Notes Chapters 12
Point of the class
Math





Growth rates
Ratio scales
Logs and exponents
Algebraic manipulation
Partial derivatives and maximization
Model building

Modifying the Solow model to include population growth and technological progress
Economic concepts and intuition

Capital/output ratio
Definitions

Rule of 70

Growth accounting
Homework Problems

On the webpage.

Capital/output ratio

Capital deepening
Properties of Growth Rates
 Growth rates: how they work
Year
𝑥𝑡+𝑛 = 𝑥𝑡 (1 + 𝑔)𝑛
o
This allows us to calculate average growth rates over long
period of time.
𝑥𝑡+𝑛 1/𝑛
𝑔=(
) −1
𝑥𝑡

1870
1929
1950
2004
US
Average
Income growth
rate
2525
7,100
1.77%
11,720
36,880 2.15%
Rule of 70
Property of logs
2𝑥𝑡 = 𝑥𝑡 (1 + 𝑔)𝑛
ln(𝑥 𝑛 ) = 𝑛 ln(𝑥)
2 = (1 + 𝑔)𝑛
ln(𝑎𝑏) = 𝑙𝑛(𝑎) + 𝑙𝑛(𝑏)
𝑙𝑛2 = 𝑙𝑛(1 + 𝑔)𝑛
𝑎
ln ( ) = 𝑙𝑛(𝑎) − 𝑙𝑛(𝑏)
𝑏
𝑙𝑛2 = 𝑛 × 𝑙𝑛(1 + 𝑔)
𝑙𝑛(1 + 𝑔) ≈ 𝑔
0.7 ≈ 𝑛 × 𝑔
𝑛≈
o
o
70
𝑔 × 100
Suppose a quantity is growing at a constant rate. To figure out the number of years for
it to double, divide 70 by the growth rate times 100.
So if it’s growing at 2% per year, it will double every 35 years, regardless of the starting
point (the xt cancelled out).
Bank account balance of $100
At times 0, 1, 2, 12, 24, 48, 60,
what is the bank balance
Assuming interest rate of 1%
Assuming interest rate of 6%

Ratio (logarithmic) scales allow one to see data more easily if their rate of growth is constant.
Plot both on a standard scale and on a ratio scale

Rules of Growth Rates
𝑥
If 𝑧 = 𝑦, then 𝑔𝑧 = 𝑔𝑥 − 𝑔𝑦
If 𝑧 = 𝑥𝑦, then 𝑔𝑧 = 𝑔𝑥 + 𝑔𝑦
If 𝑧 = 𝑥 𝑎 , then 𝑔𝑧 = 𝑎𝑔𝑥
𝑌 = 𝐴𝐾 𝛼 𝑁1−𝛼
A is growing at a rate of g(A)
K grows at g(K)
𝐾 𝛼
𝑦 = 𝐴 ( ) = 𝐴𝑘 𝛼
𝑁
Using the second rule
𝑔(𝑦) = 𝑔(𝐴) + 𝑔(𝑘 𝛼 )
N grows at g(N)
Using the first rule
Using the second rule
𝑔(𝑌) = 𝑔(𝐴) + 𝑔(𝐾 𝛼 ) + 𝑔(𝑁1−𝛼 )
𝑔(𝑦) = 𝑔(𝐴) + 𝑔(𝐾 𝛼 ) − 𝑔(𝑁 𝛼 )
Using the third rule
Using the third rule
𝑔(𝑦) = 𝑔(𝐴) + 𝛼[𝑔(𝐾) − 𝑔(𝑁)]
𝑔(𝑌) = 𝑔(𝐴) + 𝛼𝑔(𝐾) + (1 − 𝛼)𝑔(𝑁)
Suppose xt=(1.05)t and yt=(1.02)t
Calculate the growth rate of z if
z=xy, z=x/y, z=x1/3y2/3
0
5000
10000
Real GDP per capita
15000
20000
Why does GDP per worker increase?
1950
1960
1970
1980
year
1990
2000
Real GDP per capita, Korea
Real GDP per capita, Nicaragua
𝑦 = 𝐴𝑘 1/3

Increases in capital per worker.
o But this is limited by diminishing returns: Saving rates determine the level but not the
growth rate of output
Empirical Test
o
o
Suppose A=1, so only the level of capital-per-worker determined output-per-worker
 Get data for capital-per-worker and output-per-worker. Divide all the values by
the corresponding values for the US, to see if it yields commonsensical
predictions.
 Take the cubic root of capital-per-worker, and plot it.
 Why the exponent of 1/3? Because empirically, most countries are pretty close
to 1/3 contribution of capital to production.
Why do most countries underperform?
SWITZERLAND
2.097544
Predicted Real GDP
per worker, 1985
US=1
𝒚 = 𝒌𝟏/𝟑
1.28008
NORWAY
1.499282
1.144531
0.85099
LUXEMBOURG
1.419515
1.123863
0.911168
FINLAND
1.289591
1.088472
0.701536
GERMANY, WEST
1.192749
1.060514
0.806678
CANADA
1.154052
1.048919
0.921973
AUSTRALIA
1.131997
1.042194
0.857236
BELGIUM
1.108104
1.034809
0.808839
FRANCE
1.062523
1.020421
0.801113
SWEDEN
1.046817
1.015368
0.784537
NEW ZEALAND
1.019582
1.006485
0.770772
Country
Capital Stock per
worker, 1985
US=1
Actual Real GDP per
worker, 1985
US=1
0.883521
U.S.A.
1
1
1
AUSTRIA
0.992749
0.997577
0.705592
NETHERLANDS
0.97995
0.993272
0.845484
DENMARK
0.978647
0.992831
0.706302
ITALY
0.939582
0.979441
0.804813
JAPAN
0.939215
0.979313
0.557085
ISRAEL
0.74543
0.906711
0.649824
GREECE
0.740217
0.904593
0.481603
SPAIN
0.729524
0.900216
0.626617
IRELAND
0.691729
0.884393
0.568244
VENEZUELA
0.682339
0.880373
0.543528
TAIWAN
0.641404
0.862403
0.375958
U.K.
0.58934
0.838408
0.680253
PANAMA
0.548605
0.818628
0.297161
ICELAND
0.534369
0.811485
0.688394
SYRIA
0.52401
0.806207
0.508125
ECUADOR
0.519165
0.803714
0.284611
MEXICO
0.468371
0.776599
0.504277
COLOMBIA
0.427402
0.753261
0.274576
ARGENTINA
0.40381
0.739138
0.442678
KOREA, REP.
0.402206
0.738158
0.306693
POLAND
0.375439
0.721406
0.239144
HONG KONG
0.374937
0.721085
0.486843
IRAN
0.357093
0.709459
0.409881
PORTUGAL
0.317561
0.682248
0.335761
PERU
0.316792
0.681697
0.240979
SRI LANKA
0.275288
0.650523
0.165675
YUGOSLAVIA
0.257109
0.635876
0.337951
CHILE
0.235923
Predicted Real GDP
per worker, 1985
US=1
𝒚 = 𝒌𝟏/𝟑
0.617908
TURKEY
0.233551
0.615829
0.209899
Country
Capital Stock per
worker, 1985
US=1
Actual Real GDP per
worker, 1985
US=1
0.28914
BOLIVIA
0.233484
0.615771
0.166445
ZIMBABWE
0.192147
0.577047
0.096528
DOMINICAN REP.
0.176508
0.560947
0.209632
BOTSWANA
0.15462
0.536729
0.201048
HONDURAS
0.15061
0.532048
0.137702
SWAZILAND
0.150409
0.531812
0.154664
PHILIPPINES
0.136575
0.51498
0.125181
THAILAND
0.135372
0.513463
0.140633
GUATEMALA
0.133166
0.51066
0.217802
JAMAICA
0.124478
0.499303
0.139893
MOROCCO
0.093668
0.454147
0.190244
MAURITIUS
0.079967
0.430827
0.221236
ZAMBIA
0.059649
0.390722
0.071012
MADAGASCAR
0.057277
0.385472
0.050528
INDIA
0.05721
0.385322
0.080484
KENYA
0.039666
0.34104
0.059616
IVORY COAST
0.039131
0.339501
0.110707
NIGERIA
0.036859
0.332798
0.085072
PARAGUAY
0.028371
0.304994
0.184738
NEPAL
0.02533
0.293683
0.066424
MALAWI
0.015673
0.250253
0.034662
SIERRA LEONE
0.007185
0.19296
0.071367
1.2
SWITZERLAND
1
NORWAY
LUXEMBOURG
FINLAND
GERMANY,
WEST
CANADA
AUSTRALIA
BELGIUM
FRANCE
SWEDEN
NEW
ZEALAND
U.S.A.
AUSTRIA
NETHERLANDS
DENMARK
JAPAN
ITALY
.2
.4
.6
.8
ISRAEL
GREECE
SPAIN
IRELAND
VENEZUELA
TAIWAN
U.K.
PANAMA
ICELAND
SYRIA
ECUADOR
MEXICO
COLOMBIA
ARGENTINA
KOREA,
REP.
HONG
POLAND
KONG
IRAN
PORTUGAL
PERU
SRI LANKA
YUGOSLAVIA
CHILE
TURKEY
BOLIVIA
ZIMBABWE
DOMINICAN
REP.
BOTSWANA
HONDURAS
SWAZILAND
PHILIPPINES
THAILAND
GUATEMALA
JAMAICA
MOROCCO
MAURITIUS
ZAMBIA
MADAGASCAR
INDIA
KENYA
IVORY
COAST
NIGERIA
PARAGUAY
NEPAL
MALAWI
SIERRA LEONE
0
.5
1
1.5
Capital per worker (US=1)
Penn World Table 5.6. 1985 data
2

The red line is what the model would predict if countries only differed on capital stock levels.
1
1.5
Penn World Tables 5.6, in KAPW-RGDPW.dta
SWITZERLAND
0
.5
U.S.A.
CANADALUXEMBOURG
AUSTRALIA
NORWAY
NETHERLANDS
BELGIUM WEST
GERMANY,
ITALY
FRANCE
NEWSWEDEN
ZEALAND
AUSTRIA
FINLAND
ICELAND
U.K. ISRAEL DENMARK
SPAIN
IRELAND JAPAN
VENEZUELA
SYRIA GREECE
HONGMEXICO
KONG
ARGENTINA
IRAN
TAIWAN
YUGOSLAVIA
PORTUGAL
KOREA,
REP.
PANAMA
CHILE
ECUADOR
COLOMBIA
PERU
POLAND
MAURITIUS
GUATEMALA
DOMINICAN
TURKEY
REP.
BOTSWANA
MOROCCO
PARAGUAY
BOLIVIA
SRI LANKA
SWAZILAND
THAILAND
JAMAICA
HONDURAS
PHILIPPINES
IVORY
COAST
ZIMBABWE
NIGERIA
INDIA
SIERRA
LEONE
ZAMBIA
NEPAL
KENYA
MADAGASCAR
MALAWI
0
.5
1
1.5
Capital per worker (US=1)
Real GDP per worker, actual
2
RGDP, predicted
Penn World Table 5.6. 1985 data
o
Improvements in technology make a difference
𝑇𝑜𝑡𝑎𝑙 𝐹𝑎𝑐𝑡𝑜𝑟 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 𝐴 =
𝑦
𝑘 1/3
Capital Stock per
worker, 1985
US=1
Predicted Real GDP
per worker, 1985
US=1
𝒚 = 𝒌𝟏/𝟑
Actual Real GDP per
worker, 1985
US=1
𝒚 = 𝑨𝒌𝟏/𝟑
U.S.A.
1
1
1
Implied TFP
(A)
The Residual
𝒚
𝑨 = 𝟏/𝟑
𝒌
1
ITALY
0.939582
0.979441
0.804813
0.821707
U.K.
0.58934
0.838408
0.680253
0.811363
Country
SPAIN
0.729524
0.900216
0.626617
0.696074
SWITZERLAND
2.097544
1.28008
0.883521
0.690208
MEXICO
0.468371
0.776599
0.504277
0.649341
JAPAN
0.939215
0.979313
0.557085
0.568853
SIERRA LEONE
0.007185
0.19296
0.071367
0.369855
ECUADOR
0.519165
0.803714
0.284611
0.354119
INDIA
0.05721
0.385322
0.080484
0.208876
.2
.4
.6
.8
1
𝑦 = 2(𝑘)1/3
U.S.A.
CANADA
ICELANDNETHERLANDS
AUSTRALIA
ITALY
U.K.
LUXEMBOURG
FRANCE
BELGIUM
SWEDEN
NEW
ZEALAND
GERMANY, WEST
NORWAY
ISRAEL
DENMARK
AUSTRIA
SPAIN
SWITZERLAND
HONG KONG
MEXICO
IRELAND FINLAND
SYRIA
VENEZUELA
PARAGUAY
ARGENTINA
IRAN
JAPAN
YUGOSLAVIAGREECE
MAURITIUS
PORTUGAL
CHILE
TAIWAN
GUATEMALA
MOROCCO
KOREA, REP.
BOTSWANA
DOMINICAN
REP.
SIERRA LEONE
COLOMBIA
PANAMA
ECUADOR
PERU
TURKEY
POLAND
IVORY COAST
SWAZILAND
JAMAICA
THAILAND
BOLIVIA
HONDURAS
NIGERIA
SRI LANKA
PHILIPPINES
NEPAL
INDIA
ZAMBIA
KENYA
ZIMBABWE
MALAWI
MADAGASCAR
0
.5
Real GDP per worker, 1985
1

What determines TFP? (denoted by A)
o Human capital
 Average years of schooling in US is 13; 4 in poorest countries
 In US, an extra year of education raises lifetime income by 7%
 In poorest countries, because the skills are so much more basic, the rate
of return can be 10-13%
 If, then, every one of those extra 9 years of education that the average
US person receives is worth an extra 10% of income, the income
difference would be about 1.9: double.
 So the residual goes from a factor of 10 to a factor of 5.
o Technology
 Ideas
o Institutions
 Rule of law, Corruption, Property rights, Expropriation, Contract enforcement,
Separation of powers
The Steady State with Population Growth and Technological Progress
The Steady State with Population Growth
The model in chapter 11 said that capital accumulation was given by
𝐾𝑡+1 𝐾𝑡
𝑌𝑡
𝐾𝑡
− =𝑠 −𝛿
𝑁
𝑁
𝑁
𝑁
Divide both sides by 𝐾𝑡 ⁄𝑁
𝐾𝑡+1 𝐾𝑡
𝑁 − 𝑁 = 𝑔 = 𝑠 𝑌𝑡 ⁄𝑁 − 𝛿
𝐾
𝐾𝑡
𝐾𝑡 ⁄𝑁
𝑁
Since we assumed that population was stationary, we could cancel out all the “N”s and find that capital
𝑌
𝑌
grew at a rate equal to 𝑠 𝐾𝑡 − 𝛿. But population is growing, so if capital grows 𝑠 𝐾𝑡 − 𝛿, capital per
𝑡
worker will grow by 𝑔𝐾 − 𝑔𝑁 =
𝑡
𝑌
𝑠 𝐾𝑡
𝑡
− 𝛿 − 𝑔𝑁 .
We need to include a factor that we neglected in chapter 11: the price of capital!! Capital has to be
bought at a price, evidently. So it would make sense that if people save $100, and if capital costs $2
unit, firms can only buy $100/$2= 50 units. In general, if saving is a quantity of funds “X”, that quantity
of funds can only by “X/pK” units of capital. So far we’ve been (implicitly) assuming that pK=1. So we
modify the equation to take this into account.
𝑔𝐾 − 𝑔𝑁 =
𝑌
𝑠 𝑡
𝐾𝑡
𝑝𝐾
− 𝛿 − 𝑔𝑁
Growth of capital-per-worker
The Steady State with Technological Progress
Suppose technology progresses over time (a reasonable assumption in the last two hundred years or
so). We found that every improvement in technology makes the curves in the Solow diagram shift up,
so that steady-state capital-per-worker and steady-state output-per-worker grow.
The steady-state level of capital-per-worker must depend also on the rate of growth of Total Factor
Productivity, A. Since productivity grows constantly, it deserves its own separate growth rate, which we
can denote by gA.
I=sAk1/3
Well, this is fine, but it’s very unsatisfactory to talk about a “steady state” that changes all the time due
to perfectly well-known factors. We should reformulate our definition of the steady state to incorporate
the fact that productivity growth causes changes in capital-per-worker.
Imagine the world before padded horse collars. Farmers lived in a steady state with their horses, their
plows, and their houses. Then the padded horse collar is invented. Capital (the horses) become more
productive as people invest in padding. The same amount of capital produces more output out of their
soil. More can be saved and people repair their houses. But because this was a one-time increase in
productivity, eventually we come to another steady state with horses, padded collars, plows, and better
houses. People end up with more capital per unit of output produced.
So an improvement in technology changes the amount of capital-per-unit-of GDP. This suggests a new
definition. In the steady state, the capital-output ratio is constant. In the steady state, the growth rate
of capital is equal to the growth rate of output.
In the steady state,
𝐾𝑡∗
𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑌𝑡∗
∗
𝐾𝑡+1
𝐾𝑡∗
∗ − ∗
𝑌𝑡+1 𝑌𝑡
=0
𝐾𝑡∗
𝑌𝑡∗
𝑔𝐾∗ = 𝑔𝑌∗ = 0
𝑔𝐾∗ = 𝑔𝑌∗
Definition of the steady state
This definition is new, but it is consistent with the definition in
Chapter 11. There, capital accumulation stopped in the steady
state (which, remember, doesn’t make sense if population is
growing and technology is improving). If 𝑔𝐾 = 0, then we
found that 𝑔𝑌 = 0. So it was also true there that 𝑔𝐾 = 𝑔𝑌 .
Output-per-worker, Total Factor Productivity, and the Capital/Output Ratio
We can use the capital/output ratio to figure out what determines normal output growth at any point
and in the steady state. To do that, modify the “growth rate” version of the Cobb-Douglas production
function (𝑔𝑌 − 𝑔𝑁 = 𝑔𝐴 + 𝛼(𝑔𝐾 − 𝑔𝑁 )) to have the capital/output ratio on the right-hand side. If we
just plug −(𝑔𝑌 − 𝑔𝑁 ) into the parenthesis, that’s like subtracting “𝛼(𝑔𝑌 − 𝑔𝑁 )” from both sides of the
equation.
𝑔𝑌 − 𝑔𝑁 − 𝛼(𝑔𝑌 − 𝑔𝑁 ) = 𝑔𝐴 + 𝛼(𝑔𝐾 − 𝑔𝑁 − (𝑔𝑌 − 𝑔𝑁 ))
Now we simplify
(1 − 𝛼)(𝑔𝑌 − 𝑔𝑁 ) = 𝑔𝐴 + 𝛼(𝑔𝐾 − 𝑔𝑌 )
And solve for the growth of output-per-worker
(𝑔𝑌 − 𝑔𝑁 ) =
1
𝑔
(1−𝛼) 𝐴
𝛼
+ (1−𝛼) (𝑔𝐾 − 𝑔𝑌 )
output per worker
and capital deepening
𝑔𝐴
⁄(1 − 𝛼) is productivity growth divided by the contribution of labor to output, so we can
think of it as “growth in labor productivity.” Then this tells us that at any point in time, normal outputper-worker grows due to two factors: productivity growth and “capital deepening,” that is, increases in
the amount of capital-per-unit-of-GDP. This is the extent to which the economy becomes “more
industrial” in the sense that each unit of output made requires an increasing amount of capital behind it.
The factor
Since above we defined the steady state as the point where 𝑔𝐾∗ = 𝑔𝑌∗ , it follows that in the steady state
(𝑔𝑌∗ − 𝑔𝑁 ) =
1
𝛼
(0)
𝑔𝐴 +
(1 − 𝛼)
(1 − 𝛼)
(𝑔𝑌∗ − 𝑔𝑁 ) =
1
𝑔
(1 − 𝛼) 𝐴
(𝑔𝐾∗ − 𝑔𝑁 ) =
1
𝑔
(1 − 𝛼) 𝐴
So also
In the steady state, the growth rate of GDP-per-worker is given by the growth rate of labor productivity.
𝑔𝑌∗ =
𝑔𝐴
+ 𝑔𝑁
(1 − 𝛼)
In the steady state, the growth rate of GDP is given by the sum of the growth rate of labor plus the
growth rate of labor productivity.
Variable
Steady-State Growth Rate
𝑔𝐴
+ 𝑔𝑁
(1 − 𝛼)
𝑔𝐴
𝑔𝑌 =
+ 𝑔𝑁
(1 − 𝛼)
𝑔𝐾 =
Capital
Output
𝑔𝑁
Labor
𝑔𝐾 − 𝑔𝑌 = 0
Capital/output ratio
𝑔𝐴
(1 − 𝛼)
𝑔𝐴
𝑔𝐾 − 𝑔𝑁 =
Output-per-worker
(1 − 𝛼)
And output-per-worker and capital-per-worker grow by the rate of labor productivity growth. So
making workers more productive makes living standards grow.
𝑔𝐾 − 𝑔𝑁 =
Capital-per-worker
Determinants of the capital/output ratio
What determines the capital/output ratio? Above we found that capital-per worker grows by
𝑔𝐾 − 𝑔𝑁 =
𝐾
If we solve this for 𝑌𝑡
𝑡
𝑌
𝑠 𝐾𝑡
𝑡
𝑝𝐾
− 𝛿 − 𝑔𝑁
(𝑔𝐾 − 𝑔𝑁 ) + 𝛿 + 𝑔𝑁 =
𝑌
𝑠 𝐾𝑡
𝑡
𝑝𝐾
(𝑔𝐾 − 𝑔𝑁 ) + 𝛿 + 𝑔𝑁
𝑌𝑡
=
𝑠
𝐾𝑡
𝑝𝐾
We get
𝑠⁄
𝐾𝑡
𝑝𝐾
=
𝑌𝑡 (𝑔𝐾 − 𝑔𝑁 ) + 𝛿 + 𝑔𝑁
Capital output ratio
Notice that if we set 𝑝𝐾 back to 1, and 𝑔𝑁 back to zero (and we
assume no productivity growth), we get
𝑠⁄
𝐾𝑡
1
=
𝑌𝑡 (𝑔𝐾 − 0) + 𝛿 + 0
In the steady state, 𝑔𝐾 = 0, so
ratio is
𝐾𝑡∗
𝑌𝑡∗
𝐾𝑡
𝑌𝑡
s
= 0+𝛿 and the capital-output
𝑠
𝛿
= , which is exactly what we got in chapter 11.
The Capital/Output Ratio in the Steady State
Above we found that in the steady state,
(𝑔𝐾∗ − 𝑔𝑁 ) =
1
𝑔
(1 − 𝛼) 𝐴
Plugging that result into the output/capital ratio equation,1
𝐾𝑡∗
=
𝑌𝑡∗
𝑠⁄
𝑝𝐾
1
𝑔 + 𝛿 + 𝑔𝑁
(1 − 𝛼) 𝐴
Capital output ratio
in the steady-state
Phew! What’s nice about this is that historians actually have numbers for all of these variables.
Output per worker, 1865-19292
In 1865 the United States had 35 million people in it, at an average measured economic standard of
living of some $1,600 year-2008 dollars per year, at least two-thirds farmers or other small-town rural
dwellers. By 1929 farming and other small-town rural dwellers were down to one-eighth of the
population, the United States had 122 million people in it, and the average measured economic
standard of living was some $6,000 year-2008 dollars per year.
1
2
Notice that the K and the Y now have an asterisk, indicating their steady-state levels.
http://delong.typepad.com/american_economic_history/2008/09/20080927-growth.html
Using the formula for long-run growth we had earlier,
𝑥𝑡+𝑛 1/𝑛
𝑔=(
) −1
𝑥𝑡
Population grew at a rate equal to
122 1/(1929−1865)
𝑔𝑁 = (
)
− 1 = 1.9%
35
An output per worker grew at a rate equal to3
𝑔𝑌 − 𝑔𝑁 = (
6000 1/(1929−1865)
)
− 1 = 2.1%
1600
The comparable figure for pre-Civil war years is 1.4% annual output-per-worker growth. The
continuation—nay, the acceleration—of growth (compared to the Pre-Civil War period) in output per
worker alongside continued population growth is especially remarkable given that the frontier had
closed in the immediate aftermath of the Civil War: the natural resources the United States had then
conquered were all that there were. Yet growth continued: the focus shifted from expansion and
resources to industrialization. America became an industrial economy. Even farming became an
industrial occupation: much less muscle, ox, and horsepower; many more automatic reapers, harvesters,
pumps, stationary gasoline engines, tractors.
The Civil War itself brought about some of those transformations (principally in the North). As more
farmhands were off fighting the war, McCormick found a ready market for its combine harvester, which
could do the work of many farmhands. As the United States grew and transformed itself during those
years, two things happened. First, it accumulated capital: it depended more “deeply” on capital
accumulation for its production. Each unit of output relied more on capital and less on labor. Each
bushel of wheat required less sweat and more steel. Secondly, it acquired new technologies, so the
factors became more productive. In 1865
annual rate of
population
growth
𝑔𝑁
annual growth
rate of total
factor
productivity
𝑔𝐴

3%
0.6%
1/3
annual
growth rate
of labor
productivity
𝑔𝐴

(1 − 𝛼)
0.9%
annual rate of
depreciation
𝛿
rate of
national
savings
s
4%
20%
price of
capital
goods
pK
1
Thus our equation for the capital/output ratio in the steady state becomes:
3
Which means that output (not output per worker, but just output) must have growth at a rate of 4.33% every year.
𝐾𝑡
=
𝑌𝑡
𝑠⁄
𝑝𝐾
𝑔𝐴
+ 𝛿 + 𝑔𝑁
(1 − 𝛼)
=
0.20/1
= 2.53
0.006
+ .04 + .03
1 − 1/3
So there were about 2.5 units of capital per each unit of GDP produced. By 1929
annual rate of
population
growth
𝑔𝑁
annual growth
rate of total
factor
productivity
𝑔𝐴

2%
1.1%
1/3
annual
growth rate
of labor
productivity
𝑔𝐴

(1 − 𝛼)
16.5%
annual rate of
depreciation
𝛿
rate of
national
savings
s
4%
25%
price of
capital
goods
pK
0.67
So our equation becomes:
𝐾𝑡
=
𝑌𝑡
𝑠⁄
𝑝𝐾
0.25/(2/3)
=
= 4.90
𝑔𝐴
0.011
+ 𝛿 + 𝑔𝑁
+ .04 + .02
(1 − 𝛼)
1 − 1/3
The jump from 2.5 to 4.90 in the capital-output ratio (measured at 1865 prices) gives us an annual rate
of
4.90 1/(1929−1865)
1865−1929
𝑔𝐾/𝑌
=(
)
− 1 = 1.0%
2.5
The average growth rate of TFP between 1865 and 1929 was about 1.0%. Plug all this into our “outputper-worker-and-capital-deepening” equation for the growth rate of output per worker:
(𝑔𝑌 − 𝑔𝑁 ) =
1
𝛼
(𝑔 − 𝑔𝑌 )
𝑔𝐴 +
(1 − 𝛼)
(1 − 𝛼) 𝐾
(𝑔𝑌 − 𝑔𝑁 ) =
1
1
(1 − 3)
1.1% +
1
3
1
(1 − 3)
1.0%
3
1
(𝑔𝑌 − 𝑔𝑁 ) = 1.1% + 1.0%
2
2
(𝑔𝑌 − 𝑔𝑁 ) = 2.1%
1
which fits our calculations for output-per-worker growth. Between 1865 and 1929 some 1/4 (= 2 1.1%)
of American economic growth in measured economic output per capita came from capital deepening—
more capital, more produced means of production, more machines backing up each worker. And 3/4 (=
3
1.1%)
2
of American economic growth in measured economic output per capita came from
improvements in the efficiency of labor—working smarter made possible by more education,
organizational improvements, and other improvements in technology not directly related to those that
made capital goods cheaper.
Growth Accounting4
We found above that
Output-per-worker
In Levels
𝑌⁄ = 𝐴(𝐾⁄ )𝛼
𝑁
𝑁
In Rates
𝑔𝑌 − 𝑔𝑁 = 𝑔𝐴 + 𝛼(𝑔𝐾 − 𝑔𝑁 )
Output
𝑌 = 𝐴(𝐾)𝛼 (𝑁)1−𝛼
𝑔𝑌 = 𝑔𝐴 + 𝛼𝑔𝐾 + (1 − 𝛼)𝑔𝑁
Changes in the Solow residual or (the same thing) total factor productivity can come about for many
reasons. Economists often refer to total factor productivity as “technology,” but if it is technology it is
technology in the widest possible sense. Not just new ways of constructing buildings, newly-invented
machines, and new sources of power affect total factor productivity, but changes in work organization,
in the efficiency of government regulation, in the degree of monopoly in the economy, in the literacy
and skills of the workforce, and in many other factors affect total factor productivity as well.
One at-a-time changes to output growth
For the following, assume 𝛼 = 1/3.
1. Suppose the capital stock grows by 1%, while everything else remains constant.
Then output will change by
∆𝑌
𝑌
=𝛼
∆𝐾
𝐾
= ______________
2. Suppose amount of hours worked (N) grows by 1%, while everything else remains constant.
∆𝑌
∆𝑁
Then output will change by = (1 − 𝛼)
= ______________
𝑌
𝑁
3. Suppose total factor productivity (A) changes by 1%, while everything else remains constant.
Then output will change by
∆𝑌
𝑌
=
∆𝐴
𝐴
= ______________
This part is based on Bradford DeLong’s handout, available at http://www.j-bradforddelong.net/macro_online/growth_accounting.pdf
4
Calculating the Solow Residual
In 1980-2000, the growth-accounting parameters for the United States were estimated to be5
gY
gN6
gK

gA
1980-1990
3.3%
1.7%
4.9%
0.29
1990-1995
2.5%
1.3%
3.5%
0.30
0.5%
1995-2000
4.2%
1.9%
5.8%
0.30
Using the growth accounting equation gY  g N  g A   g K  g N  , calculate the growth in total factor
productivity, gA, for 1980-1990 and 1995-2000.
What caused the productivity speedup after 1995? The natural candidate is the coming of the
information age—the shifts in business organization and competition caused by the technological
revolutions in data processing and data communications. Indeed, they did play a substantial role both in
accelerating the rate of capital deepening and in boosting total factor productivity growth in hightechnology sectors. However, there is more to the story. A McKinsey Global Institute study concluded
that the jump in productivity growth was overwhelmingly driven by six sectors—computer and other
durable manufacturing, electronics, telecommunications, retail trade, wholesale trade, and securities
brokerage,. The information technology revolution was the key to the boom in the first three sectors,
and was a key but not the only key in the last three sectors. “Product, service, and process innovations…
were important causes” as well. Increased competition to spur businesses to improve productivity,
organizational changes to take advantage of the information technology revolution (like warehouse
automation and information technology-based supply chain management), and smarter government
regulation of industry played important roles as well in the acceleration of productivity growth in the
second half of the 1990s.
Calculating components of growth
Fill in the blanks in this table. Assume 𝛼 = 1/3. Notice the big estimated productivity slowdown
between 1973 and 1995. What short-run, business-cycle factors (of the kind we studied in other
chapters) do you think affected US productivity during those years?
1948-2002
1948-1973
1973-1995
1995-2002
Output per hour (Y/L)
2.5
3.3
1.5
Contribution of KL
0.9
0.9
1.3
Contribution of Labor
0.2
0.2
0.4
Contribution of TFP (A)
3.1
1.3
2.7
5
Data derived from Marcel P. Timmer, Gerard Ypma and Bart van Ark (2003), IT in the European Union: Driving
Productivity Divergence?, GGDC Research Memorandum GD-67 (October 2003), University of Groningen,
Appendix Tables, updated June 2005, http://www.ggdc.net/dseries/growth-accounting.shtml
6
Note that gN is not population growth but the growth in worker-hours. There were more people in the US between
2000-2004, and the number of workers increased by 0.4, but the recession reduced the number of hours worked
per worker by an average yearly rate of - 0.8%.
Forecasting growth
On the basis of the tools that you have and your own educated guesses, let’s make a guess for what will
happen to gY, the normal growth rate of natural output in the US, in the next generation or so. First, we
need to come up with estimates for gN, gK, , and gA in the next generation?
Growth in worker-hours
If gN is the growth in worker-hours, it must be determined by the growth in the number of people
(population growth, which is probably different for different groups). But that’s not enough. How many
people will work? (how will the labor-force participation rate change?) How will the average number of
hours worked by each worker change? So if hours worked = hours per worker x labor-force participation
rate x population, then
𝑇𝑜𝑡𝑎𝑙 𝐻𝑜𝑢𝑟𝑠
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘𝑒𝑟𝑠
𝐻𝑜𝑢𝑟𝑠 𝑤𝑜𝑟𝑘𝑒𝑑 =
×
× 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘𝑒𝑟𝑠 𝑇𝑜𝑡𝑎𝑙 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Using the same tools that we developed earlier, we can take logs and derivatives to find how “Hours
worked” will change:
g N  g hours/ wkr  glfp rate  g Pop
1. What do you think determines the number of hours each worker works? What changes it, and
is it likely to change?
2. In the next few years, will the proportion of people who are in the labor force increase or
decrease?
3. In the next few years, will population grow faster or more slowly?
Growth in capital stock
gK is the growth in capital (also known as fixed capital accumulation). But it’s not that simple. How
much capital actually gets utilized? (this might change as firms change their engineering set ups, etc.)
Capital accumulation is financed by the financial system, in one way or another. But not all the loans
that the financial system makes are for investment (some are for consumption). And the total quantity
of loans is really determined by three things: a) the efficiency with which the financial sector turns
saving into investment, b) proportion of domestic saving that is lent domestically (and by the amount of
foreign saving that is lent to domestic residents), and c) the rate at which people save their income.
1. What do you think determines the capacity utilization? What changes it, and is it likely to
change?
2. In the next few years, will consumer credit grow faster than total credit or will it slow down?
3. In the next few years, will banks become more efficient at making loans out of deposits? Will
financial disruptions cause fewer loans to be made, or lower-quality loans?
4. Will the US saving rate increase (personal saving, corporate saving, government saving)? Will
foreigners continue to place their own saving into the US?
Changes in capital intensity
 depends on the kind of products and industries in which a country specializes. Below is the evolution
of the capital/labor ratio since 1980 (distinguishing between IT and non-IT capital). Do you think the US
will continue present trends of specialization, or can you imagine reasons for change?
Growth in total factor productivity
A is technology in the broadest sense of the word. Can you imagine changes the rate of change of the
way of constructing buildings, of the invention of
Growth
 for IT for Nonmachines, and the development of new sources of
in A
capital
IT Capital
Total
power, of changes in work organization, in the
1980
0.04
0.24
0.28
-0.022
efficiency of government regulation, in the degree of
1981
0.04
0.24
0.28
0.008
1982
0.04
0.24
0.28
-0.025
monopoly in the economy, in the literacy and skills of
1983
0.05
0.23
0.28
0.022
the workforce, etc.?
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
0.06
0.06
0.06
0.05
0.05
0.05
0.05
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.25
0.24
0.24
0.23
0.23
0.23
0.24
0.25
0.29
0.30
0.29
0.29
0.29
0.29
0.29
0.29
0.29
0.30
0.30
0.30
0.31
0.31
0.30
0.30
0.29
0.29
0.29
0.30
0.31
0.020
0.005
0.010
0.002
0.010
0.005
0.004
-0.005
0.025
0.002
0.009
-0.004
0.015
0.010
0.009
0.012
0.011
0.000
0.018
0.027
0.024