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?