MACROECONOMICS
BGSE/UPF
LECTURE SLIDES SET 2
Professor Antonio Ciccone
BGSE/UPF Macroeconomics
Slide SET 2
Slide 1
5. SAVINGS, INVESTMENT AND THE CREDIT MARKET
EQUILIBRIUM—
OR FROM THE RENTAL PRICE OF CAPITAL TO THE REAL
INTEREST RATE
1. Investment and savings meet in the credit (also loan) market
Households may have a PREFERENCE to save for consumption tomorrow.
This is captured by the following extremely simple SAVINGS FUNCTION:
(E19)
S (t )  sY (t )
Households simply save a constant fraction s of their total income Y (this includes
labor and capital income).
Savings are deposited in banks that use them to make loans to firms. Firms use
credit to buy NEW machines. INVESTMENT refers to the total purchases of NEW
MACHINES and ALSO TO THE VOLUME OF LOANS IN THE ECONOMY.
Firms can therefore buy machines or instead rent them. This will give rise to the
rent-or-buy decision which determines the real interest rate.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 2
FIGURE 7
HOUSEHOLDS
(aggregate labor endowment L(t) plus property rights in firms;
preferences for consumption today and savings)
CREDIT/LOAN
MARKET
(credit/loans for
interest)
GOODS MARKET
(consumption and
investment goods)
LABOR MARKET
FIRMS (technology of production; capital
owned at the beginning of the period K(t))).
RENTAL MARKET
FOR CAPITAL GOODS
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Slide SET 2
Slide 3
How firms finance purchase of new
machinery
(1) Loans: firms ask banks for loans and banks make these loans
with the savings of households
(2) Retained earnings: firms ask their owners whether they can
retain some of their earnings in order to fund purchases of new
machinery
(3) Issues of new shares: they purchases new machines and issue
property titles to the machines (shares) directly to households
 In the Solow environment these ways of financing machinery
are all EQUIVALENT.
We can therefore just think of firms financing the purchase of new
machinery by asking banks for loans.
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Slide SET 2
Slide 4
2. The rent or buy decision
1. The user cost of capital definition in discrete time
Firm can demand credit/loans to purchase capital goods:
- in principle, this gives them an alternative to the rental market for capital
- instead of renting the capital good next year, for example, you could buy it
on credit today, use it for one year and then sell it off.
The cost of doing so is the USER COST OF CAPITAL
User Cost One-year-Period t
 1  r (t )  * pK (t )  1    pK (t  1)
-higher real interest rate and depreciation rate increase user cost
-high future price for capital goods relative to current price reduces user cost
BGSE/UPF Macroeconomics
Slide SET 2
Slide 5
2. The user cost in one-sector growth models (which
includes, among many, the Solow model)
Assume that consumption goods and investment goods can be produced
with identical technologies.
In this case, the price of the investment goods relative to the capital good
is always unity. If investment goods were more expensive only
investment goods would be produced by profit maximizing firms. And vice
versa.
User Cost One-year-Period t
(E21)
 1  r (t )   1   
 r (t )  
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Slide SET 2
Slide 6
3. The credit/loan market equilibrium
The CREDIT/LOAN MARKET is in equilibrium at time t when:
SAVINGS(t)=INVESTMENT(t)
S(t)=I(t)
As we will see, the CREDIT/LOAN market is brought into equilibrium by
adjustments of the REAL INTEREST RATE.
The rent or buy decision and the credit/loan market equilibrium:
-firms can either buy capital goods today (invest) for future use, or go to the
rental market for capital in the next period.
-the cost of renting next period is R(t  1) the cost of buying the
investment goods use it and resell what is left is r (t )  
BGSE/UPF Macroeconomics
Slide SET 2
Slide 7
We will now show that in a CREDIT MARKET EQUILIBRIUM where
INVESTMENT=SAVINGS>0:
(E22)
R(t  1)  r (t )  
PART 1 OF THE ARGUMENT:
Can it be that
R(t  1)  r (t )   and that, at the same time the credit
market is in equilibrium?
No: in this case no firm wants to invest and I<S. Given the expected rental cost
of capital, firms find it cheaper to rent some of this existing capital rather than
invest in new one. Hence, the savings of HH are greater than the investment
(which is 0). This situation will be eliminated by a fall in the real interest rate.
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Slide SET 2
Slide 8
PART 2 OF THE ARGUMENT:
Can it be that R(t  1)  r (t )  
credit market is in equilibrium?
and that, at the same time the
No: in this case firms want infinite investment because they can earn a profit by
buying capital today and rent it out tomorrow. Hence, the desired investment by
firms will be greater than the desired savings by firms, I>S.
Hence, for the credit market to be in equilibrium firms must be indifferent
by buying today and expecting to rent next period
R(t  1)  r (t )  
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Slide SET 2
Slide 9
Because the expected rental price of capital next period is equal to the expected
equilibrium MPK, see the static equilibrium condition, we obtain:
(E23)
MPK (t  1)  r (t )  
-in equilibrium firms invest to the point where the MPK is equal to the real
interest rate plus depreciation
MPK (t  1)    r (t )
- the equilibrium real interest rate is equal to the net MPK of the capital a
society has accumulate up to time t
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Slide SET 2
Slide 10
3. Summarizing the credit market equilibrium
(E24)
(E25)
MPK (t )  r (t )  
I (t )  S (t )
The STATIC EQUILIBRIUM CONDITIONS,
-in the LABOR MARKET and the RENTAL CAPITAL MARKET
and
-the CREDIT MARKET EQUILIBRIUM CONDITIONS
 DYNAMIC GENERAL EQUILIBRIUM
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Slide SET 2
Slide 11
4. The credit market equilibrium and the link between present and future
(or the capital accumulation equation in equilibrium)
As far as the economics of the Solow model are concerned we are done!
-We know how to determine output at a given moment in time given L and K; we
also how to obtain factor prices and the real interest rate.
- Now we know how to determine tomorrow’s capital stock given today’s capital
stock and employment:
Depreciation
(E26)
K (t )
K (t ) 
 I (t )   K (t )  S (t )   K (t )
t
Net Investment
Gross
Investment
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Slide SET 2
Slide 12
Making use of HH SAVINGS behavior in (E19) S(t)=sY(t)
(E27)
K (t )  sY (t )   K (t )
Making use of the fact that aggregate income=aggregate output:
Y  F ( K , AL)
(E28)
K (t )  sF ( K (t ), A(t ) L(t ))   K (t )
this is the EQUILIBRIUM CAPITAL ACCUMULATION EQUATION
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Slide 13
6. THE DYNAMICS OF THE SOLOW MODEL
1. The dynamics of capital accumulation
To solve the Solow model completely, we need to specify the evolution over time
of some EXOGENOUS factors like EFFICIENCY A and LABOR SUPPLY L
We will assume that labor supply grows at (exogenous) rate n:
(E29)
L(t )  L(0)ent
(E30)
L(t )  nL(t )
(E31)
L(t )
n
L(t )
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Slide 14
Similarly, we will assume that exogenous efficiency A we will assume growth at
(exogenous) rate a:
(E32)
A(t )  A(0)eat
(E33)
A(t )  aA(t )
(E34)
A(t )
a
A(t )
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Slide 15
SUMMARIZING THE DYNAMIC EQUATIONS OF THE SOLOW MODEL
(E35)
Kt  sF ( Kt , At Lt )   Kt
(E36)
At  aAt
(E37)
Lt  nLt
Following all 3 (so-called state) variables separately over time is somewhat
cumbersome and inconvenient.
And WE DO NOT HAVE TO, because as we have seen many things depend on
capital per efficiency worker, NOT separately on K, L, and A.
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Slide 16
More convenient to focus on the change over time of CAPITAL PER EFFICIENCY
WORKER
Kt
kt 
At Lt
WHY?
- determines output per efficiency worker through the production function in
efficiency form
Yt
yt 
At Lt
 f  kt 
- getting to the quantities we are interested in is simple
yt  yt At
(E38)
rt    f '(kt )
(E39)
(E40)
wt
 f (kt )  kt f '(kt )
At
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Slide 17
From TIME CHANGES OF K,L,A to TIME CHANGES in capital per efficiency worker


 Kt 
kt  

A
L
 t t


Kt 
Kt 
Kt
kt 

L  2 At
2 t
At Lt At Lt
At Lt




Kt Lt
K t At
Kt
kt 


At Lt At Lt Lt At Lt At

(E41)

Kt
kt 
 kt n  k t a
At Lt
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Slide 18
Using the equilibrium capital accumulation equation (E28):


(E42)
sF ( Kt , At Lt )   Kt
Kt
kt 
 kt n  kt a 
 kt n  kt a
At Lt
At Lt

sF ( K t , At Lt )
kt 
 kt   k t n  k t a
At Lt
(E43)

(E44)
kt 
 (  n  a)kt
sf (kt )
ACTUAL SAVINGS
AND INVESTMENT
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BREAK-EVEN
INVESTMENT
Slide 19
This equation gives us the change in
Therefore, given a starting point
kt
k (0)
as a function of the present
kt
, it allows to study the whole time path of
kt .
SO NOW WE ARE DONE WITH THE “MECHANICAL”
DYNAMIC ASPECTS OF THE SOLOW MODEL TOO.
Now is the time to remember: What is it we want to know
about?
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Slide SET 2
Slide 20
What we want to know:
“INTERMEDIATE” QUESTIONS
- Will capital per efficiency worker INCREASE or FALL over time?
- Will capital per efficiency worker GROW FOREVER?
- Will the GROWTH RATE of capital per efficiency worker INCREASE or
DECREASE in time?
“FINAL” QUESTIONS
- What does this imply for INCOME, WAGES, and INTEREST RATES.
The questions are most easily approached graphically.
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Slide 21
FIGURE 8a Following capital per efficiency worker in time:
THE PRODUCTION FUNCTION
y  f k 
0
k (t )
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Slide 22
FIGURE 8b Following capital per efficiency worker in time:
SAVINGS AND THEREFORE INVESTMENT
y  f k 
sf  k 
k (t )
0
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Slide 23
FIGURE 8c Following capital per efficiency worker in time:
THE EFFECTIVE DEPRECIATION LINE
(n    a )k
sf  k 
0
k (t )
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Slide 24
FIGURE 8d Following capital per efficiency worker in time:
CAPITAL GROWTH
(n    a )k
sf  k 
kt  0
0
k (t )
k (0)
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Slide 25
FIGURE 8e Following capital per efficiency worker in time:
THE CAPITAL GROWTH ZONE
(n    a )k
sf  k 
0
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k BGP
Slide SET 2
k (t )
Slide 26
FIGURE 8f Following capital per efficiency worker in time:
FALLING CAPITAL ZONE
(n    a )k
sf  k 
0
k (0)
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Slide 27
k (t )
FIGURE 8g Following capital per efficiency worker in time
(n    a )k
sf  k 
0
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k BGP
Slide SET 2
k (t )
Slide 28
Some important terminology
- BALANCED GROWTH PATH (also called STEADY STATE sometimes)
An equilibrium where all variables grow at constant rates
(this growth rate can be 0)
- GLOBALLY STABLE BGP
A BGP is globally stable if the economy ends up in the BGP in the long run NO
MATTER WHERE THE ECONOMY STARTS.
- CONVERGENCE
A somewhat fuzzy concept. Many people seem to say that there is convergence if
the growth rate of income per capital decreases as the country grows richer.
BGSE/UPF Macroeconomics
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Slide 29
The growth rate of capital per efficiency worker over time

(E45)
kt 
sf (kt )
 (  n  a)kt
ACTUAL SAVINGS
AND INVESTMENT
BREAK-EVEN
INVESTMENT

(E46)
kt
s
kt
f (kt )
 (  n  a )
kt
AVERAGE
PRODUCT
OF
CAPITAL
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Slide SET 2
Slide 30
y  f k 
y (t )
MPK
0
k (t )
k (0)
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Slide 31
y  f k 
y (t )
0
k (0)
k (t1 )
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k (t2 )
Slide SET 2
k (t )
Slide 32
y  f k 
y (t )
APK=
AVERAGE
PRODUCT
OF CAPITAL
0
k (t )
k (0)
BGSE/UPF Macroeconomics
Slide SET 2
Slide 33
y  f k 
y (t )
APK
0
k (t )
k (0)
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Slide SET 2
Slide 34
The growth rate of capital
per efficiency worker over time

(E46)
kt
s
kt
f (kt )
kt
 (  n  a )
AVERAGE
PRODUCT
OF CAPITAL
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Slide SET 2
Slide 35
f k 
s
k
k (t )
0
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Slide SET 2
Slide 36
f k 
s
k
n   a
0
k (t )
BGSE/UPF Macroeconomics
Slide SET 2
Slide 37
f k 
s
k

kt
kt
0
n   a
k (0)
BGSE/UPF Macroeconomics
k (t )
Slide SET 2
Slide 38
f k 
s
k

kt
kt
0
n   a
k (0) k (t1 )
BGSE/UPF Macroeconomics
Slide SET 2
Slide 39
f k 
s
k

kt
kt
0
n   a
k BGP
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Slide SET 2
k (0)
Slide 40
We have therefore shown the three following results:
- Result 1: Over time capital per efficiency worker tends to its balanced
growth path value, which we have denoted by k BGP (as long as the
initial capital stock is strictly positive)
- hence, the economy will end up at the same level of capital per efficiency
worker, no matter what the initial values for A,K,L
- Result 2: The closer capital per efficiency worker to its BGP value, the lower its
growth rate
-in the absence of SHOCKS to preferences or technology, the GROWTH RATE
of capital per efficiency workers is therefore falling over time
Result 3: In the balanced growth path, growth of capital per efficiency worker is
ZERO
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Slide 41
FIGURE 11 RESULT 1: over time capital per efficiency worker
tends to its balanced growth path value
k (t )
k BGP
k (0)
0
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Time t
Slide SET 2
Slide 42
FIGURE 12 RESULT 2 and 3: the closer capital per efficiency worker to its BGP
value, the lower its growth rate; in the long-run the growth rate is ZERO

k (t )
k (t )
0
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Time t
Slide SET 2
Slide 43
2. From capital accumulation to growth of output per worker
The easiest way to make the link is to assume the production function takes the
so-called Cobb-Douglas form

Yt   Kt 
(E47)
where
0   1
(E48)
 At Lt 1
is the elasticity of output with respect to capital:
Yt Kt

Kt Yt
or
Percentage increase in Yt
  * Percentage increase in K t
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Slide 44
The Cobb-Douglas production function in efficiency unit form is

(E49)
 Kt 
Yt

At Lt  At Lt 
or
(E50)
yt  kt
hence 0    1 is also the elasticity of output per efficiency worker with
respect to capital per efficiency worker.
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Slide 45
Growth in output per worker
- ouput per worker is output per efficiency worker times efficiency
yt  At yt  At kt
(E51)
- differentiating therefore yields

(E51)


y t At y t
  
a

yt At yt EFFICIENCY
ELASTICITY
OF OUTPUT
TO CAPITAL
GROWTH
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Slide SET 2

kt
kt

CAPITAL PER
EFFICIENCY GROWTH
Slide 46
FIGURE 13 Evolution of output per worker (on LN scale)
ln y *(t )  yBGP A(t )
ln y (t )
ln y (t )
ln y (0)
0
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Slide 47
Time t
FIGURE 14: growth of output per worker

y (t )
y (t )
a
0
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Time t
Slide SET 2
Slide 48
3. Real wage growth and changes in the real interest rate
Again, the easiest case is to assume the production function takes the
so-called Cobb-Douglas form

1
 Kt
At Lt
- the real wage is
wt  MPLt 
Lt
  
(E53)
(E54)

wt  (1   ) At  Kt 

 At Lt   (1   )
 Kt   At Lt 1
Lt
Yt
wt  (1   )  (1   ) yt
Lt
wt  (1   ) yt

wt yt

wt yt
the real wage is simply A CONSTANT FRACTION of income per capita and real
wage growth is EQUAL TO output per worker growth
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Slide 49
FIGURE 15a Evolution of REAL WAGE (on LN scale)
ln w *(t )  wBGP A(t )
ln w(t )
ln w(t )
ln w(0)
0
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Slide 50
Time t
- the real interest rate is the net marginal product of capital
(E55)
rt  MPKt   

  Kt 
1
 At Lt 
Lt
hence
 1
(E56)
rt    Kt 
1
 At Lt 
 1
rt    kt 
 1
 Kt 
  

A
L
 t t

NEGATIVE
NUMBER!

the real interest rate FALLS as capital per efficiency worker increases
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Slide SET 2
Slide 51
FIGURE 15b real interest rate over time
r (t )
rBGP
0
Time t
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Slide SET 2
Slide 52
7. THE EFFECTS OF AN INCREASE IN SAVINGS ON INCOME
1. Growth in the long run (in the balanced growth path)
After having analyzed the DYNAMICS of growth for all moments in time we will
now focus on the long-run, i.e. the balanced growth path
- we already have shown that

yt

yt
kt
kt
(E57)
BGP
BGP
and
therefore
(E58)
0

yt

yt
kt
kt
BGP
 a  GROWTH LABOR-EFFICIENCY
BGP
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Slide 53
Result: The long-run growth rate output per worker of a country is
determined by the GROWTH RATE OF LABOR EFFICIENCY ONLY.
-In particular, the long-run growth rate of output per worker does NOT
depend on the SAVINGS RATE at all.
-This is because of DECREASING RETURNS TO CAPITAL IN
PRODUCTION. Recall that
(E59)
kt 
sf (kt )
ACTUAL SAVINGS
AND INVESTMENT
 (  n  a)kt
BREAK-EVEN
INVESTMENT
Because of decreasing returns to capital, f ''( kt )  0 SAVINGS per
efficiency worker rises less than proportionally with capital. But BREAK-EVEN
INVESTMENT rises proportionally. So they will be eventually equal NO
MATTER what the SAVINGS RATE may be. At that point growth in income
per capita is equal to growth in labor-efficiency.
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Slide 54
2. Output per worker in the long run (in the balanced growth path)
The savings rate does, however, affect the LEVEL OF OUTPUT PER
WORKER
- having a simple expression for output per worker in the BGP is easiest with a
Cobb-Douglas production function
- note that in the BGP:
(E60)
0
 (  n  a)kt
sf (kt )
ACTUAL SAVINGS
AND INVESTMENT
(E61)
kt
yt
BGP
BREAK-EVEN
INVESTMENT
kt
s


f (kt ) BGP   n  a
BGSE/UPF Macroeconomics
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Slide 55
- the Cobb-Douglas production function in intensive forms is
which yields:
(E62)
kt
kt
BGP
yt  kt
s

 na
solving for the BGP amount of capital per efficiency worker:
1
s
1


(E63)
k BGP  

  n  a 
the amount of capital per worker can be obtained by multiplying by efficiency A
(E64)
kt , BGP
1
1
s



  n  a 
BGSE/UPF Macroeconomics
At
Slide SET 2
Slide 56
- substituting in the production function yields output per efficiency worker and
output per worker
(E65)
(E66)
yBGP

1
s



  n  a 
yt , BGP

1
s





n

a


At
kt , BGP
s

yt , BGP   n  a
BGSE/UPF Macroeconomics
Slide SET 2
Slide 57
Hence, if efficiency growth is constant in time, as assumed, we get that
income per capita growth in the BGP is constant in time.
We also get that the CAPITAL-OUTPUT (k/y) ratio is constant in time.
A constant capital-output ratio and steady growth of income per capita is
often seen to describe the U.S. well, especially for a longer time period.
(Solow developed his model thinking of the U.S. economy.)
BGSE/UPF Macroeconomics
Slide SET 2
Slide 58
JONES Slide1: US IN BGP?
BGSE/UPF Macroeconomics
Slide SET 2
Slide 59
FIGURE 16 Effect of SAVINGS RATE on capital per efficiency worker
(n    a )k
s
HIGH
f k 
s LOW f  k 
0
LOW SAVINGS
BGSE/UPF Macroeconomics
Slide SET 2
HIGH SAVINGS
Slide 60
k (t )
FIGURE 17
Effect of SAVINGS RATE INCREASE on growth (starting from BGP)

y (t )
y (t )
a
0
Time t
INCREASE IN SAVINGS RATE
BGSE/UPF Macroeconomics
Slide SET 2
Slide 61

y (t )
y (t )
a
0
Time t
INCREASE IN SAVINGS RATE
BGSE/UPF Macroeconomics
Slide SET 2
Slide 62
8. QUANTITATIVE IMPLICATIONS OF THE SOLOW MODEL
1. Effect of savings on long run income
We have seen that the effect of the savings rate on long run output per worker
can be obtained very easily when the production function is Cobb-Douglas
(E67)
yt , BGP

1
s



  n  a 
At
-- the PERCENTAGE INCREASE in long-run income that comes from a ONEPERCENT INCREASE in the savings rate is the
(E68)
yt , BGP s


1
s yt , BGP
BGSE/UPF Macroeconomics
Slide SET 2
Slide 63
- the greater the elasticity of output with respect to capital,
greater the effect of savings on long run income

the
-- HOW LARGE IS THIS ELASTICITY? It turns out that under the assumption
of the Solow model there is a simple way to estimate 
- the definition of 
is:
Yt Kt
Kt

 MPKt
Kt Yt
Yt
- equilibrium in the capital market implies that
- hence
rt    MPK t
(rt   ) Kt

 share of CAPITAL in income
Yt
- constant returns to scale implies that all income is paid to capital or labor; therefore
(E69)
  1  share of LABOR in income
BGSE/UPF Macroeconomics
Slide SET 2
Slide 64
- the LABOR INCOME SHARE in industrialized countries is around 1/3:
(E70)

1 2 / 3
1/ 3 1



1   1  (1  2 / 3) 2 / 3 2
-- hence, increasing the savings rate by 1% raises long-run income per
capita by only 0.5% under the assumptions of the Solow model
-- HOW MUCH CAN DIFFERENCES IN SAVINGS RATE OF CAPITAL
EXPLAIN?
(E71)
yt , BGP

1
s



  n  a 
BGSE/UPF Macroeconomics
Slide SET 2
At
Slide 65
- take two countries that are identical in everything but
SAVINGS/INVESTMENT rates
-denote their savings rates by s1 and s2 respectively; what is then
the difference in long-run incomes between the two countries?
(E72)
(E73)

s1 1
yCOUNTRY 1, BGP 
 
yCOUNTRY 2, BGP  s 2 
yCOUNTRY 1, BGP  0.27 1/ 2

  93
yCOUNTRY 2, BGP  0.03 
- rather small given the enormous differences in savings rates  differences in
savings rates alone cannot explain enormous differences in income between
rich and poor countries
BGSE/UPF Macroeconomics
Slide SET 2
Slide 66
2. Income per capita versus output per worker
The Solow model is about OUTPUT PER WORKER; how do we get from there
to OUTPUT PER CAPITA?
As L=NUMBER OF WORKER, we get
(E74)
Y
WORKER Y

POPULATION POPULATION L
this can be written further as ((E75))
Y

POP
WORKINGAGE POP LABORFORCE EMPLOYMENT Y
POP
WORKINGAGE POP LABORFORCE L
BGSE/UPF Macroeconomics
Slide SET 2
Slide 67
hence
(E76)
INCOME or OUTPUT per CAPITA =
DEMOGRAPHIC FACTOR
X LABOR FORCE PARTICIPATION RATE
X (1-UNEMPLOYMENT RATE)
X OUTPUT PER WORKER
Income or output per capita may therefore be low because of
- LOW output per worker
- HIGH unemployment among those who do participate
- LOW participation of the population in the labor market
- HIGH share of children and retired persons
BGSE/UPF Macroeconomics
Slide SET 2
Slide 68
With information on OUTPUT PER HOUR WORKED, we can do even better and
decompose output per worker into:
(E80)
Y
HOURS
Y


L WORKERS HOURS
where
Hours= total hours worked in the economy
Hours/Workers= hours worked per employed person
BGSE/UPF Macroeconomics
Slide SET 2
Slide 69
The next table, from
“International comparisons of labor
productivity and per capita income”
by van Ark and McGuckin, Monthly Labor Review, July 1999
illustrates the effect of the different components for the US, Japan, and the EU (all
relative to the OECD average)
BGSE/UPF Macroeconomics
Slide SET 2
Slide 70
TABLE 1
US
EU
JAPAN
OUTPUT PER HOUR
120
103
82
OUTPUT PER WORKER
118
98
92
OUTPUT PER PERSON IN LABOR FORCE
121
94
96
OUTPUT PER WORKING-AGE PERSON (age 15-64)
130
90
102
OUTPUT PER PERSON
128
90
106
FIGURES ARE RELATIVE TO OECD AVERAGE, data refer to 1997
Hence factors other than output per hour play an important role in explaining
differences in income per capita between these rich countries/regions.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 71
FIGURE 18 But the main explanation for differences in INCOME PER CAPITA
are differences in OUTPUT PER WORKER (country or region relative to US)
Gap in GDP per Capita decomposed in Participation
Gap and Labour Productivity Gap, 2004
100
80
60
40
20
0
-20
World
Western
North
Oceania
Europe America
BGSE/UPF Macroeconomics
East
Europe/
Central
Asia
Slide SET 2
Asia
Latin
America
Middle
East
Productivity gap
Slide 72
Africa
Participation gap
9. EMPIRICAL APPLICATIONS
1. Growth accounting
The aggregate production function makes clear that GROWTH in OUTPUT can be
written in terms of GROWTH in INPUTS plus GROWTH OF EFFICIENCY

Yt   Kt 
(E81)
Yt 
 At Lt 1
1

 At 
1
 Kt   Lt 
EFFICIENCY
OF ALL INPUTS
ALL INPUTS
The EFFICIENCY FACTOR multiplying all inputs is called TOTAL FACTOR
PRODUCTIVITY (TFP)

(E82)
Yt  TFPt  Kt 

BGSE/UPF Macroeconomics
 Lt 1 
Slide SET 2
Slide 73
(E83)


Y t TFP t


Yt TFPt
ELASTICITY
OF OUTPUT
TO CAPITAL
ELASTICITY
OF OUTPUT
TO LABOR

Kt
Kt


Lt
Lt
 (1   )
AGGREGATE
CAPITAL GROWTH
AGGREGATE
LABOR GROWTH
Re-arranging allows us to see how TFP growth can be estimated:

(E84)



TFPt Y t
Kt
Lt
 
 (1   )
TFPt Yt
Kt
Lt
- OUTPUT (Y), CAPITAL (K), and EMPLOYMENT (L) are easy to estimate for
many countries
BGSE/UPF Macroeconomics
Slide SET 2
Slide 74
- but to calculate TFP growth we ALSO need to know the ELASTICITY OF OUTPUT
TO CAPITAL AND TO LABOR
- we know that, in equilibrium, the MARGINAL PRODUCT of a factor is equal to the
PRICE OF THAT FACTOR and therefore
(E85)
(E86)
Yt Kt
Kt

 MPKt
 CAPITAL SHARE OF INCOME
Kt Yt
Yt
Yt Lt
Lt
1 
 MPLt  LABOR SHARE OF INCOME
Lt Yt
Yt
- data on these INCOME SHARE is available for many countries
BGSE/UPF Macroeconomics
Slide SET 2
Slide 75
TFP growth and the SOLOW RESIDUAL
We can therefore estimate TFP growth as

(E87)



TFP t Y t
Kt
Lt
   K _ SHARE 
  L _ SHARE 
TFPt Yt
Kt
Lt
SOLOW RESIDUAL
under CONSTANT RETURNS TO SCALE, K_SHARE+L_SHARE=1 and

(E88)
 
 

 
TFPt  Y t Lt 
K t Lt 



 1  L _ SHARE 

 Kt Lt 
TFPt  Yt Lt 




GROWTH OF
OUTPUT PER
WORKER
BGSE/UPF Macroeconomics
GROWTH OF
CAPITAL PER
WORKER
Slide SET 2
Slide 76
TABLE 2 macroeconomic growth data for 2 countries
AVERAGE ANNUAL GROWTH (OVER 19601990 PERIOD)
GDP
CAPITAL
STOCK
EMPLOYMENT
SHARE OF
LABOR IN
INCOME
TECHNOLISTAN
8%
10%
4%
0.5
SAVISTAN
8%
12%
4%
0.5
- note that in both countries output per worker was growing at 8%-4%=4%
Suppose that based on this figures you are asked to make best possible forecast
of LONG-RUN INCOME PER WORKER GROWTH in SAVISTAN relative to
TECHNOLISTAN
BGSE/UPF Macroeconomics
Slide SET 2
Slide 77
TABLE 3
ANNUAL CONTRIBUTION OF
ANNUAL
GDP
GROWTH
CAPITAL
STOCK
EMPLOYMENT
TFP
TECHNOLISTAN
8%
5%
2%
1%
SAVISTAN
8%
6%
2%
0%
What implications for long-run growth does this TFP growth difference have
according to the SOLOW MODEL?
(E89)
 
 
At 
 yt 


 yt 
 At 
 COUNTRY , BGP  COUNTRY , BGP
BGSE/UPF Macroeconomics
Slide SET 2
Slide 78
(E90)
1
 At 
TFPt 
EFFICIENCY
OF ALL INPUTS
(E91)
(E92)
  
 
 TFPt   LABOR SHARE   At 
 TFPt 
 At 


 
 
 yt 
 yt 
 COUNTRY , BGP
BGSE/UPF Macroeconomics
  
 TFP t 
 TFPt 



LABOUR SHARE
Slide SET 2
Slide 79
TABLE 4 LONG-RUN OUTPUT PER WORKER FORECAST
ANNUAL CONTRIBUTION OF
ANNUAL TFP
GROWTH
SHARE OF
LABOR IN
INCOME
LONG-RUN
OUTPUT PER WORKER
FORECAST
TECHNOLISTA
N
1%
0.5
2%
SAVISTAN
0%
0.5
0%
- one would expect TECHNOLISTAN to grow faster because it has proven capable
to improve efficiency (adopt or invent better technologies)
- SAVISTAN has been growing by brute force--they were savings a lot and
therefore accumulating capital rapidly; the SOLOW model says that this cannot
lead to growth in the long run because of decreasing returns to capital
BGSE/UPF Macroeconomics
Slide SET 2
Slide 80
1. Output and TFP growth of the Asian “Tigers”
The Asian “Tigers”, South Korea, Taiwan, Hong Kong, and Singapore have had
very rapid growth of OUTPUT and OUTPUT PER CAPITA
What are the PROXIMATE CAUSES of that?:
- increases in labor force participation
- capital accumulation
- TFP
IMPORTANT TO KNOW because increases in participation and capital intensity
CANNOT FUEL GROWTH FOREVER.
Alwyn Young has analyzed this issue in
“The Tyranny of Numbers: Confronting the Statistical Realities of the East-Asian
Growth Experience” Quarterly Journal of Economics, 1995
BGSE/UPF Macroeconomics
Slide SET 2
Slide 81
Young estimates:
GDP per CAPITA GROWTH  GDP GROWTH  POP GROWTH
" N " in next figures
" D " in next figures
GDP per WORKER GROWTH
 GDP GROWTH  EMPLOYMENT GROWTH
" N " in next figures
" D " in next figures
GROWTH ON INPUTS, OFTEN WEIGHTED BY “QUALITY”
TFP GROWTH
BGSE/UPF Macroeconomics
Slide SET 2
Slide 82
YOUNG SLIDE 1: growth of income and output all 4 countries
BGSE/UPF Macroeconomics
Slide SET 2
Slide 83
YOUNG SLIDE 2: TFP growth HK
BGSE/UPF Macroeconomics
Slide SET 2
Slide 84
YOUNG SLIDE 3: TFP growth SINGAPORE
BGSE/UPF Macroeconomics
Slide SET 2
Slide 85
YOUNG SLIDE 4: TFP growth in other countries
BGSE/UPF Macroeconomics
Slide SET 2
Slide 86
2. US versus EU growth: when did the EU stop to catch up (and why)?
The US has higher levels of output per worker than the EU, but the EU has been
CATCHING UP for most over the post World-War II period.
This process of CATCHING-UP has stopped in the late 1990s Has the EU
stopped catching up because of:
- TFP growth?
- CAPITAL ACCUMULATION?
Robert Gordon from Northwestern University and co-authors have researched
this in much detail; they argue that the process stopped because of both TFP
and capital growth and that new INFORMATION and COMMUNICATION
TECHNOLOGIES (ICT) played a role
BGSE/UPF Macroeconomics
Slide SET 2
Slide 87
GORDON SLIDE 1 catch up of GDP per worker stops in 1998
BGSE/UPF Macroeconomics
Slide SET 2
Slide 88
GORDON SLIDE 2 catch up TFP also stops in 1998
BGSE/UPF Macroeconomics
Slide SET 2
Slide 89
GORDON SLIDE 3 catch up capital intensity also stops in 1998
BGSE/UPF Macroeconomics
Slide SET 2
Slide 90
2. Productivity level accounting
It is also interesting to ask how much of the difference in the LEVEL of OUTPUT
PER WORKER (AVERAGE LABOR PRODUCTIVITY) are driven by:
- CAPITAL (physical and human)
- TFP
This can be done working with the Cobb-Douglas production function

Yt   Kt 
1
 At Lt 

or
 Kt 
Yt
 TFPt  
Lt
 Lt 
implies that we can estimate the level of TFP as
yt
Output per Worker at t
TFPt   
kt
(Output per Worker at t)CAP SHARE at t
BGSE/UPF Macroeconomics
Slide SET 2
Slide 91
Labour Productivity gaps
(Output per hour, Market economy 2001)
US
UK
GER
FR
140
130
120
110
100
90
80
70
60
BGSE/UPF Macroeconomics
Slide SET 2
Slide 92
Explanations for labour productivity gaps
Capital per hour, market economy
150
GER
140
US
FR
130
120
UK
110
100
90
80
70
60
BGSE/UPF Macroeconomics
Slide SET 2
Slide 93
Table 5 For US, UK, FRANCE, and GERMANY we get
US
OUTPUT per
HOUR
CAPITAL per
HOUR
BENCHMARK
BENCHMARK
FRANCE& Minus 20%
GERMANY approximately
Similar to US
UK
Minus 30%
approximately
Minus 30-35%
approximately
BGSE/UPF Macroeconomics
Slide SET 2
Slide 94
Hence:
US versus France & Germany
- the productivity gap between US on one hand and France and Germany on the
other CANNOT be explained by PHYSICAL CAPITAL
- biggest part of the gap is due to TFP
UK
- appears to be behind in TFP and PHYSICAL CAPITAL relative to both US and
France and Germany
BGSE/UPF Macroeconomics
Slide SET 2
Slide 95
3. Convergence
1. Definition and mechanisms
Convergence:
- When poorer countries grow faster than rich countries.
Mechanisms:
- Higher average and marginal productivity of capital in poor countries (the flip
side of decreasing returns to capital)
- Technological convergence
BGSE/UPF Macroeconomics
Slide SET 2
Slide 96
2. Was there convergence among today’s rich countries?
Many of today’s rich countries have data on output per person going back to the
19th century.
This allows us to ask:
Did those that started poorer grow faster since the 19th century?
For the period 1870-1980, this is the question asked by
Baumol “Productivity, Convergence, and Welfare” American Economic Review,
1986
BGSE/UPF Macroeconomics
Slide SET 2
Slide 97
Baumol Slide1: Convergence among ex-post rich
BGSE/UPF Macroeconomics
Slide SET 2
Slide 98
The figure suggests a clear pattern of convergence among today’s rich countries.
BUT there are problems with this approach:
Problem 1: The sample we have are only countries that eventually became rich.
Maybe there were some countries that were as rich as, say, Finland, in the 19th
century, but then did poorly. These countries would then BREAK the pattern of
convergence of the figure.
What countries would that be?
-Chile
-Argentina
-Portugal
-…
DeLong considers a sample consisting of the richest countries in the 19th
century (not today) and follows them over the same time period.
DeLong “Have Productivity Levels Convergence?” American Economic Review,
1988.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 99
BGSE/UPF Macroeconomics
Slide SET 2
Slide 100
Problem 2: Related to data quality. If the data for the 19th
century is worse than that of today, we may conclude
based on this data that there is convergence when in
reality there is none.
Imagine:
- In 1870: all countries really have income the same
income, equal to y(1870)
- In 1980: all countries have STILL the same income
Now imagine that the data we have is wrong and
underestimates 1870 income for some countries and
overestimates it for others. I.e. there is measurement
error.
Then we get the following figure.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 101
Figure 19: Measurement error and (fictitious) convergence
y*
1980
1870
BGSE/UPF Macroeconomics
Slide SET 2
Slide 102
Time t
DeLong argues that if this measurement problem is taken
into account than the pattern of growth 1870-1980
actually indicates DIVERGENCE among rich countries in
the 19th century.
Probably not too surprising because:
- the forces of convergence are most likely stronger in
market economies, i.e. economies with economic
freedom and protected property rights
- many economies did not conform to this pattern over long
periods of time because of communisms, expropriatory
dictatorships etc.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 103
3. Convergence among regions
We can also look at the pattern of convergence
across regions of one country. The following
two figures are about long-term income
convergence and output convergence across
the STATES of the UNITED STATES.
This question is analyzed in Barro and Sala-i-Martin
“Convergence” Journal of Political Economy, 1990.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 104
BGSE/UPF Macroeconomics
Slide SET 2
Slide 105
BGSE/UPF Macroeconomics
Slide SET 2
Slide 106
Advantages and disadvantages of looking for convergence
across regions
Advantages:
• Regions in the same country tend to share the same
political framework. Differences in income are therefore
more likely to be driven by economics.
Disadvantages:
• Income of a region is closely related to what the region
specializes in (agriculture?, gold mining?, car
manufacturing?, financial services?) and to the process
of migration of city-formation (urbanization).
• Growth theory has little to say about all that.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 107
4. Convergence world-wide after World War II
• What if we look at the relation between output
growth and initial output per worker for as broad
a sample as possible in the post WWII period?
• As has been documented by many economists,
we get no indication of convergence. Rich
countries may on average even have been
growing faster than poor countries.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 108
BGSE/UPF Macroeconomics
Slide SET 2
Slide 109
Or for the 1965-1985 period:
BGSE/UPF Macroeconomics
Slide SET 2
Slide 110
It looks like poor countries have been falling behind in output per
worker
But does this prove the absence of the two convergence mechanisms:
- decreasing returns to capital?
- technological convergence?
Not necessarily, because BAD ECONOMIC POLICIES in many poor
countries have hindered growth.
For example:
• not protecting even most basic human and economic rights
• not investing in critical factors like infrastructure (roads, electricity
etc.) and human capital (primary and secondary education)
BGSE/UPF Macroeconomics
Slide SET 2
Slide 111
1. Cross-country convergence in the Solow model
CASE 1:
-- poor country: poor only because SCARCE
CAPITAL: LOW K(0)/L(0)
-- rich country: rich only because ABUNDANT
CAPITAL: HIGH K(0)/L(0)
in all other dimensions the two countries are
identical (s,,n,a,A(t))
BGSE/UPF Macroeconomics
Slide SET 2
Slide 112
f k 
s
k
kt
kt
poor k
n   a
rich k
BGSE/UPF Macroeconomics
k (t )
k BGP
Slide SET 2
Slide 113
CASE 2:
-- poor country: poor because SCARCE
CAPITAL and LOW SAVINGS RATE
-- rich country: rich because ABUNDANT
CAPITAL and high SAVINGS RATE
BGSE/UPF Macroeconomics
Slide SET 2
Slide 114
s
LOW
f k 
k
s
HIGH
kt
kt
poor k(0)
f k 
k
n   a
k BGP, poor
BGSE/UPF Macroeconomics
rich k(0)
Slide SET 2
k BGP,rich
Slide 115
k (t )
2. Conditional convergence
The basic idea is that there is
CONDITIONAL CONVERGENCE if
poorer countries would have grown faster
than richer countries had they adopted the
same “ECONOMIC POLICIES”.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 116
How could we check in practice? By running a regression.
For example, estimate the parameters a, b, c below
ln ycountry ,1960  ln ycountry ,1985 
 a * SchoolingInvestmentcountry  b * PoliticalStabilitycountry
GROWTH “EXPLAINED”
BY “POLICIES”
 c * ln ycountry ,1960
GROWTH “EXPLAINED” BY
“CONDITIONAL CONVERGENCE”
This is done in Barro “Economic Growth in a Cross-Section of Countries”, Quarterly
Journal of Economics 1989.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 117
Growth 1960-1985 assuming same schooling
and political stability in all countries
Income in
1960
BGSE/UPF Macroeconomics
Slide SET 2
Slide 118
Approach makes sense but also has problems. Because
ECONOMIC POLICIES are not exogenous and maybe
at least in part due to low initial incomes.
For example:
- The approach says that low growth in poor countries may
be due to low investment in schooling. But maybe low
investment in schooling is a consequence (not cause) of
the poverty of countries.
- The approach says that low growth in poor countries may
be due to political instability. But maybe political
instability is a consequence (not cause) of the poverty of
countries.
- Etc.
BGSE/UPF Macroeconomics
Slide SET 2
Slide 119
5. Forecasting growth of the BRICS
1. The who?
- Brazil, Russia, India, and China
- Large countries that are still quite poor but have shown
periods of rapid growth in recent times
- Could soon represent a huge part of world GDP, leading
to important changes in international politics and
economics (which is why Goldman Sachs is interested in
forecasting their future growth)
BGSE/UPF Macroeconomics
Slide SET 2
Slide 120
2. Forecasts
Let us consider the Goldman-Sachs forecast
of when total GDP in these countries will
overtake the GDP of Germany.
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BGSE/UPF Macroeconomics
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BGSE/UPF Macroeconomics
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As you know there are many things that
must analyzed to get a forecast of total
GDP.
Among them:
- working-age population and labor-force
participation
- output per worker
….
BGSE/UPF Macroeconomics
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BGSE/UPF Macroeconomics
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BGSE/UPF Macroeconomics
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Avoid that by building in a process of
convergence (or growth slowdown):
ln ycountry ,t 1  ln ycountry ,t 
a
b * X country ,t
0.02 * ln ycountry ,1960
This can be done by building the forecasts
of growth in these countries around the
Solow model.
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BGSE/UPF Macroeconomics
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Can one check the reliability of these forecasts?
One approach is to do WITHIN SAMPLE
FORECASTING.
- Pretend it is 1960.
- Use the approach used to forecast GDP of the
BRICs to predict growth of countries 1960-2000.
- Compare to the realized 1960-2000 growth rates.
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BGSE/UPF Macroeconomics
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