Notes Chapters 10-11
Point of the class
Math



Logs and exponents
Algebraic manipulation
Partial derivatives and maximization
Model building

The Solow Growth Model
Economic concepts and intuition






Aggregate production function, CRS, diminishing returns, Cobb-Douglas function
Marginal product of capital and real interest rate
Capital comes from saving output and is used to produce more output
Diminishing returns causes capital accumulation to stop (at the steady state), so raising the
saving rate only leads to temporary increases in the level of output per capita (during the
transition period).
Consumption per worker can be maximized by picking a saving rate at which the MPK=the
depreciation rate (which equates the marginal benefit and the marginal cost of holding capital
rather than consumption goods).
The Solow Growth Model does not explain long-run growth, however, because it leaves out
technological progress
Definitions
1.
Output per capita
2.
Aggregate Production Function
3.
State of technology (Total
Factor Productivity)
4.
Constant returns to scale
5.
Decreasing returns to capital
6.
Output-per-worker
7.
Capital-per-worker
8.
Cobb-Douglas production
9.
Marginal product of capital
function
10. Real interest rate
11. Saving rate
12. Depreciation rate
13. Capital accumulation function
14. Steady-state
15. Transition dynamics (or catchup to a common steady state)
16. Convergence
Homework Problems

On the webpage.
17. Golden-rule level of capital
18. Technological progress
Facts
Facts about growth
See Notes for Ch 20, ECON 201
15000
10000
0
5000
Real GDP per capita
20000
Why does GDP per worker increase?
1950
1960
1970
1980
year
1990
2000
Real GDP per capita, Korea
Real GDP per capita, Nicaragua
1
1.5
It would seem that it has a lot to do with the amount of capital that each worker in the economy gets to
use, as we can see in the following graph:
SWITZERLAND
0
.5
U.S.A.
CANADALUXEMBOURG
AUSTRALIA
NORWAY
NETHERLANDS
BELGIUM WEST
GERMANY,
ITALY
FRANCE
SWEDEN
NEW ZEALAND
DENMARK
AUSTRIA
FINLAND
ICELAND
U.K. ISRAEL
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
Pretty obviously, having more capital-per-worker helps. But if this were the whole explanation, the
United States would be poorer than Finland or Switzerland, which is not the case. As we’ll see below,
the extra output that can be obtained from extra capital is limited by diminishing returns.
1
It turns out that improvements in technology are crucial. A measure of technology is Total Factor
Productivity. What determines TFP? Things like human capital, research and development, and
institutions.
U.S.A.
.2
.4
.6
Total Factor Productivity
.8
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
So a combination of both is what drives living standards
𝑦 = 𝐴𝑘 1/3
But this just begs the question. Where does capital come from? The following is the Solow Growth
Model, which endogenizes capital accumulation: it explains why countries accumulate capital, and the
level of capital-per-worker at which they will reach long-term equilibrium, or the steady state.
Model Building
Simplifications







A worker is a worker, no matter the qualifications (homogenous labor)
A unit of capital is a unit of capital, no matter what it actually is (homogenous capital)
Technology is just the function that connects K and N to Y. An engineering blueprint.
There’s no unemployment (or it doesn’t change)
Constant returns to scale
The economy is closed and the financial market works perfectly, so 𝐼 = 𝑆 + (𝑇 − 𝐺).
National saving equals investment.
Saving is simply a proportion of income 𝑆 = 𝑠𝑌.
Building Blocks
If we are trying to explain capital accumulation, we need to explain why capital is accumulated (to
produce output) and how capital is accumulated (by saving and purchasing new capital goods).
Aggregate production function
Output is produced with capital and labor … so that’s what capital is used for! Notice the role of A. This
is the state of technology, which allows all the factors to be productive and determines how much
output will be produced from given quantities of capital and labor.
𝑌 = 𝐴𝐹(𝐾, 𝑁)
Diminishing returns to individual inputs
2𝑌 > 𝑌′ = 𝐹(2𝐾, 𝑁)
Constant returns to scale
2𝑌 = 𝐹(2𝐾, 2𝑁)
Output-per-worker and capital-per-worker. CRS implies that if we divide the right-hand side by a
number, “N”, the left-hand side will change by the same factor
𝑌
𝐾 𝑁
𝐾
= 𝐴𝐹 ( , ) = 𝐴𝐹 ( , 1)
𝑁
𝑁 𝑁
𝑁
𝑦 = 𝐴𝑓(𝑘, 1) = 𝐴𝑓(𝑘)
𝑌
𝐾
From now on, 𝑁 = 𝑦 and 𝑁 = 𝑘. Suppose we double the number of workers and the number of units of
capital. What happens to output? It doubles, by CRS. What happens to output per capita? Nothing.
𝑌 2𝑌
2𝐾 2𝑁
=
= 𝐴𝐹 ( , )
𝑁 2𝑁
2𝑁 2𝑁
We want the “F” function to exhibit both CRS and diminishing returns to K. We can ensure CRS by
writing it as a “per worker function”, but we must pick a particular function to make it diminishing
returns. Log and square root (𝑦 = 𝐴√𝑘) both fit the bill. A special kind of very useful function is the
Cobb-Douglas function.
capital per worker, k
5000
𝒚 = √𝒌
=sqrt(5000)
Diminishing returns to capital
k
5000
6000
7000
8000
k
1000
1000
1000
y
𝒚 = √𝒌
70.71068
77.45967 6.748989
83.666 6.206336
89.44272 5.776716
Cobb-Douglas Production function
𝑌 = 𝐴𝐾 𝛼 𝑁1−𝛼 ,
Properties of exponents
0<𝛼<1
𝑥 𝑛 𝑦 𝑛 = (𝑥𝑦)𝑛
Diminishing returns to individual inputs
2𝑌 > 𝑌′ = 𝐹(2𝐾, 𝑁)
𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑛+𝑚
𝑥𝑛
= 𝑥 𝑛−𝑚
𝑥𝑚
𝑌′ = 𝐴(2𝐾)𝛼 𝑁1−𝛼 = 𝐴2𝛼 𝐾 𝛼 𝑁1−𝛼
𝑌 ′ = 2𝛼 𝑌 < 2𝑌
(𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚
Constant returns to scale
2𝑌 = 𝐹(2𝐾, 2𝑁)
𝑌 ′ = 𝐴(2𝐾)𝛼 (2𝑁)1−𝛼 = 𝐴2𝛼 (𝐾)𝛼 21−𝛼 𝑁1−𝛼
𝑌 ′ = 𝐴2𝛼 21−𝛼 (𝐾)𝛼 𝑁1−𝛼 = 𝐴2𝛼+1−𝛼 (𝐾)𝛼 𝑁1−𝛼 = 2𝐴(𝐾)𝛼 𝑁1−𝛼
𝑌 ′ = 2𝑌
Per-worker functions
𝑌 𝐴𝐾 𝛼 𝑁1−𝛼 𝐴𝐾 𝛼 𝑁1−𝛼
=
= 𝛼 1−𝛼
𝑁
𝑁
𝑁 𝑁
𝑌
𝐾 𝛼 𝑁 1−𝛼
= 𝐴( ) ( )
𝑁
𝑁
𝑁
𝑦 = 𝐴𝑘 𝛼
capital per worker, k
0
1000
y=k0.45
=(B3)^0.45
=(B4)^0.45
y=k0.5
=(B3)^0.5
=(B4)^0.5
y=k0.55
=(B3)^0.55
=(B4)^0.55
Allocating Resources
A little bit of Calculus: the Power Rule
How do firms decide how much capital and how
much labor to use? They maximize their profits.
Their profits might be given by a profit function such
as this,
𝑓(𝑥) = 𝑎𝑥 𝑛
𝑑𝑓(𝑥)
= 𝑓′(𝑥) = 𝑎𝑛𝑥 𝑛−1
𝑑𝑥
𝑓(𝑥) = 2𝑥 2 + 3𝑥 + 5
𝑃𝐹(𝐾, 𝐿) − 𝑟𝐾 − 𝑤𝑁
which simply says that profit is the difference
between revenue (price times output) minus costs
(rental for capital, wage for labor)
𝑓(𝑥) = 2𝑥 2 + 3𝑥 1 + 5𝑥 0
𝑓 ′(𝑥) = 2(2𝑥 2−1 ) + 3(1𝑥 1−1 ) + 5(0𝑥 0−1 )
𝑓′(𝑥) = 4𝑥 1 + 3
So the firm maximize this function by choosing
capital and labor.
max[𝑃𝐹(𝐾, 𝐿) − 𝑟𝐾 − 𝑤𝑁]
𝐾,𝑁
max[𝑃𝐴𝐾 𝛼 𝑁 1−𝛼 − 𝑟𝐾 − 𝑤𝑁]
𝐾,𝑁
max[𝑃𝐴𝐾 1/3 𝑁 2/3 − 𝑟𝐾 − 𝑤𝑁]
𝐾,𝑁
a function
its derivative
To do that, we simply find the slope of the profit function (the derivative) and find where the slope =0.
That would be where the function reaches a maximum.
General version
Cobb-Douglas, 𝛼 = 1/3, version
Profit = 𝐹(𝐾, 𝑁) − 𝑟𝐾 − 𝑤𝑁
Profit = 𝐴𝐾 1/3 𝑁 2/3 − 𝑟𝐾 − 𝑤𝐿
𝜕𝐹(𝐾,𝐿)
−𝑟
𝜕𝐾
=0
1
𝐴𝐾 1/3−1 𝑁 2/3
3
1 𝐴𝐾1/3 𝑁 2/3
3
𝐾
𝑀𝑃𝐾 = 𝑟
1𝑌
−𝑟 =0
=𝑟
𝑀𝑃𝐾 = 3 𝐾 = 𝑟
So for a Cobb-Douglas function, the marginal product of capital is proportional to the average amount
of output produced by K, where the factor of proportionality is =1/3.
k
MPK
1
0.333
8
0.083
27
0.037
64
0.021
125
0.013
216
0.009
To do the same for labor, take a derivative of the profit function with respect to labor and set it equal to
zero
𝜕𝐹(𝐾,𝑁)
−
𝜕𝑁
2
𝐴𝐾 1/3 𝑁 2/3−1
3
𝑤=0
2𝑌
3𝑁
𝑀𝑃𝐿 = 𝑤
−𝑤 = 0
=𝑤
So the marginal product of labor is proportional to the average amount of output produced by N, where
the factor of proportionality is (1-)=2/3.
Why does this solution make economic sense?
So the capital share of output is =1/3:
1
𝐾
=𝑟
3
𝑌
This also means that the stock market is, actually, savers’ valuation of the stock of capital. So the labor
2
𝑁
share of output is (1-)=2/3: 3 = 𝑤 𝑌
This has been found to be approximately true in practice. So we’ll keep using =1/3.
The Real Interest Rate
The Real Interest Rate is the amount of output that a person can earn by saving one unit of output.
Now, a unit of saving is used as a unit of investment, which is a new unit of capital, which produces an
extra unit of output (MPK, the marginal product of capital). So the income that can be earned from
saving – the real interest rate – is equal to the marginal product of capital.
𝑀𝑃𝐾 = 𝑟
capital stock at its optimal
𝑀𝑃𝐾 > 𝑟
firms can borrow more and earn MPK above the borrowing cost. As they
borrow more, they drive the real interest rate up.
𝑀𝑃𝐾 < 𝑟
firms that borrow to buy capital find that the borrowing cost exceeds the
returns from buying capital. As they borrow less, they drive the real interest
rate down.
Saving Behavior
Nations finance capital accumulation by giving up consumption.
Income is either saved or consumed
𝐶+𝑆 =𝑌
Output is either consumption goods or investment goods
𝐶+𝐼 =𝑌
Saving comes from households or firms (𝑆 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 ), from the government (𝑇 − 𝐺), or from foreigners,
who send us their capital inflows (𝐾𝐼), for example in the shape of foreign aid or loans. So national
saving 𝑆 = 𝑆 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 + (𝑇 − 𝐺) + 𝐾𝐼.
If the financial market works perfectly, so that every unit saved gets lent out to productive uses,
𝐼=𝑆
National saving equals investment. How much do people save? Assume that saving is simply a
proportion of income
𝑆 = 𝑠𝑌
Although saving rates (s) do vary over time and in response to the real interest rate, they don’t seem to
be related to whether the country is poor or rich, and they don’t seem to change as a country grows
richer. Combining both equations, we get
𝐼𝑡 = 𝑆𝑡 = 𝑠𝑌𝑡
Investment Allocation equation
Y
I=(0.5)Y
I=(0.6)Y
I=(0.7)Y
100
=0.5*$O3
=0.6*$O3
=0.7*$O3
200
=0.5*$O4
=0.6*$O4
=0.7*$O4
Capital Accumulation
Investment is purchases of capital, so investment = accumulation of new capital … if no capital
depreciates. But some capital does depreciate, by a rate .
−𝛿 𝐾𝑡
K
K=(0.1)K
K=(0.125)K
K=(0.15)K
1000
100
125
150
2000
200
250
300
So next period’s capital is
𝐾
⏟
𝑡+1
𝑛𝑒𝑥𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 ′ 𝑠 𝐾
=
𝐾𝑡
⏟
𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 ′ 𝑠 𝐾
+
𝐼⏟𝑡
𝑛𝑒𝑤 𝐾 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑠
−
𝛿𝐾
⏟𝑡
𝐾 𝑡ℎ𝑎𝑡 𝑑𝑖𝑠𝑎𝑝𝑝𝑒𝑎𝑟𝑠
𝐾𝑡+1 − 𝐾𝑡 = 𝐼𝑡 − 𝛿𝐾𝑡
Time
Kt
It
0
1000
200
Kt
100
Kt
100
1
1100
200
110
90
2
1190
200
119
81
3
1271
200
127.1
72.9
4
1343.9
200
134.39
65.61
5
1409.51
200
140.951
59.049
Combining the Capital Accumulation equation with the Investment Allocation equation and writing it as
a per-worker function
𝐾𝑡+1 = 𝐼𝑡 + (1 − 𝛿)𝐾𝑡
𝐾𝑡+1 = 𝑆𝑡 + (1 − 𝛿)𝐾𝑡
𝐾𝑡+1 = 𝑠𝑌𝑡 + (1 − 𝛿)𝐾𝑡
𝐾𝑡+1
𝑌𝑡
𝐾𝑡
= 𝑠 + (1 − 𝛿)
𝑁
𝑁
𝑁
𝑘𝑡+1 = 𝑠𝑦𝑡 + (1 − 𝛿)𝑘𝑡
The change of capital-per-worker is
𝑘𝑡+1 − 𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
∆𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
Capital Accumulation equation
If we were wondering at what point capital doesn’t change, that would be
∆𝑘𝑡 = 0 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
𝑠𝑦𝑡 = 𝛿𝑘𝑡
So capital accumulation stops when investment-per-worker (new capital-per-worker; the portion of
output-per-worker that is saved) is equal to the portion of old capital-per-worker that depreciates away.
This point is called the Steady State
𝑠𝑦𝑡 = 𝛿𝑘𝑡
capital stock is constant,
∆𝑘𝑡 = 0,
the economy is at the steady state
𝑠𝑦𝑡 > 𝛿𝑘𝑡
capital stock grows,
∆𝑘𝑡 > 0,
the economy is below the steady state
𝑠𝑦𝑡 < 𝛿𝑘𝑡
capital stock declines,
∆𝑘𝑡 < 0,
the economy is above the steady state
𝑦 = 𝐴𝑘 1/3
Production function
Production Function
Labor and Technology
For the moment, assume that both labor and technology are constant.
We don’t need a C=(1-s)Y equation: it’s already implied in all the above.
Putting the Building Blocks Together
Solow Diagram
What is the graph of 𝑠𝑦𝑡 , with k in the horizontal axis? Plug the production function into it, assuming
the saving rate s=0.75:
1/3
𝑠𝑦𝑡 = 𝑠𝐴𝑘𝑡
1/3
= 0.75𝐴𝑘𝑡
What is the graph of 𝛿𝑘𝑡 ? A linear function, since the depreciation rate doesn’t depend on the level of
capital.
Put them together
The Principle of Transition Dynamics – “Catch up”
∆𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
1
∆𝑘𝑡 = 𝑠𝐴𝑘𝑡3 − 𝛿𝑘𝑡
1/3
∆𝑘𝑡 = 0.75𝐴𝑘𝑡

− 0.05𝑘𝑡
When the capital stock is low,
o it is highly productive (its MPK is very high)
o high MPK means a relatively high real interest rate, which attracts saving
o lots of saving lead to lots of investment on new capital,
o lots of investment overwhelm the amount of depreciation, so capital accumulates
The farther below the steady state the economy is, the faster capital accumulates.
y
𝑨𝒌𝟏/𝟑
I
𝟎. 𝟕𝟓𝑨𝒌𝟏/𝟑
k
𝟎. 𝟎𝟓𝒌
8
20
27
2.00
2.71
3
1.5
2.04
2.25
0.4
1.0
1.35
58.09475019
3.872983346
2.90473751
2.90473751
0
64
125
216
4
5
6
3
3.75
4.5
3.2
6.25
10.8
-0.2
-2.5
-6.3
k
k
𝟏
𝒔𝑨𝒌𝟑𝒕
− 𝜹𝒌𝒕
1.1
1.04
0.9

When the capital stock is high,
o it is not very productive (its MPK is rather low)
o low MPK means a relatively low real interest rate, which fails to attract saving
o little saving leads to little investment on new capital,
o little investment fails to maintain existing capital as it depreciates,
so capital de-accumulates
The farther above the steady state the economy is, the faster capital de-accumulates.

When the capital stock is at the steady state,
o the MPK is such that
… the resulting real interest rate attracts just the right amount of saving
… that generates just the right amount of investment
… so that the amount of new capital is exactly equal to the amount of capital that
depreciates
o so the capital stock doesn’t change.
Steady State and its determinants


How do we find the steady state?
The definition of steady-state capital-per-worker (denoted as 𝑘𝑡∗) is
∆𝑘𝑡∗ = 0
o
For this reason, steady-state output-per-worker doesn’t change (𝑦𝑡∗ ).
∆𝑦𝑡∗ = 0

If the capital stock is not changing (∆𝑘𝑡∗ = 0), then
∆𝑘𝑡∗ = 𝑠𝑦𝑡∗ − 𝛿𝑘𝑡∗ = 0
∆𝑘𝑡∗ = 𝑠𝐴(𝑘𝑡∗ )𝛼 − 𝛿𝑘𝑡∗ = 0
𝑠𝐴(𝑘𝑡∗ )𝛼 = 𝛿𝑘𝑡∗
𝑠𝐴
𝑘𝑡∗
= ∗ 𝛼
(𝑘𝑡 )
𝛿
𝑠𝐴
= (𝑘𝑡∗ )1−𝛼
𝛿
1
𝑘𝑡∗

𝑠𝐴 1−𝛼
=( )
𝛿
So the steady-state level of capital-per-worker will be
o a positive function of the saving rate (s)
o a positive function of total factor productivity (A)
o a positive function of the contribution of capital to production ().
o a negative function of the depreciation rate ().
1
𝛼
𝑠𝐴 1−𝛼
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = 𝐴 [( ) ]
𝛿
1
𝛼
1−𝛼
𝑦𝑡∗ = (𝐴)1−𝛼 (𝑠⁄𝛿 )
Notice that this means that the steady-state level of output-per-worker really depends on the level of
productivity. Not only A is part of the production function, making capital more productive – it’s also a
key determinant of capital-per-worker itself.1
And yet, as important as Total Factor Productivity is in the Solow Growth Model, it is left as an
exogenous variable.
A useful concept is the “capital/output ratio”:
1
𝑘𝑡∗
=
𝑦𝑡∗
𝑠𝐴 1−𝛼
( )
𝛿
1
1−𝛼
𝛼
𝑠𝐴 1−𝛼
𝐴 [( ) ]
𝛿
𝑠𝐴 1−𝛼
( )
𝑠
= 𝛿
=
𝐴
𝛿
Interestingly, this doesn’t depend on productivity (A) or on .
1
One way to note how important TFP is in the Solow Growth Model is to note that the exponent on A in the
1
formula for steady-state output-per-worker is larger than in the plain production function:
> 1.
1−𝛼
Testing the Solow Growth Model
To test a model, we need to get it to generate a prediction. A simple, single, sharp prediction that
involves actual data that we can get our hands on.
The Capital-Output Ratio
It turns out that we can get our hands on the “capital/output ratio”, the ratio of the capital stock in an
economy to GDP.
The biggest shortcoming of the Solow model is that it doesn’t take into account TFP (which we denote as
A) . But since the capital/output ratio doesn’t depend on A, perhaps we can use it to test the Solow
Growth Model. If the model cannot explain the data even after we’ve kept A out of the picture, it’s a
pretty useless model indeed. But we will accept the Solow Growth Model if the data supports the idea
that, more or less,
𝑘𝑡∗ 𝑠
=
𝑦𝑡∗ 𝛿
The problem of this little formula, though, is that it has two variables in it. We can deal with this
through multi-variable regression, but what if the depreciation rate is very similar across the world?
Then the capital/output ratio should simply depend positively on the saving rate (or the investment
rate). This turns out to be true!
Convergence
Suppose we have two
economies with the same
level of productivity, the
same depreciation rate,
the same investment rate
and the same production
function. Then they must
have the same steadystate level capital-perworker.
Now imagine one has a
lower level capital-perworker. If an economy
grows faster if its father
away from the steady
state, then it must be the case that this economy is growing faster. So if we plot the growth rate of
countries that are pretty similar, such as the countries in the Organization for Economic Cooperation
and Development, against the actual output-per-capita a few decades ago, we should find that the
countries that were poorest have grown the fastest. This is the principle of transition dynamics at
work: it implies the convergence, or catch-up over time, of the GDP-per-capita of countries that have
similar enough technologies.
What about countries that don’t have the same technology or production function? We would expect
them to have different steady states. It’s perfectly plausible that, the economy of the United States and
the economy of
Zimbabwe are already at
their steady states, so we
would expect their
average growth rates to
be pretty darn close to
independent of how rich
they are.
Comparative Statics in the Solow Growth Model
𝛼 = 1/3, 𝑠 = 0.50, 𝛿 = 0.05, and 𝐴 = 1
∆𝑘𝑡 = 0.75(1)(𝑘𝑡∗ )1/3 − 0.05𝑘𝑡 = 0
0.75(1)(𝑘𝑡∗ )1/3 = 0.05𝑘𝑡∗
0.75(1)
𝑘𝑡∗
= ∗ 1/3
(𝑘𝑡 )
0.05
1
0.75(1)
= (𝑘𝑡∗ )1−3
0.05
1
1
𝑠𝐴 1−𝛼
0.75(1) 1−1
𝑘𝑡∗ = ( )
=(
) 3 = (15)3/2 = 58.09
𝛿
0.05
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = 𝐴[58.09]𝛼 = 1[58.09]1/3 = 3.87
Change the saving rate so now 𝑠 = 0.375.
The production function is as high as it was before: the MPK will behave just as it used to, capital will be
just as productive. But now people aren’t saving as much per unit of output – they are not thinking
about the future that much. They are perfectly content to stop capital accumulation sooner, which
makes them poorer in the long run (but since before they were barely eating and now they get to eat
more, perhaps they’ll be better off – see below).
0.375(1)(𝑘𝑡∗ )1/3 = 0.05𝑘𝑡∗
1
0.375(1)
𝑘𝑡∗
= ∗ 1/3 = (𝑘𝑡∗ )1−3
(𝑘𝑡 )
0.05
1
0.375(1) 1−1
𝑘𝑡∗ = (
) 3 = (7.5)3/2 = 20.54
0.05
1
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = (1)[20.54]3 = 2.74
Change TFP so now 𝐴 = 0.5. Return the saving rate to 𝑠 = 0.75
The production function shifts down, so each level of capital-per-worker produces much less output-perworker, so there’s less available for saving and accumulating capital. Hence capital accumulation stops
much sooner. A lower TFP makes workers less productive, so the marginal product of capital is smaller,
the real interest rate is smaller, less saving is attracted, and less capital is accumulated – which means
that the point where new investment doesn’t compensate for depreciation is reached earlier.
0.75(0.5)(𝑘𝑡∗ )1/3 = 0.05𝑘𝑡∗
1
0.75(0.5)
= (𝑘𝑡∗ )1−3
0.05
1
0.75(0.5) 1−1
𝑘𝑡∗ = (
) 3 = (7.5)3/2 = 20.54
0.05
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = 0.5[20.54]1/3 = 1.37
Is the effect of lowering the saving rate the same as the effect of lowering A? Steady-state capital-perworker, though lower than in the initial setting (𝑠 = 0.75, 𝐴 = 1) is the same as it was in the previous
example (𝑠 = 0.375, 𝐴 = 1).
A
s


k*
y*
1.00
0.75
0.05
0.33
58.09
3.87
0.50
0.75
0.05
0.33
20.54
1.37
1.00
0.375
0.05
0.33
20.54
2.74
(In the k* function, halving A or halving s give the same result.)
But changing TFP changes both the production function and steady-state capital-per-worker. Relatively
un-productive workers have less capital-per-worker, so when TPF falls by 50%, steady-state output falls
by even more.
The economy can do less with the same level of capital: the production function is lower than previously,
so this same level of k* is less productive. Output per worker is lower, even though the level of capital is
the same.
Change the depreciation rate so now 𝛿 = 0.04. Return to 𝑠 = 0.75 and 𝐴 = 1.
Because the capital stock depreciates more slowly, capital-per-worker keeps growing for a longer time
before it stops.
0.75(1)(𝑘𝑡∗ )1/3 = 0.10𝑘𝑡∗
1
0.75(1) 1−1
𝑘𝑡∗ = (
) 3
0.04
𝑘𝑡∗ = (18.75)3/2 = 81.19
1
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = (1)[81.19]3 = 4.33
Change the capital contribution to output so now 𝛼 = 0.25. Return to 𝛿 = 0.05, 𝑠 = 0.75 and 𝐴 = 1.
𝑌
Remember that the formula for the MPK (in the Cobb-Douglas function) is 𝑀𝑃𝐾 = 𝛼 𝐾. So when 
contracts, capital becomes less productive: the production function shifts in and it becomes harder to
attract savings. People’s incentive to pass up current consumption and buy (now less productive) capital
is diminished, so capital accumulation stops sooner.
0.75(1)(𝑘𝑡∗ )1/4 = 0.05𝑘𝑡∗
𝑘𝑡∗
=
1
1
1−
(2.5) 4
4
= (15)3 = 36.99
1
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = (1)[36.99]4 = 2.47
A
A2
1.00
0.50
1.00
1.00
1.00
s
B2
0.75
0.75
0.375
0.75
0.75

C2
0.05
0.05
0.05
0.04
0.05

D2
0.33
0.33
0.33
0.33
0.25
k*
E2=(A2*B2/C2)^(1/(1-D2))
58.09
20.54
20.54
81.19
36.99
y*
=A2*E2^(D2)
3.87
1.37
2.74
4.33
2.47
Steady State and Saving
1. The saving rate is positively correlated with steady-state capital-per-worker and steady-state
output-per-worker.
Experiment with different levels of the saving rate
Notice that the intersection of the saving curve with the depreciation curve happens sooner if
the saving rate is smaller: steady-state capital-per-worker is lower if the saving rate is lower.
This means that steady-state output-per-worker is lower.
s
k*
y*
0.80
64.00
4.00
0.75
58.09
3.87
0.50
31.62
3.16
0.25
11.18
2.24
0.10
2.83
1.41
A high saving rate suggests that economic agents value future consumption relatively more than
they value current consumption. This “thrifty” behavior allows them to have very tight belts
today but a large amount of consumption in the future.
2. The saving rate has no effect on the growth rate of output in the steady state.
In the steady state, capital-per-worker doesn’t change. That follows from the definition of the
steady state.
∆𝑘𝑡∗ = 0
For the same reason, output-per-worker doesn’t change.
∆𝑦𝑡∗ = 0
So what is the effect of a change in the saving rate on the growth rate of output in the steady
state? None.
The reason is diminishing returns to capital. Dedicating a greater proportion of income to
capital accumulation doesn’t change the fact that, eventually, capital stops being very
productive.
3. But changing the saving rate affects the growth rate of output in the transition to the steady
state.
Increasing the saving rate gives an economy more new capital per unit of old capital, so it
manages to keep depreciation at bay for a longer time.
Imagine two countries that have identical technologies and that start out with the same level of
capital-per-worker. Both country A and country B take their existing (for the moment identical)
amount of capital, produce (identical) output with it. The countries then save some output –
but because at the same time, some of the capital depreciates, not all the saving goes to new
capital. Some of it merely replaces the worn out capital.
The only difference between country A and country B is their saving rates. Country A saves a
greater proportion of its income, perhaps because it has a better financial system, one that
makes it easier and safer to save rather than to spend thoughtlessly.
Country A will have more capital left over after depreciation to put in more new capital, which
allows it to continue growing. Diminishing returns eventually will stop growth, but at a higher kt.
kt
y
0.75 yt
kt
kt
10.000
2.154
1.616
0.500
1.116
11.116
2.232
1.674
0.556
1.118
12.234
2.304
1.728
0.612
1.116
57.000
3.849
2.886
2.850
0.036
58.090
3.873
2.905
2.905
0.000
59.000
3.893
2.920
2.950
-0.030
kt
y
0.50 yt
kt
kt
10.000
2.154
1.077
0.500
0.577
10.577
2.195
1.098
0.529
0.569
11.146
2.234
1.117
0.557
0.560
31.000
3.141
1.571
1.550
0.021
31.620
3.162
1.581
1.581
0.000
32.000
3.175
1.587
1.600
-0.013
The Optimal Level of Saving – the Golden Rule
More saving means more capital accumulation and more output. What about consumption? A country
that saves a lot will have a lot of output, but won’t eat a whole lot – consumption will be very low. A
country that saves a little will have little output, will eat almost all of it … and consumption will be very
low, too. Viewed in a different way, a society could choose to increase its consumption for today – have
a big nice party, at the expense of consumption future generations. Or a society could also choose allow
for more consumption for future generations, but only by reducing today’s consumption.
We want to know the saving rate that gives the optimal level of consumption. Choosing an “optimum”
means choosing
saving rate  steady-state capital-per-worker  steady-state output-per-worker
 biggest steady-state consumption-per-worker
At the “optimum”, switching things around must make everyone worse off. The optimum point would
be such that everyone, present and future, is better off than in any other point.
The Golden Rule level of steady-state capital-per-worker is that which gives the same level of
consumption to current and future generations. Optimal consumption (and therefore optimal 𝑦𝑡∗ and 𝑘𝑡∗
and s) is that which makes everyone best-off.
The graph below shows three different economies in their steady states. (Notice the three economies
depicted are all in the steady state).
Now, if consumption is the difference between output and saving, and if output is denoted by the blue
line while saving (and investment) are denoted by the red line (or the maroon or the orange lines), the
vertical distance between the two lines must be equal consumption (at any level of capital).
𝐶 𝑌 𝑆
= −
𝑁 𝑁 𝑁
This is also true at the steady state.
𝐶 ∗ 𝑌∗ 𝑆 ∗
=
−
𝑁
𝑁 𝑁
Notice, too, that the level of consumption-per-worker is different at different steady states, which are
determined by the different saving rates.
Saving Rate
Steady-State
Capitalper-worker
Steady-State
Output-perworker
Steady-State
Savingper-worker
Steady-State
Consumptionper-worker
s
k*
y*
S*/N
C*/N
1.00
89.44
4.47
4.47
0.00
0.90
76.37
4.24
3.82
0.42
0.80
64.00
4.00
3.20
0.80
0.70
52.38
3.74
2.62
1.12
0.60
41.57
3.46
2.08
1.39
0.50
31.62
3.16
1.58
1.58
0.40
22.63
2.83
1.13
1.70
0.30
14.70
2.45
0.73
1.71
0.20
8.00
2.00
0.40
1.60
0.10
2.83
1.41
0.14
1.27
0.00
0.00
0.00
0.00
0.00
So if steady-state consumption-per-worker is income-per-worker minus saving-per-worker and saving
per worker is a proportion s of income, while income is determined by the production function
𝐶∗
= 𝑦𝑡∗ − 𝑠𝑦𝑡∗ = 𝑓(𝑘𝑡∗ ) − 𝑠𝑓(𝑘𝑡∗ )
𝑁
Now remember that in the steady state, ∆𝑘𝑡∗ = 0 implies 𝑠𝑓(𝑘𝑡∗ ) = 𝛿𝑘𝑡∗. Then steady-state
consumption-per-worker is given by
𝐶∗
= 𝑓(𝑘𝑡∗ ) − 𝛿𝑘𝑡∗
𝑁
We know that the saving rate is positively related to the steady-state capital-per-worker. So we can
focus on finding 𝒌𝒕∗. We do this by taking a derivative of the above function with respect to 𝑘𝑡∗ and
setting it equal to zero.
𝐶∗
𝑁 = 𝑓′(𝑘 ∗ ) − 𝛿 = 0
𝑡
𝑑𝑘𝑡∗
𝑑
𝑓′(𝑘𝑡∗ ) = 𝛿
𝑀𝑃𝐾 = 𝛿
This tells us that the consumption-maximizing steady state is one were the MPK is equal to the rate of
depreciation. This is a basic “micro” conclusion: marginal benefit must equal marginal cost.
What is the benefit of owning capital? You get to produce output. What is the benefit of an
extra unit of capital? The extra output, the MPK. So the MPK is the benefit of giving up some
current consumption to purchase a long-lived asset for the future.
What is the cost of owning capital? If you had partied away your wealth at least you’d have had
the good times. But if you hold capital, after a while you get a rusty, moth-eaten bit of junk.
Capital decays. What is the benefit of owning an extra bit of steady-state capital-per-worker? It
wears and tears, at the rate of depreciation. So the cost of saving a bit more to accumulate
capital is the depreciation rate.
So we adjust our foregoing of consumption until the marginal benefit of amount of capital we hold (the
MPK) is equal to the cost of holding capital (the depreciation rate). That is the optimal amount of rate of
saving, of foregoing consumption.
For example, for the Cobb-Douglas function
𝑀𝑃𝐾 ∗ = 𝛼
𝑦𝑡∗
𝑘𝑡∗
If the consumption-maximizing steady-stead level of capital-per-worker is given by 𝑀𝑃𝐾 = 𝛿
𝛼
𝐴(𝑘𝑡∗ )𝛼
=𝛿
𝑘𝑡∗
𝛼𝐴(𝑘𝑡∗ )𝛼−1 = 𝛿
1
And since we found above that
𝑘𝑡∗
=
𝑠𝐴 1−𝛼
(𝛿) ,
MPK= means
𝛼𝐴(𝑘𝑡∗ )𝛼−1 = 𝛼𝐴 [(
𝑠
𝑜𝑝𝑡𝑖𝑚𝑎𝑙
𝐴
𝛿
1 𝛼−1
1−𝛼
)
=𝛿
]
𝛿
𝛼𝐴 ( 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 ) = 𝛿
𝑠
𝐴
𝛿
𝛼 ( 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 ) = 𝛿
𝑠
𝑠 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 = 𝛼
1
In the specific case of 𝛼 = 1/3, 𝑠 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 = 3. So if the capital contribution to output () is about onethird across the world, more or less, the optimal saving rate should average one-third, across the world.
1
Using the definitions of the steady state, 𝑠 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 = 3 means
1
1
3
(1/3)(1) 1−1/3
𝑠𝐴 1−𝛼
20 2
𝑘𝑡∗ = ( )
=(
= ( ) = 17.21
)
(0.05)
𝛿
3
1
𝛼
𝑠𝐴 1−𝛼
𝑦𝑡∗ = 𝐴 [( ) ] = (1)(17.21)1/3 = 2.58
𝛿
𝑀𝑃𝐾 ∗ = 𝛼
𝑦𝑡∗ 1 2.58
=
= 0.05
𝑘𝑡∗ 3 17.21
So this economy accumulates capital until the extra benefit from holding it and foregoing consumption
(the MPK) is equal to the extra cost of holding capital instead of eating it (the depreciation rate).
Compare this with a low-saving economy 𝑠 = 0.10 and a high-saving economy 𝑠 = 0.60.
At a saving rate of 10%, the steady-state level of capital is 2.83. At that level, the MPK (the slope of the
production function) is larger than the depreciation rate.
1
1
3
(0.10)(1) 1−1/3
𝑠𝐴 1−𝛼
20 2
𝑘𝑡∗ = ( )
=(
= ( ) = 2.83
)
(0.05)
𝛿
10
1
𝛼
𝑠𝐴 1−𝛼
𝑦𝑡∗ = 𝐴 [( ) ] = (1)(2.83)1/3 = 1.41
𝛿
𝑦𝑡∗ 1 1.41
𝑀𝑃𝐾 = 𝛼 ∗ =
= 0.16
𝑘𝑡 3 2.83
∗
This means that, for each unit of forgone consumption in the steady-state, this economy is getting a 16%
return in terms of more output (MPK). It would make sense to keep accumulating capital, but the
citizens of this economy are kind of short-sighted and prefer to consume today. Because they don’t
save much, there are so few units of (high-productivity) capital that it is just enough to offset
depreciation.
At a saving rate of 60%, the steady-state level of capital is 41.57. At that level, the MPK (the slope of the
production function) is smaller than the depreciation rate.
1
1
(0.60)(1) 1−1/3
𝑠𝐴 1−𝛼
𝑘𝑡∗ = ( )
=(
= 41.57
)
(0.05)
𝛿
1
𝛼
𝑠𝐴 1−𝛼
𝑦𝑡∗ = 𝐴 [( ) ] = (1)(41.57)1/3 = 3.46
𝛿
𝑀𝑃𝐾 ∗ = 𝛼
𝑦𝑡∗ 1 3.46
=
= 0.0278
𝑘𝑡∗ 3 41.57
This means that, for each unit of forgone consumption in the steady-state, this economy is getting a
2.78% return: not a whole lot, given that the cost of holding capital rather than eating is 5%. But this
economy is saving a lot, so even though their capital is very unproductive, the citizens forego so much
consumption that they manage to offset depreciation.
Strengths and Weaknesses of the Solow Growth Model
Strengths
It explains why countries are rich or poor in the very long run
𝑦𝑡∗



1
𝛼
1−𝛼
= (𝐴)1−𝛼 (𝑠⁄𝛿 )
Total Factor Productivity (related perhaps to education, legal environment, etc.)
Saving and investment (related perhaps to culture and the quality of the financial system)
Low rate of depreciation (related perhaps to weather or the quality of machinery)
It explains why growth rates differ between countries with similar steady states


Countries that are closer to the steady-state (like the US) grow more slowly
Countries that are farther away from the steady-state (like Ireland) grow more slowly
Weaknesses
Leaves saving rates as exogenous. Saving rates are probably
related to how well the financial system functions, how patient
people are, or how the tax system punishes or rewards saving
Leaves Total Factor Productivity as exogenous. Because capital
accumulation cannot lead to long-run growth (eventually,
output growth stops in the steady state), the Solow Growth
Model is not a theory of long-run growth, but a theory of
transition dynamics.