Utility-Based Loss function
A Utility Based Welfare Criterion
Expected utility is
, associated with a given equilibrium where
Then using the closed economy condition C=Y and the production function
The index Y is
It follows that the elasticity of with respect to y is given by , and the elasticity of real marginal cost s with respect to Y is given by
The Efficient Level of Output Gap
The steady state level of output is given by
summarizes the overall distortion in the steady state output level as a result of both taxation and market power:
(1) is the Woodford-Rothemberg sales subsidy (financed by lump sum taxes) that aims at neutralizing the monopolistic competition inefficiency in the steady state; and
(2) is the mark up as a result of producers’ market power.
Efficient (zero mark up) output is given by
is a decreasing function of , equal to one when .
This allows the approximation
Define the efficient level of the output gap.
Quadratic Taylor series approximation for U
Derivation of (2) and (3)
Approximate . Then,
Using we get an approximation for the term: :
, where, is the mean value of across all differentiated goods, and is the corresponding variance.
Finally, going back to U, we get:
This is the end of the derivation of (2) and (3).
Price Variance in the Utility Criterion
Equation (3) is rewritten
Where, the term originates from
And the term originates from
The Dixit-Stigliz preferences over differentiated goods imply
Inflation and Relative Price Distortions
Recall the split between the goods prices that are fully flexible and the good prices that are fixed in advance. The aggregate supply is
In this model
Substituting into U, yields
Aggregate output and inflation variations matter for welfare (it is the output gap and the unexpected inflation) and the relative weight that should be placed upon the two objects is related to slope of the aggregate supply.
Derivation of a Quadratic Loss Function (Woodford (2003) Chapter 6); which is an approximation to the level of expected utility of the representative household.
We evaluate under alternative policies. A log-linear equilibrium of the endogenous variables for a policy is:
The coefficient vector () depend on policy. Taylor series approximation to exact equilibrium responses are evaluated around the point ; where, .
A similar Taylor-series approximation of the Utility function yields:
, and all partial derivatives are evaluated at .
Taking the expected value of the Utility index, and using the fact that
Now consider whether the linear approximation
to the equilibrium fluctuations in x under the given policy rule can be validly used to compute a second-order approximation to the expected utility. Substituting
for in (1) yields the estimate
This means that the criterion U(1) cannot be expected to correctly rank alternative policies among those implying the same average outcome , unless one is sure that
In this case U(1) provides a valid second-order approximation of E(U). But
is just the condition of optimality of as an outcome in the absence of disturbances. It might be thought of as a proper condition for small disturbances around the steady state.
Bt = bond holdings at the end of date t (denominated in the domestic currency)
Bt* = bond holdings at the end of date t (denominated in the foreign currency)
Mt = money holdings at the end of date t
Pt = aggregate domestic price level
Ct = consumption index
ht(j) = supply of labor of type j by the representative individual
wt(j) = wage rate of labor of type j
it* = world interest rate
Goods Market Global Equilibrium Condition:
Ignore the money balance term and write the objective function as
Express the relative quantities demanded of the differentiated good as a function of their relative price to rewrite the utility flow in terms of price dispersion factor,
A linear-Quadratic Approximate Problem
First show that there exist an optimal purely deterministic steady state where
. Next, compute a linear-quadratic problem. Begin with computing a Taylor series approximation to the welfare measure, by expanding around the steady state:
Now take a second order approximation of
and show that
there is non-zero linear term in the quadratic form. Rotemberg and Woodford avoid this problem by assuming and output subsidy to neutralize the monopolistic competition distortion.
(In the presence of distortion, do a second order approximation of the aggregate supply and substitute for the linear term. )
A Simple optimization
The first order condition is:
The optimized output gap is:
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