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.
General Principles 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