Approximate quadratic-linear optimization problem Based on Pierpaolo Benigno and Michael Woodford

The Quadratic Approximation to the Utility Function Consider the problem

The first-order condition

The second-order approximation to the utility function

The second-order approximation to the constraint

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 Subject to:

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

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:

The Quadratic Approximation to the Utility Function Consider the problem

The first-order condition

The second-order approximation to the utility function

The second-order approximation to the constraint

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.

The Micro-based Neo-Keynesian Quadratic-linear problem Based on Pierpaolo Benigno and Michael Woodford

The Micro-based Quadratic Loss Function

Welfare measure expressed as a function of equilibrium production Demand of differentiated product is a function of relative prices

The Deterministic (distorted) Steady State Maximize with respect to Subject to constraints on

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:

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 Demand of differentiated product is a function of relative prices

The Micro-based Quadratic Loss Function of Benigno and Woodford

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 Maximize with respect to Subject to constraints on

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 MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

Complete Markets = price kernel Value of portfolio with payoff D

Interest coefficient for riskless asset Riskless Portfolio

Budget Constraint Where T is the transfer payments based on the seignorage profits of the central bank, distributed in a lump sum to the representative consumer

No Ponzi Games: For all states in t+1 For all t, to prevent infinite c The equivalent terminal condition

Lagrangian

Transversality condition: Flow budget constraint:

Market Equilibrium Market solution for the transfers T

Monetary Targeting: BC chooses a path for M Fiscal policy assumed to be: Equilibrium is S.t. Euler-intertemporal condition condition FOC-itratemporal condition TVC Constraint For given

We study equilibrium around a zero-shock steady state:

Derive the LM Curve From the FOC: At the steady state:

Separable utility : Define: The “hat” variables are proportional deviations from the steady state variables.

Similar to Cagan’s semi-elasticity of money demand

We log-linearize around zero inflation define Log-linearize the Euler Equation and transform it to a Fisher equation: Elasticity of intertemporal substitution g is the “twist” in MRS between m and c

Add the identity We look for solution given exogenous shocks

Solution of the system This is a linear first-order stochastic difference equation,where, Exogenous disturbance (composite of all shocks):

given There exists a forward solution: From which we can get a unique equilibrium value for the price level: This is similar to the Cagan-Sargent-wallace formula for the price level, 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: 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 value of outsranding bonds End of period, one-period risk-less bonds

Steady state (around fix )

Is unique Can uniquely be determined! PRICE LEVEL IS INDETERMINATE: Real balances are unique Future expected inflation is 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: 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: P is not correlated to the path of M: money demand shocks affect M, but do not affect P; the LM is not used in 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, an automatic response.

Note, also that Big Small, reduces the need for accurate observation of, almost complete peg of interest rate

The path of the money supply: By using LM, we can still express But we must examine existence of a well-defined demand for money. There’s possibly liquidity trap

III. TAYLOR (feedback) RULE Steady state Assume:

Taylor principle: Is predetermined

Transitory fluctuations in Create transitory fluctuations in Permanent shifts in the price level P.

Optimizing models with nominal rigidities Chapter 3

First Order Conditions:

Firm’s Optimization: Nominal Real

Natural Level of Output

Log-linearization of real mc: Partial-equilibrium relationship?

‘where Elasticity of wage demands, wrt to output holding marginal utility of income constant Elasticity of marginal product of labor wrt output

ONE-PERIOD NOMINAL RIDIGITY Same as before, except for Y need not be equal to the natural y

C t = consumption aggregate = = gross rate of increase in the Dixit-Stiglitz price index P t A Neo-Wicksellian Framework THE IS:

Equilibrium condition: A log-linear approximation around a deterministic steady state yields the IS schedule: g=crowding out term due to fiscal shock

Equivalent to the fiscal shock Effect on fiscal shock on C

New Keynesian Phillips Curve: Taylor Rule: Inflation target Deviation of natural output due to supply shock Demand determined output deviations

Output gap: IS-curve involves an exogenous disturbance term: 3-EQUATION EQUILIBRIUM SYSTEM: Proportion of firm that prefix prices

INTEREST RULE AND PRICE STABILITY THE NATURAL RATE OF INTEREST

Percentage deviation of the natural rate of interest from its steady-state value

Inflation targeting at low, positive, inflation Composite disturbances

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 MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

Complete Markets = price kernel Value of portfolio with payoff D

Interest coefficient for riskless asset Riskless Portfolio

Budget Constraint Where T is the transfer payments based on the seignorage profits of the central bank, distributed in a lump sum to the representative consumer

No Ponzi Games: For all states in t+1 For all t, to prevent infinite c The equivalent terminal condition

Lagrangian

Transversality condition: Flow budget constraint:

Market Equilibrium Market solution for the transfers T

Monetary Targeting: BC chooses a path for M Fiscal policy assumed to be: Equilibrium is S.t. Euler-intertemporal condition condition FOC-itratemporal condition TVC Constraint For given

We study equilibrium around a zero-shock steady state:

Derive the LM Curve From the FOC: At the steady state:

Separable utility : Define: The “hat” variables are proportional deviations from the steady state variables.

Similar to Cagan’s semi-elasticity of money demand

We log-linearize around zero inflation define Log-linearize the Euler Equation and transform it to a Fisher equation: Elasticity of intertemporal substitution g is the “twist” in MRS between m and c

Add the identity We look for solution given exogenous shocks

Solution of the system This is a linear first-order stochastic difference equation,where, Exogenous disturbance (composite of all shocks):

given There exists a forward solution: From which we can get a unique equilibrium value for the price level: This is similar to the Cagan-Sargent-wallace formula for the price level, 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: 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 value of outsranding bonds End of period, one-period risk-less bonds

Steady state (around fix )

Is unique Can uniquely be determined! PRICE LEVEL IS INDETERMINATE: Real balances are unique Future expected inflation is 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: 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: P is not correlated to the path of M: money demand shocks affect M, but do not affect P; the LM is not used in 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, an automatic response.

Note, also that Big Small, reduces the need for accurate observation of, almost complete peg of interest rate

The path of the money supply: By using LM, we can still express But we must examine existence of a well-defined demand for money. There’s possibly liquidity trap

III. TAYLOR (feedback) RULE Steady state Assume:

Taylor principle: Is predetermined

Transitory fluctuations in Create transitory fluctuations in Permanent shifts in the price level P.

Optimizing models with nominal rigidities Chapter 3

First Order Conditions:

Firm’s Optimization: Nominal Real

Natural Level of Output

Log-linearization of real mc: Partial-equilibrium relationship?

‘where Elasticity of wage demands, wrt to output holding marginal utility of income constant Elasticity of marginal product of labor wrt output

ONE-PERIOD NOMINAL RIDIGITY Same as before, except for Y need not be equal to the natural y

C t = consumption aggregate = = gross rate of increase in the Dixit-Stiglitz price index P t A Neo-Wicksellian Framework THE IS:

Equilibrium condition: A log-linear approximation around a deterministic steady state yields the IS schedule: g=crowding out term due to fiscal shock

Equivalent to the fiscal shock Effect on fiscal shock on C

New Keynesian Phillips Curve: Taylor Rule: Inflation target Deviation of natural output due to supply shock Demand determined output deviations

Output gap: IS-curve involves an exogenous disturbance term: 3-EQUATION EQUILIBRIUM SYSTEM: Proportion of firm that prefix prices

INTEREST RULE AND PRICE STABILITY THE NATURAL RATE OF INTEREST

Percentage deviation of the natural rate of interest from its steady-state value

Inflation targeting at low, positive, inflation Composite disturbances

Evolution of money supply: The only exogenous variables in the system are: = the natural interest rate =nominal rate consistent with inflation target