Production Functions
Students Should Be Able To
Use the Cobb-Douglas production function
to calculate:

1.
2.
3.

Output as a product of inputs
marginal and average factor products as a
product of inputs or output and inputs
Total Factor Productivity Growth
Construct input demand curve using
marginal products.
HK vs. USA




In 1998, USA’s real GDP per capita was
about 1/3 greater than Hong Kong.
But average US growth rate over the
preceding 50 years was about 2% per year.
Average HK growth rate was 4.5% per year.
If these two growth performances continue, in
50 years HK GDP per Capita would be 2.5
times that in the USA.
Will this occur?
Sources of Growth



Because dividends are limited by capital
income, dividend growth is determined by
GDP growth.
Nominal GDP growth can be divided into two
parts: 1) inflation; 2) real GDP growth.
Real GDP growth can be divided into two
parts: 1) population growth; 2) growth in real
GDP per capita.
Chinese GDP per Capita by
Dynasty (1990 US$ per person)
Year
Dynasty China
Europe
50AD
Han
400
450
960AD
Tang
400
350
1280
Sung
600
450
1400
Ming
600
450
1820
Qing
600
1122
Industrial Age




In Britain in late 1700’s a new economic
began to take shape
Key characteristic of this age was use of
machinery (or capital) to augment labor.
Relatively large growth in output
Population grows more slowly than output
GDP per % of
capita,
ACNZUS
1950
GDP per % of
capita,
ACNZUS
1992
ACNZUS
9255
100%
20,850
100%
W. EUROPE
5126
55.4
17,387
83.9
LATIN
AMERICA
2487
26.8
4,820
23.1
ASIA
765
8.3
3,252
15.6
JAPAN
1873
20.4
19425
93.2
HONG KONG
1962
21.1
17,120
82.1
AFRICA
830
8.9
1284
6.1%
Post-War Facts


Large Income Differences Across Countries
Convergence to World Leaders in Two Areas:
Europe and East Asia



Low initial level of Japan and Europe due to destruction of
capital stock
Divergence from World Leaders in Africa and Latin
America
Small Gains in Asia as Whole

Interesting dynamics amongst East Asian economies.
Population Growth:
Hong Kong and Singapore
6000
5000
4000
3000
2000
1000
0
Hong Kong
19
90
19
50
Singapore
19
13
18
70
Thousands
Population
Population:
China and India
1200
1000
800
600
400
200
0
China
19
90
19
50
India
19
13
18
70
Millions
Population
GDP per Capita
GDP per Capita
6000
China
4000
India
2000
0
19
13
19
32
19
50
19
73
19
90
1990 US$1000
8000
Burma
Philippines
Thailand
GDP per Capita pt. 2
GDP per Capita
20000
Hong Kong
15000
Singapore
10000
S. Korea
5000
Taiwan
0
19
13
19
50
19
73
19
90
1990 US$1000
25000
Production Functions
Production Function

An economy’s value added is produced by its




Stock of capital equipment denoted Kt
Labor force denoted Lt
Technology/Worker Efficiency denoted Zt
Cobb-Douglas production function
GDPt   K t  ( Z t Lt )
a

1 a
The parameter, a, is sometimes referred to as
capital intensity, i.e., the greater is a, the more
important capital is in production.
Advantages of Cobb-Douglas
Production Function

Constant Returns to Scale
If you increase both capital and labor by a factor
of N, then you will also increase output by a factor
of N
a
a
1 a
GDPt   Kt  ( Z t Lt )  N  GDPt   N  K t  ( Z t N  Lt )1 a


Implications for Country Size: Output per
capita depends only on capital per capita and
labor per capita, not on population size itself.
a
GDPt  K t 
Lt 1 a

)
 (Zt
POPt  POPt 
POPt
Marginal Product


The marginal product of a factor is the extra
output that results from the extra use of the
factor relative to the size of the increase in
factor use.
GDP
GDP
MPL 
MPK 
L
K
Marginal products of very small increases in
factor use can be derived with derivatives
GDP
MPL 
 (1  a )  K a X 1 a L a
L
GDP
MPK 
 a  K a 1 ( XL)1 a
K
Production as a function of
labor (holding capital fixed)
GDP
GDP
L
GDP
L
L
Marginal Product of Labor
GDP
L
L
Advantages of Cobb-Douglas
Production Function Pt.2

Diminishing returns


Holding capital & technology constant, the
marginal product of labor is a decreasing function
of labor.
Holding labor & technology constant, the marginal
product of capital is a decreasing function of
capital.
Average Product

We define average productivity of a factor as
the ratio of output to the level of factor use
GDP
APL 
L

GDP
APK 
K
Under Cobb-Douglas, the marginal product is
proportional to average product.
GDP
MPL  (1  a )  K X L  (1  a )
L
GDP
MPK  a  K a 1 ( XL)1 a  a
K
a
1 a
a
Marginal Product =
Marginal Cost



Profit maximization
suggests that the marginal
product of a factor should
equal its real cost.
The real cost of labor is the
real wage, the dollar wage
rate divided by the price
level.
Wt
MPL 
Pt
A firm can raise its
profits by increasing
labor as long as the
cost of the extra labor is
less than the extra
goods produced. Since
the extra goods
produced drops as
more labor is added,
firms will hire more
labor until the marginal
product flls as low as
the real wage.
Factor Shares


Under a Cobb-Douglas production function,
labor compensation is a constant share of
value added.
Labor compensation is the product of the
wage rate and the quantity of labor WtLt.
Wt
GDPt
 (1  a)
 Wt Lt  (1  a ) PGDP
t
t
Pt
Lt

Capital income is also a constant share of
value added.
EDIBTA  PGDP
t
t  Wt Lt  aPGDP
t
t
Growth Rate Rules of Thumb
1.
If Xt = Yt x Zt then g  g  g
2.
If X t  (Yt ) then
3.
X
t
a
If
Yt
Xt 
Zt
then
Y
t
Z
t
gtX  a  gtY
g g g
X
t
Y
t
Z
t
Productivity Growth
When economists study productivity, they
often decompose output into two parts:

1.
2.

F: output due to the accumulation of the factors
of production, capital and labor;
TFP: total factor producivity or output due to
advances in technology.
GDPt  TFPt  Ft
Using Cobb-Douglas, it is easy to do this
Ft   Kt  ( Lt )
a
1 a
TFPt  Z
1 a
t
TFP Growth

Total factor productivity is implicitly defined as
the ratio of output to a combination of the
factors of production.
GDPt
TFPt 

Ft
TFP growth is the difference between output
growth and the growth of the combined factor.
g
TFP
t
g
GDP
t
g
F
t
Measuring F

1.
Measuring the growth in F has three parts
Measuring a. Under Cobb-Douglas, we can
measure a, from labor’s share of income.
Wt  Lt
1  a  
PGDPt
2.
3.
Measuring L. Government statistical bodies
periodically measure the stock of labor using
surveys of employers or households.
Measuring K: Perpetual Inventory Method. Guess
at initial capital stock. Use constant dollar
measures of investment and estimates of
depreciation to recursively calculate investment.
Kt 1  Kt  DPNt  It
TFP Growth

The growth rate of factor is
F
K
L
gt  a  gt  (1  a)  gt

TFP growth can be calculated as
gtTFP  gtGDP   a  gtK  (1  a )  gtL 

Growth accounting attributes those parts of
growth that are due to its different elements.
Growth Due to
Capital
(a) gk
Labor
(1-a)gL
TFP
gTFP
TFP Growth in HK & Singapore