Spring 2007
Birkbeck MSc Economics
Economic Theory and Its Applications II (Advanced Macroeconomics)
Course structure:
The long run: Capital accumulation and technological progress modelled. Typical questions:
What explains cross-country differences in the standard of living? Why have living standards
improved over the past century? Which institutions are most conducive to growth?
The medium run: Stock of capital and state of technology given. Money prices are flexible.
Equilibrium in product, capital and labour markets analysed. Typical questions: Why is
unemployment on average higher in some countries than others? Why was unemployment in
Europe higher in the seventies and eighties than in the fifties and sixties? What are the
employment implications of imperfect competition in good markets?
The short run: Prices fixed (in New Keynesian models but not in RBC models!). What
causes cyclical changes in output from one quarter to another? Why are prices and wages
sticky in the short run? What are the transmission mechanisms of monetary policy? Issues in
monetary and fiscal policy.
I. The Long Run
Stylized facts of growth
1. Average real incomes in US and Western Europe are between 10 and 30 times larger than
a century ago.
2. Average growth rates in industrialised countries higher in the 20th century than in the 19th
century and higher in the 19th than in the 18th.
3. Productivity slowdown in industrialised nations from early 1970s to mid 1990s followed
by a rebound.
4. Average real incomes in such countries as the US, Germany and Japan appear to exceed
those in the developing world by a factor of 20.
5. Growth miracles: South Korea, Taiwan, Singapore, Hong Kong, Japan.
6. Growth disasters: Argentina, Africa (Sub-Saharan).
7. Widening of cross-country income differences.
Solow growth model
This is a benchmark model that provides a natural starting point for the study of economic
growth. It provides important insights but also through its apparent limitations and
weaknesses guides the way to more recent contributions.
Technology
Yt  F K t , At Lt 
Where Y is output, K is capital, L denotes labour and A is the effectiveness of labour, closely
related to the state of technology.
Constant returns to scale.
Diminishing marginal productivity of capital and labour
Inada conditions.
Using constant returns to scale we can rewrite the production function in intensive form:
yt  f k t 
where yt  Yt
 At Lt  and
kt  K t
 At Lt  .
The product AL is a measure of labour input, sometimes called effective labour or augmented
labour units.
Assumptions
L
 n and is exogenous to the model.
L
A
The growth of worker effectiveness (technology)  g is also exogenous.
A
The saving rate is denoted by s, S=sY, and is exogenous.
Population growth is
The economy is closed, so that savings equal investment: S=sY=I.
Dynamics
K t  sYt  K t
Using the quotient and product rules of differentiation we can derive a differential equation
for k;
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kt  sf kt     n  g kt
Note that the population growth rate and the rate of growth of worker effectiveness appear
alongside the depreciation rate!
Steady state solution
(+n+g)k
I/AL
sf(k)
k*
k
Note that sf(k) gives actual investment (per efficiency unit of labour) while (+n+g)k gives the
level of investment that is required to keep the (normalised) capital stock k constant. Clearly,
investment is needed to replace the capital that is used up (depreciation) but investment is also
needed because of population growth (the aggregate capital stock has to growth so that each
worker has the same capital to work with) and the same applies to the growth of worker
effectiveness.
The existence of a steady state depends crucially on the assumption of diminishing marginal
product of capital.
When the economy finds itself at k* it is travelling along the so-called balanced growth path.
Here K (and Y) are growing at rate n+g and K/L (and Y/L) are growing at the rate g. This
gives us one key implication of the Solow growth model:

In steady state (along the balanced growth path) output per unit of labour only grows
because labour is becoming more effective over time at rate g.
Note that the growth of the effectiveness of labour is exogenous to the model so that the
Solow growth model says nothing about the causes of long run growth of Y/L (or K/L or C/L!).
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Effect of a rise in the saving rate s
An increase of the saving rate s would shift upwards the curve sf(k) which makes growth rise
temporarily but having a long-run effect only on the level of capital k.
(+n+g)k
I
AL
sf(k)
k*
k**
k
A second implication of the Solow growth model follows:

An increase of the saving rate only affects the level of capital, output and consumption in
the long run; the long-run rate of growth of these variables is unaffected.
The implied time paths for the growth of capital and consumption per effective labour unit are
shown in the figure below.
s
k
time
time
c
k
?
time
time
4
Note the lower plot on the right-hand side. Clearly consumption falls when the saving rate
goes up and then recovers as the capital stock starts to increase and hence also output.
However, it is not clear whether consumption will eventually increase its initial level. This
brings us to the so-called Golden rule of capital accumulation.
Consumption (per labour unit) in steady state can be written as follows
 
 
c *  1  s  f k *  f k *  n  g   k
 
because, clearly, we have sf k *  n  g   k in steady state. Note that the steady state level
of capital is a function of the saving rate; k * s  , so we can maximise consumption in steady
state c* w.r.t. s. This gives the following first order condition:
 
*
f ' k GR
 n  g 
*
where kGR
stands for the Golden rule level of capital. There is also a corresponding Golden
rule level of the saving rate sGR.
The condition says that consumption is maximised when the slope of the production function
is equal to the slope of the required-investment line. Consumption is clearly equal to the gap
between the production function and the intersection of the actual investment curve and the
required investment line.
Another way of looking at this result is to realise that f’(k)-  can be looked at as the real
interest rate in the model. We can then formulate a third implication of the model:

Consumption is maximised in steady state when the real interest rate equals the sum of
population growth and the rate of growth of worker effectiveness.
The textbook has a discussion of empirical work intended to test whether actual market
economies follow this rule.
The last implication can be seen from the figure above. The gap between actual and required
investment is greater the further we find ourselves from steady state. This implies:


Absolute convergence: The rate of growth of capital is greater the further below its steady
state we find ourselves; poor countries should grow faster than the richer ones,
ASSUMING that they have the same steady states. A variant relaxes this assumption:
Conditional convergence: The rate of growth of capital is greater the further below ONE’S
OWN steady state a country finds itself; poor countries should ONLY grow faster than the
richer ones if they have the same steady state level of capital, output etc, per effective
labour.
This last implication should be clear from the differential equation in k, which can be
rewritten as follows;
f  kt 
kt
s
   n  g 
kt
kt
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The first term on the right hand side has the product of s and the average product of capital
while the last term is a constant. Due to diminishing marginal product of capital we also have
the average product of capital falling in k, hence the rate of growth of k is falling in k!
Limitations of Solow growth model

Saving rate exogenous  Ramsey growth model, overlapping generations models

Technological progress exogenous 
endogenous growth models.
We now turn to the Ramsey growth model.
Ramsey model – saving made endogenous
How much should a nation save? In the Solow model we found the level of s that maximised
consumption in steady state. But we did not find an optimal path to the steady state. Nor was
it clear that utility would be maximised. Now imagine a social planner who decides how
much a nation should save at every instant. He solves the following problem where we have
assumed no population growth nor technological progress to simplify the exposition and
normalised L=1 so labour can be omitted from the model;

max  u Ct  exp  t dt
C
0
subject to
K t  F K t   Ct  K t
The social planner maximises the sum of discounted utility from time zero to infinity, where 
is the discount rate and exp(-t) the corresponding discount factor. Utility function has
diminishing returns to consumption. The constraint tells us that the output that is not
consumed, F(K)-C, is invested and added to the capital stock. The net increase in K is then the
difference between this investment and depreciation K.
Note: the textbook adds population growth and technological progress. Also, it solves the
model by looking at the behaviour of households and firms in a perfect market economy.
However, it can be shown that given a set of assumptions, the solution is identical to that
attained be a fictional social planner, i.e. the market economy is dynamically efficient. 1
Finally, the textbook avoids using the optimal control theory which you have seen in your
mathematics course in the autumn.2
Utility function is of the constant-relative-risk-aversion kind: U 
1
C 1
1 
These assumptions are; constant returns to scale production function; all consumers being alike; perfect
competition in all markets; and, importantly, households have infinite horizons.
2
For a refresher: Alpha Chian, Dynamic Optimization, McGraw Hill.
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Where  is the coefficient of relative risk aversion which is a measure of the concavity of the
utility function (hence risk aversion).
There are two stages to solving the problem. First, write the Hamiltonian;
H  uCt exp  t   F K t   Ct  K t 
where C is the control variable, K is the state variable and  is the costate variable. Then write
down the necessary conditions:
(1) H C  uC Ct exp  t     uC Ct    exp t 
The left hand has the marginal utility of consumption while the right-hand has the value of
wealth at time t, that is how much future discounted utility would increase where the
consumer to refrain from consuming the last unit. Clearly, the two have to be the same along
the optimal path.
(2) H K     FK   
Taking the time derivative of condition (1) we can combine the two equations to give;
FK K      
u C
uC
This is the so-called Keynes Ramsey rule which also has to satisfied on the optimal path. The
left-hand side has the marginal benefit of saving more while the right-hand side gives the
marginal cost, which is the sum of the pure rate of time preference and the anticipated change
of marginal utility (bad idea to save for a sunny day!!).
Using the utility function above we can rewrite the Keynes-Ramsey equation as
C 1
 FK K      
C 
where 1/ is the elasticity of inter-temporal substitution.
We now have a system of two differential equations in two variables, C and K:
C 1
 FK K      
C 
K t  F K t   Ct  K t
which we can solve using a phase diagram (which you have seen in the maths course).
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C
Saddle path
K
K MGR K GR
Modified golden rule: FK     : Maximises utility in steady sate.
Golden rule: FK    0 : Maximises consumption in steady state.
Note that as long >0 the Modified golden rule capital stock – where utility is maximised in
steady state – is smaller than the Golden rule capital stock – where consumption is maximised.
The reason is that in order to reach the Golden rule level consumers need to sacrifice current
consumption in order to increase their consumption in the long run. Because of their
impatience – reflected in >0 – the social planner does not do this because utility would not
be maximised. If =0 then the two would be identical.
Exercise. An increase in the rate of time preference .
C
K MGR K GR
K
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Here consumption jumps immediately and we go straight to a new saddle path. We then
converge to the new steady state along this saddle path.

k
time
k
time
c
?
time
time
The assumption of infinite horizons gives us the result that market economies are dynamically
efficient in that they save at the utility-maximising level but this assumption is clearly not
very realistic (note, however, the 1974 paper by Barro on Ricardian dynasties). We now come
to a model where consumers do have finite lives. This is the so-called overlappinggenerations model of Diamond and Samuelson.
Overlapping-generations model
The basic textbook version of this model has consumers/workers living for two periods.
During the first period of their lives they work, consume and save for retirement. During the
second period they run down their savings, consume and then die. They only decision they
need to make is how much to save and how much to consume when young.3 During each
period there are two generations alive, the young and the old.4
Notation
Lt = number of young individuals alive at time t
n = rate of population growth
c t1 = consumption of a young individual at time t
3
The model can be extended to include a bequest motive so that the utility of the currently living generation
depends on the expected utility of the following generations. In this way the model in the extreme give us
infinite horizons as we saw in the Ramsey model!
4
We could also have three or more generations but in those cases we could not solve the model graphically.
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ct21 = consumption of an old individual at time t+1
Consumers/workers
Individuals decide on their consumption profile in order to maximise their lifetime utility.
 
 
max
u ct1  1    u ct21
1 2
ct , ct 1
1
such that
wt  ct1  st
ct21  1  rt 1 st
This is a simple constrained optimisation. We can write the Lagrangian for the problem;

c2 
1
l  u ct1  1    u ct21   wt  ct1  t 1 
1  rt 1 

 
 
and derive the first-order conditions for a maximum.
First order conditions:
 
u ' ct1  
1   1 u ' ct21  

1  rt 1
Combining gives;
 
 
u ' ct1
 1  rt 1
u ' ct21 1   
Here the left-had side hast he slope of the indifference curve in the c1—c2 space and the righthand side the slope of the budget line. The solution corresponds to the point of tangency
between the indifference curve and a budget line in the figure below:
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ct21
c t1
wt
This solution should be familiar to you from your course in microeconomics.
This solution yields a saving function, where savings are a positive function of wages and an
indeterminate function of interest rates (income and substitution effects!):
(*) swt , rt 1 
Firms
Firms rent capital from the older generation and hire the services of labour from the younger
generation. They choose the level of each to maximise profits
 t  F  Kt , Lt   rt Kt  wt Lt
Normalising with L gives
 t  Lt f  kt   rt Kt  wt Lt
where f(k) is the production function in intensive form (output per members of the working
age generation).
FOC
(**) K: f '  kt   rt
(***) L: f  kt   f '  kt  Kt  wt
These two conditions also describe equilibrium in the labour and in the capital market.
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Dynamics
Finally we have a difference equation that describes the evolution of the capital stock
Kt 1  Kt  s  wt , rt 1  Lt  Kt
Dividing through by Lt gives
(****) 1  n  kt 1  s  wt , rt 1 
We can now combine equations (*), (**), (***) and (****) which gives
kt 1 
s  f  kt   kt f '  kt  , f '  kt 1  
1 n
This is a nonlinear difference equation in k. In the well-behaved case we can solve the
equation in the following way
kt 1
kt 1  kt 
k*
kt
In this case the steady state k* is both unique as well as stable.
There remains the question whether k* is also the “best” steady state in the sense that utility is
maximised. In order to find the optimal steady state we bring back the social planner. We
assume that he does not discriminate between generations and so maximises the utility of a
representative generation in steady state.
The budget constraint faced by the social planner is:
Kt 1  Lt ct1  Lt 1ct2  Kt  F  Kt , Lt 
This equation can be rewritten as follows;
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nk  c1  c2 1  n   f  k 
The social planner’s problem is then the following;
 
 
max
u c1  1    u c 2
1 2
c ,c
1
subject to the constraint
nk  c1  c2 1  n   f  k 
FOC
 
u ' c1  
1    u '  c 2    1  n 
f 'k   n  0
1
This implies
 
u 'c 
u ' c1 1   
2
 1 n
f 'k   n
It follows that if r<n the level of capital in the market solution is inefficient in the sense that
there is too much capital in the market economy compared to the social optimum; the social
planner would choose to have less. This is a case of Pareto inefficiency.
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