Review Notes on Business Cycles - Boston College · Review Notes on Business Cycles David Schenck...

Review Notes on Business Cycles

David Schenck

May 7, 2012; version 1.1.32

Abstract

Still preliminary. Use with caution. Comments and corrections welcome. For each of sevenmodels, I provide (1) statements of the actors’ problems and first order conditions, (2) solutionsystems in both levels and log-deviations, and (3) IRFs for supply and demand shocks.

Good luck!

Brief Contents

0 The Goal 4

1 RBC model, No Capital 5

2 RBC with Capital 7

3 RBC with Imperfect Competition (Static Markup) 10

4 RBC with Imperfect Competition (Cyclical Markup) 14

5 Monetary Model, No Capital, Flexible Prices 16

6 Monetary Model, No Capital, Sticky Prices 19

7 Monetary Model, Capital, Sticky Prices 25

A RBC2: A Derivation of the RatEx Solution 28

B RBC2: A Sample Matlab file 30

1

Contents

0 The Goal 4

1 RBC model, No Capital 51.1 Consumer UMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Firm PMP and Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The Solution System, RBC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 OPT/ADD Analysis with RBC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 RBC with Capital 72.1 Consumer UMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Firm PMP and Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 The Solution System, RBC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Impulse Responses, RBC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 RBC with Imperfect Competition (Static Markup) 103.1 Consumer UMP for the composite good . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Consumer CMP for individual goods . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Firm PMP and Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 The Solution System, RBC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Impulse Responses, RBC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 RBC with Imperfect Competition (Cyclical Markup) 144.1 The Solution System, RBC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Impulse Responses, RBC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Monetary Model, No Capital, Flexible Prices 165.1 Consumer UMP for the composite good . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Consumer CMP for individual goods . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Firm PMP and Central Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 The Solution System, MM1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Monetary Model, No Capital, Sticky Prices 196.1 Consumer UMP for the composite good . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Consumer CMP for individual goods . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Firm PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 Rotemberg Pricing and the NK Phillips Curve . . . . . . . . . . . . . . . . . . . . . 206.5 NK Analysis with MM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 Wicksellian Analysis with MM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.7 The Solution System, MM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.8 Impulse Responses, MM2, Monetary Rule . . . . . . . . . . . . . . . . . . . . . . . . 236.9 Impulse Responses, MM2, Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2

7 Monetary Model, Capital, Sticky Prices 257.1 Consumer UMP for the composite good . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Consumer CMP for the individual goods . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Firm PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 Rotemberg Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 Central Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 Equilibrium and Solution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A RBC2: A Derivation of the RatEx Solution 28

B RBC2: A Sample Matlab file 30

3

0 The Goal

Macroeconomists care about the determinants of Yt. We care about other things, like interest rates,inflation, wages, etc., only to the extent that they affect Yt. Growth economists care about thedeterminants of the trend in Yt; business cycle macroeconomists care about the determinants offluctuations in Yt. This is a course in business cycles. We are going to write down a series of generalequilibrium models of the economy. Throughout, we are interested in identifying shock processeswhose effects qualitatively match business cycle facts. The key stylized facts of business cycles arethat

1. Consumption, output, hours worked, and the real wage all co-move positively

2. Consumption is less volatile than output, and output less volatile than investment

The first fact deserves two notes. First, the positive co-movement of wages and hours has strongimplications for the labor market. If labor markets always clear, and we are always on the laborsupply and labor demand curves, positive net comovement between hours and wages implies thatlabor demand is more volatile than labor supply.1 We hence will focus an enormous amount ofeffort on the labor market and the labor demand curve. We begin with a model which predicts thattechnology shocks are the only process which can affect hours and wages in the desired manner.We will then investigate models in which other sorts of shocks can have the desired properties.

Second, getting hours and consumption to move together is challenging when the consumer isoptimizing. This is because, generally speaking, leisure and consumption are both normal goods,so wealth effects should cause both to move in the same direction (pushing hours and consumptionin opposite directions). This problem is so pervasive that we can classify business cycle models interms of how they get consumption and hours to comove.

Throughout we focus on two shocks: “technology” shocks as a catch-all for “supply” shocks,and “government spending” shocks as a catch-all for “demand” shocks. We are not necessarilysaying that government spending is a significant driver of the business cycle or that it is the onlyalternative to technology shocks; rather we use government shocks as a vehicle to explore non-technology shocks generally. When we get to the monetary section, the government spending shockwill be replaced with a monetary shock, but substantively both G and M shocks are “aggregatedemand” shocks.

In light of the these considerations, the notes follow a symmetric pattern for all models. First, Ipresent the actors’ problems and their first-order conditions. Second, I present the solution systemfor the model in levels and, where appropriate, in log-deviations from the steady state. Third, Iinclude impulse response functions for demand and supply shocks, to see if they match the twostylized facts above.

Nowhere in the course do we talk directly about recessions and what (if anything) the govern-ment should do about them. Indeed there is no unemployment in many of these models and henceno underutilization of resources: properly speaking, recessions don’t exist in the RBC1-RBC4 mod-els. Output might be below trend, but that is because of a bad draw of technology. The consumeris still optimizing and there is no room for the government to make the consumer better off. Inthe monetary models we can have periods where output is below its flexible-price potential, whichlooks sort of like a recession. These are times where the consumer’s utility is lower than it couldhave been, if only prices would adjust. Note, however, that in all of these models the labor marketclears continuously: there is no unemployment in any model we will study herein.

1Of course, if the labor market doesn’t always clear, then we need not be on the labor supply curve, and theseconsiderations don’t matter as much.

4

1 RBC model, No Capital

Call this RBC1.

1.1 Consumer UMP

The household is taken forward from our study of consumption, with one change: the addition ofleisure in the utility function. The houshold maximizes:

max{Ct,Ht,Bt+1}

Et

{ ∞∑s=0

[1

1 + ρ

]s [ln(Ct+s) + V (H −Ht+s)

]}s.t. Bt+1+s = (1 + rt)Bt+s +Wt+sHt+s + Πt+s − Ct+s − Tt+s; λt

V (·) is continuously twice differentiable, increasing, and strictly concave. λt is the induced Lagrangemultiplier. For simplicity, I take s = 0, but note that there is a whole sequence of FOCs.

1

Ct= λt (1.1)

V ′(H −Ht) = λtWt (1.2)

Et

[1 + rt1 + ρ

λt+1

λt

]= 1 (1.3)

The three equations are output demand, labor supply and the Euler equation. The labor supplycurve can be shown to be increasing in (W,H) space, as desired.

1.2 Firm PMP and Government

The firm produces output Yt from labor input Ht and a fixed capital stock K. Technology issummarized by Z and is assumed to be stationary.

The firm’s problem is:

maxY,H

Yt −WtHt

s.t. Yt = Ztf(Ht); ϕt

and FOC are:

Yt = Ztf(Ht) (1.4)

Ztf′(Ht) = Wt (1.5)

The first equation is the FOC with respect to the induced Lagrange multiplier; the second is theFOC with respect to Ht. Note in particular that the FOC are: (1) the economy is on its productionfunction, and (2) the economy is on its labor demand curve.

The government is constrained to

Gt = Tt (1.6)

(weaker but identical assumption: assume full Ricardian equivalence). Finally we have the aggre-gate demand condition

Yt = Ct +Gt (1.7)

that is, consumption plus government expenditure cannot exceed output.

5

1.3 The Solution System, RBC1

We care about C, H, λ, Y , W , T , r, G and Z. We thus have nine variables; we also have sevenequations plus two unspecified “shock processes” for G and Z. Compactly:

1

Ct= λt

V ′(H −Ht) = λtWt

Et

[1 + rt1 + ρ

λt+1

λt

]= 1

Yt = Ztf(Ht)

Ztf′(Ht) = Wt

Gt = Tt

Yt = Ct +Gt

Use repeated substitution to boil everything down into two equations in four unknowns, theoptimality condition and the budget constraint:

Ct =Ztf

′(Ht)

V ′(H −Ht)(OPT)

Ztf(Ht) = Ct +Gt (ADD)

and analyze in (C, (H−H)) space. To get this: (1) into (2) to eliminate λ, (5) into (2) to eliminateW gets you the OPT condition. (4) into (7) gets you the ADD condition. The optimality conditionis upward-sloping; the budget constraint is downward-sloping. The above system of two equationsin four unknowns, together with specifications for the shock processes, is the model solution.

1.4 OPT/ADD Analysis with RBC1

We obtain:Ct

(H −Ht)

OPT

ADD

A G shock shifts the ADD schedule inward, reducing consumption and reducing leisure (in-creasing hours). The G shock predicts that C and H move in opposite directions, and hence is apoor replicator of business cycle facts. Meanwhile, the Z shock shifts ADD out and OPT back.Depending on the strength of the shifts, it could increase both C and H, along with Y and W ,thus replicating business-cycle facts. The key of RBC1 is Technology shocks can re-create businesscycle facts with appropriate calibration; government shocks cannot.

6

2 RBC with Capital

Call this RBC2 for short. Reference is King and Rebelo, “Resuscitating RBC,” NBER WP 7534.

2.1 Consumer UMP

Virtually identical except that instead of bonds, we have capital. Technically there are still bonds,but they are perfect substitutes for capital to the consumer and in equilibrium they are zero anyway.

max{Ct,Ht,Kt+1}

Et

{ ∞∑s=0

[1

1 + ρ

]s [C1−σt+s

1− σ+ V (H −Ht+s)

]}s.t. Kt+1+s = (1− δ)Kt+s +Rt+sKt+s +Wt+sHt+s + Πt+s − Ct+s − Tt+s; λt

with λ the associated Lagrange multiplier. The FOC are:

C−σt = λt (2.1)

V ′(H −Ht) = λtWt (2.2)

Et

[1 +Rt+1 − δ

1 + ρ

λt+1

λt

]= 1 (2.3)

This is exactly identical to the equilibrium conditions in RBC1, with rt = Rt+1− δ. The consumerhasn’t changed, but the firm will.


The representative firm continues to max profits, but now has maximization problem

maxY,L,K

Yt −WtHt −RtKt

s.t. Yt = F (Kt, ZtHt); ϕt

with ϕ the associated Lagrange multiplier. The firm’s FOC are

F1(Kt, ZtHt) = Rt (2.4)

ZtF2(Kt, ZtHt) = Wt (2.5)

Yt = F (Kt, ZtHt) (2.6)

As before, the government is fully Ricardian and pays for itself out of current taxes. The nationalincome accounting identity now includes the investment expenditure category.

Gt = Tt (2.7)

Yt = Ct + It +Gt (2.8)

We may also add an equation for investment:

It = Kt+1 − (1− δ)Kt (2.9)

2.3 Equilibrium

Relative to RBC1, we have one new FOC for capital, and we’ve added investment to the nationalincome identity. We can log-linearize, get a linear system of ten equations in ten unknowns, toss itinto Matlab and get a solution. Ignoring (2.7), which is just an identity to obtain T , we have noweight equations plus two shock processes.

7


Start with eight equations in ten unknowns, plus two shock processes: we care about nine variablesplus two shock variables: again C, H, λ, Y , W , T , r, G and Z; but now also K and I.

C−σt = λt


Et

[1 +Rt+1 − δ

1 + ρ

λt+1

λt

]= 1

Yt = Ct + It +Gt

Yt = F (Kt, ZtHt)

F1(Kt, ZtHt) = Rt

ZtF2(Kt, ZtHt) = Wt

Kt+1 = It + (1− δ)Kt

These equations are: the three FOC for consumers; the national income identity; the productionfunction; the two FOC for firms; and the capital accumulation equation.

These can be re-written into a loglinear system as:

Ct = − 1

σλt

Ht = εHW (λt + Wt)

R∗

1 + ρEtRt+1 + Etλt+1 − λt = 0

Yt = sCCt + sI It + sGGt

Yt = sKKt + sH(Ht + Zt)sHεKH

(Ht + Zt − Kt) = Rt

Z +sKεKH

(Kt − Zt − Ht) = Wt

Kt+1 = δIt + (1− δ)Kt

This is a nice linear system that can be thrown into any matrix algebra solver with no difficulty.Relative to RBC1, in levels,

1. The first three equations are basically unchanged; we have relaxed σ = 1 in the first eqn.

2. The fourth equation adds It as an expenditure category

3. The fifth equation uses the new production function

4. The sixth equation is new

5. The seventh equation uses the new production function

6. The eighth equation is new.

We have, in addition to the above eight equations, two equations for shock processes (left unspecified,usually autoregressive) and an equation for each expectations operator. The latter is typicallythe “Rational Expectations” assumption, which uses a fixed point between current and expectedfuture equilibria to derive its solution. I present the full Rational Expectations equilibrium in theAppendix.

8

2.5 Impulse Responses, RBC2

I choose to calibrate the elasticity parameters as εHW = 1 and εKH = 1. Share information comesfrom the NIPA and is calibrated as sK = 0.333, sI = 0.15, sG = 0.20.

5 10 15 200

2

4

6

8x 10

−3 c

5 10 15 20−5

0

5

10

15x 10

−4 h

5 10 15 200

2

4

6

8x 10

−3 y

5 10 15 200

2

4

6

8x 10

−3 w

5 10 15 200

0.01

0.02

0.03i

5 10 15 200

0.005

0.01

0.015z

Figure 1: RBC2 model, Z shock

Z shock: consumption, hours, output, wages, and investment all co-move positively. Consump-tion is less volatile than otupt, which is less volatile than investment.

5 10 15 20−1.5

−1

−0.5

0x 10

−3 c

5 10 15 200

2

4

6

8x 10

−4 h

5 10 15 20−2

0

2

4

6x 10

−4 y

5 10 15 20−4

−3

−2

−1

0x 10

−4 w

5 10 15 20−6

−4

−2

0x 10

−3 i

5 10 15 200

0.005

0.01

0.015g

Figure 2: RBC2 model, G shock

G shock: output and hours rise, but consumption and wages fall. The G shock does notreproduce key business cycle facts.

9

3 RBC with Imperfect Competition (Static Markup)

For reference call this RBC3. We add imperfect competition to the RBC2 model. This induces amarkup wedge between price and marginal cost, and between factor demand and factor supply.

3.1 Consumer UMP for the composite good

Consumers are standard:

max{C,H,K}

Et

{ ∞∑s=0

βt+s

[C1−σt+s

1− σ+ V (H −Ht+s)

]}s.t. Kt+s+1 = (1− δ)Kt+s +Rt+sKt+s +Wt+sHt+s + Πt+s − Ct+s − Tt+s; λt

with λ the Lagrange multiplier. This gives the FOC:

C−σt = λt (3.1)

V ′(H −Ht) = λtWt (3.2)

Et [β(1 +Rt − δ)λt+1] = λt (3.3)

These are the FOC with respect to Ct, the FOC with respect to Ht, and the intertemporal Eulerequation, respectively. The units C, K(?), and G are Dixit-Stiglitz aggregators of differentiatedcommodities, as explained presently.

We have the composite output demand function, the labor supply function, and the Eulerequation.

3.2 Consumer CMP for individual goods

A continuum of firms indexed i ∈ [0, 1] produce differentiated goods. These goods are aggregatedinto composites of the form:

Yt =

[∫ 1

0y(θ−1)/θit di

]θ/(θ−1)and all users of the goods (firms, consumers, and households) first solve the minimization problem:

min

∫ 1

0pityitdi

s.t. Yt =

[∫ 1

0y(θ−1)/θit di

]θ/(θ−1); ψt

where the pi are prices posted in terms of a unit of account and with ψ the Lagrange multiplier. Thesolution to this sub-problem yields the ideal price index Pt (think of it as a deflator?). In the end,all that’s going to matter are relative prices, so don’t think about it too hard. The sub-problemalso yields the output demand curve for good yit. In particular,

Pt = ψt =

[∫ 1

0p1−θit

]1/(1−θ)(3.4)

yit =

[pitPt

]−θYt (3.5)

We add to the above: the equation determining the composite price index and the individualoutput demand function.

10


Each little firm in [0, 1] is the sole producer of variety i and is a profit maximizer. Each firmmaximizes

maxy,k,h

pit(yit)yit −Rtkit −Wthti

s.t. yit = F (kit, Zthit)− Φ; ϕit

with ϕit the Lagrange multiplier and Φ a fixed cost parameter set so that steady-state pure profitvanishes. The FOC are:

pit =θ

θ − 1ϕit; let µ ≡ θ

θ − 1(3.6)

F1(Kt, ZtHt)ϕit = Rt =⇒ F1(Kt, ZtHt) = µRtpit

(3.7)

ZtF2(Kt, ZtHt)ϕit = Wt =⇒ ZtF2(Kt, ZtHt) = µWt

pit(3.8)

These are the FOC with respect to y, k, and h, respectively. They are the output supply curve(written as a pricing equation), the capital demand curve, and the labor demand curve. Relativeto the no-markup case, the only difference is that there is now a wedge µ in all three equilibriumconditions. Prices are set as a markup (µ) over marginal cost (ϕ); in the special case of perfectcompetition µ = 1 and there is no markup in equilibrium. If the markup is not time-varying, thiswedge disappears in a first-order approximation and is inconsequential; however if it is time-varying,the markup will be important for macroeconomic fluctuations.

In aggregate output is produced according to

Yt = F (Kt, ZtHt)− Φ (3.9)

which we hinted at earlier in the firm’s PMP but is now stated in aggregate form. The governmentand resource constraints are:

Yt = Ct + It +Gt (3.10)

Gt = Tt (3.11)

3.4 Equilibrium

We also have symmetry conditions. Since firms are identical except in the type of good theyproduce, and the consumer regards different types symmetrically, we have yit = Yt, hit = Ht,pit = Pt, et cetera.

The final trick is to note that while relative prices matter, the scale of prices is unimportant,so we may normalize the price index:

Pt = 1 (3.12)

The unknowns we care about are: C, H, λ, ϕ, ψ, Y , W , R, T , and K; we also have two shockprocesses G and Z. We can do some simple manipulation (G = T and ψ = Pt, for example) so thelist of variables we “really” care about can be reduced somewhat from the long list into a moretractible form. In the end there are still eight equations in eight unknowns, plus the two shocksprocesses, the same size as RBC2.

11


Gather all of the equations together:

C−σt = λt


Et

[β(1 +Rt − δ)

λt+1

λt

]= 1

Yt = Ct + It +Gt

Yt = F (Kt, ZtHt)− Φ

F1(Kt, ZtHt) = µRt

ZtF2(Kt, ZtHt) = µWt

Kt+1 = It + (1− δ)Kt

and this is the set of equations you log-linearize and toss into Matlab. The level equations are:three consumer FOC, the national income identity, the production function, two firm FOC, andthe capital accumulation equation. The log-linear system is:

Ct = − 1

σλt


(βR∗)EtRt+1 + Etλt+1 − λt = 0


Yt = µsKKt + µsH(Ht + Zt)sHεKH


Z +sKεKH

(Kt − Zt − Ht) = Wt

Kt+1 = δIt + (1− δ)Kt

Let’s take a step back and think about what we’ve done. What are the differences betweenRBC2 and RBC3?In levels:

1. The first four equations are identical

2. The fifth equation is augmented with an ad hoc fixed cost to eliminate pure profit

3. The sixth and seventh equations feature a constant markup wedge

4. The eighth equation is unchanged.

Log-linear:

1. The first four equations are identical

2. the fifth equation is amplified by the markup

3. The sixth through eighth equations are unchanged

12


I choose to calibrate the elasticity parameters as εHW = 1 and εKH = 1. The steady-state markupis assumed to be 1.2. Share information comes from the NIPA and is calibrated as sK = 0.333,sI = 0.15, sG = 0.20.

5 10 15 200

0.005

0.01c

5 10 15 20−15

−10

−5

0

5x 10

−4 h

5 10 15 200

0.005

0.01y

5 10 15 200

2

4

6

8x 10

−3 w

5 10 15 200

0.01

0.02

0.03i

5 10 15 200

0.005

0.01

0.015z


Relative to RBC2, we see amplification in the consumption, output, and investment graphs.Strikingly, the hours response is now negative on impact; before, it only went negative after tenor so periods. Hence in the model with markups, technology shocks are not good replicators ofbusiness cycle facts.

5 10 15 20−1

−0.5

0x 10

−3 c

5 10 15 200

2

4

6

8x 10

−4 h

5 10 15 20−2

0

2

4

6x 10

−4 y

5 10 15 20−4

−3

−2

−1

0x 10

−4 w

5 10 15 20−6

−4

−2

0x 10

−3 i

5 10 15 200

0.005

0.01

0.015g


G shocks fare no better. Again consumption, investment, and wages are countercyclical; thisshock fails to reproduce business cycle facts at a very basic level.

13

4 RBC with Imperfect Competition (Cyclical Markup)


Call this RBC4. Make a single change: suppose markups are cyclical. The levels solution is now:

C−σt = λt


Et

[β(1 +Rt − δ)

λt+1

λt

]= 1

Yt = Ct + It +Gt

Yt = F (Kt, ZtHt)− Φ

F1(Kt, ZtHt) = µtRt

ZtF2(Kt, ZtHt) = µtWt

Kt+1 = It + (1− δ)Kt

µt = g(Yt)

where the last equation ad-hoc. The loglinear system is:

Ct = − 1

σλt


(βR∗)EtRt+1 + Etλt+1 − λt = 0


Yt = µsssKKt + µsssH(Ht + Zt)sHεKH

(Ht + Zt − Kt) = Rt + µt

Z +sKεKH

(Kt − Zt − Ht) = Wt + µt

Kt+1 = δIt + (1− δ)Kt

µt = εµYt

with µss the steady-state markup. This is a nine-equation system that we can toss into Matlab. If,for example, εµ < 0, then we have a countercyclical markup, and this endogenously pushes aroundthe labor demand and capital demand schedules.

We could do this far more carefully, but also far more painfully, by modifying the firm sideyet again. Bring in some kind of IO model that has markups depend on aggregate output, crunchthrough the algebra, and get the relevant g(Y ). In the end, though, all that matters is the εµ, andwhether or not it’s negative.

It is useful to keep some persepctive. Recall that the markup is a function of the consumer’selasticity of substitution between goods. Thus, theories of the markup are also theories of consumerpreferences. If you want to understand markups, study preferences; if you want to have preferenceshocks, those show up as markup shocks, not as some kind of “demand” shock.

14


I choose to calibrate the elasticity parameters as εHW = 4 and εKH = 1. The steady-state markupis assumed to be 1.2 and εµ is calibrated to −0.8. Share information comes from the NIPA and iscalibrated as sK = 0.333, sI = 0.15, sG = 0.20. These IRFs are highly sensitive to the labor supplyelasticity. Be careful when generalizing the results.

5 10 15 200

0.02

0.04

0.06c

5 10 15 200

0.01

0.02

0.03h

5 10 15 200

0.02

0.04

0.06y

5 10 15 200

0.02

0.04

0.06w

5 10 15 200

0.05

0.1

0.15

0.2i

5 10 15 200

0.005

0.01

0.015z


With variable markups, the Z shock is able to match the business cycle facts. All four keyvariables comove positively, and investment, output and consumption follow the proper relativevolatilities.

5 10 15 200

1

2

3x 10

−3 c

5 10 15 200

2

4

6

8x 10

−3 h

5 10 15 200

2

4

6x 10

−3 y

5 10 15 200

1

2

3

4x 10

−3 w

5 10 15 200

0.01

0.02

0.03i

5 10 15 200

0.005

0.01

0.015g


The G shock is, for the first time, satisfactory. Output, wages, consumption and hours all movein the right direction on impact and over time.

15

5 Monetary Model, No Capital, Flexible Prices

Call this MM1. The vast majority of the elements are familiar from the RBC1 and RBC3 models.


The consumer’s problem is essentially the same:

maxC,H,B

Et

{βt+s

[ ∞∑s=0

C1−σt+s

1− σ− χ

H1+ηt+s

1 + η

]}

s.t.Bt+sPt+s

=Wt+s

Pt+sHt+s + (1 + it+s−1)

Bt+s−1Pt+s−1

+Πt+s

Pt+s− Ct+s; λt

There is no government and no capital, hence no taxes. Timing conventions have been alteredsomewhat: the old “Bt+1” is the new “Bt”. The timing convention has no substantive effects. Thesubstantive change is that bonds are now nominal and are a promise to pay (1 + it) units of cash inthe next period, instead of (1 + rt) units of the consumption good. The safe interest rate continuesto be known at time t. Wages are paid in cash. Consumption is still measured in real units. Theconsumer has a cash-in-advance constraint (???) and must obtain cash before purchasing goods.In particular he cannot barter directly his labor for goods.

The consumer FOC are

C−σt = λt (5.1)

χHηt = λt

Wt

Pt(5.2)

Et

[β(1 + it)

λt+1

λt

PtPt+1

]= 1 (5.3)

Notice that the FOC are very, very close to the FOC in RBC3. The difference is that now theconsumer has to worry about an uncertain real rate of return, due to the inflation term.


As usual Ct is a Dixit-Stiglitz aggregate. We next turn to the CMP for obtaining that aggregate.The consumer solves

minc

∫ 1

0pitcitdi

s.t. Ct =

[∫ 1

0c(θ−1)/θit

]θ/(θ−1); ψt

Solving this problem yields:

ψt =

[∫ 1

0p(1−θ)it

]1/(1−θ)≡ Pt (5.4)

cit =

[pitPt

]−θCt (5.5)

which are the ideal price index and the individual demand curve respectively. Our monopolisticallycompetitive firms will take the demand curve as an argument in their profit max problem.

16

5.3 Firm PMP and Central Bank

We have two equivalent ways of solving the firm’s problem: real and nominal. Because we set upthe consumer’s problem in real terms, let us do the same for the firm.

maxy,h

pit(yit)

Ptyit −

Wt

Pthit

s.t. yit = Zthit; ϕit

with ϕ, the induced multiplier, being the real marginal cost of producing an additional unit ofoutput. The function p(y) is given by the inverse of (5.5). Relevant FOC are:

pitPt

=θ


θ − 1so pit/Pt = µϕit (5.6)

ϕit =(Wt/Pt)

Zt=⇒ Zt

µ=Wt

Pt(mod Pit/Pt which cancels in eq) (5.7)

yit = Zthit (5.8)

These are FOC of the induced Lagrangian with respect to y, h, and ϕ respectively. The firstequation is an output supply curve expressed as a pricing equation: prices exceed marginal costby a constant fraction. We define that fraction to be the markup. The second is the (horizontal)labor demand curve; note the markup wedge. The final equation is the production function.

We finally turn to the central bank’s decision-making. Money demand is ad hoc:

Mt = PtYγt (1 + it)

ν (5.9)

and the money supply is chosen as a shock processes. We can justify a particular money demandfunction, Mt = PtYt, as a cash-in-advance constraint.

The national income identity is Yt = Ct as there is no government or investment.

5.4 Equilibrium

We care about C, H, i, λ, W , Y , and P with M and Z serving as shock variables. So far wehave no government and no capital. In particular, this means that there is no government shock tocompare against the technology shock. Instead we have the money stock as an instrument of policy.It turns out that in the benchmark model, changes in the money stock is entirely uninteresting: itfeeds one-for-one into nominal prices and nominal wages, leaving real variables unchanged.

17

5.5 The Solution System, MM1

We care about C, H, λ, Y , W , P , and i along with M and Z. Seven equations. In levels, we have

C−σt = λt

χHηt = λt

Wt

Pt

Et

[β(1 + it)

λt+1

λt

PtPt+1

]= 1

Yt = Ct

Yt = ZtHt

Ztµ

=Wt

Pt

Mt = PtYγt (1 + it)

ν

plus two shock processes for Z and M . In order, the equations are: three FOC for consumers, thenational income identity, the production function, the firm’s labor FOC, and the money demandfunction. We have enough structure on the problem (specific functional forms) that we may solvefor the flexible-price equilibrium output level:

Yt =

(1

χµ

)1/(σ+η)

Z(1+η)/(η+σ)t ≡ Y f

t (5.10)

How to get this: we need (1), (2), (4), (5), and (6). (4) into (1) eliminates C, (1) into (2) eliminatesλ, (6) into (2) eliminates W/P , (5) into (2) eliminates H, then rearrange. We can do comparativestatics without even tossing the system into Matlab. This object will play a key role in the sticky-price model to follow. Note immediately that Y f is independent of M . In the flexible-price world,money doesn’t matter.

The log-linear system is:

Ct = − 1

σλt

ηHt = λt + Wt − Pt(βi∗)Et(it + Pt+1 − Pt) + Etλt+1 = λt

Yt = Ct

Yt = Zt + Ht

Zt = Wt − PtMt = Pt + γYt + νit

which we toss into Matlab and can draw impulse responses for Z and M shocks. Actually showingthe IRFs is uninteresting. The M shock immediately feeds 1-for-1 into P and w; the Z shock hasits usual effects on C, H, and W/P .

18

6 Monetary Model, No Capital, Sticky Prices

Call this MM2 or DNK. The only thing we’re doing is adding a cost of changing prices.


We aren’t changing the consumer. Recall that she solves:

maxC,H,B

Et

{βt+s

[ ∞∑s=0

C1−σt+s

1− σ− χ

H1+ηt+s

1 + η

]}

s.t.Bt+sPt+s

=Wt+s

Pt+sHt+s + (1 + it+s−1)

Bt+s−1Pt+s

+Πt+s

Pt+s− Ct+s; λt

with λ that reliable old Lagrange multiplier. The consumer FOC continue to be:

C−σt = λt (6.1)

χHηt = λt

Wt

Pt(6.2)

Et

[β(1 + it)

λt+1

λt

PtPt+1

]= 1 (6.3)

These FOC should be immediately familiar. They are the usual marginal utility of wealth condition,labor supply curve, and Euler equation.


It’s again the same, and we get the same results. The consumer attempts to minimize the cost ofbuying a Dixit-Stiglitz aggregate of the underlying goods:

minc

∫ 1

0pitcitdi

s.t. Ct =

[∫ 1

0c(θ−1)/θit

]θ/(θ−1); ψt

The FOCs which must hold in equilibrium are:

ψt =

[∫ 1

0p(1−θ)it

]1/(1−θ)≡ Pt (6.4)

cit =

[pitPt

]−θCt (6.5)

We obtain the ideal price index as the Lagrange multiplier, and we obtain a constant elasticitydemand function for each little firm i. Each firm will take into account this demand curve whenchoosing the level of output to supply and the markup it charges over marginal cost.

6.3 Firm PMP

Firms still maximize real profit. The problem is:

maxy,h

pit(yit)

Ptyit −

Wt

Pthit

s.t. yit = Zthit; ϕit

19

with ϕ, the induced multiplier, being the real marginal cost of producing an additional unit ofoutput. The function p(y) is given by the inverse of (6.5). Relevant FOC are:

pitPt

=θ



ϕit =(Wt/Pt)

Zt=⇒ Zt

µ=Wt

Pt(6.7)

yit = Zthit (6.8)

6.4 Rotemberg Pricing and the NK Phillips Curve

We have a complete labor market; let’s gather up those equations and derive the aggregate supplycurve. Suppose that for whatever reason, prices are sluggish to adjust. Let the price one would setin the flexible-price equilibrium be P ∗t . Then equation (6.6) says:

P ∗tPt

= µϕt =⇒ P ∗t − Pt = ϕt

in log-deviations. Now bring forward the labor demand and labor supply curves:

ϕt = (Wt − Pt)− Zt (Hd = µMPL)

(Wt − Pt) = ηHt + σCt = ηHt + σYt (Hs = MRS)

and plug labor supply into labor demand to solve out both real wages and hours:

ϕt = ηHt + σYt − Zt= η(Yt − Zt) + σYt − Zt= (η + σ)Yt − (1 + η)Zt

and plugging this expression for marginal cost back into our original equation:

P ∗t − Pt = (η + σ)

[Yt −

1 + η

η + σZt

]= (η + σ)(Yt − Y f

t )

That is, the price gap is proportional to the output gap. We are almost to the NKPC. The nextstep is to assume that (correct up to a second order approximation):

Πt ≈ −δ[P ∗t − Pt]2

(that is, profit deviations are proportional to squared price deviations). Finally, formalize thesluggish price change with an ad-hoc assumption that price changes have quadratic cost to the firm:

ct = φ[Pt − Pt−1]2 (6.9)

and assume that the firm maximizes profits again at the level of the deviation,

max∞∑t=0

βt(Πt − ct)

for the FOC

δ[P ∗t − Pt]− φ[Pt − Pt−1] + βφ[EtPt+1 − Pt] = 0

which we rewrite as

πt = βEtπt+1 +(σ + η)δ

φ(Yt − Y f

t ) =⇒ πt = βEtπt+1 + κ(Yt − Y f

t

)(6.10)

which is the desired expression. This is the New Keynesian Phillips Curve.

20

6.5 NK Analysis with MM2

Second, we can do NK analysis in (X,π, i) space. Bring forward the NKPC:

πt = βEtπt+1 + κ (Xt) (NKPC)

with Xt ≡ Yt− Y ft . Next log-linearize the consumption Euler equation, substituting λ = −σC and

C = Y , and obtain:

Xt = EtXt+1 −1

σ

(it − Etπt+1

)+ ut (DIS)

with ut ≡ EtY ft+1−Y

ft , which is the “Dynamic IS equation”. Add an interest-rate rule for monetary

policy, which can be derived using the money demand equation as long as ν 6= 0:

it = δππt + δxXt (TR)

and you have the canonical 3-equation New Keynesian model in inflation, interest rate, and output-gap terms. Notice that the NK solution is primarily interested in rates, particularly the interest rateand inflation rate. The money demand equation is subsumed, and replaced with an interest-rateequation: the money supply is endogenous and set so as to achieve the interest-rate rule. Dynamicstability is ensured if δπ > 1, the “Taylor Principle”.

6.6 Wicksellian Analysis with MM2

Let us now define the natural real rate of interest rnt . Start with the Euler equation:

Yt = EtYt+1 −1

σ(it − Etπt+1)

Invert the object and evaluate at flexible-price output to obtain:

rnt ≡ σ(EtY

ft+1 − Y

ft

)(6.11)

then we may rewrite the output-gap DIS equation as

Xt = EtXt+1 −1

σ(rt − rnt ) (6.12)

that is, the output gap is a function of the expected future output gap and the deviation of theinterest rate from its natural rate. Holding expectations fixed, a rise in the natural rate causes theinterest rate to be below the natural rate, and the output gap rises (a boom); a secular fall in thenatural rate causes a negative output gap (recession). Note that the natural rate is time-varyingand in particular varies with Z shocks.

This suggests a monetary policy rule to keep the interest rate at its natural rate at all times:

it = rnt + πt

but note that this fails the Taylor principle, since the implied δπ = 1! Instead we need a rule like

it = δr(rnt + πt)

with δr > 1, i.e. the Fed has a strong preference to keep the interest rate at its natural rate, andwill respond over-aggressively to deviations from that state.

21

6.7 The Solution System, MM2

We care about C, λ, H, Y , W , P , i, and ϕ, along with M and Z. The log-linearized system is:

Ct = − 1

σλt

ηHt = λt + Wt − PtEt [it − πt+1] + Etλt+1 = λt

Yt = Ct

Yt = Zt + Ht

Zt + ϕt = Wt − PtMt = Pt + γYt + νit (or it = δππt + εit)

πt = βEtπt+1 +δ

φϕt

πt = Pt − Pt−1

That is, the three FOC for consumers, the national income identity, the production function, thelabor FOC for firms, the money demand function/Taylor Rule, and the marginal profit FOC.Relative to MM1, notice that the labor demand function has a wedge equal to marginal cost, andwe’ve added a new marginal profit condition. Technically one could plug (8) into (6) and eliminateϕ entirely.

The key parameter here for the elasticity of “price stickiness” is δ/φ. I have calibrated thiselasticity to 0.20, a quite small but still empirically plausible value, for the purposes of showing offthe model. With the solution system in hand I turn to its impulse responses.

22

6.8 Impulse Responses, MM2, Monetary Rule

With the full model in hand we may graph impuse responses. I calibrate η = 0.25, δ(η+σ)/φ = 0.20,ν = −0.5, and γ = 1.

5 10 15 200

0.005

0.01y

5 10 15 20−0.01

−0.005

0p

5 10 15 200

0.005

0.01c

5 10 15 20−10

−5

0

5x 10

−3 h

5 10 15 200

0.005

0.01rw

5 10 15 200

0.005

0.01

0.015z

Figure 7: M2 model, Z shock

On impact of a Z shock, output, consumption and the real wage rise while hours fall; as pricesadjust hours rise to its old steady-state value from below.

5 10 15 200

2

4

6

8x 10

−3 y

5 10 15 200

0.005

0.01p

5 10 15 200

2

4

6

8x 10

−3 c

5 10 15 200

2

4

6

8x 10

−3 h

5 10 15 200

0.005

0.01rw

5 10 15 200

0.005

0.01

0.015m

Figure 8: M2 model, M shock

On impact of a (permanent nominal) monetary shock, output, hours, consumption, and realwages all rise. As prices adjust, the output and consumption responses vanish and all effects areabsorbed into the nominal price level. In the new steady-state, the price level is permanently higherbut real variables return to their pre-shock steady-states.

23

6.9 Impulse Responses, MM2, Taylor Rule

5 10 15 200

0.005

0.01y

5 10 15 20−6

−4

−2

0x 10

−4 yg

5 10 15 200

0.005

0.01

0.015yf

5 10 15 20−1.5

−1

−0.5

0x 10

−3 pi

5 10 15 200

0.005

0.01c

5 10 15 20−6

−4

−2

0x 10

−4 h

5 10 15 200

0.005

0.01rw

5 10 15 200

0.005

0.01

0.015z

5 10 15 20−2

−1.5

−1

−0.5

0x 10

−3 i

Figure 9: M2 model with Taylor Rule, Z shock

If monetary policy is set by a Taylor Rule, an output shock continues to boost output, con-sumption and the real wage while reducing hours. Flexible-price output rises, and an output gapdevelops and closes asymptotically. The interest rate endogenously falls and returns to the oldsteady-state asymptotically.

5 10 15 20−0.05

0

0.05

0.1

0.15y

5 10 15 20−0.05

0

0.05

0.1

0.15yg

5 10 15 20−0.01

0

0.01

0.02

0.03pi

5 10 15 20−0.05

0

0.05

0.1

0.15c

5 10 15 20−0.05

0

0.05

0.1

0.15h

5 10 15 20−0.1

0

0.1

0.2rw

5 10 15 20−0.2

−0.15

−0.1

−0.05

0i

Figure 10: M2 model with Taylor Rule, i shock

This interest-rate shock is new. A decrease in the interest rate causes output, consumption,hours and the real wage to rise on impact, but they return to their steady-state values quickly.Flexible-price output is unaffected I calibrate the central bank’s response to inflation as 1.5, andits response to the output gap as 0.5.

24

7 Monetary Model, Capital, Sticky Prices

This model molds together RBC3 with the pricing frictions of MM2.


Consumers maximize

max{Ct,Ht,Kt+1.Bt,λt}

E0

{ ∞∑t=0

βs

[C1−σt

1− σ+ χ

H1+ηt

1 + η

]}

s.t. Kt+1 + Ct +BtPt

= (1 +RtPt− δ)Kt + +

Wt

PtHt + (1 + it)

Bt−1Pt

+Πt

Pt− Ct − Tt; λt

where I am trying to be as careful with the notation as possible. there are two assets, (nominal)bonds and capital (implicitly, there’s a third asset: the medium of exchange). R/P is the real netrental rate on capital, with (1 + R/P − δ) being the real gross return on capital. B is a quantity ofnominal bonds so B/P is a quantity of real bonds. W/P is the real wage. Capital K, consumptionC and taxes T are already in real terms and don’t need P denominators. λ is the real MU ofwealth. Define R = R/P and W = W/P to be consistent with Basu. The four FOC are:

C−σ = λ (7.1)

χHηt = Wtλt (7.2)

Et[β(1 +Rt+1 − δ)λt+1] = λt (7.3)

Et[β(1 + rt)λt+1] = λt (7.4)

where (1 + rt) ≡ (1 + it)/(1 + πt+1) is the real rate of return on a (nominally) riskless bond. I haveexplicitly defined the two Euler equations; note that they are identical to the Euler equations inRBC3. (In RBC3, I left the second Euler equation implicit.)

7.2 Consumer CMP for the individual goods

Same song and dance as we’ve had since RBC3. The problem is:

minc

∫ 1

0pitcitdi

s.t. Ct =

[∫ 1

0c(θ−1)/θit

]θ/(θ−1); ψt

The FOCs which must hold in equilibrium are:

ψt =

[∫ 1

0p(1−θ)it

]1/(1−θ)≡ Pt (7.5)

cit =

[pitPt

]−θCt (7.6)

and the same is true for government and investment demand for the individual goods as well. Hencethe output demand curve (7.6) holds for cit, git, iit and yit. We haven’t explicitly stated it before,but in all of these models output can be costlessly transformed into the consumption good, theinvestment good, and the government-purchases good.

25

7.3 Firm PMP

The firm takes as given its output demand curve and seeks to maximize real profits:

maxy,h

pit(yit)

Ptyit −Wthit −Rtkit

s.t. yit = kαit(Zthit)1−α − Φ; ϕit

with ϕ, the induced multiplier, being the real marginal cost of producing an additional unit ofoutput. Recall that W and R are already in real units, so we don’t need to divide by P . Thefunction p(y) is given by the inverse of (7.6). Relevant FOC are:

pitPt

=θ



Rt = ϕt

[αYt + Φ

Kt

](7.8)

Wt = ϕt

[(1− α)

Yt + Φ

Ht

](7.9)

where I have already substituted k = K, h = H and y = Y . As before, interpret pit = P ∗t , theideal nominal price, the nominal price one would set in a flexible-price setting. That implies

P ∗t − Pt = ϕt

7.4 Rotemberg Pricing

Again there is sluggish price adjustment, so the firm has:

Πt = −δ[P ∗t − Pt]2

ct = φ[Pt − Pt−1]2

the marginal-profit-deviation and price-changing cost respectively. The firm’s goal is to

maxE0

∞∑t=0

{Πt − ct

}which yields FOC

δ[P ∗t + Pt]− φ[Pt − Pt−1] + βφ[EtPt+1 − Pt] = 0

which rearranges to

δϕt − φπt + βEtπt+1 = 0

πt = βEtπt+1 +δ

φϕt (7.10)

and we recover the same “marginal cost” Phillips Curve that we had in MM2.

7.5 Central Bank

The Fed faces money demand:

Mt = PtYγ(1 + it)

ν (7.11)

and supplies money exogenously. Alternatively one could specify monetary policy in terms of aninterest-rate target.

26

7.6 Equilibrium and Solution System

Gather up all the sheep and pray that we have them all. We care about C, H, λ, T , R, r, i,W , Y , K, I, P , π, Y f , X, and ϕ along with (stochastic processes for) M , G, and Z. That’ssixteen endogenous variables and three exogenous variables, the largest model we’ve built yet. Ourequations are the following.

We have four consumer FOC:

C = − 1

σλ

ηHt = Wt + λt

Etλt+1 + EtRt+1 = λt

Etλt+1 + rt = λt

namely, the MU of wealth condition, the labor supply function, the Euler for capital, and the Eulerfor nominal bonds. σ is the (inverse) IES. η is the (inverse) Frisch labor supply elasticity. rt isknown at time t (?) and hence has no expectation. We have four firm-side equations:

Rt = ϕt + µssYt − Kt

Wt = ϕt + µssYt − Kt

Yt = µ∗αKt + µ∗(1− α)(Zt + Ht)

πt = βEtπt+1 +δ

φϕt

namely, capital demand, labor demand, output supply, and Rotemberg/Calvo pricing. µ∗ is thesteady-state markup. We have additional macro-aggregate equations:

Yt = scCt + sI It + stGt

Mt − Pt = γYt + νit

Tt = Gt

Kt+1 = (1− δ)Kt + δIt

namely, output demand, money demand, government budget balance, and the capital accumulationequation. Finally we have four equations defining and linking “residual” variables

Xt = Y ft − Yt

rt = it − Etπt+1

Y ft = ζ(Zt, Gt)

πt = Pt − Pt−1

namely, the definition of the output gap; the definition of the riskless bond rate; the definition offlexible-price output; and the definition of inflation. I have log-linearized the equations that werenot already log-linear, with the exception of flex-price output, because I haven’t yet calculated therelevant elasticities. I don’t have any markups in any equations because I find markups confusing,but µt = P ∗t − Pt − ϕt. I think.

27

A RBC2: A Derivation of the RatEx Solution

Just showing that it can be done. Begin with the full-blown model.

Ct = − 1

σλt (A.1)

Ht = εHW (λt + Wt) (A.2)

R∗

1 + ρEtRt+1 + Etλt+1 − λt = 0 (A.3)

Yt = sCCt + sI It + sGGt (A.4)

Yt = sKKt + sH(Ht + Zt) (A.5)sHεKH

(Ht + Zt − Kt) = Rt (A.6)

Z +sKεKH

(Kt − Zt − Ht) = Wt (A.7)

Kt+1 = δIt + (1− δ)Kt (A.8)

Sub (1) into (4) to eliminate C. Sub (8) into (4) to eliminate I. Sub (5) into (4) to eliminate Y .Sub (7) into (2) to eliminate W . Obtain:

Ht = εHW

[λt +

(Z +

sKεKH

(Kt − Zt − Ht)

)]R∗

1 + ρEtRt+1 + Etλt+1 − λt = 0

sKKt + sH(Ht + Zt) = −sC1

σλt + sI [Kt+1 − (1− δ)Kt] + sGGt

sHεKH


We are left with K, R, λ, and H plus Z, G. This is four equations in four unknowns plus twoshocks. Start renaming things to make our lives easier.

Ht = a11λt + a12Zt + a13Kt (A.9)

a21EtRt+1 + Etλt+1 − λt = 0 (A.10)

sKKt + sHHt + shZt = a31λt + a32Kt+1 + a33Kt + a34Gt (A.11)

a41Ht + a42Zt + a43Kt = Rt (A.12)

Sub (9) into (11) and (12) to eliminate H; obtain a system of three equations. Then we need onlyget rid of R to have a system of two expectational equations.

a21EtRt+1 + Etλt+1 − λt = 0 (A.13)

b21Kt + b22λt + b23Zt = a31λt + a32Kt+1 + a33Kt + a34Gt (A.14)

b31Kt + b32λt + b33Zt = Rt (A.15)

We need to wipe out R, then we’re home free. Idea: iterate (15) forward once and stick it in (13).Let’s do this carefully:

a21EtRt+1 + Etλt+1 − λt = 0

b21Kt + b22λt + b23Zt = a31λt + a32Kt+1 + a33Kt + a34Gt

b31Kt+1 + b32λt+1 + b33Zt+1 = Rt+1

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then:

a21Et

[b31Kt+1 + b32λt+1 + b33Zt+1

]+ Etλt+1 − λt = 0

b21Kt + b22λt + b23Zt = a31λt + a32Kt+1 + a33Kt + a34Gt

and rearrange:

A11Kt+1 +A12Etλt+1 = λt + C11EtZt+1 (A.16)

A21Kt+1 = B21λt +B22Kt +D21Zt +D22Gt (A.17)

or

AEtXt+1 = BXt + CEtUt+1 +DUt (A.18)

with Xt = [λt,Kt]′ and Ut = [Zt, Gt]

′. This is the desired expression. I think. One can diagonalizeand check for stability.

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B RBC2: A Sample Matlab file

A complete .mod file for producing the RBC2 IRFs would be:

var c g h i lambda k r w y z;

varexo ez eg ;

parameters SIGMA, EHW, BETA, R, SH, EKH, SI, SG, RHOG, RHOZ, VOLG, VOLZ;

SIGMA = 1.00; EHW = 1.000; BETA = 0.99; R = 0.04;

SH = 0.67; EKH = 1.000; SI = 0.15; SG = 0.20;

RHOG = 0.90; VOLG = 0.01;

RHOZ = 0.98; VOLZ = 0.01;

model(linear);

lambda = -SIGMA*c;

h = EHW*(lambda + w);

(BETA * R)*r(+1) - lambda + lambda(+1) = 0;

y = (1-SI-SG)*c + SI*i + SG*g;

y = (1 - SH)*k(-1) + SH*z + SH*h;

r = (SH / EKH) * (h + z -k(-1));

w = ((1 - SH) / EKH)*(k(-1) - z - h) + z;

k = (1-DELTA)*k(-1) + DELTA*i;

z = RHOZ*z(-1) + ez;

g = RHOG*g(-1) + eg;

end;

initval;

c = 0; g = 0; h = 0; i = 0; k = 0;

r = 0; w = 0; y = 0; z = 0; lambda = 0;

end;

steady;

shocks;

var ez; stderr VOLZ ;

var eg; stderr VOLG ;

end;

stoch_simul(periods=200,order=1,irf=20) c h y w i z g;

Save as “rbc.mod” and run in Matlab with “dynare rbc.mod”. The file has several parts. First,define the endogenous cariables, the exogenous variables, and the model parameters. Second, pro-vide values for the model parameters. Third, define the log-linear model. Fourth, set initial values.Finally, Matlab finds the steady-state, shocks the system, and produces the stochastic simulation(the IRF graphs). The other models are similar: just add the required parameters, calibrate them,add any new model equations, and add any new variables of interest to the simulation. Note thetiming convention: k(−1) is capital at the start of the period and is our Kt. This is simply theconvention of how Matlab deals with state variables.

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