Economics 101 Lecture 9
Economic Growth
Solow Model
Equilibrium
Steady State
Economics 101 Lecture 9 Jennifer Milosch Summer Session B
August 22, 2011
Milosch
Econ 101-Lecture 9
1/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Growth Models
Important topic of study in macroeconomics is economic growth and how to sustain it What is it? Growth in GDP per capita over time There are several facts seen in the data concerning economic growth We will use these facts in analyzing a model of economic growth In the previous model, no growth since there was only one period
Milosch
Econ 101-Lecture 9
2/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Solow Growth Model
Solow growth model developed by Robert Solow in the 1950s It is an exogenous growth model, or the growth seen in the model is caused by forces outside of the model We will be able to analyze some potential causes of growth, but by itself growth will not arise in the model Instead we will change exogenous variables to get economic growth The model presents some interesting implications on how we can have sustained increases in the standard of living
Milosch
Econ 101-Lecture 9
3/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Economic Growth Facts 1. Before the Industrial Revolution in about 1800, standards of living varied little over time and across countries 2. Since the Industrial Revolution, per capita income growth has been sustained in the richest countries 3. There is a positive correlation between the rate of investment and output per worker across countries 4. There is a negative correlation between the population growth rate and output per worker across countries
Milosch
Econ 101-Lecture 9
4/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Economic Growth Facts
5. Differences in per capita incomes increased dramatically among countries of the world between 1800 and 1950, with the gap widening between the countries of Western Europe, US, Canada, Australia, New Zealand, and the rest of the world 6. There is essentially no correlation across countries between the level of output per worker in 1960 and the average rate of growth in output per worker for the years of 1960-2000 7. Richer countries are much more alike in terms of rates of growth of real per capita income than are poor countries
Milosch
Econ 101-Lecture 9
5/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.2-Real Income Per Capita vs. Investment Rate 3. There is a positive correlation between the rate of investment and output per worker across countries
Milosch
Econ 101-Lecture 9
6/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.3-Real Income Per Capita vs. Pop. Growth Rate 4. There is a negative correlation between the population growth rate and output per worker across countries
Milosch
Econ 101-Lecture 9
7/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.4-Real Income Per Capita for Various Countries 7. Richer countries are much more alike in terms of rates of growth of real per capita income than are poor countries
Milosch
Econ 101-Lecture 9
8/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Description of the Model
Unlike the previous model, to study growth we need many periods Focus on just the “current” period and the “future” period Actors: consumers, firms (no government) Goods to be traded: capital, labor Preferences: consumers save a constant fraction of their income, consume the rest Do not care about leisure, so work all of the time available Production technology: Y = zF (K , N) Capital market, labor market
Milosch
Econ 101-Lecture 9
9/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
Population of consumers grows over time N 0 = (1 + n)N N is the current period population, also the labor force N 0 is the future period population n > −1 is the population growth rate, fixed across periods Each consumer has 1 unit of time available, and does not care about leisure So N is also the amount of labor supply in the current period
Milosch
Econ 101-Lecture 9
10/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
To simplify, we assume that the optimal choice for the consumers is to save some constant fraction of their income C = (1 − s)Y C is aggregate current period consumption 0 ≤ s ≤ 1 is the savings rate Y is aggregate current period income, current period output Since there is no government, consumers receive all of the output in income through taxes or dividends Then aggregate savings for the consumers is S = sY
Milosch
Econ 101-Lecture 9
11/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
Y = (1 − s)Y + sY = C + S The amount of total income will be determined by how much the firm produces C , S, Y are aggregate values, for all of the consumers Instead, put everything into per worker or per capita terms Divide by the number of people, N Uppercase letters indicate aggregate values, lowercase letters indicate per worker values c = (1 − s)y
Milosch
Econ 101-Lecture 9
12/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Firms As before, the firm has a CRS production function, Y = zF (K , N) All N consumers work their 1 hour of available time The labor market will adjust wages so that N is also the amount of labor demand Convert the production function into per worker terms: Y zF (K , N) = N N Because the production function is CRS, it is the case that K Y = zF ,1 N N Milosch
Econ 101-Lecture 9
13/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Firms Rewriting the production function in per worker terms we get y = zF (k, 1) If we define F (k, 1) = f (k) our per worker production function becomes y = zf (k) Note that unlike the previous model, the capital input is the input that the firm is choosing In this model the slope of the production function is the marginal product of capital The same features of the marginal product apply here: Increasing k decreases MPK Increasing N increases MPK
Milosch
Econ 101-Lecture 9
14/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Fig. 6.12-Per Worker Production Function
Milosch
Econ 101-Lecture 9
15/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital
Over time the capital stock depreciates at a rate of d per period So it must be replenished in each period by investing in capital This gives rise to a law of motion of capital that describes how the capital stock changes across periods Just as with labor supply, K is the current period capital stock and K 0 is the future period capital stock
Milosch
Econ 101-Lecture 9
16/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital
Tells us the amount of capital tomorrow, K 0 , given the amount that depreciates and the amount invested K0 =
K |{z}
−
Current capital
dK |{z}
Amount Depreciated
+
I |{z}
Amount Invested
Rearranging, K 0 = I + (1 − d)K If we are investing more than is depreciating in each period, K will be growing over time
Milosch
Econ 101-Lecture 9
17/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Competitive Equilibrium
Consumers and firms behave as described Consumers work and consume and save Firms produce output and invest in new capital The two markets must clear in equilibrium Amount of labor in labor market determined by inelastic supply by consumers Capital market clears when what the consumers want to save equals what the firms want to invest
Milosch
Econ 101-Lecture 9
18/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Note About Capital
Recall the Solow residual and relationship to average labor productivity Capital is not measured in a timely manner The model captures this by fixing capital in the current time period Or, investment in capital does not affect the capital stock today, only the capital stock tomorrow
Milosch
Econ 101-Lecture 9
19/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Income-Expenditure Identity
In this model, no government and no foreign sector Add investment in the capital stock Y =C +I Equilibrium in the capital market says that supply of funds for capital equals demand of funds for capital Consumers supply funds by saving, firms demand funds for investment ⇒ S = I Recalling the equation for aggregate savings, sY = I
Milosch
Econ 101-Lecture 9
20/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Law of Motion of Capital
K 0 = I + (1 − d)K = S + (1 − d)K sY = I , Plugging in for investment in terms of output: K 0 = sY + (1 − d)K Finally, replace Y with the production function K 0 = szF (K , N) + (1 − d)K
Milosch
Econ 101-Lecture 9
21/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital: Per Worker Write the law of motion to be in per worker terms Divide through by N, the number of workers K0 zF (K , N) K =s + (1 − d) N N N This can be rewritten as: K0 = szf (k) + (1 − d)k N Need to get the left side of this equation in per worker terms k 0 (1 + n) = szf (k) + (1 − d)k k0 =
szf (k) + (1 − d)k 1+n
Milosch
Econ 101-Lecture 9
22/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Variables
Exogenous variables: 1 2 3 4
z, TFP d, depreciation rate s, savings rate n, population growth rate
Endogenous Variables 1 2 3
c, consumption per worker y , output per worker k, capital per worker
Milosch
Econ 101-Lecture 9
23/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Evolution of Capital Over Time
Capital will be changing from one period to the next as described by the law of motion k0 =
szf (k) + (1 − d)k 1+n
Focus on the point where k is constant across periods This point is the steady state, a long run equilibrium or rest point in the economy Once steady state is reached, capital per worker is the same in every period, holding the exogenous variables constant
Milosch
Econ 101-Lecture 9
24/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Fig. 6.13-Steady State Capital per Worker
Milosch
Econ 101-Lecture 9
25/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Steady State
Once the economy has reached steady state, k = k 0 = k ∗ in every period Notice that y = zf (k) so in steady state y ∗ = zf (k ∗ ) Thus output per worker is constant in steady state And, c = (1 − s)y so in steady state c ∗ = (1 − s)y ∗ = (1 − s)zf (k ∗ ) Consumption per worker is constant in steady state as well
Milosch
Econ 101-Lecture 9
26/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Steady State
Steady state does NOT mean that nothing is changing It simply means that the ratios of K /N, Y /N, and C /N are constant over time Consider this example: N = 50 n = 0.05 K = 100 K 0 = 105
Milosch
Econ 101-Lecture 9
27/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Example- Cobb-Douglas Production Function
Consider a firm with a CRS Cobb-Douglas Production function Y = zK a N 1−a Converting to per worker terms, y = zk a = zf (k) Note: a is also an exogenous variable in this model
Milosch
Econ 101-Lecture 9
28/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Example
The law of motion for capital is k0 =
szf (k) + (1 − d)k 1+n
So plugging in for f (k), k0 =
szk a + (1 − d)k 1+n
Now we can solve for steady state capital Steady state is the point where k = k 0 = k ∗
Milosch
Econ 101-Lecture 9
29/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Example Replace k and k 0 with k ∗ and solve for k ∗
∗
k =
sz n+d
1 1−a
Then we can solve for y ∗ and c ∗ ∗
y = zk
∗
∗a
=z
∗
sz n+d
c = (1 − s)y = (1 − s)z
Milosch
a 1−a
sz n+d
Econ 101-Lecture 9
a 1−a
30/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Next Time
Chapter 6, p. 212-228 Working with the Solow Model NOTE: Skip the section Growth Accounting Exercise starting on page 228 in the 4th edition Problem Set 4, Problem Set 3 answers posted Today
Milosch
Econ 101-Lecture 9
31/ 31
Solow Model
Equilibrium
Steady State
Economics 101 Lecture 9 Jennifer Milosch Summer Session B
August 22, 2011
Milosch
Econ 101-Lecture 9
1/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Growth Models
Important topic of study in macroeconomics is economic growth and how to sustain it What is it? Growth in GDP per capita over time There are several facts seen in the data concerning economic growth We will use these facts in analyzing a model of economic growth In the previous model, no growth since there was only one period
Milosch
Econ 101-Lecture 9
2/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Solow Growth Model
Solow growth model developed by Robert Solow in the 1950s It is an exogenous growth model, or the growth seen in the model is caused by forces outside of the model We will be able to analyze some potential causes of growth, but by itself growth will not arise in the model Instead we will change exogenous variables to get economic growth The model presents some interesting implications on how we can have sustained increases in the standard of living
Milosch
Econ 101-Lecture 9
3/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Economic Growth Facts 1. Before the Industrial Revolution in about 1800, standards of living varied little over time and across countries 2. Since the Industrial Revolution, per capita income growth has been sustained in the richest countries 3. There is a positive correlation between the rate of investment and output per worker across countries 4. There is a negative correlation between the population growth rate and output per worker across countries
Milosch
Econ 101-Lecture 9
4/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Economic Growth Facts
5. Differences in per capita incomes increased dramatically among countries of the world between 1800 and 1950, with the gap widening between the countries of Western Europe, US, Canada, Australia, New Zealand, and the rest of the world 6. There is essentially no correlation across countries between the level of output per worker in 1960 and the average rate of growth in output per worker for the years of 1960-2000 7. Richer countries are much more alike in terms of rates of growth of real per capita income than are poor countries
Milosch
Econ 101-Lecture 9
5/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.2-Real Income Per Capita vs. Investment Rate 3. There is a positive correlation between the rate of investment and output per worker across countries
Milosch
Econ 101-Lecture 9
6/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.3-Real Income Per Capita vs. Pop. Growth Rate 4. There is a negative correlation between the population growth rate and output per worker across countries
Milosch
Econ 101-Lecture 9
7/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Fig 6.4-Real Income Per Capita for Various Countries 7. Richer countries are much more alike in terms of rates of growth of real per capita income than are poor countries
Milosch
Econ 101-Lecture 9
8/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Description of the Model
Unlike the previous model, to study growth we need many periods Focus on just the “current” period and the “future” period Actors: consumers, firms (no government) Goods to be traded: capital, labor Preferences: consumers save a constant fraction of their income, consume the rest Do not care about leisure, so work all of the time available Production technology: Y = zF (K , N) Capital market, labor market
Milosch
Econ 101-Lecture 9
9/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
Population of consumers grows over time N 0 = (1 + n)N N is the current period population, also the labor force N 0 is the future period population n > −1 is the population growth rate, fixed across periods Each consumer has 1 unit of time available, and does not care about leisure So N is also the amount of labor supply in the current period
Milosch
Econ 101-Lecture 9
10/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
To simplify, we assume that the optimal choice for the consumers is to save some constant fraction of their income C = (1 − s)Y C is aggregate current period consumption 0 ≤ s ≤ 1 is the savings rate Y is aggregate current period income, current period output Since there is no government, consumers receive all of the output in income through taxes or dividends Then aggregate savings for the consumers is S = sY
Milosch
Econ 101-Lecture 9
11/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Consumers
Y = (1 − s)Y + sY = C + S The amount of total income will be determined by how much the firm produces C , S, Y are aggregate values, for all of the consumers Instead, put everything into per worker or per capita terms Divide by the number of people, N Uppercase letters indicate aggregate values, lowercase letters indicate per worker values c = (1 − s)y
Milosch
Econ 101-Lecture 9
12/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Firms As before, the firm has a CRS production function, Y = zF (K , N) All N consumers work their 1 hour of available time The labor market will adjust wages so that N is also the amount of labor demand Convert the production function into per worker terms: Y zF (K , N) = N N Because the production function is CRS, it is the case that K Y = zF ,1 N N Milosch
Econ 101-Lecture 9
13/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Firms Rewriting the production function in per worker terms we get y = zF (k, 1) If we define F (k, 1) = f (k) our per worker production function becomes y = zf (k) Note that unlike the previous model, the capital input is the input that the firm is choosing In this model the slope of the production function is the marginal product of capital The same features of the marginal product apply here: Increasing k decreases MPK Increasing N increases MPK
Milosch
Econ 101-Lecture 9
14/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Fig. 6.12-Per Worker Production Function
Milosch
Econ 101-Lecture 9
15/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital
Over time the capital stock depreciates at a rate of d per period So it must be replenished in each period by investing in capital This gives rise to a law of motion of capital that describes how the capital stock changes across periods Just as with labor supply, K is the current period capital stock and K 0 is the future period capital stock
Milosch
Econ 101-Lecture 9
16/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital
Tells us the amount of capital tomorrow, K 0 , given the amount that depreciates and the amount invested K0 =
K |{z}
−
Current capital
dK |{z}
Amount Depreciated
+
I |{z}
Amount Invested
Rearranging, K 0 = I + (1 − d)K If we are investing more than is depreciating in each period, K will be growing over time
Milosch
Econ 101-Lecture 9
17/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Competitive Equilibrium
Consumers and firms behave as described Consumers work and consume and save Firms produce output and invest in new capital The two markets must clear in equilibrium Amount of labor in labor market determined by inelastic supply by consumers Capital market clears when what the consumers want to save equals what the firms want to invest
Milosch
Econ 101-Lecture 9
18/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Note About Capital
Recall the Solow residual and relationship to average labor productivity Capital is not measured in a timely manner The model captures this by fixing capital in the current time period Or, investment in capital does not affect the capital stock today, only the capital stock tomorrow
Milosch
Econ 101-Lecture 9
19/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Income-Expenditure Identity
In this model, no government and no foreign sector Add investment in the capital stock Y =C +I Equilibrium in the capital market says that supply of funds for capital equals demand of funds for capital Consumers supply funds by saving, firms demand funds for investment ⇒ S = I Recalling the equation for aggregate savings, sY = I
Milosch
Econ 101-Lecture 9
20/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Law of Motion of Capital
K 0 = I + (1 − d)K = S + (1 − d)K sY = I , Plugging in for investment in terms of output: K 0 = sY + (1 − d)K Finally, replace Y with the production function K 0 = szF (K , N) + (1 − d)K
Milosch
Econ 101-Lecture 9
21/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Law of Motion of Capital: Per Worker Write the law of motion to be in per worker terms Divide through by N, the number of workers K0 zF (K , N) K =s + (1 − d) N N N This can be rewritten as: K0 = szf (k) + (1 − d)k N Need to get the left side of this equation in per worker terms k 0 (1 + n) = szf (k) + (1 − d)k k0 =
szf (k) + (1 − d)k 1+n
Milosch
Econ 101-Lecture 9
22/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Variables
Exogenous variables: 1 2 3 4
z, TFP d, depreciation rate s, savings rate n, population growth rate
Endogenous Variables 1 2 3
c, consumption per worker y , output per worker k, capital per worker
Milosch
Econ 101-Lecture 9
23/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Evolution of Capital Over Time
Capital will be changing from one period to the next as described by the law of motion k0 =
szf (k) + (1 − d)k 1+n
Focus on the point where k is constant across periods This point is the steady state, a long run equilibrium or rest point in the economy Once steady state is reached, capital per worker is the same in every period, holding the exogenous variables constant
Milosch
Econ 101-Lecture 9
24/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Fig. 6.13-Steady State Capital per Worker
Milosch
Econ 101-Lecture 9
25/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Steady State
Once the economy has reached steady state, k = k 0 = k ∗ in every period Notice that y = zf (k) so in steady state y ∗ = zf (k ∗ ) Thus output per worker is constant in steady state And, c = (1 − s)y so in steady state c ∗ = (1 − s)y ∗ = (1 − s)zf (k ∗ ) Consumption per worker is constant in steady state as well
Milosch
Econ 101-Lecture 9
26/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Steady State
Steady state does NOT mean that nothing is changing It simply means that the ratios of K /N, Y /N, and C /N are constant over time Consider this example: N = 50 n = 0.05 K = 100 K 0 = 105
Milosch
Econ 101-Lecture 9
27/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Example- Cobb-Douglas Production Function
Consider a firm with a CRS Cobb-Douglas Production function Y = zK a N 1−a Converting to per worker terms, y = zk a = zf (k) Note: a is also an exogenous variable in this model
Milosch
Econ 101-Lecture 9
28/ 31
Economic Growth
Solow Model
Equilibrium
Steady State
Example
The law of motion for capital is k0 =
szf (k) + (1 − d)k 1+n
So plugging in for f (k), k0 =
szk a + (1 − d)k 1+n
Now we can solve for steady state capital Steady state is the point where k = k 0 = k ∗
Milosch
Econ 101-Lecture 9
29/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Example Replace k and k 0 with k ∗ and solve for k ∗
∗
k =
sz n+d
1 1−a
Then we can solve for y ∗ and c ∗ ∗
y = zk
∗
∗a
=z
∗
sz n+d
c = (1 − s)y = (1 − s)z
Milosch
a 1−a
sz n+d
Econ 101-Lecture 9
a 1−a
30/ 31
Cobb-Douglas
Economic Growth
Solow Model
Equilibrium
Steady State
Cobb-Douglas
Next Time
Chapter 6, p. 212-228 Working with the Solow Model NOTE: Skip the section Growth Accounting Exercise starting on page 228 in the 4th edition Problem Set 4, Problem Set 3 answers posted Today
Milosch
Econ 101-Lecture 9
31/ 31
