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ContentContent

Content

- Overview
- Solow Model I (accumulation of capital and population growth )
- Solow Model II (considering technological progress)
- Endogenous growth models
- Economic growth accounting

Intermediate Macroeconomics

The Importance of Economic Growth

- For poor countries, a stagnant economy means a persistent absolute poverty.
- In relative terms, a slight but persistent difference in growth rate results in huge income gaps.
- The following table illustrates how three different growth rates (from the same income per cap, 100 ) lead to starkly different outcomes.

Intermediate Macroeconomics

Growth Performances of 143 Economies

Intermediate Macroeconomics

Where Economic Growth Comes From

- Increases in factor inputs
- Capital
- Labor
- Technology progress

Intermediate Macroeconomics

Content

- Overview
- Solow Model I (accumulation of capital and population growth )
- Solow Model II (considering technological progress)
- Endogenous growth models
- Economic growth accounting

Intermediate Macroeconomics

Solow Model I

- The first Solow model characterizes the role of factor inputs in economic growth.
- We assume:
- Closed economy (), no government spending ().
- Fixed production function, , a constant-return-to-scale technology.
- The saving rate is constant:
- Capital depreciates at constant rate .
- Population grows at constant rate ,

Intermediate Macroeconomics

The Per Capita Output

- Let and is the output per cap. And is the amount of capital per cap. We have
- Define We have
- f(k) isthe“per worker production function,” how much output one worker could produce using k units of capital. We assume

Note that is the marginal product of capital (MPK).

- We also assume:

Intermediate Macroeconomics

The Per Capita Demand

- Without government spending and net export, the demand for goods and services is composed of consumption () and investment ().
- In per cap terms, we have

where and

- The per cap investment is a constant fraction of the out

Intermediate Macroeconomics

Accumulation of Capital

- Investment causes capital to rise and depreciation causes capital to wear out.
- The aggregate capital accumulation is described by
- Similarly, the evolution of the population is characterized by
- Let , the capital per cap at time We then characterize the dynamics of capital accumulation by

.

Intermediate Macroeconomics

Steady-State Level of Capital

- As capital accumulates, it will reach a point where new investment equals depreciation,
- At this level of capital, , the economy reaches a steady state, where capital does not increase or decrease. We call the stead-state level of capital.

Intermediate Macroeconomics

A Graphic Illustration ()

Intermediate Macroeconomics

An Example

- Suppose . Then we have
- Let Each year (), the capital stock changes by

.

- Solving , we obtain

Intermediate Macroeconomics

Approaching the Steady State: A Numerical Illustration

Assumptions: , s=0.3, ,

Intermediate Macroeconomics

Implications of Solow Model I

- If the economy is already at the steady state, per capita income () ceases to grow. However, the total income continues to grow as the population grows,
- If the initial level of capital is below the steady-state level, there will be a convergence (or, catch-up) period to the steady-state level.

Intermediate Macroeconomics

The Role of Saving

- To see the effect of a change in the saving rate, , we examine the equation characterizing the steady state,
- Fix and , let . Using the implicit function theorem, we have
- Due to the assumption , we must have , otherwise the curve cannot cross with the line . Hence must be positive.
- An increase in saving rate would lead to a higher level of steady-state capital and income.

Intermediate Macroeconomics

A Graphic Illustration ()

Intermediate Macroeconomics

The Golden-Rule Level of Capital

- At steady states, the consumption is given by
- The level of capital that corresponds to the maximum consumption, which we call the golden-rule level of capital, must satisfy the first-order condition:
- At the golden-rule level, marginal product of capital (MPK) equals the depreciation rate plus the population growth rate.

Intermediate Macroeconomics

A Graphic Illustration ()

Intermediate Macroeconomics

How The Golden-Rule Level of Capital Might Be Achieved

- Recall that the steady-state level of capital is a function of the saving rate, .We might adjust to achieve the golden-rule level of capital.
- Suppose that . Since , we might increase the saving rate to achieve the golden-rule level.
- If the initial level of capital is higher than the golden-rule level, then we might decrease the saving rate to achieve the golden-rule level.

Intermediate Macroeconomics

An Example

- Suppose Then solving for the steady-state level

we obtain the function .

- Suppose we obtain the steady-state level of capital in this economy.
- The golden-rule level, however, is obtained from

which gives =25.

- In this economy, the steady-state level of capital is too low. We might increase the saving rate to achieve the golden rule. Which saving rate corresponds to the golden rule?
- We solve and obtain

Intermediate Macroeconomics

Approaching The Golden-Rule

Intermediate Macroeconomics

Dealing with Too Much Capital

Intermediate Macroeconomics

Case Studies

- The economic miracle of Japan and Germany since the end of the World War II.
- The economic miracle of China since 1978.
- Why East-Asian economies grew so fast?

Intermediate Macroeconomics

The Effect of Population Growth

- To see the effect of a change in , we still examine the equation characterizing the steady state,
- Fix and , let . Using the implicit function theorem, we have
- Hence higher population growth leads to lower steady-state capital per cap.

Intermediate Macroeconomics

A Graphic Illustration

Intermediate Macroeconomics

Case Study: Population Growth v.s. Per Capita Income

Intermediate Macroeconomics

Content

- Overview
- Solow Model I (accumulation of capital and population growth )
- Solow Model II (considering technological progress)
- Endogenous growth models
- Economic growth accounting

Intermediate Macroeconomics

Solow Model II

- The first Solow model fails to allow sustainable growth in per capita output/income.
- The second Solow model introduce technological progress as the engine for sustainable growth.
- We assume:
- Closed economy (), no government spending ().
- Labor-augmenting production function, , where
- is a constant-return-to-scale function,
- is technology level, which grows at a constant rate and
- denotes population, which grows at a constant rate .
- The saving rate is constant:
- Capital depreciates at constant rate .

Intermediate Macroeconomics

Output Per Effective Worker

- Let and is called the output per effective worker. And is the amount of capital per effective worker. We have
- As in the first Solow model, we define and write
- f(k) isthe“per effective worker production function,” how much output one effective worker could produce using k units of capital. We assume

Note that is the marginal product of capital (MPK).

- We also assume:

Intermediate Macroeconomics

The Accumulation of Capital

- Note that And as previously, we have
- The the dynamics of capital accumulation is characterized by

.

Intermediate Macroeconomics

The Steady State

- The steady state capital per effective worker, , is then characterized by the following equation,
- At steady state, the capital per effective worker is a constant,
- We have the following implications:
- Total output, , grows at the constant rate
- Per capita output, , grows at the constant rate .
- Thus the Solow Model II, incorporating the factor of technological progress, is able to explain sustained growth in per capita terms.

Intermediate Macroeconomics

Exercises

- What would be the impact of an increase in saving rate on the steady-state capital per capita?
- What would be the Golden-rule level of capital per effective worker?

Intermediate Macroeconomics

Real Wage and Real Rental Price of Capital

- Recall that in a competitive economy, the real wage equals marginal product of labor () and the real rental price of capital equals the marginal product of capital ():
- Hence the Solow Model II implies that the real wage grows with the technological progress and that the real return to capital remains constant.

Intermediate Macroeconomics

- Overview
- Solow Model I (accumulation of capital and population growth )
- Solow Model II (considering technological progress)
- Endogenous growth models
- Economic growth accounting

Intermediate Macroeconomics

What Endogenous Models Do

- In the Solow model, technological progress is assumed.
- Endogenous models treat the technology progress as an outcome of economic activities, or in other words, as an endogenous process.
- In this part of the lecture, we introduce two simple endogenous models.
- A basic model
- Two-sector model

Intermediate Macroeconomics

A Basic Model

- The basic model assumes
- Constant population.
- A linear technology,

where is output, is capital stock that includes “knowledge”, and is a constant.

- The capital accumulation follows
- It is obvious that

Intermediate Macroeconomics

The Implication

- As long as , sustained growth is achieved. The sustained growth depends on high saving and investment.
- This is possible because this model enjoys the constant return to capital, while the Solow model assumes diminishing return to capital.
- The capital stock in this model should be understood to include “knowledge”.

Intermediate Macroeconomics

A Two-Sector Model

- The two-sector model assumes
- Production function in manufacturing

where is the fraction of the labor force in universities.

- Production function in research universities

where is describes how the growth in knowledge depends on the fraction of labor force in universities.

- Capital accumulation

Intermediate Macroeconomics

Exercise

- Solve this two-sector model and obtain the stead state.
- Does this model exhibits constant return to capital?
- Does this model allows sustained growth?

Intermediate Macroeconomics

- Overview
- Solow Model I (accumulation of capital and population growth )
- Solow Model II (considering technological progress)
- Endogenous growth models
- Economic growth accounting

Intermediate Macroeconomics

Accounting for Economic Growth

- Where does economic growth come from?
- Increases in the factors of production
- Capital (K)
- Labor (L)
- Technological progress
- How much does each element contribute to economic growth?

Intermediate Macroeconomics

Increases in Factors

- We first assume that growth comes from increases in factors only.
- Assume , which is to say that technology does not change over time.We obtain:
- Dividing the equation by , we have

where is capital’s share and is labor’s share.

Intermediate Macroeconomics

Technological Progress

- Assume , where is called total factor productivity. measures current level of technology.
- Similar derivation yields the equation of growth accounting:
- ,which is called the Solow residual, measures technological progress. It is the changes in output that cannot be explained by changes in inputs.

Case Study: Source of Growth in US

Intermediate Macroeconomics

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