The Real Business Cycle Model
Macroeconomics II 2
1 The real business cycle model1.1 Introduction
• This model explains the comovements in thefluctuations ofaggregate economic variables around their trend.
• It is a competitive model with perfect markets:
– No externalities
– Symmetric information
– Complete markets
– No other imperfections
• The real business cycle model ”builds up” on the Solow growthmodel, which generates an economy which converges to a”balanced growth path” and then grows smoothly.
• We modify this model in order to generate:
– Fluctuations of aggregate output around trend
– Comovements of output and other aggregate economicvariables around their respective trends
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• The two ingredients used are:
I Shocks to the economy’s technology (changes in theproduction function from period to period. Anotherpossible source of shocks is the unexpected changes ingovernment purchases. )
II An optimising household that decide how much to consumeand to work. The cost of work is the loss in leisure time.
• Therefore we follow the Brock and Mirman ’72 idea thatGrowth and Fluctuations are not distinct phenomena, to bestudied with separate data and different analytical tools.
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• Note 1: since markets are perfect, there are no market failures,andfluctuations are the optimal responses of agents to theexogenous shocks. Therefore:
– There is no deterministic cycle (in the Mitchell sense).
– There is no scope for government intervention.
• Note 2: here we consider a walrasian model of the aggregateeconomy wherefluctuations are generated by real shocks.
– The current debate in the economic theory is about the factthat walrasian models with real shocks are insufficient inexplaining aggregate economicfluctuations.
– Later we will consider non-walrasiam models of aggregateeconomic activity wherefluctuations are generated bynominal shocks.
– Other strands of macroeconomics consider models with realshocks and with non walrasian imperfections.
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2 The baseline real business cyclemodel
• The economy is populated by:
I A large number of identical, price-taking firms
II A large number of identical, price-taking households
III A government which each period purchases an amount ofgoodsGt and finances itself using lump sum taxes
• Since all agents are identical and price taking, we can aggregateand consider an economy with one representative firm and onerepresentative household.
• The ricardian equivalence holds
• Note: The government is only a source of real shocks in thismodel.
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2.1 The firm
• In each period the firm produces outputYt using capitalKt
and labourLt.
• The units of labourLt are multiplied byAt, the labouraugmenting technology.
• ThereforeAtLt is the effective labour input.
• The production function is aCRTS Cobb Douglas function:
Yt = Kαt (AtLt)
1−α (1)
0 < α < 1
• Capital depreciates at the rateδ :
Kt+1 = δKt + It+1 (2)
• WhereIt+1 is investment.
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• The technologyAt is determined by the following equation:
lnAt = A + gt + At (3)
• A andg are positive constants. Therefore without the last termwe would have an economy growing smoothly along the trend.
• The last term is the random disturbance:
At = ρAt−1 + εt (4)
−1 < ρ < 1
• εt is a white noise:
E (εt) = 0 (5)
cov (εt, εs) = 0 for anyt �= s (6)
• The binomial process we considered in the example last weekis an example of a stochastic process that satisfies (5) and (6).
• If ρ = 0 thenAt = εt. The technological shock is a whitenoise.
• If ρ > 0, it means that the shock in technology disappearsgradually over time.At is persistent.
• In the last week example we saw that ifρ is close to 1,At is sopersistent that it seem to have a cyclical pattern
• This is the shock that determines the business cyclefluctuations
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• The firm observesAt and choosesKt andLt in order tomaximise the profits at timet.
• LabourLt is paid with the wagewt, while the opportunity costof capital is(rt + δ) , wherert is the real interest rate.
MAXKt,Lt
Πt =MAXKt,Lt
Yt − wtLt − (rt + δ)Kt (7)
• We use (1) to substituteYt in (7). The First Order Conditions(FOC):
•∂Πt
∂Kt
= αKα−1
t (AtLt)1−α − (rt + δ) = 0 (8)
=⇒ rt = αKα
t (AtLt)1−α
Kt
− δ (9)
∂Πt
∂Lt
= (1− α)AtKαt (AtLt)
−α − wt = 0 (10)
=⇒ wt = (1− α)Kα
t (AtLt)1−α
Lt
(11)
• The firm solves (8) and (10) with respect toKt andLt.
• Instead we substituteYt back in (8) and (10) and derive theequilibrium interest rates and wages:
rt = αYt
Kt
− δ (12)
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wt = (1− α)Yt
Lt
(13)
• Also useful is to rearrange the two equations as follows:
(rt + δ)Kt
Yt
= α (14)
wtLt
Yt
= 1− α (15)
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2.2 The household
• The representative household is infinitely lived.
• It is endowed with a certain amount of time each period(normalised to one unit), which can be used either to work oras leisure time.
• Therefore with respect to the optimal consumption problemanalysed last week, here labour supply is endogenous.
• The household maximises the expected value of the intertem-poral utility function:
U0 = maxct,At,lt
E0
[∞∑t=0
βtu (Ct, 1− Lt)
](16)
0 < β ≤ 1 (17)
• Ct is the level of consumption.
• Lt is the amount of time worked.
• 1− Lt is the amount of leisure time.
• β is the intertemporal discount factor.
• The lower isβ, the less future consumption and leisure arevalued with respect to present ones.
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• The utility function is assumed to be strictly concave in botharguments:
u1 > 0; u11 < 0; u2 > 0; u22 < 0 (18)
• We use the following notation:
u1,t =∂u (Ct, 1− Lt)
∂Ct
; u2,t =∂u (Ct, 1− Lt)
∂Lt
(19)
u11,t =∂2u (Ct, 1− Lt)
(∂Ct)2
; u22,t =∂2u (Ct, 1− Lt)
(∂Lt)2
(20)
• The household maximises the intertemporal utility functionsubject to the budget constraint.
• We introduce, like last week, the notion of the stock of netassetsAt :
At+1 = (1 + rt+1) (At + wtLt − Ct) (21)
• wtLt is the labour income of the household.
• Note 1: now we consumeCt at the beginning of periodt.
• Note that you can consume more than your salary: ifCt > wtLt
then you reduce your net wealth.
• This means thatAt+1 can become negative, but you cannotborrow infinitely:
limt→∞
At∏tj=0
(1 + rj)≥ 0 (22)
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• We can once again use the lagrangean solution method:
L = E0{∞∑t=0
βt {u (Ct, 1− Lt) + λt+1 [(1 + rt+1) (At + wtLt − Ct)−At+
(23)
2.3 Optimal household choices with certainty
• In this case the lagrangean is without the expectation term:
L ={∞∑t=0
βt {u (Ct, 1− Lt) + λt+1 [(1 + rt+1) (At + wtLt − Ct)−At+
(24)
• The first order conditions are given by the first derivatives ofLwith respect toCt, Lt andAt equal to zero.
∂L
∂Ct
= βt [u1,t − (1 + rt+1)λt+1] = 0 (25)
∂L
∂Lt
= βt [−u2,t + wt (1 + rt+1)λt+1] = 0 (26)
∂L
∂At
= βtλt+1 (1 + rt+1)− βt−1λt = 0 (27)
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2.3.1 Intertemporal substitution in consumption
• Like we did last week, we consider (25) and (27):
u1,t = (1 + rt+1)λt+1 (28)
λt = β (1 + rt+1)λt+1 (29)
• First we substitute (28) in (29):
λt = βu1,t (30)
• Then we forward by one period:
λt+1 = βu1,t+1 (31)
• Finally we substitute (30) and (31) back in (29):
u1,t = β (1 + rt+1) u1,t+1 (32)
• (32) is the euler equation for consumption, which has also thefollowing interpretation:
u1,tβu1,t+1
= 1 + rt+1 (33)
u1,tβu1,t+1
=SUBJECTIVE VALUE of present consumptionwith respect to futute consumption
1 + rt+1 =MARKET PRICE of present consumptionwith respect to futute consumption
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2.3.2 Intertemporal substitution in labour supply
• We consider now equations (26) and (27):
u2,t = wt (1 + rt+1)λt+1 (34)
λt = β (1 + rt+1)λt+1 (35)
First we substitute (34) in (35):
λt = βu2,twt
(36)
• Then we forward by one period:
λt = βu2,t+1
wt+1
(37)
• Finally we substitute (36) and (37) back in (35):u2,twt
= β (1 + rt+1)u2,t+1
wt+1
(38)
• (38) has also the following interpretation:u2,t
βu2,t+1
=wt
wt+1/ (1 + rt+1)(39)
u2,tβu2,t+1
=SUBJECTIVE VALUE of present leisurewith respect to future leisure
wt
wt+1/ (1 + rt+1)=
OPPORTUNITY COST of present leisurewith respect to the one of futute leisure
•
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2.3.3 Intratemporal substitution between consumption andleisure
• We consider now (25) and (26), the FOCs with respect toconsumption and labour:
u1,t = (1 + rt+1)λt+1 (40)
u2,t = wt (1 + rt+1)λt+1 (41)
• Interpretation:λt+1 is the increase in the value function(intertemporal utility) if we increase the net assets by one unit.
• (41) means that the loss in utility in decreasing leisure by oneunit is equal to the gain we have by:
– working and gainingwt
– savingwt and increasing our net assets bywt (1 + rt+1)
• Therefore the trade off between consumption and leisure is thefollowing:
u2,tu1,t
= wt (42)
• wt is the relative price of leisure with respect to consumption
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2.4 An example
• We consider the logarithmic utility function:
u (Ct, 1− Lt) = lnCt + b ln (1− Lt) (43)
b > 0
• Therefore:
u1,t =1
Ct
; u2,t =b
1− Lt
(44)
• The euler equation for consumption:u1,t
βu1,t+1
= 1 + rt+1 (45)
• Becomes:Ct
Ct+1
=1
β (1 + rt+1)(46)
• And similarly:u2,t
βu2,t+1
=wt
wt+1/ (1 + rt+1)(47)
• Becomes:1− Lt
1− Lt+1
=1
β (1 + rt+1)
wt+1
wt
(48)
• Therefore ifwt increases with respect towt+1, we have thatwt+1
wt
↓
• Therefore 1−Lt
1−Lt+1↓ and henceLt
Lt+1↑ . We decrease our leisure
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and increase our labour supply at timet.
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• This is because the substitution effect (higher opportunity costof leisure) more than compensates the income effect (higherwage means that we are richer and want to consume more)
• This is clear from the intratemporal optimal condition:u2,tu1,t
= wt (49)
• Which becomes:Ct
1− Lt
=wt
b(50)
•
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2.5 Optimal household choices with uncertainty
• In this case the lagrangean is with the expectation term:
L = E0{∞∑t=0
βt {u (Ct, 1− Lt) + λt+1 [(1 + rt+1) (At + wtLt − Ct)−At+
(51)
• The first order conditions are similar to before:∂L
∂Ct
= βtEt [u1,t − (1 + rt+1)λt+1] = 0 (52)
∂L
∂Lt
= βtEt [−u2,t + wt (1 + rt+1)λt+1] = 0 (53)
∂L
∂At
= βtEt
[λt+1 (1 + rt+1)− β−1
t λt
]= 0 (54)
• Which can be written as:
u1,t = Et [(1 + rt+1)λt+1] (55)
λt = βEt [(1 + rt+1)λt+1] (56)
u2,t = wtEt [(1 + rt+1)λt+1] (57)
• Now we can derive the intertemporal optimal conditions in thesame way as before.
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• Consider for example the euler equation for consumption:
u1,t = βEt [(1 + rt+1)u1,t+1] (58)
• Using again the logarithmic utility function we have:
1
Ct
= βEt
[(1 + rt+1)
1
Ct+1
](59)
• From this point onwards things are different with respect to thecertainty case, because we have that:
Et
[(1 + rt+1)
1
Ct+1
]=
Et (1 + rt+1)Et
(1
Ct+1
)+ cov
(1 + rt+1,
1
Ct+1
) (60)
• Using (60) in (59):
Et
(Ct
Ct+1
)=
1− Ctβcov(1 + rt+1,
1
Ct+1
)βEt (1 + rt+1)
(61)
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• If cov(1 + rt+1,
1
Ct+1
)= 0 then we have the same result than
in the certainty case:
Et
(Ct
Ct+1
)=
1
βEt (1 + rt+1)(62)
• The ratio of current to expected consumption is equal to therelative prices.
• Now suppose thatcov(1 + rt+1,
1
Ct+1
)< 0
• This means that marginal utility of consumption(
1
Ct+1
)tends
to be lower when the interest rate is higher.
• In this case the household is less incentive to save for futureconsumption:
• In fact from (61) it follows that ifcov(1 + rt+1,
1
Ct+1
)decreases thenEt
(Ct
Ct+1
)increases!
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