Approximating the flow of Period Utility with a Sec- ond Order Taylor Approximation

9.1 Approximating the flow of Period Utility with a Sec- ond Order Taylor Approximation

As noted in the previous section, the optimisation by the central bank requires the use of a welfare measure, which this section will be devoted to deriving. The analysis is heavily reliant on that in Woodford (2003) with the modification to the model as outlined in earlier sections.

Additionally, the form of the welfare function will illustrate the economic costs that are introduced with the addition of noise to the information set of the agents.

Consider that the utility flow of the household in every period is:

t ) 1−δ

U t = − M t L t N t where log(N t )˜N (0, σ n 2 ) log(M t )˜N (0, σ 2 m ) 1−δ

Here C t is the Dixit-Stiglitz consumption aggregate, which is equal to the overall level of output Y t . The Taylor approximation of this expression is around the equilibrium where the shocks equal to their expected values. Terms without subscripts represent equilibrium values.

MY γ (N t −N)− γ 1 γ Y (M t −M)+

+ 1−δ Y − 1 γ

+ −δ −Y γ −1 1 0 Y t (i) − Y (i) di − Y γ −1 (N t − N ) (M t −M)+

2 − (γ − 1) Y γ −2 0 (Y t (i) − Y (i)) di + + 1−δθ Y −1−δ − (γ − 1) Y γ −2 1 θ R 0 Y 1 t (i) − Y (i) 0 Y t (j) − Y (j) djdi+

The first term of the above expansion is just the steady state level of utility, while the next three terms are linear deviations of both shocks and individual outputs from the steady state. The way that this economy has been set-up means that in the absence of shocks the economy settles to an equilibrium level of output that is inefficiently small, which provides incentives for a welfare max- imizing central banker to attempt to push output beyond the steady state level for protracted periods of time. In the welfare function the degree of the dis- crepancy is essentially caused by the fact that the first derivative of the utility function evaluated at the steady state is non-zero. Essentially, the marginal benefits of an additional unit of consumption outweigh the labour costs as given by the disutility of producing that unit of output. It is caused by the constraint that firms are monopolistically competitive, hence enjoy a degree of market power, which allows them to restrict the level of output produced. This ineffi- ciency vanishes in the limiting case where θ− > ∞. In more realistic settings, the inefficiency remains. Several authors have proposed a variety of methods which could allow to deal with this issue. One is to imagine a fiscal policymaker which imposes a proportional sales subsidy, which restores the equilibrium output to the socially optimal one. In this case, stabilisation will be the sole objective of monetary policy. Alternatively, following Woodford it is possible to argue that the distortion, if it exists is in any case small, so can be set to zero without im- posing a large loss in the accuracy of the approximation of the welfare function. This is the approach taken here due to it’s expositional simplicity. Essentially, this amounts to a parametric restriction on the policy rule of the central bank to exclude any constant terms. In this case, again the best symmetric rule would

be solely concerned with stabilisation of output and inflation rate around the steady state.

The second term on the third line is the interaction term between the two sources of shock, which in expectation will be zero due to the assumption of independence between the supply shock (M t ) and the demand shock (N t ). The quadratic term in the individual outputs of the firms is a measure of the dis- persion of individual outputs from their steady state values. As this is caused by the formation of atoms of firms which are unable to change their prices ev- ery period leading to asymmetries in the production of different goods. Due to symmetry the most efficient production plan involves all the firms producing identical quantities equalised at marginal costs. Due to nominal frictions the asymmetry in the production of firms causes a reduction in welfare. The term on the fourth line of the above equation refers to the covariance between one firm’s output with that of all the others. Since the firms in this economy are all directly competing against all the others so this term is non-zero as any adjustment by one firm will cause the outputs of others to change. The terms on lines 5 and 6 are simply the covariance between the level of production and the effect of the shocks. These terms similarly have a non-negligible effect on welfare as the level of output has already been shown to depend directly on the magnitude of shocks, so these are not independent from the each other.

Also, note that the individual outputs are integrated around the steady state values for output. Since firms are located on the unit interval, their individual steady state outputs are just the same as the steady state level of output in the economy. This is shown below as well as the calculation of the value of the total steady state level of labour.

as due to symmetry in the steady state Y (i) = Y (j)∀i, j then

Y θ = Y (..) θ

Y (i)di

The above function represents a second order approximation to the utility function and is therefore only accurate in regions near to the steady state. For small shock variances this approximation will work reasonably well, though the Calvo constraint implies that there will always remain a small proportion of prices which depart arbitrarily far from the current price level due to not having the opportunity to adjust.

In steady state when N,M =1, total output can be shown to equal Y = ( θ −1

1 θ ) δ +γ−1 < 1 and Y (i) = Y (j)∀i, j

using the fact that Y t Y =1+ˆ y t + 1 2 y ˆ 2 t Y ... where ˆ y t = log( t Y )

N =1+n t + 1 2 N 2 t n t ... where n t = log( N )

Y t (i) 2 Y (i)

Y (i) =1+ˆ y t (i) + 1 2 y ˆ t (i) ... where ˆ y t (i) = log( t Y t )

M =1+m t + 1 2 m 2 t M ... where m t = log( t M )

The deviations of individual outputs for firms as expressed in the last equa- tion can be re-written in the following way:

Y t (i) − Y (i) di

(Y t (i) − Y t ) + (Y t − Y (i)) di

Now, using the fact that Y (i) = Y and splitting the integral into the con- stituent parts gives:

(Y t (i) − Y t ) di + Y t −Y

The above equation allows to differentiate between aggregate shifts in output and the dispersion of individual firms’ production plans from the current level of output. These two are both sources of inefficiency, however the causes of these losses vary. The losses that occur due to the dispersion of outputs around the current level are caused by price variations, which in turn are the consequence of the Calvo pricing constraint. The variability of the aggregate level of output is caused by the two types of shock which prevail in the economy. By separating the two sources of loss in this way allows the inflationary and output gap objec- tives to be pinned down. Using this expansion means that the period-by-period approximation to welfare comes to:

+ − Y − γ t − Y m t +Y 1−δ y ˆ t + (109) 1−δ

i (ˆ y t (i))+

+Y 1−δ +Y γ t y ˆ t −Y γ m t y ˆ t + O |..| 3

Here, the expectation and variance are taken over the distribution of outputs by all firms in the economy. As mentioned above the expansion of individual outputs is around the current level of output Y t . This means that the expecta- tion of the integral of the output gaps is approximately 0, apart from any small deviations that could arise through the non-linearity of the aggregator. Nev- ertheless there is no reason for these to have systematic deviations upwards or downwards and so in expectation these terms have no effect on period welfare.

The second line of the above expansion also contains terms for the quadratic deviation of the shock from it’s steady state. This occurs due to the fact that the approximation has been recast into the logarithms of the variable rather than the actual value for the shock. This has the effect that, while there are no quadratic shock terms in the initial expansion, they reappear as the second order approximation to the linear shock deviation in the final expression. Es- sentially, by converting all variables into logarithms, leads to a rescaling, an approximation to which causes the second order shock terms to reappear. This The second line of the above expansion also contains terms for the quadratic deviation of the shock from it’s steady state. This occurs due to the fact that the approximation has been recast into the logarithms of the variable rather than the actual value for the shock. This has the effect that, while there are no quadratic shock terms in the initial expansion, they reappear as the second order approximation to the linear shock deviation in the final expression. Es- sentially, by converting all variables into logarithms, leads to a rescaling, an approximation to which causes the second order shock terms to reappear. This

Loss t =E t

This means that the loss function can be rewritten more completely as:

− (γ − 1) Y

i (ˆ y t (i))+ 

γ m y (111)

3   +t.i.s. + O |..| 

Here t.i.s. refers to terms which are independent of the structure of the economy (in the sense of the levels of noise in relation to the actual shocks) as well as any policy actions. These contain the quadratic shock deviations as well as the constant welfare level in addition to the linear terms.

The above expression for the welfare losses based on the representative con- sumer utility function shows that the terms relevant to changing the structure of the economy will be those which are non-constant as the informational regime is adjusted. Therefore, the function above can be adjusted to give the following:

− (γ − 1) Y

i (ˆ y t (i))+

Loss t =E t β 

+t.i.s. + O |..| 3

Essentially, the terms that matter for welfare evaluation are the variance of aggregate output and the variance of output between firms in the economy, which is determined by the degree of price dispersion. As the demand curve given by Dixit-Stiglitz preferences is:

log Y t (i) = log Y t − θ(log P t (i) − log P t )

(114) In any given period the degree of price dispersion can be related to the series

var (log Y t (i)) = θ 2 var (log P t (i))

of past rates of inflation. This is because in any period only a portion of firms given by α are free to adjust their prices. This means that after each period there is an atom of firms who kept their prices at the ”old” level and did not adjust in response to the shock. With every period a single atom will remain as the general price level moves in one direction or the other. Following Woodford (2003) the degree of price dispersion can be shown to be:

var (p t (i)) = ∆ t

Then iterating backwards gives:

This is then combined into the infinite sum of discounted expected utility. For this reason, the term for current inflation enters into every periods welfare flow as follows:

E i (W π )= [(Σ ∞ β π t ) + (Σ ∞ β +1 s =0 +s i =0 π t −(i+1) )] (118) 1−α 1 − αβ

Here only the first term in square brackets denotes future levels of infla- tion and therefore affects the future flow of utility, while the second term is a constant, which reduces welfare based on pre-existing price dispersion before entry into the period. As it is not affected by actions in any period, it cannot

be assigned as a flow utility in any period, hence will be excluded from what immediately follows.

Rewriting the Welfare function gives:

X ∞   − θ 1−α 1−αβ

t +s −   Loss t =E 0 β

t +s y ˆ t +s +Y γ m t +s y ˆ t +s  (119)

In this highly simplified form the objectives of the central bank are clearly reduced to minimising the deviations of inflation and output gap. In addition, the objective function contains terms for the covariance between the output gap and the shock itself. This is partly due to the construction of the shock process. As these cause fluctuations to arise in the utility function, a household would like these deviations to be mitigated by the movements in output. This leads to the utility function to be supplemented with these terms.