Solow completa - updated.pptx

ECONOMIC GROWTH: CONTEXT
We take it for granted but its new in terms of human history
While it is common to think about growth today as being somehow natural, even
expected – in fact, if world growth falls from 3.5 to 3.2%, it is perceived as a big crisis –
it is worthwhile to acknowledge that this was not always the case.
Pretty much until the end of the 18th century growth was quite low, if it happened at
all.
In fact, it was so low that people could not see it during their lifetimes
Then, towards the turn of the 18th century, something happened that created
explosive economic growth as the world had never seen before.

Source: Campante, F., Sturzenegger, F. y Velasco, A. (1) (2021). Advanced Macroeconomics: An Easy Guide LSE Press

Source: Campante, F., Sturzenegger,
F. y Velasco, A. (1) (2021). Advanced
Macroeconomics: An Easy Guide
LSE Press

MACROECONOMÍA
AVANZADA: SOLOW
GROWTH MODEL
Profesora: Sarah carrington

SOLOW GROWTH MODEL -
INTRODUCTION
• Traditional model
• Other models (more advanced) are often best understood
in relation to the Solow model (often called Solow-Swan
model)
• The Solow model has no optimization in it:
• It takes the savings rate as exogenous and constant
• (later we will relax these assumptions and add other
endogenous elements to the model).

ASSUMPTIONS
INPUTS AND OUTPUTS
• Model focuses on 4 variables: Y, K, L and A
• A is knowledge, or the “effectiveness of labour”
• At a point in time, the economy has some inputs of K, L and A – and
these are combined to produce Y:
𝑌 𝑡 = 𝐹 𝐾 𝑡 , 𝐴 𝑡 𝐿(𝑡)
• Labour’s value in output depends on its effectiveness, A.
• t denotes time
• So output changes over time as inputs change

ASSUMPTIONS
INPUTS AND OUTPUTS
• Output obtained from a given stock of K and L can only be
augmented if there is technological progress => A↑
• A x L is multiplicative
• A.L = effective labour
• Technological progress enters into the model as labour
augmenting.
• The implication of this is that the ratio of K/Y eventually “settles
down” in the longer run (we will see this in the phase diagrams):
• It basically makes the analysis simpler

ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION
FUNTION
• Central assumptions relate to the properties of the
production function (and the evolution of inputs).
• We assume Constant Returns to Scale (CRS) in A.L and
K
• That means that doubling K and AL (that is, 2xL with
A fixed) doubles the output
• Generally:

ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION FUNTION
• Why CRS?
• This is really a combination of 2 separate assumptions
1. The Economy is big enough that the gains from specialization have been exhausted (smaller
economies might still gain)
2. Inputs other than K and L and A are relatively unimportant (E.g.: no land etc).
• Assumption of CRS allows us to work with the production function in its
intensive form:
• Setting 𝑐 =
1
𝐴𝐿
:
• 𝑭
𝑲
𝑨𝑳
, 𝟏 =
1
𝐴𝐿
𝐹 𝐾, 𝐴𝐿

ASSUMPTIONS
FUNTION
•
1
𝐴𝐿
𝐹 𝐾, 𝐴𝐿 = 𝑭
𝑲
𝑨𝑳
, 𝟏
•
𝑲
𝑨𝑳
is the amount of capital per unit of effective labour
•
𝐹 𝐾,𝐴𝐿
𝐴𝐿
=
𝑌
𝐴𝐿
= the output per unit of effective labour
• If we define 𝓀 =
𝑲
𝑨𝑳
, 𝑦 =
𝒀
𝑨𝑳
, and 𝒇 𝓀 = 𝑭 𝓀, 𝟏
• Then we get the production function in its intensive form:
𝑦 = 𝒇 𝓀

ASSUMPTIONS
FUNTION
𝑦 = 𝒇 𝓀
• What is this?
• This is just the OUTPUT PER UNIT OF EFFECTIVE LABOUR
• Output per unit of effective labour is a function of capital
per unit of effective labour
• This makes sense when you think of it.
• It just means that the output that we are capable of
producing is proportional to the ratio of the main inputs

ASSUMPTIONS
FUNTION
• Note that these new variables are not of interest in
and of themselves, but they are tools to learn about
the variables that we are interested in
• We will see that the easiest way to analyse the model
is to focus on the behaviour of 𝓀, rather than on K,
and AL
• Example: the behaviour of Y/L (per worker) = A(Y/AL) or
Af(𝓀)

ASSUMPTIONS
• INTUITION?
• Imagine that the economy can be divided into little mini
economies, AL of them (we can do this because there are CRTS):
ECONOMY (total)
1,
𝐾
𝐴𝐿
1,
𝐾
𝐴𝐿 1,
𝐾
𝐴𝐿
1,
𝐾
𝐴𝐿
Each mini economy
is for one unit of
effective labour
Amount of effective
labour per mini
economy
Amount of capital
per mini economy
Each mini economy produces
1/AL as much as the big economy
So output depends only on the
amount of K per unit of AL, and
nothing more (i.e.: not on the size
of the economy)

SOLOW GROWTH
MODEL -
ASSUMPTIONS
ASSUMPTIONS
CONCERNING THE
PRODUCTION
FUNTION
• What are the qualities
of this intensive form
of production
function?
• f(𝓀) satisfies f(0)=0,
f’(𝓀)>0, f’’(𝓀)<0:

ASSUMPTIONS
MATHEMATICALLY:
• Since 𝐹 𝐾, 𝐴𝐿 = 𝐴𝐿 𝐹 𝐾/𝐴𝐿 the marginal product of capital,
𝜕𝐹 𝐾,𝐴𝐿
𝜕𝐾
:
𝜕𝐹 𝐾, 𝐴𝐿
𝜕𝐾
= 𝐴𝐿. 𝑓
′
𝐾
𝐴𝐿 ×
1
𝐴𝐿
= 𝒇′ 𝓀
• What is the implication?
• The MPK is positive, but is declining in 𝐾/𝐴𝐿 (think of a typical
factory example)
• A typical example of this type of function is a COBB-DOUGLAS
function:
𝐹 𝐾, 𝐴𝐿 = 𝐾∝(𝐴𝐿) 1−∝
Where 0 < α < 1

SOLOW GROWTH MODEL - ASSUMPTIONS
• This gives us CRS:
𝐹 𝑐𝐾 𝑡 , 𝑐𝐴 𝑡 𝐿(𝑡) = 𝑐𝐾 ∝(𝑐𝐴𝐿) 1−∝
= 𝑐𝛼 𝑐1−𝛼𝐾𝛼(𝐴𝐿)1−𝛼
= 𝑐𝐹(𝐾, 𝐴𝐿)
• To get the intensive form, divide both sides by AL:
𝒇 𝓀 = 𝑭
𝑲
𝑨𝑳
, 𝟏
=
𝑲
𝑨𝑳
𝛼
= 𝓀𝛼
𝒇′ 𝓀 = 𝛼𝓀 𝛼−1
(meets assumptions)

EVOLUTION OF INPUTS INTO
PRODUCTION
• How do L, A and K evolve?
• The model is set in continuous time; variables are defined at every
point of time
• Initial levels are exogenous (taken as given)
• Labour and Knowledge grow at constant rates:
𝐿 𝑡 = 𝑛𝐿 𝑡
𝐴 𝑡 = 𝑔𝐴 𝑡
• ‘n’ and ‘g’ are exogenous (we cannot choose them in this model)
• And a dot above a variable signifies it’s variance over time: 𝑋 𝑡 =
𝑑𝑋(𝑡)
𝑑𝑡

PRODUCTION
• What is a “growth rate”?
• A proportional rate of change:
𝑋 𝑡
𝑋(𝑡)
• Implication?
•
𝐿 𝑡
𝐿 𝑡
= 𝑛 and
𝐴 𝑡
𝐴 𝑡
= 𝑔.
• As they are both constants, we have constant growth rates
• Another important piece of information:
• Growth rate of a variable = rate of change of its natural log
•
𝐿 𝑡
𝐿 𝑡
=
𝑑 ln 𝐿 𝑡
𝑑𝑡
• See Romer for the proof using the chain rule.

PRODUCTION
• What is the evolution path of L and A?
• Because the variables’ growth rates are ∆𝑙𝑛:
𝑙𝑛𝐿 𝑡 − 𝑙𝑛𝐿 0 = 𝑛𝑡
𝑙𝑛𝐴 𝑡 − 𝑙𝑛𝐴 0 = 𝑔𝑡
• Applying the exponent to both sides (to get rid of the “ln”’s)
𝑙𝑛𝐿 𝑡 − 𝑙𝑛𝐿 0 = 𝑙𝑛
𝐿 𝑡
𝐿 0
= 𝑛𝑡 → 𝐿 𝑡 = 𝐿 0 𝑒𝑛𝑡
𝑙𝑛𝐴 𝑡 − 𝑙𝑛𝐴 0 = 𝑙𝑛
𝐴 𝑡
𝐴 0
= 𝑔𝑡 → 𝐴 𝑡 = 𝐴 0 𝑒𝑔𝑡
Thus, 𝐿 and 𝐴 grow exponentially

PRODUCTION
• Output can be used for consumption or investment.
• In this model, the fraction of output devoted to Investment (through
savings = s) is exogenous and constant
• And one unit of output devoted to investment => one unit of capital.
• In addition, capital depreciates at the rate 𝛿. So:
𝐾 𝑡 = 𝑠𝑌 𝑡 − 𝛿𝐾 𝑡
• Although there are no restrictions placed on n, g, and 𝛿 individually,
their SUM (𝑛 + 𝑔 + 𝛿) is assumed to be positive.

A SUMMARY OF THE MODEL
• Very simplified
• Only a single good
• No government
• Fluctuations in employment are ignored
• Production only incorporates 3 inputs
• 𝑠, 𝑛, 𝑔 and 𝛿 are all assumed to be constant
• However, while simple, this is not always a defect
• A model’s job is to provide insights about CERTAIN ASPECTS of the world
• We cannot use a model that is too complicated to understand
• If the simplification does not give INCORRECT answers, then the lack of realization can be a
virtue.

THE DYNAMICS OF THE MODEL
• The evolution of two of the three inputs into production (L and A) is
exogenous
• The main analysis of the dynamics of the model is then focused on K as this
is the “DRIVING VARIABLE” in the model.
• THE DYNAMICS OF K
• We will focus on the capital stock per unit of effective labour, 𝓀
• Since 𝓀=K/AL, we can use the quotient rule to find:
• 𝓀 𝑡 =
𝐾(𝑡)
𝐴 𝑡 𝐿(𝑡)
−
𝐾 (𝑡)
[𝐴 𝑡 𝐿 𝑡 ]2 𝐴 𝑡 𝐿 𝑡 + 𝐿 𝑡 𝐴 𝑡
• =
𝐾(𝑡)
−
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
𝐿 𝑡
𝐿 𝑡
−
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
𝐴 𝑡
𝐴 𝑡
.
ℎ 𝑥 =
𝑓(𝑥)
𝑔(𝑥)
ℎ′
𝑥 =
𝑔 𝑥 𝑓′
𝑥 − 𝑓 𝑥 𝑔′(𝑥)
𝑔(𝑥)2

•
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
is just 𝓀.
𝐿 𝑡
𝐿 𝑡
is 𝑛 and
𝐴 𝑡
𝐴 𝑡
is 𝑔. And we know that 𝐾 𝑡 = 𝑠𝑌 𝑡 −
𝛿𝐾 𝑡 .
• We can substitute these elements into the equation to get:
• 𝓀 𝑡 =
𝑠𝑌 𝑡 −𝛿𝐾 𝑡
− 𝓀 𝑡 𝑛 − 𝓀 𝑡 𝑔 = 𝑠
𝑌 𝑡
𝐴 𝑡 𝐿 𝑡
− 𝛿𝓀 𝑡 −𝑛𝓀 𝑡 − 𝑔𝓀 𝑡
• And, because we know that
𝑌 𝑡
𝐴 𝑡 𝐿 𝑡
= 𝑦 = 𝑓(𝓀):
𝓴 𝒕 = 𝒔𝒇(𝓴 𝒕 ) − 𝜹𝓴 𝒕 − 𝒏𝓴 𝒕 − 𝒈𝓴 𝒕

𝓴 𝒕 = 𝒔𝒇(𝓴 𝒕 ) − 𝒏 𝒕 + 𝒈 𝒕 + 𝜹 𝒕 𝓴
This equation is the key equation of the Solow model
It says that the rate of change of the capital stock per unit of effective labor is the difference
between 2 terms:
• 𝑠𝑓(𝓀 𝑡 ): the fraction of output per unit of effective labour that is saved – this is ACTUAL INVESTMENT
• 𝓀 𝑛 + 𝑔 + 𝛿 : the amount of investment that must be done just to keep 𝓀 at it’s existing level – this is
BREAK EVEN INVESTMENT
There are 2 reasons that some investment is needed to keep 𝓀 from falling:
• Existing capital is depreciating: 𝛿𝓀 𝑡
• The quantity of effective labour is growing: 𝑛𝓀 𝑡 + 𝑔𝓀 𝑡
• To keep 𝓀 constant requires that the growth in population and technology as well as
depreciation is accounted for.

In other words, the rate of capital must grow at a rate larger than the sum of
the:
• depreciation rate,
• the population growth rate and
• the rate of growth in technology
In order to maintain the same K/AL ratio which is the key driver of growth in the
model.
When actual investment > break even investment, 𝓀 is rising
When actual investment < break even investment, 𝓀 is falling
In equality, 𝓀 is constant.

GRAFICAR:
• 𝒔𝒇 𝓴 𝒕
• 𝒏 𝒕 + 𝒈 𝒕 + 𝜹 𝒕 𝓴
• En el espacio: horizontal
= k,
• vertical = inversión/AL,
Y/AL
•
s, y
k
(n+g+δ)k

THE
DYNAMICS
OF THE
MODEL
At f(0)=0, actual and break even
investment are equal at zero.
ACTUAL INVESTMENT:
We know from our assumptions
regarding the production
function that when 𝓀 is close to
zero, f’(𝓀) is very large.
But as 𝓀 increases, the marginal
product of 𝓀 decreases until it
falls towards zero.

At some point the slope of the ACTUAL INVESTMENT line falls below that
of the BREAK EVEN INVESTMENT line (which has a constant slope).
So the two lines must cross.
The fact that f’’(𝓀) < 0 implies that the two lines only intersect ONCE for
𝓀 > 0.
We call the level of 𝓀 the point of intersection 𝓀*.
𝓀* is the point at which break even and actual investment are equal and
we are in equilibrium.

THE DYNAMICS
OF THE MODEL
This is the phase diagram for 𝓀
What is a phase diagram?
It is when we map the change in
the variable as a function of the
level of the same variable (𝓀 in
terms of 𝓀).
If 𝓀 < 𝓀*, 𝓀 is positive
If 𝓀 > 𝓀*, 𝓀 is negative and capital
per unit of effective labour is
falling.
𝓀 = 𝓀*, 𝓀 is zero.
Thus we have convergence to 𝓀*.

DYNAMICS: THE BALANCED GROWTH
PATH
• Since 𝓀 converges to 𝓀*, it is of value to know how the other variables behave when 𝓀= 𝓀*.
• By assumption, L and A are growing at their constant rates of 𝑛 and 𝑔, respectively.
• The capital stock?
• Becuase K = AL𝓀, and 𝓀 is constant at 𝓀*, K is growing at a rate of 𝑛 + 𝑔. (i.e.:
𝐾
𝐾
= 𝑛 + 𝑔)
• So we have K and AL growing at the same rate of 𝑛 + 𝑔.
• And becuase we have constant returns to scale, we know that Y is also growing at 𝑛 + 𝑔.
• What about K/L and Y/L?
• They are growing at rate 𝑔 (the rate of growth of technology).

DYNAMICS: THE BALANCED GROWTH
PATH
• The implication is that the economy, regardless of where it starts from,
will end up converging to a point where all variables grow at the same
constant rate.
• This is called the BALANCED GROWTH PATH.
• Importantly, the key implication of this is that on this path, the only way
that we can have an increase in the output per worker is by achieving
technological progress.

IMPACT OF A CHANGE IN THE SAVING
RATE
• The most policy relevant parameter in the model is the savings rate.
• Why policy relevant?
• Because in practice, government policies can impact on the savings rate in
many ways, for eg.:
• the division of its budget between consumption and investment,
• it’s funding source for the budget: taxes or borrowing
• How savings and invesment are treated for tax purposes
• All of these policy settings will impact on the fraction of output that is
invested, that is s.
• So it is natural to examine the impact of a change in s on the model.

RATE
• To investigate this, we will first assume that the economy is
initially on a balanced growth path
• The shock will be a permanent increase in the rate of savings.
• We can then investigate how the model behaves when it has
been “tipped off” it’s balanced growth path.
IMPACT ON OUTPUT
• The increase in s shifts the actual investment line upwards, and
so 𝓀* rises.
• But 𝓀 does not immediately jump to the new value of 𝓀*….

IMPACT OF A
CHANGE IN THE
SAVING RATE
• When 𝓀= 𝓀*OLD, with the new
savings rate, actual investment
exceeds break even investment:
• 𝑠𝑓(𝓀 𝑡 ) > 𝓀 𝑛 + 𝑔 + 𝛿
• That is, more resources are
devoted towards investment
than are needed just to keep 𝓀
constant.
• 𝓀 is now positive and 𝓀 begins
to rise –it rises until it reaches
𝓀*NEW.
• At this point it will remain
constant again.

IMPACT OF A
CHANGE IN THE
SAVING RATE
• These results are summarized in
these 3 panels.
• t0 denotes the time that there is
a jump in the savings rate. By
assumption it remains constant
thereafter.
• Since the jump in s causes
actual investment to exceed
break-even investment by a
strictly positive amount, 𝓀
jumps from zero to a strictly
positive amount.
• 𝓀 rises gradually from the old
value of 𝓀 *to the new value,
and 𝓀 falls gradually back to
zero

IMPACT OF A
CHANGE IN THE
SAVING RATE
• What about the behavior of output per worker, Y/L = Af (𝓀)?
• When 𝓀 is constant, Y/L grows at rate g, the growth rate of A.
• When 𝓀 is increasing, Y/L grows both because A is increasing
and because 𝓀 is increasing.
• Thus its growth rate exceeds g.
• When 𝓀 reaches the new value of 𝓀*, however, again only the
growth of A contributes to the growth of Y/L, and so the
growth rate of Y/L returns to g.
• Thus a permanent increase in the saving rate produces a
temporary increase in the growth rate of output per worker:
• 𝓀 is rising for a time, but eventually it increases to the
point where the additional saving is devoted entirely
to maintaining the higher level of 𝓀.

RATE
• In sum, a change in the saving rate has a level effect but
not a growth effect:
• it changes the economy’s balanced growth path, and
thus the level of output per worker at any point in time
• But it does not affect the growth rate of output per
worker on the balanced growth path.
• Indeed, in the Solow model only changes in the rate of
technological progress have growth effects; all other
changes have only level effects.

THE IMPACT ON CONSUMPTION
• As a household’s welfare is also dependent on
consumption, we are interested in the behaviour of
consumption as well.
• Consumption may be more important for households
than output.
• Consumption, C, per unit of effective labour (AL) =
𝑓 𝓀 ∗ 1 − 𝑠 .
• This is, C/AL is equal to the amount of capital per unit
of effective labour multiplied by the fraction of output
that is not saved.
• Thus, since s changes discontinuously at t0 and 𝓀 does
not, initially consumption per unit of effective labor
jumps downward.
• Consumption then rises gradually as k rises and s
remains at its higher level.

• Whether consumption eventually exceeds its level before the rise in s is not immediately clear.
• Let c∗ denote consumption per unit of effective labor on the balanced growth path.
• c∗ equals output per unit of effective labor, 𝑓 𝓀∗ , minus investment per unit of effective labor,
s𝑓 𝓀∗
• On the balanced growth path, actual investment [s𝑓 𝓀∗ ] equals break-even investment, (𝑛 + 𝑔 +

This implies:
𝜕𝑐∗
𝜕𝑠
=
𝜕𝑐∗
𝜕𝓀
.
𝜕𝓀∗
𝜕𝑠
𝜕𝑐∗
𝜕𝑠
= 𝑓′(𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 ) − (𝑛 + 𝑔 + 𝛿) .
𝜕𝓀∗
𝑠, 𝑛, 𝑔, 𝛿
𝜕𝑠
What does this tell us?
• We know that
𝜕𝓀∗ 𝑠,𝑛,𝑔,𝛿
𝜕𝑠
is positive.
So to know whether
𝜕𝑐∗
𝜕𝑠
is positive or not, we need to know whether
𝑓′(𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 ) − (𝑛 + 𝑔 + 𝛿) is positive, or:
𝑓′
𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 > (𝑛 + 𝑔 + 𝛿)

Intuitively, when 𝓀 rises, the marginal increase in intensive capital
must be sufficient to compensate the marginal reduction in
intensive capital due to the impacts of 𝑛 + 𝑔 + 𝛿 on the level of
K/AL.
• If f’(𝓴∗
) is less than 𝑛 + 𝑔 + 𝛿, then the additional output from the
increased capital is not enough to maintain both consumption and
capital at their higher levels
• The proportionate level of C/AL must fall to allow for the higher
K/AL.
• If f’(𝓴∗) exceeds 𝑛 + 𝑔 + 𝛿, there is more than enough
additional output to maintain 𝓀 at its higher level, and so
consumption rises.

• In reality, 𝑓′(𝓀∗ 𝑠, 𝑛, 𝑔, 𝛿 ) can be greater or less than (𝑛 +
𝑔 + 𝛿).
• We can look at the effects in the diagram.
• The figure in the next slide shows not only (𝑛 + 𝑔 + 𝛿)𝓀
and 𝑠𝑓(𝓀), but also 𝑓(𝓀).
• Since consumption on the balanced growth path equals
output less breakeven investment, 𝑐∗
is the distance
between 𝑓 𝓀 and (𝑛 + 𝑔 + 𝛿)𝓀 at 𝓀 = 𝓀∗

TRANSLATING THE MATHS TO GRAPHS….
sf(k)
f(k)
k

The figure shows the determinants of c*
for three different values of s (relating to
three different levels of 𝓴∗
.
A: s is high, and so 𝓀∗ is high and f’(𝓀∗) is
less than 𝑛 + 𝑔 + 𝛿.
As a result, an increase in the saving rate
lowers consumption even when the
economy has reached its new balanced
growth path
s is low, 𝓀∗ is
low, f’(𝓀∗
) is
greater than
𝑛 + 𝑔 + 𝛿, and
an increase in s
raises
consumption in
the long run.
s is at the level that causes f’(𝓀∗
)
to just equal 𝑛 + 𝑔 + 𝛿 —that is,
the tangent of the two lines are
parallel at𝓀 = 𝓀∗
. In this case, a
marginal change in s has no effect
on consumption in the long run,
and consumption is at its
maximum possible level among
balanced growth paths.

• This value of 𝓀∗
is known as the golden-rule level of
the capital stock.
• Of course, in the Solow model, where saving is
exogenous, there is no more reason to expect the
capital stock on the balanced growth path to equal
the golden rule level than there is to expect it to equal
any other possible value.

IMPLICATIONS OF THE SOLOW MODEL
• The Solow model shows that capital accumulation by itself cannot sustain growth in per capita
income in the long run: diminishing marginal returns.
• At some point the capital stock becomes large enough that a given savings rate can only provide
just enough new capital to replenish ongoing depreciation and increases in labour force.
• Alternatively, if we introduce exogenous technological change (productivity), we can generate long-
run growth in income per capita, but we do not really explain it – we are not explaining those
differences, we are just assuming them!
• As a result, nothing within the model tells you what policy can do about growth in the long run.
• We do learn a lot about growth in the transition to the long run, about differences in income levels,
and how policy can affect those things.
(i) convergence – the model predicts conditional convergence;
(ii) dynamic inefficiency – it is possible to save too much in this model; and
(iii) long-run differences in income – they seem to have a lot to do with differences in productivity.

Solow completa - updated.pptx

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