I NTERMEDIATE M ACROECONOMIC T HEORY
L ECTURE 2
Simon Fraser University
Instructor: Marieh Azizirad
Summer 2023
Our plan in this lecture
Long-run economic growth
Production functions
Solow growth model
ECON 305 D100 Summer 2023, Lecture 2
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Long-run economic growth
some facts:
No growth in the long history.
Modern economic growth only
emerges in the most recent three
centuries.
’The great divergence’.
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Long-run economic growth
some facts:
A growth rate in some variable x is the percentage change in that variable between
time t and t + 1:
gx =
xt+1 −xt
xt
Examples:
Yt+1 −Yt
, or
Yt
Lt+1 −Lt
n = Lt ,
Output growth: gY =
Yt+1 = Yt (1 + gY )
Population growth:
or Lt+1 = Lt (1 + n)
At a constant growth rate,
Yt = Y0 (1 + gY )t
Lt = L0 (1 + n)t
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Long-run economic growth
some facts:
Ratio scale (or log scale):
start with
Lt = L0 (1 + n)t
in natural logs (ln, sometimes economists write it as log),
ln (Lt ) = ln (L0 ) + t ∗ ln (1 + n)
for small growth rates n,
ln (1 + n) ≈ n
therefore:
ln (Lt ) = ln (L0 ) + n ∗ t
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Long-run economic growth
some facts:
Ratio scale (or log scale): Rule of 70
→ If a variable x is growing at a constant rate equal to gx then it takes
variable x to be doubled.
70
gx
years for the
→ The time it takes for income to double depends only on the growth rate, not on the
current level of income.
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Long-run economic growth
some facts:
Ratio scale (or log scale):
a: Lt = L0 (1 + n)t
b: ln (Lt ) = ln (L0 ) + n ∗ t
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Long-run economic growth
some facts:
Ratio scale (or log scale):
Some important growth rate rules:
Suppose there are three variables x and y with constant growth rates given by gx
and gy , then:
if z = x ∗ y, then gz = gx + gy ;
if z = xy , then gz = gx − gy ;
if z = xa , then gz = a ∗ gx ;
what if z = a?
Note
These are easily derived from the log growth equations.
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Our plan in this lecture
Long-run economic growth
Production functions
Solow growth model
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A model of production
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A model of production
Production function: Y = F(K, L).
A reasonable production function has the following properties so that it fits the
real-world production better:
More is better: Y increases in K and L (holding the other argument constant);
dY
MPK = dK
> 0 and MPL = dY
dL > 0.
2
2
Concavity: Diminishing MPK and MPL; ∂∂ KY2 < 0, ∂∂ LY2 < 0.
Duplicable: Constant returns to scale (CRS): for any parameter c > 0,
F(cK, cL) = cF(K, L).
→ Learn the difference between IRS and CRS in the textbook.
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A model of production
A frequently used example: Cobb-Douglas production function
Y = AK α L1−α
parameters: A > 0, 0 < α < 1.
Exogenous variable:
L: labor input (the total number of person-hours worked).
K: capital input (the real value of all machinery, equipment, and other capital input).
Endogenous variable:
Y: total production/output (the real value of all goods produced).
Question:
Does it satisfy the three properties above?
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A model of production
Cobb-Douglas production function:
Y = AK α L1−α
A = 1;
α = 0.5
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A model of production
Cobb-Douglas production function:
Y = AK α L1−α refers to the aggregate variables.
Sometimes we use the per capita version of the production function by dividing
both sides of the equation by L (why?):
L1−α
Y
= AK α
L
L
→ y = Akα
y = YL : output per capita or ’living standard’.
k = KL : capital per capita.
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A model of production
Cobb-Douglas production function:
What is A?
Y = AK α L1−α
A is called "Total factor productivity". It is interpreted as efficiency or productivity. It
measures how productive countries are using their K and L.
There is no measures to collect data on A, therefore it is also referred to as "the
residual" or "a measure of our ignorance".
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A model of production
Cobb-Douglas production function:
What is A?
y = Akα
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A model of production
Cobb-Douglas production function:
What is α in Y = AK α L1−α ?
If input markets are characterized by perfect competition, then profit maximization
conditions are:
MPL = w →
− (1 − α)AK α L−α = w
MPK = r →
− αAK α−1 L1−α = r
w ∗ L = (1 − α)AK α L−α
Therefore,
∗ L = (1 − α)Y. So, (1 − α) is the share of total income
going to workers, or the labor share of income.
Similarly, α is the capital share of income.
Empirically, α ≈ 1/3.
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Our plan in this lecture
Long-run economic growth
Production functions
Solow growth model
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Solow growth model
Goal:
Study how capital accumulates over time, and how the limitations of capital
accumulations leave long-run economic growth unexplained.
The Solow model allows us to consider the accumulation of capital as a possible engine
of long-run economic growth.
Features:
Take A (TFP) and L (labor) as given, and endogenize the accumulation of K (capital).
Base on the Cobb-Douglas production function.
K can be produced through "investment", but also depreciates over time.
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Solow growth model
Model setup:
Assume a closed economy with no government.
NX = 0 and G = 0.
So, Yt = Ct + It .
Assume a constant saving rate at every time period, s ∈ (0, 1).
It = s ∗ Yt .
Ct = (1 − s) ∗ Yt .
Assume a constant depreciation rate, δ ∈ (0, 1).
So that total depreciation at time t is δ Kt .
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Solow growth model
Dynamics of K: Change in K over time;
∆Kt+1 ≡ Kt+1 − Kt
= It − δ Kt
A side note: flow v.s. stock variables:
A flow variable is an economic variable that is
measured per period of time.
→ E.g. Investment, output/income/GDP.
A stock variable is an economic variable that is
measured at a particular point of time.
→ E.g. Capital, wealth.
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Solow growth model
Dynamics of K: Change in K over time;
∆Kt+1 ≡ Kt+1 − Kt
= It − δ K t
= sYt − δ Kt
= sAKtα Lt1−α − δ Kt
= [sALt1−α ]Ktα − δ Kt
→ now we have a dynamic model of capital accumulation:
∆K is a function of K.
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Solow growth model
Solving the model: The Solow diagram.
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Solow growth model
Solving the model: The Solow diagram.
There is only one point where this dynamic system becomes stable (i.e., K does
not change over time anymore).
It is when investment equals depreciation and is called the "Steady State" shown
by K ∗ .
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Solow growth model
How about output and consumption?
Recall that,
It = s ∗ Yt ,
and Ct = (1 − s) ∗ Yt .
There is a steady state Y ∗ and C∗
accordingly.
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Solow growth model
Solving for K ∗ ,
I − δ K∗ = 0
sAK ∗α L1−α − δ K ∗ = 0
sAL1−α
= K ∗(1−α)
δ
→
K∗ =
ECON 305 D100 Summer 2023, Lecture 2
1
1
s 1−α
· A 1−α · L
δ
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Solow growth model
Solving for Y ∗ and C∗ ,
Plug K ∗ in the production function,
Y ∗ =AK ∗α L1−α
s α
α
1−α
· A 1−α · Lα L1−α
=A
δ
α
1
s 1−α
=A 1−α
L
δ
Exercise: Using C = (1 − s)Y solve for C∗ .
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Solow growth model
Lessons about the steady state:
Why does the Solow model eventually reaches a constant level of K ∗ and Y ∗ , no
matter where the initial state is?
→ Diminishing MPK!
Diminishing returns to capital accumulation means that each increment of K increases
production (and investment) by less and less, but the amount of depreciation rises
one-for-one with K.
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Solow growth model
Lessons about the steady state:
y∗ =
Y∗
L
1
= A 1−α
s
δ
α
1−α
living standards = efficiency * capital intensity
∗
capital intensity: At the S.S.,sY ∗ = δ K ∗ , and thus KY ∗ = δs .
power of A,
rewrite the equation as,
α
Y∗
s 1−α
= A· A·
L
δ
the direct productivity effect on Y ∗ ;
the indirect effect on Y ∗ through K ∗ . (Recall that K ∗ =
ECON 305 D100 Summer 2023, Lecture 2
s
δ
1
1−α
1
· A 1−α · L)
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Solow growth model
Lessons about the long-run economic growth:
There is NO long-run growth via capital accumulation.
→ Every variable (K, Y, C) eventually converges to a constant level or ’steady
state’.
Long-run living standards depend on A, s, and δ , not on initial capital or the size of
the population. Recall,
α
1
s 1−α
y∗ = A 1−α
δ
All long-term growth comes from a constant growth in TFP (A), not even a one
time increase in A.
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Solow growth model
Lessons about the long-run economic growth: Some experiments on A, s, and δ , and
their effect on the economy.
A permanent increase in s:
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Solow growth model
Lessons about the long-run economic growth: Some experiments on A, s, and δ , and
their effect on the economy.
A permanent increase in s increases the level of income permanently, but affect
income growth only temporarily. The long-run economic growth eventually goes
back to 0.
Exercise: Change other parameters and see what happens.
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Solow growth model
Lessons about the short-run economic growth (transition dynamics):
When K < K ∗ , there is a short-run growth before the economy hits the steady
state.
The further away K is from K ∗ , the faster this growth will be. This is called
"catch-up" growth.
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Solow growth model
Limitations of the Solow Model:
Assumptions are too strict. e.g. People choose and change their saving rate all
the time in the real world.
It does not explain the long-run economic growth endogenously,
→ It all contributes to this mysterious A.
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Readings
Jones: Ch3, Ch4.1-4.3, Ch5
Andolfatto: Ch9.1, 9.7, Appendix 9.1
https://www.pitt.edu/ mgahagan/Solow.htm
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