A New Keynesian Model with Noisy
Observations of Aggregate Variables
Dennis Tatarkov
September 21, 2011
Abstract
1
Introduction
A feature of the real world which is often overlooked in economic modeling is the presence of uncertainty regarding the true economic variables. The aim of this paper is to incorporate elements of this uncertainty through additive noise into a DSGE model that has become standard in the literature. This is done through re-examination of the first order conditions under partial information. The results are then used to assess the welfare costs which are imposed on society by a lack of access to complete and accurate information the economy.
The effects of this uncertainty have been recognized by policymakers at the Bank of England, who in their output projections have placed error bars onto past observations of GDP. Retrospective revisions of GDP data are also frequent, some of these occur after several years of the release of the initial estimate. Similarly, it is also widely recognized that measures of inflation have inherent problems, such as an inability to take into account the substitution effect or an insufficient adjustment for the quality of manufactured goods. Unlike the data for GDP, inflation data is never revised. However, due to substitution in the consumption basket, introduction of new goods and changes in the quality of goods already being consumed, the measured rate of inflation is never a perfectly accurate measure of the price changes in the basket of consumption goods and it is reasonable to assume that some noise is present in this measure. This thesis focuses on the effects of this uncertainty in a standard DSGE model, and leading from this onto the effects that uncertainty has on welfare as measured by the level of utility of a representative household.
In terms of the DSGE model, the measure of the inflation rate is relevant to producers to determine the pricing strategy that they follow since in the im-perfectly competitive setting that is considered here, the relevant price statistic is the ratio of own price to the general price level. Since a Dixit-Stiglitz con-sumption aggregate is used, the overall price level is the relevant metric for competitor prices. Therefore, the noise in the measure of inflation could be seen as the effect of not knowing how the economy-wide inflation measure translates into a relevant metric of competitor prices.
The way that this paper is organized is as follows. The first section provides a literature review and places this model among the recent developments in macroeconomics, the next section details the model and calculates welfare losses as well as providing some numerical estimates. The final section concludes.
2
Literature Review
structure has been the Lucas islands model (1972, 1973, 1975). The model is based on an idea that the economy consists of some number of islands, which in one formulation is a continuum between 0 and 1. These Islands are taken to be separate, which allows for the markets to be both physically and informa-tionally separate. There is only one consumption good, which is produced and traded competitively on each island. In this setting, each island is subject to a mean zero idiosyncratic shock, while at the same time the economy experiences a shock to the level of the overall money supply. After each period individuals are randomly reassigned to another island. Since the cash balances that they hold depreciate at the rate of inflation in the overall economy, this means that despite being physically separate from the rest of the economy, their decisions are affected by the aggregate price level that prevails in the set of islands which comprise the economy. As each individual is allocated to another island at ran-dom, in expectation all goods are perfectly substitutable and are defined as a good belonging to a particular island.
This model focuses on the aggregate supply side of the economy and is a attempt at an explanation behind the apparent failure of monetary neutrality in the short run. The model can be summarized using a few key equations. The essential idea is that rational agents are placed into a setting where they are concerned with relative prices, yet unable to distinguish relative from general price movements. At the simplest level, Lucas assumes that the supply yt in
each marketi depends on the ratio of the price in that market to the general price level as:
yt(i) =κ(pt(i)−E(pt|It)) (1)
Here, p denotes the (log of) the price level and y denotes output with indi-vidual market price being given by terms with the market index i as a functional definition. To complete the model, Lucas assumed that the general price level is unobserved and follows a stochastic path, the distribution of which is known to all participants. Furthermore, there is an additional shock process on the individual prices follows the following:
pt(i) =pt+z (2)
E(pt|It) =E(pt|pt(i)) (3)
=κpt(i)
Here, κis the signal-to-noise ratio which has the form of:
κ= σ
2
z
σ2
z+σ2τ
(4)
The variance of the general price level is given byσ2τwhich stands as a magni-tude of the size of the general shock affecting the economy. In this case,σ2zis the idiosyncratic shock to the individual firm’s markets. Putting all this together and summing across the distribution of firms acrossi allows the derivation of the well-known Lucas supply function:
ˆ
yt=κκpt (5)
In keeping with notation later, the hat over the output term signifies a gap from the competitive allocation. This states that in response to a price increase in the local market, the firm will only raise price a portion of the way of that increase. The structure of theκmultiplier is such that if all shocks are due to the aggregate andσ2
zis large relative toσ2τ,then most of the shock will be taken
up by prices and vice versa.
This model is one of the first which uses incomplete information as a cause behind incomplete adjustment, although the particular focus of the problem is not framed in terms of informational errors, but rather a lack of it altogether. Following it’s publication this model attracted criticism for it’s inability to ex-plain deviations from the flexible price equilibrium which last longer than the delay in publishing the economy-wide aggregate data. As this is released with a lag of only a month to a quarter, this model fails in explaining deviations with a periodicity of a year or more.
relevant are the assumptions of monopolistic competition between firms which allows price-setting to take place as well as a limit on the agents’ ability to process information. The general result is that since second-order beliefs are more sticky, hence leading to greater internal model persistence. It is found that this mechanism can in some cases approximate that of the more standard Calvo contracts.
Similarly, Aoki (2007) looks at the effects of uncertainty about the bank’s inflation target as a key informational handicap for economic agents. In the presence of stochastic disturbances, private agents cannot determine whether a shift in the real rate is caused by a shock, or whether that signifies a change in the bank’s target rate, both of which are unobservable. Without a credible commitment mechanism, a situation arises where shocks are imperfectly ob-served, and the variation in the natural rate is only partially attributed to these shocks. This imperfect credibility leads to incomplete adjustment in response to shocks in a similar vein to the Lucas model, however rather than depending on informational imperfections the key mechanism here is learning. Learning on behalf of the private sector which leads to greater credibility of the infla-tion target rate, which means that some amount of time is needed before the target becomes more credible, diminishing the volatility of transmitted shocks. Both these models continue to employ information as the driving mechanism for incomplete adjustment.
Instead of letting the informational imperfections play a key role in breaking monetary neutrality, it has become more common to assume the presence of nominal rigidities, which force producers to set prices away from the flexible-price equilibrium. This has been motivated firstly by the observation that flexible-prices are never set in a continuous manner, with revisions only occurring at regular intervals. Furthermore, it is commonly observed that wages are set at intervals of 1-2 years. To provide rigorous micro-foundations, this behaviour is usually rationalised by taking into account the existence of small menu costs, which nevertheless induce agents to only change prices infrequently. This strand has separated into dependent and time-dependent pricing rules. While a state-dependent pricing rule is obviously superior in terms of it’s micro-foundations, the complexity of such rules prohibits a model from achieving parsimony. This stems from the fact that state-dependent pricing rules result in a strong history dependence, with each successive pricing decision becoming dependent on a string of past realisations.
a constant hazard rate is appropriate for a situation where inflation is largely stable, this would not be the case for countries which experience periods of very high and volatile inflation. In this case, price changes would be carried out with greater frequency, i.e. the parameter for the hazard rate could no longer be approximated to be constant.
The basic model itself owes much of it’s structure to the model exposited in Chapter 3 of Woodford (2003), while assuming some particular, though com-mon functional forms for simplicity of exposition. Solution is then achieved by log-linear approximations around the steady state of the economy. Woodford presents a general model of an economy with a representative household, which is subjected to a vector of stochastic disturbances. Using the utility function of the representative consumer, the author is able to derive a second order ap-proximation to the welfare function of the economy. This welfare function is outwardly similar to the more standard ad hoc assumption of a loss function (Clarida, Gali, Gertler, 1999) for the central bank dependent on the square of the deviations of the level of output and the inflation rate. The presence of the squared output gap term results from the curvature of the utility function, with greater variance leading to reductions in the average level of utility. In-flation penalizes the welfare function due to distortions in the price level. The derivation of the welfare function shows this result to be highly dependent on the time-dependent pricing rule that the model employs. The essential intuition states that higher rates of inflation lead to larger gaps in the pricing between firms who are able to adjust in the current period and those that set their prices in one of the previous periods. This causes some resource mis-allocation as prices set by firms in the past period no longer reflect the marginal costs of producing those goods. This micro-foundation for the costs of inflation depends on the form of the pricing rigidity. A situation could be envisaged where if all firms adjust their prices simultaneously, even with some delay, then inflation causes no utility loss. Furthermore, this derivations for the costs of inflation draws some criticism from the observation that higher inflation is associated with greater volatility in the inflation. As this makes long term contracts (such as employment contracts) more risky, this imposes costs on individuals. This source of disutility is not adequately reflected in the standard DSGE model.
In making the welfare derivation Woodford goes on to assume that the econ-omy’s inefficiency tends to zero, so that the central bank does not have an incentive to try to push output beyond it’s natural rate in the sense of Gordon and Barro model. In another paper Benigno and Woodford (2005) explore an approximation which does not rely on either making an assumption regarding the size of the distortion, or an assumption that such a distortion is offset by an appropriate fiscal policy. They show that this results in a modification of the weights attributed to inflation and output in the welfare function.
with imperfect data. The paper, which has provided much of the motivation be-hind the interest in informational imperfections has been a paper by Orphanides (1997). In this paper, the author looks at a specification of the popular Taylor rule using contemporaneous data estimates and finds that this differs from the rule that emerges from data of varying vintages. While this paper does not di-rectly test whether initial indicator estimates are subject to noise and the degree of error that this entails, the paper does indicate a large discrepancy between the policy interest rate which is set according to contemporaneous and revised data.
It can be argued that from the outset, the baseline model considered here is insufficiently detailed to be able to handle information uncertainty, as the assumption of a representative consumer rules out any heterogeneity in the response of the private sector. Aoki (2006b) explores the micro-foundations be-hind informational constraints. The author considers a model of islands, where information is dispersed among agents in the economy. The essential structure means that each agent is only able to observe the activity of those islands that are nearest to him rather than aggregate information. This leads to a situation where none of the agents have reliable information about the true state of the economy. It is then shown that this model behaves in an identical manner to a model with a representative consumer, where the consumer has full information while the monetary authority only has only partial information about the econ-omy. This can be seen as an argument against the assumption used here about the symmetry of informational noise, however the type of uncertainty considered in this paper does not correspond to the idea of errors in aggregate data, which is relevant for economy-wide decision making. On the other hand, following this model, shows that the effects of idiosyncratic uncertainty are likely to be small and the assumption of a representative consumer is warranted, which is inherited from the full information baseline model.
is not considered.
Additionally, McCallum (2001) has written on the implications of output gap mis-measurement if the Central bank follows a policy rule, but instead of using the true economic output gap, defined as the difference between the current level of output and the level of output that prevails in steady state, the bank has an incorrect measure. This possibility translates to model uncertainty as the optimal policy in any particular case depends on the specification of the model economy, it is necessary to have some general policy prescriptions which would be applicable to a large set of candidate models. For this reason, McCallum argues against a strong response to output data, partially grounding his argument in the observed uncertainty of the output gap.
With this, I have chosen to simplify the set-up as much as possible, while retaining the key elements of the DSGE model. This means that I will not be concentrating on the possible gains from commitment and will rather follow a simpler discretionary policy framework. Similarly, the shock structure will also be simplified from that adopted in Clarida, Gali and Gertler as I will only focus on one-period transitory shocks. With most of the set-up following the lines of Woodford chapter 2 model, this work will take particular functional forms for ease of exposition.
3
Households
The model is based around a set of standard assumptions and this section details the description of the household. The individual household’s utility, U is given by:
U =E0Σ∞s=0βsUt+s (6)
where
Ut=Nt(
Ct1−δ
1−δ − Lγt
γ ); (7)
Ct= (
Z 1
0
Ct(i)
θ−1
θ di)θ−θ1;L
t=
Z 1
0
Lt(i)di (8)
Here, C denotes consumption and L denotes labour input, with functional arguments referring to the quantity for a given good. Consumption is modeled using a Dixit-Stiglitz aggregator where consumption goods are located on the continuum between 0 and 1. Each of the goods is produced by a separate firm, which employs labour from the household. The overall labour disutility of the household is determined by a simple sum of the labour supplied to each firm.
The form of the utility function implies that it is additively separable over time, which serves to rule out some realistic features such as habit formation. In the event that habit formation was included in the description, the house-hold optimization would yield a more persistent pattern for consumption, which would have served as an integral mechanism for generating persistence. At the same time, there is a preference shock modeled byNt, which has a log-normal
distribution around zero with a variance of σ2n.This term denotes a demand
shock that is introduced into the economy.
The household also faces a budget constraint, where At, is the holding of
government assets with a nominal interest rate set by the central bank at time t.
Maximizing the households utility subject to the constraint results in the following set of first order conditions with a Lagrange multiplierλt:
At:−λt
Nt Ct−δ
Since there is no capital in the model, total output must equal total con-sumption, therefore this can be re-written as:
Yt(i) =Yt
According to this equation, demand for individual goods is proportional to the total level of demand as well as the ratio of the price of the good relative to the overall price level. The parameterθ,determines elasticity of substitution between goods, and as such controls the degree of monopoly power that each individual producer enjoys.
3.1
Euler Equation
Using the demand curves above to substitute forλt, givesλt, =NtCt−δ.Hence
it becomes possible to derive the following Euler equation for this household:
NtCt−δ
correspond to this steady state, let them be denoted by letters with no sub-scripts. Using lower-case letters to denote the log of a variable and using hats to denote gaps, such that for any variable, say X:
Xt−X
X ≃log (Xt)−log (X) =xt−x= ˆxt (21)
A log-linearised relation of this type leads to the following form of the IS curve, which has become standard in the literature.
ˆ
yt=Etyˆt+1−
1
δ(it−Etπt+1) +nt (22)
Essentially this relation states that the current output gap depends entirely on the expectation of the output gap in the next time period, the gap between the current interest rate and the inflation rate expected to prevail in the future and lastly, the current output gap depends on the value of the ”preference” shock Nt, which affects the marginal value of utilities between periods, hence
leads the household to shift consumption between time periods.
One of the shortcomings of the structure of an IS curve such as that presented above is that decisions on consumption in the current period depend almost entirely on expectations of the future. On the scale that time periods represent, which in this case corresponds to quarters, adjustment is instantaneous. Some more recent models incorporate the effects of habits on the form of the above equation, which leads to some dependence on lagged values and smooths the adjustment process for households. This step has been avoided in this case to simplify the overall set-up to focus on the effects of informational problems alone.
3.2
Labour Supply
Using the definition ofλt,=NtCt−δ,the real marginal cost of labour is equated
to the real wage:
Wt
Pt
=Lγt−1Ctδ (23)
Lt(i). Since the production function for firm i is Yt(i) = Lt(i), and Lt =
R1
0 Lt(i)di,then to a first approximation ˆlt= ˆyt. Due to there being no capital
accumulation,Ct=Yt.The real marginal cost of output then depends entirely
on the output gap.
3.2.1 Supply Shocks
The way that the model is formulated allows supply shocks to be introduced with relative ease into the model. Suppose that the per-period utility function has the following form:
Ut=Nt(
Ct1−δ
1−δ − MtLγt
γ ) log (Mt) ˜N 0, σ
2
m
(24)
The only difference from the baseline case is that there is an additional mul-tiplicative term on the dis-utility of labour. As with the time preference shock, suppose that the logarithm of Mt(denoted by mt) has a normal distribution
with a zero mean and a known variance ofσ2
m.
This shock is motivated to take the form of a cost-push shock by creating a gap between the wage rate and the marginal dis-utility of labour. It is important to note, that while in form this can be seen as a real shock, the effect ofMtdoes
not shift the natural rate of output in the same way that a productivity shock or preference shock would. In that sense this appears to be similar in form to those shocks, but must be seen as conceptually different.
This modification leads to the following first order condition for labour:
Lt:NtMtLγt−1−λt
Wt
Pt
= 0 (25)
As before, λt =Ct−δNt.Therefore, the level of the real wage must now be
equal to the following expression:
Wt
Pt
=MtLγt−1Ctδ (26)
This expression shows that the form of the real wage is broadly the same, although there is now a stochastic component to the level of the dis-utility of labour. The baseline model is simply nested within this modification as a model whereσ2
of labour to households now also contains a stochastic component. This can roughly be interpreted as the variation in the real production costs. Therefore, this is a cost-push shock, which means that firms will have an incentive to pass it onto prices, generating a burst of inflation.
4
Steady State
This section characterises the steady state of the economy described here, a point around which all of the log-linear expansions are calculated. As described in the previous sections, all firms are symmetrical, hence in the steady state they all produce output at the same level as one another. Both the inflation rate and the output gap are equal to zero in the steady state defined here. In this symmetric steady state all firms demand identical quantities of labour from the household, face identical costs and set identical prices as a mark-up over the marginal cost determined by their degree of market power. Using this characterisation of the steady state the optimal pricing equation becomes:
N = 1, M= 1
P∗ P = 1 =
θ θ−1
W
P (27)
Using the labour supply equation from above, the real wage rate must satisfy:
W
P =L
γ−1Cδ (28)
Now, as the model features no capital, C = Y. Also, consider the steady state equivalent of L, given the fact that L(i) is a constant.
L=
Z 1
0
L(i)di=L
Z 1
0
di (29)
This then leads to the following result concerning the steady state level of output.
Y = (θ−1
θ )
1
This is the level of output that prevails when the economy is not subject to any shocks. The way that the supply shock has been introduced in the first section means that the shock to the dis-utility of labour does not have an effect on the natural output level of the economy. If this took the form of a productivity shock, then, the stochastic term would enter into the specification of the steady state. As this would shift the natural rate by the same amount as the actual output expansion, this would not enter the Phillips curve additively. Under the cost-push structure of the shock described above, the steady state level of output Y, remains the competitive equilibrium of the economy.
5
Firms
Firms are assumed to optimize their expected level of profit in every period. Consider the two cases: one where firms are free to set prices every period and another case where firms are constrained with a Calvo pricing scheme.
5.1
Flexible Pricing
Firstly, consider a situation where all firms can change their prices in every pe-riod to maximize the discounted present value of profits. It is assumed that firms are located on the unit interval indexed by i and hire labour from a competitive labour market. Capital is absent and labour is the only factor of production. Labour required between firms is not specialized and the index i denoted the demand for labour by firm i. Firms face a technological constraint in the form of the production function:
Yt(i) =Lt(i) (31)
As profit maximization is independent between time-periods, optimization entails period-by period profit maximisation. Given the production function above and the demand function derived from household optimization results in the following profit function for firm i:
Πt= (Pt(i)−Wt)
Yt(i)
Pt
(32)
Yt(i) =Yt
Pt(i)
Pt
−θ
(33)
This expression is optimized with respect to the price set by an individual firm to give the first order condition of the form:
Pt(i) =
θ
θ−1Wt (34)
The first order condition shows that the price set by an individual firm will be held as a constant mark-up over costs, as typified by the wage rate. At the same time, since all firms are assumed to be symmetrical, then this first order condition applies to all firms, and the equilibrium of this economy must feature the same price set by all firms in the economy. Hence, taking the labour supply relationship gives the result that output with competitive firms is always equal to the natural rate.
Pt=
θ
θ−1Wt (35)
Wt
Pt
=θ−1
θ (36)
Hence, taking the labour supply relationship gives the result that output with competitive firms is always equal to the natural rate.
Yt=
θ−1
θ
δ+γ−1 1
(37)
5.2
Calvo Pricing
Now suppose that firms are also subject to a Calvo (1983) time dependent pricing constraint with a constant hazard rate of changing prices: 1−α.Using the demand curves derived previously, the maximization problem for the firm is:
The first order condition is given by:
∂πt(i)
From the first order condition it can be seen that the pricing decision for the firm depends entirely on three variables. These consist of the general price level, the level of demand in the economy and the wage rate, which is the only cost of production for the firm. Since the firm is an imperfect competitor, the price level acts as a measure for the prices set by the other firms, which determines the demand of a particular firm in a continuous fashion. An alternative assumption would have been to treat firms as perfect competitors, although in this case, the demand changes for prices would not have been continuous as the entire market demand would have been split between those firms, which charge the lowest prices. In conjunction with Calvo price setting it would lead to a situation with discontinuous supply curves, and no possibility of price setting by producers.
As mentioned earlier the firm’s decision depends entirely on the general price level, the size of the overall demand in the economy, and the level of the wage rate in the economy. However, wages are determined in a perfectly competitive labour market, with the current wage rate reflecting the real cost of labour as reflected by both the disutility of working and the value of forgone consumption as well as the amount of output that this additional unit of labour produces as determined by the production function. It can be shown that the size of all of these factors can be summarized by the size of the output gap in time t. Using labour supply first order condition:
Wt
Pt
Since no capital accumulation occurs, all production must go towards con-sumption, hence it is easy to show that: Ct = Yt. Also, using the production
function, the overall level of labour supply is a direct integral over all of the firms production level. Hence,
Lt=
At the same time, the Dixit-Stiglitz consumption aggregate in the absence of capital accumulation can be rewritten as:
Yt= (
Then using a second order Taylor approximation around the steady state of the model, denoted by variables without any subscripts this expression becomes:
Y
Note that the steady state level of output that is assumed here implies that all firms are located on the unit interval and due to symmetry means thatYi=Y
for all i. Hence the above expression can be further simplified to give:
Yt−Y
Now, as before using lower case letters to denote the logs of variables and us-ing hats to signify the percentage deviation from steady state gives the followus-ing simplification.
All this implies that the deviation of the wage rate from the steady state can be approximated sufficiently well by an estimate of the output gap in any time period. Using φt to denote the (log) level of marginal costs in period t, such thatφt= log(Wt
Pt).gives the following approximation to the level of costs
ˆ
φt= (δ+γ−1) ˆyt (46)
As such, the implication is that the firm’s decision is dependent only on two variables, namely the general price level and the level of the overall output gap. This can be explained in the following way. The demand for a firm’s good, derived in the previous section depends on the general output gap due to the constancy of the elasticity of substitution between the firm’s and other goods, so an increase in the overall level of demand raises the demand for all firms in a symmetric fashion. This also holds for the costs to each firm of producing an extra unit of output, as the wage rate that they have to pay for a unit of labour depends only on the output gap. There are no effects arising due to the curvature of the production function, ie the marginal product of labour does not depend on the level of output already being produced. The ratio between the price set by an individual firm and the overall price level in the economy determines it’s relative share of the overall demand. As such, the variations in the price level between firms is the only source of heterogeneity in production decisions and is the source of the costs which arise from inflation.
Hence the first order condition can be further rearranged to give the optimum price to be set whenever the firm has the chance to adjust prices, with Pt
denoting the price level prevailing when the decision has to be made.
Pt∗
Pt
=
θ θ−1
EtP∞
s=0(αβ)
sPt+s
Pt
θ
Wt+s
Pt+sYt+s EtP∞s=0(αβ)
s
Pt+s
Pt
θ−1
Yt+s
(47)
This equation can be log-linearised to give the familiar pricing equation:
p∗−pt= (1−αβ)Et
∞
X
s=0
[pt+s−pt+ (δ+γ−1) ˆyt+s] (48)
Due to Calvo pricing with a hazard rate of changing prices of αthe law of motion for the price level is given by:
pt=αp∗+ (1−α)pt−1 (49)
Letting the desired mark-up by a firm in period t be denotedqt, means that
qt=p∗−pt=
α
1−α
πt (50)
Now, combining this with the log-linearised pricing equation allows the derivation of the full information Phillips curve for this economy in the standard way.
qt=αβEtπt+1+ (1−αβ) (δ+γ−1) ˆyt+αβEtqt+1 (51)
Finally, substituting for qtproduces the standard Phillips curve:
πt=βEtπt+1+
(1−α) (1−αβ) (δ+γ−1)
α yˆt (52)
This Phillips curve corresponds to a standard relationship between inflation and prices and has been used extensively in the literature. It links the current rate of inflation to that expected in the next period and the current output gap. This is due to the dependence of the wage rate on the current output gap. Since labour is also the sole factor of production, the output gap is a direct determinant of production costs in this model.
5.2.1 Supply Shock
In the case where the utility function also has a stochastic element in the disu-tility of labour, this leads to several changes in the profit maximizing condition for firms. Now, the real cost of production is given by:
Wt
Pt
=MtYtδ+γ−1 (53)
Therefore, the extra stochastic term comes through into the log-linearised version of this pricing equation as follows:
p∗−pt= (1−αβ)Et
∞
X
s=0
[pt+s−pt+ (δ+γ−1) ˆyt+s+mt+s] (54)
Here, lower case mt = log MMt
qt=αβEtπt+1+ (1−αβ) (δ+γ−1) ˆyt+ (1−αβ)mt+αβEtqt+1 (55)
Then, using the substitution thatqt=
α
1−α
πt,gives:
πt=βEtπt+1+
(1−α) (1−αβ) (δ+γ−1)
α yˆt+ (1−αβ)
(1−α)
α mt (56)
Therefore, this version of the model also contains a stochastic disturbance to the inflation rate as well as the output gap. To simplify notation, let the shock be relabeled into: µt= (1−αβ)(1−αα)mt.This new variable is also a randomly
distributed variable asµt˜N 0, σ2
µ
, where σ2
µ =
h
(1−αβ)(1−αα)i2σ2
m. This
means that the Phillips curve can be written as:
πt=βEtπt+1+
(1−α) (1−αβ) (δ+γ−1)
α yˆt+µt (57)
It is possible to consider this shock in the case of competitive pricing, how-ever, the symmetry result reappears again, since the shock affects prices sym-metrically across firms it leads to a situation, where this shock would be entirely costless as despite causing significant variation in inflation rates, this does not affect the utility of the representative household.
6
Shock with no information noise
The economy can be represented by the following set of equations in the absence of any information noise:
ˆ
yt=Etyˆt+1+
1
δ(it−Etπt+1) +nt (58)
πt=βEtπt+1+
(1−α) (1−αβ) (δ+γ−1)
α yˆt+µt (59)
nt˜N(0, σ2n)µt˜N 0, σ
2
µ
andσ2µ =
(1−
α) (1−αβ)
α
2
The Phillips curve is derived from the optimal pricing equation given in the previous section.
Then the path of the economy given that the central bank follows a passive policy with no biases to inflation and output, which amounts to keeping the interest rate at it’s long run steady state value of 0, will lead to the following solution for the motion of the economy expressed in terms of the state variable (the shock):
ˆ
yt=nt (61)
πt=
(1−α) (1−αβ) (δ+γ−1)
α
nt+µt (62)
This demonstrates the effect of the demand shock is spread onto both the inflation rate and the output gap, however, the cost-push shock only affects the current rate of inflation. In this formulation the effect of the supply shock is to change the marginal disutility of labour. As the household adjusts the quantity of its labour supply, this shifts the wage rate. Since the wage rate equals to the marginal disutility of labour in this setting, the cost-push shock affects firms through the wage rate labour is the sole factor of production.
7
Information Structure
The information set of all the agents in the economy is assumed to be imperfect. The firms (and the households) are assumed to be unable to observe current macro-indicators with certainty, i.e. the current observed variables are observed with some random noise. This noise disappears in the next period and all agents know past variables with certainty. This informational constraint also applies to the policy-maker in the economy and means that they must form an expectation of the current conditions.
Prior to period t a set of variables (p′t,yˆ
′
t), which are subject to some random
noise are observed by all agents in the economy. This can be viewed as a forecast of the relevant economic indicators which contains some errors. The forecast of i′t also includes the expectation of the action of the central bank. Formally
let the set Ωt = (pt,yˆt, pt−1,yˆt−1, pt−2,yˆt−2, ...). Then the information set of
agents at time t is
where
ˆ
yt′ = ˆyt+ωtwhereωt∼N(0, σ2y) (64)
p′t=pt+εt whereεt∼N(0, σ2p) (65)
This means that all expectations above must be taken with respect to this information set. The additive stochastic component of the forecast variables (the errors) are also assumed to be uncorrelated with both each other and with the demand shock.
The link between inflation and the output gap in any period as determined through the Phillips curve represents the best response of all the firms to their available information. Therefore, in making a decision of it’s own level of output a firm must consider that every other firm will follow it’s own best policy and will not deviate from it. In effect, in setting it’s price the Phillips curve already exists, such that the firm is able to assume some degree of covariance between these two variables. This means that the Phillips curve allows inference to be drawn about the value of one variable from the observation of the other.
It is possible to imagine that given some structure for the Phillips curve, the demand shock only moves the economy along this curve to another location on the same curve. Given this fact, the problem for the firm is always estimating the size of the demand shock rather than the value of the relevant variables per se, although perfect knowledge about either of these indicators would give a perfect indicator for the size of the shock. To avoid this problem, the non-trivial solution requires that the noise in both of these indicators is non-zero, since if one is perfectly known it allows the other to be determined.
With the introduction of a set of signal variables it now becomes necessary to differentiate between 3 sets of distinct, but related variables which operate in the solution for this economy. The first set is the set of variables described here, the signals which can be thought of as arriving before period t is in effect. The set of forecast variables (π′
t,yˆ′t) are connected to the actual observations of
inflation and the output gap in the way that is described above. At the same time there exists an extra set of variables : (Et|Itπt, Et|Ityˆt). These variables
are distinct from both the realisation and the forecast. Since the noise (both the shock and informational noise) are normal, these variables are also random variables, although they are the expectation of the conditional distribution of the output gap and the inflation rate given the observation of the forecast set. It is then these variables that are used for decision-making by firms in this model.
the aggregated pricing decision from the set of firms existing in the economy. the signal, which arrives at the beginning of the period is constructed to be a signal on the realised value of the future variables. In that sense, this should represent a forecast value. An individual firm makes it’s own decision on ob-serving the signals from these forecasts and the realised aggregate value is just the aggregation of the decisions of the individual firms.
In assuming that the forecast variables are rational, this means that based on observing a particular signal, the economy rationally responds in such a way that the expected value of the forecast is still the true value of the aggregate variable.
8
Phillips Curve
Given the fact that the forecast of economic conditions includes an expectation on the use of policy by the central bank, the Phillips curve depends on the way that policy is carried out. Here, an inactive policy mode is considered first, followed by an active policy procedure.
8.1
Passive policy with Imperfect Information
8.1.1 Demand Shock
Under an inactive policy the central bank does not adjust interest rates and therefore the private sector must form it’s own expectations about the future given the forecast. Also, in this section consider only the effects of the demand shock. This means that M=1 for all t or equivalently σ2m = 0. Based on the assumption that the central bank will not respond to the shock, firms must estimate the size of this shock and adjust their optimal price accordingly. Firstly, consider a relationship between output and inflation in period t to be of the form:
πt=kyˆt (66)
Given the way that the economy functions, the only complication that arises from the information structure is the fact that the agents must construct a forecast of the demand shock that hits the economy.
Then, using the IS relationship above and the postulated relationship, these variables relate to the shock in the following way:
ˆ
yt=nt (67)
πt=knt (68)
This gives the following joint distribution for the states of the economy:
This 3-d normal distribution describes the sigma-space of the economy. This description of the joint distribution takes account of the fact that inflation and output gap deviations are linked via the equation above. The i.i.d. demand shock pushes the economy away from equilibrium as detailed in the IS curve. As this is expected to raise costs, firms pass on some of the shock into higher prices. In this sense, the demand shock affects both the inflation rate and the output gap. Actual inflation rates and output gap influence what the forecasts say, hence the extra variance on these terms is due to informational noise. Since the demand shock has a non-zero effect on both inflation and output, then the covariances must necessarily be non-zero.
In every period, the agents in the economy make an estimate of the size of the shocknt,based on the observable set of macro-indicators available to them. In
this section it is possible to consider the joint distribution of the state variables and the observed indicators to find the optimal estimate given observed values of the indicators. By substituting the optimal estimate into the firms’ pricing equation, it becomes possible to determine the Phillips curve by finding the fixed point of the correlation between the output gap and the inflation rate. This is done to ensure that forecasts of the inflation and output gap are rational.
The distribution of interest is the conditional distribution of (nt|yˆ′t, π′t).To
evaluate this, consider the covariance matrix of the above distribution:
Now partition this matrix in the following manner:
Then, the mean of the conditional distribution is given by:
E(nt|yˆt′, π
Therefore, let the estimate of the demand shock in period t be:
ˆ
Based on this the estimates of the output gap and inflation are:
Et|It(ˆyt) = ˆnt (78)
p∗t = (1−αβ)Et|Itpt+ (1−αβ) (δ+γ−1)Et|Ityˆt+αβEt|Itpt+1+αβEt|Itqt+1
(80)
subtractingptfrom both sides yields:
qt= (1−αβ) (δ+γ−1)Et|Ityˆt+αβEt|Itpt+1+αβEt|Itqt+1 (81)
Now, use the expectations derived previously:
qt= (1−αβ) (δ+γ−1)Et|It
Now expand the term in square brackets:
qt= (1−αβ) (δ+γ−1)Et|It
Therefore, use the initial condition that πt = kyˆt, to give the following
expression:
This leads to a Phillips Curve with the following form:
πt=(1−α) (1−αβ) (δ+γ−1)
(1−α) (1−αβ) (δ+γ−1)
α
σ2n σ2p k2σ2
n σ2y+σ2n σ2p+σ2y σ2p
+ k
2σ2
n σ2y k2σ2
n σ2y+σ2n σ2p+σ2y σ2p
=k (85)
These solutions have been plotted for a variety of values ofσ2p andσ2y.I also assume a set of values for exogenous parameters as shown in the table below:
α β γ θ δ σ2
η
0.75 1.04−1
4 1.5 7.88 0.6 0.03Y
These assumptions imply that the average mark-up per firm will be approx-imately 14%, and that the variance of the shocks to output is approxapprox-imately 3% of steady state output as denoted by Y. Furthermore, the assumptions on the utility parameters are such that the discount rate is 4% per annum and the inter-temporal elasticity of substitution is -1.67, while the assumption on
γ corresponds to a labour supply elasticity of γ−11,here is equal to 2. The as-sumption thatαis set to 0.75, corresponds to an average length of a contract of 4 periods, hence if this model is taken to quarterly data, this implies an average length of price setting of 1 year.
As the solution is plotted for a set of noise levels, some results are immedi-ately apparent. Firstly, along the axes (where either σ2
previously. This is the result that if one of the variables is forecast perfectly, then the other also becomes revealed and firms behave in an identical fashion to the situation where there was no complications arising from information.
As the noise level is increased, or departs from zero in both the inflation and output forecast dimensions, the responsiveness of price-setters to the output gap is diminished. The effect of improving the inflation forecast is much greater than that for the output gap as this signifies a forecast of the actual price level change which will be observed. Since inflation moves by a lesser amount than output in relation to the underlying shock, then improving the inflation forecast represents a greater improvement in information on the shock than an identical improvement in the precision of the estimate of the output gap. This can be seen most easily in the noise-to-signal ratios of the inflation and the output gap forecasts. For output this is trivially σ2n
σ2
which means that the inflation observation is more informative for given values ofσ2y andσ2p.
The general negative slope of the surface in th direction of increasing noise represents a form of noise-to-signal ratio. As the informativeness of the forecasts diminishes and they become more dominated by pure noise, so the responsive-ness of producers who set their price levels falls, a fact illustrated by a lesser magnitude of the k coefficient in the Phillips curve equation.
8.1.2 Supply Shock under Passive Policy
Now consider the effects of both a supply and a demand shock with non-zero variances affecting an economy with a non-zero level of informational noise. In this case the observation of inflation is the only variable which allows the size of the supply shock to be estimated as it does not show up in the output gap. It is therefore reasonable to expect that more of the inference burden will be placed on inflation and that the output gap will serve more as a statistic for the demand shock on it’s own. More formally, the economy with 4 stochastic terms can now be represented in the following way with the same functional relationship between inflation and output gap as before:
The covariance matrix can be partitioned to evaluate the conditional dis-tribution of the unobserved shocks dependent on the observed values for the forecast in the same way as above.
Letting: Σ =
the partitions themselves are equal to:
Σ11=
Exploiting the convenient properties of normal distributions allows the con-ditional density to be expressed as:
E(nt, µt|yˆt′, π′t) = Σ12Σ−221
In the imperfect information case the pricing equation introduced previously must be modified in the following way:
qt=αβEt|Itπt+1 + (1−αβ) (δ+γ−1)Et|Itytˆ + (1−αβ)Et|Itmt+αβEt|Itqt+1 (88)
Using the result that ˆyt=nt, and µt= (1−αβ)
(1−α)
α mtand combining
this with the conditional expectation above gives:
+ (1−αβ)
Now using the relationship between the forecast variables and the actual outcomes allows the above equation to be restated in terms of realised values of inflation and output gap as well as the underlying information and preference shocks.
While having a complex structure some features of the equation above have intuitive explanations. Firstly note that with an extra shock, there are far more stochastic elements in the system. This makes any inference less accurate, so the level of responsiveness to the output gap is necessarily reduced. Note that this shows up as a much larger term in the fraction, which itself is just the inverse of the determinant of the covariance matrix for the forecast terms.
The shock itself enters the equation with a multiplier which depends on the relative variances of the other shocks and of the noise terms. This means that any shock will not be transmitted one-for-one by firms. Any signal that they receive in the form of the forecasts means could be due entirely to variation in the noise term alone, which means that they are not going to respond to the shock sufficiently.
The last two terms in the equation are similar to the previous section as reflecting error forecasts. These terms are both weighted by their variances, so the error with a larger variance receives less weight and vice versa.
8.2
Active Central Bank Policy
8.2.1 Demand Shock
If the central bank adopts an active policy stance it will be able to respond to demand shocks as far as it can estimate the size of these shocks. With demand shocks only, there is no trade-off between output and inflation stabilisation so no explicit optimization is required. Under perfect information σ2
p, σ2y
= (0,0),
the monetary policy authority is able to perfectly offset the effects of the shock, which leads to a situation where both the output gap and inflation rate are perfectly stabilised around the steady state of the economy. In this case the bank interest rate follows:
it=δEt|Itnt (91)
Under imperfect information, the role of stabilisation policy becomes more complicated as the central bank must form it’s own estimate of the size of the shock before enacting this policy. In this case as the forecast includes the interest rate to be set by the central bank. Since this forecast represents the best available information at the time when decisions have to be made, it leads the private sector to ”trust” in the action of the central bank with all of the fluctuations which occur in practice due to mistakes in the way the shock was forecast. In this case the Phillips curve is the same as that which would prevail without any noise.
πt=βEtπt+1+kyˆt+Wyεt+Wπωt (92)
By construction, all of the losses that accrue in this case are attributable to data errors. Essentially, the central bank responds to all the loss that it can respond to, leaving the error in observations to have some residual demand shock on the economy. By construction these are given by the noise terms themselves.
8.2.2 Supply Shock
In the case where the central policy-maker faces supply shock, the problem becomes non-trivial as it becomes necessary to find the optimal trade-off between inflation and the output gap. The welfare function is derived in a later section, but by jumping forwards it is possible to present a form of this objective function and by postulating a benevolent policymaker who acts to optimize this concept of social welfare allows the closed form solution to the behavior of the economy to be solved. Take the welfare function to be:
Losst=E0
Essentially, take the simplified version of the above function to be:
Lt=E0
Solving this equation gives an equation for the optimal output gap/inflation trade off that the central bank is prepared to accept. Rational agents in the economy recognize the objectives and optimization problem of the central bank, so fully expect it to choose a value on the optimal trade-off path as given by the maximization problem of the central bank. This implies the following trade-off between the output gap and the inflation rate:
πt=
The set of coefficients{A, B, C, D} correspond to exogenously given values in the utility function. For ease of exposition, the policy is stated in terms of these newly defined coefficients. Essentially, this trade-off can be restated as a single coefficient, relating the response of the interest rate to the supply shock. While the maximum of the welfare function is obtained by setting the response to the demand shock to be complete, for the supply shock a partial response is preferred as this spreads the welfare consequences of the supply shock onto both the output gap and the inflation rate.
of commitment rules. Due to the fact that the system has been log-linearized, the optimal rule, which also delivers an expected interest rate of zero belongs to the class:
it=φ1Et|Itnt+φ2Et|Itmt (96)
Note, that policy considered here is entirely discretionary and postulating the above rule as a description of policy rules out a commitment solution. Given this relationship between the interest rate set by the central bank and the underlying shocks. This has implications for the form of the distribution of the economic observables and the underlying shocks as follows:
Here is the form of the variance-covariance matrix:
Σ =
and partitioning it as
Σ =
Σ11 Σ12
Σ21 Σ22
the partitions themselves are equal to:
Σ22=
Again, as shown previously, the expectations of the underlying shocks can be calculated and a Phillips curve can be derived in the usual way.
E(nt, µt|yˆt′, π′t) = Σ12Σ−221
Unlike the previous case, it has not been possible to compute a closed form solution for the expectations above. Numerical solutions remain viable however. These are presented in the sections that follow.
9
Welfare Function
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:
Ut=
Here Ct is the Dixit-Stiglitz consumption aggregate, which is equal to the
overall level of output Yt. The Taylor approximation of this expression is around
Ut=
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 (Mt) and the demand shock (Nt). 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.
Y = (
Z 1
0
Y(i)θ−θ1di) θ
θ−1 (100)
Yθ−θ1 =
Z 1
0
Y(i)θ−θ1di (101)
as due to symmetry in the steady state Y(i) =Y(j)∀i, j then
Yθ−θ1 =Y(..) θ−1
θ
Z 1
0
di (102)
Y =Y(i) for alli
L=
Z 1
0
L=Y
Z 1
0
di (104)
L=Y (105)
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−θ1)δ+1γ−1 <1 andY(i) =Y(j)∀i, j
using the fact that Yt
Y = 1 + ˆyt+
1 2yˆ
2
t...where ˆyt= log(YYt)
Nt
N = 1 +nt+
1 2n
2
t...where nt= log(NNt) Yt(i)
Y(i) = 1 + ˆyt(i) +12yˆt(i)2... where ˆyt(i) = log(YtY(ti))
Mt
M = 1 +mt+
1 2m
2
t...wheremt= log(MMt)
The deviations of individual outputs for firms as expressed in the last equa-tion can be re-written in the following way:
Z 1
0
Yt(i)−Y(i)di (106)
Z 1
0
(Yt(i)−Yt) + (Yt−Y(i))di (107)
Now, using the fact that Y(i) = Y and splitting the integral into the con-stituent parts gives:
Z 1
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:
Ut=
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 outputYt. 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.
is also true for all other variables, although as they all have non-zero second derivatives, the effect of the logarithmic expansion is to supplement the coeffi-cients in the welfare function. Alternative cases can now be analysed as the loss function at time t is given by:
Losst=Et
∞
X
s=0
βsUt+s (110)
This means that the loss function can be rewritten more completely as:
Losst=Et
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:
Losst=Et
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:
var(logYt(i)) =θ2var(logPt(i)) (114)
In any given period the degree of price dispersion can be related to the series of past rates of inflation. This is because in any period only a portion of firms given by αare 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(pt(i)) = ∆t (115)
∆t=α∆t−1+
α
1−απt (116)
Then iterating backwards gives:
∆t= lim
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:
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:
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.
9.2
Calibrating the Welfare Losses using numerical values
9.2.1 Passive policy case
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 most striking feature of the above result is that in this case only im-provements in the gathering of output data lead to imim-provements 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:
k= (1−α)
α (1−αβ) (δ+γ−1)
(
σ2n σ2p+σ2µ
+σ2nσ2yk2 k2σ2
nσ2y+ σ2n+σ2y
σ2
p+σ2µ
)
(120)
Letting σ2
increasingσ2
µis analogous to increasingσ2p.In effect, by considering a case with
variability of the supply shock is the same as considering a range for noise which is translated away from 0. Then similarly to the case of only a demand shock, the responsiveness of inflation to the output gap declines, the more noisy that the setting is. Furthermore, at higher levels of noise, the change in responsiveness declines as can be seen from the strong non-linearity of the curve for the solution of k in figure 1. This holds for all non-negligible levels of supply shock noise, since at this level, the area where inflation rate improvements would have much more effect than equivalent improvements in output gap data are pushed out by the presence of the supply shock.
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:
it=δEt|Itnt (121)
the private sector has no additional information. This means that all variations in the forecast of inflation are taken to be the result of information errors in equilibrium, while the error in estimating the output gap is just the noise term on that indicator.
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.
the predictable shifts in the output gap, the presence of active policy orthog-onalises inflation and the output gap since there is no relevant information on the output gap for the private sector once policy is taken into account.
The central bank sets interest rates according to the rule shown previously, i.e.
it=φ1Et|Itnt+φ2Et|Itmt (122)
The private sector takes this into account, which means that the value of
φ2 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 beforeφ1 is set
equal toδ to offset the predictable part of the demand shock completely, while
φ2 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:
