Download Notes Solow Growth Model

Notes Solow Growth Model
Point of the class
Math
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Logs and exponents
Algebraic manipulation
Partial derivatives and maximization
Model building
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The Solow Growth Model
Economic concepts and intuition
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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
Output per capita
Aggregate Production Function
State of technology (Total Factor
Productivity)
Constant returns to scale
Decreasing returns to capital
Output-per-worker
Capital-per-worker
Cobb-Douglas production function
Marginal product of capital
Real interest rate
Saving rate
Depreciation rate
Capital accumulation function
Steady-state
Transition dynamics (or catch-up to
a common steady state)
Convergence
Golden-rule level of capital
Technological progress
Homework Problems
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On the webpage.
1
Korea was able to achieve a much faster long-run rate
of growth than Nicragua. Why does GDP per worker
increase?
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:
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. The extra output
that can be obtained from extra capital is limited by
diminishing returns.
0
See Notes for Economic Growth, Principles of
Economics.
1950
1960
1970
1980
year
0
.5
𝑦 = 𝐴𝑘 1/3
SWITZERLAND
2
1
1.5
Capital per worker (US=1)
RGDP, predicted
1
Penn World Table 5.6. 1985 data
U.S.A.
Total Factor Productivity
.4
.6
.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
YUGOSLAVIA GREECE
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
.2
A combination of (physical and human) capital,
denoted by K, and of the state of technology
(denoted by A) is what drives living standards
2000
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
REP.
TURKEY
BOTSWANA
MOROCCO
PARAGUAY
BOLIVIA
SRI LANKA
SWAZILAND
JAMAICA
THAILAND
HONDURAS
PHILIPPINES
IVORY
COAST
ZIMBABWE
NIGERIA
INDIA
SIERRA
LEONE
ZAMBIA
NEPAL
KENYA
MADAGASCAR
MALAWI
Real GDP per worker, actual
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.
1990
Real GDP per capita, Korea
Real GDP per capita, Nicaragua
Real GDP per worker (US=1), actual and predicted
0
.5
1
1.5
Facts about growth
Real GDP per capita
5000
10000
15000
20000
Facts
0
.5
Real GDP per worker, 1985
1
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.
See Robert Solow’s Nobel Autobiography at
http://nobelprize.org/nobel_prizes/economics/laureates/1987/solow-autobio.html
2
Model Building
Simplifications
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A worker is a worker. Assume equal 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: 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)
We can pick almost any function to describe the relation between inputs and output … almost any
function. Of course, the function must be positive (more inputs mean more output), and it would be
nice if it were easy to work with. “Easy to work with” means that it should be monotonic (the relation
shouldn’t ever change signs), continuous (no breaks), differentiable (no kinks).
Diminishing returns to individual inputs
A very important feature of the production function is that it must exhibit diminishing marginal returns
to individual inputs
2𝑌 > 𝑌′ = 𝐹(2𝐾, 𝑁)
2𝑌 > 𝑌′ = 𝐹(2𝐾, 𝑁)
So, for example, the production function
𝑌 = 𝐴√𝐾√𝐿
Properties of exponents
𝑥 𝑛 𝑦 𝑛 = (𝑥𝑦)𝑛
𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑛+𝑚
𝑥𝑛
= 𝑥 𝑛−𝑚
𝑥𝑚
(𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚
3
exhibits diminishing marginal returns to individual inputs
𝐴√2𝐾√𝐿 = 𝐴�√2√𝐾√𝐿� < 2𝐴√𝐾√𝐿
The assumption of diminishing marginal returns to individual inputs is
important for two reasons
1. It is an important characteristic of nearly any production process we
can think of, once the process has been going on for a while, and
2. It is the key assumption of the Solow Model.
Constant returns to scale
CRS sometimes applies, sometimes it doesn’t. CRS means that increase all factors will double
production. While this may or may not be the case in reality, it makes the math much easier. And it
might pay off to see whether we can sustain long-run growth if returns to scale are constant (as
opposed to, say, increasing returns to scale).
2𝑌 = 𝐹(2𝐾, 2𝑁)
The production function 𝑌 = 𝐴√𝐾√𝐿 exhibits constant returns to scale
𝐴√2𝐾√2𝐿 = 𝐴�√2√2√𝐾√𝐿� = 2𝐴√𝐾√𝐿
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
From now on,
𝑌
𝑁
= 𝑦 and
𝐾
𝑁
= 𝑘.
𝑌
𝐾 𝑁
𝐾
= 𝐴𝐹 � , � = 𝐴𝐹 � , 1�
𝑁
𝑁 𝑁
𝑁
𝑦 = 𝐴𝑓(𝑘, 1) = 𝐴𝑓(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.
capital per worker, k
𝒚 = √𝒌
𝑌
𝑁
=
2𝑌
2𝑁
2𝐾 2𝑁
, �
2𝑁 2𝑁
= 𝐴𝐹 �
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
5000
=sqrt(5000)
Diminishing returns to capital
k
5000
6000
7000
8000
∆k
1000
1000
1000
∆y
70.71068 𝒚 = √𝒌
77.45967 6.748989
83.666 6.206336
89.44272 5.776716
4
returns. Log (𝑦 = 𝐴𝑙𝑜𝑔(𝑘)) and square root (𝑦 = 𝐴√𝑘) both fit the bill.
A special kind of function is the Cobb-Douglas function, which turns out to be a very useful functional
form.
Cobb-Douglas Production function
𝑌 = 𝐴𝐾 𝛼 𝑁1−𝛼 ,
0<𝛼<1
Diminishing returns to individual inputs
2𝑌 > 𝑌′ = 𝐹(2𝐾, 𝑁)
𝑌′ = 𝐴(2𝐾)𝛼 𝑁1−𝛼 = 𝐴2𝛼 𝐾 𝛼 𝑁1−𝛼
Constant returns to scale
𝑌 ′ = 2𝛼 𝑌 < 2𝑌
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
Output per worker
𝑌 ′ = 2𝑌
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
5
capital per worker, k
0
1000
y=k0.45
0
22.38721
y=k0.5
0
31.62278
y=k0.55
0
44.66836
6
Saving Behavior
Nations finance capital accumulation by giving up consumption.
Output is either consumption goods or investment goods
Income is either saved or consumed
𝐶/𝑁 + 𝐼/𝑁 = 𝑦
𝐶/𝑁 + 𝑆/𝑁 = 𝑦
Combining these two equations, we get that 1
𝐼/𝑁 = 𝑆/𝑁
How much do people save? Assume that saving is simply a proportion of income
𝑆/𝑁 = 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. So we’ll take it as an exogenous parameter. Combining both equations, we get
𝑰𝒕 /𝑵 = 𝒔𝒚𝒕
Investment Allocation equation
Depreciation and Capital Accumulation
Investment adds to capital. But some capital depreciates (wears out). Assume that a proportion δ of
capital wears out every period, so the capital that wears out is -δ kt. Then next period’s capital is
𝑘�
𝑡+1
𝑛𝑒𝑥𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 ′ 𝑠 𝐾
=
𝑘⏟𝑡
𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 ′ 𝑠 𝐾
+
𝐼�
𝑡 /𝑁
𝑛𝑒𝑤 𝐾 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑠
𝑘𝑡+1 − 𝑘𝑡 = 𝐼𝑡 /𝑁 − 𝛿𝑘𝑡
−
𝛿𝑘
�𝑡
𝐾 𝑡ℎ𝑎𝑡 𝑑𝑖𝑠𝑎𝑝𝑝𝑒𝑎𝑟𝑠
Capital Accumulation equation
Time
kt
It/N
0
1000
200
δkt
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
1
National Saving comes from households or firms (𝑆 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 ) or from the government (𝑇 − 𝐺). So national saving
𝑆 = 𝑆 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 + (𝑇 − 𝐺). If the financial market works perfectly, so that every unit saved gets lent out to
productive uses, and total national saving equals total investment. An additional source of funds is foreigners, who
send us their capital inflows (𝐾𝐼), for example in the shape of foreign aid, loans, stock-market purchases, or bank
deposits. We’ll ignore foreigners for the moment.
7
Saving Behavior
S/N = sy
y
I/N =(0.5)y
I/N =(0.6)y
I/N=(0.7)y
100
50
60
70
200
100
120
140
Depreciation
-δ kt
k
δk=(0.1)k
δk=(0.125)k
δk=(0.15)k
1000
100
125
150
2000
200
250
300
8
By how much does the capital stock change every period? It grows thanks to investment, which comes
from people’s saving out of income; it shrinks due to depreciation. So we can combine the Capital
Accumulation equation with the Investment Allocation equation
to find the change of capital-per-worker
The Steady State
𝑘𝑡+1 = 𝑠𝑦𝑡 + (1 − 𝛿)𝑘𝑡
∆𝑘𝑡 = 𝑘𝑡+1 − 𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
We have an equation for how the capital stock changes. When does capital no longer change?
0 = ∆𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
𝒔𝒚
�𝒕
𝐬𝐚𝐯𝐞𝐝 𝐨𝐮𝐭𝐩𝐮𝐭−𝐩𝐞𝐫−𝐰𝐨𝐫𝐤𝐞𝐫
=
𝑠𝑦𝑡 = 𝛿𝑘𝑡
𝜹𝒌
�𝒕
𝐝𝐞𝐩𝐫𝐞𝐜𝐢𝐚𝐭𝐞𝐝 𝐩𝐨𝐫𝐭𝐢𝐨𝐧 𝐨𝐟 𝐨𝐥𝐝 𝐜𝐚𝐩𝐢𝐭𝐚𝐥−𝐩𝐞𝐫−𝐰𝐨𝐫𝐤𝐞𝐫
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,
𝑠𝑦𝑡 < 𝛿𝑘𝑡
capital stock declines,
𝑠𝑦𝑡 > 𝛿𝑘𝑡
capital stock grows,
∆𝑘𝑡 = 0,
∆𝑘𝑡 > 0,
∆𝑘𝑡 < 0,
the economy is at the steady state
the economy is below the steady state
the economy is above the steady state
Production Function
y = Ak1/3
Production function
Labor and Technology
For the moment, assume that both labor and technology are constant.
9
𝒚𝒕 , 𝒌𝒕+𝟏 ,
𝑪/𝑵, 𝑺/𝑵, 𝑰/𝑵
Unknowns/endogenous variables
Equations
5. Allocation of Resources
𝐶/𝑁 + 𝑆/𝑁 = 𝑦
3. Resource Constraint
𝐶/𝑁 + 𝐼/𝑁 = 𝑦
4. Saving Function
𝑆/𝑁 = 𝑠𝑦
2. Capital Accumulation
𝑘𝑡+1 − 𝑘𝑡 = 𝐼𝑡 /𝑁 − 𝛿𝑘𝑡
𝒚𝒕 = 𝐴𝑘𝑡𝛼
1. Production Function
���0
�, 𝑘
𝐴, 𝑁, 𝑠, 𝛿, 𝑁
Parameters
Combining (1) and (3) implies 𝐶/𝑁 = (1 − 𝑠)𝑦: consumption is whatever is left over, after households
have decided to save. Combining (1) and (2) implies that saving is equal to investment. Combining (1),
(2), and (3) tells us how investment is financed (it’s the investment allocation function)
Putting the Building Blocks Together
Solow Diagram
The graph of 𝑦𝑡 with 𝑘 in the horizontal axis (that is, the graph of the production function) would have
to be
a) upward sloping
b) not a straight line, but a line that becomes flatter and flatter as we add 𝑘: extra units of capital
produce less and less extra output.
1/3
Suppose that the production function is 𝑦𝑡 = 𝐴𝑘𝑡 . Then assume that 𝐴 = 1, so that the equation that
1/3
you graph is 𝑦𝑡 = 𝑘𝑡 .
The graph of 𝑠𝑦𝑡 , with 𝑘 in the horizontal axis would have to be
a) also upward sloping and exhibiting diminishing returns
b) below the graph of the production function (if 𝑠 < 100%, 𝑠𝑦 is less than 𝑦)/
1/3
If the production function is, then the investment function is 𝐼/𝑁 = 𝑠𝑦𝑡 = 𝑠𝑘𝑡 . If we assume that the
1/3
saving rate is 𝑠 = 0.75, the equation that you graph is 𝐼/𝑁 = 0.75𝑘𝑡 .
The graph of 𝛿𝑘𝑡 , on the other hand, would have to be a straight line, since the depreciation rate
doesn’t depend on the level of capital. Let’s assume 𝛿 = 5%.
Put them together, and you get the Solow Diagram
10
The production function and the
investment function
The depreciation function
The Solow Diagram
11
The Principle of Transition Dynamics – “Catch up”
∆𝑘𝑡 = 𝑠𝑦𝑡 − 𝛿𝑘𝑡
1
∆𝑘𝑡 = 𝑠𝐴𝑘𝑡3 − 𝛿𝑘𝑡
1/3
k
8
20
27
58.09475019
•
∆𝑘𝑡 = 0.75𝐴𝑘𝑡
− 0.05𝑘𝑡
y
𝑨𝒌𝟏/𝟑
I
𝟎. 𝟕𝟓𝑨𝒌𝟏/𝟑
δk
𝟎. 𝟎𝟓𝒌
3.872983346
2.90473751
2.90473751
2.00
2.71
3
1.5
2.04
2.25
0.4
1.0
1.35
∆k
𝟏
𝒔𝑨𝒌𝟑𝒕
− 𝜹𝒌𝒕
1.1
1.04
0.9
0
64
4
3
3.2
-0.2
125
5
3.75
6.25
-2.5
216
6
4.5
10.8
-6.3
When the capital stock is low,
o it is highly productive (its MPK is very high)
… which means a relatively high real interest rate, which attracts saving
… which finances lots of investment on new capital,
… which more than compensates the depreciation of capital
o so capital accumulates
The farther below the steady state the economy is, the faster capital accumulates.
•
When the capital stock is high,
o it is not very productive (its MPK is rather low)
… which means a relatively low real interest rate, which fails to attract saving
… which leads to little investment on new capital,
… which fails to maintain the level of capital as it depreciates,
o 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.
12
13
Steady State and its determinants
How do we find the steady state? The definition of steady-state capital-per-worker (denoted as 𝑘𝑡∗) is
∆k ∗t = 0
Steady State Capital
For this reason, steady-state output-per-worker doesn’t change (𝑦𝑡∗ ).
∆yt∗ = 0
Steady State Output
If the capital stock is not changing (∆𝑘𝑡∗ = 0), then
General version
∆𝑘𝑡∗ = 𝑠𝑦𝑡∗ − 𝛿𝑘𝑡∗ = 0
Cobb-Douglas, 𝛼, version
∆𝑘𝑡∗ = 𝑠𝐴𝑓(𝑘𝑡∗ ) − 𝛿𝑘𝑡∗ = 0
∆𝑘𝑡∗ = 𝑠𝐴(𝑘𝑡∗ )𝛼 − 𝛿𝑘𝑡∗ = 0
𝑠𝐴𝑓(𝑘𝑡∗ ) = 𝛿𝑘𝑡∗
𝑠𝐴(𝑘𝑡∗ )𝛼 = 𝛿𝑘𝑡∗
∗
∗
𝑠𝐴� = 𝑘𝑡� ∗ = (𝑘 ∗ )1−𝛼
𝑡
𝛿
(𝑘𝑡 )𝛼
𝑠𝐴� = 𝑘𝑡� ∗
𝛿
𝑓(𝑘𝑡 )
𝑠𝐴𝑓(𝑘𝑡∗ )
=
1
1
1−𝛼
1−𝛼
�𝑠𝐴�𝛿 �
= ((𝑘𝑡∗ )1−𝛼 )1−𝛼 = (𝑘𝑡∗ )1−𝛼
𝛿𝑘𝑡∗
𝑘𝑡∗
1
1−𝛼
= �𝑠𝐴�𝛿 �
Steady-state level of capital-per-worker is
a) a positive function of the saving rate (𝑠)
b) a positive function of total factor productivity (𝐴)
c) a positive function of the contribution of capital to production (𝛼).
d) a negative function of the depreciation rate (𝛿).
To find the steady-state level of output-per-worker, we just plug 𝑘𝑡∗ into the production function:
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼
𝑦𝑡∗
𝑦𝑡∗
=
𝐴𝑓(𝑘𝑡∗ )
=
1 𝛼
𝑠𝐴 1−𝛼
𝐴 �� � �
𝛿
𝑦𝑡∗
=
𝛼
𝑠𝐴 1−𝛼
𝐴� �
𝛿
1
1−𝛼
= (𝐴)
𝛼
1−𝛼
= 𝐴(𝐴)
𝛼
𝛼
𝑠 1−𝛼
� �
𝛿
1−𝛼
�𝑠�𝛿 �
14
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. And yet, as important as Total Factor Productivity is in the
Solow Growth Model, it is left as an exogenous variable.
The Capital/Output Ratio
A useful concept is the “capital/output ratio”. It tells us how many units of capital are used to produce a
unit of output. One of the reasons why it is such a useful concept is that it is (comparatively) easy to
measure and to use in empirical tests of the theory.
𝑘𝑡∗
𝑦𝑡∗
=
1
1−𝛼
�𝑠𝐴�𝛿 �
1
(𝐴)1−𝛼 �𝑠�
𝛼
1−𝛼
𝛿�
⇒
𝑘𝑡∗
𝑦𝑡∗
=
1
1−𝛼
�𝑠�𝛿 �
𝛼
1−𝛼
�𝑠�𝛿 �
⇒
1−𝛼
𝑘𝑡∗
1−𝛼
= �𝑠�𝛿 �
𝑦𝑡∗
⇒
𝑘𝑡∗ 𝑠
=
𝑦𝑡∗ 𝛿
Interestingly, this doesn’t depend on productivity (𝐴) or on α. That’s convenient, because it’s difficult to
get empirical estimates of 𝐴.
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. 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. We can also get our hands
on the saving rate of an economy, and we can estimate the depreciation rate.
The biggest shortcoming of the Solow model is that it doesn’t take into account total factor productivity
(𝐴). But since the Solow Model’s capital/output ratio doesn’t depend on A, the model should be able to
predict it independently of the state of technology of the economy.
That is, 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 on the right-hand side. We can deal with this
through multi-variable regression, of course. On second
thought, what if the depreciation rate is very similar across
the world? Then there’s only one variable on the right-hand
side. Then the capital/output ratio should simply depend
positively on the saving rate. This turns out to be true!
15
Convergence
Suppose we have two economies with the same level of productivity (𝐴), the same depreciation rate (𝛿),
the same saving rate (𝑠) and the same production function (governed by 𝛼). Then they must have the
same steady-state level capital-per-worker.
𝑘𝑡∗
1
1−𝛼
= �𝑠𝐴�𝛿 �
If one of the economies has a lower level capital-per-worker, it is farther from its steady state, and it
must be growing faster,
on average. 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-percapita a few decades ago,
we should find that the
countries that were
poorest have had the
highest average growth
16
rates. This is the principle
of transition dynamics at
work: it implies the
convergence, or catch-up
over time, of the GDPper-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
17
The Baseline
𝛼 = 1/3, 𝑠 = 0.75, 𝛿 = 0.05, and 𝐴 = 1
1
𝑠𝐴 1−𝛼
𝑘𝑡∗ = � �
𝛿
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼
⇒
⇒
1
0.75(1) 1−1
𝑘𝑡∗ = �
� 3 = (15)3/2 = 58.09
0.05
𝑦𝑡∗ = 𝐴[58.09]𝛼 = 1[58.09]1/3 = 3.87
Low Saving
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).
1
0.375(1) 1−1
𝑘𝑡∗ = �
� 3 = (7.5)3/2 = 20.54
0.05
1
𝑦𝑡∗ = (1)[20.54]3 = 2.74
Low TFP
𝐴 = 0.5, 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 than in the baseline. A lower TFP makes workers less productive, so the marginal product
of capital is smaller and investment in physical capital is less attractive. So less saving is attracted, and
less capital is accumulated – which means that the point where new investment just barely manages to
compensate for depreciation is reached earlier.
1
0.75(0.5) 1−1
𝑘𝑡∗ = �
� 3 = (7.5)3/2 = 20.54
0.05
𝑦𝑡∗ = 0.5[20.54]1/3 = 1.37
18
Lower Saving Rate
Lower State of Technology
A
s
δ
α
k*
y*
Baseline
1.00
0.75
0.05
0.33
58.09
3.87
Low Saving
1.00
0.375
0.05
0.33
20.54
2.74
Low TFP
0.50
0.75
0.05
0.33
20.54
1.37
19
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 baseline is the same as it was in the previous example. In the k*
function, halving A or halving s give the same result:
1
𝑠𝐴 1−𝛼
𝑘𝑡∗ = � �
𝛿
But changing TFP changes both the production function and steady-state capital-per-worker. Relatively
unproductive workers have less capital-per-worker, so when TPF falls by 50%, steady-state output falls
by even more.
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼
The economy produces less even though it has the same level of capital as in the low-saving case: 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.
𝛿 = 0.04,
Low 𝜹
𝑠 = 0.75,
𝐴=1
Because the capital stock depreciates more slowly, capital-per-worker keeps growing for a longer time
before it stops.
𝑘𝑡∗ = (18.75)3/2 = 81.19
1
𝑦𝑡∗ = (1)[81.19]3 = 4.33
Low 𝜶
𝛼 = 0.25, 𝛿 = 0.05, 𝑠 = 0.75, 𝐴 = 1
Remember that the formula for the Marginal Productivity of Capital we found above. In the CobbDouglas function,
𝑌
𝑀𝑃𝐾 = 𝛼 .
𝐾
When α contracts, capital becomes less productive: the production function shifts in and it becomes
harder to attract savings. People are less likely to pass up current consumption and buy (now less
productive) capital, so capital accumulation stops sooner.
𝑘𝑡∗
=
1
1
1−
(2.5) 4
4
= (15)3 = 36.99
1
𝑦𝑡∗ = 𝐴(𝑘𝑡∗ )𝛼 = (1)[36.99]4 = 2.47
20
Lower Depreciation Rate
Lower Contribution of Capital
A
Baseline
1.00
Low delta
1.00
Low alpha
1.00
s
k*
y*
0.75
δ
0.05
α
0.33
58.09
3.87
0.75
0.04
0.33
81.19
4.33
0.75
0.05
0.25
36.99
2.47
21
Steady State and Saving
1. The saving rate is positively correlated with steady-state capital-per-worker and steady-state
output-per-worker.
The formula that defines the steady-state level of capital-per-worker and steady-state level of
output -per-worker shows that these two values depend on the saving rate. Higher levels of
saving generate higher steady states; less saving means lower steady states.
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.
22
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 happens to the growth rate of output in the steady state if the saving rate changes?
Nothing. Output becomes steady, so its growth rate is zero. Between steady states, however…
Output per worker , y
y1
With saving rate s1>s0
y0
With saving rate s0
Time
Although the level of output in the steady state depends on the saving rate, its growth rate
doesn’t. 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 –
23
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.
Period
kt
y
0.75 yt
δkt
∆kt
0
10.000
2.154
1.616
0.500
1.116
11.116
2.232
1.674
0.556
1.118
2
12.234
2.304
1.728
0.612
1.116
99
57.000
3.849
2.886
2.850
0.036
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.
100
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
δkt
∆kt
10.000
2.154
1.077
0.500
0.577
10.577
2.195
1.098
0.529
0.569
2
11.146
2.234
1.117
0.557
0.560
99
31.000
3.141
1.571
1.550
0.021
1
0
1
100
Country A will have more
31.620
3.162
1.581
1.581
0.000
capital left over after
32.000
3.175
1.587
1.600
-0.013
depreciation to put in more
new capital, which allows it to continue growing. Diminishing returns eventually will stop
growth, but at a higher kt.
a function
its derivative
24
Allocating Resources
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,
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)
So the firm maximizes this function by choosing
capital and labor.
General version
max𝐾,𝑁 [𝑃𝐹(𝐾, 𝐿) − 𝑟𝐾 − 𝑤𝑁]
A little bit of Calculus: the Power Rule
𝑓(𝑥) = 𝑎𝑥 𝑛
𝑑𝑓(𝑥)
= 𝑓′(𝑥) = 𝑎𝑛𝑥 𝑛−1
𝑑𝑥
𝑓(𝑥) = 2𝑥 2 + 3𝑥 + 5
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
Cobb-Douglas, 𝛼 = 1/3, version
max𝐾,𝑁 [𝑃𝐴𝐾 𝛼 𝑁1−𝛼 − 𝑟𝐾 − 𝑤𝑁]
max𝐾,𝑁 �𝑃𝐴𝐾1/3 𝑁 2/3 − 𝑟𝐾 − 𝑤𝑁�
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
Profit = 𝐹(𝐾, 𝑁) − 𝑟𝐾 − 𝑤𝑁
𝜕𝐹(𝐾,𝐿)
−𝑟
𝜕𝐾
=0
𝑀𝑃𝐾 = 𝑟
Cobb-Douglas, 𝛼 = 1/3, version
Profit = 𝐴𝐾 1/3 𝑁 2/3 − 𝑟𝐾 − 𝑤𝐿
1
𝐴𝐾 1/3−1 𝑁 2/3
3
−𝑟 =0
1 𝐾1/3 2/3
𝐴 𝐾 𝑁
3
1 𝐴𝐾1/3 𝑁 2/3
𝐾
3
𝑀𝑃𝐾 =
1𝑌
3𝐾
=𝑟
=𝑟
=𝑟
So for a Cobb-Douglas function, the marginal product of capital is proportional
to the average amount of output produced by 𝐾 (that is, 𝑌/𝐾), and the factor of
proportionality is 𝛼 = 1/3. In plainer English: the productivity of capital
depends on whether there’s a lot of it or a little of it.
How about the marginal productivity of labor? Take a derivative of the profit
function with respect to labor and set it equal to zero
k
1
8
27
64
125
216
MPK
0.333
0.083
0.037
0.021
0.013
0.009
25
𝜕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 − 𝛼�) = 2/3.
We can use the formula for the MPK that we derived above to calculate the capital share of output,
which would be 𝑟 (the income earned by capital) times 𝐾 (the capital stock) divided by 𝑌 (in order to
express it as a %). Then, the capital share is equal to … 𝛼.
𝑀𝑃𝐾 = 𝛼
𝑌
=𝑟
𝐾
⇒
𝛼=𝑟
𝑁
𝑌
𝐾
𝑌
So the labor share of output is (1 − 𝛼�), because (1 − 𝛼�) = 𝑤 . In practice, the share of GDP
earned by owners of capital is about 1/3, and the share of wages in GDP is about 2/3 … for most
countries in most time periods. So we’ll keep using 𝛼 = 1/3.
If the value of a company in the financial market is the value that savers give to their expected income
from the stock of capital, then the fundamental value of the stock market should be related to 𝑟, 𝐾/𝑌,
and 𝛼.
The Real Interest Rate
The real interest rate is the amount of output that a person can earn by foregoing consumption and
saving one unit of output. A unit of saving is used as a unit of investment, which is a new unit of capital,
which produces an extra MPK of output. 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.
26
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.
Remember that consumption-per-worker is
This is also true at the steady
state.
𝑆∗
𝐶∗
= 𝑦∗ −
𝑁
𝑁
The level of consumption-perworker is different at different
steady states, which are
determined by the different
saving rates.
We want to know the saving
rate that gives the optimal
level of consumption.
Choosing an “optimum” means
choosing
𝑆
𝐶
=𝑦−
𝑁
𝑁
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
optimum saving rate  optimum steady-state capital-per-worker
 optimum steady-state output-per-worker  biggest steady-state consumption-per-worker
At the “optimum”, any change must make society 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.
27
The graph below shows three different economies in their steady states. (Notice the three economies
depicted are all in the steady state).
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).
Low levels of saving produce little output – which is consumed almost entirely. High levels of saving
produce a lot of output, but little of it is consumed. Somewhere in the middle there’s a saving rate that
gives maximum consumption.
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
28
𝐶∗
= 𝑦𝑡∗ − 𝑠𝑦𝑡∗
𝑁
⇒
𝐶∗
= 𝑓(𝑘𝑡∗ ) − 𝑠𝑓(𝑘𝑡∗ )
𝑁
In the steady state, ∆𝑘𝑡∗ = 0 means that 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 cost of owning an extra bit of steady-state capital-per-worker? Its
wear-and-tear, its depreciation. So the cost of saving a bit more to accumulate one more unit of
capital is the depreciation rate.
So we forgo consumption until the marginal benefit of holding capital (the MPK) is equal to the marginal
cost of holding capital (the depreciation rate). That is the optimal amount of rate of saving, of foregoing
consumption.
Notice also that MPK is the slope of the production function: the additional output-per-worker out of
the additional capital-per-worker. 𝛿 is, well, the slope of the depreciation function. So the largest
consumption-per-worker (the largest distance between the amount produced and the amount saved in
the steady state) is found at the level of capital where the production function and the depreciation
function are parallel.
𝑀𝑃𝐾 = 𝛿
Slope of production function = slope of depreciation function
29
30
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, then the
golden-rule level of steady-state capital per worker 𝑘 𝐺𝑅 and the golden-rule output per worker 𝑦 𝐺𝑅 are
given by
𝛼
𝑦 𝐺𝑅
=𝛿
𝑘 𝐺𝑅
𝛼𝑦 𝐺𝑅 = 𝛿𝑘 𝐺𝑅
In the steady state, 𝛿𝑘𝑡∗ = 𝑠𝑦 ∗. Is this true at the Golden Rule level? Of course! It’s the golden-rule level
of steady-state capital per worker. This is just the best steady state. Then we can say, 𝛿𝑘 𝐺𝑅 = 𝑠 𝐺𝑅 𝑦 𝐺𝑅 .
𝛼𝑦 𝐺𝑅 = 𝑠 𝐺𝑅 𝑦 𝐺𝑅
𝑠 𝐺𝑅 is what we are looking for, the consumption-maximizing saving rate
𝛼 = 𝑠 𝐺𝑅
1
3
In the specific case of 𝛼 = 1/3, 𝑠 𝐺𝑅 = . So if the capital contribution to output (α) is about one-third,
more or less, the golden-rule saving rate 𝑠 𝐺𝑅 should average one-third.
Numerical exercise
Suppose that 𝐴 = 1, and 𝛿 = 5% and 𝛼 = 25%. Show that, at the Golden-Rule saving rate,
consumption-per-worker is maximized. We know that the Golden Rule says that 𝛼 = 𝑠 𝐺𝑅 and so in this
case 𝑠 𝐺𝑅 = 0.25.
Given the parameters, let’s first find the MPK (make sure that you understand the derivation)
𝑀𝑃𝐾 ∗ = 𝛼
𝑦𝑡∗
=𝛼
𝑘𝑡∗
1
𝑠𝐴 1−𝛼
𝐴 �� � �
𝛿
1
1−𝛼
𝑠𝐴
� �
𝛿
𝑀𝑃𝐾 ∗ =
𝛼
=
𝛼𝐴
1
1−𝛼
𝑠𝐴 1−𝛼
�� 𝛿 � �
=
𝛼𝐴
𝛼𝛿
=
𝑠𝐴
𝑠
� �
𝛿
(0.25)0.05 1
= (0.0125)
𝑠
𝑠
It will be handy to have formulas for steady-state capital-per-worker, steady-state output-per-worker,
and steady-state consumption-per-worker.
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1
1
𝑦𝑡∗
=
Saving Rate
0.10
0.25
0.40
𝑠𝐴 1−𝛼
𝑠(1) 1−0.25
𝑘𝑡∗ = � �
=�
= (𝑠)4⁄3 (20)4⁄3
�
(0.05)
𝛿
1 𝛼
𝑠𝐴 1−𝛼
𝐴 �� � �
𝛿
0.25
= (1)�(𝑠)4⁄3 (20)4⁄3 �
MPK
(1⁄𝑠)(0.0125)
0.125
0.050
0.031
1/4
= �(𝑠)4⁄3 (20)4⁄3 �
𝒌∗𝒕
⁄3
4
(𝑠) (20)4⁄3
2.520
8.550
16.000
= (𝑠)1⁄3 (20)1⁄3
𝒚∗𝒕
⁄3
1
(𝑠) (20)1⁄3
1.260
1.710
2.000
𝑪 ∗ ⁄𝑵
− 𝑠𝑦𝑡∗
1.134
1.282
1.200
𝑦𝑡∗
Increasing its saving rate makes this society’s steady-state capital-per-worker grow dramatically. Fastrising capital must be maintained, and that means that this country is devoting fast-rising amounts of
resources to fighting off depreciation.
On the other hand, due to diminishing returns, output is not rising that fast. At pretty low levels of
capital, there’s not much capital to maintain, so most output goes to consumption. This means that a
growing share of (slow-growing) output goes to pay for (fast-rising) depreciation, leaving less for
consumption.
Notice how, in the low-saving economy, the MPK is very high, much higher than the depreciation rate.
Recall that this means that the marginal benefit of holding an additional unit of capital exceeds the
marginal cost of that unit of capital – it would make sense for this economy to hold more capital, to save
more. But the citizens of this economy are short-sighted and prefer to consume today. Because they
don’t save much, there are so few units of (high-productivity) capital that the overall return is barely
enough to offset depreciation and the economy is at the steady state.
In the high-saving economy, on the other hand, the MPK is very low, far below the depreciation rate.
This society is holding so much capital that its marginal benefit doesn’t justify the marginal cost – it
would make sense for this economy to save less. Nevertheless, the citizens forego so much
consumption that they manage to offset the depreciation.
For this reason, in both the low-saving and the high-saving economies
consumption is lower than it could be if the society brought its saving rate
to equality with 𝛼, that is, to 25%.
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Strengths and Weaknesses of the Solow Growth Model
Strengths
It explains why countries are rich or poor in the very long run
•
•
•
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 quickly
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. A full model
would explain why people make individual choices that are (or aren’t) consistent with the optimum.
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.
33