The Solow Model

Transcript The Solow Model

Growth, Capital Accumulation and the Economics of Ideas: Catching Up vs. The Cutting Edge

Chapter 7

Chapter Outline

• • • • • •

The Solow Model and Catch-Up Growth The Solow Model- Details and Further Lessons (Optional Section) Growing on the Cutting Edge: the Economics of Ideas The Future of Economic Growth Takeaway See the Invisible Hand Blog (click) for more examples

Introduction

• • •

In 2006 China: GDP per capita grew by 10%

–

U.S: GDP per capita grew by 2.3 % United States has never grown as fast as the Chinese economy is growing today.

Why is China growing more rapidly than the U.S.?

–

Is there something wrong with the U.S. economy?

Introduction

•

There are

two types of growth

–

Catch-up growth

•

takes advantage of ideas, technologies, or methods of management already in existence

•

focuses on capital accumulation

–

Cutting-edge growth

•

developing new ideas

•

focuses on developing new technology for resources.

• • •

The Solow Model and Catch-Up Growth

Robert Solow (Nobel Prize Laureate) Total Output,

, of an economy depends on:

–

Physical capital:

–

Human capital: education x Labor =

–

Ideas:

This can be expressed as the following “production function”:



F(A, K, eL)

•

The Solow Model and Catch-Up Growth

For now, ignore changes in ideas, education, and labor so that

A, e,

and

are constant. The production function becomes:



F(K)

• •

is constant, then increases in

capital per worker mean more

MP K

: marginal product of capital : The additional output resulting from using an additional unit of capital.

–

diminishes the more capital is added.

Output, Y

The Solow Model and Catch-Up Growth

Assume a production function like Y



K Y



5 4

MP K

 5 .

0  4 .

7 10  9  0 .

3 3 2

MP K

 2  0 1  0  2

Conclusion: as more capital is added, MP

declines.

Capital, K

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Growth in China and United States

•

The “iron logic of diminishing returns” largely explains why…

–

The Chinese economy is able to grow so rapidly.

It turned toward markets which increased incentives.

The capital stock was low.

The MP

was high.

–

China will not be able to achieve these high growth rates indefinitely.

•

The Solow Model and Catch-Up Growth

Why Bombing a Country Can Raise Its Growth Rate:

– –

Much of the capital stock was destroyed during WWII. Therefore the MP

was high.

Following the war, both Germany and Japan were able to achieve much higher growth rates than the U.S. as they “caught-up.”

SEE THE INVISIBLE

The first $500 spent on a computer for each of these kids yields a lot. The second $500? Not as big an increase.

The Solow Model and Catch-Up Growth

• •

Capital Growth Equals Investment minus Depreciation

–

Capital is “output that is saved and invested.”

–

Let

be the fraction of output that is invested in new capital.

The next figure shows how output is divided between consumption and investment when

= 0.3.

(30% of additional output is saved and put into new capital)

12.11

The Solow Model and Catch-Up Growth

Output, Y 20 Capital Growth Equals Investment Minus Depreciation When K = 100, Output = 10



15 10 5 3 2 0 0 Consumption = (1- 0.3) x 10 = 7 Investment = 0.3∙Y Investment = (0.3) x 10 = 3 100 200 300 400 Capital, K 12

The Solow Model and Catch-Up Growth

•

Capital Growth Equals Investment minus Depreciation (cont.).

–

Depreciation: amount of capital that wears out each period

–

Let

be the fraction of capital that wears out each period. This is called the depreciation rate so that:



depreciati on K

12.13

The Solow Model and Catch-Up Growth

Capital Depreciation Depends on the Amount of Capital Depreciation Depreciation = 0.02

∙

8 6 4

Slope

 4  2 200  100

2 0 0 100 200 300 400 Capital, K 14

•

The Solow Model and Catch-Up Growth

Capital Alone Cannot be the Key to Economic Growth

–

As capital increases,

• •

depreciation increases at a

constant

output increases at a

diminishing

rate.

rate of

•

Because investment is a constant fraction of output, at some point depreciation will equal investment.

–

The capital stock will stop growing.

–

Output will stop growing.

2 8

The Solow Model and Catch-Up Growth

Capital Increases or Decreases Until Investment = Depreciation GDP, Y Depreciation = 0.02

∙

At K = 400, Inv. < Dep. → ↓ K 6 4.5

4 3 At K = 100, Inv. > Dep.

→ ↑ K Investment = 0.3∙Y Result: equilibrium at K = 225, Y = 4.5

investment = depreciation =4.5

0 0 100 200 225 300 400 Capital, K 16

Check the Math

• •

At K = 100,

Y = √100 = 10 • Depreciation = 0.02∙100 = 2 Investment = 0.3x10 = 3 • Investment > Depreciation

Result: K and Y grow.

Check the Math

Capital Adjusts Until Investment = Depreciation

Check the Math

• • • • At K = 225, Y = √225 =15 Depreciation = 0.02x225 = 4.5

Investment = 0.3x15 = 4.5

Investment = Depreciation Result: 1. Investment = Depreciation 2. K and Y are constant.

• • • •

At K = 400,

Y = √400 = 20 Depreciation = 0.02x400 = 8 Investment = 0.3x20 = 6 Investment < Depreciation

Result: K and Y decrease.

This is steady state.

• • •

The Solow Model and Catch-Up Growth

The logic of diminishing returns means that eventually capital and output will cease growing.

Therefore, other factors must be responsible for long run economic growth. Consider:

–

Human capital: knowledge, skills, experience

–

Technological knowledge: better ideas

Human Capital Investment Pays Off

The Solow Model and Catch-Up Growth

•

Better Ideas Drive Long Run Economic Growth

–

Human Capital

• •

Like capital, it is subject to diminishing returns and it depreciates.

Logic of diminishing returns also applies to human capital.

•

Conclusion: Human capital also cannot drive long-run economic growth.

–

(What about technological knowledge?)

•

The Solow Model and Catch-Up Growth

Better Ideas Drive Long Run Economic Growth

–

Technological knowledge

•

A way of getting more output from the same input (an increase in productivity).

•

We can include technological knowledge in our model by letting

stand for ideas that increase productivity. Now the production function is:



A K

The Solow Model and Catch-Up Growth

Increasing technology (even while holding K constant) creates a higher growth rate.

•

The Solow Model and Catch-Up Growth

An Increase in

A K

Constant Increases Output Holding

–

Conclusion:

•

Technological knowledge / better ideas are the key to long run economic growth.

•

Solow estimated that better ideas are responsible for ¾ of our increased standard of living.

Roman aquaducts: ushering in an era of economic growth… Aquaducts were sophisticated feats of engineering that enabled population and industry to thrive.

-Roman Aquaduct in Segovia, Spain, built in 2 nd century A.D./C.E.

• •

The Solow Model

–

Details and Further Lessons

What we know so far:

– – –

If Investment > Depreciation → K and Y grow.

If Investment < Depreciation → K and Y fall.

If Investment = Depreciation → K and Y are constant.

Two important conclusions

– –

Steady state equilibrium occurs when investment equals depreciation.

When K is in steady state equilibrium, Y is in steady state equilibrium.

6 4.5

4 3 2 0

The Solow Model

–

Details and Further Lessons

When K is in steady state equilibrium, Y is in steady state equilibrium.

Output, Y 8 Depreciation = 0.02

∙

Investment = 0.3∙Y The Steady State K is found where

Investment = Depreciation

0 100 200 225 300 400 Capital, K 26

The Solow Model

–

Details and Further Lessons

When K is in steady state equilibrium, Y is in steady state equilibrium.

Output, Y 20 Steady state output



15 Depreciation = 0.02

∙

10 5 Investment



0.3

K Steady state capital stock Capital, K 0 100 200 225 300 400 27

•

The Solow Model

–

Details and Further Lessons

The Solow Model and an Increase in the Investment Rate

–

What happens when

, (the fraction of output that is saved and invested) increases?

– •

↑

g  K 

↑ Y

Conclusion: an increase in the investment rate increases a country’s steady state level of GDP.

•

Countries with higher rates of investment will be wealthier.

The Solow Model

–

Details and Further Lessons

GDP per Capita is Higher in Countries with Higher Investment Rates

Output, Y 20 An Increase in the Investment Rate Increases Steady State Output

Y  K

15 10 5 Depreciation = 0.02

∙

Inv.

 .4

K Inv.

 0.3

0 100 200 225 300 400 Capital, K 30

The Solow Model

–

Details and Further Lessons

• •

Note:

–

An increase in the investment rate = ↑ steady state level of output.

–

As the economy moves from the lower to the higher steady state output = ↑ growth rate of output.

–

This higher growth rate is temporary.

Conclusion: ↑ investment rate = ↑ steady state level of output

but not its long-run growth rate

•

The Solow Model

–

Details and Further Lessons

The Case of South Korea

– – – – –

In 1950, South Korea was poorer than Nigeria.

1950s: the investment rate was < 10%.

1970s: Investment rate more than doubled.

1990s: Investment rate increased to over 35%.

South Korea’s GDP increased rapidly.

–

As GDP reached Western levels, the growth rate has slowed…

What economic growth looks like at night 33

The Solow Model

–

Details and Further Lessons

•

The Solow Model and Conditional Convergence

–

Conditional Convergence: Among countries with similar steady state levels of output, poorer countries tend to grow faster than richer countries, and so converge in income.

–

The Solow model predicts that a country will grow faster the farther its capital stock is below its steady state value.

•

Conditional convergence is a prediction of the Solow model

The Solow Model

–

Details and Further Lessons

The poorer the OECD country in 1960, the faster its growth between 1960-2000

•

The Solow Model

–

Details and Further Lessons

Solow and the Economics of Ideas in one diagram

– – •

Generation of ideas results in long run economic growth. Let’s see how this works: We begin at steady state equilibrium.

• • •

New ideas → ↑A → ↑Output at every level of K ↑ Output → ↑Investment → Investment > Depreciation →↑ K→ ↑ Output (movement along new production function).

As ideas continue to grow, output continues to grow.

Output, Y 33.7

Output ↑

Solow and the Economics of Ideas in One Diagram

Effect of ↑A from 1 to 1.5



1 .

b Better Ideas



15 a Old steady state capital stock 225 Depreciation = 0.02

∙

Investment



0.3(1.5) K Investment



0.3(1) 506 New steady state capital stock K Capital, K 37

•

The Solow Model

–

Details and Further Lessons

The Big Question: What Determines High Investment Rates?

– –

Incentives

• •

Low real interest rates Low marginal tax rates Institutions which include

• • •

which include Honest government Secure property rights Effective financial intermediaries (banks)

Growing on the Cutting Edge: The Economics of Ideas

•

The Economics of Ideas 1. Ideas for increasing output are primarily researched, developed, and implemented by profit-seeking firms.

2. Spillovers mean that ideas are underprovided.

3. Government has a role in improving the production of ideas.

4. The larger the market, the greater the incentive to research and develop new ideas.

Growing on the Cutting Edge: The Economics of Ideas

1. Research and development is investment for profit.

– – •

keys to increasing technological knowledge: Incentives

•

Institutions that encourage investment in physical and human capital and R&D. 70% of scientists and engineers in the U.S. work for private firms.

Growing on the Cutting Edge: The Economics of Ideas

1. Research and development is investment for profit

– • •

All kinds of people come up with new ideas.

Business culture and institutions are also important.

Appreciation of entrepreneurs is a relatively recent phenomenon.

–

Institutions that are especially important:

•

Commercial settings that help innovators to connect with capitalists

• •

Intellectual property rights A high-quality education system

SEE THE INVISIBLE

John Kay, “destroyer of jobs.” John Kay invented the “flying shuttle” in cotton weaving, the single most important invention launching the industrial revolution.

(1704-1780) used Kay was rewarded for his efforts destroyed afraid of job loss. He man.

by having his house by “machine breakers,” died a poor

Growing on the Cutting Edge: The Economics of Ideas

•

Institutions that increase R & D

–

A commercial setting that helps innovators connect with capitalists.

• • •

Ideas without financial backers are dead.

The U.S. is good at connecting innovators with businessmen and venture capitalists.

American culture supports entrepreneurs.

Growing on the Cutting Edge: The Economics of Ideas

– •

Intellectual property rights

•

New processes, products, and methods can be copied by competitors.

• •

World’s first MP3 player was the Eiger Labs MPMan introduced in 1998.

Eiger Labs lost out to competition.

Patents

• •

Grant temporary monopoly.

But: can slow down the spread of technology.

SEE THE INVISIBLE

Profits provide incentive to invest in R&D:

Property rights, honest government, political stability, a dependable legal system, and competitive open

markets create profit and so

help drive the generation of technological knowledge.

“The patent system…added the fuel of interest to the fire of genius”

-Abraham Lincoln, 1859 (only U.S. President to have been granted a patent)

–

Growing on the Cutting Edge: The Economics of Ideas

A high-quality education system

•

Important at all levels of education.

• •

Creates necessary talent.

Universities generate basic and applied research.

Growing on the Cutting Edge: The Economics of Ideas

2. Spillovers, and why there aren’t enough good ideas

– – – –

Ideas are non-rivalrous.

•

Ideas can be used simultaneously.

Use of an idea by one individual does mean less of the idea available to someone else.

The spillover (or “diffusion”) of new ideas generates widespread economic growth.

Implication: Spillovers mean that the generator of the idea doesn’t get all of the benefits.

•

Result?

Too few ideas are produced.

– – – – –

Growing on the Cutting Edge: The Economics of Ideas

Optimal social investment in R&D occurs where: MSB = MSC (Marginal Social Benefit = Marginal Social Cost) Optimal private investment occurs where: MPB = MPC (Marginal Private Benefit = Marginal Private Cost) With spillover benefits: MSB = MPB + spillovers and MSC = MPC Conclusion: Optimal Private Investment in R&D < Optimal Social Investment in R&D Implication: Spillovers result in too little investment in research and development.

Spillovers Mean Too Little Investment in Research and Development $ MPB = MPC Spillover benefits I P = optimal private investment in R&D I S =optimal social investment in R&D I P I S MSB = MSC MPC = MSC MSB Assumes there are no spillover costs MPB Quantity of R&D 49

Growing on the Cutting Edge: The Economics of Ideas

Government’s Role in the Production of New Ideas

– •

Ideas in mathematics, physics, and molecular biology have many applications so

spillovers

can be large.

Problem: Even if the social benefits are large, the

private

benefits can be small.

•

Solution: Subsidize the production of new ideas or give tax breaks for R&D expenditures.

–

Both shift the MC of R&D curve down → ↑ R&D investment.

Growing on the Cutting Edge: The Economics of Ideas

–

Large spillovers to basic science suggest a role for government subsidies to universities.

• •

Especially those parts of the universities that produce innovations and the basic science behind those innovations.

Universities produce scientists.

–

Most of the 1.3 million scientists were trained in government-subsidized universities.

–

Growing on the Cutting Edge:

Development expenditures.

The Economics of Ideas

4. Market Size and Research and Innovations like pharmaceuticals, new computer chips, software, and chemicals require large R&D

– – – –

Companies will avoid investing in innovations with small potential markets.

Larger markets mean increased rewards (thus incentives) for R&D.

As the world market grows some companies get bigger and will increase their R&D investments.

Click here for Alex Tabarrok’s TED talk, “How Ideas Trump Crises.”

The Future of Economic Growth

– – – – – –

Dawn of civilization to about 1500 A.D./C.E.: growth = 0% AD 1500 – 1760: growth = 0.08% Growth doubled in next 100 years Increased even further during the 19 th centuries and 20 th Today: world wide growth of per capital GDP = 2.2% Economic growth can be even faster. How?

The Future of Economic Growth

A (ideas) = Population

– –

x Incentives x Ideas/Hour Population

•

↑population → ↑ number of people with new ideas Much of the world is poor:

•

thousands of potentially great scientists are laboring in menial jobs.

–

As the world gets richer → ↑ production of ideas → everyone benefits

The Future of Economic Growth

• •

More Hopeful Signs: If A (ideas) = Population x Incentives

–

x Ideas/Hour

•

Incentives Appear to be increasing in many places:

– – – – – – –

Consumers are richer Markets are expanding due to trade World wide improvement in institutions Property rights Honest government Political stability Dependable legal system

The Future of Economic Growth

• •

More Hopeful Signs: If A (ideas) = Population x Incentives x Ideas/Hour

– • •

Ideas per Hour New ideas do not experience diminishing returns.

Two reasons why: 1. Many ideas make creating new ideas easier.

2. The field of ideas that can be explored is so large that diminishing returns may not set in for a very long time.

The Future of Economic Growth

•

Recap: Economic growth might be even faster in the future than it has been in the past…

– – –

There are more scientists and engineers in the world than ever before, and their numbers are also increasing as percentage of the population.

Incentives are increasing due to growing markets resulting from

• •

Increasing trade Increasing wealth in developing countries Better institutions and more secure property rights are spreading throughout the world.

• • • •

Marginal Product of Capital Steady State Conditional Convergence Non-Rivalrous

Key Concepts

Try it!

In the Solow model, if the first unit of capital increases output by one unit, then the second unit of capital will cause total output to a) increase, but by less than one unit.

b) double.

c) remain the same as with one unit of capital.

d) increase exponentially.

Try it!

In the Solow model, if the first unit of capital increases output by one unit, then the second unit of capital will cause total output to a) increase, but by less than one unit.

b) double.

c) remain the same as with one unit of capital.

d) increase exponentially.

Try it!

The marginal product of capital I. refers to the cost of purchasing one extra unit of capital.

II. is expected to be higher for very poor countries relative to wealthy countries.

III. increases as capital accumulation occurs.

a) I only b) II only c) II and III only d) III only

Try it!

The marginal product of capital I. refers to the cost of purchasing one extra unit of capital.

II. is expected to be higher for very poor countries relative to wealthy countries.

III. increases as capital accumulation occurs.

a) I only b) II only c) II and III only d) III only

Try it!

Country Germany Japan United States 1950–1960 6.6% 6.8% 1.2% 1980–1990 1.9% 3.4% 2.3%

Table: GDP Growth Average Annual Growth Rate of Per Capita GDP

Which of the following answers describes why Germany and Japan experienced such high growth rates just after the Second World War?

a) Trade agreements between Germany and Japan greatly enhanced economic growth.

b) High MP

growth.

levels contributed to significant per capita output c) Their technological advances at this time were higher than those of the U.S.

d) All of the answers are correct.

Try it!

Country Germany Japan United States 1950–1960 6.6% 6.8% 1.2% 1980–1990 1.9% 3.4% 2.3%

Table: GDP Growth Average Annual Growth Rate of Per Capita GDP

Which of the following answers describes why Germany and Japan experienced such high growth rates just after the Second World War?

a) Trade agreements between Germany and Japan greatly enhanced economic growth.

b) High MP

growth.

levels contributed to significant per capita output c) Their technological advances at this time were higher than those of the U.S.

d) All of the answers are correct.

Try it!

If a country is at its steady state level of capital, which of the following will not result in economic growth in future years ceteris paribus?

a) a technological advancement b) producing capital at its current rate c) an increase in the savings rate d) a decrease in the depreciation rate

Try it!

If a country is at its steady state level of capital, which of the following will not result in economic growth in future years ceteris paribus?

a) a technological advancement b) producing capital at its current rate c) an increase in the savings rate d) a decrease in the depreciation rate

Try it!

Which of the following should lead to an increased rate of economic growth due to increased development of ideas?

a) increased population b) increased consumer wealth and larger markets c) political stability and honest governments d) All of the answers are correct

Try it!

Which of the following should lead to an increased rate of economic growth due to increased development of ideas?

a) increased population b) increased consumer wealth and larger markets c) political stability and honest governments d) All of the answers are correct

Appendix

•

Excellent Growth

–

Using a spreadsheet, you can easily explore the Solow model and duplicate all the graphs.

First, calculate the increasing capital stock using the formula in A2 and let the spreadsheet do the rest.

Note: Clicking on the lower right corner of a cell and dragging it down will duplicate the formula in the lower cells.

Appendix

•

Excellent Growth (cont.) Second, calculate output, Y, using the formula:

Y  K 12.70

•

Excellent Growth (cont.)

Appendix

Fourth, graphs can be created using the data generated In the steps one through three.

•

Excellent Growth (cont.)

Appendix

Fifth, You can experiment with different investment shares in E2 or the depreciation rates in F2.

Appendix

• • •

The Mathematics of Economic Growth along the Transition Path Objective: To see how economic growth varies along the transition path to a new steady state equilibrium.

We will do two things:

–

Outline the mathematics

–

Use a spreadsheet to visualize our results.

Appendix

•

The Mathematics

Recall

Investment



Depreciati on



γ K



γK

1 2

(

e.g., 0.3

  d

( e.g., 0.02

 K ) thus

ΔK



Investment - Depreci ation

 g

1 2  d

K ) The growth rate of the capital stock is given by

ΔK K

 Growth rate of K  g

K K

1 2  d

K K

 g

2 1 Implicatio g If 1 n :  d  Growth rate of K is positive

2 g  d  Growth rate of K is negative

2 1  d

By plotting these two expressions separately on a graph, we can see how the steady state changes with the values of the investment rate and depreciation rate.

Appendix

•

The Mathematics d, g/K 1/2 0.08

0.07

Difference is the growth rate of the capital stock. The bigger the difference the faster K grows.

0.06

0.05

0.04

0.03

0.02

0.01

400 d = 0.02

0.4/K 1/2 Capital, K 75

•

The Spread Sheet

Appendix

Plotting Y against time shows the transition to steady state

Appendix

•

The Spread Sheet

Output, Y 16.00

14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.00

0 Output, Y 100 200 300

Time

400 500 600

Result: The transition to steady state proceeds at a decreasing rate. As K approaches 400 growth slows down.