econ problem set
Weeks 2-3: Economic Growth
Saki Bigio
UCLA
Spring 2022
Growth in the Very Long Run (Maddison Data)
GDP per capita: France, Germany, Italy, the United Kingdom, and the United States
Figure: Very Long-Run Growth
Very Long Run (UK)
Figure: UK Long-Run
Other Countries
Figure: Various Countries
Kaldor Facts
Figure: Nicholas Kaldor, Baron Kaldor
Kaldor Facts
1 That GDP per capita grows at a constant exponential rate
2 That capital per worker grows over time
3 That the capital/output ratio is constant
4 That GDP share of capital and labor is constant over time
5 That the return on capital is constant
6 That real wage grows over time
Linear vs. Exponential Growth
Let t be time after an initial date t0
A variable grows linearly if:
x (t) = x (t0) + g ∗(t−t0)
A variable grows exponentially if:
x (t) = x (t0) (1 + g) (t−t0)
Linear vs. Exponential Growth
Example of linear growth:
Example
You have an account with US$ 100. You add US$ 10 every month
Example
You have an account with US$ 100. You earn 10% return every year.
Linear vs. Exponential Growth
Fact 1: if a variable grows linearly, its graph over time is a line
Fact 2: If a variable grows exponentially, it's logarithm grows linearly:
log x (t) = log x (t0) + (t−t0) log (1 + g)︸ ︷︷ ︸ new G
Combining 1-2: If a variable grows exponentially, the graph of its logarithm is a line
Levels
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Year
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
55000
G D
P p
e r
ca p ita
( P
P P
) -
2 0 1 1 U
S $
United States
Peru
Log Scale
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Year
2000
5000
10000
20000
30000
40000
50000
G D
P p
e r
ca p ita
( P
P P
) -
2 0 1 1 U
S $ (
L o g S
ca le
)
United States
Peru
Kaldor Facts
1 That GDP per capita grows at a constant exponential rate
2 That capital per worker grows over time
3 That the capital/output ratio is constant
4 That GDP share of capital and labor is constant over time
5 That the return on capital is constant
6 That real wage grows over time
Kaldor Fact 2: Capital Per Worker over time
Figure: Output per worker
Kaldor Fact 3: Capital Per Output
Figure: Capital / Output Ratio
Kaldor Fact 4: Constant Labor/Capital Share
Figure: Labor-Capital Shares
Kaldor Fact 5: Return on Capital
Figure: Return on Capital
Kaldor Fact 6: Real Wages
Figure: Real Wages US
Kaldor Fact 6: Real Wages
Figure: Real Wages
The Solow-Swan Model
Figure: Robert Solow
The Solow-Swan Model
Three fundamental equations: I neoclassical production function I aggregate demand identity I capital accumulation equation
Savings Rule I savings-investment identity
Neo-classical Production Function
Production Yt = AtF (Kt,Lt ) (1)
I function of capital Kt I function of labor Lt I function of technology At
Neo-classical Production: Properties
Constant Returns (all factors)
λ F (K,L) = F (λ K,λ L)
Decreasing Returns (individual factors)
∂ F
∂ K ≡FK > 0;
∂ F
∂ L ≡FL > 0
and ∂ 2F
∂ K2 ≡FKK < 0;
∂ 2F
∂ L2 ≡FLL < 0
Inada Conditions:
lim K→0
FK = lim L→0
FL = ∞
lim K→∞
FK = lim L→∞
FL = 0.
Neo-classical Production Function: Slices
Figure: Production Function - Cuts
Neo-classical Production Function
3 Dimensions
O u tp
u t Y
t
1
Capital K t Labor L
t
Capital Accumulation and Aggregate Demand - Supply
Aggregate Demand (De�nition).
Yt = Ct + It. (2)
Capital Accumulation (Stock Equation).
Kt+1 = Kt − δ Kt︸︷︷︸ dep.
+It. (3)
I Capital at t+1 will be used in production tomorrow.
Investment-Savings Identity and Savigngs Rule.
It = sYt = s AtF (Kt,Lt )︸ ︷︷ ︸ Yt
. (4)
I s, constant savings rule I note that
Ct = (1−s) AtF (Kt,Lt ) .
Growth in Exogenous factors factors
Exogenous Growth Rates.
At+1 = (1 + x) At and (5)
Lt+1 = (1 + n) Lt.
Notation Alert
Per capita variabes
We express variables in per capita terms with lower cases
Thus
yt = Yt/Lt
and kt = Kt/Lt
De�nitions: Steady-State Balanced Growth
De�nition.
An equilibrium growth path is a sequence for quantities {Kt,Ct,It} from t = 0 to t → ∞ such that, given an initial level K (0), capital satis�es the law of motion (3), investment is given by (4) and output is given by (1) and (2) also holds.
Summary equation:
Kt+1 = Kt −δ Kt + sAtF (Kt,Lt ) .
De�nition.
A steady-state equilibrium are values {Kss,Css,Iss} for which, given an initial level K (0), variables satis�es (3), investment is given by (4) and output is given by (1) and (2) and capital does not grow.
De�nition.
A balanced growth path is an equilibrium growth path in which all variables grow at the same rate.
Solution: Model without pop or tech growth
Here: set x = 1 and A0 = 1 as in textbook, but for now x = n = 0.
Target: GDP per capita.
Lt = L0 = Lss and At = A0 = Ass where Lss and Ass are constants.
KEY: kt ≡Kt/Lt.
Solution: Model without pop or tech growth
Divide 3 by Lss and replacing in 4 yields:
kt+1 = Kt Lss
= (1−δ ) Kt Lss
+ sAtK
α t L
1−α t
Lss
= (1−δ ) kt + sAtK
α t L
1−α t
Lαt L 1−α t
= (1−δ ) kt + sAtkαt .
Note: that kt+1 = Kt+1/Lt+1 but since Lt+1 = Lt we have that kt+1 = Kt+1/Lt.
I Later, cannot make this substitution.
Solution: Model without pop or tech growth
Summary Equation: ∆kt ≡kt+1−kt = sAsskαt −δ kt (6)
k0 we can determin capital in every date
Changes
We express changes is variables with a ∆.
Solution: Graphical Device Working with our summary equation:
kt+1 = s Assk α t︸ ︷︷ ︸
yt
+ (1−δ ) kt
Figure: The Solow diagram
Solution: Graphical Device
Working with our summary equation:
∆kt ≡kt+1−kt = syt −δ kt
i, δk
klow0 kss k high 0 k
sy
δk
Figure: The Solow diagram
Solution: Graphical Device
Working with our summary equation:
∆kt ≡kt+1−kt = syt −δ kt
∆k
klow0 kss k high 0 k
∆k = sy − δk
Figure: Change in capital per capita
Solution: Graphical Device Working with our summary equation:
∆kt ≡kt+1−kt = syt −δ kt
I Steady state: ∆kt = 0⇐⇒kt = kss for any t.
∆k k
δ
kss k
∆k k
sy/k = skα−1
Figure: Rate of change of capital per capita
Reminder: What is the steady state? Working with our summary equation:
∆kt ≡kt+1−kt = syt −δ kt
I Steady state: ∆kt = 0⇐⇒kt = kss for any t.
∆k k
δ
kss k
∆k k
sy/k = skα−1
Figure: Rate of change of capital per capita
Steady State of the Model By de�nition, in a steady state we must have the condition that the variables determined by the model do not grow:
0 = sAssk α ss −δ kss.
Clear kss from this equation to obtain:
kss =
( sAss
δ
) 1 1−α
. (7)
Steady-State GDP per capita: I we know:
yt = Assk α t . (8)
Thus:
yss = Ass
( sAt δ
) α 1−α
= A 1
1−α ss
( s δ
) α 1−α
= A 1
1−α ss
( s δ
) α 1−α
Solution: Dynamics
Working with our summary equation:
∆kt ≡kt+1−kt = syt −δ kt
Can show that convergence is from above of below.
klow0
kss
k high 0
k
time
Figure: Evolution of Capital Through Time
Question
Does the Solow model, with x and n zero, explain growth?
What happens if a country has a higher savings rate than another?