L3 Technical Change & Convergence

Question 1

How is the production function once you introduce technological change / technology?

Answer 1

Y = B (Ak * K)^α * (Al * L)^1−α

B = neutral technology (increases productivity and the effective rates of both K and L)
Ak = capital- augmenting technology (if Ak increases, the effective amount of K increases)
Al = labor-augmenting technology (if Al increases, the effective input of L increases)

Question 2

What is technological (or technical) change?

Answer 2

The improvement in the instructions for mixing together raw materials.

Question 3

Is technological change an endogenous or exogenous variable?

Answer 3

Technological change is exogenous. It is just given (by nature) and it does not come from anywhere (investment, etc,), so this model does not explain it but its effects.

Question 4

How is the production function adding just labor-augmenting technology (A)?

Answer 4

Y = K^α * (AL) ^ 1-α

AL = effective rate of L (A * L)

Question 5

At which rate does the labor-augmenting technology grow in the Solow Model?

Answer 5

At a constant rate (g):

A· / A = g = A(0) * e ^ gt

Question 6

Rewrite this production function (with AL) in per worker terms.

Answer 6

y = k^α * A ^ 1-α

The better the technology for a given amount of per worker capital (k), the better the output per worker (y).

Question 7

Convert this per worker production function (with A) to growth rates.

Answer 7

y· / y = α(k· / k) + (1 - α) (A· / A)

Both k and A growth rates are positively correlated with y.
Therefore, an increase in y can be explained by a capital increase or a technological progress.

Question 8

Explain the economy functioning according to the Solow Model using all exogenous and endogenous variables, parameters, and technology (A).

Answer 8

Firms combine L and K to produce Y.
A fraction (s) of the output Y is invested / saved and therefore added to K stock. The other fraction (1-s) is consumed (C) and therefore leaves the economy.
A fraction δ of K is depreciated every period, so it also leaves the economy.
New workers enter the economy at a rate (n) - population growth.
There is also a technology index (A), which makes L and K more productive, and increases exogenously over time at a rate g.

Question 9

Define the “per effective worker” ratios of k and y.

Answer 9

1. Capital per effective worker (k~) = K/AL = k/A

2. Output per effective worker (y~) = Y/AL = y/A

Question 10

Rewrite the new production function with the “per effective worker” ratios (or “intensive form”)

Answer 10

y~ = k~ ^ α

Question 11

Define the growth rate of k~.

Answer 11

k·~ / k~ = K· / K - A· / A - L· / L

growth rate of k~ = K· / K - g - n

Question 12

Rewrite the capital accumulation equation with the “per effective worker” ratios and growth rates.

Answer 12

k·~ = sy~ - (δ + n + g)*k~

δ + n + g -> negative correlation with k·~
sy~ -> positive correlation

Question 13

Which is the steady steate and how do you solve it?

Answer 13

The steady state is k·~ = 0 (capital accumulation equation = 0)

k~* = [s / (δ + n + g)] ^ 1/(1-α)
y~* = [s / (δ + n + g)] ^ α/(1-α)

Question 14

Which growth rates do we have in the steady state for y, r, and w?

Answer 14

(y· / y)* = g
(r· / r)* = 0
(w· / w)* = g

Question 15

What results have changed in the model after adding technology?

Answer 15

• Per capita income grows (y) at rate g in the long run (same as data)
• The real wage (w) grows at rate g in the long run (same as data)

Therefore, now the model replicates long-run behavior of industrial economies.

Question 16

Does capital accumulation generate long-run growth?

Answer 16

Capital accumulation (& investment in physical and human capital) generates only transitional growth of per capita income, but it is insufficient to create long-run growth on its own.

Question 17

What is the actual engine of long-run growth according to the Solow Model?

Answer 17

Technological progress.

Question 18

What is the formula that describes per capita income in the long run?

Answer 18

y* = A [ s / (δ + n + g)] ^ α/1-α

• Changes on (s) and (n) affect the long run LEVEL of y (s increases y, and n decreases y), but do not affect the long-run GROWTH RATE of y.

Question 19

Why would we extend also our model with human capital?

Answer 19

To account for international differences in education and skills of the labor force (e.g., strong correlation between GDP/capita and years of schooling).

Question 20

What does the production function look like when you add human capital?

Answer 20

Y = K^α (AH)^(1-α)

Question 21

Which is the formula for Human Capital (H)?

Answer 21

H = e ^ (ψu) * L

ψ > 0
u = years of schooling

(e.g., If ψ=0.10 and workers spend 1 additional year studying, then H increases by 10%)

Question 22

The constant (u) is endogenous or exogenous?

Answer 22

Exogenous.

Question 23

Redefine the “per effective worker” ratios adding human capital (H).

Answer 23

k~ = K/AH = k/Ah
y~ = Y/AH = y/Ah

Question 24

Do the production function, capital accumulation equation, and output per effective worker (in the steady state) change?

Answer 24

No, they remain the same.

y˜ = ˜k^α
˜k· = sy˜ − (δ + n + g)˜k
y~* = [s / (δ + n + g)] ^ α/(1-α)

Question 25

After adding human capital, which is now the long-run income per worker formula?

Answer 25

y* = hA [ s / (δ + n + g)] ^ α/1-α

Question 26

Why some countries become richer than others?

Answer 26

• They have a higher savings rate (s)
• They have a lower population growth rate (n)
• They have a higher level of technology (A)
• They spend a larger fraction of time accumulating skills (H)

Question 27

How do you calculate relative income per capita (to US values)?

Answer 27

y’* (relative) = y* / y*US

```
x = δ + n + g
y’* = h’A’ (s’ / x’) ^ α/1-α
~~~

Question 28

Once you calculate relative incomes with the Solow Model formula to compare countries, does the Model fit reality? Why?

Answer 28

If we compare the correlation between relative per capita incomes with the predicted steady-state values of relative per capita incomes, the model does not get the magnitudes right. It predicts that poorest countries should be richer than they actually are.
It is because although we took into account technology, we did not take into account DIFFERENCES in technology between countries.

Question 29

How can we represent differences in technology in the production function?

Answer 29

We have to isolate the variable of technology (A) of the production function, becuase it is a measure of “total factor productivity” (TFP) and represents any difference in y (among countries) not attributable to difference in inputs (such as quality of education, health…).

y = k^α (Ah)^ (1−α) ⇒ A = (y / k) ^ (α/1−α) * y/h

There is a strong positive correlation between A and Y/L, which means that still a large part of differences between poor and rich countries remains unexplained (“TPF”).

Question 30

What does the unconditional convergence hypothesis state?

Answer 30

That developing countries will ultimately catch up with the industrially advanced countries because poor countries tend to grow faster than rich countries. Therefore, in the long run, the standards of living throughout the world become more or less the same.

Question 31

Does the Solow Model predict the unconditional convergence hypothesis?

Answer 31

It works well within the OECD (negative correlation between GDP/capita and growth rates), but at the world level there is no indication of unconditional convergence.

Question 32

Why does the Solow Model predict conditional convergence?

Answer 32

In the Solow Model, countries converge to their steady state and the OECD is a homogeneous group with similar steady states.
Convergence is therefore conditional on the savings rate, population growth rate, technological level, and education level.