中级宏观-Chapter 7
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Chapter Seven 9
The saving rate s determines the allocation of output between consumption and investment. For any level of k, output is f(k), investment is s f(k), and consumption is f(k) – sf(k).
The steady-state value of k that maximizes consumption is called the Golden Rule Level of Capital. To find the steady-state consumption per worker, we begin with the national income accounts identity: y-c+i and rearrange it as: c = y - i. This equation holds that consumption is output minus investment. Because we want to find steady-state consumption, we substitute steady-state values for output and investment. Steady-state output per worker is f (k*) where k* is the steady-state capital stock per worker. Furthermore, because the capital stock is not changing in the steady state, investment is equal to depreciation dk*. Substituting f (k*) for y and dk* for i, we can write steady-state consumption per worker as: c* = f (k*) - dk*.
Tutorial written by:
Mannig J. Simidian
Chapter Seven B.A. in Economics with Distinction, Duke University 1 M.P.A., Harvard University Kennedy School of Government M.B.A., Massachusetts Institute of Technology (MIT) Sloan School of Management
®
CHAPTER 7 Economic Growth I: Capital Accumulation and Population Growth
A PowerPointTutorial
To Accompany
MACROECONOMICS, 7th. Edition N. Gregory Mankiw
4)
i = sy
Investment = savings. The rate of saving s is the fraction of output devoted to investment.
8
Chapter Seven
Here are two forces that influence the capital stock: • Investment: expenditure on plant and equipment. • Depreciation: wearing out of old capital; causes capital stock to fall. Recall investment per worker i = s y. Let’s substitute the production function for y, we can express investment per worker as a function of the capital stock per worker: i = s f(k ) This equation relates the existing stock of capital k to the accumulation of new capital i.
y
Output, f (k) c (per worker) Investment, s f(k) i (per worker)
y (per worker)
Chapter Seven
k
10
Impact of investment and depreciation on the capital stock: Dk = i –dk Change in capital stock
7
1)
y=c+i
consumption per worker
2)
c = (1-s)y
Output per worker depends on
investment per worker
consumption per worker
savings rate (between 0 and 1)
3)
y = (1-s)y + i
Chapter Seven
k1
k*
k2
Capital per worker, k
12
The Solow Model shows that if the saving rate is high, the economy will have a large capital stock and high level of output. If the saving Investment rate is low, the economy will have a small capital stock and a and Depreciation, dk depreciation low level of output. Investment, s2f(k)
k
11
Investment astment equals depreciation and capital will not change over time.
Depreciation, dk
i* = dk*
Below k*, investment exceeds Investment, s f(k) depreciation, so the capital stock grows. Above k*, depreciation exceeds investment, so the capital stock shrinks.
Chapter Seven
3
Chapter Seven
4
Let’s analyze the supply and demand for goods, and see how much output is produced at any given time and how this output is allocated among alternative uses.
Investment dk
Depreciation dk
Remember investment equals savings so, it can be written: Dk = s f(k) – dk
Depreciation is therefore proportional to Chapter the capital stock. Seven
The Production Function
The production function represents the transformation of inputs (labor (L), capital (K), production technology) into outputs (final goods and services for a certain time period). The algebraic representation is: L) z Y = F (zK , z
Chapter Seven
2
The Solow Growth Model is designed to show how growth in the capital stock, growth in the labor force, and advances in technology interact in an economy, and how they affect a nation’s total output of goods and services. Let’s now examine how the model treats the accumulation of capital.
Income
is some function of our given inputs5
Key Assumption: The Production Function has constant returns to scale.
Chapter Seven
This assumption lets us analyze all quantities relative to the size of the labor force. Set z = 1/L. This is a constant Y/ L = F ( K / L , 1 ) that can be ignored. the amount of is some function of Output capital per worker Per worker Constant returns to scale imply that the size of the economy as measured by the number of workers does not affect the relationship between output per worker and capital per worker. So, from now on, let’s denote all quantities in per worker terms in lower case letters. Here is our production function: y = f ( k ) , where f (k) = F (k,1).
i* = dk*
Investment, s1 f(k) An increase in the saving rate causes the capital stock to grow to a new steady state.
k1*
k 2*
Capital per worker, k
13
Chapter Seven
Chapter Seven 6
MPK = f(k + 1) – f (k)
y
MPK
1
Chapter Seven
k
The production function shows how the amount of capital per f(k) worker k determines the amount of output per worker y = f(k). The slope of the production function is the marginal product of capital: if k increases by 1 unit, y increases by MPK units.
• • • • •
Data evidence Solow growth model Why output growth? Why growth differently? This chapter focus on the roles of saving and population growth.
The saving rate s determines the allocation of output between consumption and investment. For any level of k, output is f(k), investment is s f(k), and consumption is f(k) – sf(k).
The steady-state value of k that maximizes consumption is called the Golden Rule Level of Capital. To find the steady-state consumption per worker, we begin with the national income accounts identity: y-c+i and rearrange it as: c = y - i. This equation holds that consumption is output minus investment. Because we want to find steady-state consumption, we substitute steady-state values for output and investment. Steady-state output per worker is f (k*) where k* is the steady-state capital stock per worker. Furthermore, because the capital stock is not changing in the steady state, investment is equal to depreciation dk*. Substituting f (k*) for y and dk* for i, we can write steady-state consumption per worker as: c* = f (k*) - dk*.
Tutorial written by:
Mannig J. Simidian
Chapter Seven B.A. in Economics with Distinction, Duke University 1 M.P.A., Harvard University Kennedy School of Government M.B.A., Massachusetts Institute of Technology (MIT) Sloan School of Management
®
CHAPTER 7 Economic Growth I: Capital Accumulation and Population Growth
A PowerPointTutorial
To Accompany
MACROECONOMICS, 7th. Edition N. Gregory Mankiw
4)
i = sy
Investment = savings. The rate of saving s is the fraction of output devoted to investment.
8
Chapter Seven
Here are two forces that influence the capital stock: • Investment: expenditure on plant and equipment. • Depreciation: wearing out of old capital; causes capital stock to fall. Recall investment per worker i = s y. Let’s substitute the production function for y, we can express investment per worker as a function of the capital stock per worker: i = s f(k ) This equation relates the existing stock of capital k to the accumulation of new capital i.
y
Output, f (k) c (per worker) Investment, s f(k) i (per worker)
y (per worker)
Chapter Seven
k
10
Impact of investment and depreciation on the capital stock: Dk = i –dk Change in capital stock
7
1)
y=c+i
consumption per worker
2)
c = (1-s)y
Output per worker depends on
investment per worker
consumption per worker
savings rate (between 0 and 1)
3)
y = (1-s)y + i
Chapter Seven
k1
k*
k2
Capital per worker, k
12
The Solow Model shows that if the saving rate is high, the economy will have a large capital stock and high level of output. If the saving Investment rate is low, the economy will have a small capital stock and a and Depreciation, dk depreciation low level of output. Investment, s2f(k)
k
11
Investment astment equals depreciation and capital will not change over time.
Depreciation, dk
i* = dk*
Below k*, investment exceeds Investment, s f(k) depreciation, so the capital stock grows. Above k*, depreciation exceeds investment, so the capital stock shrinks.
Chapter Seven
3
Chapter Seven
4
Let’s analyze the supply and demand for goods, and see how much output is produced at any given time and how this output is allocated among alternative uses.
Investment dk
Depreciation dk
Remember investment equals savings so, it can be written: Dk = s f(k) – dk
Depreciation is therefore proportional to Chapter the capital stock. Seven
The Production Function
The production function represents the transformation of inputs (labor (L), capital (K), production technology) into outputs (final goods and services for a certain time period). The algebraic representation is: L) z Y = F (zK , z
Chapter Seven
2
The Solow Growth Model is designed to show how growth in the capital stock, growth in the labor force, and advances in technology interact in an economy, and how they affect a nation’s total output of goods and services. Let’s now examine how the model treats the accumulation of capital.
Income
is some function of our given inputs5
Key Assumption: The Production Function has constant returns to scale.
Chapter Seven
This assumption lets us analyze all quantities relative to the size of the labor force. Set z = 1/L. This is a constant Y/ L = F ( K / L , 1 ) that can be ignored. the amount of is some function of Output capital per worker Per worker Constant returns to scale imply that the size of the economy as measured by the number of workers does not affect the relationship between output per worker and capital per worker. So, from now on, let’s denote all quantities in per worker terms in lower case letters. Here is our production function: y = f ( k ) , where f (k) = F (k,1).
i* = dk*
Investment, s1 f(k) An increase in the saving rate causes the capital stock to grow to a new steady state.
k1*
k 2*
Capital per worker, k
13
Chapter Seven
Chapter Seven 6
MPK = f(k + 1) – f (k)
y
MPK
1
Chapter Seven
k
The production function shows how the amount of capital per f(k) worker k determines the amount of output per worker y = f(k). The slope of the production function is the marginal product of capital: if k increases by 1 unit, y increases by MPK units.
• • • • •
Data evidence Solow growth model Why output growth? Why growth differently? This chapter focus on the roles of saving and population growth.