INTEREST AND PRICES

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INTEREST AND PRICES

• MICHAEL WOODFORD

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MICROFOUNDED CAGAN-SARGENT PRICE LEVEL
DETERMINATION UNDER MONETARY TARGETING
FLEX-PRICE, COMPLETE-MARKETS MODEL
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Complete Markets
Value of portfolio with payoff D
price kernel
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Interest coefficient for riskless asset
Riskless Portfolio
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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
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No Ponzi Games
For all states in t1
For all t, to prevent infinite c
The equivalent terminal condition
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Lagrangian
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Transversality condition
Flow budget constraint
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Market Equilibrium
Market solution for the transfers T
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Monetary Targeting BC chooses a path for M
Fiscal policy assumed to be
Equilibrium is
S.t. Euler-intertemporal condition condition FOC-i
tratemporal condition TVC Constraint
For given
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We study equilibrium around a zero-shock steady
state
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Derive the LM Curve
From the FOC
At the steady state
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Separable utility
Define
The hat variables are proportional deviations
from the steady state variables.
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Similar to Cagans semi-elasticity of money
demand
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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
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Add the identity
We look for solution given exogenous shocks
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Solution of the system
This is a linear first-order stochastic
difference equation ,where,
Exogenous disturbance (composite of all shocks)
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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.
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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
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Steady state (around
)
fix
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PRICE LEVEL IS INDETERMINATE
Real balances are unique
Future expected inflation is unique
Is unique
But, neither
Can uniquely be determined!
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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!
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II. Wicksellian Rules interest rate is a
function of endogenous variables (feedback rule)
Vcontrol error of CB
Fiscal Policy
Exogenous
Endogenous
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Steady State
Log-linearize
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We can find two processes
Add the identity
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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.
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FEATURES

• Forward looking
• Price is not a function of i rather , a function
  of the feedback rule and the target
• suppose

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Additionally

• If

Price level instability can be reduced by raising
, an automatic response.
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Note, also that

• Big
• Small

, reduces the need for accurate observation of
, almost complete peg of interest rate
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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. Theres possibly
liquidity trap
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III. TAYLOR (feedback) RULE

• Steady state

Assume
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Taylor principle
Is predetermined
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Transitory fluctuations in
Create transitory fluctuations in
Permanent shifts in the price level P.
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Optimizing models with nominal rigidities

• Chapter 3

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First Order Conditions
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Firms Optimization
Nominal
Real
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Natural Level of Output
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Log-linearization of real mc
Partial-equilibrium relationship?
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where
Elasticity of marginal product of labor wrt output
Elasticity of wage demands, wrt to output
holding marginal utility of income constant
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ONE-PERIOD NOMINAL RIDIGITY
Same as before, except for
Y need not be equal to the natural y
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A Neo-Wicksellian Framework
THE IS
Ct consumption aggregate

gross rate of increase in the Dixit-Stiglitz
price index Pt
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Equilibrium condition
A log-linear approximation around a deterministic
steady state yields the IS schedule
gcrowding out term due to fiscal shock
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Effect on fiscal shock on C
Equivalent to the fiscal shock
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New Keynesian Phillips Curve
Deviation of natural output due to supply shock
Demand determined output deviations
Taylor Rule
Inflation target
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Output gap
3-EQUATION EQUILIBRIUM SYSTEM
Proportion of firm that prefix prices
IS-curve involves an exogenous disturbance term
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INTEREST RULE AND PRICE STABILITY
THE NATURAL RATE OF INTEREST
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Percentage deviation of the natural rate of
interest from its steady-state value
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Inflation targeting at low, positive, inflation
Composite disturbances
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Evolution of money supply
The only exogenous variables in the system are
the natural interest rate
nominal rate consistent with inflation target