Calibrating the Welfare Losses using numerical values

9.2 Calibrating the Welfare Losses using numerical values

9.2.1 Passive policy case

Firstly, consider a situation with only a demand shock. This means that the system consists of a single filtering problem, that is finding an estimate for the size of the demand shock in this period. Using the values above and substitut- ing into the welfare function allows the to estimate the welfare current value associated with this set-up. The expectation is taken in period zero, i.e. this is not conditional on the outcome of the current period.

The above graph is plotted to show the percentage of steady state welfare that the loss function takes under current settings for a variety of values of the noise levels. Higher levels of noise correspond to a larger variance on the forecast error terms and it is in terms of this variance that the results are shown. As can

be seen from the graph, as the levels of noise approach the magnitudes equivalent to the variance of the shock, then the cost in terms of steady state consumption rises by approximately 0.2%. While on the scale of steady state welfare this may appear to be an insignificant, this should be be seen as amplified by the scale of the entire economy when compared to the costs of information gathering.

A broad feature of the result is that greater information uncertainty leads to larger welfare losses. This is caused both by the fact that higher uncertainty leads to lesser responsiveness to indicators since these are less informative. Fur- thermore, greater incidence of error lead to larger deviations from the efficient allocation due to the aggregation of decisions made on seeing errors in the fun- damental variables.

Moving along the axes of the graph, the losses from a lack of information have a similar shape, in fact they are almost identical, save for the fact that along the inflation axis, greater noise results in the inflation rates causes the loss curve to rise much more steeply along this axis and flattens out much earlier than a similar interval along the axis of output noise. This means that for low levels of noise, at least, an improvement in inflation data would have a greater effect on welfare than a similar one for the output gap. This is caused by the fact that as inflation only reacts to a demand shock through a response to the change in the output gap, a greater precision in this variable has a more than one for one effect on the precision of backing out the required output gap.

The above result can also be recomputed for the case where the economy is affected by both the demand and the supply shock as detailed in previous sections. Using the same values for exogenous variables as those used in the above computation leads to the result as shown on the graph below.

The most striking feature of the above result is that in this case only im- provements in the gathering of output data lead to improvements in welfare. Data on inflation becomes nearly irrelevant for decisions and welfare is driven mainly by the level of noise in output gap data. The cause of this is that re- gardless of the precision with which inflation data is gathered, in this setting inflation is driven by two sources of shock as well as two sources of noise. This is caused entirely by the additional noise that a supply shock imposes on inflation data and the result can be understood in terms of the case without a supply shock. Consider the responsiveness of the inflation rate to fluctuations in the output gap as given by the fixed point of the equation:

Letting σ 2 µ = 0 returns this expression to the case discussed previously as one without a supply shock. Note also that in this expression, the effect of Letting σ 2 µ = 0 returns this expression to the case discussed previously as one without a supply shock. Note also that in this expression, the effect of

The essential point here is that, once the inflation rate is subjected to the influence of more stochastic terms, this variable declines in it’s usefulness as a forecast. The most important finding is that the decline is extremely dramatic, so as to render any improvements in that variable’s accuracy null once the supply shock is taken into account. This applies for a specification of the supply shock of a similar magnitude to the demand shock. It is possible to envisage a situation where the variability of the supply shock is so small, that the case would return to a situation shown previously where the supply shock was excluded.

Similarly to the case considered previously, the loss due to informational uncertainty is approximately 0.2% of steady state consumption. Due to there being an extra source of shock in this case, the baseline losses are greater relative to the first case.

9.2.2 Welfare Calculations under Active Policy Regimes

Having looked at the situation where the central bank did not respond to shocks it is possible to turn attention to the case of active policy. Firstly, consider the case where the only stochastic disturbance prevalent in the economy is due to demand shocks. In an environment with perfect information the central bank would be able to completely offset the shock and perfectly stabilise the economy as described earlier. Once some informational noise is allowed to affect the system, the policy followed by the monetary authority would follow the pattern given by:

(121) In this way the central bank uses all available information to offset the shock.

i t = δE t |I t n t

The only part of the shock which remains is given by n t −E t |I t n t , on which The only part of the shock which remains is given by n t −E t |I t n t , on which

This means that the presence of policy orthogonalises variations in inflation and the output gap, so now welfare is entirely dependent on the error in esti- mating the size of the output gap. The effect of this on welfare is illustrated on the graph below:

As noted earlier, it is only improvements in the quality of output gap data that cause improvements in welfare. Due to a loss of one source of informa- tion and such a large burden placed on the output gap variable leads welfare to degrade in a much more rapid fashion. At the point where the noise in the information variable reaches the same magnitude as the shock, the loss due to policy errors amounts to approximately 1-1.3% of steady state consumption. This contrasts from the previous case by being much larger than the case which ignored policy, however this should be seen in the light of the fact that the in- formational set from which inference is drawn about the shock as being reduced from 2 to just 1 dimension. Essentially, this is not a fair comparison, rather a consequence of the set-up employed here.

When the possibility of supply shock is introduced on top of the already mentioned demand shocks, a similar picture emerges. As, before, by wiping out When the possibility of supply shock is introduced on top of the already mentioned demand shocks, a similar picture emerges. As, before, by wiping out

The central bank sets interest rates according to the rule shown previously, i.e.

(122) The private sector takes this into account, which means that the value of

i t =φ 1 E t |I t n t +φ 2 E t |I t m t

φ 2 leads for a new relationship between the output gap and the inflation rate, this time led by the reaction function of the central bank. As before φ 1 is set equal to δ to offset the predictable part of the demand shock completely, while φ 2 equals to a value prescribed by the welfare function as an optimal trade-off between inflation and the output gap. The effects on the loss function are shown in the graph below: