MACROECONOMICS BGSE/UPF

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MACROECONOMICSBGSE/UPF

• LECTURE SLIDES
• Professor Antonio Ciccone

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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 RW 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 dynamics macro (including business-cycle
  macro).

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• 4. Many things are left out of the Solow model
• Including
• - learning externalities
• - RD
• - 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 wages w(t)

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FIGURE 1

• 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
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)
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 I(t-D) is now the investment flow, i.e. the
investment per (very small) unit of time and d
is depreciation per unit of time.
Rewriting yields
(E3)
Taking the limit as D goes to zero becomes
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(E4)

• - 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
• 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 doesnt matter.
• 2. Production
• Production of investment as well as consumption
  goods
• Firms produce investment and consumption goods
  using the following technology
• Where
• K is the capital they use in production
• L is the labor they use in production

(E5)
Slide 23
ITFD Growth and Development Slide SET 1
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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.

Slide 24
ITFD Growth and Development Slide SET 1
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• Assumptions about the production function F
• 1. Constant returns to scale (CRTS) to the
  inputs K and L
• (E6)
• 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 b1/L
• (E7)

Slide 25
ITFD Growth and Development Slide SET 1
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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)
Slide 26
ITFD Growth and Development Slide SET 1
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• 2. Positive but decreasing MARGINAL PRODUCTS (MP)
  to capital and labor taken separately

(E9)
(E10)
Slide 27
ITFD Growth and Development Slide SET 1
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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
Hence, the MPK will not change over time if is
constant
Slide 28
ITFD Growth and Development Slide SET 1
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CRTS also implies that MPL depends on and A
(E12)
Hence, the MPL will increase over time with A
even if constant
Slide 29
ITFD Growth and Development Slide SET 1
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• 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)

Slide 30
ITFD Growth and Development Slide SET 1
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3. So-called Inada conditions
(E14)
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
Slide 31
ITFD Growth and Development Slide SET 1
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FIGURE 3 The production function in
labor-efficiency units
-
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3. Market structure and equilibrium All markets
are assumed to be perfectly competitive HH
satisfy their budget constraint DemandSupply 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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• Labor market
• - Labor supply is inelastic, as assumed under
  preferences
• - Labor demand
• For any given capital stock firms hire labor to
  maximize profits

(E15)
(bars to make clear what is taken as given by
firms) which gives rise to labor demanded as a
function of w
(E16)
for any w, firms demand labor to equalize MPL and
w.
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- FIGURE 4 Labor market equilibrium
Equilibrium employment L Equilibrium real wage
wMPL
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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)
(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)
for any R, firms demand labor to equalize MPK and
R.
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FIGURE 5 Rental capital market equilibrium
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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 are
see E11
see E12
or
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FIGURE 6 The production function in intensive
form and wages/rental price of capital
Capital per Efficiency Worker