Production Functions &
Productivity
Long-term
Economic Value Added & GDP
• An individual production unit (typically a firm)
calculates its value added as the market value of its
sales plus change in its inventories minus the cost of
material inputs which includes energy, raw materials
and unfinished products but does not include the cost
of durable machinery and structures or labor.
• The sum of the value added of all the production units
located within the areas of a region is called (current
price) Gross Domestic Product.
• Designate this value at time t, PYt.
Real GDP
•
We calculate constant price or real GDP by :
1. summing the value added of small sub-groups of
goods (such as shoes, TV’s etc.)
2. adjusting the value added of each sub-group by
multiplying by the ratio of the prices of those goods
relative to some base year
3. Then adding up the adjusted sum of the value added
across different groups.
•
We designate real GDP as Yt.
GDP, Income and Welfare
• Value added generated by firms is equal to the
revenue they have available to pay to workers, renters,
creditors.
• Income available to people in the economy is
equivalent to the income they have to purchase goods
and services.
• GDP per person closely connected to the lifestyle
people can enjoy. Perhaps the dominant fact in
macroeconomics is the wide variation at the national
level in GDP per person.
GDP per Capita 2002
Source: Groningen Global Development Center http://www.ggdc.net/
Guatemala
Kenya
India
China
Thailand
Brazil
South Korea
Hong Kong
USA
Japan
Germany
0
5,000
10,000
15,000
20,000
1990 US$
25,000
30,000
35,000
World Distribution of Income
GDP per Capita 2002
GDP per Capita 2002
Middle Income
Philippines
40,000
35,000
30,000
China
Thailand
Malaysia
Mexico
Brazil
0
2,000
4,000
6,000
8,000
10,000
2002 US$
20,000
15,000
GDP per Capita 2002
10,000
Sub-Saharan
Africa
5,000
0
USA
Hong
Kong
Japan
Germany
High Incom e Countries
Korea
Portugal
Low Income
2002 US$
25,000
Burma
India
0
500
1,000
1,500
2002 US$
2,000
2,500
3,000
Why are some countries rich and some
poor?
10 Richest, 2000
1. Luxembourg
2. USA
3. Norway
4. Canada
5. Singapore
6. Denmark
7. Switzerland
8. Hong Kong
9. Ireland
10. Australia
48,968
35,619
32,057
28,731
28,644
28,539
28,209
27,893
27,197
27,193
10 Poorest, 2000
Tanzania490
Burundi
Ethiopia 720
Sierra Leone
Guinea-Bissau
Malawi
Nigeria
Zambia
Madagascar
Niger
619
734
738
808
826
841
877
902
Unit: US$
Source: Penn World Tables, http://pwt.econ.upenn.edu
Why are some countries poor and some
countries rich?
• Large differences in the income per capita are
associated with differences in living standards
and other measures such as longevity, child
mortality, literacy etc. (these may also be
associated with income distribution)
Human Development Index, 2002
GDP and United Nations Human Development Index
1.2
1
HDI
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
GDP
Source: UN Human Development Reports, http://hdr.undp.org/statistics/data/
Social Welfare
Development and Welfare
120
100
80
60
40
20
0
Life Expectancy
Adult Literacy
High human development
Low human development
Enrolment Rate
Medium human development
Malthusian Economy
• Prior to 1700, the vast majority of the
economy was in agriculture.
• There were many advances in techniques for
producing goods and the amount of food
grown per acre of land increased substantially.
• All extra food went to additional people, not to
improving living standards.
GDP per capita through history
Year
-5000
-1000
1
1000
1500
1800
Population
5
50
170
265
425
900
GDP per Capita
130
160
135
165
175
250
Macroeconomics by J. Bradford DeLong, Chap. 5
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
The World Economy, A Millienial Perspective by Angus Madisson
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
Growth by Region
GDP 1950
ACNZUS
9,288
% of ACNZUS GDP 1998
100%
26,146
% of ACNZUS Growth Rate
100%
2.16%
W. EUROPE
4,594
55.4
17,921
69%
LATIN
AMERICA
2,554
26.8
4,531
17%
ASIA
AFRICA
635
852
8.3
8.9
2,936
1,368
11%
5%
The World Economy, A Millienial Perspective by Angus Madisson
2.84%
1.19%
3.19%
0.99%
East Asia
% of
GDP 1950
% of
ACNZUS
GDP 1999
ACNZUS
Growth
Rate
Japan
South
Korea
1,926
21%
20,431
78%
4.82%
770
8%
13,317
51%
5.82%
Hong Kong
2,218
24%
20,352
78%
4.52%
936
10%
15,720
60%
5.76%
2,219
24%
23,582
90%
4.82%
Thailand
817
9%
6,398
24%
4.20%
Malaysia
1,559
17%
7,328
28%
3.16%
Taiwan
Singapore
The World Economy, A Millienial Perspective by Angus Madisson
Production Function
• To describe the determinants of production,
economists use as a tool an algebraic function
which maps inputs into output.
• Macroeconomists use an aggregate production
function which maps aggregate inputs
typically including capital, K, labor, L, and
sometime other inputs.
Yt  Ft ( Kt , Lt )
Inputs
• Capital, Kt : Sum total of the structures
(residential and non-residential) used to produce
goods and services.
– Sometimes, especially in short-term applications, we
might adjust capital input for utilization.
• Labor, Lt : Sum total of labor units. Ideally, we
would use labor hours worked, but due to lack
of measurement, we sometimes use # of workers.
Productivity of Inputs
Average
Marginal (Extra
(Output per Unit Output per Unit
of Input)
of Input)
Capital
Productivity
Y
Labor
Productivity
Y
K
L
MPK  Y
MPL Y
K
L
Capital Productivity
Capital Productivity
0.63
0.62
0.61
0.6
0.59
0.58
0.57
0.56
0.55
0.54
USA
EU
Productivity Catch Up: Europe
Source: Groningen Growth & Development Center
U.S.A
% of
1950 USA
% of
2003 USA
Growth
Rate
12.00 100.0%
33.97 100.0%
2.00%
France
5.63
46.9%
37.75
111.1%
3.46%
Germany
4.36
36.3%
30.01
88.3%
3.95%
UK
7.49
62.4%
28.01
82.5%
2.91%
Spain
2.60
21.7%
22.21
65.4%
4.94%
1990 US$, Average Output per Hour (Y/L)
Productivity Catch Up:
Latin America
Source: Groningen Growth & Development Center
1950
U.S.A
12.00
% of
2003
USA
100.0% 33.97
% of
Growt
USA
h Rate
100.0% 2.00%
Argentina
6.16
51.4%
10.57
31.1%
1.04%
Brazil
2.48
20.7%
7.81
23.0%
2.21%
Chili
4.66
38.9%
14.07
41.4%
2.12%
Mexico 3.56
29.7%
10.24
30.1%
2.03%
Productivity Catch Up: East Asia
Source: Groningen Growth & Development Center
1950
% of
USA
2003
% of
USA
Growth
Rate
U.S.A
12.00
100.0%
33.97
100.0%
2.00%
Japan
2.30
19.2%
24.78
73.0%
4.57%
1973
% of
USA
2003
% of
USA
Hong
Kong
7.49
35.0%
22.28
65.6%
4.74%
Korea
3.64
17.0%
14.25
42.0%
5.93%
Singapor 6.80
e
31.8%
19.63
57.8%
4.61%
Taiwan
20.4%
18.77
55.2%
6.33%
4.37
Workhorse Production Function
• Cobb-Douglas Function
Yt
 (Kt )  ( At Lt )1
0   1
• Technology, At, represents the way the
production possibilities of a country change
over time through the development of new
inventions and techniques for production
Constant Factor Intensities
• Marginal Product is proportional to average
productivities.
Kt
MPL  (1   )
Lt

At1
Kt
MPK  
Lt
 1
( Kt )  ( At Lt )1
 (1   )
Lt
At1
Yt
 (1   )
Lt
( Kt )  ( At Lt )1

Kt
Yt

Kt
Constant Returns to Scale
• CRTS means that if you multiply all of your
inputs by some factor, κ, you will also multiply
your output by the same factor.
Yt  Ft ( Kt ,  Lt )

1


) ( A  L )
( Kt
t t
    1  ( Kt )  ( At Lt )1

1


  ( Kt )  ( At Lt )
Implications of Constant Returns to
Scale
1. GDP per Capita depends only on inputs per
capita and not the size of the economy
1
Yt  Ft ( Kt ,  Lt ),   POP
t
Yt
POPt
 Ft (
Kt
POPt
,
Lt
POPt
)
2. Capital and labor productivity are functions
of the capital labor ratio.

 1
Yt Kt
K
Yt
1



At
 t
At1
Lt Lt
K t Lt
Properties of Cobb-Douglas
Production Function
• Inputs have positive effects on production.
Marginal product of capital and labor are
positive.

 1
K
Y
K
Y
1


t
 MPL  (1   )
At
 t
At1
L
Lt
K
Lt
0
• Inputs have diminishing returns
 2Y MPL
(1   )





L2
L
Lt
Kt 
Lt
 2Y MPK
(1   )





K 2
K
Kt
At1  0,
Kt  1 1
At
Lt
0
Diminishing Marginal Returns
Y
ΔY’
ΔK
ΔY
ΔK
K
Marginal Product of Capital
MPK = ΔY
ΔK
MPK
K
Positive Cross Products
• Marginal product of labor is increasing in
capital and marginal product of capital is
increasing in labor.
MPL MPK
Y
(1   )


 
K
L
L K
Kt
2
Kt
Lt

At1  0,
Marginal Product Curves, Shifted by
Technology and Other Factor
MPL
MPK
K↑, A↑
L
L↑, A↑
K
Labor Shares
• Assume Price Taking by competitive firms in
the economy. Firms sell goods at a competitive
price Pt.. Firms hire workers at a dollar wage
rate Wt . Also capital is assumed to rent in
capital rental market at rate Rt.
• Profits are
 t  PY
t t  Wt Lt  Rt K t
 t  PF
t t ( K t , Lt )  Wt Lt  Rt K t
Marginal Analysis
• The firms must choose a level of capital and
labor to maximize profits.
• Optimal labor: Marginal cost is the wage
payment. Marginal benefit of hiring labor is
the extra revenue generated which is goods
produced multiplied by the price at which
goods are sold.
Yt
Wt
Pt
Lt
 Wt  MPL 
Pt
Factor Demand Curves
• The marginal cost of hiring capital is rental
price, Rt. The marginal benefit is price times
marginal product of capital.
Yt
Rt
Pt
 Rt  MPK 
K t
Pt
• We can use the marginal product curves as
demand curves to graphically analyze the
optimal demand for capital or labor.
Marginal Product Curves, Shifted by
Technology and Other Factor
MPL
MPK
K↑, A↑
L↑, A↑
R
W
P
P
L*
L
K*
K
Constant Factor Shares
• Under a Cobb-Douglas production function
and price-taking, factor shares of income are
constant.
• Labor share of income
Wt Lt

PY
t t
Wt
Pt
Y
 L
Yt
Lt
(1   )
Yt
Lt

Yt
Lt
Yt
Lt
 (1   )
• Capital Share of Income
Rt K t

PY
t t
Rt
Pt
Yt
Kt
Y
 K
Yt
Kt

Yt

Kt
Yt
Kt

Mar-98
Mar-96
Mar-94
Mar-92
Mar-90
Mar-88
Mar-86
Mar-84
Mar-82
Mar-80
Mar-78
Mar-76
Mar-74
Mar-72
1- α ≈2/3
Labor Share of Income
0.8
0.7
0.6
0.5
0.4
USA
JAPAN
0.3
0.2
0.1
0
Implications of Constant Returns to
Scale
• Add up factor shares
Wt Lt Rt K t Wt Lt  Rt K t


   (1   )  1
PY
PYt Qt
PY
t t
t t
Profits, Π = 0
• This is true for PC & CRTS in general,
Analytical Exercise
• Q: How does an increase in immigration increase or
reduce real wages.
– Depends how capital is supplied?
• Treat labor supply as fixed. Then assume that there is
an increase in labor supply through immigration. If
capital is supplied inelastically, this will reduce wages.
• However, if capital supply is fixed, the marginal
product of capital will shift up, increasing the rental
price of capital.
Labor Supply increases, capital supply fixed,
wages fall and capital rental prices rise.
L
MPL
L↑
MPK’
W*
W **
R*
P
P
P
MPK
L
K
K
Capital Supply
• If the high capital rental price induces
investors to increase their ownership of capital.
• Assume that capital is supplied perfectly
elastically. That is more capital would become
available if the rental rate ever raised above .
R
P
R
P
Labor Supply increases, capital supply elastic,
wages and capital rental prices unchanged.
MPL
L'
L
L↑
K↑
MPK
W
P
MPK’
R
P
*
W **
P
K
L
K*
K**
Why are wages unchanged.
• More capital will be supplied as long as capital-labor
ratio is below target level.
• As labor supply increases, capital increases to bring
capital-labor ratio back to target.
• But capital-labor ratio will determine the productivity
of labor and real wage.
• Thus, if new workers come into the market place,
they will make it more profitable to add capital until
wages reach their previous level.
• Under capital accumulation, real wages are
determined by required rental price for capital and
technology level.