Unemployment, the business cycle and monetary policy: augmenting a medium sized new keynesian DSGE model with labor market dynamics / Alexander Koll

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Unemployment, the Business Cycle and

Monetary Policy

Augmenting a Medium Sized New Keynesian DSGE Model

with Labor Market Dynamics

Master’s Thesis

to confer the academic degree of

Master of Science

in the Master’s Program

Economics

Author: Alexander Koll Submission: Department of Economics Thesis Supervisor:

Prof. Dr. Michael Landesmann

Assistant Thesis Supervisor:

Dr. Jochen Güntner

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1

Sworn Declaration

“I hereby declare under oath that the submitted Master's degree thesis has been written solely by me without any third-party assistance, information other than provided sources or aids have not been used and those used have been fully documented. Sources for literal, paraphrased and cited quotes have been accurately credited.

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2

I develop and simulate a medium sized New Keynesian DSGE model that incorporates a variant of the Diamond Mortensen Pissarides Search and Matching model of the labor market. Conventional New Keynesian models struggle to account for involuntary unemployment. In contrast, the model presented here is able to capture the reaction of unemployment to a monetary policy shock. The framework also accounts for the observed inertia and persistence in several aggregate quantities. More work is required to improve the response of inflation in the model.

2. Introduction

The primary motivation for this thesis is twofold. Firstly, in conventional New Keynesian frameworks, nominal wage rigidity is the key feature to replicate the empirically observed inertial, persistent and hump-shaped response of inflation and aggregate variables to a monetary policy shock. I want to ascertain if staggered wage contracts are still an important friction that is essential for the augmented model’s performance.

Secondly, fluctuations in involuntary unemployment are an integral part of the business cycle and an unpleasant fact of everyday life. However, many macroeconomic models struggle to account for this component. As a result, this thesis extends the now conventional New Keynesian model developed by Christiano et al. (2005) to a more realistic framework that incorporates involuntary equilibrium unemployment and delivers predictions for labor market flows.

Chapter 3 and 4 describe the two models in detail. Chapter 5 explains the way the frameworks are linked together. The only significant modification to the Christiano et al. (2005) model is the different treatment of the labor market, which integrates the DMP model. This thesis discusses the issues and considerations that arise in New Keynesian modelling in general, and specifically in the combination of the two frameworks. The model presented here seeks to combine the most plausible considerations of various papers to reach a credible solution. Chapter 6 simulates the augmented model with respect to an expansionary monetary policy shock, as well as a positive technology shock. 1_{For this purpose, the model is calibrated using}

parameter values from the existing literature. Specifically, the parameterizations provided by Christiano et al. (2013) as well as Gertler et al. (2008) are utilized. Furthermore, the importance of several parameters and their implications for the performance of the model are explained. Before proceeding to the conclusion in chapter 7, a short discussion about potential future research is provided. The appendix includes the derivations for all the important equations as well as the respective log-linearizations. The working paper of Christiano et al. (2001) contains any aspect of the New Keynesian model that is not described in this thesis. Moreover, the appendix includes the complete Dynare (Adjemian, et al., 2011) and MATLAB codes needed to replicate the simulations and the corresponding graphs used throughout this thesis.

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3. The Christiano Eichenbaum Evans Model

This chapter explains the model and summarizes the key parts and findings of the paper written by Christiano et al. (2005). The authors, CEE henceforth, develop and simulate a New Keynesian model that is used to examine the combination of real and nominal rigidities that help to replicate the empirically observed inertia and persistence in real variables in response to a monetary policy shock

3.1. Empirical Estimation of a Monetary Policy Shock

Prior to building a model one needs to determine how macroeconomic variables actually respond in reality. CEE describe monetary policy as

𝑅𝑡 = 𝑓(𝛀𝑡) + 𝜖𝑡, 3.1-1

where 𝑅_𝑡 is the federal funds rate, 𝑓 is a linear function of the time 𝑡 information set 𝛀_𝑡 and 𝜖_𝑡 is the monetary policy shock. The Federal Reserve Bank is assumed to allow money growth to be whatever is required to ensure that equation 3.1-1 holds. The maintained assumption for identification states that 𝜖𝑡 is orthogonal to the entries in 𝛀𝑡.

The authors use an identified nine-variable vector autoregressive (VAR) model for estimating the impulse response functions of eight major macroeconomic variables. The VAR has the following structure: 𝒀𝑡 = 𝑌_1𝑡 𝑅_𝑡 𝑌_2𝑡 , 𝒀1𝑡 = [ 𝑅𝑒𝑎𝑙 𝐺𝐷𝑃 𝑅𝑒𝑎𝑙 𝐶𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝐺𝐷𝑃 𝐷𝑒𝑓𝑙𝑎𝑡𝑜𝑟 𝑅𝑒𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑅𝑒𝑎𝑙 𝑊𝑎𝑔𝑒 𝐿𝑎𝑏𝑜𝑟 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦] , 𝒀2𝑡 = [ 𝑅𝑒𝑎𝑙 𝑃𝑟𝑜𝑓𝑖𝑡𝑠 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑀2] 3.1-2

The ordering of the variables entails two key identification assumptions. Firstly, the vector 𝒀_1𝑡 consists of those variables that are assumed to respond slowly to a monetary policy shock. Therefore, they are contained in the time 𝑡 information set. Secondly, 𝒀2𝑡 contains variables

that are allowed to respond contemporaneously with a monetary policy shock and are thus not contained in 𝛀𝑡. For this reason, only past values of 𝒀2𝑡 are included in the time 𝑡 information

set.

The choice of variables in each vector is consistent with the timing assumptions made in the model described below. All variables, except money growth, have been transformed using the natural logarithm but were kept in levels. Since several variables are growing over time, this could theoretically affect the estimation results. However, CEE argue that alternative specifications, that account for potential cointegration relationships, have been tested and that the results are unaffected.

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5 The sample period in the paper spans from the third quarter 1965 to the third quarter 1995.2

Ignoring the constant term, the VAR(4) model, as estimated by CEE, has the following form 𝒀𝑡 = 𝑨1𝒀𝑡−1+ ⋯ + 𝑨4𝒀𝑡−4+ 𝑪𝜼𝑡. 3.1-3

The vector 𝜼_𝑡 is a nine dimensional zero-mean, serially uncorrelated shock with a diagonal variance-covariance matrix. 𝑪 is a 9x9 lower triangular matrix with diagonal terms equal to unity. The seventh element of the vector 𝜼𝑡 is the monetary policy shock denoted by 𝜖𝑡. This

results from the fact that there are six variables in 𝒀_1𝑡.

Note that a contractionary monetary policy shock corresponds to a positive shock to 𝜖_𝑡. The dynamic behavior of 𝒀_𝑡, after a one standard deviation shock to 𝜖_𝑡, is computed by ordinary last squares. The resulting path gives the coefficients in the impulse response functions that CEE are interested in. The authors argue that the ordering of the variables within 𝒀_𝑖𝑡 does not alter the results.

Figure 3.1-1, taken from CEE, shows the impulse response functions of all variables in 𝒀𝑡 following an expansionary one-standard-deviation shock in monetary policy. Lines

marked with plus signs correspond to the VAR based point estimates. Grey areas are the 95 percent confidence intervals from the VAR.3_{Solid lines are the DGE model’s impulse}

responses. The asterisk specifies the period in which the policy shock occurs.

Units on the horizontal axes are quarters, whereas units on the vertical axes denote percentage deviations from the unshocked path, except for inflation, the interest rate as well as the growth rate of money which are denoted in annualized percentage point deviations (APR) from their unshocked path.

Based on the VAR results, CEE point out several interesting responses to an expansionary monetary policy shock.

 Output, consumption as well as investment respond in a hump-shaped manner, reaching their peak after around one and a half years. All three variables return to pre-shock levels roughly three years later.

 Inflation also responds in a sluggish way, while reaching its peak after about two years.  The interest rate decreases for about a year.

 Real profits, real wages and labor productivity rise.  The growth rate of money increases instantaneously.

2_{The sample period is identical to Christiano et al. (1999).}

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Figure 3.1-1 Model and VAR based Impulse Responses (CEE 2005)

Table 3.1-1 Percentage Variance due to Monetary Policy Shocks (CEE 2005)

The authors remark that their estimation strategy focuses solely on the share caused by a monetary policy shock. Table 3.1-1 seeks to explain how large that factor is in relation to the aggregate variation in the data. With the exemption of inflation and the real wage, monetary policy shocks appear to explain a nontrivial fraction of the variation in the variables. However, the sizeable confidence intervals reveal that these point estimates should be interpreted with caution. Moreover, unlike the impulse response functions discussed above, variance decompositions are generally not insensitive to alternative specifications. This thesis abstracts from issues that arise from potential misspecifications.

3.2. The New Keynesian Model

CEE estimate a dynamic general equilibrium model (DGE) with a mixture of five different real and nominal rigidities. Calvo-style nominal price and wage contracts are implemented as nominal rigidities, whereas habit formation in the utility function, convex investment adjustment costs, variable capital utilization as well as working capital loans are considered as

The variance decompositions show how much of the k-step ahead forecast error variance of each of the variables in 𝒀_𝑡 can be explained by the exogenous monetary policy shock, for 𝑘 = 4, 8 and 20 quarters.

The boundaries of the associated 95% confidence intervals are shown in parenthesis and are calculated from the estimated VAR via bootstrapping.

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7 real rigidities. The upcoming sections explain the model economy as developed by CEE.4_Some

additional comments not provided by CEE are added whenever it facilitates the understanding of the model.

3.3. Overview of the Model

Before explaining the parts of the model in detail, it helps to provide a crude overview of the core elements. To facilitate the understanding of the basic structure of the model, Figure 3.3-1 sketches out how the various agents are interconnected. The following sections systematically formulate the problems firms and households face in detail.

A representative, perfectly competitive firm produces a final consumption good that consists of a continuum of intermediate goods, which in turn are produced by monopolists that employ homogenous labor and capital services rented in perfectly competitive factor markets. At the end of each period, profits generated by intermediate goods firms are distributed to a continuum of households, who face several decisions during each period. They choose how much to consume, how many units of capital services they accumulate and supply, how to divide their financial assets into deposits, cash holdings, and so forth.

Figure 3.3-1 The Model Economy

4_{To provide a more comprehensive picture, the description of the model combines parts from the published paper}

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3.4. Final Goods Firm

A final consumption good is produced by a representative, perfectly competitive firm that bundles a continuum of intermediate goods, indexed by 𝑗 ∈ (0,1), using the technology

𝑌_𝑡 = (∫ 𝑌_𝑗𝑡 1 𝜆𝑓 1 0 𝑑𝑗) 𝜆𝑓 , 3.4-1

where 1 ≤ 𝜆_𝑓 < ∞ denotes the amount of substitutability between the different intermediate goods, 𝑌_𝑗𝑡, because 𝜆_𝑓 is related to the elasticity of substitution, denoted by 𝜀, via the relationship 𝜆_𝑓 = 𝜀/(𝜀 − 1). Therefore, if 𝜆_𝑓 = 1, the intermediate goods are perfectly substitutable because in that case 𝜀 → ∞. The linear homogenous function used in equation 3.4-1 is a standard variant of the so-called Dixit-Stiglitz aggregator.5

Since the final goods producer is competitive, it takes its output price, 𝑃𝑡, as well as its input

prices, 𝑃_𝑗𝑡, as given. Profit maximization thus implies the Euler equation, which describes the optimal demand for intermediate good 𝑗,

𝑌𝑗𝑡 = ( 𝑃_𝑡 𝑃_𝑗𝑡) 𝜆𝑓 𝜆𝑓−1 𝑌𝑡, 3.4-2

and the aggregate price index for the final good, which equals

𝑃_𝑡 = (∫ 𝑃_𝑗𝑡 1 1−𝜆𝑓 1 0 𝑑𝑗) 1−𝜆𝑓 . 3.4-3

3.5. Intermediate Goods Firms

Intermediate good 𝑗 is produced by a monopolistically competitive firm, which supplies a differentiated input factor to the final goods producing firm. Capital and labor is rented in perfectly competitive factor markets. More details about the functioning of these markets can be found in the next section that describes the household problem. The production technology is

𝑌𝑗𝑡= {

𝑘_𝑗𝑡𝛼𝑙_𝑗𝑡1−𝛼− 𝜙, 𝑖𝑓 𝑘_𝑗𝑡𝛼𝑙_𝑗𝑡1−𝛼 ≥ 𝜙

0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, 3.5-1

where 0 < 𝛼 < 1. Due to the Cobb-Douglas type production function, the parameter 𝛼 corresponds to the capital share used in the production process. Time 𝑡 labor and capital services are denoted by 𝑙_𝑗𝑡 and 𝑘_𝑗𝑡 respectively. The parameter 𝜙 denotes fixed costs of production. Therefore, 𝜙 > 0 and the value is set to guarantee that profits are zero in a

5_{This framework, named after the authors of the seminal paper written by Dixit and Stiglitz (1977), is used heavily}

in economics for its appealing properties. The function is a CES production function because it exhibits constant elasticity of substitution between the various input factors. A widely referenced source of information about this function is the appendix in Baldwin et al. (2005).

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9 state. Profits are distributed to households at the end of each period. Naturally, production only takes place whenever fixed costs are covered, as represented by 𝑘_𝑗𝑡𝛼_𝑙

𝑗𝑡1−𝛼 ≥ 𝜙.

Entry and exit is ruled out to keep the analysis technically feasible. Since profits are stochastic and zero on average, they must be negative at times. If firms were allowed to exit, companies must also be able to enter, otherwise the economy would end up with no firms, and thus no production. However, once the model allows for entry, the firms cannot remain monopolists because the apparent profit opportunities would result in firms wanting to enter the market and exploit the profitable intermediate goods sectors. Consequently, monopoly power would vanish and a more complicated analysis that endogenously determines entry and exit dynamics would be necessary.

The nominal wage, 𝑊_𝑡, does not have a firm specific subscript because it is chosen to be the same across households that can reoptimize.6_{Moreover, the nominal wage bill, 𝑊}

𝑡𝑙𝑗𝑡, must be

paid at the beginning of each period. As a result, the firms must borrow from financial intermediaries. Repayment, denoted by 𝑊_𝑡𝑙_𝑗𝑡𝑅_𝑡, occurs at the end of each period, where 𝑅_𝑡 denotes the gross interest rate. The authors refer to this setup as working capital loans. They are important because these loans generate a reduction in the firms’ marginal cost whenever the interest rate drops due to an expansionary monetary policy. This in turn leads to a decline in inflation.

Let 𝑅_𝑡𝑘_{be the nominal rental rate on capital. Therefore, total period 𝑡 costs for an intermediate}

goods firm are 𝑆𝑡(𝑌𝑡, 𝑅𝑡𝑘, 𝑊𝑡𝑅𝑡) = 𝑅𝑡𝑘𝑘 + 𝑊𝑡𝑙𝑗𝑡𝑅𝑡. Given Cobb-Douglas technology, cost

minimization implies that real marginal costs have the form

𝑠_𝑡 = ( 1 1 − α) 1−𝛼 (1 𝛼) 𝛼 (𝑟_𝑡𝑘)𝛼(𝑤_𝑡𝑅_𝑡)1−𝛼_, _3.5-2 where 𝑟_𝑡𝑘 _{= 𝑅}

𝑡𝑘/𝑃𝑡 and 𝑤𝑡 = 𝑊𝑡/𝑃𝑡 denote the real rental rate on capital services and the real

wage rate. As usual, capital letters correspond to nominal variables, whereas lower case letters correspond to the respective real terms. The firms’ profits, putting aside fixed costs, are given by the relation [𝑃_𝑗𝑡/𝑃_𝑡 − 𝑠_𝑡 ]𝑃_𝑡𝑌_𝑗𝑡, where 𝑃_𝑗𝑡 denotes firm 𝑗’s price.

Price-setting is assumed to follow a variation of the mechanism suggested in Calvo (1983). The Calvo model of staggered price adjustment is probably the most popular price-setting framework in modern macroeconomics. Only a constant, exogenously determined fraction of firms is allowed to reoptimize its nominal price. Thus, reoptimization is independent across firms and time and each firm faces a constant probability, 1 − 𝜉_𝑝, of being able to optimize prices. CEE assume that firms that can reoptimize its price do this before the realization of the time 𝑡 growth rate of money.

The evident shortcoming of this theory is that firms cannot influence the timing of price adjustments. Nonetheless, empirical evidence suggests that prices are not fully flexible. A more realistic framework results in an inflation equation that becomes complex and hard to solve analytically. For this reason, the Calvo model is a convenient way of generating empirically plausible results within an operational framework. Moreover, CEE explain that one objection to the Calvo style staggered price setting is that the standard formulation implies that inflation

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10 leads output, which is empirically counterfactual, as highlighted by Fuhrer and Moore (1995). However, Gali and Gertler (2000) point out that this criticism does not apply to frameworks in which 𝑠_𝑡 represents real marginal costs rather than the output gap. Therefore, according to CEE, this criticism of the Calvo pricing scheme does not apply to their model.

CEE interpret the Calvo price setting as capturing various costs of changing prices, such as costs that arise from collecting data, from negotiating and communicating new prices or from decision making itself. However, they do not allow for so-called menu cost interpretations because they would apply to all price changes, including the ones associated with the simple lagged inflation indexation scheme explained above. The authors reference microeconomic evidence provided by Zbaracki et al. (2000) that suggests that expenses associated with reoptimization are significantly more important than menu costs.

Firms that cannot reoptimize prices engage in lagged inflation indexation:

𝑃_𝑗𝑡 = 𝜋_𝑡−1𝑃_𝑡−1 3.5-3

In other words, these firms change prices to match the past inflation rate. Therefore, the current inflation rate is characterized by 𝜋𝑡 = 𝑃𝑡+1/𝑃𝑡. Lagged inflation indexation implies that

inflation itself becomes sticky. Given a proper choice of parameters, the model is able to account for the empirically observed degree of serial correlation in inflation. Moreover, let 𝑃̂𝑡

denote the value of 𝑃_𝑗𝑡 chosen by firms that can reoptimize prices at time 𝑡. Just as in the case of wages mentioned above, the nominal reoptimized price does not have a firm specific subscript because it is identical across firms. The firm chooses 𝑃̂𝑡 to maximize

Ε𝑡−1∑(𝛽𝜉𝑝) 𝑙 𝑣𝑡+𝑙(𝑃̂𝑡𝑋𝑡𝑙− 𝑠𝑡+𝑙𝑃𝑡+𝑙)𝑌𝑗,𝑡+𝑙, ∞ 𝑙=0 3.5-4

with 𝑣𝑡 being the marginal value of a dollar to the household, which is due to the assumption

of state contingent securities identical across households. Naturally, 𝑣𝑡 is outside the firms’

control and thus taken as exogenous. The lagged expectations operator, Ε𝑡−1, is conditional on

lagged growth rates of money, denoted by 𝜇_𝑡−𝑙, with 𝑙 ≥ 1. This specification is used because of the assumption that firms set 𝑃̂_𝑡 prior to the time 𝑡 growth rate of money.

The optimization problem maximizes 3.5-4, subject to 3.5-2 and 3.4-2 as well as 𝑋_𝑡𝑙 = {𝜋𝑡× 𝜋𝑡+1× ⋯ 𝜋𝑡+𝑙+1 𝑓𝑜𝑟 𝑙 ≥ 1

1 𝑓𝑜𝑟 𝑙 = 0 . 3.5-5 The reoptimized price 𝑃̂_𝑡 changes firm 𝑗’s profit only as long as it cannot optimize prices itself, in which case 𝑃_{𝑗,𝑡+𝑙} = 𝑃̂_𝑡𝑋_𝑡𝑙. The probability that a firm is forced to engage in lagged inflation indexation is denoted by (𝜉_𝑝 )𝑙.

The first order condition of the optimization problem reads

Ε𝑡−1∑(𝛽𝜉𝑝) 𝑙 𝑣𝑡+𝑙𝑌𝑗,𝑡+𝑙(𝑃̂𝑡𝑋𝑡𝑙− 𝜆𝑓𝑠𝑡+𝑙𝑃𝑡+𝑙) ∞ 𝑙=0 = 0. 3.5-6

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11 In the absence of Calvo style staggered price setting, all firms can optimize prices and 𝜉_𝑝 = 0. Consequently, 3.5-6 reduces to the standard condition that firms set prices as a constant markup, denoted by 𝜆_𝑓, over marginal costs. As already shown in Calvo (1983), equation 3.4-3 can be rewritten as 𝑃𝑡 = [(1 − 𝜉𝑝)𝑃̂_𝑡 1 1−𝜆𝑓 + 𝜉𝑝(𝜋𝑡−1𝑃𝑡−1) 1 1−𝜆𝑓_] 1−𝜆𝑓 . 3.5-7

Log-linearization of equation 3.5-7 in real terms and slightly rearranging yields7

𝑝̂̃_𝑡= 𝜉𝑝

1 − 𝜉_𝑝(𝜋̃𝑡− 𝜋̃𝑡−1). 3.5-8

Given these definitions, log-linearization of equation 3.5-6 results in

𝑝̂̃𝑡 = Ε𝑡−1[𝑠̃𝑡+ ∑(𝛽𝜉𝑝) 𝑙 [(𝑠̃𝑡+𝑙− 𝑠̃𝑡−𝑙−1) + (𝜋̃𝑡+𝑙− 𝜋̃𝑡+𝑙−1)] ∞ 𝑙=1 ], 3.5-9

Equation 3.5-9 together with 3.5-8 can be combined to the inflation Phillips Curve8

𝜋̃𝑡 = 1 1 + 𝛽𝜋̃𝑡−1+ 𝛽 1 + 𝛽𝐸𝑡−1𝜋̃𝑡+1+ (1 − 𝛽𝜉𝑝)(1 − 𝜉𝑝) 𝜉_𝑝 𝐸𝑡−1𝑠̃𝑡. 3.5-10 As outlined in section 3.1, one of the key identification assumptions of the VAR model is that the price level does not respond contemporaneously with a monetary policy shock and is thus not contained in time 𝑡 information set Ω_𝑡. Under this Phillips curve specification, the time 𝑡 inflation rate does not respond to a time 𝑡 monetary policy shock either.

3.6. Households

A continuum of households face a number of decisions during each period. Unlike CEE, this thesis uses 𝑖 ∈ (0,1) instead of 𝑗 ∈ (0,1) to avoid confusion.9_{Each household decides upon}

consumption versus capital accumulation and how many units of capital services to supply. Furthermore, it acquires state contingent securities, which are conditional on whether it can reoptimize its wage decision. These securities guarantee that households are homogenous with respect to consumption and asset holdings. However, they are heterogeneous with respect to the wage rate and the hours they work, as is reflected in the notation, an issue already mentioned in section 3.5.

Households that can reoptimize wages set them according to a Calvo framework that is similar to the one used for price setting. Households are also assumed to receive a lump sum transfer

7_{Section A.2 of the appendix explains the method of log-linearization and the notation. The following derivation}

is already provided here to facilitate a coherent characterization of the intermediate goods sector problem.

8_{Named after the economist Alban W. Phillips, who was the first to observe and describe the short run inverse}

relationship between unemployment and inflation.

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12 from the monetary authority. Furthermore, they decide upon the amount of financial assets they hold in the form of deposits with a financial intermediary or in the form of cash.

The utility function for the 𝑖𝑡ℎ_{household has the form}10

𝛦_𝑡−1𝑖 ∑ 𝛽𝑙[𝑙𝑜𝑔(𝑐𝑡+𝑙− 𝑏𝑐𝑡+𝑙−1) − 𝜓0(𝑛𝑖,𝑡+𝑙) 2 + 𝜓𝑞 𝑞_𝑡+𝑙1−𝜎𝑞 1 − 𝜎𝑞 ] ∞ 𝑙=0 , 3.6-1

where 𝑐_𝑡 denotes time 𝑡 real consumption and 𝑛_𝑖,𝑡 symbolizes time 𝑡 hours worked, which CEE denote as ℎ_𝑖𝑡. Since this quantity denotes labor as measured in the data, the standard notation for employment used in the literature, 𝑛_𝑡, seems more functionally adequate than ℎ_𝑡.11_The

functional form implies a standard disutility of work term with 𝜓₀ set to ensure a steady-state value of labor equal to unity, thus full employment. Furthermore, 𝑞𝑡 = 𝑄𝑡/𝑃𝑡 denotes real cash

balances and 𝑄𝑡 represents nominal cash balances.12

The value for parameter 𝜓_𝑞 is set to guarantee that 𝑄/𝑀 = 0.44 in steady-state, which is the ratio of the money aggregates M1 over M2 at the beginning of the data set. 𝑀 denotes the steady-state stock of money in the model. According to CEE, different values of 𝜓_𝑞 only change the estimate of the elasticity of money demand, 𝜎𝑝. The lagged expectation operator has the

same purpose as described in section 3.5.

The parameter 𝑏 introduces non-separability of preferences over time. In other words, an increase in time 𝑡 consumption lowers marginal utility of current consumption and increases marginal utility of next period’s consumption. Intuitively, old habits are hard to break and new habits are difficult to form. This feature is called habit formation in the literature and is essentially a statement about the behavioural pattern of individuals.

Figure 3.6-1 Habit Formation in the Utility Function

10_{The representation of the utility function is different from CEE because the authors describe the functional form}

of the terms separately. This thesis combines the equations to facilitate notation.

11_{The Search and Matching model discussed in chapter 4 also uses 𝑛}

𝑖𝑡.

12 _{Including money in the utility function (MIU) is a frequently used framework in macroeconomics that was}

initially developed by Sidrauski (1967). This setup models the opportunity costs of holding money with respect to foregone interest payments. Therefore, the MIU term encourages households to optimize money holdings.

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13 Figure 3.6-1 shows how consumption evolves over time in response to a positive monetary policy shock. Habit formation is important for replicating the observed hump-shaped rise in consumption. Standard utility functions conventionally used in economic models cannot generate this pattern.

The households’ budget constraint in nominal terms has the form 𝑀_𝑡+1= 𝑅_𝑡[𝑀_𝑡 − 𝑄_𝑡+ (𝜇_𝑡− 1)𝑀_𝑡𝛼_{] + 𝐴}

𝑖,𝑡 + 𝑄𝑡+ 𝑊𝑖,𝑡𝑛𝑖,𝑡 + 𝑅𝑡𝑘𝑢𝑡𝑘̅𝑡

+ 𝐷𝑡− 𝑃𝑡[𝑖𝑡+ 𝑐𝑡+ 𝑎(𝑢𝑡)𝑘̅𝑡].

3.6-2 The period 𝑡 + 1 money stock of households, denoted by 𝑀_𝑡+1, has to equal the sum of the expressions on the right-hand side. Deposits held at financial intermediaries are defined as [𝑀_𝑡 − 𝑄_𝑡+ (𝜇_𝑡− 1)𝑀_𝑡𝛼]. These deposits earn the gross nominal interest rate 𝑅_𝑡. Moreover, 𝐴𝑖,𝑡 denotes net cash flows from state contingent securities, and 𝑄𝑡 stands for nominal cash

balances. Labor income is given by 𝑊_𝑖,𝑡𝑛_𝑖,𝑡, whereas earnings from supplying capital services are denoted by 𝑅𝑡𝑘𝑢𝑡𝑘̅𝑡. Firm profits are represented by 𝐷𝑡, and nominal consumption is

specified as 𝑃𝑡𝑐𝑡. Lastly, 𝑃𝑡[𝑖𝑡+ 𝑎(𝑢𝑡)𝑘̅𝑡] denotes the stock of installed capital, which is owned

by households and evolves according to

𝑘̅_𝑡+1= (1 − 𝛿)𝑘̅_𝑡+ [1 − 𝜙_𝑖( 𝑖𝑡

𝑖_𝑡−1 − 1)

2

] 𝑖_𝑡. 3.6-3

In other words, next period’s physical capital stock equals the sum of current period’s capital stock adjusted for capital depreciation, denoted by 𝛿, and current and past investment that is transformed via a technology that adds to next period’s installed capital. CEE only discuss the properties of this function without stating the exact functional form, as can be seen in equation 3.6-15. Thanks to my supervisor Dr. Güntner, I was able to write down an operational function that fulfils the properties stated in CEE.

Moreover, the physical capital stock, 𝑘̅_𝑡, is associated with capital services via the relationship 𝑘_𝑡 = 𝑢_𝑡𝑘̅_𝑡, with 𝑢_𝑡 symbolizing the utilization rate of capital, which is effectively a control variable of the households. Finally, (𝜇_𝑡− 1)𝑀_𝑡𝛼_{is a lump sum payment that the}

monetary authority is assumed to pay to households. The variable 𝜇_𝑡 denotes the gross growth rate of the economy-wide per capita stock of money, 𝑀_𝑡𝛼_.

The budget constraint in real terms can be expressed as

𝜋_𝑡+1𝑚_𝑡+1= 𝑅_𝑡(𝑚𝑡− 𝑞𝑡) + (𝜇𝑡− 1)𝑀𝑡𝛼 𝑃_𝑡 + 𝑎𝑖𝑡+ 𝑞𝑡+𝑤𝑖𝑡𝑛𝑖𝑡+ 𝑟𝑡 𝑘_𝑢 𝑡𝑘̅𝑡 + 𝑑_𝑡− 𝑖_𝑡− 𝑐_𝑡+ 𝑎(𝑢_𝑡)𝑘̅_𝑡, 3.6-4

where 𝜋_𝑡 = 𝑃_𝑡/𝑃_𝑡−1 denotes the gross inflation rate of the general price level. Household 𝑖 is assumed to

max

𝑐𝑡, 𝑞𝑡, 𝑚𝑡+1, 𝑢𝑡, 𝑖𝑡, 𝑘̅𝑡+1

utility function 3.6-1

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14 Therefore, the Lagrangean has the form

ℒ = 𝛦_𝑡−1𝑖 ∑ 𝛽𝑙−𝑡 ∞ 𝑙=0 {[𝑙𝑜𝑔(𝑐_𝑡+𝑙− 𝑏𝑐_{𝑡+𝑙−1}) − 𝜓0(𝑛𝑖,𝑡+𝑙) 2 + 𝜓_𝑞 𝑞𝑡+𝑙 1−𝜎𝑞 1 − 𝜎_𝑞] + 𝜓_{𝑐,𝑡+𝑙}[𝑅_𝑡+𝑙(𝑚_𝑡+𝑙− 𝑞_𝑡+𝑙) +(𝜇𝑡+𝑙− 1)𝑀𝑡+𝑙 𝛼 𝑃_𝑡+𝑙 + 𝑎𝑖𝑡+𝑙+𝑞𝑡+𝑙+ 𝑤𝑖𝑡+𝑙𝑛𝑖𝑡+𝑙 + 𝑟𝑡+𝑙𝑘 𝑢𝑡+𝑙𝑘̅𝑡+𝑙+ 𝑑𝑡+𝑙− 𝑖𝑡+𝑙 − 𝑐𝑡+𝑙+ 𝑎(𝑢𝑡+𝑙)𝑘̅𝑡+𝑙− 𝜋𝑡+𝑙+1𝑚𝑡+𝑙+1]}, 3.6-6

where 𝜓_𝑐,𝑡 = 𝑣_𝑡𝑃_𝑡 and 𝑣_𝑡 denotes the marginal value of a dollar to the household, which is due to the assumption of state contingent securities identical across households. As a result, the Lagrange multiplier 𝜓_𝑐,𝑡 denotes the marginal utility of 𝑃_𝑡 units of currency.

The corresponding FOCs are given by: 𝜕ℒ 𝜕𝑐𝑡 : 1 𝑐𝑡− 𝑏𝑐𝑡−1 − 𝛽𝑏 𝑐𝑡+1− 𝑏𝑐𝑡 − 𝜓𝑐,𝑡 = 0. 3.6-7

In CEEs timing convention, the equation has the form

Ε_𝑡−1𝑢_𝑐,𝑡 = Ε_𝑡−1𝜓_𝑐,𝑡, 3.6-8

where 𝑢𝑐,𝑡 denotes the marginal utility of consumption at time 𝑡. Moreover, 𝜓𝑐,𝑡 corresponds

to the value of a dollar in the current period.

Equation 3.6-9 describes the household’s FOC for nominal cash balances, 𝜕ℒ

𝜕𝑞_𝑡: 𝜓𝑞𝑞𝑡 −𝛿𝑞

− 𝜓_𝑐,𝑡(𝑅𝑡− 1) = 0, 3.6-9

which holds irrespective of the realization of the contemporary money growth rate because the cash balance decision is carried out afterwards. Hence, the marginal utility of dollar assigned to cash balances must correspond to the marginal utility of a dollar allocated to the financial intermediary.

𝜕ℒ

𝜕𝑚_𝑡+1: Ε𝑡𝛽𝜓𝑐,𝑡+1𝑅𝑡+1− Ε𝑡𝜓𝑐,𝑡𝜋𝑡+1= 0. 3.6-10

Rewriting this equation results in

Ε𝑡𝛽𝜓𝑐,𝑡+1

𝑅_𝑡+1 𝜋𝑡+1

= Ε𝑡𝜓𝑐,𝑡,

which shows that the expected present discounted value of the cash acquired by depositing a dollar in next period’s financial market matches the value of a dollar in the current period.

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15 𝜕ℒ

𝜕𝑢_𝑡: Ε𝑡−1𝜓𝑐,𝑡[𝑟𝑡

𝑘_{− 𝑎}′_(𝑢

𝑡) ] = 0. 3.6-11

Equation 3.6-11 is the Euler equation that characterizes the household’s capital utilization decision. Accordingly, the expected marginal cost of raising the capital utilization rate must equal the corresponding marginal benefit.

A few further remarks are necessary to determine the FOC with respect to time 𝑡 investment as well as time 𝑡 + 1 physical capital. As already remarked in section 3.3, intermediate goods firms employ homogenous labor and capital services rented in perfectly competitive factor markets. The following considerations are based on lecture notes from the PHD course “Advanced Economics” of my supervisor Dr. Güntner.

Assume that intra-temporal investment is implemented by a competitive capital goods producer that purchases (1 − 𝛿)𝑘𝑡 at the market price, 𝑃𝑘′𝑡, obtains 𝑖𝑡 investment goods, and sells 𝑘𝑡+1

at the same market price, taking equation 3.6-3 into consideration. Since firms are also owned by households, the same discount factor can be used and the competitive capital goods producer maximizes Ε𝑡−1∑ 𝛽𝑙 ∞ 𝑙=0 [𝑃𝑘′,𝑡+𝑙𝑘𝑡+𝑙+1− 𝑃𝑘′,𝑡+𝑙(1 − 𝛿)𝑘𝑡+𝑙− 𝑖𝑡+𝑠] = Ε_𝑡−1∑ 𝛽𝑙 ∞ 𝑙=0 [𝑃_{𝑘′,𝑡+𝑙}(1 − 𝜙_𝑖( 𝑖𝑡+𝑙 𝑖𝑡+𝑙−1 − 1) 2 ) 𝑖_𝑡− 𝑖_𝑡+𝑙]. 3.6-12

Consequently, the FOC with respect to time 𝑡 investment is 𝜕ℒ 𝜕𝑖_𝑡: 𝜓𝑐,𝑡{1 + 𝜙𝑖𝑃𝑘′𝑡[( 𝑖_𝑡 𝑖_𝑡−1) 2 − 𝑖𝑡 𝑖_𝑡−1] + 𝑃𝑘′𝑡 𝜙_𝑖 2 ( 𝑖_𝑡 𝑖_𝑡−1 − 1) 2 } − 𝛽Ε𝑡−1𝑃𝑘′,𝑡+1𝜓𝑐,𝑡+1𝜙𝑖[( 𝑖_𝑡+1 𝑖𝑡 ) 3 − (𝑖𝑡+1 𝑖𝑡 ) 2 ] − 𝜓𝑐,𝑡𝑃𝑘′𝑡 = 0. 3.6-13

CEE do not explicitly state the functional form 𝐹(𝑖_𝑡, 𝑖_𝑡−1) that describes how the capital stock evolves, but rather use the expression provided in equation 3.6-15. Therefore, the FOC in their paper has the shortened, yet equivalent form

Ε_𝑡−1𝜓_𝑐,𝑡 = Ε_𝑡−1[𝜓_𝑐,𝑡𝑃_𝑘′𝑡𝐹_1,𝑡+ 𝛽𝜓_𝑐,𝑡+1𝑃_𝑘′_𝑡+1𝐹_2,𝑡+1], 3.6-14

where 𝐹_𝑗,𝑡 is the partial derivative of 𝐹(𝑖_𝑡, 𝑖_𝑡−1) with 𝑗 = 1,2. This function characterizes how current and past investment can be converted into installed capital in the following period and is specified by

𝐹(𝑖_𝑡, 𝑖_𝑡−1) = [1 − 𝑆 (_𝑖𝑖𝑡

𝑡−1)] 𝑖𝑡. 3.6-15

CEE only restrict the function to the following properties: 𝑆(1) = 𝑆′_{(1) = 0 and the}

investment adjustment cost parameter 𝜂_𝑘 ≡ 𝑆′′_{(1) > 0. The right-hand side of equation 3.6-14}

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16 an extra unit of the investment goods, whose value is denoted by 𝑃_𝑘′𝑡𝐹_1,𝑡Ε_𝑡−1𝜓_𝑐,𝑡. Moreover, increasing period 𝑡 investment affects next period’s quantity of installed capital by 𝐹2,𝑡+1. The

respective discounted value is given by 𝛽𝜓_𝑐,𝑡+1𝑃_𝑘′_𝑡+1𝐹_2,𝑡+1. Note that the price of investment

goods in terms of consumption is equal to unity. Hence, equation 3.6-14 states that the marginal cost of one unit of investment corresponds to the sum of these values.

Likewise, for the FOC with respect to time 𝑡 + 1 physical capital, the household is assumed to purchase 𝑘_𝑡 in period 𝑡 − 1 at the competitive market price 𝑃_𝑘′_,𝑡−1. In period 𝑡, the household

rents out the capital stock at the real capital rental rate, 𝑟_𝑡𝑘_{, and sells the depreciated capital}

stock at the end of the period at the current market price 𝑃_𝑘′_𝑡. As a result, the respective terms

in the budget constraint read

Ε_𝑡−1∑ 𝛽𝑙 ∞ 𝑙=0 𝜓_{𝑐,𝑡+𝑙}[𝑟_𝑡+𝑙𝑘 𝑢_𝑡+𝑙𝑘̅_𝑡+𝑙+ 𝑃_𝑘′_,𝑡+𝑙(1 − 𝛿)𝑘̅_𝑡+𝑙 − 𝑃_𝑘′_{,𝑡+𝑙−1}𝑎(𝑢_𝑡+𝑙)𝑘̅_𝑡+𝑙], 3.6-16

and hence the Euler equation for next period’s physical capital stock 𝑘̅𝑡+1 has the form

𝜕ℒ 𝜕𝑘̅_𝑡+1: 𝛽Ε𝑡−1𝜓𝑐,𝑡+1[𝑢𝑡+1𝑟𝑡+1 𝑘 _{+ 𝑃} 𝑘′,𝑡+1(1 − 𝛿) − 𝑃𝑘′,𝑡𝑎(𝑢𝑡+1)] − Ε_𝑡−1𝜓_𝑐,𝑡𝑃_𝑘′_,𝑡 = 0. 3.6-17

To obtain the expression found in CEE, equation 3.6-17 needs to be divided by 𝑃_𝑘′_,𝑡. Note that

there is a typo in the equation provided in the working paper of the authors, because 𝑃_𝑘′_,𝑡𝑎(𝑢_𝑡+1) misses 𝑃_𝑘′_,𝑡. As will be shown in the appendix, this typo is irrelevant in

the log-linearization of the equation, since the term cancels out either way.

3.7. The Wage Decision

Based on Erceg et al. (2000), CEE adopt a wage setting decision that takes place between households and a representative, competitive firm that converts labor into an aggregate labor input, 𝑙𝑡, which is used in the intermediate goods sector, as described in section 3.5.

The following technology converts labor into the aggregated labor input 𝑙𝑡 = (∫ 𝑛_𝑖𝑡 1 𝜆𝑤 1 0 𝑑𝑖) 𝜆𝑤 , 3.7-1

where 𝑛_𝑖𝑡 denotes a differentiated labor service that is supplied by household 𝑖. The CES production function is technically identical to the one used in the final goods production, as described in section 3.4. Therefore, the optimal demand for labor supplied by household 𝑖 equals 𝑛_𝑖𝑡 = (𝑊𝑡 𝑊_𝑖𝑡) 𝜆𝑤 𝜆𝑤−1 𝑙_𝑡, 3.7-2

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17 with 1 ≤ 𝜆_𝑤 ≤ ∞. The nominal price of aggregate labor is denoted by 𝑊_𝑡 and related to the wage set by the 𝑖𝑡ℎ_{household via the function}

𝑊_𝑡 = [∫ 𝑊_𝑖𝑡 1 1−𝜆𝑤 1 0 𝑑𝑖] 1−𝜆𝑤 . 3.7-3

Households take 𝑙𝑡 as well as 𝑊𝑡 as given. Wage setting is done in the fashion of the Calvo

price setting decision. Consequently, households face a constant probability, 1 − 𝜉_𝑤, of being capable to reoptimize their nominal wage, which is independent across time and households. Households being unable to optimize their wage at time 𝑡 change wages to match the past inflation rate. This is called lagged inflation indexation, as described in section 3.5. Therefore, these wages are set as

𝑊_𝑖,𝑡 = 𝜋_𝑡−1𝑊_{𝑖,𝑡−1}. 3.7-4

In line with the discussion from the price setting decision, 𝑊̂_𝑡_{denotes the value of 𝑊}_𝑖𝑡_chosen by households that can reoptimize prices at time 𝑡. This wage is identical across households that can reoptimize.

The FOC with respect to 𝑊̂_𝑡_{is, in real terms,}

Ε𝑡−1∑(𝛽𝜉𝑤)𝑙𝜓𝑐,𝑡+𝑙( 𝑊̂_𝑡𝑋_𝑡𝑙 𝑃𝑡+𝑙 − 𝜆𝑤 2𝜓₀(𝑛_{𝑖,𝑡+𝑙}) 𝜓𝑐,𝑡+𝑙 ) 𝑛𝑖,𝑡+𝑙 = 0, ∞ 𝑙=0 3.7-5

where 𝑋𝑡𝑙 is defined just like in equation 3.5-5. CEE write 𝑧𝑛,𝑡+𝑙 instead of 2𝜓0(𝑛𝑖,𝑡+𝑙), because

they do not explicitly state the functional form of the disutility of labor term in the household utility function 3.6-1. Similar to the findings in section 3.5, assuming fully flexible wages, ergo 𝜉_𝑤 = 0, reduces equation 3.7-5 to

𝑊̂𝑡

𝑃𝑡

− 𝜆_𝑤 𝑧𝑛,𝑡 Ε𝑡−1𝑢𝑐,𝑡

= 0. 3.7-6

Therefore, in the absence of wage rigidities, households set real wages equal to a constant markup 𝜆𝑤, times the expected marginal rate of substitution between consumption and leisure.

Note that equation 3.6-8 was used to get from 3.7-5 to 3.7-6. The “wage Phillips curve” is derived by log-linearizing equation 3.7-5, together with 3.7-2. Therefore,

𝑤̃_𝑡−1 =𝑏𝑤(1 + 𝛽𝜉𝑤 2_{) − 𝜆} 𝑤 𝑏_𝑤𝜉_𝑤 𝐸𝑡𝑤̃𝑡− 𝛽𝐸𝑡𝑤̃𝑡+1 − 𝐸_𝑡[𝛽(𝜋̃_𝑡+1− 𝜋̃_𝑡) − (𝜋̃_𝑡− 𝜋̃_𝑡−1)] −1 − 𝜆𝑤 𝑏_𝑤𝜉_𝑤 𝐸𝑡(𝜓̃𝑐,𝑡− 𝑙̃𝑡), 3.7-7 where 𝑏_𝑤 = (2𝜆_𝑤 − 1)/[(1 − 𝜉_𝑤)(1 − 𝛽𝜉_𝑤)].

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18

3.8. Monetary and Fiscal Policy

Unlike the standard monetary policy representation used in CEE, this thesis uses a Taylor rule of the form 𝑅_𝑡 𝑅 = ( 𝑅_𝑡−1 𝑅 ) 𝜌𝑟 [(Ε𝑡𝜋𝑡+1 𝜋 ) 𝜌𝜋 (𝑌𝑡 𝑌) 𝜌𝑌 ] 1−𝜌𝑟 𝑒𝜖𝑡_, 3.8-1

where 0 ≤ 𝜌_𝑟 ≤ 1 determines the central bank’s pursuit to smooth interest rates over time.13 A value different from zero reflects interest rate inertia in the Taylor rule. Moreover, 𝜌_𝜋 gauges the central bank’s reaction to deviations of inflation from gross steady-state inflation, which is defined by 𝜋 = 𝑃/𝑃 = 1. The parameter 𝜌_𝑌 regulates the central bank’s response to deviations of output of the final goods from steady-state output. The mean zero, identically and independently distributed (i.i.d) random error term 𝜀_𝑡 accounts for the monetary policy shock described in equation 3.1-1. Please refer to section 6.2 for more details about the Taylor rule. The government is assumed to impose non distortionary lump sum taxes. Furthermore, it follows a Ricardian fiscal policy. Consequently, fiscal policy need not be specified and inflation is unaffected by government actions, as CEE explain.

3.9. Market Clearing and Equilibrium

According to the authors, financial intermediaries obtain 𝑀_𝑡− 𝑄_𝑡 from households and a transfer (𝜇𝑡− 1)𝑀𝑡 from the monetary authority. In equilibrium, 𝑀𝑡 = 𝑀𝑡𝑎, where 𝑀𝑡𝑎 defines

the economy-wide per capita stock of money. Likewise, the variable 𝜇𝑡 denotes the gross

growth rate of 𝑀_𝑡𝛼_{. Therefore, the total amount of money that financial intermediaries receive}

is 𝜇_𝑡𝑀_𝑡− 𝑄_𝑡.

Loan market clearing implies

𝜇_𝑡𝑀_𝑡− 𝑄_𝑡 = 𝑊_𝑡𝑙_𝑡, 3.9-1

where 𝑊𝑡𝑙𝑡 denotes the nominal wage bill, which intermediate goods producers must pay at the

beginning of each period. Consequently, these firms must borrow from financial intermediaries. Finally, the aggregate resource constraint reads

𝑐𝑡+ 𝑖𝑡+ 𝑎(𝑢𝑡) ≤ 𝑌𝑡. 3.9-2

4. The Search and Matching Model

As mentioned in the introduction, the frictional equilibrium unemployment model presented here is based on papers written by Diamond (1982) as well as Mortensen and Pissarides (1994). The underlying idea is that firms and workers are simultaneously searching for matching counterparts in the labor market. Therefore, the literature generally refers to the model as the Diamond-Mortensen-Pissarides Search and Matching model of unemployment (the DMP

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19 model, hereafter). This chapter develops a standard DMP model. Any considerations regarding the integration of the DMP model into the CEE framework are left for chapter 5.

The model seeks to explain the existence of involuntary unemployment via frictional search unemployment. In other words, there are workers that are willing to work at the current wage but cannot find jobs. Search efforts by workers and firms are coordinated via a so-called matching function that determines flows from unemployment to employment. On the other hand, current matches are destroyed with an exogenous separation probability that can be interpreted as firing or turnover dynamics which generate flows from employment to unemployment.

The core assumptions of the basic model are as follows:

‒ Workers and firms are heterogeneous with respect to their skills, skill requirements as well as their location.

‒ Workers and firms are risk neutral and all have imperfect information regarding their potential counterparts.

‒ Constant returns to scale (CRS) in production and the matching function: no need to distinguish between firms and jobs.

‒ Free entry of firms ensures that the value of posting a vacancy is zero in equilibrium. ‒ Search is costly but each match generates a surplus that is shared between workers and

firms via generalized Nash bargaining.

‒ Workers always participate in the job market and their reservation wage is lower than the productivity of any offered job. Thus, workers always accept the offer.

To find new workers and to hire 𝑛_𝑗𝑡 employees, each firm post vacancies, denoted by 𝑣_𝑗𝑡. The total number of vacancies is calculated as 𝑣𝑡 = ∫ 𝑣𝑗𝑡𝑑𝑗

1

0 and the total number of employed

workers is 𝑛𝑡 = ∫ 𝑛𝑗𝑡𝑑𝑗 1

0 respectively.

14_{It is assumed that all unemployed workers search for a}

job. Moreover, the total work force is normalized to unity. Therefore, the pool of unemployed workers coincides with the unemployment rate, which is calculated as the difference between unity and employment.

𝑢𝑡 = 1 − 𝑛𝑡. 4-1

Two different timing assumptions are used in the literature. Firstly, newly employed workers have to wait until next period before they start working. This setup is in line with the baseline DMP model and for example used by Krause and Lubik (2007). Secondly, newly hired workers are expected to meet with firms, to bargain and to start to work immediately in the same period. This framework is for instance used by Gertler et al. (2008) as well as Christiano et al. (2013). As mentioned in section 3.1, CEE’s sample period for their VAR estimations is 1965Q3- 1995Q3. The U.S. labor market has a median duration of unemployment of roughly 7 weeks in the model’s time period (Fred Database, 2015). Christiano et al. (2013) reason that using quarterly data is comparatively long with respect to the US unemployment duration. Therefore, the authors argue that it seems more plausible to use their employed timing assumption.

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20

Figure 4-1 Dice-DFH Mean Vacancy Duration Measure

On the other hand, not all workers apply for jobs and bargain only at the beginning of each period. It seems more realistic that potential employees and firms continuously try to find suitable matches. This in turn implies that the workforce would also fluctuate intra-temporally. Moreover, using more up-to-date data reveals that employers are recently waiting longer to fill vacant positions. According to the Fred database, the median duration of unemployment is slightly above twelve weeks since the turn of the millennium.

Furthermore, the Dice-DFH Vacancy Duration Measure, as developed by Davis et al. (2010), quantifies the average number of working days taken to fill vacant job positions. As can be seen in Figure 4-1, the index ranges approximately between 15 and 26 working days over the last 14 years (Dice Holding, 2015). Consequently, this thesis assumes that newly hired workers have to wait until next period before they begin working. As a result, it is assumed that newly hired workers remain in the work force for at least one period, which is in line with Christiano et al. (2013) as well as Gertler et al. (2008).15

The matching function 𝑚_𝑡, which determines the number of newly hired workers, is given by

𝑚_𝑡 = 𝛾𝑢_𝑡𝜉𝑣_𝑡1−𝜉, 4-2

with 0 < 𝜉 < 1.16_{This Cobb-Douglas type function describes the result of the matching process}

which depends on the constant efficiency parameter 𝛾, the time 𝑡 unemployment rate 𝑢_𝑡, as well as the available job vacancy rate 𝑣_𝑡, which is computed as the quantity of unfilled jobs expressed as a proportion of the labor force. Furthermore, the matching function’s diminishing marginal return implies a congestion externality because each worker searching for a job decreases the likelihood of others finding a job.

15_{Other authors like Krause and Lubik (2007) assume that new employees are immediately subject to job}

separation.

16_{The name matching function originates from newly hired workers being termed matches.} 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Jan -01 A u g-01 Mar -02 O ct -02 May-03 De c-03 Ju l-04 Fe b -05 Se p -05 A p r-06 N o v-06 Ju n -07 Jan -08 A u g-08 Mar -09 O ct -09 May-10 De c-10 Ju l-11 Fe b -12 Se p -12 A p r-13 N o v-13 Ju n -14 Jan -15 Wor ki n g Day s

Source: Dice Hiring Indicators

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21 Labor market tightness 𝜃, from the worker’s perspective, is defined as vacancies over unemployment

𝜃𝑡≡

𝑣_𝑡

𝑢𝑡. 4-3

The job filling rate of firms is denoted by17

𝑞_𝑡 = 𝑞(𝜃_𝑡) =𝑚𝑡

𝑣_𝑡. 4-4

Using the matching function this can be shown to be decreasing in labor market tightness

𝑞(𝜃_𝑡) = 𝑚𝜃_𝑡−𝜉, 4-5

with – 𝜉 being the elasticity of the job filling rate with respect to the labor market tightness, defined as 𝜕𝑞𝑡

𝜕𝜃𝑡 𝜃𝑡 𝑞𝑡.

On the other hand, the job finding rate of workers is increasing in labor market tightness 𝑓𝑡 = 𝜃𝑡𝑞(𝜃𝑡) =

𝑚_𝑡 𝑢𝑡

= 𝑚𝜃𝑡1−𝜉. 4-6

Both the job finding and the job filling rate depend on aggregate variables and are thus taken as given by individual firms and workers and can be interpreted as probabilities.

The law of motion for employment is described by the function

𝑛_𝑡= (1 − 𝜌)𝑛_𝑡−1+ 𝑥_𝑡−1𝑛_𝑡−1, 4-7 where 𝑛_𝑡 denotes employment and 𝜌 the exogenous job separation rate. The presumed stability of job separations is based on findings by Hall (2005) and Shimer (2005), who argue that the job separation rate is relatively acyclical, or weakly countercyclical. Hall explains that it is a common, yet erroneous belief that the sharp rise of unemployment during recessions were the consequence of increased job separation rates. In fact, unemployment increases mainly because of reduced hiring rates during downturns and changes in the job separation rate are miniscule relative to the observed movements in employment.

Moreover, Shimer (2005) deduces that a time varying job separation cannot be important because it would imply a positively sloped Beveridge curve, which plots the relationship between the unemployment and the vacancy rate.18_{The latter is typically depicted on the}

vertical axes. Empirically, the Beveridge curve is negatively sloped, which suggests that high unemployment rates are generally accompanied by a low vacancy rate, and vice versa.

As a result of the stable job separation rate, fluctuations in unemployment are due to cyclical variation in hiring in this model. Moreover, workers who lose their job are not allowed to search

17_{The job filling rate is also called the job matching rate in the literature.} 18_{The curve is named after the economist William H. Beveridge.}

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22 for a new occupation until the next period. The hiring rate is defined as the ratio of new hires to the existing workforce

𝑥_𝑖𝑡 = 𝑞𝑡𝑣𝑖𝑡

𝑛_𝑖𝑡 . 4-8

Given this setup, the hiring rate is effectively a firm’s control variable since the likelihood that each posted vacancy will be filled is known to be the job filling rate 𝑞_𝑡. Note that the firm’s problem is equivalent to choosing how many vacancies to post, since one implies the other. Moreover, using the definition of the hiring rate together with the job matching rate, equation 4-7 can be rewritten as

𝑛_𝑡= (1 − 𝜌)𝑛_𝑡−1+ 𝑚_𝑡−1. 4-9

4.1. Workers

The worker’s choices depend on the value of employment in period 𝑡 which equals19

𝐻_𝑡 = 𝑤_𝑡+ Ε_𝑡𝛽[(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1]. 4.1-1 The real wage, denoted by 𝑤𝑡, is the result of the Nash Bargaining process described below.

The term in parenthesis describes the expected discounted value of being employed or unemployed in period 𝑡 + 1, weighted by the relevant probabilities.

The value of being unemployed is given by20

𝐼_𝑡 = 𝑜 + Ε_𝑡𝛽[𝑓_𝑡𝐻_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1], 4.1-2 with 𝜊 being the worker’s outside option which can be interpreted as an umbrella term for all sorts of different non-market activities and options considered in the literature.21_{Since the}

outside option is constant in this thesis, no exact interpretation is provided. Otherwise, all possible explanations would always have to exactly offset each other. As explained in section 3.8, the government is assumed to impose non distortionary lump-sum taxes. Insofar, fiscal policy need not be specified, as CEE explain.

Section 6.4 explains the importance of the outside option for the model’s performance. A more detailed discussion of the labor market in general can be found in section 5.4. As before, the expression in parenthesis of equation 4.1-2 describes the expected discounted value of being employed or unemployed in period 𝑡 + 1. This time, weighted by the job finding rate and its complement, respectively.

19_{The capital letter 𝐻 was chosen because 𝑁 already denotes employment and 𝐿 denotes the aggregate labor input}

as used in the intermediate goods production process. Therefore, the value of employment is denoted as 𝐻 to avoid confusion. It helps to think of 𝐻 as standing for “hired”, thus denoting the value of being hired.

20_{The capital letter 𝐼 was used because 𝑈 already denotes unemployment. 𝐼 can be interpreted as idleness.} 21_{For instance, the outside option can be understood as the disutility of working (Lubik, 2009), the value of home}

production (Walsh & Ravenna, 2007), the value of being unemployed (Christiano et al., 2013) or the value of leisure (Shimer, 2005).

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23

4.2. Firms

Firms can post vacancies (or job openings, positions) at a constant cost 𝜅 per period and vacancy. The value of a vacancy is therefore given by

𝑃_𝑡 = −𝜅 + Ε_𝑡𝛽[𝑞(𝜃_𝑡)𝐽𝑡+1+ (1 − 𝑞(𝜃𝑡))𝑃𝑡+1], 4.2-1

with the weight 𝑞(𝜃𝑡) being the aforementioned job matching rate, or the probability of filling

a job. The value of job 𝐽 is described via the relationship

𝐽_𝑡= 𝐴_𝑡− 𝑤_𝑡+ Ε_𝑡𝛽[(1 − 𝜌)𝐽_𝑡+1+ 𝜌𝑃_𝑡+1], 4.2-2 where 𝐴_𝑡 is the productivity of firms. Constant returns to scale in the production function as well as the matching function imply that this relationship can also be interpreted as the value of a firm. The value of next period’s job 𝐽_𝑡+1 is weighted by the probability of still having a job in that period, denoted by 1 − 𝜌. Likewise, the value of posting a vacancy in period 𝑡 + 1 is weighted by the probability of being laid-off (the job separation rate). This accounts for the fact that unemployed people increase the value of a vacancy for firms.

Free entry of firms ensures that, in equilibrium, no producer can generate excess returns. Hence, the value of posting a vacancy must be zero (𝑃_𝑡= 0) and 4.2-1 can be rewritten as

𝜅

𝑞(𝜃_𝑡) = Ε𝑡𝛽𝐽𝑡+1, 4.2-3

where 1/𝑞(𝜃_𝑡) = 𝑣_𝑡/𝑚_𝑡 can be interpreted as the expected duration of a posted vacancy. Combining this with 4.2-2, again using 𝑃_𝑡 = 0, yields the job creation condition

𝜅

𝑞(𝜃_𝑡)= Ε𝑡𝛽 [𝐴𝑡+1− 𝑤𝑡+1+ (1 − 𝜌) 𝜅

𝑞(𝜃_𝑡+1)]. 4.2-4 The equality in 4.2-4 states that 𝜅/𝑞(𝜃𝑡) , the expected cost of posting a vacancy, equals the

expected benefit from a filled vacancy. Further considerations are left unexplained in this section. Refer to section 5.2 for a complete treatment of the firms in the DMP model within the CEE framework.

4.3. Wage determination

The equilibrium real wage 𝑤_𝑡 is the outcome of an axiomatic Nash Bargaining process that maximizes the generalized Nash product denoted by

max

𝑤𝑡 (𝐽𝑡− 𝑃𝑡)

1−𝜂_(𝐻

𝑡− 𝐼𝑡)𝜂, _4.3-1

with 𝜂 being the workers bargaining power, or the share of the joint surplus that goes to the workers.22_{The final solution to this bargaining process is}

𝑤𝑡= 𝜂(𝐴𝑡+ 𝜅𝜃𝑡) + (1 − 𝜂)𝑜. 4.3-2

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24 As a result, the real wage is the weighted average of productivity, labor market tightness and the outside option of workers. Moreover, by the definition of the labor market tightness, the more firms look for workers (the higher 𝑃_𝑡, given 𝐼_𝑡) or the fewer workers look for jobs (the lower 𝐼_𝑡, given 𝑃_𝑡), the higher the wage the worker can get.

5. Augmenting the CEE Model with the Search and Matching Model

There are five points of contact that link the two models together. The first point is the redefined discount factor of firms in the DMP model. The second point regards the way the firm problem is formulated in the DMP model in order to introduce output, which is neglected in traditional Search and Matching frameworks. This also slightly changes the wage bargaining process, which is the third point of contact. The fourth point regards the DMP firms producing the aggregate labor input that is used in the intermediate goods production process. The final point of contact concerns the resource constraint.

The augmented model is known as the New Keynesian Search and Matching model in the literature, or NKSM model for short. This chapter discusses the issues and considerations that arise in New Keynesian modelling in general, and specifically in the combination of the two models. Overall, several related, yet competing frameworks exist and more research is needed to determine the preferable model. The model presented here seeks to combine the most plausible considerations of various papers to reach a credible solution.

The first papers incorporating DMP frameworks into DSGE models date back to Merz (1995) and Andolfatto (1996). However, both papers used Real Business cycle models instead of New Keynesian DSGE models. Seminal papers that combine contemporary New Keynesian models with Search and Matching frameworks are Walsh (2003), Krause and Lubik (2007), Gertler et al. (2008) and Christoffel et al. (2009). Furthermore, Christiano et al. (2013) build a model that uses a setup of the labor market that is close to the one considered in this thesis.

Contrary to CEE, Gertler et al. (2008) assume that final goods producers are monopolistically competitive, whereas intermediate goods producers (wholesalers in Gertler et al.) are assumed to be competitive. This circumvents bargaining spillovers amid employees, which is needed because Gertler et al. (2008) incorporate the labor market directly into the intermediate goods producers’ problem. More details can be found in the authors’ paper. Due to the separated labor market in CEE, there is no need to worry about bargaining spillovers in this framework, despite monopoly power of intermediate goods producers.

The question of which of the two variants is more realistic remains unanswered in this thesis, which attempts to change as little as possible from the CEE model. As a result, it assumes that the labor market is separated from the intermediate goods sector.

5.1. The Discount Factor of Firms

The first point of contact arises via a redefined discount factor of the firms operating in the DMP model, which is now given by

Ε𝑡𝛽 ( 𝑈_𝑐,𝑡+1 𝑈𝑐,𝑡 ) = Ε𝑡𝛽 ( 𝜓_𝑐,𝑡+1 𝜓𝑐,𝑡 ), 5.1-1

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25 where 𝑈_𝑐,𝑡 denotes the marginal utility of consumption from the CEE model. This construction of a stochastic pricing Kernel is consistent with other general equilibrium models that account for labor market variables.23

The original DMP framework does not model production because the main interest lies in employment dynamics. In this thesis, companies operating in the DMP part of the model economy bundle the workforce to a homogenous aggregate labor input using the following production function24

𝑙_𝑗𝑡= 𝐴_𝑡𝑛_𝑗𝑡. 5.1-2

Therefore, technology is linear in labor input and driven by an aggregate labor augmenting technology shock that is identical to all firms. This shock follows the stationary law of motion

𝐴𝑡 = 𝐴𝑡−1 𝜌

𝐴1−𝜌eϵt_, _5.1-3

with 𝜖_𝑡∼ 𝑁(0, 𝜎_𝜖2_{) and 𝜌 < 1 being the degree of autocorrelation. A more detailed discussion}

of technology shocks can be found in section 6.5.

Given the linear production function, firms are indifferent to the quantity of labor hired at a given wage rate. Due to the separation of the labor market and the intermediate goods producers in the CEE model, there is only a small change to the Nash bargaining setup used in the standard DMP model, which is due to a modification in the marginal product of a worker.

5.2. Firms in the DMP Framework

Recall from section 3.5 that intermediate goods firms rent labor and capital in perfectly competitive factor markets. Consequently, each labor market firm faces a perfectly elastic demand curve and the market price for the aggregate labor input used in the intermediate goods production process, denoted by 𝑃𝑡𝑤, is taken as given by individuals.

The equilibrium capacity of the labor output of a single labor market firm will be entirely determined by the quantity of output the firm decides to supply. Therefore, equation 5.1-2 represents the output constraint, which can be written as

𝐴_𝑡𝑛_𝑗𝑡≥ 𝑙_𝑗𝑡, 5.2-1

where the left-hand side characterizes the potential maximum output of the homogenous aggregate labor input, because 𝑛_𝑗𝑡 corresponds to the labor force as measured in the data. As usual, profits are calculated as revenues (𝑃_𝑡𝑤_/𝑃

𝑡 )𝑙𝑗𝑡 minus costs, which consist of the real wage

bill 𝑤_𝑗𝑡𝑛_𝑗𝑡 and vacancy posting costs 𝑘𝑣_𝑗𝑡.

23_{Examples of papers that employ this setup are Christiano et al. (2013), Lubik (2009) or Gertler et al. (2008).}

More details about the usefulness of a stochastic discount factor can be found in Hansen and Renault (2010).

24_{Eventually, all 𝑗 subscripts can be erased due to the fact that constant returns to scale in the DMP firms}

production and matching functions imply that there is no need to distinguish between firms. Recall that subscript 𝑗 is used to denote firms, whereas subscript 𝑖 characterizes households. Thus, 𝑛𝑗𝑡 = 𝑛𝑖𝑡 must hold in equilibrium.

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26 As a result, the intertemporal profit maximization problem of labor market firms has the following form: max 𝑙𝑖𝑡, 𝑛𝑡, 𝑣𝑖𝑡 𝐸₀∑ 𝛽𝑡𝜓𝑐,𝑡 𝜓_𝑐,0 ∞ 𝑡=0 [(𝑃𝑡 𝑤 𝑃_𝑡) 𝑙𝑗𝑡− 𝑤𝑗𝑡𝑛𝑗𝑡− 𝑘𝑣𝑗𝑡]. 5.2-2 Consequently, the Lagrangean becomes

ℒ = 𝐸0∑ 𝛽𝑡 𝜓_𝑐,𝑡 𝜓𝑐,0 ∞ 𝑡=0 [(𝑃𝑗𝑡 𝑤 𝑃𝑡 ) 𝑙𝑗𝑡− 𝑤𝑗𝑡𝑛𝑗𝑡− 𝑘𝑣𝑗𝑡] + 𝐸₀∑ 𝛽𝑡𝜓𝑐,𝑡 𝜓_𝑐,0 ∞ 𝑡=0 𝑚𝑐_𝑗𝑡𝑤[𝐴_𝑡𝑛_𝑗𝑡− 𝑙_𝑗𝑡] + 𝐸₀∑ 𝛽𝑡𝜓𝑐,𝑡 𝜓_𝑐,0 ∞ 𝑡=0 𝜇_𝑗𝑡[(1 − 𝜌)𝑛_𝑗𝑡−1+ 𝑞(𝜃_𝑡−1)𝑣_𝑗𝑡−1− 𝑛_𝑗𝑡], 5.2-3

where the Lagrange multiplier 𝜇_𝑗𝑡 denotes the value of a job, as will be explained below. For the second constraint, equation 4-4 was used to rewrite the employment accumulation equation, denoted by equation 4-9. Note that marginal costs act as a Lagrange multiplier in this setup.25

This is feasible because the contribution of an additional unit of output to the firm’s revenue equals marginal costs.

The FOCs have the following form: 𝜕ℒ 𝜕𝑙𝑖𝑡 : 𝑃𝑗𝑡 𝑤 𝑃𝑡 = 𝑚𝑐_𝑗𝑡𝑤 5.2-4 𝜕ℒ 𝜕𝑛_𝑖𝑡: 𝑤𝑡 = 𝑚𝑐𝑗𝑡 𝑤_𝐴 𝑡− 𝜇𝑗𝑡+ (1 − 𝜌)𝐸𝑡𝛽 𝜓𝑐,𝑡+1 𝜓_𝑐,𝑡 𝜇𝑗𝑡+1 5.2-5 𝜕ℒ 𝜕𝑣𝑖_𝑡: 𝜅 = 𝐸𝑡𝛽 𝜓𝑐,𝑡+1 𝜓_𝑐,𝑡 𝜇𝑗𝑡+1𝑞(𝜃𝑡) 5.2-6

The third FOC can be rewritten as 𝜅

𝑞(𝜃_𝑡)= 𝐸𝑡𝛽

𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 𝜇𝑗𝑡+1.

Moreover, following the discussion of footnote 24, all 𝑗 subscripts can be erased. Therefore, ∫ 𝜇𝑗𝑡𝑑𝑗 = 𝐽𝑡

1

0 , denotes, just like in the standard DMP model, the value of all firms.

25_{The notation 𝑚𝑐}

𝑖𝑡𝑤 was chosen to distinguish these marginal costs from the ones in the New Keynesian baseline

Unemployment, the business cycle and monetary policy: augmenting a medium sized new keynesian DSGE model with labor market dynamics / Alexander Koll