# ECO2143 Chapter 1-7: Review for Midterm 1 on Growth

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15 Feb 2016

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Professor

Economic growth is the change in GDP per capita.

There was virtually no growth in most countries before the industrial revolution.

Inequalities between countries are increasing.

Convergence is the idea that all countries converge towards the same GDP per

capita and that, as they approach this figure, their growth slows.

Capital is a physical object that let’s us work. It is

oProductive: its purpose is to create something.

oProduced: it comes from investment1. Since investment=savings (IS-LM),

this investment is symbolized by I=y�, where � is the saving rate.

oLimited.: only a limited amount of people can use it at the same time2.

oEarns a return: people are willing to pay to use it. It’s the incentive for its

production. For example, taxis are capital because the driver rents them. A

factory is also capital because it makes profit for the owner.

oDepreciates: It wears out. Symbolized by δ.

Human capital (labour) has all of these previously mentioned characteristics

except for limitation. Human capital also depreciates much slower than physical

capital and can only earn a return when there is work done

Productivity (A) is the effectiveness to which capital can be transformed into

output.

The two sources of growth are capital accumulation (increasing output per

worker) and productivity (increasing how much output is created for a fix level of

capital)

The production function, under its Cobb-Douglas form, is

F

(

K , N

)

=A Kα(h L)1−α

, where A is productivity, K is capital, N is the number

of workers and is the capital shareɑ3. 4If we divide by N, output per worker is

then

F

(

K

N,1

)

=f

(

K

N

)

=y=A kαh1−α

.

1 Land is not considered to be capital

2 An idea is not capital.

3 A number that gives the proportion of output that stems from

capital. In this course, we assume it is 1/3.

4 In the Cobb-Douglas model,

Output=Productivity × factors

1

Normal

Higher Productivity (A)

Lower productivity (A)

Capital per worker (k)

Output per worker (y)

The Solow Model links output per worker (y) and capital per worker (k):

∆ k =γ y−δk

Steady state occurs when capital doesn’t vary (∆k=0, or

γ y =δk

).

Replacing y by its Cobb-Douglas function, we have that capital at the

steady state is

γ A

(

kss

)

α=δ kss

⇒kss=

(

γ A

δ

)

1

1−α

Replacing this in the Cobb-Douglas function, we have that output at

the steady state equilibrium is

yss=A1/(1−α)×

(

γ

δ

)

α

1−α

In the previous model, we held labour output constant. However, if

we take in account the change in productivity output

A=A h1−α

,

then this becomes

yss=h × A1/(1−α)×

(

γ

δ

)

α

1−α

2

oIf � and δ are held constant, then, whatever k, output will always

tend towards the steady state. Therefore, the Solow model cannot

explain long-term growth (weakness).

oIf we have two countries i and j with the same productivity and

depreciation, then

yi

ss

yj

ss =

(

γi

γj

)

α/(1−α)

oThe golden-rule level of capital is the � that maximizes

consumption per worker5 in steady state. It’s the � for which the

distance between the output curve and the depreciation curve is the

farthest.

oLet n be population growth. Then

yss=h × A1/(1−α)×

(

γ

n+δ

)

α

1−α

Therefore, if the saving rate and depreciation rate are the same

between two countries,

yi

ss

yj

ss =

(

δ+nj

δ+ni

)

α/(1−α)

5 Consumption per worker :

c=f

(

kss

)

−δ k ss

3