DSGE Models for Monetary Policy

Lawrence Christiano

Outline• Basic New Keynesian (NK) model

– Two fundamental frictions in the model.

• Implications of NK model for monetary policy:– Clarifying the concepts of ‘excess and inadequate aggregate demand’.

– The Taylor principle and inflation targeting.– Cases where ‘overzealous inflation targeting’ can go awry:

• News shocks and the relationship between monetary policy and stock market volatility

• The working capital channel and the Taylor principle.

Outline, cnt’d• Adjustments to NK model to enable it to account for empirical estimates of the monetary transmission mechanism (the ‘David Hume puzzle’).

• Costly State Verification model of financial frictions: microeconomics.

• Introducing csv model into otherwise standard NK DSGE model– The importance of a ‘risk shock’ in economic fluctuations.– Using the model to assess a monetary policy that reacts to the interest rate premium.

Basic New Keynesian Model• Description of the economic environment

– Preferences and technology.– The best possible allocation of labor and consumption, regardless of how they are brought about.

• How ideally functioning markets can bring about the best allocations (the ‘invisible hand’).

• What happens with markets that resemble actual real world markets where prices don’t adjust instantly.

Model

• Household preferences:

E0∑t0

t logCt − exp tNt

1

1 ,

t t−1 t, t~iidN0,2

Production• Final output requires lots of intermediate inputs:

• Production of intermediate inputs:

• Constraint on allocation of labor:

0

1Nitdi Nt

Yt 0

1Yi,t

−1 di

−1

, 1

Yi,t eatNi,t, Δat Δat−1 ta, ta~iidN0,a2

Efficient Allocation of Total Labor• Suppose total labor,      , is fixed.

• What is the best way to allocate       among the various activities,                ? 

• Answer: – allocate labor equally across all the activities

0 ≤ i ≤ 1

Nit Nt, all i

Nt

Nt

Suppose Labor Not Allocated Equally

• Example:

• Note that this is a particular distribution of labor across activities:

Nit 2Nt i ∈ 0, 1

2

21 − Nt i ∈ 12 , 1

, 0 ≤ ≤ 1.

0

1Nitdi 1

2 2Nt 12 21 − Nt Nt

Labor Not Allocated Equally, cnt’d

Yt 0

1Yi,t

−1 di

−1

0

12 Yi,t

−1 di

12

1Yi,t

−1 di

−1

eat 0

12 Ni,t

−1 di

12

1Ni,t

−1 di

−1

eat 0

12 2Nt

−1 di

12

121 − Nt

−1 di

−1

eatNt 0

12 2

−1 di

12

121 −

−1 di

−1

eatNt 12 2

−1 1

2 21 − −1

−1

eatNtf

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.88

0.9

0.92

0.94

0.96

0.98

1

f

Efficient Resource Allocation Means Equal Labor Across All Sectors

6

10

f 12 2

−1 1

2 21 − −1

−1

Economy with Efficient N Allocation• Efficiency dictates

• So,  with efficient production:

• Resource constraint:

• Preferences:

Yt eatNt

Ct ≤ Yt

E0∑t0

logCt − exp tNt

1

1 , t t−1 t, t~iid,

Nit Nt all i

Efficient Determination of Labor• Lagrangian:

• First order conditions:

• or:

maxCt ,Nt

E0∑t0

t

logCt−exp t Nt

1

1

uCt,Nt, t teatNt − Ct

ucCt,Nt, t t, unCt,Nt, t teat 0

un,t uc,teat 0

marginal cost of labor in consumption units− dudNtdudCt

dCtdNt

−un,tuc,t

marginal product of laboreat

Efficient Determination of Labor, cont’d• Solving the fonc’s:

• Note:– Labor responds to preference shock, not to tech shock

−un,tuc,t eat

Ct exp tNt eat

eatNt exp tNt eat

→ Nt exp − t1

→ Ct exp at − t1

Response to a Jump in a

Leisure, 1-N

consumptionHigher a

Indifference curve

Production frontier

Case Where Markets Work Beautifully(triumph of the ‘invisible hand’)

• Give households budget constraints, put them in markets and let them pursue their individual interests.

• Give the production functions to firms and suppose that they seek to maximize profits.

• There is monopoly power….extinguish the effects of that by a subsidy.

• Let markets work perfectly: prices and wages adjust instantly all the time, to clear markets.

Households• Solve:

• Subject to:

• First order conditions:

maxE0∑t0

t logCt − exp tNt

1

1 ,

Ct bonds purchases in t

Bt1 ≤

wage ratewt Nt

profits t

(real) interest on bondsrt−1 Bt

−un,tuc,t Ct exp tNt

wt ‘marginal cost of working equals marginal benefit’

uc,t Etuc,t1rt ‘marginal cost of saving equals marginal benefit’

Profits, net of government taxes

Final Good Firms• Final good firms buy       ,               ,  at given prices,        , to maximize profits:

• Subject to

• Fonc’s:   

Yt 0

1 Yi,t−1 di

−1

Yi,t i ∈ 0,1Pi,t

Yt − 0

1Pi,tYi,tdi

Pi,t YtYi,t

1

→ Yi,t Pi,t−Yt, 1 0

1Pi,t1−di

Intermediate Good Firms• For each         there is a single producer who is a monopolist in the product market and hires labor,       in competitive labor markets.  

• Marginal cost of production:

• Subsidy will be required to ensure markets work efficiently.

Yi,tNi,t

(real) marginal costst dCostdwor kerdoutputdwor ker

1 −subsidy payment to firm

wt

expat

Intermediate Good Firms

Yi,t

Pi,t Yi,t Pi,t−Yt

Marginal revenue

Marginal cost,

Demand curve

st

ith Intermediate Good Firm• Problem:

• Subject to demand for       :

• Problem:

• Note: all prices are the same, so resources allocated efficiently across intermediate good firms.

Yi,t Yi,t Pi,t−Yt

maxNit PitYit − stYit

maxNitPitPi,t−Yt − stPi,t−Yt

fonc : 1 − Pit−Yt stPi,t−−1Yt 0

Pit − 1 st ‘price is markup over marginal cost’

Pi,t Pj,t 1, because 1 0

1Pi,t1−di

Equilibrium• Pulling things together:

• If proper subsidy is provided to monopolists, employment is efficient:

1 − 1 st

− 1

1 − wtexpat

household fonc 1 −

− 1

−un,tuc,t

expatif 1−

−1

−un,tuc,t

expat.

if 1 − − 1 , then −un,tuc,t expat

=1

Equilibrium Allocations• With efficient subsidy,

• Bond market clearing implies:

Bt 0 always

−un,tuc,t

functional form Ct exp tNt

resource constraint

expatexp tNt1 expat

→ Nt exp − t1

Ct eatNt exp at − t1

Interest Rate in Equilibrium• Interest rate backed out of household intertemporal Euler equation:uc,t Etuc,t1rt → 1

Ct Et 1

Ct1rt

→ rt 1Et CtCt1

1Et expct − ct1

1Et exp at − at1 − t−t1

1

1exp Et −Δat1 − t−t1

1 12 V

, Vt a2 11

22

logrt − log Et

ct1−ct

Δat1 − t1 − t1 1

2 V

using assumptions about Δat and t − log Δat − −1t

1 12 V

Dynamic Properties of the Model

2 4 6 8 10 12 14 16 18 20 22 240

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Response to .01 Technology Shock in Period 1

(at - at-1) = 0.75(at-1 - at-2) + epsatlog

Ct

t

2 4 6 8 10 12 14 16 18 20 22 24

0.008

0.009

0.01

0.011

0.012

0.013

0.014

interest rate

log,

r t

t

log rt = -log + 0.75at-1

2 4 6 8 10 12 14 16 18 20 22 24-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

x 10-3

log

Ct

t

t = 0.5t-1 + epstaut

Response to .01 Preference Shock in Period 1

2 4 6 8 10 12 14 16 18 20 22 24

7.5

8

8.5

9

9.5

x 10-3

log,

r t

t

log rt = -log - (0.5 - 1)t/(1 + 1)

interest rate

logrt − log 0.75Δat

Key Features of Equilibrium Allocations• Allocations efficient (as long as monopoly power neutralized)

• Employment does not respond to technology– Improvement in technology raises marginal product of labor and marginal cost of labor by same amount.

• First best consumption not a function of intertemporal considerations– Discount rate irrelevant.– Anticipated future values of shocks irrelevant.

• Natural rate of interest steers consumption and employment towards their natural levels, ‘as if guided by an invisible hand’.

Introducing Price Setting Frictions(Clarida‐Gali‐Gertler Model)

• Households maximize:

• Subject to:

• Intratemporal first order condition:

E0∑t0

logCt − exp tNt

1

1 , t t−1 t, t~iid,

PtCt Bt1 ≤ WtNt Rt−1Bt Tt

Ct exp tNt Wt

Pt

Profits, net of taxes raised byGovernment to finance subsidies.

Household Intertemporal FONC• Condition:

– or

– take log of both sides:

– or

1 Etuc,t1uc,t

Rt1 t1

1 Et CtCt1

Rt1 t1

Et explogRt − log1 t1 − Δct1

≃ explogRt − Et t1 − EtΔct1 , ct ≡ logCt

0 log rt − Et t1 − EtΔct1, rt logRt

ct − log − rt − Et t1 ct1

NK IS Curve• Euler equation in two equilibria:

• Subtract: Output gap

xt −rt − Et t1 − rt∗ Etxt1

Actual equilibrium: yt −rt − Et t1 − rr Etyt1

Natural equilibrium: yt∗ −rt∗ − rr Etyt1∗

Output equals consumption

Final Good Firms• Buy                     at prices       and sell      for • Take all prices as given (competitive)• Profits:

• Production function:

• First order condition:    

Yt 0

1Yi,t

−1

−1di, 1,

PtYt − 01 Pi,tYi,tdi

Yi,t, i ∈ 0,1 Pi,t Yt Pt

→ Pt 0

1Pi,t1−di

11−Yi,t Yt Pi,t

Pt

−

Intermediate Good Firms• Each ith good produced by a single monopoly producer. 

• Demand curve:

• Technology:

• Calvo Price‐setting Friction

Yi,t expatNi,t, Δat Δat−1 ta,

Yi,t YtPi,tPt

−

Pi,t P̃t with probability 1 − Pi,t−1 with probability

Marginal Cost

real marginal cost st dCostdwor kerdOutputdwor ker

1 − Wt/Ptexpat

−1 in efficient setting

1 − Ct exp tNt

expat

Marginal Cost• Marginal cost:

• Then,

st 1 fCt exp tNt

expatCtexpat Nt holds only as a first order approximation

≃1fCt exp t Ct

expat

expat

1 fCt

1 exp t1

expat

1

1 f

CtCt∗

1

ŝta hat indicates log-deviation from steady state

≡ logst f 1 ct − ct∗ 1 xt

1f

− 1 ~ tax subsidy, 1 − , to ‘trick’ firms into

setting price equal to marginal cost

Marginal Cost• Marginal cost:

• Then,

st 1 fCt exp tNt

expatCtexpat Nt holds only as a first order approximation

≃1fCt exp t Ct

expat

expat

1 fCt

1 exp t1

expat

1

1 f

CtCt∗

1

ŝta hat indicates log-deviation from steady state

≡ logst f 1 ct − ct∗ 1 xt

=1/s

The Intermediate Firm’s Decisions• ith firm is required to satisfy whatever demand shows up at its posted price.

• Its only real decision is to adjust price whenever the opportunity arises.

• It sets its price as a function of current and expected future marginal cost.– Inflation evolves (to first order approximation) as follows:

t 1 − 1 −

ŝt Etŝt1 2Etŝt2 . . .

ŝt ≡ st − ss ≃ log sts , t Pt − Pt−1

Pt−1

NK Phillips Curve• Repeating from previous slide:

• Also,

• Then,

t 1−1−

ŝt Etŝt1 2Etŝt2 . . .

Et t1 1 − 1 −

Etŝt1 2Etŝt2 3Etŝt3 . . .

t Et t1 1 − 1 −

ŝt

Back to the Phillips Curve• Phillips curve summarizes price setting by intermediate good firms:

• or, substituting out for marginal cost:

t Et t1 1 − 1 −

ŝt

t Et t1 1−1−

1 xt

Equations of Actual Equilibrium, with Monetary Policy Rule

t Et t1 xt (Calvo pricing equation)

xt −rt − Et t1 − rt∗ Etxt1 (intertemporal equation)

rt rt−1 1 − t xxt ut (policy rule)

rt∗ Etyt1∗ − yt∗ Et Δat1 − 1

1 Δ t1 (natural rate)

yt∗ at − 11 t (natural output), xt yt − yt∗ (output gap)

also, time series representations for shocks listed above

Solving the Model• Express the equations in matrix form:

• Solution: A and Bmatrices such that (*) is satisfied and

zt Azt−1 Bst

zt

txtrtrt∗

, st tatut

, t t

ta

tu

∗ 0Etzt1 1zt 2zt−1 0Etst1 1st 0st Pst−1 t

Solving the model….• Iterating forward one period on the solution:

• Then,

zt1 Azt Bst1

A2zt−1 ABst Bst1

Et0zt1 1zt 2zt−1 0st1 1st Et0A2zt−1 ABst Bst1 1Azt−1 Bst 2zt−1 0st1 1st 0A2 1A 2 zt−1 Et0B 0 st1 0AB 1B 1st

0

0A2 1A 2 zt−1 0

0B 0P 0AB 1B 1 st 0, all st, zt−1

Finding solution for A is only hard part. Choose A that solves problem if there is only onewith eigenvalues less than unity in absolute value.If more than one A like this: multiple solutions.

Impulse Response Function• Law of motion of endogenous and exogenous variables:

• Set one element of      non‐zero and let  

• Set                            and compute

zt Azt−1 Bstst Pst−1 t

0

t 0, t 0s−1 z−1 0

s0, s1, s2, . . .z0, z1, z2, . . .

1 2 3 4 5 6 7

0.01

0.02

0.03

0.04

Inflation

1 2 3 4 5 6 7

0.05

0.1

0.15

Ouput Gap

1 2 3 4 5 6 7

0.05

0.1

0.15

0.2Nominal Rate

Natural Nominal Rate

Actual Nominal Rate

1 2 3 4 5 6 7

0.05

0.1

0.15

0.2Real Rate

NaturalActual

1 2 3 4 5 6 7

1.05

1.1

1.15

1.2

log Technology

1 2 3 4 5 6 7

1.05

1.1

1.15

1.2

Output

Natural OutputActual Output

1 2 3 4 5 6 70

0.05

0.1

0.15

Employment

Natural EmploymentActual Employment

Cost of workingBenefit of working

Dynamic Response to a Technology Shock

1 2 3 4 5 6 7

1.1

1.2

1.3

x 0, 1.5, 0.97, 1, 0.2, 0.75, 0, 0.5.

1 2 3 4 5 6 7

0.02

0.04

0.06

0.08

0.1Inflation

1 2 3 4 5 6 7

0.05

0.1

0.15

0.2

0.25

Ouput Gap

1 2 3 4 5 6 7

0.05

0.1

0.15

0.2

0.25Nominal Rate

Natural Nominal RateActual Nominal Rate

1 2 3 4 5 6 7

0.05

0.1

0.15

0.2

0.25Real Rate

NaturalActual

1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1Preference Shock

1 2 3 4 5 6 7-0.5

-0.4

-0.3

-0.2

-0.1

Output

Natural OutputActual Output

1 2 3 4 5 6 7-0.5

-0.4

-0.3

-0.2

-0.1

Employment

Natural EmploymentActual Employment

Cost of workingBenefit of working

Dynamic Response to a Preference Shock

1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5