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Long-run Economic Growth
Part I: Production and Solow’s Growth Model
Chris EdmondNYU Stern
Spring 2007
1
Agenda
• Two classes on ‘long-run economic growth’
– the aggregate production function, a fundamental tool (today)– Robert Solow’s growth model– growth accounting (next class)– productivity and institutions
• To start with, what do we mean by ‘long-run growth’?
– in contrast with business cycle fluctuations– a key conceptual distinction in macroeconomics
2
US real GDP
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
1929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2004
in 2000 dollars
Source: Department of Commerce, BEA.
3
Trend
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
1929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2004
log real GDP
y = 2.02 + 0.036x
Source: Department of Commerce, BEA.
4
Business cycle fluctuations
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
annual percentage change in real GDP
Source: Department of Commerce, BEA.
5
Long-run economic growth
• Questions
– why does a country’s income per capita grow over time?– why is there so much cross-country variation in income per capita?
• Data
– Penn World Tables, careful cross-country measurements
6
Cross-country GDP di!erences
Country GDP per person (2000 dollars at ‘PPP’)
Argentina 10,938Brazil 7,204China 5,332Egypt 4,759France 26,168India 2,990Ireland 28,956Japan 24,661Korea 18,423Mexico 8,165United States 36,098Zimbabwe 2,438
Source: Penn World Tables, 6.2
7
GDP per person
0
5000
10000
15000
20000
25000
30000
35000
40000
1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002
USA
FRA
CHN
IND
KOR
ARG
IRL
JPN
in 2000 dollars at ‘PPP’
Source: Penn World Tables, 6.2
8
Production functions
GDP
Capital and LaborProductivity
(''Black Box'')
Output
Inputs
9
Production functions
• Relate output to inputs, e.g., capital and labor
• Mathematical version
Y = AK!L1!!
(‘Cobb-Douglas’)
• Definitions
Y = quantity of outputK = quantity of physical capital used (plant and equipment)L = quantity of labor usedA = ‘total factor productivity,’ i.e., everything left out! = 0.33 in US data
10
Properties
• More inputs give more output
– positive marginal products of capital and labor"Y
"K> 0,
"Y
"L> 0
• Output e!ect of additional inputs falls
– diminishing marginal products
"2Y
"K2< 0,
"2Y
"L2< 0
• If we double all inputs, we double output
– constant returns to scale
#Y = AF (#K,#L), # > 0
11
Production function
0 50 100 150 200 250 300 350 40020
40
60
80
100
120
140
160
Y = AF (K,L), holding labor fixed
K
Y
12
Capital
• Meaning: physical capital, plant and equipment
• Why does it change?
– depreciation– destruction– new investment
• Mathematical version
Kt+1 !Kt = It ! $Kt
for simplicity, a constant rate of depreciation, $
• Adjustments for quality?
13
Capital per worker
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002
USA
FRA
CHN
IND
KOR
ARG
IRL
JPN
in 2000 dollars at ‘PPP’
Source: Penn World Tables, 6.2
14
Labor
• Meaning: units of work e!ort
• Why does it change?
– population growth– fraction of population employed, hours worked
• Adjustments for quality?
– skills? education? other factors?– if call this H for human capital, then
Y = AF (K, HL)= AK!(HL)1!!
= (AH1!!)K!L1!!
(‘augmented production function’)
15
Employed as fraction of population
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002
USA
FRA
CHN
IND
KOR
ARG
IRL
JPN
Source: Penn World Tables, 6.2
16
Human capital?
0
2
4
6
8
10
12
14
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004
USA
CHN
IND
KOR
ARG
IRL
JPN
average years schooling, 15+ population
Source: Barro-Lee educational attainments database
17
Productivity
• Standard number
– average product of labor, YL
• Our preferred number
– total factor productivity, A = YF (K,L)
• How do we measure it?
– use the production function to solve for A
F (K, L) = K!L1!! " A =Y
L÷
!K
L
"!
• US example
– YL = 67, 865 and K
L = 177, 007 and ! = 13 so A = 1, 208
18
Simple productivity calculations
Country GDP/person GDP/worker Capital/worker TFP
Argentina 10,170 24,284 57,858 627Brazil 7,204 15,461 37,603 461China 4,969 8,283 18,015 315Egypt 4,759 12,051 9,593 567France 25,663 56,908 166,636 1,034India 2,990 6,724 7,891 337Ireland 28,248 65,924 131,131 1,297Japan 24,036 45,030 180,283 797Korea 17,596 33,783 113,711 697Mexico 7,938 18,627 47,089 515United States 36,098 67,865 177,007 1,208Zimbabwe 2,438 5,416 14,064 224
Source: Penn World Tables, 6.2
19
What have we learned so far?
• links output to inputs and productivity
Y = AK!L1!!
• Capital input, K
– plant and equipment– a consequence of investment, I
• Labor input, L
– population growth– participation and hours– skills etc, H
• TFP, A, can be inferred from data on output and inputs
20
Rest of this class
• Robert Solow’s growth model
– pieces of the model– dynamics– ‘steady state’
• Main implication: productivity is key
– long-run growth due to productivity increases, not capital investment– cross-country variation in income per capita due to variation in productivity
21
Solow’s growth model
• Just four equations
– production function
Y = AF (K, L) = AK!L1!!
– national income accounting
C + I = Y
– constant savings rate s
S = I = sY
– capital accumulation
Kt+1 !Kt = It ! $Kt
22
Capital accumulation dynamics
• Put it all together to get
Kt+1 !Kt = sAK!t L1!!
t ! $Kt
• Suppose constant labor force, Lt = L. Let k = K/L, y = Y/L, etc. Then
kt+1 ! kt = sAk!t ! $kt
• Given parameters (s,A, !, $) and initial capital k0, see how kt moves through time
– capital increasing if investment is bigger than depreciation– capital decreasing if investment is smaller than depreciation
• A picture might help . . .
23
Savings versus depreciation
0 50 100 150 200 250 300 350 4000
20
40
60
80
100
120
140
160
y = Ak!
sAk!
!k
k
y
24
‘Steady state’
• If new investment equals depreciation, capital per worker is constant
0 = sAk! ! $k
• Solving this we get
k =!
sA
$
" 11!!
• This is the ‘steady state’ level of capital per worker
– higher when s,A, ! are higher– lower when $ is higher
• Capital stock ‘gravitates’ to this steady state value
• To see why, let’s look at that picture again . . .
25
Gravitation to ‘steady state’
0 50 100 150 200 250 300 350 4000
5
10
15
20
25
30
35
40
k
sAk!
!k
k
26
Convergence hypothesis
• Capital per worker converges to steady state
k =!
sA
$
" 11!!
• Therefore output per worker converges to steady state too
y = Ak!
= A1
1!!
#s
$
$ !1!!
• Implication
– capital investment does not explain long run growth of output per worker(that we see in data)
– higher savings rate s increases long run level of output, not growth rate
27
Higher savings rate
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
saving per worker
higher saving per worker
depreciation per worker
k
k!
28
A historical lesson
• Soviet economic growth very impressive in late 1950s – early 1960s
– 8-9% per year, roughly three times faster than US– concern about strategic implications of Soviet industrial ‘might’
• But ‘perspiration not inspiration’
– too much reliance on physical capital accumulation– generates growth in short run– but cannot generate sustained long run growth– diminishing returns
29
But what about long-run growth
• In this theory, capital investment alone cannot generate long run growth
– why? diminishing returns
• What can generate long run growth in output per worker?
– growth in total factor productivity A
– if A grows at 2%, then in the long run output per worker will also grow at 2%
• But where does TFP come from?
– in the Solow model, TFP is a ‘black box’– guess what we will talk about next?
30
Long run growth across countries
0
5000
10000
15000
20000
25000
30000
35000
40000
1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002
USA
FRA
CHN
IND
KOR
ARG
IRL
JPN
in 2000 dollars at ‘PPP’
Source: Penn World Tables, 6.2
31
Levels of output across countries
0.000
20.000
40.000
60.000
80.000
100.000
120.000
140.000
160.000
180.000
GDP per capita as percentage of US
Source: United Nations
32
Levels of output
• Solow model predicts output per worker
y = A1
1!!
#s
$
$ !1!!
– cross-country variation in y explained by four parameters (!, s, $, A)– we can measure (!, s, $)– they do not vary enough across countries to explain enormous variation in y
– again, leaves TFP as explanation
33
What have we learned today?
• Production
– production function links output to inputs and productivity– ‘total factor productivity’ can be inferred from data on output and inputs
• Solow model
– countries with higher savings rates have higher levels of output per worker– but capital investment cannot generate long run growth in output per worker– need productivity growth if output per worker is to grow in long run– need cross-country variation in productivity levels to explain enormous dis-
parities in data
34
