Real business cycles
Prof. A. Pommeret
I. Jaccard
HEC Lausanne, 2003/2004
Problem set 6: Sketch of Answers
(June, 29th 2004)
Part I: Open Questions (Exams 2002 and 2003)
Question 1 (10 points)
Explain the role of the intertemporal substitution in the RBC literature.
Answer:
According to the RBC approach, intertemporal substitution lies at the heart of labor market
fluctuations and cyclical fluctuations in employment are one of the leading forces behind the
business cycle. The main idea is that workers, as rational maximizing agents, compare actual
and expected future real wages and adjust their labor supply accordingly. Morevoer, if due to
an increase in the real interest rate, workers expect the future real wage to decrease relatively
to the present real wage and they increase their labor supply and vice and versa.
Question 2 (10 points)
Explain how the RBC literature has developed in order to take unemployment into account.
Answer:
It has long been the opinion of many that the variations in employment have nothing to do
with workers' preferences but rather with the fact that labor supply is rationed at the labor
market equilibrium with the consequences that any increase in demand will be essentially
reflected in the exchanged quantities. This has caused some authors to abandon the
assumption that labor fluctuations are caused exclusively by the intertemporal substitution
mechanism and to incorporate non-walrasian considerations on labor market (efficiency wage
and matching models for example).
Question 3 (10 points)
Justify, explain and appraise the results of the two steps the RBC literature has followed in
order to take money into account
Answer:
The first step is the introduction of cash-in-advance constraints. This constraint means that the
payment of consumption goods must be done with money transferred from the last period.
Therefore the household must hold money at each period if it wants to consume.
Secondly, nominal rigidities were introduced in the hope to be able to account for the
propagation of monetary shocks in the economy. As a result, wage contracts were introduced
in RBC models.
Question 4 (20 points)
According to the benchmark RBC model, what are the qualitative responses of the major
macroeconomic aggregates after a technological shock? How are they affected by the
introduction of money through a cash-in-advance constraint?
Answer:
The effects of a transitory technology shock are the following:
Output instantaneously increases. The underlying mechanism has 2 levels:
1/ An increase in total factor productivity generates a direct increase in output
2/ By generating an increase in the productivity of labor and capital, a positive technology
shock also has an indirect effect on output. The high productivity of capital induces interest
rates to rise on impact and therefore agents have a strong incentive to save which implies an
increase in the capital stock and therefore allow the economy to produce a larger amount of
output. Moreover, the substitution effect implies that agents will decide to work harder in
response to such a shock since they understand that wages are temporarily higher and
therefore more labor input will also contribute to increase output.
Consumption
The response of consumption is explained through mechanisms generated by both substitution
and wealth effects.
-The positive technology shock leads to an increase in income through both a higher product
and a higher wage. This positive wealth effect generates an increase in consumption
-On impact, this positive effect is however weakened by the rise in interest rate which means
a decrease in the discounted price of future consumption.
Investment
A positive technology shock tends to increase the marginal productivity of capital, thus
encouraging investment. There is thus an instantaneous increase in investment compared to
the steady state, which implies more capital accumulation.
Labor
On impact, the number of hours worked are positively affected by the shock. Several
mechanisms explain this:
- The shock implies a positive wealth effect through the increase in output and in wage.
Thus the agent can sustain its wealth level while working less.
- But it is mainly substitution effect that determines the behavior of labor in this model:
the instantaneous increase in the wage rate implies an intra-temporal substitution
between leisure and consumption (since the price of leisure increases, agents rather
consume) ; this tends to introduce a substitution between labor and leisure. This effect
is largely compensated by the intertemporal substitution mechanism due to the rise in
the interest rate. Indeed, an increase in the interest rate generates a decrease in the flow
of discounted future wages. Household understanding that the rise in wage is only
temporary decide to work harder during the first periods following the shock. This
intertemporal substitution effect prevails, at a macroeconomic level, on the first effect
due to intra-temporal substitution. The final effect on labor is therefore positive.
How are they affected by the introduction of money through a cash-in-advance constraint?
The only difference is that we now have inflation in the model which is negatively affected by
technology shocks.
Part II: Problem
Note that each question can be answered independently.
1/ Objective of this problem
Explain what are the problems of the benchmark model as far as these correlations are
concerned.
Answer:
The benchmark model does not allow to distinguish labor productivity from real wages and
fails to reproduce the close to zero correlation between hours worked and labor productivity.
The correlation generated by the benchmark model is strongly positive.
2/ Presentation of the problem
a/ The households
The representative household has the following utility function:
1
1
u(Ct , N t ) 
Ct1  d (1  nt )1 1
1
with   0,1 and   0,1  1, and d  0 . The representative household has a time
endowment normalized at 1, shared between labor n and leisure ; C represents its
consumption.


The money supply follows the following stochastic process: M t 1  gt M t
g t is a stochastic stationary AR(1) process: log g t   m log( g t 1 )  (1   m ) log( g )   mt
where  m  1 for stationary purposes and log( g ) gives the unconditional mean of the
process.
 mt
,
the
innovation,
is
a
Gaussian
white
noise:
E ( mt )  0, E ( mt ,  mt )   , E ( mt ,  m )  0, t   .
2
m
Households are subject to the following constraint : Ct 
1
M t  ( g t  1) M t 1  .
pt
Explain the meaning of this constraint.
Answer:
Households being supposed to hold in advance the money necessary to purchase a
consumption good, they are subject to a cash-in-advance constraint.
b/The firms
A representative firm produces output using capital K and labor L. Investment I and hiring H
are subject to interrelated adjustment costs. The production technology, F, net of the
adjustment costs is described by the function:

I H 
a I t2 c H t2
F ( K t , N t , I t H t ; z t )  1  s t t  z t K t1 N t 

 It
Kt Nt 
2 Kt 2 Nt

where a and c are positive ; s is a parameter that may be positive or negative
z t is an exogenous productivity shock which is assumed to follow a stationary AR(1) process :
with   1 and  t  N (0,   ) .
log( zt )   log( zt 1 )  (1   ) log( z )   t
Capital and employment are accumulated according to:
K t 1  (1   ) K t  I t
N t 1  (1  ) N t  H t
where  is the depreciation rate and  represents the quit rate of labor force.
Explain the second equation.
Answer:
Labor can only be adjusted according to the above law of motion. The fact that labor is now a
state variable (like capital) will imply that it will not be possible for the firms to adjust their
level of labor instantaneously and that the response of employment to aggregate shocks will
be more gradual.
 
We also assume: log wtC  Et  j log wt .
Explain the meaning of this constraint.
Answer:
The wage determination is no more walrasian and nothing ensured that the labor demand
matches the supply. Wt c is the nominal contractual wage and Wt the wage that would allow to
balance the market in the absence of contracts
c/ The government
The government expenditures are denoted by Gt which follows a stationary process defined
by : log Gt   g log( Gt 1 )  (1   g ) log( G )   gt
where log( G ) is the unconditional
mean of the process,  g  1 and  gt  N (0,  g ) . At each period, government expenditures
are financed by a lump-sum tax exactly equal to their level.
d/ Solving the problem
Why do we have to solve separately the program of the household and that of the firm?
Answer:
The introduction of market imperfection caused by the introduction of non-walrasian labor
market feature implies that the equivalence between the planner's problem and the
competitive equilibrium does not hold any more.
d.1: the program of the household:

max U  E0   t u Ct ,1  nt 
Ct ,nt
t 0



M t 1
w
1
 Et  t 1  t 1    t  t nt  M t  ( g t  1) M t 1   Gt
(1) : Ct 
pt
pt
pt
 t



1
s.t.(2) : Ct  M t  ( g t  1) M t 1 
pt

(3) : nt  N t


Where nt is the labor quantity the household wishes to supply and N t is that demanded by


the firm. Et  t 1  t 1  gives the value at time t of the financial portfolio which provides the
 t

agent with  t 1 units of goods at period t+1.
Explain this program.
Answer:
The innovative feature of this problem is that the representative household program
corresponds to the walrasian one with the new constraint on employment:
nt  N t
which reflects the fact that the household provides exactly the labor quantity requested by the
firm. nt denotes the labor quantity that the household wishes to supply, which usually differs
from the one demanded by the firm. Thus, nt represents the the employment level desired by
the household.
Note  t the multiplier relative to the constraint (2),  t the multiplier relative to the constraint
(3) and Vt M the value of the program of the household.
Give the optimality conditions.
Answer:
Vt
M




M
U (ct ,1  nt )  EtV t 1 

 



w
M
1

 max ct ,M t ,nt t  t  t nt  M t  ( g t  1) M t 1   G  Ct  t 1  Et  t 1  t 1   
pt
pt
pt
 t

 


   1 M  ( g  1) M  C    N  n 

t
t
t 1
t 
t
t
t
 t  pt


ct :
U c  t   t
nt :
U l  t
wt
t  0
pt
 t 1 :
 t 1
 Et t 1
t
M t 1 :
t E t
t


1
1 
1
1 
    Et  t 1

   Et t 1
 Et t 2 ( g t 2  1)
 Et  t 2 ( g t 2  1)
pt
pt 1
pt 2 
pt 1
pt 2 


d.2: the program of the firm



w
max E0   t  F K t , N t , I t , H t , z t   t N t 
It ,H t
pt 
t 0

 K t 1  (1   ) K t  I t

s.t. N t 1  (1  ) N t  H t

C
log wt   Et  j log wt 
Explain this program.
Answer:
The intertemporal maximization problem of the firm is the same as in the competitive model,
except that a constraint corresponding to the wage contract is added and that labor is now
predetermined.
We note Vt F the value of the firm.
Give the optimality conditions.
Answer:



wt
 t 1
F
F
(
K
,
N
,
I
H
;
z
)

N

E
V
(
K
,
N
,
z
)

F
t
t
t
t
t
t
t
t 1
t 1
t 1 
Vt ( K t , N t , zt )  max It , H t , Nt 1 , Kt 1 
pt
 t t 1


q (1   ) K  I  K   x (1   ) N  H  N 

t
t
t 1
t
t
t
t 1
 t

It :
FI  qt  0
Ht :
FH  xt  0
K t 1:

qt  Et t 1 VK (t  1) 
t
N t 1 :

xt  Et t 1 VN (t  1) 
t
Enveloppe:
VK  FK (t )  qt (1   )
w
VN  FN (t )  t  xt (1   )
pt
d.3. the equilibrium
It is characterized by the equilibrium on the money market and:
F K t , N t , I t , H t ; zt   Ct  Gt
 t  V F (t )
Explain.
Answer:
These are the market clearing conditions imposed on the good market and on the financial
market. In contrast to the three other markets, the labor market does not clear.
3/ Results
a/ How do you expect the correlation between productivity and employment and the
correlation between employment and wage to be affected by
- the introduction of adjustment costs
Answer:
After a technology shock, the response of employment will be gradual because of the the
constraint on labor accumulation, N t 1  (1  ) N t  H t , implying that labor cannot be adjusted
instantaneously, and humped shaped, thanks to the introduction of labor adjustment cost. In
contrast, labor productivity, (Y/N), will instantaneously jump in response to a technological
shock and then monotically decrease to eventually converge back to the steady state. As a
result, the fact that employment is now predetermined should help us to reduce the correlation
between these two variables.
- the constraint log wtC  Et  j log wt 
Answer:
The introduction of wage contracts will firstly allow us to distinguish real wages from labor
productivity and secondly, combined with monetary shocks, will also help us to reduce the
correlation between real wages and employment. Since a monetary shock implies an increase
of prices and that the nominal wage do not adjust immediately, such a shock will cause real
wages to decline. As a result, firms will find profitable to increase their demand of labor in
order to benefit from this reduction in the cost of labor. Employment being determined by the
demand of firms, this will cause an increase in the number of hours worked and therefore
monetary shocks will generate a negative correlation between employment and real wages.
Combined with technology shocks, that imply a strong positive correlation between real
wages and employment, monetary shocks will therefore help to improve the performances of
the model by lowering this correlation.
 