Transcript ITFD-SET1 - Antonio Ciccone`s Webpage

ITFD Growth and
Development
LECTURE SLIDES
Professor Antonio Ciccone
ITFD Growth and Development
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I. THE SOLOW MODEL
1. WHY HAVE PRODUCTIVITY LEVELS
BEEN RISING? PROXIMATE CAUSES
1. (Physical) Capital per worker
Physical capital refers to all durable inputs into
production: machine tools, motor vehicles,
computer hardware and software etc.
One reason for higher productivity today is that
workers have on average much more capital to
work with.
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2. Technology or the Accumulation
of Knowledge (“Ideas”)
Over the course of time, people have
accumulated more and more ideas that
allow them to get more output using the
same inputs. That is economies have
become more EFFICIENT.
For example, three field crop rotation.
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The mechanisms through which knowledge is
accumulated include:
• Learning externalities: learn how to do things
more efficiently based on your own experience
or that of others. This experience accumulates
over time.
• Specialization: larger markets allow people to
focus on a more limited set of tasks; in the past
for example the same people would be both
dentists and hairdressers.
• Research and development: some people do
nothing but try to come up with ideas all day; for
profit, they hope to be able to patent them.
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3. Human capital
• Today, almost everybody can read and write in
industrialized countries while in the past these
capacities were limited to very limited segments
of society only. The capacity to R&W makes
people more productive.
• In some industrialized countries almost half of
the students go on to get a higher education.
This makes them able to produce more
efficiently and also be more involved in
generating new ideas.
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2. WHY THE SOLOW MODEL
1. Focus on the accumulation of physical
capital
Capital accumulation evidently and always part
of the growth process.
- Makes it empirically relevant
- Necessary ingredient in growth models who
focus on other drivers of economic growth.
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2. Capital accumulation and savings alone
cannot explain long-run growth
Its main result was at the time “counterintuitive”
but is based on solid fundamentals and has
been show to be consistent with many empirical
observations:
Capital accumulation and savings alone
CANNOT explain why income per capita keeps
growing.
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3. A dynamic general equilibrium model
Solow model is the first (reasonable)
general equilibrium model about how the
economy evolves over time.
And it is still the backbone of the models
used in macro.
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4. Many things are left out of the Solow
model
Including:
- learning externalities
- R&D
- human capital
 Modern growth theories
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3. STATIC AND DYNAMIC GENERAL
EQUILIBRIUM MODELS
1. A GE model is simply a model of the
economy as a whole
This means that it treats together markets that in microeconomics
would be dealt with separately. Economic growth has implications
for many different but related markets and studying it therefore
requires a GE model
For example, if firms have access to better technologies:
• this will affect their labor demand, and therefore the labor market
• through the labor market this will affect the wage/income workers
earn and therefore their capacity to save
• the savings of workers will affect how much new investment firms
are able to do
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Tractable GE models:
- have to focus on the INTERACTING
MARKETS that appear essential for the
question asked
- otherwise there is no way one could make
progress given the many markets and
market participants that are part of even a
small economies like Luxembourg
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2. Static GE models
Are snapshots of an economy at one
moment in time.
For example, the following extremely
simple model to determine:
- output Y(t) of an economy
- real real wages w(t)
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FIGURE 1
HH (preference for leisure and consumption;
aggregate labor endowment L(t))
LABOR MARKET
GOODS MARKET
FIRMS (technology of production)
To determine PRICES and ALLOCATIONS what is going on we
therefore need to specify:
 preferences
 technology
 markets that exist and their structure
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3. Capital
An dynamic general equilibrium model (growth
model) needs:
• a way to transfer resources from the present to
the future
This will be accomplished by having a production
factor called CAPITAL in the model:
FOREGO CONSUMPTION TODAY
 BUILD UP NEW CAPITAL
 PRODUCE MORE GOODS TOMORROW
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4. Snapshot of economy with capital as a production factor
FIGURE 2
HH (preference for leisure;
aggregate labor endowment L(t) plus property rights in firms)
LABOR MARKET
GOODS MARKET
FIRMS (technology of production; capital owned
at the beginning of the period K(t))).
RENTAL MARKET
FOR CAPITAL GOODS
The rental market for capital goods determines the RENTAL PRICE OF CAPITAL
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5. From the static to the dynamic model
• The static model determines aggregate
output Y(t) and the real wages w(t) and
real rental prices of capital R(t) for a given
labor supply L(t) and the capital stock K(t).
• The Solow model tells us how to
determine the whole evolution of capital
stocks and output levels over time, from
time 0 to infinity.
• As a result it tells about factors prices,
income distribution, and income/output.
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The key to going from the static to the
dynamic model:
- understanding the evolution of the capital
stock over time
- imagine time going from period 0 to period
1, 2, 3, 4, and so on
- what is the link then between the capital
stock at period t and the period before that
t-1?
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Machines available for production at time t =
Machines available for production at time t-1
MINUS Machines that broke during production in period t-1
PLUS New Machines produced by firms at time t-1
(E1)
K [t ]  [ K [t  1]   K [t  1]]  I [t  1]  (1   ) K [t  1]  I [t  1]
CAPITAL ACCUMULATION EQUATION, a key equation in the Solow model
-- “delta” fraction of capital stock that breaks in use
-- I investment (machines produced)
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We will work in continuous time. The capital accumulation will therefore look a bit different.
It can be derived from the equation above by letting the time between periods becomes
smaller and smaller. Denote the time between periods by D then:
(E2)
where
K [t ]  K [t  D]   K [t  D]D  I [t  D]D
I(t) is now the investment flow, i.e. the investment per (very small)
unit of time and

is depreciation per unit of time.
Rewriting yields:
(E3)
K [t ]  K [t  D]
 I [t  D]  K [t  D]
D
Taking the limit as D goes to zero becomes:
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(E4)
K [t ]

K [t ] 
 I [t ]  K [t ]
t
- From now on DOTS over symbols will denote TIME
DERIVATIVES.
- (E4) says that the change in the capital stock over time
(net investment) is equal to (gross) investment minus the
capital that depreciates while being used in production.
- This is the capital accumulation equation in continuous
time, which will end up linking different time periods.
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4. THE SOLOW MODEL AT A MOMENT IN TIME
1. A model of output and factor prices given
factor stocks
The goal is to understand the determination of
output (the precise economic statistic is called
gross domestic product, GDP), wages, the rental
price of capital, and the distribution of income
among factors of production (functional income
distribution).
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To do that we need to:
(1)
(2)
(3)
(4)
specify preferences of HH
specify technology of production of
firms
specify structure of the labor market,
the goods market, and the labor market
define an equilibrium
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1. Preferences
Households supply all their labor L(t) to the labor market,
whatever wages may be (they supply L(t) inelastically).
For the dynamics we will have to specify how they save,
for what we do now it doesn’t matter.
2. Production
Production of investment as well as consumption goods
Firms produce investment and consumption goods using
the following technology
(E5)

1


Y  K ( EL)
Where:
• K is the capital they use in production
• L is the labor they use in production
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Hence there are two inputs in production that the firm
can control.
• E is a factor that will capture technological progress or
improvements in efficiency
The greater E, the more the firm produces with a given
amount of resources K and L (the “more efficient” is the
firm).
Technological progress is taken as given; the firm cannot
control it. Technological progress multiplies labor, it is as
if it increased the efficiency of labor. This is called
LABOR-AUGMENTING TECHNICAL PROGRESS.
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Assumptions about the production function F
1. Constant returns to scale (CRTS) to the inputs K and L
(E6) (bK ) ( E bL )1  b( K ) ( E  L )1  bY
So, if you double inputs, you double output. This make
sense in long run because you can always at least
“replicate”.
An important implication of this is that output per worker
only depends on capital per worker. To see this take
b=1/L
(E7)
Y  K   EL 1
 E1
y    

k

L L  L 
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Hence CRTS implies that large firms produce as much
output per worker as small firms if they have the same
K/L.
It will be useful to introduce the notation of output per
efficiency worker (which because of CRTS depends only
on capital per efficiency worker)
(E8)
Y
y k E1
y
 
 k E   (k / E )
EL E
E
y  k
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2. Positive but decreasing MARGINAL PRODUCTS (MP)
to capital and labor taken separately
(E9)
MPK 
Y F ( K , EL)

0
K
K
MPK  2Y  2 F ( K , EL)


0
K
K 2
K 2
MPL 
Y F ( K , EL)

0
L
L
(E10)
MPL  2Y  2 F ( K , EL)


0
L
L2
L2
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An important implication of CRTS is that the MPK
ONLY DEPENDS ON capital per efficiency worker
To see this note:
E11
F ( K , EL)
MPK 
  K  1( EL)1
K
 1
K


 1
MPK   


k

EL


y k 
MPK 

  k  1
k
k
Hence, the MPK will not change over time if is
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k
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constant
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CRTS also implies that MPL depends on
(E12)
k and A
F ( K , AL)
MPL 
 (1   ) EK  ( EL)
L

 K 
MPL  (1   ) E 

 EL 
MPL  (1   ) Ek 
Hence, the MPL will increase over time with A even if
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k constant
-Finally, CRTS also implies that if firms pay the MP to their
inputs K and L, there will be no (pure/economic) profit left,
i.e. all output will be paid to production factors.
(E13)
MPL * L  MPK * K  Y
MPL
K
Y
 MPK *

E
EL EL
MPL
 y  MPK * k
E
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3. So-called Inada conditions
(E14)
MPK   when k  ( K / EL)  0
and
MPK  0 when k  ( K / EL)  .
Which say that the marginal produce of capital is very high
when there is little of it and very low when there is a lot
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y
y  k
k
-
FIGURE 3 The production function in labor-efficiency units
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FIGURE 6 The production function in intensive form and
wages/rental price of capital
MPK
yk
y

MPL
E
k
Capital per Efficiency Worker
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3. Market structure and equilibrium
All markets are assumed to be perfectly competitive
HH satisfy their budget constraint
Demand=Supply in all market
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2. The static equilibrium
Output depends on K and L employed in
production, i.e. on factor use.
Factor use is determined in factor markets.
Let us take a look at the two markets:
- labor market
- rental market for capital
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1. Labor market
- Labor supply is inelastic, as assumed under preferences
- Labor demand
For any given capital stock firms hire labor to maximize profits
(E15)
max F ( K , EL)  wL
L
(bars to make clear what is taken as given by firms) which gives rise to labor
demanded as a function of w
(E16)
MPL  w
for any w, firms demand labor to equalize MPL and w.
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- FIGURE 4 Labor market equilibrium
Wage
LABOR SUPPLY CURVE
W
LABOR DEMAND
CURVE=MPL SCHEDULE
L
Employment
Equilibrium employment L; Equilibrium real wage w=MPL
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2. Rental market for capital
Firms own their capital, but that does not prevent them from renting it
out if they think they can make money doing so
- Capital supply is inelastic at a given moment in time
- Capital demand
For any given level of employment firms rent capital to maximize profits
(E17)
max F ( K , EL )  RK
K
(bars to make clear what is given) which gives rise to capital demanded
as a function of rental cost of capital R (use one period then return what is left)
(E17)
MPK  R
for any R, firms demand labor to equalize MPK and R.
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FIGURE 5 Rental capital market equilibrium
Rental
Price
CAPITAL GOODS SUPPLY CURVE
R
CAPITAL GOODS DEMAND
CURVE=MPK SCHEDULE
K
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CAPITAL GOODS
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3. Summarizing the static equilibrium
The factor market (static) equilibrium conditions for given K(t), L(t), A(t)
and hence k (t ) are :
Rt  MPKt   (kt ) 1
wt  MPLt  Et (1   )(kt )
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see E11
see E12
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FIGURE 6 The production function in intensive form and
wages/rental price of capital
R(t )
yk

y (t )
w( t )
E (t )
k (t )
Capital per Efficiency Worker
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