 4 model:

this results because a higher productivity parameter raises output as in the production model.

however, higher productivity also implies the economy accumulates additional capital.

the level of the capital stock itself depends on productivity

http://www.webpages.ttu.edu/vvalcarc

looking at data through the lens of the solow model
Looking at Data through the Lens of the Solow Model

The Capital-Output Ratio

  • the capital to output ratio is given by the ratio of the investment rate to the depreciation rate:
  • while investment rates vary across countries, it is assumed that the depreciation rate is relatively constant

http://www.webpages.ttu.edu/vvalcarc

empirically countries with higher investment rates have higher capital to output ratios
Empirically, countries with higher investment rates have higher capital to output ratios:

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differences in y l
Differences in Y/L
  • the Solow model gives more weight to TFP in explaining per capita output than the production model does
  • Just like we did before with the simple model of production, we can use this formula to understand why some countries are so much richer
  • take the ratio of y* for a rich country to y* for a poor country, and assume the depreciation rate is the same across countries:

45

=

18

x

2.5

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slide37
Now we find that the factor of 45 that separates rich and poor country’s income per capita is decomposable into:

A 103/2 = 18-fold difference in this productivity ratio term

A (30/5)1/2 = 61/2 = 2.5-fold difference in this investment rate ratio

In the Solow Model, productivity accounts for 18/20.5 = 90% of differences!

45

=

18

x

2.5

http://www.webpages.ttu.edu/vvalcarc

understanding the steady state
Understanding the Steady State
  • the economy will settle in a steady state because the investment curve has diminishing returns
    • however, the rate at which production and investment rise is smaller as the capital stock is larger
    • a constant fraction of the capital stock depreciates every period, which implies depreciation is not diminishing as capital increases

http://www.webpages.ttu.edu/vvalcarc

slide39
In summary, as capital increases, diminishing returns implies that production and investment increase by less and less, but depreciation increases by the same amount .
  • Eventually, net investment is zero and the economy rests in steady state.
    • There are diminishing returns to capital: less Yt per additional Kt
    • That means new investment is also diminishing: less sYt = It
    • But depreciation is NOT diminishing; it’s a constant share of Kt

http://www.webpages.ttu.edu/vvalcarc

economic growth in the solow model
Economic Growth in the Solow Model
  • there is no long-run economic growth that holds forever in the Solow model
  • in the steady state: output, capital, output per person, and consumption per person are all constant and growth stops

both constant

http://www.webpages.ttu.edu/vvalcarc

slide41
empirically, economies appear to continue to grow over time

thus capital accumulation is not the engine of long-run economic growth

saving and investment are beneficial in the short-run, but diminishing returns to capital do not sustain long-run growth

in other words, after we reach the steady state, there is no long-run growth in Yt (unless Lt or A increases)

http://www.webpages.ttu.edu/vvalcarc

some economic experiments
Some Economic Experiments
  • while the Solow model does not explain long-run economic growth, it does help to explain some differences across countries
  • economists can experiment with the model by changing parameter values

An Increase in the Investment Rate

  • the investment rate increases permanently for exogenous reasons
  • the investment curve rotates upward, but the deprecation line remains unchanged

http://www.webpages.ttu.edu/vvalcarc

an increase in the investment rate

Investment, depreciation

New investment

exceeds depreciation

Depreciation: d K

Old investment: s Y

K*

K**

Capital, K

An Increase in the Investment Rate

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slide44
the economy is now below its new steady state and the capital stock and output will increase over time by transition dynamics

the long run, steady-state capital and steady-state output are higher

What happens to output in response to this increase in the investment rate?

the rise in investment leads capital to accumulate over time

this higher capital causes output to rise as well

output increases from its initial steady-state level Y* to the new steady state Y**

http://www.webpages.ttu.edu/vvalcarc

the behavior of output following an increase in s

Investment, depreciation, and output

New

investment:

s ‘Y

Y**

Y*

Depreciation: d K

Old

investment:

s Y

K*

K**

Capital, K

The Behavior of Output Following an Increase in s

Output: Y

(a) The Solow diagram with output.

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the behavior of output following an increase in s cont

Output, Y

(ratio scale)

Y**

Y*

2000

2020

2040

2060

2080

2100

Time, t

The Behavior of Output Following an Increase in s (cont.)

(b) Output over time.

http://www.webpages.ttu.edu/vvalcarc

a rise in the depreciation rate
A Rise in the Depreciation Rate
  • the depreciation rate is exogenously shocked to a higher rate
  • the depreciation curve rotates upward and the investment curve remains unchanged
  • the new steady state is located to the left: this means that depreciation exceeds investment
  • the capital stock declines by transition dynamics until it reaches the new steady state
    • note that output declines rapidly at first but less rapidly as it converges to the new steady state

http://www.webpages.ttu.edu/vvalcarc

a rise in the depreciation rate1

Investment, depreciation

Old

depreciation:

d K

New

depreciation:

d ‘K

Depreciation

exceeds

investment

Investment: s Y

K**

K*

Capital, K

A Rise in the Depreciation Rate

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slide49
What happens to output in response to this increase in the depreciation rate?

the decline in capital reduces output

output declines rapidly at first, and then gradually settles down at its new, lower steady-state level Y**

http://www.webpages.ttu.edu/vvalcarc

the behavior of output following an increase in d

Investment, depreciation,

and output

Investment:

s Y

Y**

Y*

New depreciation: d‘K

Old depreciation: dK

K**

K*

Capital, K

The Behavior of Output Following an Increase in d

Output: Y

(a) The Solow diagram with output.

http://www.webpages.ttu.edu/vvalcarc

the behavior of output following an increase in d cont

Output, Y

(ratio scale)

Y**

Y*

2000

2020

2040

2060

2080

2100

Time, t

The Behavior of Output Following an Increase in d (cont.)

(b) Output over time.

http://www.webpages.ttu.edu/vvalcarc

experiments on your own
Experiments on Your Own
  • Try experimenting with all the parameters in the model:
    • Figure out which curve (if either) shifts.
    • Follow the transition dynamics of the Solow model.
    • Analyze the steady-state values of capital, output, and output per person.

http://www.webpages.ttu.edu/vvalcarc

the principle of transition dynamics
The Principle of Transition Dynamics
  • when the depreciation rate and the investment rate were shocked, output was plotted over time on a ratio scale
    • ratio scale allows us to see that output changes more rapidly the further we are from the steady state
    • as the steady state is approached, growth shrinks to zero

http://www.webpages.ttu.edu/vvalcarc

slide54
the principle of transition dynamics says that the farther below its steady state an economy is, in percentage terms, the faster the economy will grow

similarly, the farther above its steady state, in percentage terms, the slower the economy will grow

this principle allows us to understand why economies may grow at different rates at the same time

http://www.webpages.ttu.edu/vvalcarc

strengths and weaknesses of the solow model
Strengths and Weaknesses of the Solow Model
  • The strengths of the Solow model are:
    • It provides a theory that determines how rich a country is in the long run.
    • The principle of transition dynamics allows for an understanding of differences in growth rates across countries.
  • The weaknesses of the Solow model are:
    • It focuses on investment and capital, while the much more important factor of TFP is still unexplained.
    • It does not explain why different countries have different investment and productivity rates.
    • The model does not provide a theory of sustained long-run economic growth.

http://www.webpages.ttu.edu/vvalcarc

summary
Summary
  • The starting point for the Solow model is the production model of Chapter 4. To that framework, the Solow model adds a theory of capital accumulation. That is, it makes the capital stock an endogenous variable.
  • The capital stock is the sum of past investments. The capital stock today consists of machines and buildings that were bought over the last several decades.

http://www.webpages.ttu.edu/vvalcarc

slide59
The goal of the Solow model is to deepen our understanding of economic growth, but in this it’s only partially successful. The fact that capital runs into diminishing returns means that the model does not lead to sustained economic growth. As the economy accumulates more capital, depreciation rises one-for-one, but output and therefore investment rise less than one-for- one because of the diminishing marginal product of capital. Eventually, the new investment is only just sufficient to offset depreciation, and the capital stock ceases to grow. Output stops growing as well, and the economy settles down to a steady state.

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slide60
There are two major accomplishments of the Solow model. First, it provides a successful theory of the determination of capital, by predicting that the capital-output ratio is equal to the investment-depreciation ratio. Countries with high investment rates should thus have high capital-output ratios, and this prediction holds up well in the data.

http://www.webpages.ttu.edu/vvalcarc

slide61
The second major accomplishment of the Solow model is the principle of transition dynamics, which states that the farther below its steady state an economy is, the faster it will grow. While the model cannot explain long-run growth, the principle of transition dynamics provides a nice theory of differences in growth rates across countries. Increases in the investment rate or total factor productivity can increase a country’s steady-state position and therefore increase growth, at least for a number of years. These changes can be analyzed with the help of the Solow diagram.

http://www.webpages.ttu.edu/vvalcarc

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