MACROECONOMICS BGSE/UPF

BGSE/UPF Macroeconomics Slide SET 1 Slide 1

MACROECONOMICSBGSE/UPF

LECTURE SLIDES

Professor Antonio Ciccone


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.


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.


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.


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.


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.


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.


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).


4. Many things are left out of the Solow model

Including:

- learning externalities

- R&D

- human capital

Modern growth theories


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


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


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)


FIGURE 1

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


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 TODAYBUILD UP NEW CAPITAL PRODUCE MORE GOODS TOMORROW


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)

FIRMS (technology of production; capital owned at the beginning of the period K(t))).

GOODS MARKET LABOR MARKET

RENTAL MARKETFOR CAPITAL GOODS

The rental market for capital goods determines the RENTAL PRICE OF CAPITAL


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.


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?


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) ( ) ( 1) ( 1) ( 1) 1 ( 1) ( 1)K t K t K t I t K t I t

CAPITAL ACCUMULATION EQUATION, a key equation in the Solow model

-- “delta” fraction of capital stock that breaks in use-- I investment (machines produced)


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 then:

(E2) ( ) ( ) ( ) ( )K t K t K t I t

where 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

I t K t

Taking the limit as goes to zero becomes:


(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.

( )( ) ( ) ( )

K tK t I t K t

t


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).


To do that we need to:

(1) specify preferences of HH

(2) specify technology of production of firms

(3) specify structure of the labor market, the goods market, and the labor

market

(4) define an equilibrium


1. PreferencesHouseholds 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. ProductionProduction of investment as well as consumption goodsFirms 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) ( ) ( ( ), ( ) ( ))Y t F K t A t L t


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.


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)

( ) ([ ( )], ( )[ ( )])bY t F bK t A t bL t

, ( , )Y K

y F A F k AL L


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) ,1Y K K

y F f f kAL AL AL


2. Positive but decreasing MARGINAL PRODUCTS (MP) to capital and labor taken separately

2 2

2 2

( , )0

( , )0

Y F K ALMPK

K K

MPK Y F K ALK K K

(E9)

2 2

2 2

( , )0

( , )0

Y F K ALMPL

L L

MPL Y F K ALL L L

(E10)


An important implication of CRTS is that the MPK ONLY DEPENDS ON capital per efficiency worker

To see this note:

1 1

1

,1( , )

1, ,1

( ,1) '( )

KALF

F K AL ALMPKK KK K

MPK AL F A FL AL AL

MPK F k f k

Hence, the MPK will not change over time if is constantk

E11


CRTS also implies that MPL depends on and Ak

(E12)

1 2

1

( ,1)( , )

( ,1) ( , )

( ,1) ( ,1)

( ) '( ) ( ) '( )

KALF

F K AL ALMPLL LK K K

MPL AF ALF AAL L AL

K K KMPL AF AF

AL AL AL

MPL Af k Akf k A f k kf k

Hence, the MPL will increase over time with A even if constant k


-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

LA f k kf k Kf k

LAf k

F K AL

Y


3. So-called Inada conditions

'(0) and '( ) 0.f f

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

(E14)


( )k t

y f k

( )y t

- FIGURE 3 The production function in labor-efficiency units


3. Market structure and equilibrium

All markets are assumed to be perfectly competitive

HH satisfy their budget constraintDemand=Supply in all market


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


1. Labor market

- Labor supply is inelastic, as assumed under preferences- Labor demand

For any given capital stock firms hire labor to maximize profits

max ( , )L

F K AL wL

(bars to make clear what is taken as given by firms) which gives rise to labor demanded as a function of w

2 ( , )AF K AL MPL w

for any w, firms demand labor to equalize MPL and w.

(E15)

(E16)


Wage LABOR SUPPLY CURVE

LABOR DEMANDCURVE=MPL SCHEDULE

L(t)

W(t)

Employment

- FIGURE 4 Labor market equilibrium

Equilibrium employment L; Equilibrium real wage w=MPL


2. Rental market for capital

Firms own their capital, but that does not prevent them from renting itout 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

max ( , )K

F K AL RK

(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)

1 ( , )F K AL MPK R

for any R, firms demand labor to equalize MPK and R.

(E17)

(E17)


RentalPrice

CAPITAL GOODS SUPPLY CURVE

CAPITAL GOODS DEMANDCURVE=MPK SCHEDULE

K(t)

R(t)

CAPITAL GOODS

FIGURE 5 Rental capital market equilibrium


3. Summarizing the static equilibriumThe factor market (static) equilibrium conditions for given K(t), L(t), A(t) and hence are :( )k t

'( )t t tR MPK f k see E11

( ) '( )t t t t t tw MPL A f k k f k see E12

( ) '( )tt t t

t

wf k k f k

A

or


Capital per Efficiency Worker ( )k t

y f k

( )y t

( )( )w tA t

( )R tFIGURE 6 The production function in intensive form and wages/rental price of capital

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