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Title: real business cycles
Description: ALL ECONOMICS ,MATHEMATICS AND BUSINESS NOTES

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LECTURE 5
REAL BUSINESS CYLCE THEORY

1
...
Real business cycles are recurrent fluctuations in an
economy’s incomes, products, and factor inputs – especially labour – that are due to
nonmonetary sources
...

Modern economies undergo significant short – run variations in aggregate output and
employment
...

Understanding the causes of aggregate fluctuations is a central goal of macroeconomics
...
Because
output movements are not regular, modern macroeconomics has generally turned
away from attempts to interpret fluctuations as combinations of deterministic cycles
of different lengths; efforts to discern regular Kitchin (3 – year), Juglar (10 – year),
Kuznets (20 – year), and Kondratiev (50 – year) cycles have been largely abandoned
as unproductive
...
Where the major
macroeconomic schools of thought differ is in their hypotheses concerning these
shocks and propagation mechanisms
...


iii)

Asymmetries in output movements
...
There does, however, appear to be asymmetry of a second type: output seems
to be characterized by relatively long periods when it is slightly above its usual path,
interrupted by brief periods when it is relatively far below
...
Employment falls and unemployment rises during recessions
...
The declines in employment and the declines in
hours in the economy as a whole are generally small relative to the falls in output
...
The conjunction of the declines in productivity and hours implies that
movements in the unemployment rate are generally smaller than the movements in
output
...


The Basic Real Business Cycle Framework

Real business cycle models view aggregate economic variables as the outcomes of the decisions
made by many individual agents acting to maximize their utility subject to production
possibilities and resource constraints
...
More explicitly, real business cycle models ask the question: how do rational
maximizing individuals respond over time to changes in the economic environment and what
implications do those responses have for the equilibrium outcomes of aggregate variables?
To address these questions, it is necessary to specify the economic environment and how it
evolves over time
...
It is
important in developing a model of this sort to recognize that business cycles are fundamentally
phenomena that are characterized by their behavior through time
...

The objective of any model of the business cycle is to generate a coherent understanding of how
and why these characteristics arise
...



 

3
...


ii)

A large number of identical, price – taking households
...


Since all agents are identical and price taking, we can aggregate and consider an economy with
one representative firm and one representative household
...
1 The Firm
and labour
In each period the firm produces output using capital
are multiplied by , the labour augmenting technology
...



...
Thus, output in
period is given by
, 0
Capital depreciates at the rate

1
...
Thus the capital stock in period
,

1 is given by
(2)


...
Therefore, without the last term we would have an
economy growing smoothly along the trend
...
It is assumed to follow a first – order
autoregressive process
...


If
0 then

...
If, however,
this means that the effects of a shock to technology disappear gradually over time
...
This is the shock that determines the business cycle fluctuations
...
Labour
is paid the wage , while the opportunity cost of capital is
, where is the real interest
rate
...


,

We use equation (1) to substitute

(7)

in equation (7)
...


,

(8)

The First Order Conditions (FOC) are given by:
0
...


(10)

Equation (9) implies that:

...


(12)

Equation (11) can be rearranged as follows:

...


(14)


 

3
...
It is endowed with a certain amount of time each
period (normalized to one unit), which can be used either to work or as leisure time
...


The lower is , the less future consumption and leisure are valued relative to present one
...


,

(17)

where
,

,

;
,

,

,

,

;

,


...


(19)

The household maximizes the intertemporal utility function subject to the budget constraint
...

We can therefore use the lagrangean solution method:


,1

1


 


...
3 Optimal Household Choices With Certainty
In this case, the lagrangean is without the expectation term:


1

,1

The First Order Condition (FOC) are given by the first derivatives of
, ,
equal to zero
...


(24)
(25)

3
...
1 Intertemporal Substitution In Consumption
Consider equations (23) and (25) which can be rewritten as follows:
,

1


...


(27)

First we substitute (26) into (27):
,


...


(30)

Equation (30) is the Euler equation for consumption, which has also the following interpretation:
,
,

1


...



 

1

market price of present consumption with respect to future consumption
...
3
...


1

(32)


...


(34)

Forwarding (34) by one period gives:
,


...


(36)

Equation (36) has also the following interpretation:
,



,


...


,



opportunity cost of present leisure with respect to the one of future leisure


 

3
...
3 Intratemporal Substitution Between Consumption and Leisure
We consider now equations (23) and (24), the First Order Conditions with respect to
consumption and labour:
1

,


...


(39)

Here,
is the increase in the value function (intertemporal utility) if we increase the net assets
by one unit
...


i) Working and gaining
ii) Saving

1

and increasing our net assets by


...


,

Thus,

(40)

is the relative price of leisure with respect to consumption
...
4 An Example
Consider the logarithmic utility function:
ln

,1

ln 1

,

0

(41)

For this utility function:
;

,


...


(44)


 

Similarly,
,

(45)



,

becomes:

...


increases with respect to

decreases and hence

(46)
, we have that

decreases
...


We therefore decrease our leisure and increase our labour supply at time
...

This is clear from the intratemporal optimal condition:
,

(47)

,

which becomes:

...
5 Optimal Household Choices With Uncertainty
In this case the lagrangean is with the expectation term:


1

,1


...


(53)

1


...


(55)

We can derive the intertemporal optimal conditions in the same way as before
...


,

(56)

Using again the logarithmic utility function we have:
1


...


(58)

Using equation (58) in equation (57):
,


...


(60)

The ratio of current to expected consumption is equal to the relative prices
...
This means that marginal utility of consumption

tends to be lower when the interest rate is higher
...
59) it follows that if

1

,

11 
 

decreases then

increases
Title: real business cycles
Description: ALL ECONOMICS ,MATHEMATICS AND BUSINESS NOTES