1. MACROECONOMICS BGSE/UPF
LECTURE SLIDES
Professor Antonio Ciccone
BGSE/UPF Macroeconomics Slide SET 1

2. 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.
For example, the most important help even the best off farmers a few
centuries ago would have gotten is from animals. Today, the best-off farmers work with air-conditioned tractors with radio, television, as well as internet.
BGSE/UPF Macroeconomics Slide SET 1

3. 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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4. 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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5. 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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6. 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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7. 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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8. 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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9. 4. Many things are left out of the Solow model
Including:
- learning externalities
- R&D
- human capital
 Modern growth theories
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10. 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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11. 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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12. Are snapshots of an economy at one moment in time.
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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13. FIGURE 1 LABOR MARKET GOODS MARKET preferences technology
HH (preference for leisure and consumption;
aggregate labor endowment L(t))
FIRMS (technology of production)
GOODS MARKET
LABOR MARKET
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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14. 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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15. 4. Snapshot of economy with capital as a production factor
HH (preference for leisure;
aggregate labor endowment L(t) plus property rights in firms)
FIRMS (technology of production; capital owned at the beginning of the period K(t))).
GOODS MARKET
LABOR MARKET
RENTAL MARKET
FOR CAPITAL GOODS
FIGURE 2
The rental market for capital goods determines the RENTAL PRICE OF CAPITAL
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16. 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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17. 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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18. Machines available for production at time t =
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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19. unit of time and d is depreciation per unit of time.
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(tt-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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20. - 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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21. 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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22. 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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23. Production of investment as well as consumption goods
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
Where:
K is the capital they use in production
L is the labor they use in production
(E5)
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24. Hence there are two inputs in production that the firm can control.
A is a factor that will capture technological progress or improvements in efficiency
The greater A, 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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25. Assumptions about the production function F
1. Constant returns to scale (CRTS) to the inputs K and L
(E6) for any b>0
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)
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26. Hence CRTS implies that small 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)
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27. 2. Positive but decreasing MARGINAL PRODUCTS (MP) to capital and labor taken separately
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28. 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
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29. CRTS also implies that MPL depends on and A
Hence, the MPL will increase over time with A even if constant
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30. 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)
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31. 3. So-called Inada conditions
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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32. FIGURE 3 The production function in labor-efficiency units
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33. 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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34. 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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35. - Labor supply is inelastic, as assumed under preferences
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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36. Equilibrium employment L; Equilibrium real wage w=MPL
- FIGURE 4 Labor market equilibrium
Wage
LABOR SUPPLY CURVE
LABOR DEMAND
CURVE=MPL SCHEDULE
L(t)
W(t)
Employment
Equilibrium employment L; Equilibrium real wage w=MPL
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37. 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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38. Rental Price K(t) R(t) CAPITAL GOODS
FIGURE 5 Rental capital market equilibrium
Rental
Price
CAPITAL GOODS SUPPLY CURVE
CAPITAL GOODS DEMAND
CURVE=MPK SCHEDULE
K(t)
R(t)
CAPITAL GOODS
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39. 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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40. Capital per Efficiency Worker
FIGURE 6 The production function in intensive form and wages/rental price of capital
Capital per Efficiency Worker
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