Intermediate Macroeconomics
Lecture 3 - The Solow model
Zsófia L. Bárány
Sciences Po
2014 February
Recap of last week
I
two distinct phases of growth:
1. pre-industrial revolution - stagnation: growth in total output is
offset by population growth, constant per capita consumption
2. post-industrial revolution - sustained growth: around 1.5% to
2% growth per year sustained over 150 years
I
different models can explain the different stages:
1. Malthusian model - main assumptions: population growth
depends on consumption per capita & land is available in
limited supply
2. Solow model - the topic of today’s lecture
The Solow model
I
key: the description of the saving - investment - capital
accumulation - growth process
I
the model has stark predictions relating to the growth facts
I
main prediction: technological progress is necessary for
sustained growth in living standards
I
note: not a micro-founded model, i.e. the goals and
constraints of the agents are not explicit
I
the Solow model is the basis for the modern theory of
economic growth
Assumptions on Consumers/Workers
Labor supply
I
there are N consumers
I
who do not make a labor-leisure choice
I
⇒ each provides his one unit of labor
I
⇒ N is also the size of the labor force
I
the population is growing at exogenous constant rate n
N 0 = (1 + n) · N
Consumers/Workers
Income and consumption
I
consumers are assumed to save an exogenous, constant
fraction s of their income, and consume the rest
I
the total income of consumers is Y , which is equal to total
output, as there is no government
I
note: we do not need to distinguish between labor income and
capital income or profits (since households own the firms and
do not make a labor supply decision)
I
the consumer’s budget constraint is Y = C + S
I
⇒ total saving in a period is s · Y
I
⇒ total consumption in a period is (1 − s) · Y
Production
Production function
Y = z · F (K , N)
I
Y total output
I
z total factor productivity (TFP)
I
K stock of capital
I
N labor input
Important features:
I
output depends on the quantity of labor and capital, and TFP
I
no other inputs, such as land or natural resources, matter
Production function is assumed to be neoclassical
I
positive, but diminishing marginal products
I
Inada conditions
I
constant returns to scale (CRS)
Characteristics of the neoclassical production function I.
positive, but diminishing marginal products:
I z · F (K , N) is monotone increasing in K and N
→ positive marginal product of each inputs
I
I
I
MPK = ∂z·F
∂K = z · FK > 0: if K increases by 1, z · F (K , N)
increases by MPK
MPN = ∂z·F
∂N = z · FN > 0: if N increases by 1, z · F (K , N)
increases by MPN
z · F (K , N) is concave - decreasing returns to K and N
→ decreasing marginal product of both inputs
∂2F
∂K 2
=
FK
∂K
< 0 and FNN =
∂2F
∂N 2
=
FN
∂N
<0
I
FKK =
I
as K increases more and more, F (K , N) increases by less and
less
I
as N increases more and more, F (K , N) increases by less and
less
Characteristics of the neoclassical production function II.
Inada conditions:
I
the marginal product of an input goes to zero as that input
goes to infinity (holding all other inputs constant):
limK →∞ FK (K , N) = 0 and limN→∞ FN (K , N) = 0
I
the marginal product of an input goes to infinity as that input
goes to zero (holding all other inputs constant):
limK →0 FK (K , N) = ∞ and limN→0 FN (K , N) = ∞
Characteristics of the neoclassical production function III.
constant returns to scale:
F (λK , λN) = λF (K , N) for any λ > 0
I
e.g. λ = 2: if K and N double, then F (K , N) and Y doubles
consequence of CRS:
I
since we can pick λ = 1/N, output per worker can be written
as:
y=
Y
N
K
N
= z · F(K
N , N ) = z · F ( N , 1) = z · f (k)
| {z }
≡f (k)
→ allows us to work with per capita quantities
K
N,
capital per capita; y =
Y
N
I
denote: k =
output per capita
I
f inherits the properties from F : fk (k) > 0, fkk (k) < 0
Shape of the production function
Y
z · F (K , N)
K
For a given labor supply (N) Y as a function of capital (K ) looks
something like this.
Shape of the production function
Y
N
z · F(K
N , 1)
K
In per capita terms, Y /N looks something like this.
Shape of the production function
y=
Y
N
z · f (k)
k=
The per capita production function, f (k) = F ( K
N , 1) looks
therefore something like this.
K
N
Example: the Cobb-Douglas production function
Y = z · K α N 1−α
I
has constant returns to scale as the coefficients sum to 1:
zF (λK , λN)
= z(λK )α (λN)1−α = λα+1−α zK α N 1−α
=λzK α N 1−α
=λF (K , N)
Per capita output is
y = Y /N = zk α
I
positive, but diminishing marginal products, which are given
by:
MPK = αzK α−1 N 1−α
MPN = (1 − α)zK α N −α
I
Inada conditions hold
The capital accumulation equation
Capital stock
I
increases - due to investment
I
decreases - due to depreciation, d
Capital evolves according to:
K 0 = (1 − d)K + I
How do we determine I ?
This is part of the equilibrium condition of the model.
Competitive equilibrium
Two options:
1. Income - expenditure identity holds as an equilibrium condition
i.e. the goods market is in equilibrium
Y =C +I
combine this with the consumer’s budget constraint: Y = C + S
⇒ gives us market clearing on the capital market:
S =I
2. Capital market is in equilibrium
total saving = total investment
S =I
combine this with the consumer’s budget constraint: Y = C + S
⇒ gives us the income-expenditure identity:
Y =C +I
Equilibrium dynamics of capital per worker/per capita
The equilibrium dynamics of capital is then:
K 0 = I +(1−d)K = S+(1−d)K = s·Y +(1−d)K = s·z·F (K , N)+(1−d)K
Transforming this into per capita terms:
K0
F (K , N)
K
=s ·z ·
+ (1 − d)
N
N
N
Equilibrium dynamics of capital per worker/per capita
The equilibrium dynamics of capital is then:
K 0 = I +(1−d)K = S+(1−d)K = s·Y +(1−d)K = s·z·F (K , N)+(1−d)K
Transforming this into per capita terms:
K0
F (K , N)
K
=s ·z ·
+ (1 − d)
N
N
N
N0 K 0
F (K , N)
K
=s ·z ·
+ (1 − d)
0
N N
N
N
|{z}
=1
Equilibrium dynamics of capital per worker/per capita
The equilibrium dynamics of capital is then:
K 0 = I +(1−d)K = S+(1−d)K = s·Y +(1−d)K = s·z·F (K , N)+(1−d)K
Transforming this into per capita terms:
K0
F (K , N)
K
=s ·z ·
+ (1 − d)
N
N
N
N0 K 0
F (K , N)
K
=s ·z ·
+ (1 − d)
0
N N
N
N
|{z}
=1
(1 + n)N K 0
F (K , N)
K
=s · z ·
+ (1 − d)
N
N0
N
N
Equilibrium dynamics of capital per worker/per capita
The equilibrium dynamics of capital is then:
K 0 = I +(1−d)K = S+(1−d)K = s·Y +(1−d)K = s·z·F (K , N)+(1−d)K
Transforming this into per capita terms:
K0
F (K , N)
K
=s ·z ·
+ (1 − d)
N
N
N
N0 K 0
F (K , N)
K
=s ·z ·
+ (1 − d)
0
N N
N
N
|{z}
=1
(1 + n)N K 0
F (K , N)
K
=s · z ·
+ (1 − d)
N
N0
N
N
(1 + n)k 0 =s · z · f (k)+(1 − d)k
the key equation of the Solow model
k0 =
s · z · f (k) (1 − d)k
+
1+n
1+n
→ determines capital per worker in the next period as a function
of capital per worker today
capital per capita in the next period is
I
investment in the current period
I
plus whatever is left of capital after depreciation
I
adjusting for the fact that the population is higher in the next
period
Finding the steady state
the steady state k ∗ is where k = k 0
I
if k < k ∗ ⇒ k 0 > k
I
if k > k ∗ ⇒ k 0 < k
The steady state
Equation determining the steady state of the economy:
k∗ = k0 = k
in the steady state capital per worker is constant
s · z · f (k ∗ ) (1 − d)k ∗
+
1+n
1+n
∗
∗
(1 + n)k = s · z · f (k ) + (1 − d)k ∗
k∗ =
(n + d)k ∗
| {z }
break-even investment
= s · z · f (k ∗ )
| {z }
actual investment
In the steady state
I
the amount of actual investment is exactly such that
I
it compensates for depreciation and the growth in population,
i.e. for the break-even investment
⇒ the level of capital per worker stays constant
A useful graphical representation:
The ‘savings curve’ (actual investment) and the ‘effective
depreciation line’ (break-even investment)
break-even investment
actual investment
steady state: where the
break-even investment =
= actual investment
This graph is very useful for studying the effects of n, s and z
Three important questions about the Solow model:
1. does the steady state exist?
2. is the steady state unique?
3. is the steady state stable, i.e. does the economy converge to
k ∗ from any initial k0 ?
Three important questions about the Solow model:
1. does the steady state exist?
YES, if the two curves cross
2. is the steady state unique?
3. is the steady state stable, i.e. does the economy converge to
k ∗ from any initial k0 ?
Three important questions about the Solow model:
1. does the steady state exist?
YES, if the two curves cross
2. is the steady state unique?
YES, if the two curves only cross once
3. is the steady state stable, i.e. does the economy converge to
k ∗ from any initial k0 ?
Three important questions about the Solow model:
1. does the steady state exist?
YES, if the two curves cross
2. is the steady state unique?
YES, if the two curves only cross once
3. is the steady state stable, i.e. does the economy converge to
k ∗ from any initial k0 ?
YES, if the actual investment curve is above the break-even
investment curve for k < k ∗ , and if the actual investment
curve is below the break-even investment curve when k > k ∗
Three important questions about the Solow model:
1. does the steady state exist?
YES, if the two curves cross
2. is the steady state unique?
YES, if the two curves only cross once
3. is the steady state stable, i.e. does the economy converge to
k ∗ from any initial k0 ?
YES, if the actual investment curve is above the break-even
investment curve for k < k ∗ , and if the actual investment
curve is below the break-even investment curve when k > k ∗
This is due to the neoclassical production function.
Example 1.
i
(d + n)k
k
Example 1.
i
(d + n)k
szf (k)
k
Which of the assumptions on a neoclassical production function is
violated?
Example 1.
i
szf (k)
(d + n)k
k
Which of the assumptions on a neoclassical production function is
violated?
Example 2.
i
(d + n)k
szf (k)
k
Which of the assumptions on a neoclassical production function is
violated?
Example 3.
i
szf (k)
(d + n)k
k
Which of the assumptions on a neoclassical production function is
violated?
With a neoclassical production function
A unique and stable steady state exists.
break-even investment
actual investment
steady state:
break-even investment=actual
investment
k∗ = k0 = k
Implications 1.
In the steady state
I
k ∗ is constant
I
y ∗ = f (k ∗ ) is constant as well
I
c ∗ = (1 − s)y ∗ is constant as well
I
K ∗ = N · k ∗ grows at rate n
I
Y ∗ = N · y ∗ grows at rate n
I
C ∗ = N · c ∗ grows at rate n
in the long-run no growth in output per worker
Implications 2.
I
if two countries have the same n, s, and d, as well as the same
production function ⇒ they have the same steady state capital
per worker, k ∗ ⇒ they converge to the same steady state
⇒ conditional convergence
along this convergence path poorer countries grow faster
(why?)
Implications 3.
I
what happens if two countries have different saving rates, i.e.
sA < sB ?
I
⇒ their steady state per capita capital is different, i.e.
kA∗ 6= kB∗
I
both countries converge to their own steady state, but these
are different
⇒ no absolute convergence
when saving rates are different poorer countries don’t
necessarily grow faster
Saving rate and the steady state per capita capital
i
(d + n)k
k
Saving rate and the steady state per capita capital
i
(d + n)k
sA zf (k)
k
Saving rate and the steady state per capita capital
i
(d + n)k
sA zf (k)
kA∗
k
Saving rate and the steady state per capita capital
i
(d + n)k
sB zf (k)
sA zf (k)
kA∗
k
Saving rate and the steady state per capita capital
i
(d + n)k
sB zf (k)
sA zf (k)
kA∗
kB∗
k
Saving rate and the steady state per capita capital
i
(d + n)k
sB zf (k)
sA zf (k)
kA∗
kB∗
The steady state capital per worker is higher, when the savings
rate is higher.
sA < sB ⇒ kA∗ < kB∗
k
Data: Real income per capita and the investment rate
→ per capita GDP and the investment rate are positively correlated
Income per capita and saving rate in the model
What is the implication of a higher saving rate in the Solow
model?
I
intuitively, a higher saving rate implies that the saving curve,
the actual investment curve is higher
I
⇒ higher steady state k ∗ ⇒ higher y ∗
What is the effect of an increase in the saving rate?
I
long-run increase in the level of per capita capital
I
long-run increase in the level of per capita output
I
temporary increase in the growth rate of per capita capital
and per capita output
I
no effect on the long-run growth rate of capital per worker
and output per worker, which are equal to zero
A change in the savings rate has a long-run level effect, but does
not have a long-run growth effect.
Saving rate and the steady state per capita capital
I
so a higher saving rate, s, leads to higher per capital capital in
the steady state, k ∗ , and thus to a higher steady state per
capita income, y ∗
I
⇒ should people increase their saving rate in order to increase
output? would this be good for them?
I
put another way: Does it mean that a higher saving rate is
always better?
I
what is the ’best’ steady state?
Saving rate and the steady state per capita capital
I
so a higher saving rate, s, leads to higher per capital capital in
the steady state, k ∗ , and thus to a higher steady state per
capita income, y ∗
I
⇒ should people increase their saving rate in order to increase
output? would this be good for them?
I
put another way: Does it mean that a higher saving rate is
always better?
I
what is the ’best’ steady state?
I
define the ’best’ steady state as the one that maximizes
consumption per capita, c ∗
Consumption and steady state capital
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ )
i
zf (k)
szf (k)
k
Consumption and steady state capital
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ )= zf (k ∗ )−(n + d)k ∗
zf (k)
i
(d + n)k
szf (k)
k∗
k
Consumption and steady state capital
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ )= zf (k ∗ )−(n + d)k ∗
zf (k)
i
(d + n)k
c∗
szf (k)
k∗
k
To maximize c ∗ , need to find the biggest distance between zf (k)
and (d + n)k.
The golden rule
I
what are the effects of a higher s on c ∗ ?
I
higher s ⇒ higher k ∗ ⇒ higher y ∗
I
higher s ⇒ smaller consumption share in output
c ∗ = (1 − s) y ∗
|{z}
| {z } |{z}
?
↓
↑
The golden rule
I
what are the effects of a higher s on c ∗ ?
I
higher s ⇒ higher k ∗ ⇒ higher y ∗
I
higher s ⇒ smaller consumption share in output
c ∗ = (1 − s) y ∗
|{z}
| {z } |{z}
?
I
↓
net effect in general is ambiguous
↑
The golden rule
I
what are the effects of a higher s on c ∗ ?
I
higher s ⇒ higher k ∗ ⇒ higher y ∗
I
higher s ⇒ smaller consumption share in output
c ∗ = (1 − s) y ∗
|{z}
| {z } |{z}
?
I
I
↓
↑
net effect in general is ambiguous
due to the assumptions made on the production function
The golden rule
I
what are the effects of a higher s on c ∗ ?
I
higher s ⇒ higher k ∗ ⇒ higher y ∗
I
higher s ⇒ smaller consumption share in output
c ∗ = (1 − s) y ∗
|{z}
| {z } |{z}
?
I
I
I
↓
↑
net effect in general is ambiguous
due to the assumptions made on the production function
c ∗ first increases, then decreases in s
The golden rule
I
what are the effects of a higher s on c ∗ ?
I
higher s ⇒ higher k ∗ ⇒ higher y ∗
I
higher s ⇒ smaller consumption share in output
c ∗ = (1 − s) y ∗
|{z}
| {z } |{z}
?
I
I
I
I
I
↓
↑
net effect in general is ambiguous
due to the assumptions made on the production function
c ∗ first increases, then decreases in s
golden rule of saving: s such that c ∗ is the highest
how to find it? by choosing s maximize
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ )= zf (k ∗ )−(n + d)k ∗
note: the last equality makes use of the steady state condition
(szf (k ∗ ) = (n + d)k ∗ )
The golden rule
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ ) = zf (k ∗ ) − (n + d)k ∗
→ take the derivative with respect to s, and find s where
where does s enter the right hand side?
∂c ∗
∂s
=0
The golden rule
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ ) = zf (k ∗ ) − (n + d)k ∗
→ take the derivative with respect to s, and find s where
∂c ∗
∂s
=0
where does s enter the right hand side? only through k ∗ , as the
steady state level of per capita capital increases in s
∂c ∗
∂f (k ∗ )
∂k ∗
∂k ∗
∂k ∗
=z
− (n + d)
= zf 0 (k ∗ )
− (n + d)
∂s
∂s
∂s
∂s
∂s
∗
∂k
= zf 0 (k ∗ ) − (n + d)
=0
∂s
⇒ zf 0 (k ∗ ) =
n
+ d}
| {z
| {z }
MPk
effective depreciation rate
The golden rule
c ∗ = (1 − s)zf (k ∗ ) = zf (k ∗ ) − szf (k ∗ ) = zf (k ∗ ) − (n + d)k ∗
→ take the derivative with respect to s, and find s where
∂c ∗
∂s
=0
where does s enter the right hand side? only through k ∗ , as the
steady state level of per capita capital increases in s
∂c ∗
∂f (k ∗ )
∂k ∗
∂k ∗
∂k ∗
=z
− (n + d)
= zf 0 (k ∗ )
− (n + d)
∂s
∂s
∂s
∂s
∂s
∗
∂k
= zf 0 (k ∗ ) − (n + d)
=0
∂s
⇒ zf 0 (k ∗ ) =
n
+ d}
| {z
| {z }
MPk
effective depreciation rate
∗
∂c
→ First find the k ∗ that maximizes c ∗ (where ∂k
∗ = 0).
∗
→ Then find the s that achieves this k using the steady state
condition szf (k ∗ ) = (n + d)k ∗ .
The golden rule
I
so the marginal product of capital should equal n + d in the
steady state to achieve the highest per capita consumption
I
→ we can find the saving rate that results in this specific
steady state: sgr
I
if the government, central planner would prescribe this saving
rate, the economy would reach the steady state and the
maximum possible consumption
I
in practice: we can estimate MPK , we know n, we can
calculate d ⇒ we can actually calculate sgr
should the government do something, try to impose this?
I
1. redistribution across generations
2. consumers save optimally (given their preferences)
⇒ is there a market failure that prevents them from achieving
the correct trade-off between current consumption and
savings?
Data: Income per capita and the population growth rate
→ GDP per capita and pop growth rate negatively correlated
Income per capita and the population growth rate in the
model
What is the effect of a higher population growth rate in the Solow
model?
I
a higher n increases the break-even investment, rotates the
effective depreciation line up
I
lower steady state per capita capital and output: k ∗ , y ∗ ↓
I
temporary decrease in the growth rate of output per worker
I
long-run decrease in the level of output per worker
I
no effect on the long-run growth rate of capital per worker
and output per worker, which are equal to zero
I
of course the growth rate of aggregate variables increases
Income per capita and the population growth rate in the
model
Income per capita and technology in the model
What is the effect of higher productivity in the Solow model?
I
a higher productivity, z increases actual investment in the
model
→ the savings curve pivots up
I
higher steady state per capita capital and output: k ∗ , y ∗ ↑
I
temporary increase in the growth rate of output per worker
I
long-run increase in the level of output per worker
I
no effect on the long-run growth rate of capital per worker
and output per worker, which are equal to zero
Income per capita and technology in the model
Income per capita and technology in the model
but what is the effect of a sustained productivity growth?