Optimal inflation target : an analysis

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University of Amsterdam Faculty of Economics and Business

OPTIMAL INFLATION TARGET: AN ANALYSIS

Bachelor Thesis Econometrics by Sean Bagcik 10277528 Supervisor: dr. T.A. Makarewicz 27 June, 2017 Abstract

This thesis investigates an optimal inflation target within a nonlinear New Keynesian framework in which economic agents have homogeneous naive expectations and the nominal interest rate follows a backward-looking Taylor rule. Given the

parametrisation of the model, it shows that only a small domain of target inflation rates is stable and therefore optimal. Inflation targets outside of this domain result in a dynamic system with either ever-increasing or ever-decreasing inflation rates. Keywords: monetary policy, expectations, Taylor rule, zero lower bound

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1 Introduction

The nominal interest rate is an important tool in monetary policy used by central banks. By lowering the interest rate, the central bank increases the money supply in order to stimulate the economy in times of low aggregate demand and vice versa in times of economic highs. The key objective of monetary policy is to ensure price stability by setting an inflation target and to ensure gradual economic growth. The question then arises: what is the optimal inflation target?

The New Keynesian framework has emerged as the decision-feedback mechanism in monetary policy-design. The analytical structure is formed by (but not limited to)1 three equations: the interest-savings curve describing output gap, a Phillips curve describing inflation, and a rule for the nominal interest rate in monetary policy. The framework’s microfoundations are clearly elaborated by Woodford (2003). He shows how the equations can be derived from optimising behaviour on the part of the central bank, as well as firms and households in the presence of some nominal imperfections. In the literature on interest rate rules by far the most common one is the so-called ‘Taylor rule’ created by J.B. Taylor (1993). An interest rate rule provides the central bank with guidance as to its optimal response to an inflation, a demand or supply shock, or a combination of these factors. Nevertheless, a more recent alternative has been proposed by Hommes & Lustenhouwer (2015). Pursuing the two main goals of price stability and gradual economic growth, the central bank aims to minimise a loss function consisting of the expectations of all future deviations of inflation from its target and all future output gaps, since there is a trade-off between the inflation target and output target. This theoretical reasoning leads to an alternative interest rate rule.

Another important aspect of the New Keynesian framework is the expectation formation process in economic agents. Traditionally, all agents are assumed to be homogeneous and rational so that they behave according to perfect model-consistent expectations that allow for the forecast of future inflation and output. However, in light

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of empirical evidence, other behavioural models appear to be operative as well, such as the heterogeneous expectations switching model (Assenza, et al, 2013). These models find that economic agents tend to base their predictions on past observations, following simple forecasting rules so that the equilibria should be seen as the long-run outcome of some kind of learning and updating process. In this case, convergence to a steady state is slow, as agents continually update their beliefs as they keep making adjustments, while still holding on to certain fundamental economic principles. In the presence of heterogeneity, these models suggest a monetary policy that reacts aggressively to deviations from the inflation target to lead the economy to the desired outcome. The same suggestion is made by Anufriev, et al. (2011): after investigating inflation dynamics when agents have heterogeneous expectations, this theory concludes that when agents have a continuum of beliefs, the standard policy recommendation of the framework is no longer sufficient to guarantee uniqueness and stability of the rational steady state. When the central bank responds only mildly to inflation, heterogeneous agents, trying to learn from their forecast errors, receive misleading signals from the central bank’s policy, which creates a cumulative process of accelerating inflation or deflation. Exogenous shocks to economic fundamentals are reinforced by self-fulfilling expectations. Therefore, in the presence of heterogeneity the ‘Taylor principle’ states that the central bank can avoid the cumulative process by reacting aggressively to inflation in order to induce stable dynamics converging to the rational steady state.

A theoretical analysis of the New Keynesian framework under heterogeneous expectations follows from Massaro (2012). He adapted the microfoundations for when expectations are heterogeneous and performed a numerical analysis of the parameter outcomes. He concluded that in this case the framework does not lead to a unique equilibrium, something that should be a certainty in optimal monetary policy design. One may then raise the question whether this rational-agent assumption is too restrictive to accurately reflect reality and thus may lead to faulty policy recommendations (Hommes & Lustenhouwer, 2016).

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A second point of contention is the effectiveness of traditional monetary policy in the context of a zero lower bound on interest. Ever since the financial crisis of 2007 triggered by the housing bubble in the United States, interest rates have been historically low. The Eurozone is currently experiencing a near-zero nominal interest rate, in combination with economic stagnation and low inflation. This liquidity trap is a situation in which central banks cannot encourage spending by lowering interest rates. Hommes and Lustenhouwer (2015) have investigated the theoretical consequences of a liquidity trap in a simulated economy. They argue that according to rational expectations models shocks to the fundamentals of the economy can lead to a temporary liquidity trap after which the situation stabilizes again. However, in practice, a self-fulfilling deflationary spiral may occur, because the nominal interest rate will never be set at a negative value. Empirical evidence on solving the liquidity trap under learning has been proposed by Hommes, Massaro and Salle (2015). They argue that monetary policy alone is insufficient in preventing the economy from falling into a liquidity trap, but that such a policy augmented with a fiscal switching rule does succeed in avoiding and escaping liquidity traps. A similar situation occurred in the 1990’s in Japan, which resulted in prolonged economic stagnation. Krugman (2000) has extensively investigated Japan’s situation and the liquidity trap and proposed alternatives to the conventional monetary policy such as fiscal policy, quantitative easing and unconventional open-market operations.

Continuing in the line of these cited papers and taking into account recent economic developments, this thesis will research a nonlinear New Keynesian model with homogeneous naive expectations. Secondly, there will be a brief discussion on the zero lower bound case and its consequences for the central bank’s monetary policy. Computations will be done in MATLAB®.

The rest of this paper is organized as follows: Section 2 presents the necessary theory for the analysis. Section 3 will cover the main techniques behind the numerical analysis in MATLAB®; and section 4 will contain the results. Finally, section 5 will conclude this thesis and point out certain flaws in the model setting.

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2 Theory

This section contains five subparts. The first subpart starts with an explanatory as well as a brief theoretical reasoning on the existence of a zero lower bound, its relation with a liquidity trap and the consequences for monetary policy. The second subpart introduces the main model of interest within this thesis: a nonlinear version of the New Keynesian model. The third subpart discusses the two most commonly used approaches to model firms’ price-setting behaviour and how it is related to the model. The fourth subpart presents the interest rate rule for monetary policy and the section ends with the introduction of naive expectations.

2.1 Zero Lower Bound

Söderström and Westermark (2008) give an intuitive explanation for the existence of a zero lower bound on the nominal interest rate and the way a liquidity trap occurs.

Economic agents choose between holding money or investing in real assets and financial assets. Money has the advantage that it is directly interchangeable for goods and services, but without providing interest. When the interest rate on financial assets becomes zero, there is no longer a reason to invest in such assets, as the return is equal for currency, while money still has the advantage of being directly interchangeable for goods and services. In theory, when the interest rate is zero, agents will hold just as much money as they need, and nothing more. Lowering the interest rate no longer affects the demand for money. The nominal interest rate cannot become negative, because investors could then earn money by borrowing and investing in currency, putting an upward pressure on the interest rate. This phenomenon is called a liquidity trap: a situation in which an increase in money supply has no longer the effect of increasing aggregate demand.

However, the true variable that drives consumption is the real interest rate: the nominal interest rate adjusted for expected inflation. Agents balance the benefit of saving today and consuming in the future versus the opposite. An important deciding factor in

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saving, is that agents want stable purchasing power over time. During economic stagnation, inflation expectations are low and therefore the real interest rate is low as well. A sufficiently low real interest rate stimulates demand and could lead to economic growth again. However, the real interest may still be too high to encourage consumption and since the nominal interest rate is already zero, the central bank loses an important tool of its traditional monetary policy.

There is also a theoretical explanation for the existence of such a zero lower bound on nominal interest rates as given by McCallum (2002). Without going in too much of detail, his main reasoning takes into account for the ‘transaction-facilitating properties of money’, the economy’s medium of exchange. The conclusion of his analysis lies in establishing the relationships among the marginal service yield from holding (real) money balances, the quantity of money held and spending (2002, pp. 6-7). Additional empirical evidence of the occurrence of liquidity traps is given by Hommes, Massaro and Salle (2015). They argue that under adaptive learning, liquidity traps and deflationary spirals are triggered by large pessimistic expectations shocks. Liquidity traps in the form of a deflationary spiral can emerge as a result of severely pessimistic expectations. An adverse shift in expectations creates the possibility of a self-reinforcing feedback loop in which sufficiently pessimistic expectations result in low output and inflation, leading to high real interest rates because of the zero lower bound, which then causes a downward revision of expectations, strengthening the downward pressure on output and inflation (2015, p. 34).

2.2 A Nonlinear New Keynesian Model

The basic analytical structure of the New Keynesian framework consists of three equations: an IS curve describing the output gap, a Phillips curve describing inflation and a monetary policy rule for the nominal interest rate. This basic framework has many variations and is highly adaptable to different economic circumstances. In order to answer the main question of what is an optimal inflation target, this thesis focuses on a nonlinear version of the New Keynesian model. One of the advantages using a nonlinear

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version, is that it is able to deliver absolute values of the steady state variables in contrast to the models that are linearized around the steady state. Linearized models only operate in a small area around those variables. The main equations describing the nonlinear model follow from Hommes, Massaro and Salle (2015) and Evans et al. (2008):

(2.1) 𝑐" = 𝑐"$%& (()*+ , -.)) + 0 (2.2) 𝜋_" 𝜋_"− 1 = 𝛽𝜋_"$%& _𝜋 "$%& − 1 +₇₈6 𝑐"+ 𝑔" +*: ; +%<6 8 𝑐"+ 𝑔" 𝑐" <=

Equation (2.1) describes the dynamics of net consumption 𝑐", where 𝑐"$%& and 𝜋"$%& are

defined as respectively expectations of future net consumption and inflation, 𝑖_" is the nominal interest rate as chosen by the central bank, 𝛽 is the discount factor and is set2_at

0.99. Finally, 𝜎 is the elasticity of intertemporal substitution, a measure of responsiveness of the growth rate of consumption to the real interest rate and equals 1.

The second equation, equation (2.2), is a Philips curve describing the dynamics of inflation 𝜋", where 𝑔" is government spending and is set to 0.2, 𝜀 is the marginal

disutility of labour set at 1, 𝛼 is the return to labour in the production function set at 0.7, 𝛾 is the cost of deviating from the inflation target under Rotemberg price adjustment costs and, finally, 𝜈 is the elasticity of substitution between differentiated goods and set at 21.

This part briefly discusses the microfoundations of the theoretical model underlying the thesis. It is assumed that each consumer wants to maximise the expected value of its discounted utility function (2.3) while being constrained by (2.4). The utility function in (2.5) is assumed to have a parametric form:

(2.3) 𝐦𝐚𝐱 𝐸_N[ 𝛽"_𝑈 ",.(𝑐",. R "ST ,U)V+,W_X ) , ℎ",., X),W X)V+,W− 1)]

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(2.4) 𝐬. 𝐭. 𝑐_",.+ 𝑚_",.+ 𝑏_",.+ Υ_",. = 𝑚_"<%,.𝜋_"<%_{+ 𝑖} "<%𝜋"<%𝑏"<%,.+X_X),W ) 𝑦",. (2.5) 𝑈_",. = a),W+V0 %<= + b %<=c( U)V+,W X) ) %<=c−d),W +*: %$e − 8 f( X),W X)V+,W−1) f

Where 𝑐_",. is the consumption aggregator, 𝑀_"<%,. and 𝑚_",. define nominal and real money balances, 𝑃_" is the aggregate price level, ℎ_",. is the labour input, 𝑃_",. is the price of consumption good 𝑖 and, the last term in (2.3), X)

X)V+, is the gross inflation rate 𝜋".

In (2.4) Υ",. is the lump-sum tax collected by the government, 𝑏",. is the real

quantity of risk-free, one period nominal bonds held by the consumer, 𝑖"<% stands for the

nominal interest rate between 𝑡 − 1 and 𝑡 and lastly, 𝑦_",. is the production of good 𝑖. Equation (2.5) has two more undefined parameters, namely 𝜎_f and 𝜒 which play a role in the money demand function, but is not of importance in this thesis. Firms are assumed to have production function (2.6) with decreasing returns to scale. Each firm operates under monopolistic competition and faces demand curve (2.7) for its differentiated good 𝑖. 𝑃_",. is then its profit maximising price and 𝑌_" is the aggregate output: (2.6) 𝑦_",. = (ℎ_",.)7_, _{0 < 𝛼 < 1} (2.7) 𝑃_",. = (m),W n)) V+ o𝑃_"

The last part of these microfoundations is formed by the government’s flow budget constraint (2.8), which ensures that an increase in real government debt leads to an increase in taxes sufficient to cover the increased interest and some increased principal; the passive fiscal policy rule (2.9); and the market clearing condition (2.9), implying price frictionless:

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(2.8) 𝑔_"+ 𝑏_"+ 𝑚_"+ Υ_" = 𝑚_"<%𝜋_"<%_{+ 𝑅} "<%𝜋"<%𝑏"<% (2.9) Υ" = 𝜅T+ 𝜅𝑏"<%, 𝛽<%− 1 < 𝜅 < 1 (3.0) 𝑔" = 𝑦"− 𝑐"

2.3 Rotemberg versus Calvo Price Setting

In the standard New Keynesian framework of monopolistically competitive firms there are two commonly used approaches to model firms’ price-setting behaviour: the Rotemberg quadratic price adjustment costs and the Calvo random price adjustment signal. One of the key differences between the two models is the presence of price dispersion, defined as the variation in prices among sellers of the same items, often attributed to consumers’ search costs or other unmeasured attributes. The log linearized New Keynesian model, in line with Woodford (2003), incorporates Calvo’s model for price adjustments, while in this thesis the Rotemberg quadratic price adjustment costs are used. This explains the presence of the quadratic term 𝜋_" 𝜋_"− 1 in (2.2). In the Rotemberg setting firms face a symmetric optimisation problem and therefore choose the same price-level explaining why there is no price dispersion (Ascari & Rossi, 2008, pp. 5-6).

2.4 Interest Rate Rule

Given that the central bank wants to reach its optimal inflation target, the CB needs an interest rate rule in order to do so. In addition, the central bank wants the output to be at its most efficient level, which corresponds to the output being as close as possible to its potential or ‘natural’ level. Therefore, in addition to equations (2.1) and (2.2) the model is augmented with a forward-looking Taylor rule. This interest rate rule tells the central bank how to set the nominal interest rate at time 𝑡 given the steady state solution of (2.2) and given the (observable) expectations formation process of 𝜋_"$%& _{and 𝑐}

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(3.1) 𝑖_" = 1 + (𝑖∗− 1)(()*+ , (∗ ) stW∗ W∗V+(a)*+ , a∗ ) suW∗ W∗V+ 𝚤 𝑖𝑓 𝜋_" > 𝜋 𝑖𝑓 𝜋_" < 𝜋

In (3.1) 𝑖_" is the nominal interest rate, 𝑖∗_{is the steady state nominal interest rate,}_𝜋∗_is

the target steady state inflation and 𝑐∗_{the corresponding steady state net consumption.}

Reaction coefficients are given by 𝜙₍ and 𝜙_m, and equal to respectively 2 and 0.5. Each time inflation drops below a certain threshold 𝜋, the central bank could preventively set the nominal interest rate to the zero lower bound. Note that in this thesis there is no separate analysis for the zero lower bound case. Therefore, in the case of 𝜋_"< 𝜋 the interest rate rule still follows a ‘normal’ Taylor rule. Nevertheless, it must be noted that this model is still suitable for incorporating a zero lower bound on the interest rate. Hommes, Massaro and Salle (2015) in their model assume that in the case inflation drops below a threshold of 𝜋 = 1.016, the central bank preventively sets the interest rate at the zero lower bound, 𝚤 = 1.0001.

2.5 Naive Expectations

In the literature on behavioural models there are several models that are subjected to theoretical analysis as an alternative to the rational expectations benchmark, such as adaptive expectations, the heuristics switching model (Anufriev et al., 2013, p. 1581), in which agents use a number of simple forecasting rules, and naive expectations. Evans et al. (2008) show that in model (2.6) – (2.7), given expectations 𝑐_"$%& _{and 𝜋}

"$%& , any level

of inflation 𝜋_" can be achieved setting the government spending 𝑔_" sufficiently high. Therefore, the actual inflation dynamics depends strongly on the forecasting rule(s) that agents use. In line with these suggestions, this thesis has the naive expectations model as main focus point, that is, when agents expect high persistence in inflation and output dynamics, and believe future inflation or output to be equal to their last observed values. Expectations are then formed according to:

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(1) 𝜋_"$%& _{= 𝜋} "<%

(2) 𝑐_"$%& _{= 𝑐} "<%

Substituting this in (2.1), (2.2) and (3.1) leads to a nonlinear New Keynesian framework under naive expectations:

(3.2) 𝑐_" = 𝑐_"<%(()V+ -.)) + 0 (3.3) 𝜋_" 𝜋_"− 1 = 𝛽𝜋_"<% 𝜋_"<%− 1 + 6 78 𝑐"+ 𝑔" +*: ; +%<6 8 𝑐"+ 𝑔" 𝑐" <= (3.4) 𝑖_" = 1 + (𝑖∗− 1)((()V+∗ ) s+W∗ W∗V+(a)V+ a∗ ) scW∗ W∗V+ 𝚤 𝑖𝑓 𝜋_" > 𝜋 𝑖𝑓 𝜋" < 𝜋 𝜙_%, 𝜙_f = 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝑣𝑎𝑙𝑢𝑒𝑑 𝑎𝑡 2 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦 0.5

Notice that the Taylor rule in (3.4) now became backward-looking as its dynamics depend on 𝜋_"<% and 𝑐_"<%.

3 Elaboration of the Model

In the previous section the nonlinear model and the naive expectations model were introduced. The goal of this section is to analyse this framework in MATLAB® and look at the equilibrium outcomes for different targeted inflation rates, as well as their stability.

In order to numerically analyse the steady state outcomes, the 𝑡 indices drop out of the equation, which is equivalent to looking at an infinite time horizon. The goal is to consider a grid of values for the target inflation, analyse its effects within the model on the steady state consumption and determine the optimal choice. Equation (2.2) then simplifies to:

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(3.4) 𝜋∗ _𝜋∗_{− 1 = 𝛽𝜋}∗ _𝜋∗_{− 1 +} 6 78 𝑐 + 𝑔 +*: ; +%<6 8 𝑐 + 𝑔 𝑐 <=

Substituting the parameters with their calibrations in (3.4) results in the following equation for numerical analysis:

(3.5) 𝜋∗ _𝜋∗_{− 1 = 0.99𝜋}∗ _𝜋∗_{− 1 +} f% f‡ˆ 𝑐 ∗_{+ 0.2} c‰_Š ₋ fT ‹ˆT 𝑐 ∗_{+ 0.2 (𝑐}∗₎<%_⟺ 𝜋∗₌ •T Ž (𝑐∗+ 0.2) c‰ Š −‡T Ž 𝑐∗+ 0.2 (𝑐∗)<%+ % ‡+ % f

To have a meaningful model it is a necessary condition for 𝜋 ≥ %

f. Furthermore, it is

assumed that 𝜋 ∈ [0.8, 1.2] for the reason that it is unrealistic that the central bank has such large fluctuations in target inflation. A graphical representation of equation (3.5) is given in the next section.

The second step is to derive the Jacobian matrix and to finds its eigenvalues at the steady state solutions of (3.5). For the steady state to be locally stable, it is required that all the eigenvalues are within the unit circle in the case of real eigenvalues. If the eigenvalues are complex, then the same inequality holds but for its moduli. By solving equation (3.3) explicitly for 𝜋_" and addressing only positive values it follows that: (3.6) 𝜋" = 0.99𝜋"<% 𝜋"<%− 1 +_f‡ˆf% 𝑐"+ 0.2 c‰ Š + fT ‹ˆT 𝑐"+ 0.2 𝑐" <%₊% ‡+ % f

Next periods output and inflation (𝑐_"$% and 𝜋_"$%) are determined by the current values of these variables (𝑐" and 𝜋"). In this model, these two variables are enough to determine

the future dynamics of the system, since 𝑐_"$f and 𝜋_"$f depend on 𝑐_"$% and 𝜋_"$%, and so forth. The system is two dimensional and the state vector can be written as:

(1) _𝜋𝑐"

" = 𝐹

𝑐_"<% 𝜋_"<%

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Substituting the parameters’ calibrations in equation (3.1) and implementing naive expectations results in the following backward-looking Taylor rule:

(3.7) 𝑖_" = 1 + (𝑖∗_{− 1)(}()V+ (∗ ) cW∗ W∗V+(a)V+ a∗ ) ‰.“W∗ W∗V+

To find the Jacobian it is necessary to substitute 𝑖_" in 𝑐_", and to substitute this new equation 𝑐_"”•–_{in 𝜋} ": (3.8) 𝑐_"”•– ₌ a)V+()V+ -$-(.∗_<%)(t)V+ t∗ ) cW∗ W∗V+(—)V+_—∗ )‰.“W∗W∗V+ (3.9) 𝜋"= 0.99𝜋"<% 𝜋"<%− 1 +_f‡ˆf% a)V+_a()V+ ) ˜™š + 0.2 c‰ Š +_‹ˆTfT a)V+()V+ a)˜™š + 0.2 a)V+()V+ a)˜™š <% +%_‡+%_f

With equations (3.8) and (3.9) and given the fact that the system is two dimensional, the Jacobian is derived as:

(1) 𝐽 = œa) œa)V+ œa) œ()V+ œ() œa)V+ œ() œ()V+ (2) œa) œa)V+= ()V+ T.••$T.•• .∗<%(𝜋𝑡−1_𝜋∗ ) 2𝑖∗ 𝑖∗−1(𝑐𝑡−1_𝑐∗ ) 0.5𝑖∗ 𝑖∗−1 <T.‡•ˆ()V+a)V+.∗.∗<%_—∗+(t)V+_t∗ ) cW∗ W∗V+₍—)V+ —∗ ) ‰.“W∗ W∗V+V+ (T.••$T.•• .∗_{<% (}t)V+ t∗ ) cW∗ W∗V+(—)V+_—∗ )‰.“W∗W∗V+)c (3) œa) œ()V+= a)V+ T.••$T.•• .∗<%(𝜋𝑡−1_𝜋∗ ) 2𝑖∗ 𝑖∗−1(𝑐𝑡−1_𝑐∗ ) 0.5𝑖∗ 𝑖∗−1 <T.‡•ˆ()V+a)V+.∗.∗<%_t∗+(—)V+_—∗ ) ‰.“W∗ W∗V+₍t)V+ t∗ ) cW∗ W∗V+V+ (T.••$T.•• .∗_{<% (}t)V+ t∗ ) cW∗ W∗V+(—)V+_—∗ )‰.“W∗W∗V+)c

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The partial derivatives on the last row are too large to mention here and are derived within MATLAB®. In the next section this matrix is evaluated at all steady state solutions (𝜋∗_{, 𝑐}∗_{, 𝑖}∗_{) and a graphical representation is given to show the relationship}

between the two eigenvalues and each target inflation rate 𝜋∗_.

4 Results

Fig. 1: Steady state consumption at different inflation targets

Figure 1 shows the solutions of equation (3.5). It shows that there is a strictly positive relationship between steady state inflation 𝜋∗ _{∈ [0.8 ; 1.2] and steady state net}

consumption. Albeit the effect is small: a percentage increase of 50% in inflation leads to an increase of only 2.2% in net consumption.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.736 0.738 0.74 0.742 0.744 0.746 0.748 0.75 0.752 0.754 c

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The next step is the evaluation of the symbolic Jacobian of the previous section at each steady state solution (𝜋∗_{, 𝑐}∗_{, 𝑖}∗_{) and determining the eigenvalues:}

Figure 2: Eigenvalues of the Jacobian at different inflation targets

Figure 2 shows the eigenvalues computed at 50 different steady state inflation targets 𝜋∗ _{∈ [0.8 ; 1,2]. As mentioned previously, only the eigenvalues that are within the unit}

circle imply stability of the steady state targets. All eigenvalues were real except for the two eigenvalues at 𝜋∗ _{= 0.8. The moduli of the eigenvalues at this target were within the}

unit circle, so 𝜋∗ _{= 0.8 is also within the stable domain. Figure 2 therefore implies that}

only inflation targets 𝜋∗ _{∈ [0.8 ; 0.86] are stable within this particular framework with}

corresponding steady state net consumption 𝑐∗ _{∈ 0.7365 ; 0.7381 and interest 𝑖}∗_∈

0.81 ; 0.87]. 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

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5 Conclusion

This thesis researched an optimal inflation target within a nonlinear New Keynesian framework under homogeneous naive expectations. This was done by solving the main equations of the framework for a host of steady state target inflation rates. Furthermore, the stability condition for each of these values was checked by evaluating the Jacobian and determining its eigenvalues. The parameters’ calibrations used in this thesis were such that the steady state target inflation rates 𝜋∗_{∈ [0.8 ; 0.86] were stable.}

It was shown that in these cases the eigenvalues of the Jacobian fell within the unit circle, implying stability. Inflation targets outside of this domain results in a dynamic system with either ever-increasing or ever-decreasing inflation rates. These results are in line with what the traditional New Keynesian framework justifies: a small, but positive inflation rate. However, there were certain flaws in the model setting; it was highly stylized. To name a few shortcomings: the setting did not account for random shocks, it operated in a closed economy, fiscal policy was assumed to be passive, and the parameters’ calibration values might not sufficiently reflect the current reality enough. What’s more, the New Keynesian seems to be ineffective in guiding monetary policy during liquidity trap episodes, especially when taking into consideration non-rational economic agents. Nevertheless, even though the model setting was not optimal, the results shown give policy-makers a reasonable indication of what is an optimal inflation target.

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A

Matlab

A.1 Steady State Algorithm

beta = 0.99; v = 21; gamma = 350; eps = 1; g = 0.2; sigma = 1; alpha = 0.7; c_sol = zeros(1,50); p = linspace(0.8,1.2,50); for k = 1 : numel(p) pi_actual = p(k); fun = @(c) (1 - beta)*pi_actual*(pi_actual-1) - (v/(alpha*gamma))*(c+g).^((1+eps)/alpha) - ((1-v)/gamma)*(c+g)*c.^(-sigma); c_sol(k) = fzero(fun,0.5); end plot(p,c_sol) A.2 Jacobian syms cpRc2p2 outp = (c2*p2)/(((99*R)/100 - 99/100)*(p2/p)^((2*R)/(R - 1))*(c2/c)^(R/(2*(R - 1))) + 99/100); infl = (((2*c2*p2)/(35*(((99*R)/100 - 99/100)*(p2/p)^((2*R)/(R - 1))*(c2/c)^(R/(2*(R - 1))) + 99/100)) + 2/175)/(((99*R)/100 - 99/100)*(p2/p)^((2*R)/(R - 1))*(c2/c)^(R/(2*(R - 1))) + 99/100) - (99*p2)/100 + (99*p2^2)/100 + (3*((c2*p2)/(((99*R)/100 - 99/100)*(p2/p)^((2*R)/(R - 1))*(c2/c)^(R/(2*(R - 1))) + 99/100) + 1/5)^(20/7))/35 + 1/4)^(1/2) + 1/2;

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J = matlabFunction([jacobian(outp, c2), jacobian(outp, p2) ; jacobian(infl, c2), jacobian(infl, p2)]);

% A.1 repeated

beta = 0.99 ; v = 21 ; gamma = 350 ; eps = 1 ; g = 0.2 ; sigma = 1; alpha = 0.7; c_sol = zeros(1,50); p = linspace(0.8,1.2,50); for k = 1 : numel(p) pi_actual = p(k); fun = @(c) (1 - beta)*pi_actual*(pi_actual-1) - (v/(alpha*gamma))*(c+g).^((1+eps)/alpha) - ((1-v)/gamma)*(c+g)*c.^(-sigma); c_sol(k) = fzero(fun,0.5); end c = c_sol; c2 = c_sol; p2 = p; R = p/beta; % interest % eigenvalues plot eigVals = zeros(2,numel(p)); figure for k = 1:numel(p) A = J(c(k), p(k), R(k), c2(k), p2(k)); eigVals(:,k) = eig(A); hold on; plot(p(k), eigVals(:,k)', '*', ); end

Optimal inflation target : an analysis