A New Keynesian Model with Noisy Observa (1)

Dennis Tatarkov September 21, 2011

Abstract

This thesis looks at the effects of data uncertainty on the representative consumer DSGE model. Several cases are considered: demand/supply shock as well as the effects on policy. A measure of welfare is approximated from the assumed utility function. Using a set of parameter values, the welfare effects of a range of noise levels is calculated. The results imply

a strongly non-linear effect of uncertainty on welfare. Optimal filtering places more weight on the variable which is subject to less noise. Hence in a regime where the inflation rate is affected by both the cost-push shock and an informational error term, the utility of this indicator is found to decline very rapidly. Policy acts on the perceived shocks, so that in the presence of policy the two indicators become orthogonalized.This leads to

a situation where only improvements in the measurement of the output gap positively affect welfare.

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

The most well-known model to deal with an imperfect information among mar- ket participants, and the macroeconomic implications of such an information The most well-known model to deal with an imperfect information among mar- ket participants, and the macroeconomic implications of such an information

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 y t in each market i depends on the ratio of the price in that market to the general price level as:

(1) Here, p denotes the (log of) the price level and y denotes output with indi-

y t (i) = κ (p t (i) − E (p t |I t ))

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:

(2) Assuming that the z term is stochastic and follows a known distribution

p t (i) = p t +z

allows for a filtering problem to be set up for the expectation of the general price level. In that case, the responsiveness to that expectation will depend on the signal-to-noise ratio of relative price to aggregate shifts. Therefore, the expectation in the firm supply function is given by:

(3) = κp t (i)

E (p t |I t ) = E (p t |p t (i))

Here, κ is the signal-to-noise ratio which has the form of:

z +σ 2 τ

The variance of the general price level is given by σ 2 τ which stands as a magni- tude of the size of the general shock affecting the economy. In this case, σ 2 z is the idiosyncratic shock to the individual firm’s markets. Putting all this together and summing across the distribution of firms across i allows the derivation of the well-known Lucas supply function:

(5) In keeping with notation later, the hat over the output term signifies a gap

y ˆ t = κκp t

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 the κ multiplier is such that if all shocks are due to

the aggregate and σ 2 z is large relative to σ 2 τ , 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. Since the publication of the Lucas model, several others have proposed al-

ternative monetary policy transmission mechanisms based on uncertainty and beliefs, while responding to the criticism of insufficient persistence. These have tended to depart from the original idea of uncertainty about current data, as it tends to be released relatively frequently. This does not lead to any sig- nificant sources of internal persistence of shocks apart from those generated through assumptions on the shock process itself. Woodford (2001) proposes an idea that given that information is incomplete means that second order beliefs would be relevant to decision making. The key changes that allow this to be ternative monetary policy transmission mechanisms based on uncertainty and beliefs, while responding to the criticism of insufficient persistence. These have tended to depart from the original idea of uncertainty about current data, as it tends to be released relatively frequently. This does not lead to any sig- nificant sources of internal persistence of shocks apart from those generated through assumptions on the shock process itself. Woodford (2001) proposes an idea that given that information is incomplete means that second order beliefs would be relevant to decision making. The key changes that allow this to be

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 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 state-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.

The introduction of Calvo pricing means that the model has an appearance which is similar to more contemporary formulations. This sets a constant hazard rate for pricing decisions and while potentially some prices can be set for an unlimited time period, the average duration of a price contract is given by the inverse of the hazard rate. With this value calibrated to that observed in the wider economy would allow for a realistic portrayal of the rigidities present in the economy. This approach also has limitations. While it can be argued that The introduction of Calvo pricing means that the model has an appearance which is similar to more contemporary formulations. This sets a constant hazard rate for pricing decisions and while potentially some prices can be set for an unlimited time period, the average duration of a price contract is given by the inverse of the hazard rate. With this value calibrated to that observed in the wider economy would allow for a realistic portrayal of the rigidities present in the economy. This approach also has limitations. While it can be argued that

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.

The baseline model, as presented by Woodford, is then supplemented in my setting by a set of informational imperfections, essentially making agents work The baseline model, as presented by Woodford, is then supplemented in my setting by a set of informational imperfections, essentially making agents work

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.

Several papers also look at the theoretical implications of informational noise for policy formulation. Aoki (2006a) makes an assumption that the private sector is fully informed, while the central bank has only partial information about current indicators. In this setting data is revised within one period to it’s true values. This is the opposite assumption to the setting considered previously and focuses on the filtering and optimisation problem of the monetary authority. Aoki derives the optimal commitment solution using a Kalman filter to take account of both filtering and optimal control. The findings show that initial responses are likely to overshoot the value in the next period. Policy here is found to closely resemble that under price-level targeting. Similarly, Svensson and Woodford (2002) consider the form of the solution in a model where partial information is available. They detail the conditions where certainty equivalence holds for the optimal reaction function, as well as a situation where optimization and filtering cannot be made separate. These papers impose the condition that the information set of the central bank is a subset of the information set of the public. This means that the effect of partial information on household decisions Several papers also look at the theoretical implications of informational noise for policy formulation. Aoki (2006a) makes an assumption that the private sector is fully informed, while the central bank has only partial information about current indicators. In this setting data is revised within one period to it’s true values. This is the opposite assumption to the setting considered previously and focuses on the filtering and optimisation problem of the monetary authority. Aoki derives the optimal commitment solution using a Kalman filter to take account of both filtering and optimal control. The findings show that initial responses are likely to overshoot the value in the next period. Policy here is found to closely resemble that under price-level targeting. Similarly, Svensson and Woodford (2002) consider the form of the solution in a model where partial information is available. They detail the conditions where certainty equivalence holds for the optimal reaction function, as well as a situation where optimization and filtering cannot be made separate. These papers impose the condition that the information set of the central bank is a subset of the information set of the public. This means that the effect of partial information on household decisions

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:

log(N t )=n t ˜N (0, σ 2 n )

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 by N t, which has a log-normal

distribution around zero with a variance of σ 2 n . This term denotes a demand shock that is introduced into the economy.

The household also faces a budget constraint, where A t , is the holding of government assets with a nominal interest rate set by the central bank at time t.

di − =0 (10) P t

+Π t −

Maximizing the households utility subject to the constraint results in the following set of first order conditions with a Lagrange multiplier λ t :

since λ t is the same for all goods it is possible to derive demand curves from these equations as follows.

for all i ∈ [0, 1] (16) P t (i)

Since there is no capital in the model, total output must equal total con- sumption, therefore this can be re-written as:

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 parameter θ, 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 for λ t, gives λ t, =N t C −δ t . Hence it becomes possible to derive the following Euler equation for this household:

This can then be used to derive the IS curve for this model economy. In order to derive the IS curve, log-linearise the model around the steady state with a value for the shock equal to it’s expectation. For all the variables which This can then be used to derive the IS curve for this model economy. In order to derive the IS curve, log-linearise the model around the steady state with a value for the shock equal to it’s expectation. For all the variables which

X t −X ≃ log (X t ) − log (X) = x t −x=ˆ x t

A log-linearised relation of this type leads to the following form of the IS curve, which has become standard in the literature.

y ˆ t =E t y ˆ t +1 − (i t −E t π t +1 )+n t

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 N t , 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 of λ t, =N t C t −δ , the real marginal cost of labour is equated to the real wage:

γ =L −1

The labour market functions competitively with both the household and the firms acting as price-takers. Households supply an amount of undifferentiated labour equal to L t which is employed by firms on the unit interval in amounts

L t (i). Since the production function for firm i is Y t (i) = L t (i), and L t = R 1

0 L t (i) di, then to a first approximation ˆl t =ˆ y t . Due to there being no capital accumulation, C t =Y t . 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:

t =N t (

) log (M t ) ˜N 0, σ 2 (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 M t (denoted by m t ) has a normal distribution

with a zero mean and a known variance of σ 2 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 of M t does 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:

As before, λ t =C −δ t N t . Therefore, the level of the real wage must now be equal to the following expression:

L γ −1 =M δ t t C t

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

where σ 2 m → 0. This now means that firms’ costs which consist of the real cost where σ 2 m → 0. This now means that firms’ costs which consist of the real cost

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:

Using the labour supply equation from above, the real wage rate must satisfy:

=L γ −1 C δ

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.

This then leads to the following result concerning the steady state level of output.

Y 1 =( ) δ +γ−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:

(31) As profit maximization is independent between time-periods, optimization

Y t (i) = L t (i)

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:

Y t (i)

Π t = (P t (i) − W t )

where

This expression is optimized with respect to the price set by an individual firm to give the first order condition of the form:

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.

Hence, taking the labour supply relationship gives the result that output with competitive firms is always equal to the natural rate.

In this case, the demand shock has no effect on the equilibrium of the econ- omy as a frictionless adjustment fully stabilises the economy. Since the changes in prices are still driven by the period-by-period realisation of the shock, infla- tion will not in general be stable. However, in a symmetric equilibrium that is described above there is no possibility of resource mis-allocation due to price distortions. In this case inflation is not correlated with the variance of outputs across the individual firms and any magnitude of the realisation of inflation is entirely costless.

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:

(39) ∂P t (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:

=L δ

Since no capital accumulation occurs, all production must go towards con- sumption, hence it is easy to show that: C t =Y t . Also, using the production function, the overall level of labour supply is a direct integral over all of the firms production level. Hence,

At the same time, the Dixit-Stiglitz consumption aggregate in the absence of capital accumulation can be rewritten as:

Then using a second order Taylor approximation around the steady state of the model, denoted by variables without any subscripts this expression becomes:

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 that Y i =Y for all i. Hence the above expression can be further simplified to give:

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 following simplification.

ˆ y t (i) di

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 φ

W t to denote the (log) level of marginal costs in period t, such that φ t = log( t P t ). gives the following approximation to the level of costs

at any time:

(46) As such, the implication is that the firm’s decision is dependent only on two

φ ˆ t = (δ + γ − 1) ˆ y t

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 P t denoting the price level prevailing when the decision has to be made.

E t s =0 (αβ)

This equation can be log-linearised to give the familiar pricing equation:

p ∗ −p t = (1 − αβ) E t [p t +s −p t + (δ + γ − 1) ˆ y t +s ] (48)

s =0

Due to Calvo pricing with a hazard rate of changing prices of α the law of motion for the price level is given by:

(49) Letting the desired mark-up by a firm in period t be denoted q t , means that

p t = αp ∗ + (1 − α) p t −1

it will have the following relationship with the inflation rate in that period:

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.

(51) Finally, substituting for q t produces the standard Phillips curve:

q t = αβE t π t +1 + (1 − αβ) (δ + γ − 1) ˆ y t + αβE t q t +1

(1 − α) (1 − αβ) (δ + γ − 1) π t = βE t π t +1 +

y ˆ t (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:

=M δ

t Y +γ−1 t

Therefore, the extra stochastic term comes through into the log-linearised version of this pricing equation as follows:

p −p t = (1 − αβ) E t [p t +s −p t + (δ + γ − 1) ˆ y t +s +m t +s ] (54)

s =0

Here, lower case m t = log M t M shock. Now taking a similar path as before allows the full information Phillips curve to be derived without any added difficulty as: Here, lower case m t = log M t M shock. Now taking a similar path as before allows the full information Phillips curve to be derived without any added difficulty as:

Then, using the substitution that q t =

π t , gives:

(1 − α) (1 − αβ) (δ + γ − 1) (1 − α) π t = βE t π t +1 +

ˆ y t + (1 − αβ) m t (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: µ (1−α) t = (1 − αβ) α m t . This new variable is also a randomly

h (1−α) i 2 distributed variable as µ t ˜N 0, σ µ 2 µ 2 = (1 − αβ) α σ 2 m . This means that the Phillips curve can be written as:

(1 − α) (1 − αβ) (δ + γ − 1) π t = βE t π t +1 +

y ˆ t +µ t (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:

y ˆ t +µ t (59)

n t ˜N (0, σ 2 )µ t ˜N 0, σ 2 µ and σ 2 n 2 µ = σ

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):

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 , ˆ y 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 Ω t = (p t , y ˆ t ,p t −1 , y ˆ t −1 ,p t −2 , y ˆ t −2 , ... ) . Then the information set of agents at time t is

I t = (p ′ t , y ˆ ′ t , Ω t −1 )

y ˆ ′ t =ˆ y t +ω t where ω t ∼ N (0, σ 2 y ) (64)

(65) This means that all expectations above must be taken with respect to this

p ′ t =p t +ε t where ε t ∼ N (0, σ 2 p )

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 , y ˆ ′ 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 : (E t |I t π t ,E t |I t y ˆ t ). 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 reason behind the existence of a distinct set of variables becomes clear in the following section, but in short the Phillips curve relationship depends on The reason behind the existence of a distinct set of variables becomes clear in the following section, but in short the Phillips curve relationship depends on

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 σ 2 m = 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:

(66) This equation has the form of the Phillips curve that it proposes to solve,

π t =kˆ y t

however, this equation is only intended as a proposed correlation between in- flation and the output gap, and as such does not require that expected future inflation to be included. It is possible to specify this more fully, but as terms other than inflation and the output gap will not enter the inference problem of the firm, then at this point they would be redundant. Note, that the policy under consideration here also assumes that expected inflation is zero in any case.

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:

This gives the following joint distribution for the states of the economy:

2 2 2  n  t 0 σ n σ n kσ n

 y ˆ ′ t  ˜N   0  ,  σ 2 n σ 2 n +σ 2 y

kσ 2 n   (69)

0 kσ n 2 kσ 2 n

k 2 σ 2 n +σ p 2

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 shock n t , 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 (n t |ˆ y t ′ ,π ′ t ). To evaluate this, consider the covariance matrix of the above distribution:

kσ 2 n

Σ=  σ 2 n σ 2 n +σ 2 y

kσ n 2  (70)

kσ 2 n

kσ 2 n

kσ 2 n +σ p 2

Now partition this matrix in the following manner:

Σ =σ 2 11 n where Σ 12 = σ 2 n kσ 2 n (72)

kσ 2 n +σ 2 p (74)

Then, the mean of the conditional distribution is given by:

k 2 σ 2 σ 2 +σ 2 σ 2 +σ 2 σ 2 t n (76) y n p y p n y n p y p

E (n t |ˆ y ′ ,π ′ t t )= 2 2 2 2 2 2 2 y ˆ ′ +

Therefore, let the estimate of the demand shock in period t be:

Based on this the estimates of the output gap and inflation are:

E t |I t (ˆ y t )=ˆ n t

(79) Using these measures derive the Phillips curve in the following way:

E t |I t (π t ) = kˆ n t E t |I t (π t ) = kˆ n t

subtracting p t from both sides yields:

q t = (1 − αβ) (δ + γ − 1) E t |I t y ˆ t + αβE t |I t p t +1 + αβE t |I t q t +1 (81) Now, use the expectations derived previously:

kσ2 nσ y 2 ′ qt = (1 − αβ) (δ + γ − 1) Et|It

σ2 nσ p 2 ′

k2 σ2 nσ 2 y+σ 2 nσ 2 p+σ 2 2 ˆ yσ y p t+ k2 σ2 nσ 2 y+σ 2 p+σ +αβEt|I 2 π nσ  yσ 2 2 p t t pt+1+αβEt|It qt+1 (82)

Now expand the term in square brackets:

qt = (1 − αβ) (δ + γ − 1) Et|It 

kσ2 nσ y 2    k2 σ2 nσ 2 y +σ nσ 2 2 p +σ yσ 2 p 2 ωt + k2 σ2 nσ 2 y +σ nσ 2 2 p +σ 2 yσ 2 p εt  + αβEt|I t pt+1 + αβEt|It qt+1

Therefore, use the initial condition that π t =kˆ y t , to give the following expression:

σ2 nσ 2 p

k2 σ2 nσ 2 

qt = (1 − αβ) (δ + γ − 1) Et|It  k2 σ2

2    k2 σ2 nσ 2 y +σ nσ 2 p +σ 2 yσ 2 2 p ωt + k2 σ2 nσ 2 nσ y +σ y nσ 2 2 p +σ 2 yσ 2 p εt  + αβEt|I t pt+1 + αβEt|It qt+1

This leads to a Phillips Curve with the following form:

(84) y+σ nσ p+σ yσ p

k2 σ2 nσ 2 2 2 2 2  yt+ ˆ

 k2 σ2 nσ 2 y +σ 2 nσ 2 p +σ 2 yσ p 2 ωt + k2 σ2 nσ 2 nσ y +σ y 2 nσ 2 p +σ 2 yσ 2 p εt  + βEt|I t πt+1

σ2 nσ p 2 + 1−  kσ2 2 