Approximate quadratic-linear optimization problem

1
Approximate quadratic-linear optimization problem

• Based on
• Pierpaolo Benigno and Michael Woodford

2
The Quadratic Approximation to the Utility
Function

• Consider the problem

3
The first-order condition
4
The second-order approximation to the utility
function
5
The second-order approximation to the constraint
6

• 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

7
The approximate optimization problem
Subject to
8
Which is supposed to be(?) a first order
approximation of
9
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

10
The Quadratic Approximation to the Utility
Function

• Consider the problem

11
The first-order condition
12
The second-order approximation to the utility
function
13
The second-order approximation to the constraint
14
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.

15
The Micro-based Neo-Keynesian Quadratic-linear
problem

• Based on
• Pierpaolo Benigno and Michael Woodford

16
The Micro-based Quadratic Loss Function
17
Welfare measure expressed as a function of
equilibrium production
Demand of differentiated product is a function
of relative prices
18
The Deterministic (distorted) Steady State
Maximize with respect to
Subject to constraints on
19

• 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.

20
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

21

• 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.

22
Welfare measure expressed as a function of
equilibrium production
Demand of differentiated product is a function
of relative prices
23
The Micro-based Quadratic Loss Function of
Benigno and Woodford
24

• 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.

25
The Deterministic (distorted) Steady State
Maximize with respect to
Subject to constraints on
26

• 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.

27
MICROFOUNDED CAGAN-SARGENT PRICE LEVEL
DETERMINATION UNDER MONETARY TARGETING
28
MICROFOUNDED CAGAN-SARGENT PRICE LEVEL
DETERMINATION UNDER MONETARY TARGETING
FLEX-PRICE, COMPLETE-MARKETS MODEL
29
Complete Markets
Value of portfolio with payoff D
price kernel
30
Interest coefficient for riskless asset
Riskless Portfolio
31
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
32
No Ponzi Games
For all states in t1
For all t, to prevent infinite c
The equivalent terminal condition
33
Lagrangian
34
Transversality condition
Flow budget constraint
35
Market Equilibrium
Market solution for the transfers T
36
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
37
We study equilibrium around a zero-shock steady
state
38
Derive the LM Curve
From the FOC
At the steady state
39
Separable utility
Define
The hat variables are proportional deviations
from the steady state variables.
40
Similar to Cagans semi-elasticity of money
demand
41
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
42
Add the identity
We look for solution given exogenous shocks
43
Solution of the system
This is a linear first-order stochastic
difference equation ,where,
Exogenous disturbance (composite of all shocks)
44
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.
45
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
46
Steady state (around
)
fix
47
PRICE LEVEL IS INDETERMINATE
Real balances are unique
Future expected inflation is unique
Is unique
But, neither
Can uniquely be determined!
48
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!
49
II. Wicksellian Rules interest rate is a
function of endogenous variables (feedback rule)
Vcontrol error of CB
Fiscal Policy
Exogenous
Endogenous
50
Steady State
Log-linearize
51
We can find two processes
Add the identity
52
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.
53
FEATURES

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

54
Additionally

• If

Price level instability can be reduced by raising
, an automatic response.
55
Note, also that

• Big
• Small

, reduces the need for accurate observation of
, almost complete peg of interest rate
56
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
57
III. TAYLOR (feedback) RULE

• Steady state

Assume
58
Taylor principle
Is predetermined
59
Transitory fluctuations in
Create transitory fluctuations in
Permanent shifts in the price level P.
60
Optimizing models with nominal rigidities

• Chapter 3

61
(No Transcript)
62
(No Transcript)
63
First Order Conditions
64
Firms Optimization
Nominal
Real
65
Natural Level of Output
66
Log-linearization of real mc
Partial-equilibrium relationship?
67
where
Elasticity of marginal product of labor wrt output
Elasticity of wage demands, wrt to output
holding marginal utility of income constant
68
ONE-PERIOD NOMINAL RIDIGITY
Same as before, except for
Y need not be equal to the natural y
69
A Neo-Wicksellian Framework
THE IS
Ct consumption aggregate

gross rate of increase in the Dixit-Stiglitz
price index Pt
70
Equilibrium condition
A log-linear approximation around a deterministic
steady state yields the IS schedule
gcrowding out term due to fiscal shock
71
Effect on fiscal shock on C
Equivalent to the fiscal shock
72
New Keynesian Phillips Curve
Deviation of natural output due to supply shock
Demand determined output deviations
Taylor Rule
Inflation target
73
Output gap
3-EQUATION EQUILIBRIUM SYSTEM
Proportion of firm that prefix prices
IS-curve involves an exogenous disturbance term
74
INTEREST RULE AND PRICE STABILITY
THE NATURAL RATE OF INTEREST
75
Percentage deviation of the natural rate of
interest from its steady-state value
76
Inflation targeting at low, positive, inflation
Composite disturbances
77
(No Transcript)
78
Evolution of money supply
The only exogenous variables in the system are
the natural interest rate
nominal rate consistent with inflation target
79
MICROFOUNDED CAGAN-SARGENT PRICE LEVEL
DETERMINATION UNDER MONETARY TARGETING
FLEX-PRICE, COMPLETE-MARKETS MODEL
80
Complete Markets
Value of portfolio with payoff D
price kernel
81
Interest coefficient for riskless asset
Riskless Portfolio
82
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
83
No Ponzi Games
For all states in t1
For all t, to prevent infinite c
The equivalent terminal condition
84
Lagrangian
85
Transversality condition
Flow budget constraint
86
Market Equilibrium
Market solution for the transfers T
87
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
88
We study equilibrium around a zero-shock steady
state
89
Derive the LM Curve
From the FOC
At the steady state
90
Separable utility
Define
The hat variables are proportional deviations
from the steady state variables.
91
Similar to Cagans semi-elasticity of money
demand
92
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
93
Add the identity
We look for solution given exogenous shocks
94
Solution of the system
This is a linear first-order stochastic
difference equation ,where,
Exogenous disturbance (composite of all shocks)
95
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.
96
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
97
Steady state (around
)
fix
98
PRICE LEVEL IS INDETERMINATE
Real balances are unique
Future expected inflation is unique
Is unique
But, neither
Can uniquely be determined!
99
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!
100
II. Wicksellian Rules interest rate is a
function of endogenous variables (feedback rule)
Vcontrol error of CB
Fiscal Policy
Exogenous
Endogenous
101
Steady State
Log-linearize
102
We can find two processes
Add the identity
103
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.
104
FEATURES

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

105
Additionally

• If

Price level instability can be reduced by raising
, an automatic response.
106
Note, also that

• Big
• Small

, reduces the need for accurate observation of
, almost complete peg of interest rate
107
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
108
III. TAYLOR (feedback) RULE

• Steady state

Assume
109
Taylor principle
Is predetermined
110
Transitory fluctuations in
Create transitory fluctuations in
Permanent shifts in the price level P.
111
Optimizing models with nominal rigidities

• Chapter 3

112
(No Transcript)
113
(No Transcript)
114
First Order Conditions
115
Firms Optimization
Nominal
Real
116
Natural Level of Output
117
Log-linearization of real mc
Partial-equilibrium relationship?
118
where
Elasticity of marginal product of labor wrt output
Elasticity of wage demands, wrt to output
holding marginal utility of income constant
119
ONE-PERIOD NOMINAL RIDIGITY
Same as before, except for
Y need not be equal to the natural y
120
A Neo-Wicksellian Framework
THE IS
Ct consumption aggregate

gross rate of increase in the Dixit-Stiglitz
price index Pt
121
Equilibrium condition
A log-linear approximation around a deterministic
steady state yields the IS schedule
gcrowding out term due to fiscal shock
122
Effect on fiscal shock on C
Equivalent to the fiscal shock
123
New Keynesian Phillips Curve
Deviation of natural output due to supply shock
Demand determined output deviations
Taylor Rule
Inflation target
124
Output gap
3-EQUATION EQUILIBRIUM SYSTEM
Proportion of firm that prefix prices
IS-curve involves an exogenous disturbance term
125
INTEREST RULE AND PRICE STABILITY
THE NATURAL RATE OF INTEREST
126
Percentage deviation of the natural rate of
interest from its steady-state value
127
Inflation targeting at low, positive, inflation
Composite disturbances
128
(No Transcript)
129
Evolution of money supply
The only exogenous variables in the system are
the natural interest rate
nominal rate consistent with inflation target