Linearization around this particular (optimal) steady state is extremely convenient, since our approximate measure of W takes an especially simple form in that case. In particular, in this case our second-order approximation for W depends only upon a first-order approximation to the equilibrium responses of inflation and output to the exogenous shocks. This means that we can solve a log-linear approximation to the model’s equilibrium conditions using standard linear methods, as sketched in the previous subsection, and obtain an approximation to W that neglects only terms of third order and higher in the deviations from the steady state. This result depends upon the absence of any first-order contribution to our welfare measure from changes in the average level of output under alternative rules (as a result of the optimality of the level Y relative to which we consider deviations); for if W contained a term of first order in the average level of output, then second-order terms in the equations determining output would matter for a second-order approximation to W. If quick loans cash can fix your financial problems right now, then you should get that money from a lender you can trust without waiting any further. We will be glad to help you by offering a loan for as long as you need. Come and find our offer at Link and see how affordable those loans are.
In fact, in the calculations reported here, we furthermore assume that the tax rate r actually varies depending on the monetary policy rule, so as to ensure that E[\ogYt) = log Y in any event. This allows us to obtain a measure of the deadweight loss associated with price-level instability that abstracts from any effects of alternative monetary policies upon the long-run average level of output. While many analyses of the welfare effects of monetary policy have emphasized exactly such effects, we think there is good reason to abstract from them. Our primary reason is that there exist other policy instruments, such as the general level of and structure of taxation, which allow the government to influence the average level of output while, at the same time, being much less well-suited for the achievement of stabilization objectives, since they cannot be adjusted quickly and precisely in response to shocks.
It thus makes sense to assume that, in an optimal policy regime, the other instruments are chosen to achieve the desired average level of output for a given monetary policy, while the monetary policy rule is chosen to minimize those contributions to deadweight loss that are independent of the economy’s average level of output. We do this by choosing the monetary policy rule that maximizes W under the assumption that the other instruments are adjusted in the manner stated in response to any change in the monetary policy rule.
Abstracting from these effects also has the advantage of making our results independent of a feature of our model about which we are especially uncomfortable, namely its predictions about the effects of sustained inflation upon the long-run level of output. One might think that sustained inflation should result in adaptations that eliminate any effects of the average inflation rate upon average output. One such adaptation would be price commitments that specify a constant rate of price increase of n* between the occasions upon which the commitments are modified, as assumed in Yun (1996). With this modification, our model would come to satisfy the “natural rate hypothesis”. In the modified model, the correct second-order approximation to W would be exactly the one that we report here, but then it would apply to small fluctuations in the rate of inflation around any average value 7Г*.