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Introducing
Advanced
Macroeconomics:
Chapter 5 – first
lecture
Growth and business
cycles
TECHNOLOGICAL
PROGRESS AND
GROWTH: THE
GENERAL SOLOW
MODEL
©The McGraw-Hill Companies, 2005
The general Solow model
• Back to a closed economy.
• In the basic Solow model: no growth in GDP per worker in
steady state. This contradicts the empirics for the Western
world (stylized fact #5). In the general Solow model:
• Total factor productivity, Bt, is assumed to grow at a
constant, exogenous rate (the only change). This implies a
steady state with balanced growth and a constant, positive
growth rate of GDP per worker.
• The source of long run growth in GDP per worker in this
model is exogenous technological growth. Not deep, but:
– it’s not trivial that the result is balanced growth in steady state,
– reassuring for applications that the model is in accordance with a
fundamental empirical regularity.
• Our focus is still:
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what creates economic progress and prosperity…
The micro world of the Solow model
… is the same as in the basic Solow model, e.g.:
• The same object (a closed economy).
• The same goods and markets. Once again, markets are
competitive with real prices of 1, rt and wt , respectively.
There is one type of output (one sector model).
• The same agents: consumers and firms (and
government), essentially with the same behaviour,
specifically: one representative profit maximising firm
decides K td and Ldt given rt and wt .
• One difference: the production function. There is a
possibility of technological progress:
Yt  Bt Kt L1t , 0    1.
The full sequence  Bt  is exogenous and Bt  0 for all t .
Special case is Bt  B (basic Solow model). ©The McGraw-Hill Companies, 2005
The production function with technological
progress
with a given sequence,  Bt  
1 / 1 
1

A
,
A

B
.
Yt  K t  At Lt  with a given sequence,  t  t
t
• With a Cobb-Douglas production function it makes no
difference whether we describe technical progress by a
sequence,  Bt  , for TFP or by the corresponding
sequence,  At  , for labour augmenting productivity.
• In our case the latter is the most convenient. The
exogenous sequence,  At  , is given by:
Yt  Bt Kt L1t
At 1  1  g  At , g  1
 At  1  g  A0 , g  1
t
• Technical progress comes as ”manna from heaven” (it
requires no input of production).
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• Remember the definitions: yt  Yt / Lt and kt  Kt / Lt .
1

Y

K
A
L
• Dividing by Lt on both sides of t
gives the per
t  t t
capita production function:

yt  kt At
1
.
• From this follows:
lnyt  lnyt 1    lnkt  lnkt 1   1    lnAt  lnAt 1  
gty   gtk  1    gtA   gtk  1    g.
Growth in yt can come from exactly two sources, and gt is
k
the weighted average of g t and g with weights  and 1   .
y
g
k
/
y
• If, as in balanced growth, t t is constant, then t  g !
y
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The complete Solow model

Yt  K t
 At Lt 
1
 1
 Kt 
rt   Kt  At Lt    

A
L
 t t

 Kt 
  1
wt  1    Kt Lt At  1    
 At
 At L t 
St  sYt
 1
1
Kt 1  Kt  St   Kt , K0 given
Lt 1  1  n  Lt , L0 given
At 1  1  g  At , A0 given
• Parameters:  ,s, ,n,g . Let g  0.
• State variables: K t ,Lt and At .
• Full model? Yes, given K 0 ,L0 and A0 the model determines
the full sequences  Kt  , Lt  , At  ,Yt  , rt  , wt©The
 , SMcGraw-Hill
t .
Companies, 2005
• Note:
rt Kt   Kt  At Lt 
1
wt Lt  1    Kt  At Lt 
1
 Yt
 1    Yt .
That is: capital’s share   , labour’s share  1   , pure
profits  0 . Our  should still be around 1 / 3.
• Also note: defining ”effective labour input” as Lt  At Lt:
Lt 1  1  n 1  g  Lt  1  n  Lt ,
The model is matematically equivalent to the basic Solow
model with Lt taking the place of Lt , and n taking the
place of n , and with B  1 !
We could, in principle, take over the full analysis from the
basic Solow model, but we will nevertheless be…
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Analyzing the general Solow model
• If the model implies convergence to a steady state with
balanced growth, then in steady state kt and yt must
grow at the same constant rate (recall again that kt / yt is
constant under balanced growth). Remember also:
gty   gtk  1    gtA .
Hence if g ty  g tk , then gt  gt  gt . If there is convergence
towards a steady state with balanced growth, then in this
steady state kt and yt will both grow at the same rate as At ,
and hence kt / At and yt / At will be constant.
• Furthermore: from the above mentioned equivalence to
the basic Solow model, Kt / Lt  Kt /  At Lt   kt / At and
Yt / Lt  Yt /  At Lt   yt / At converge towards constant steady
state values.
• Each of the above observations suggests analyzing the
model in terms of:
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y
k
A
1.
kt 
kt
K
y
Y
 t and yt  t  t .
At At Lt
At At Lt
2. From Yt  K t  At Lt  we get yt  kt .
3. From Kt 1  Kt  St   Kt and St  sYt we get

1
Kt 1  sYt  1    Kt
4. Dividing by At 1Lt 1  1  g 1  n  At Lt on both sides gives
kt 1 
sy  1    k  .

1  n 1  g 
1
t
t

5. Inserting yt  kt gives the transition equation:
1
kt 1 
skt  1    kt .
1  n 1  g 


6. Subtracting kt from both sides gives the Solow equation:
1
kt 1  kt 
skt   n  g    ng  kt .
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1  n 1  g 


Convergence to steady state:
the transition diagram
• The transition equation is:
1
kt 1 
skt  1    kt .
1  n 1  g 


• It is everywhere increasing and passes through (0,0).
• The slope of the transition curve at any kt is:
 1
dkt 1 s kt  1   

.
dkt
1  n 1  g 
• We observe: limk 0 dkt 1 / dkt   . Furthermore,
t
limk  dkt 1 / dkt  1  n  g    ng  0 . We assume that
t
the latter very plausible stability condition is
fulfilled.
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• The transition equation must then look as follows:
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• Convergence of kt to the intersection point k * follows from
the diagram. Correspondingly: yt  y*  k * .
 
Some first conclusions are:
• In the long run, kt  kt / At and yt  yt / At converge to
constant levels, k * and y*, respectively. These levels
define steady state.
• In steady state, kt and yt both grow at the same rate as At ,
that is, at the rate g and the capital output ratio,
Kt / Yt  kt / yt , must be constant.
©The McGraw-Hill Companies, 2005
Steady state
• The Solow equation
kt 1  kt 
sk

1  n 1  g 
1

t
  n  g    ng  kt

together with kt 1  kt  k * gives:


s
k 

n

g



ng


*
1
1



s
y 

n

g



ng


*

1
.
• Using kt  kt / At and yt  yt / At we get the steady state
growth paths:


s
k  At 

n

g



ng


*
t
1
1
,


s
yt*  At 

n

g



ng



1
.
©The McGraw-Hill Companies, 2005
• Since ct  1  s  yt ,


s
ct*  At 1  s  

n

g



ng



1
.
• It also easily follows from
 
rt   kt
 1
 
and wt  1    At kt

that


s
r*   

n

g



ng


1


s
and wt*  At 1    

n

g



ng



1
.
• There is balanced growth in steady state: kt , yt and wt
grow at the same constant rate, g , and rt is constant.
• There is positive growth in GDP per capita in
steady state (provided that g  0 ).
©The McGraw-Hill Companies, 2005
Structural policies for steady state
• Output per capita and consumption per capita in steady state

are:
1


s
y  A0 1  g  
and

 n  g    ng 


1
s
t
*
ct  A0 1  g  1  s  
 .
 n  g    ng 
• Golden rule: the s, that maximises the entire path, ct*.
Again: s**   .
*
y
• The elasticities of t wrt. s and n  g   are again  / 1  a 
and  / 1  a , respectively.
*
t
t
• Policy implications from steady state are as in the basic Solow
model: encourage savings and control population growth.
• But we have a new parameter, g ( A0 corresponds to B ). We
want a large g , but it is not easy to derive policy conclusions
wrt. technology enhancement from our model ( g is
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exogenous).
Empirics for steady state


s
y  At 

n

g



ng


*
t

1


lny  lnAt 
lns  ln  n  g    ng   .
1
*
t
• Assume that all countries are in steady state in 2000!
• It’s hard to get good data for At , so make the heroic
assumption that At is the same for all countries in 2000.
• Set (plausibly) g    0.075.
i
• If y00 is GDP per worker in 2000 of country i , the above
equation suggests the following regression equation:
i
lny00
  0   1 lnsi  ln  ni 0.075 ,
i
with s i and n measured appropriately (here as averages
over 1960-2000), and where  1   / 1    .
©The McGraw-Hill Companies, 2005
An OLS estimation across 86 countries gives:
i
lny00
 8.812  1.47 lns i  ln  ni 0.075  , adj. R 2  0.55
 se 0.14
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• High significance! Large R2! Even though we have
assumed that A00 is the same in all countries!
• But always remember the problem of correlation
vs. causality.
• Furthermore: the estimated value of  is not in
accordance with the theoretical (model-predicted)
value of 1 / 2 . Or:

 1.47    0.60.
1
• The conclusion is mixed: the figure on the previous
slide is impressive, but the figure’s line is much
steeper than the model suggests.
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