Approximate quadratic-linear optimization problem

transcript

Based on

Pierpaolo Benigno and Michael Woodford

The Quadratic Approximation to the Utility Function

•Consider the problem

)},({,

yxuMaxyx

The first-order condition

)}),(({max yyx

The second-order approximation to the utility function

22 )(2

1)( yyuyyuxxuxxu yyyxxx

The second-order approximation to the constraint

1)( yyFyyFx yyy

•Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the utility function, using the FOC, yields a quadratic objective function

The approximate optimization problem

)({max

Subject to:

)( yyFx y

xFuyyFu

yxxyxx

Which is supposed to be(?) a first order approximation of

0 yyx uFu

A Linear-Quadratic Approximate Problem

•Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:

uuYYuYEuYconstU

ttttyttyyt

The Quadratic Approximation to the Utility Function

•Consider the problem

)},({,

yxuMaxyx

The first-order condition

)}),(({max yyx

The second-order approximation to the utility function

22 )(2

1)( yyuyyuxxuxxu yyyxxx

The second-order approximation to the constraint

1)( yyFyyFx yyy

Approximate optimization

•Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the linear term of the second-order approximation of the utility function, using the first-order conditions, yields a quadratic objective function.

•The approximate optimization is to maximize the quadratic objective function, subject to the first-order approximation of the constraint. The first-order condition is equal to the first order approximation of the FOC of the original problem.

yyuyyuxxuxyyFyyFu

yyuyyuxxuxxu

yyyxxyyyx

yyyxxx

The Micro-based Neo-Keynesian Quadratic-linear problem

Based on

Pierpaolo Benigno and Michael Woodford

The Micro-based Quadratic Loss Function

ttjtttt

djHvCuEU

));(();(

Welfare measure expressed as a function of equilibrium production

));(();(

ttjtttt

djyvYuEU

Demand of differentiated product is a function of relative prices

The Deterministic (distorted) Steady State

)0;;(0

),,( tttY Maximize with respect to

Subject to constraints on

),,( tttY

•BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.

A Linear-Quadratic Approximate Problem

•Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:

uuYYuYEuYconstU

ttttyttyyt

•There is a non-zero linear term in the approximate welfare measure, unless

•As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.

Welfare measure expressed as a function of equilibrium production

));(();(

ttjtttt

djyvYuEU

Demand of differentiated product is a function of relative prices

The Micro-based Quadratic Loss Function of Benigno and Woodford

ttjtttt

djHvCuEU

));(();(

•There is a non-zero linear term in the approximate welfare measure, unless

•As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.

The Deterministic (distorted) Steady State

)0;;(0

),,( tttY Maximize with respect to

Subject to constraints on

),,( tttY

•BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.

MICROFOUNDED CAGAN-SARGENT

PRICE LEVEL DETERMINATION

UNDER MONETARY TARGETING

FLEX-PRICE, COMPLETE-MARKETS MODEL

tttttttt

cpTypWBM

;;(max0

MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

Complete Markets

);()(),(

)(),()(

ssttss

ssttst

zzprobzDzzQ

zzprob

zzprobzDzzq

zDzzqzq

= price kernel

Value of portfolio with payoff D

Interest coefficient for riskless asset

Riskless Portfolio

Budget Constraint

tttttttttt

ttttttt

WQEQEcp

)())(1(

1,1,1,

Where T is the transfer payments based on theseignorage profits of the central bank, distributedin a lump sum to the representative consumer

No Ponzi Games:

0)(lim

tTTTTTtt

tTTTTTttt

TypQEW

For all states in t+1

For all t, to prevent infinite c

The equivalent terminal condition

Lagrangian

)((();;((

TypQEwp

ttttttt

cpTypWBM

0)(lim ,

Transversality condition:

Flow budget constraint:

Market Equilibrium

sttttt

sttjttt

iTWWQE

11,,111

Market solution for the transfers T

Monetary Targeting: BC chooses a path for M

st MMTMWB 110

Fiscal policy assumed to be:

Equilibrium is tt ip ; S.t.Euler-intertemporal conditionconditionFOC-itratemporal conditionTVCConstraint

For given sttt My ;;

We study equilibrium around a zero-shock steady state:

Derive the LM Curve

From the FOC:

At the steady state:

);();();;( mvcumcu Separable utility :

Define:

The “hat” variables are proportional deviations from the steady state variables.

1Similar to Cagan’ssemi-elasticity of money demand

We log-linearize around zero inflation1_

define 1logloglog tttt PPLog-linearize the Euler Equation and transform it to a Fisher equation:

tttttt

gygyEr

)]()([

Elasticity of intertemporal substitution

g is the “twist” in MRS between m and c

Add the identity

tttt mm

We look for solution

given exogenous shocks

ttt im ;;

ttt y ;;

Solution of the system

))(1( 11

ttitttt EumEm This is a linear first-order stochastic difference equation ,where,

Exogenous disturbance (composite of all shocks):

)]()([ 111

tttttt

titymtt

gygyEr

100 iThere exists a forward solution:

)()1(0

jtijttj

From which we can get a unique equilibrium value for the price level:

log)(log)1(logj

t muMEP

This is similar to the Cagan-Sargent-wallace formula for the pricelevel, but with the exception that the Lucas Critique is taken care of and it allows welfare analysis.

I. Interest Rate Targeting based on exogenous shocks

Choose the path for i; specify fiscal policy which targets D:

st BMD Total end of period public sector liabilities.

Monetary policy affects the breakdown of D between M and B:

No multi-period bonds

Beginning of period valueof outsranding bonds

End of period, one-periodrisk-less bonds

Steady state (around 1

endogenous

exogenous

tttttt

tmttityt

Is unique

Can uniquely be determined!

PRICE LEVEL IS INDETERMINATE:

Real balances are unique

Future expected inflationis unique

But, neither

To see the indeterminancy, let “*” denote solution value:

v is a shock, uncorrelated with(sunspot), the new triple is also a solution, thus:

ttt iy

Price level is indeterminate under the interest rule!

II. Wicksellian Rules: interest rate is a function of endogenous variables (feedback rule)

V=control error of CB

Fiscal Policy

Exogenous

Endogenous

Steady State:

Log-linearize:

We can find two processes

Add the identity

1), 2) and 3) yield:

)((log)1(log

)*()1(

)*()()1(

jjtjtpjtt

jjtjtjtt

tttttttp

vErPEP

P is not correlated to the path of M:money demand shocks affect M, butdo not affect P; the LM is not usedin the derivation of the solution to P.

FEATURES:

• Forward looking

• Price is not a function of i; rather , a function of the feedback rule and the target

• suppose

Additionally:

• If

Price level instability can be reduced by raising

p , an automatic response.

Note, also that

• Big

• Smallp , reduces the need for accurate observation of tr

p , almost complete peg of interest rate

The path of the money supply:

);;*;;( ttttt

s vyPPMM

By using LM, we can still express

But we must examine existence of a well-defineddemand for money. There’s possibly liquidity trap

III. TAYLOR (feedback) RULE

1)0,1(

• Steady state

Assume:

loglog

Taylor principle:

Is predetermined1tP

Transitory fluctuations in

Create transitory fluctuations in t

Permanent shifts in the price level P.

Optimizing models with nominal rigidities

Chapter 3

))(()(

ihfAiy

));(();;(

)()()(

tttttttt

diihvP

diidiihiwyP

diicipcP

cPTyPwBM

First Order Conditions:

tttttc

11,111

Firm’s Optimization:

)(()()(

()()()(

iwismc

iyiwiSMC

iyfiwihiwVC

Nominal

));(();(

););(())(

Natural Level of Output

Log-linearization of real mc:

ttssti

tissii

yyiyis

iyiyiy

)()()(

)())()((

1)log()log(

Partial-equilibrium relationship?

‘where

)());((

Elasticity of wage demands, wrt to output holdingmarginal utility of incomeconstant

Elasticity of marginal product oflabor wrt output

ONE-PERIOD NOMINAL RIDIGITY

0;;([);(

0)()(([0)(

()()([

,11,11

ttttttct

tttttttt

tttttttttt

yysyyuE

iSipPyQEip

ipPyfiwipPyQMaxE

iQMaxE

Same as before, except for 1tE

Y need not be equal to the natural y

ttctt Cu

Ct = consumption aggregate

P= = gross rate of increase in

the Dixit-Stiglitz price index Pt

A Neo-Wicksellian Framework

THE IS:

ttctt Yu

Equilibrium condition:

A log-linear approximation arounda deterministic steady state yields the ISschedule:

)()ˆ(ˆ111 ttttttt EigYEgY

g=crowding out term due to fiscal shock

)ˆ(lim TTtT

01)(ˆ

jjtjtttt iEgY

Equivalentto the fiscalshock

Effect on fiscalshock on C

New Keynesian Phillips Curve:

1)ˆˆ( tn

ttt EYY Taylor Rule:

tyttt Yii ˆ)( ** Inflation target

Deviation of natural outputdue to supply shock

Demand determinedoutput deviations

Output gap:

nttt YYx ˆˆ

]ˆ)ˆ[( 11nttttttt rEixEx

1 tttt Ex

)()( *** xxii txttt IS-curve involves an exogenous disturbance term:

)]ˆ()ˆ[(ˆ 111 n

ttnt YgEYgr

3-EQUATION EQUILIBRIUM SYSTEM:

Proportion offirm that prefixprices

INTEREST RULE AND PRICE STABILITY

THE NATURALRATE OF INTEREST

log)1log(1

1logˆ

Percentage deviation of the natural rate of interestfrom its steady-state value

Inflation targeting at low, positive,inflation

*ˆˆ ntt ri

ˆ ttnt

Composite disturbances

])1()1)(1(

)1)(1(ˆ)1[(1

))1((1

))1(ˆ(ˆ

tcGtGnt

mttityt

ˆˆ*loglog

ˆˆloglog

Evolution of money supply:

The only exogenous variables in the system are:

= the natural interest rate

=nominal rate consistent with inflation target

FLEX-PRICE, COMPLETE-MARKETS MODEL

tttttttt

cpTypWBM

;;(max0

MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

Complete Markets

);()(),(

)(),()(

ssttss

ssttst

zzprobzDzzQ

zzprob

zzprobzDzzq

zDzzqzq

= price kernel

Value of portfolio with payoff D

Interest coefficient for riskless asset

Riskless Portfolio

Budget Constraint

tttttttttt

ttttttt

WQEQEcp

)())(1(

1,1,1,

Where T is the transfer payments based on theseignorage profits of the central bank, distributedin a lump sum to the representative consumer

No Ponzi Games:

0)(lim

tTTTTTtt

tTTTTTttt

TypQEW

For all states in t+1

For all t, to prevent infinite c

The equivalent terminal condition

Lagrangian

)((();;((

TypQEwp

ttttttt

cpTypWBM

0)(lim ,

Transversality condition:

Flow budget constraint:

Market Equilibrium

sttttt

sttjttt

iTWWQE

11,,111

Market solution for the transfers T

Monetary Targeting: BC chooses a path for M

st MMTMWB 110

Fiscal policy assumed to be:

Equilibrium is tt ip ; S.t.Euler-intertemporal conditionconditionFOC-itratemporal conditionTVCConstraint

For given sttt My ;;

We study equilibrium around a zero-shock steady state:

Derive the LM Curve

From the FOC:

At the steady state:

);();();;( mvcumcu Separable utility :

Define:

The “hat” variables are proportional deviations from the steady state variables.

1Similar to Cagan’ssemi-elasticity of money demand

We log-linearize around zero inflation1_

define 1logloglog tttt PPLog-linearize the Euler Equation and transform it to a Fisher equation:

tttttt

gygyEr

)]()([

Elasticity of intertemporal substitution

g is the “twist” in MRS between m and c

Add the identity

tttt mm

We look for solution

given exogenous shocks

ttt im ;;

ttt y ;;

Solution of the system

))(1( 11

ttitttt EumEm This is a linear first-order stochastic difference equation ,where,

Exogenous disturbance (composite of all shocks):

)]()([ 111

tttttt

titymtt

gygyEr

100 iThere exists a forward solution:

)()1(0

jtijttj

From which we can get a unique equilibrium value for the price level:

log)(log)1(logj

t muMEP

This is similar to the Cagan-Sargent-wallace formula for the pricelevel, but with the exception that the Lucas Critique is taken care of and it allows welfare analysis.

I. Interest Rate Targeting based on exogenous shocks

Choose the path for i; specify fiscal policy which targets D:

st BMD Total end of period public sector liabilities.

Monetary policy affects the breakdown of D between M and B:

No multi-period bonds

Beginning of period valueof outsranding bonds

End of period, one-periodrisk-less bonds

Steady state (around 1

endogenous

exogenous

tttttt

tmttityt

Is unique

Can uniquely be determined!

PRICE LEVEL IS INDETERMINATE:

Real balances are unique

Future expected inflationis unique

But, neither

To see the indeterminancy, let “*” denote solution value:

v is a shock, uncorrelated with(sunspot), the new triple is also a solution, thus:

ttt iy

Price level is indeterminate under the interest rule!

II. Wicksellian Rules: interest rate is a function of endogenous variables (feedback rule)

V=control error of CB

Fiscal Policy

Exogenous

Endogenous

Steady State:

Log-linearize:

We can find two processes

Add the identity

1), 2) and 3) yield:

)((log)1(log

)*()1(

)*()()1(

jjtjtpjtt

jjtjtjtt

tttttttp

vErPEP

P is not correlated to the path of M:money demand shocks affect M, butdo not affect P; the LM is not usedin the derivation of the solution to P.

FEATURES:

• Forward looking

• Price is not a function of i; rather , a function of the feedback rule and the target

• suppose

Additionally:

• If

Price level instability can be reduced by raising

p , an automatic response.

Note, also that

• Big

• Smallp , reduces the need for accurate observation of tr

p , almost complete peg of interest rate

The path of the money supply:

);;*;;( ttttt

s vyPPMM

By using LM, we can still express

But we must examine existence of a well-defineddemand for money. There’s possibly liquidity trap

III. TAYLOR (feedback) RULE

1)0,1(

• Steady state

Assume:

loglog

Taylor principle:

Is predetermined1tP

Transitory fluctuations in

Create transitory fluctuations in t

Permanent shifts in the price level P.

Optimizing models with nominal rigidities

Chapter 3

))(()(

ihfAiy

));(();;(

)()()(

tttttttt

diihvP

diidiihiwyP

diicipcP

cPTyPwBM

First Order Conditions:

tttttc

11,111

Firm’s Optimization:

)(()()(

()()()(

iwismc

iyiwiSMC

iyfiwihiwVC

Nominal

));(();(

););(())(

Natural Level of Output

Log-linearization of real mc:

ttssti

tissii

yyiyis

iyiyiy

)()()(

)())()((

1)log()log(

Partial-equilibrium relationship?

‘where

)());((

Elasticity of wage demands, wrt to output holdingmarginal utility of incomeconstant

Elasticity of marginal product oflabor wrt output

ONE-PERIOD NOMINAL RIDIGITY

0;;([);(

0)()(([0)(

()()([

,11,11

ttttttct

tttttttt

tttttttttt

yysyyuE

iSipPyQEip

ipPyfiwipPyQMaxE

iQMaxE

Same as before, except for 1tE

Y need not be equal to the natural y

ttctt Cu

Ct = consumption aggregate

P= = gross rate of increase in

the Dixit-Stiglitz price index Pt

A Neo-Wicksellian Framework

THE IS:

ttctt Yu

Equilibrium condition:

A log-linear approximation arounda deterministic steady state yields the ISschedule:

)()ˆ(ˆ111 ttttttt EigYEgY

g=crowding out term due to fiscal shock

)ˆ(lim TTtT

01)(ˆ

jjtjtttt iEgY

Equivalentto the fiscalshock

Effect on fiscalshock on C

New Keynesian Phillips Curve:

1)ˆˆ( tn

ttt EYY Taylor Rule:

tyttt Yii ˆ)( ** Inflation target

Deviation of natural outputdue to supply shock

Demand determinedoutput deviations

Output gap:

nttt YYx ˆˆ

]ˆ)ˆ[( 11nttttttt rEixEx

1 tttt Ex

)()( *** xxii txttt IS-curve involves an exogenous disturbance term:

)]ˆ()ˆ[(ˆ 111 n

ttnt YgEYgr

3-EQUATION EQUILIBRIUM SYSTEM:

Proportion offirm that prefixprices

INTEREST RULE AND PRICE STABILITY

THE NATURALRATE OF INTEREST

log)1log(1

1logˆ

Percentage deviation of the natural rate of interestfrom its steady-state value

Inflation targeting at low, positive,inflation

*ˆˆ ntt ri

ˆ ttnt

Composite disturbances

])1()1)(1(

)1)(1(ˆ)1[(1

))1((1

))1(ˆ(ˆ

tcGtGnt

mttityt

ˆˆ*loglog

ˆˆloglog

Evolution of money supply:

The only exogenous variables in the system are:

= the natural interest rate

=nominal rate consistent with inflation target

Approximate quadratic-linear optimization problem

Documents