Toulouse School of Economics, 2012-2013Macroeconomics 1 – Franck Portier

Problem set 8: Real Business Cycles - Solution

Problem I – A Simplified Real-Business-Cycle Model with Additive Technology Shocks

Consider an economy consisting of a constant population of infinitely-lived individuals. The representative in-dividual maximizes the expected value of

∑∞t=0 u(Ct)/(1 + ρ)t, ρ > 0. The instantaneous utility function,u(Ct), is

u(Ct) = Ct − θC2t , θ > 0. Assume that C is always in the range where u′(C) is positive.

Output is linear in capital, plus an additive disturbance: Yt = AKt + et. There is no depreciation; thus Kt+1 =Kt + Yt − Ct, and the interest rate is A. Assume A = ρ. Finally, the disturbance follows a first-order autoregressiveprocess: et = φet−1 + εt, where −1 < φ < 1 and where the εt’s are mean-zero, i.i.d. shocks.

1 – Find the first-order condition (Euler equation) relating Ct and expectations of Ct+1.The Lagrangian of the Social Planner problem is

L = E0

∞∑t=0

(1

1 + ρ

)t (Ct − θC2

t + λt ((A+ 1)Kt + et −Kt+1 − Ct))

FOC with respect to Ct and Kt+ are:

1− 2θCt = λt (1)

λt =1

1 + ρEt(1 + 1)λt+1 (2)

Substituting in (2) λt by its expression in (1), we obtain

1− 2θCt =1 +A

1 + ρEt (1− 2θCt+1)

Using A = ρ and simplifying, one gets the Euler equation

Ct = EtCt+1

2 – Guess that consumption takes the form Ct = α+ βKt + γet. Given this guess, what is Kt+1 as a function of Kt

and et?The resource constraint implies Kt+1 = (A + 1)Kt − Ct + et. Replacing Ct by its expression α + βKt + γet, one

getsKt+1 = (A+ 1− β)Kt − α+ (1− γ)et (?)

3 – What values must the parameters α, β, and γ have for the first-order condition in part (1) to be satisfied for allvalues of Kt and et?

Let’s compute EtCt+1 using equation (?):

EtCt+1 = α+ βEtKt+1 + γEtet+1

= α(1− β) + β(A+ 1− β)Kt + (β(1− γ) + γφ)et

The Euler equation implies Ct = EtCt+1. Knowing that Ct = α+ βKt + γet, one obtains the conitions

α = α(1− β)

β = β(A+ 1− β)

γ = β(1− γ) + γφ

from which we obtain

α = 0

β = A

γ =A

A+ 1− φ

1

The equilibrium then writes

Kt+1 = Kt +1− φ

A+ 1− φet

Ct = AKt +A

A+ 1− φet

4 – What are the effects of a one-time shock to ε on the paths of Y , K, and C?

Figure 1: Impulse response to a one time shock ε

t

1

et

0 t0

Kt

t

Ct

0

Problem III – An analytic model with log-linear depreciation

Consider a model economy populated with a representative household and a representative firm. The firm has aCobb-Douglas technology:

Yt = ZtKγt N

1−γt (3)

where Kt is capital, Nt labor input, and Zt a stochastic technological shock. All profits of the firm are distributed tothe household. Capital evolves according to the log linear relation

Kt+1 = AK1−δt Iδt 0 < δ ≤ 1 (4)

where δ is the rate of depreciation and where It is investment in period t.

The representative household works Nt, consumes Ct and invests It in period t. Preferences are given by:

U = E0

∞∑t=0

βt [logCt − V (Nt)] (5)

where V is a convex function. Capital is accumulated by the household and rented to the firm.Let κ denote the real rental rate of capital, P the price of the final good and W the nominal wage.

We first assume that δ = 1 (full depreciation).

1 – Write down the budget constraint of the household and the profit function of the firm

2 – Derive FOCs of the utility and profit maximization

3 – Define a competitive equilibrium of this economy

4 – Solve the model and show that Nt is constant along an equilibrium path.

5 – Derive a AR(1) process for log of output. Draw the Impulse Response Function of y to a unit shock to logZ = z,assuming that z is iid. What are the determinants of the size and persistence of this IRF?

?

2

We assume now that δ ∈]0, 1[.

6 – What is the economic meaning of equation (2)?

7 – We still denote κt the real return in t on investment It−1:

κt =∂Yt∂It−1

(6)

and note that the real price, in terms of current output, of capital to be transmitted to the next period is not one aswhen δ = 1. We denote it as qt (like Tobin’s q), and it is equal to:

qt =∂Kt+1/∂Kt

∂Kt+1/∂It(7)

8 – Comment the new budget constraint of the household

Ct + It =Wt

PtNt + κt(It−1 + qt−1Kt−1) (8)

9 – Solve for the competitive equilibrium of this economy and compute ItCt

.

10 – Compute the constant level of N .

11 – Derive the new process for log of output. Draw output typical IRF and discuss of its shape as a function ofparameters.

The first part of the exercise (full depreciation) has been solve in class. In each period the firm demands laborcompetitively so that the real wage is equal to the marginal productivity of labor:

Wt

Pt=∂Yt∂Nt

= (1− γ)YtNt

(9)

Also the real return on capital is simply the marginal productivity of capital :

κt =∂Yt∂Kt

= γYtKt

(10)

The household maximizes the expected value of his discounted utility subject to the sequence of budget constraints.Denote by λt the marginal utility of real wealth in period t (i.e. the Lagrange multiplier associated with the corre-sponding budget constraint. Then the usual optimality conditions for the consumer’s program yield:

1

Ct= λt (11)

V ′(Nt) = λtWt

Pt(12)

λt = βEt(λt+1κt+1), (13)

Combining (11), (13), the condition Yt = Ct + It and the definition of κt in (10), we obtain:

ItCt

= βγ + βγEt

(It+1

Ct+1

)(14)

which, by solving forward, yields :ItCt

=βγ

1− βγ(15)

so that:Ct = (1− βγ)Yt (16)

It = Kt+1 = βγYt (17)

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Now combining condition (12) with the expression of the real wage (9) and the value of consumption (16), we findthat Nt is constant and equal to N, where N is given by:

NV ′(N) =1− γ

1− βγ(18)

Then formula (18) yields a Walrasian quantity of labor equal to:

N =

[1− γ

ξ (1− βγ)

]1/ν(19)

To sum up, the dynamics is given byNt = N (20)

Yt = ZtKγt N

1−γ (21)

Wt

Pt= (1− γ)

YtN

(22)

Kt+1 = βγYt (23)

Consider now the case where capital evolves according to the log linear relation

Kt+1 = AK1−δt Iδt 0 < δ ≤ 1 (24)

where δ is the rate of depreciation.A first thing to note is that the price, in terms of current output, of capital to be transmitted to the next period isnot one. We denote it as qt (like Tobin’s q), and it is equal to:

qt =∂Kt+1/∂Kt

∂Kt+1/∂It=

1− δδ

ItKt

(25)

We still call κt the real return in t on investment It−1. We have:

κt =∂Yt∂It−1

=∂Yt∂Kt

∂Kt

∂It−1= γδ

YtIt−1

(26)

The household’s budget constraint is now replaced by

Ct + It =Wt

PtNt + κt(It−1 + qt−1Kt−1) (27)

The household maximizes his utility: ∑t

βt [logCt − V (Lt)] (28)

subject to his budget constraints and the capital accumulation equations. The Lagrangean for this program is:∑t

βt [logCt − V (Lt)] +∑t

βtλt

[Wt

PtNt + κt(It−1 + qt−1Kt−1)− Ct − It

](29)

+∑t

βtζt(AK1−δ

t−1 Iδt−1 −Kt

)The first order conditions for investment and capital yield:

λt = βEt (λt+1κt+1) + βδEt

(ζt+1Kt+1

It

)(30)

ζt = βEt (λt+1κt+1qt) + β (1− δ)Et(ζt+1Kt+1

Kt

)(31)

Comparing the two, and using the definition of qt, we find:

ζt =1− δδ

ItKt

λt (32)

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Inserting this into either (30) or (31) we obtain:

ItCt

= βγδEt

(Yt+1

Ct+1

)+ β (1− δ)Et

(It+1

Ct+1

)(33)

= βγδ + [βγδ + β (1− δ)]Et(It+1

Ct+1

)(34)

which solves as:ItCt

=βγδ

1− β (1− δ + γδ)(35)

so that:

Ct =1− β (1− δ + γδ)

1− β + βδYt (36)

It =βγδ

1− β + βδYt (37)

Finally, equations (9), (11), (12) and (36) yield a Walrasian quantity of labor N now given by :

N V ′(N) =(1− γ) (1− β + βδ)

1− β + βδ − βγδ(38)

Problem IV – Technological shocks, Preference shocks and the endogeneity of TFP

We consider here a variation of the simple analytical RBC model. We study a model economy A populated witha representative household and a representative firm. The firm has a Cobb-Douglas technology:

Yt = eztKγt N

1−γt (39)

where Kt is capital, Nt labor input, and ezt the stochastic Total Factor Productivity (TFP). All profits of the firmare distributed to the household. Capital evolves according to

Kt+1 = It (40)

where It is investment in period t.

The representative household works Nt and consumes Ct. Preferences are given by

U = E0

∞∑t=0

βt [logCt − eχtNt] (41)

where χt is a preference shock. Capital is accumulated by the household and rented to the firm.Let κt denote the real rental rate of capital, Pt the price of the final good and Wt the nominal wage. It is assumed

that γ and β are between 0 and 1.

1 – Write down the budget constraint of the household and the profit function of the firm

Budget constraint: PtCt + PtKt+1 = WtNt + κtKt (f)Profit: Πt = PtYt −WtNt − κtKt = eztKγ

t N1−γt −WtNt − κtKt

2 – Derive FOCs of the utility and profit maximization

Household Problem:

maxL = E0

∞∑t=0

βt [logCt − eχtNt − λt (PtCt + PtKt+1 −WtNt − κtKt)]

with K0 given.FOCs are:

w.r.t. Ct : 1/Ct = λt (a)w.r.t. Nt : eχt = λtWt (b)w.r.t. Kt+1 λtPt = βEtλt+1κt+1 (c)

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Firm problem:max Πt = eztKγ

t N1−γt −WtNt − κtKt

FOCs are:w.r.t. Nt : Wt/Pt = (1− γ)Yt/Nt (d)w.r.t. Kt : κt/Pt = γYt/Kt (e)

3 – Define a competitive equilibrium of this economy

A competitive equilibrium is a sequence of prices {Pt,Wt, κt} and quantities {Ct,Kt+1, Nt+1} such that (i) quantitiesmaximize profits and utility given prices (meaning that equations (a) to (f) hold) and (ii) markets clear, meaningthat Yt = Ct +Kt+1 (g) hold. Note that because of Walras law, one price can be chosen to 1 (for example Pt). Theequilibrium will only determine relative prices.

4 – Solve the model and show that the equilibrium process of output is yt = zt + γyt−1 − (1 − γ)χt (1) (droppingconstants)

Using (a) and (c), one gets 1/Ct = Etλt+1Pt+1κt+1/Pt+1. Using (e), one obtains 1/Ct = βγEt1/Ct+1Yt+1/Kt+2.Using (f), one gets 1/Ct = βγEt1/Ct+1(Ct+1 +Kt+2)/Kt+2 which implies Kt+1/Ct = βγEt(1 +Kt+2/Ct+1).

Solving forward this equation, we obtain Kt+1/Ct = βγ/(1 − βγ). Using again (g), we obtain Kt+1 = βγYt andCt = (1− βγ)Yt. Using (b), we obtain eχt = λtPt ×Wt/Pt = 1/Ct × (1− γ)Yt/Nt which gives Nt = 1−γ

1−βγ e−χt .

Plugging the preceding results into the production function gives

YteztKγ

t N1−γt = ezt (βγYt−1)

γ

(1− γ

1− βγe−χt

)1−γ

Taking logs and dropping constants gives

yt = zt + γyt−1 − (1− γ)χt (1)

5 – Assume y−1 = 0, zt = 0 ∀t, χt = 0 ∀t, except χ0 = 1%. Draw the time path on χt, zt and yt. Explain why yt ispersistent.

Figure 2: Impulse response to a one time shock χ, economy A

t

0 χt

0

1

t

0 zt

t

0 yt

01− γ

?

We now consider an economy B, in which the TFP is not exogenous at the aggregate level, but given by ezt = Yθ

t ext ,

where x is the exogenous part of TFP and Yθ

t acts as an externality. More precisely, Y t is taken as given by firms andhouseholds, but one has at the competitive equilibrium Y t = Yt. I is assumed that θ is between 0 and 1.

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6 – What is the economic interpretation of this externality?This externality implies that inputs are individually (at the level of one firm) more productive when aggregate pro-duction is large. On can think of network externalities or thick market externalities (it is easier to do business in aboom).

7 – Solve for the competitive equilibrium and give the equilibrium process of output (again in logs). Comment

As Y enters as an externality, all individual decisions are kept the same, except that ezt has now a different expression.We can therefore directly get at the competitive equilibrium

YteztKγ

t N1−γt = eztY θt (βγYt−1)

γ

(1− γ

1− βγe−χt

)1−γ

Solving for Yt, taking logs and dropping constants, we obtain:

yt =1

1− θxt +

γ

1− θyt−1 −

1− γ1− θ

(2)

and zt = θyt + xt

8 – Assume that an economist observes the economy B, with externality, but thinks that he is observing economy A,and is therefore using equation (1) to measure TFP. Draw the response of observed TFP zt, of yt and χt if y−1 = 0,xt = 0 ∀t, χt = 0 ∀t, except χ0 = 1%.

Figure 3: Impulse response to a one time shock χ, economy B wrongly measured as an economy A

t

0 χt

0

1

t

0 zt

0

θ(1−γ)1−θ

t

0 yt

0

(1−γ)1−θ

9 – How to interpret the positive correlation between observed TFP and output? Could such an economist believe(wrongly) that technological shocks are driving part of the response of the economy? Discuss.

The correlation between TFP and output could lead the economist to attribute that fluctuation to technological shocks.In reality, the output movements are due to the preference shocks. Because the economist mismeasures productivity(zt instead of xt), the measure of TFP is contaminated by preference shocks.

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