The Real Business Cycle Model - Ш . ЖИГЖИД ... · PDF fileMacroeconomics II 2 1 The real business cycle model 1.1 Introduction • This model explains the comovements in the

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

Department of Economics

Macroeconomics II 3

• 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.


Macroeconomics II 4

• 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.


Macroeconomics II 5

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.


Macroeconomics II 6

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.


Macroeconomics II 7

• 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


Macroeconomics II 8

• 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)


Macroeconomics II 9

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)


Macroeconomics II 10

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.



• 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)



• 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


(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)



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



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

•



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



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



and increase our labour supply at timet.



• 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)

•



2.5 Optimal household choices with uncertainty

• In this case the lagrangean is with the expectation term:

L = E0{∞∑t=0


(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.



• 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)



• 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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The Real Business Cycle Model - Ш . ЖИГЖИД ... · PDF fileMacroeconomics II 2 1 The real business cycle model 1.1 Introduction • This model explains the comovements in the