Growth and Development
LECTURE SLIDES SET 2
Professor Antonio Ciccone
Slide 1
5. SAVINGS, INVESTMENT AND THE CREDIT MARKET
EQUILIBRIUM—
OR FROM THE RENTAL PRICE OF CAPITAL TO THE REAL
INTEREST RATE
5.1. Investment and savings meet in the credit (also loan) market
•Households may not want to consume all their current income
•Instead households may want to save some of their current income for future
consumption
This gives rise to a dynamic economy, as the future (future consumption) is
linked to current economic behavior (savings)
•In the Solow model this is captured by am extremely simple savings function
(E19)
S = Aggregate Savings
= (savings rate)*(Aggregate Income)=s*Y
• Households are assumed to deposit savings in banks
• Banks use household savings to make loans to firms that buy investment goods
Slide 2
FIGURE
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
Slide 3
How firms finance purchase of new
machinery and how households save
(1) Banks: firms ask banks for loans and household deposit savings in banks
(2) Debt obligations: firms issue debt directly and households buy debt
(3) Retained earnings (stock market): firms ask their owners whether they
can retain some of their earnings in order to fund purchases of new
machinery and household save by reinvesting their earnings
(4) Issues of new shares (stock market): firms issue new shares and
households save by buying shares
 In the Solow environment (and wider economic environments as
explained in Modigliani-Miller theorem) these ways of financing
machinery are all EQUIVALENT
 We can therefore just think of savings/financing occuring through
bank loans only
Slide 4
5.2. Neoclassical investment theory
5.2.1. The decision to buy investment goods: a one-step-at-a-time approach
5.2.2. The user cost of capital definition in discrete time
5.2.3. The user cost of capital in one-sector growth models
5.2.4. The user cost of capital and the desired future capital
5.2.5. The user cost of capital, investment, and the demand for credit/loans
5.2.6. Investment, the demand for credit/loans, and the real interest rate
Slide 5
5.2.1. The decision to buy investment goods: a onestep-at-a-time approach
Invest or not invest?
•To decide whether to buy an investment good, you can compare the cost of
the loan you need to buy this good to the future extra revenue generated by the
good
•To do that, you need to look into the future quite a bit, to forecast revenue
A one-step-at-a-time approach
•Instead you can look at it this way
•Invest of next year’s extra revenue generated by an investment good exceeds
the cost of
1)Get a loan, buy the investment good
2)Produce for a year
3)Sell the investment good, liquidate the loan
The cost of (1)-(3) is called the user cost of capital
Slide 6
One step at a time investment
Extra revenue from producing with
one more investment good for year
On 1 January:
- Get loan
- Buy investment good
On 31 December:
- Pay off loan
- Sell (used) investment good
User cost of capital
Invest if: USER COST OF CAPITAL <= EXTRA REVENUE
Slide 7
5.2.2. The user cost of capital definition in discrete time
The annualized cost of buying investment goods in terms of consumption goods
is the user cost of capital
User Cost One-year-Period t
 1  r (t  1) * pK (t )  1    pK (t  1)
•
•
r is the real interest rate
pK is the price of capital good relative to consumption goods
 higher real interest rate and depreciation rate increase user cost
 high future price for investment goods relative to current price reduces user cost
Slide 8
5.2.3. The user cost in one-sector growth models (which
includes, among many, the Solow model)
•In one-sector models, consumption goods and investment goods are the
same type of goods (or more generally can be produced with identical
technologies)
•Example: cars used both for consumption and for production
•In this case, the price of the investment goods relative to the capital
good is always unity
•Follows from a law of one price in models with perfect competition:
identical goods cost everybody the same. Therefore the relative price of
investment relative to consumption goods is 1, pK=1. And
User Cost One-year-Period t
(E21)
 1  r (t  1)   1   
 r (t  1)  
Slide 9
5.2.4. The user cost of capital and the desired
future capital stock
Consider a firm that faces a user cost of capital r+
How much does the firm want to invest for
producing next year?
Depends on extra revenue of investment goods
In our model this extra revenue is given by the
MPK next year
This is because the MPK next year gives us how
much more given workforce can produce if given
one more unit of capital
Slide 10
EXTRA REVENUE FROM INVESTMENT NEXT YEAR
= MPK NEXT YEAR
MPK(t+1)
MPK NEXT
YEAR
=
MPK(t+1)
Extra revenue from
one more investment
good of a firm that
has future capital
stock K1
MPK1
0
K1
K(t+1)
Slide 11
THE DESIRED (PROFIT MAXIMIZING) FUTURE CAPITAL STOCK
MPK(t+1)=MARGINAL REVENUE FROM INVESTMENT
r(t+1)+δ
0
K*(t+1)
K(t+1)
Slide 12
5.2.5. The user cost of capital, investment, and the demand for credit/loans
MPK(t+1)
r(t+1)+δ
(1- δ )K(t)
0
K*(t+1)
Investment demand
K(t+1)
=
Demand for
credit/loans
ITFD Growth and Development
Slide SET 2
Slide 13
5.2.6. Investment, the demand for credit/loans, and
the real interest rate
INVESTMENT DEMAND AS WELL AS DEMAND
FOR CREDIT/LOANS
r(t+1)
0
INVESTMENT,
DEMAND FOR
CREDIT/LOANS
Slide 14
5.3. INVESTMENT AND THE CREDIT/LOAN MARKET EQUILIBRIUM
INVESTMENT, DEMAND FOR CREDIT/LOANS
r(t+1)
SAVINGS, SUPPLY OF CREDIT/LOANS
EQUILIBRIUM
EQUILIBRIUM
INTEREST
RATE
0
INVESTMENT,
SAVINGS
S*=I*
Slide 15
5.3.1. Summarizing the credit market equilibrium
Investment to point
where MPK equals
user cost of capital
(E24)
(E25)
MPK  r  
I S
Links equilibrium real
interest rate to the
agggregate capital
stock
Investment financed by
credit/loans whose
supply is given by
savings
Slide 16
Summarizing ALL equilibrium conditions at time t
(E24)
(E25)
MPK (t )  r (t )  
I (t )  S (t )
PLUS EQUILIBRIUM CONDITIONS,
-in the LABOR MARKET
MPL(t )  w(t )
- in RENTAL CAPITAL MARKET
MPK (t )  R(t )
 GENERAL EQUILIBRIUM AT TIME t
Slide 17
5.3.2. The equilibrium capital accumulation equation
-- We already knew how to determine output at a given moment in time given L and K
-- Now we can also determine the future equilibrium capital stock
Step 1) The capital accumulation equation (capital accounting not equilibrium)
Depreciation

(E26)
K (t ) 
Net Investment
K (t )
 I (t )   K (t )  S (t )   K (t )
t
Gross
Investment
Slide 18
Step 2) Equilibrium and savings behavior
1) Equilibrium: Investment=Savings (I=S)
2) Savings behavior: S=sY
3) Income=Output: Y=F(K,EL)
Slide 19
Equilibrium capital accumulation equation

(E27)
K (t )  sY (t )   K (t )
Making use of the fact that aggregate income=aggregate output:
Y  F ( K , EL)

(E28)
K  sF ( K , EL)   K
• This is the EQUILIBRIUM CAPITAL ACCUMULATION EQUATION
• It links the future capital stock to current capital stock, labor force and
technology, as well as savings and depreciation rates
• This is the key dynamic equation of the Solow model
Slide 20
6.1. THE DYNAMICS OF THE SOLOW MODEL
6.1.1. The three main state variables, exogenous (E,L) and endogenous (K)
There are 3 variables in the Solow growth model that
1)Change over time
2) If one knows the value of these variables one can
a) determine the static equilibrium (given model parameters like s, , a etc.)
b) determine where the economy is going next
(Such variables of a dynamic system are often called STATE VARIABLES)
These variables are
1)Size of the labor force or aggregate labor supply (L)
2)Level of labor-augmenting technology (E)
3)Capital stock (K)
Slide 21
• Of these 3 state variables, 2 change exogenously
• These are labor force/supply L and technology E
• Both are taken to grow at constant rates, called n and a
Labor force L (aggregate labor
supply) grows at growth rate n
Labor augmenting technology E
grows at growth rate a
nt
(E32)
E (t )  E (0)eat
(E30)
L(t )  nL(t )
(E33)
E (t )  aE (t )
(E31)
L(t )
n
L(t )
(E29)
L(t )  L(0)e
(E34)
E (t )
a
E (t )
Slide 22
The third state variable is the most interesting
one, as it is endogenous
 Equilibrium capital accumulation equation
(E35)
Kt  sF ( Kt , Et Lt )   Kt
Slide 23
• Hard to believe, but now we are done with the
‘economics’ of the Solow model
• Still some work required to get the ‘mechanics’ in a form
so that we can analyze the economics in a insightful way
• This involves identifying the key variables of the dynamic
system (there turn out to be 3)
• And simplify that dynamic system (turns out we can
simplify to 1 state variable
Slide 24
The 3 state variables together define a DYNAMIC SYSTEM
•Given a point in space (K,L,E)
•We know where the system goes next
•And can therefore calculate all state variables going forward forever
(E35)
Kt  sF ( Kt , Et Lt )   Kt
(E36)
Et  aEt
(E37)
Lt  nLt
• But following all 3 (so-called state) variables separately over time is difficult
• Solow was the first to realize that we can avoid this complication when we
focus on capital per efficiency worker K/(EL)
• Rather than looking separately on K, L, and E
Slide 25
More convenient to focus on the change over time of CAPITAL PER
EFFICIENCY WORKER
Kt
kt 
Et Lt
WHY?
 determines output per efficiency worker through the production
function in efficiency form
Yt
a
yt 
Et Lt
 f  kt    kt 
 getting to the quantities we are interested in is simple
(E38)
(E39)
(E40)
yt  yt Et
rt    f '(kt )  a (kt )a 1
wt
a
 (1  a ) f (kt )  (1  a )  kt 
Et
Slide 26
6.1.3.
The simple math of percentage
changes
Or if you know the growth rate of
and the growth rate of variables x
and of z, what is the growth rate of
x*z and of x/z?
Slide 27
Growth of a product of two variables and
growth of a ratio of two variables
Slide 28
From TIME CHANGES of K,L,E to TIME CHANGES in capital per efficiency worker


k t  Kt

kt  Kt

  Et Lt 
 

E
L
  t t
The growth rule for ratios as
K/(EL) is K divided by EL


 

k t K t  Lt E t 

 
Lt Et 
kt Kt




The growth rule for products as
EL is E times L


kt Kt

 (n  a )
kt Kt
Substitute n and a for growth
rates of L and o E respectively


Kt
kt 
 (n  a )kt
Et Lt
Multiply both sides of equation
by k_tilde=K/(EL)
Slide 29
6.1.4. The equilibrium capital accumulation equation for
capital per efficiency worker K/(LE)
Slide 30
•
Gives us the future change in
• Therefore, given a starting point
kt
as a function of the present
k (0)
kt
it allows to study the whole time path of
• Mathematically, this is the simplest possible dynamic equation system: a
one-dimensional differential equation
• This is a dynamic equation that just moves along a line
•
Now we are done with the ‘mechanics’ of the dynamics
•
Time to recall what it is we want to know about?
•
And find a intuitive way to answering these questions
Slide 31
kt
6.1.5. Key questions about the dynamics of economic growth and
how to answer them
‘INTERMEDIATE’ QUESTIONS (about K/(EL))
•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’ OR ‘ULTIMATE’ QUESTIONS (about economic variables of interest)
•- What does this imply for INCOME, WAGES, and INTEREST RATES?
Slide 32
A graphical analysis of the implication of the
equilibrium capital accumulation equation for K/(EL)
Slide 33
6.1.6. A graphical analysis of the dynamics of
economic growth: the ‘SOLOW A’ GRAPH
y  f k 
0
k
Slide 34
FIGURE Following capital per efficiency worker in time:
SAVINGS AND THEREFORE INVESTMENT
y  f k 
sf  k 
0
k
Slide 35
FIGURE Following capital per efficiency worker in time:
THE EFFECTIVE DEPRECIATION LINE
(n    a )k
sf  k 
0
k
Slide 36
FIGURE Following capital per efficiency worker in time:
CAPITAL PER EFFICIENCY WORKER GROWTH
(n    a )k
sf  k 

kt  0
0
k
k (0)
Slide 37
FIGURE Following capital per efficiency worker in time:
THE CAPITAL GROWTH ZONE
(n    a )k
sf  k 
0
k BGP
k
Slide 38
FIGURE Following capital per efficiency worker in time:
FALLING CAPITAL ZONE
(n    a )k
sf  k 
0
k BGP
k (0)
Slide 39
k
FIGURE Following capital per efficiency worker in time
(n    a )k
sf  k 
0
k BGP
k
Slide 40
Some useful terminology
• BALANCED GROWTH PATH (BGP) also called STEADY STATE (SS) sometimes
An equilibrium where all variables grow at constant rates (growth rate can be 0)
• GLOBALLY STABLE BGP OR SS
A BGP or SS is globally stable if the economy ends up in the BGP in the long run NO
MATTER WHERE THE ECONOMY STARTS OUT (in terms of K,L,E). ‘History doesn’t
matter’ in the long run
• GLOBAL CONVERGENCE to BGP or SS
If the BGP or SS is globally stable there is global convergence
•ECONOMY STARTING BELOW/ABOVE THE BGP OR SS
An economy starts below the BGP or SS if it has less capital per efficiency worker than
it will have in the BGP/SS. It starts above the BGP or SS if it has higher capital per
efficiency worker than in BGP/SS
Slide 41
6.1.6. A graphical analysis of the dynamics of economic growth:
the ‘SOLOW B’ GRAPH for the ANALYSIS OF GROWTH RATES
Obtaining the growth rate of capital per efficiency worker over time:

(E45)
kt 
sf (kt )
ACTUAL SAVINGS
AND INVESTMENT
 (  n  a)kt
BREAK-EVEN
INVESTMENT

(E46)
f (kt )
kt
s
 (  n  a )
kt
kt
AVERAGE
PRODUCT
OF
CAPITAL
Slide 42
The ‘SOLOW B’ GRAPH FOR THE GROWTH RATE OF
CAPITAL PER EFFICIENCY WORKER K/EL

(E46)
kt
s
kt
f (kt )
kt
 (  n  a )
AVERAGE
PRODUCT
OF CAPITAL
Slide 43
f k 
s
k
0
k
Slide 44
f k 
s
k
n   a
0
k
Slide 45
f k 
s
k

kt
kt
0
n   a
k (0)
k
Slide 46
f k 
s
k

kt
kt
0
n   a
k (0) k (t1 )
k
Slide 47
f k 
s
k

kt
kt
0
n   a
k BGP
k (0)
Slide 48
k
6.1.7. Three main results
Result 1: Over time capital per efficiency worker tends to a balanced
growth path (BGP) or steady state (SS) value. The BGP/SS does not
depend on the initial values of K,L,E (the BGP/SS does depend on
model parameters like s,n, etc.)
 economy is globally stable (history doesn’t matter in the long run)
Result 2: The closer capital per efficiency worker to its BGP value, the lower the
growth rate of K/(EL) and therefore the growth rate of K (recall that E,L grow
at constant rates)
 in the absence of changes to preference or technology parameters, the
growth rate of capital (per efficiency worker) is therefore falling over time, as
economy approaches its BGP/SS over time
Result 3: In the BGP/SS, growth of capital per efficiency worker is zero. In
economies starting below BGP/SS the growth of K/(EL) is positive. In
economies starting above BGP/SS the growth of K/(EL) is negative
Slide 49
6.2. From capital accumulation to growth of output per worker
6.2.1. More simple math of percentage changes with application to Cobb-Douglas
production function
Once we know the growth of K,E,L, and therefore the growth rate of K/(EL), we
can determine the growth rate of Y,Y/L, and Y/(EL) using the production function
a
(E47)
where
0  a 1
Yt   Kt 
1a
 Et Lt 
is the elasticity of output with respect to capital
To do so, some more simple math of percentage changes is useful.
Growth rule for variables raised to a power:
Slide 50
For our Cobb-Douglas production function the growth rules imply
1) When the capital stock K grows but L and E are constant
ALPHA tells us the percentage growth in Y
when K grows by 1%
This is called the ELASTICITY OF OUTPUT
with respect to CAPITAL
Slide 51
For our Cobb-Douglas production function the growth rules imply
2) When the labor force L grows but K and E are constant
1 MINUS ALPHA tells us the percentage
growth in Y when L grows by 1%
This is called the ELASTICITY OF OUTPUT
with respect to LABOR
Slide 52
For our Cobb-Douglas production function the growth rules imply
3) When the labor force L and capital both grow but E is constant
Slide 53
6.2.2. The Cobb-Douglas production function in per worker form
a
(E49)
(E50)
 Kt 
Yt

Et Lt  Et Lt 
yt  Et yt  Et kta
Slide 54
Our growth rules therefore yield the following expression
for output per worker growth
yt  Et yt  Et kta

(E51)


yt E t yt

 
a

yt Et yt EFFICIENCY
GROWTH
Apply growth
rule for
products
ELASTICITY
OF OUTPUT
TO CAPITAL

kt
kt
a
CAPITAL PER
EFFICIENCY GROWTH
Apply growth rule for
variables raised to power
Slide 55
FIGURE growth of output per worker for economy starting
below BGP/SS

y (t )
y (t )
a
0
Time t
Slide 56
FIGURE Evolution of output per worker on LN scale (or ratio
scale) for economy starting below BGP/SS
BGP/SS output per worker
ln y (t )
ln y (t )
ln y (0)
0
Slide 57
Time t
6.3. Real wage growth and changes in the real interest rate
The real wage is determined by w=MPL
a
(E52)
(E53)
(E54)
wt  MPLt 
a
wt  (1  a ) Et  Kt 
  Kt 
1a
 Et Lt 
Lt
 Et Lt a  (1  a )
 Kt a  Et Lt 1a
Lt
Yt
wt  (1  a )  (1  a ) yt
Lt
product rule for growth rates 
wt yt

wt yt
The real wage is simply A CONSTANT FRACTION of income per capita and
real wage growth is therefore EQUAL TO output per worker growth
Slide 58
The real interest rate is determined by r+=MPK
(E55)
(E56)
rt  MPKt   
a 1
rt  a  Kt 
a
  Kt 
 Et Lt 
rt  a  kt 
 Et Lt 
Lt
1a
a 1
1a
a 1
 Kt 
 a 

E
L
 t t

NEGATIVE
NUMBER!

 the real interest rate FALLS as capital per efficiency worker increases
 the real interest rate INCREASES as capital per efficiency worker falls
Slide 59
FIGURE: real interest rate over time for economy
starting below BGP/SS
r (t )
rBGP
0
Time t
Slide 60
7. THE EFFECTS OF AN INCREASE IN SAVINGS ON INCOME
7.1. Growth in the long run (in the balanced growth path)
This was the dynamics of economic growth
Now let’s see where the economy ends up in the long run
But when does the long run start? It is when the economy has reached the
BGP/SS
The economy is in the BGP/SS when there is no longer growth per
efficiency worker

(E57)

yt

yt
kt
kt
BGP
0
BGP
 This is the condition we need to work with to find out about the long run
economics
Slide 61
Key result about the determinants of long run growth of capital per worker
and output per worker (the rate of economic growth of the economy once it
has reached the BGP/SS)


yt

yt
kt
kt
BGP
0
BGP
Implies once we recognize that K/(EL)=(K/L)*E and that Y/(EL)=(Y/L)*E
and apply the growth rule for products


(E58)
yt

yt
kt
kt
BGP
 a  GROWTH LABOR-EFFICIENCY
BGP
Slide 62
Key result for long run economic growth: The long-run growth rate of output per
worker of a country is determined by the GROWTH RATE OF LABOR EFFICIENCY
ONLY
This implies in particular that the long-run growth rate of output per worker does
NOT depend on the SAVINGS RATE at all
INTUITION: The result is driven by DECREASING RETURNS TO CAPITAL IN
PRODUCTION. Recall that
(E59)
Because of decreasing returns to capital, 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.
Slide 63
FIGURE: Effect of SAVINGS RATE on capital per efficiency worker
(n    a )k
s
HIGH
f k 
s LOW f  k 
0
LOW SAVINGS BGP
HIGH SAVING BGP
Slide 64
k (t )
FIGURE: Effect of SAVINGS RATE INCREASE on output per worker growth
starting from in a BGP

y (t )
y (t )
a
0
Time t
INCREASE IN SAVINGS RATE
Slide 65
7.2. Output per worker in the long run (in the BGP/SS)
The savings rate does, however, affect the LEVEL OF OUTPUT PER
WORKER
This came out in the previous slides as the growth rate of output per worker
increased for a while when the savings rate increased
To determine the effect of the savings rate on BGP/SS output per worker,
we have to work with the equation that defines the BGP/SS
0
(E60)
 (  n  a)kt
sf (kt )
ACTUAL SAVINGS
AND INVESTMENT
BREAK-EVEN
INVESTMENT
An immediate implication of the equation is the that BGP/SS capital/output
ratio is increasing in the savings rate
(E61)
kt
yt
BGP
s

 na
Slide 66
For the Cobb-Douglas production function we can work things out in more detail
(E62)
kt
kta
BGP
s

 na
Solving for the BGP amount of capital per efficiency worker yields
(E63)
k BGP
1
1a
s





n

a


Hence a higher savings rate implies a higher K/(EL) in the BGP/SS
The amount of capital per worker can be obtained by multiplying by efficiency E
(E64)
kt , BGP
1
1a
s





n

a


Et
Slide 67
Substituting in the production function yields output per efficiency worker and
output per worker
(E65)
(E66)
yBGP
a
1a
s



  n  a 
yt , BGP
a
1a
s





n

a


Et
Hence a higher savings rate implies a higher Y/(EL) and Y/L in the BGP/SS
Slide 68
8. QUANTITATIVE FEATURES OF THE SOLOW MODEL
8.1. 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
a
1a
s



  n  a 
Et
Using the growth rule for variables raised to a power we therefore get
(E68)
Elasticity of long-run income level
with respect to savings rate
Slide 69
Quantitative effect of savings on long run income
growth long-run output
increase in savings rate
growth savings rate
alpha=0.33
alpha=0.66
elasticity=0.5
elasticity=2
15% to 16%
0.07
0.03
0.13
15% to 17%
0.13
0.07
0.27
15% to 18%
0.20
0.10
0.40
15% to 19%
0.27
0.13
0.53
15% to 20%
0.33
0.17
0.67
Slide 70
How large is a and therefore the ELASTICITY OF LONG-RUN OUTPUT
WITH RESPECT TO THE SAVINGS RATE?
a In the Solow model is the ELASTICITY OF OUTPUT WITH
RESPECT TO THE CAPITAL STOCK. Formally
Yt Kt
Kt
a
 MPKt
Kt Yt
Yt
Equilibrium in the capital market implies that rt
Hence
   MPK t
(rt   ) Kt
a
 share of CAPITAL in income
Yt
Slide 71
In practice one usually doesn’t measure the capital income share directly
Instead one measures the labor income share and uses the formula
CAPITAL INCOME SHARE=1 minus LABOR INCOME SHARE
We have seen that the CAPITAL SHARE IN INCOME in industrialized
countries is around 1/3 (and labor share around 2/3). Hence
(E70)
a
1 2 / 3
1/ 3 1



1  a 1  (1  2 / 3) 2 / 3 2
Slide 72
NEXT QUESTION ABOUT SAVINGS RATES: How much of cross-country
differences in output per worker can be explained by cross-country
differences in savings/investment rates?
Back to our equation for output per worker in BGP/SS
(E71)
yt , BGP
a
1a
s



  n  a 
Et
Take two countries, country1 and country2, that are identical in everything
except their savings rates s1 and s2.
What is the long-run output per worker difference?
Slide 73
In this case we get that output of country1 relative to country1 in the
BGP/SS is
(E72)
(E73)
yCOUNTRY 1, BGP
yCOUNTRY 2, BGP
yCOUNTRY 1, BGP
yCOUNTRY 2, BGP
a
s1 1a
  
 s2 
1/ 2
0.27 



 0.03 
 93
Hence rather small output difference given the enormous differences in savings
rates  differences in savings rates alone cannot explain enormous differences
in income between rich and poor countries
Slide 74
8.2. The speed of convergence to the BGP/SS
How quickly do economies close the gap to their BGP/SS?
How quickly does the economic growth rate drop as the
economy approach its BGP/SS?
How quickly do transitional growth – driven by capital
deepening – vanish as the economy approaches its BGP/SS?
Slide 75
Closed form solution for Y/(EL) in Solow model with CobbDouglas technology
1,2
1
0,8
0,6
0,4
0,2
0
0
20
40
60
80
100
120
140
Jones 2010 Notes on closed form Solow
Slide 76
160
When the economy starts with a level of output per
efficiency worker that is PHI (f) times the BGP/SS level
 Output relative to BGP/SS only a function of relative initial condition
and parameters l,a
Slide 77
Convergence of BGP/SS and growth illustrated for alpha=0.33
Growth rates of output per worker over time (years)
0,1
0,09
Parameters
0.33
alpha (a)
0.02
a
0.03
delta ()
0.02
n
0.5
phi (f)
0,08
0,07
0,06
0,05
0,04
Total growth
0,03
0,02
0,01
Transitional growth
0
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105
Closes half the gap to BGP/SS
after 12 years
Slide 78
Convergence of BGP/SS and growth illustrated for alpha=0.66
Growth rates of output per worker over time (years)
0,1
0,09
Parameters
0.66
alpha (a)
0.02
a
0.03
delta ()
0.02
n
0.5
phi (f)
0,08
0,07
0,06
0,05
0,04
Total growth
0,03
0,02
Transitional growth
0,01
0
0
20
40
60
80
100
Closes half the gap to BGP/SS
after 33 years
Slide 79
120
A slightly different perspective on the rate of
convergence (the predicted future growth path)
Parameters
unknown
0.02
0.03
0.02
unknown
alpha (a)
a
delta ()
n
phi (f)
We see a high initial growth rate
of around 9%, for example
(seen at the moment of ‚take off‘
in China, Taiwan, South Korea,
and Singapore for example)
Question: What is the path of future growth rates that we would
predict based on the Solo model
Answer: depends on the value for a
Slide 80
Path of future growth based on the Solo model
for different vales of alpha (a)
0,1
0,09
0,08
0,07
0,06
alpha=0.9
alpha=0.8
alpha=0.66
alpha=0.33
0,05
0,04
0,03
0,02
0,01
0
0
10
20
30
40
Slide 81
8.3. 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
Slide 82
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
Slide 83
With information on OUTPUT PER HOUR WORKED, we can do even better and
decompose output per worker into
(E76)
Y
HOURS
Y


L WORKERS HOURS
where
•Hours= total hours worked in the economy
•Hours/Workers= hours worked per employed person
Slide 84
The role of demographics, labor force participation, unemployment, and
productivity for income per capita in US, EU, and Japan relative to OECD
US rel
OECD
EU rel
OECD
JAPAN rel
OECD
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
Data from “International comparisons of labor productivity and per capita income” by van Ark
and McGuckin, Monthly Labor Review, July 1999
 Hence factors other than output per hour play an important role in explaining
differences in income per capita between these rich countries/regions
Slide 85
FIGURE : Around the world the mai factor accounting for differences in INCOME PER
CAPITA (vertical axis) are differences in OUTPUT PER WORKER (horizontal axis)
From PWT8.1
Slide 86