Excerpt

## Table of contents

1 Introduction

2 Literature review

3 Data

4 Stylized Facts of U.S. Business Cycles

4.1 The Hodrick-Prescott Filter

4.2 Features of U.S. business cycles

5 Baseline Cash-in-Advance Model

5.1 The Structure

5.2 The Full Model

5.3 The Stationary State

5.4 Calibration

5.5 Impulse Responses for Cash-in-Advance Model

5.5.1 Response to a technology shock

5.5.2 Response to a money growth shock

5.6 Assessing the baseline Cash-in-Advance Model

6 Working Capital Model

6.1 The Structure

6.1.1 Households

6.1.2 Firms

6.1.3 Financial Intermediaries

6.1.4 Monetary Policy

6.2 The Optimization Problem

6.2.1 The Representative Household’s Problem

6.2.2 The Representative Firm’s Problem

6.3 The Competitive Equilibrium

6.4 The Stationary State

6.5 Impulse Responses for Working Capital Model .

6.5.1 Response to a technology shock

6.5.2 Response to a money growth shock when *η* = 0

6.5.3 Response to a money growth shock when *η* = 1

6.6 Assessing the Working Capital Model when *η* = 0

6.7 Assessing the Working Capital Model when *η* = 1

7 Discussion

8 Conclusion

A Tables

B Figures

C The Log Linearization

C.1 The Baseline Cash-in-Advance Model

C.1.1 Solving the Log-Linear System

C.2 The Working Capital Model

C.2.1 Solving the log-linear system

D Data Sources

## LIST OF FIGURES

1 Output Construction (Y vs. C+I)

2 HP filtering

3 Response of cash-in-advance model to a technology shock

4 Response of cash-in-advance model to a money growth shock

5 Response of the working capital model to a money growth shock (*η* = 0)

6 Response of the working capital model to a money growth shock (*η* = 1)

7 Response of the working capital model to a technology shock (*η* = 0)

8 Response of the working capital model to a technology shock (*η* = 1)

## LIST OF TABLES

1 U.S. data

2 Stationary state values of variables for the baseline cash-in-advance model

3 Parameter Values

4 Summary Statistics: Actual data vs. Technology shock vs. Both shocks

5 Stationary state values of variables for working capital model

6 Summary Statistics (*η* = 0): Actual data vs. Technology shock vs. Both shocks

7 Summary Statistics (*η* = 1): Actual data vs. Technology shock vs. Both shocks

8 U.S. economy statistics

9 Simulated statistics from cash-in-advance model with only technology shock and constant money growth

10 Simulated statistics from cash-in-advance model with both technology and money growth shocks

11 Simulated statistics from working capital model with only technology shock (*η* = 0)

12 Simulated statistics from working capital model with both technology and money growth shocks (*η* = 0)

13 Simulated statistics from working capital model with only money growth shocks (*η* = 0)

14 Simulated statistics from working capital model with only technology shock (*η* = 1)

15 Simulated statistics from working capital model with both technology and money growth shocks (*η* = 1)

16 Simulated statistics from working capital model with only money growth shocks (*η* = 1)

## 1 Introduction

As economies grow over time, they exhibit short-run fluctuations in various economic aggregates. Business cycle research focuses on the causes and consequences of this peri- odic expansion and contraction in economic activity. However, during the last century, exploration of the causes of business cycles has itself undergone periods of fluctuation. During the early years of the 20 *th* century and before the Great Depression, studies fo- cused on real theories. With the onset of the Great Depression and the publication of ‘ *The General Theory of Employment, Interest, and Money* ’ by John Maynard Keynes and the subsequent rise in Keynesian Macroeconomics, economists were influenced to be more sceptical about real factors and focused more on monetary factors. This led to the development of macro-econometric models which were widely used by policymakers in the 1960s

The idea that economic fluctuations are driven by demand shocks was further strength- ened by the empirical findings of Friedman & Schwartz (1963) who documented strong association between periods of economic decline and declines in the stock of money. Their findings influenced the construction of equilibrium business cycle models where unantic- ipated changes in the money supply played an important role in generating fluctuations (Lucas Jr 1972).

The fatal breakdown of macro-econometric models in the 1970s left many economists and policymakers in despair and the subsequent rational expectations revolution (Lucas Jr 1976) cautioned against using historically observed data to draw conclusions about the effects of potential economic policy changes that have not been previously used in the prevailing economic environment. This led to a desperate search for a general equilibrium model with rational forward looking agents who understood the cross equation restrictions inherent in their economic environment and led to the subsequent development of Real Business Cycle (RBC) theory based on ground-breaking work by Kydland & Prescott (1982) and Long Jr & Plosser (1983).

The core of the RBC methodology is the neoclassical growth model that resembles a stable economy, assumed to be following its long-term growth trend. When hit by exogenous shocks that affect its environment, the model generates fluctuations that re- semble business cycles. The model is termed real because it ignores nominal factors such as money and bonds. It is a dynamic general equilibrium model in that it studies an economy that evolves over time. However, a remarkable feature of the business cycle in many industrialized countries is the striking association between movements in mone- tary aggregates and aggregate output. In fact, the strength of this association has been sufficiently persuasive in the U.S. that M2 has long been included in the Commerce De- partment’s Index of Leading Economic Indicators (LEI). Although correlation does not imply causality, this coherence is interpreted by authors as an evidence that monetary forces are important for fluctuations in economic aggregates. Allocations in equilibrium RBC models are Pareto optimal due to the presence of a complete set of contingent claims markets and perfectly flexible prices. Hence, equilibrium RBC models implicitly assume that monetary shocks do not have significant impact on real variables. However, the absence of money in the real business cycle model has been a source of discomfort for many macroeconomists and influenced studies focusing on the effect of money on output by Bernanke (1986) and Eichenbaum & Singleton (1986) among others.

The purpose of this study is two-fold. First, it attempts to illustrate how the basic neoclassical business cycle model can be modified to incorporate money in an attempt to construct monetary business cycle models of the US economy. Traditional RBC research have focused on technology shocks calculated from the Solow residuals as the main driver of economic fluctuations. This study, however, explores whether and how monetary forces can be an important cause of business cycle fluctuations over and above technology shocks in a world where agents are assumed to behave rationally. Second, it attempts to show that how money enters an economy matters for the impact of monetary shocks. In this regard, this thesis offers a quantitative assessment of how the fluctuations and co- movements that we observe in the data compare with those displayed by the artificial economies that are constructed.

The paper is organized as follows. Section 2 provides a review of literature focusing on money and the role of monetary shocks in business cycles. Section 3 describes the sources and explains the measurement of data such that it matches with the structure and assumptions of the model. Section 4 provides some stylized facts of the US business cycle where business cycles are defined, following Lucas Jr (1977), as recurrent deviations of aggregate variables from their long-term trend and measured by the Hodrick-Prescott filter (Hodrick & Prescott 1997). In subsequent sections, we consider two different ar- tificial economies. In the first model environment, section 5, agents simply hold money because cash is required to purchase consumption goods. In this model, monetary shocks enter as direct lump-sum transfers to households. In the second model, section 6, we add a financial intermediary to the previous model and allow monetary shocks to enter into the economy through the financial system rather than directly through households. In our analysis, we will focus on the extent to which these basic neoclassical models with money can explain the observed features of business cycles. Section 7 then provides a brief discussion and finally section 8 concludes.

## 2 Literature review

Almost all economists believe that money is neutral in the long-run as long-run effects of money fall almost entirely on prices with little impact on real variables. However, many also believe that monetary factors can have effects on real variables in the short-run. The most influential evidence that money does matter for business cycle fluctuations is the comprehensive historical research by Friedman & Schwartz (1963). Based on almost 100 years of data from the United States, they documented that faster money growth tends to be followed by increases in output above the trend and slowdowns in money growth tend to be followed by declines in output. However, evidence based on timing patterns and simple correlations may not indicate the true causal role of money. Tobin (1970) was the first to formally model the idea that timing evidence as empirical proof of propositions about causation does not imply that money ‘causes’ output and the causality could in fact run in the opposite direction. With the development of time-series econometrics, identified vector autoregressions (VARs) have been employed to estimate the impact of monetary policy. The consensus from the empirical literature on the short-run effects of money is that exogenous monetary policy shocks produce hump-shaped movements in real economic activity with the peak effects occurring after a lag of several quarters.^{1}

The neoclassical growth model proposed by Solow (1956) is a non-monetary model. Although transactions take place, there is no medium of exchange and hence no role for money. Employing the neoclassical framework to analyze monetary issues require a role for money to be specified so that agents will wish to hold positive amounts of money in equilibrium. This leads to the fundamental question of how we should model the demand for money.

Sidrauski (1967) introduced money by treating it symmetrically with other goods in assuming that holdings of real cash balances generate a flow of services per unit of time and incorporated real money balances into the utility function. This came to be known as the *money-in-the-utility function* model. In this model the growth rate of money and hence the inflation rate have no effect on steady state values of real variables and the model displays what is called *superneutrality*. Since inflation reduces real money balances, an increase in the rate of monetary expansion generates a welfare loss. In terms of the dynamics of the model, the real variables of the economy are not affected by money growth shocks for the case of log-separable utility function. Modeling money in this fashion, however, implies that there is no clear purpose of money in this model other than giving utility from its possession. For instance, no trade ever takes place and money is never used in model economies with only one good and identical agents.

Clower (1967) identified that the role of money is indistinguishable from that of any other commodity when money is treated symmetrically and argued that a precise distinc- tion between money and non-money commodities is required for a theory of monetary phenomena. He puts forth the role of money as a medium of exchange by requiring explicitly that money be used for certain types of transactions. This idea was later de- veloped formally by Lucas Jr (1980) where each household consists of two members - a shopper and a worker. The shopper spends each day shopping at different stores while the worker works at the same store. A *cash-in-advance* constraint requires households to bring in money from the previous period which they use in the current period to make purchases.

Clower (1967, pp.5) also stated that ‘‘Money buys goods and goods buy money; but goods do not buy goods”. Motivated by this real world phenomenon we intend to model money in this study as a medium of exchange requiring explicitly that money be used for the purchase of consumption goods. The requirement that money be used to purchase goods is simply imposed. Nothing in the model explains why money is used but rather it is a social convention. If, for some reason, everyone else uses money for transactions, then it is in one’s own interest to use money as well.

Early attempts to juxtapose the long-run neutrality of money and the short-run effects of money were made by Friedman (1968) and Lucas Jr (1972). Friedman (1968) distinguished between actual and perceived real wages and argued that actual real wages are important for firms hiring decisions whereas perceived real wages are important for workers labor-supply decisions. Lucas Jr (1972) constructed Friedman’s idea by creating information problems for rational economic agents. He showed that monetary shocks could result in real fluctuations if they created confusion among economic agents as to whether changes in observed prices reflect changes in relative prices or changes in the aggregate price level. The paper is considered to be a ground-breaking work as it is the first equilibrium business cycle model in which agents have rational expectations, all markets clear, and monetary shocks are impulses leading to aggregate fluctuations.

A second way of exploring the effects of monetary shocks on real activity is to intro- duce nominal rigidities. For instance, Cho & Cooley (1995) examined the quantitative implications of multi-period wage contracts for business cycle fluctuations. First, they showed that monetary shocks, propagated by nominal contracts, are not the major source of business cycle fluctuations. Second, they further showed that monetary shocks combined with technology shocks do not account for business cycle fluctuations as monetary shocks seem too strong in their results.

In stark contrast, Christiano et al. (2005) showed that a model embodying moderate amounts of nominal rigidity accounts well for their estimate of the dynamic response of the U.S. economy to a monetary policy shock. The impulse responses of key macroe- conomic variables were estimated using structural vector autoregression (VAR). A key finding is that stickiness in nominal wages is crucial for the model’s performance.

It is worthy of notice that the effect of a positive money growth shock results in two opposing effects. One is known as the *liquidity effect* in which the extra money pushes down interest rates and stimulates economic activity. The other is known as the *anticipated inflation effect* in which people expect more increases in money growth and higher inflation in the future. According to Fisherian fundamentals^{2}, this results in higher nominal interest rates and thus depresses economic activity. A conventional view held by most economists and monetary policymakers is that central banks can reduce short-term nominal interest rates by employing policies that lead to faster growth in the money supply and by doing so can lead to a persistent increase in the level of employment and output.^{3}

Christiano (1991), Christiano & Eichenbaum (1992) and Fuerst (1992) among others introduced the *liquidity effect* by distinguishing between households, firms and financial intermediaries. Households in these models allocate resources between bank deposits and money balances used to finance consumption. Financial intermediaries lend out their deposits to firms who either borrow to finance purchases of only labor services as in Christiano & Eichenbaum (1992) and Christiano (1991) or to finance purchases of both labor services and capital goods as in Fuerst (1992). Money is injected into the economy through financial intermediaries. A key feature of these models which lets them generate a substantial *liquidity effect* is the assumed sluggishness of household saving decisions. Thus a monetary shock affects households and firms asymmetrically as firms have to absorb a disproportionately larger share of a money injection which causes nominal interest rates to decline. Firms then borrow more and increase production as their costs are now lower. This puts upward pressure on employment and economic activity. Christiano (1991) further analyzed the implications of sluggish investment. An important point to note is that some of these models allow current period wage income to be used for current consumption (Christiano & Eichenbaum 1992) while others do not (Christiano 1991, Fuerst 1992).

It is well perceived that the question of why money matters and how monetary shocks generate real effects are critical for any normative analysis of monetary policy since designing good policy requires understanding of how monetary policy affects the real economy. Therefore, we believe that there is strong motivation to focus our research on construction of monetary business cycle models in order to gain deeper understanding about the monetary transmission mechanism. In this regard this study is an addition to research focused on the role of monetary shocks.

## 3 Data

In order to analyze the performance of our artificial economies, quarterly data is required to represent the equivalent of the variables in the model. The variables are output, con- sumption, investment, labor, price level, inflation, money supply and nominal interest rate. The purpose of this section is to discuss how we have matched our data measure- ments to the structure of the models which will be discussed in section 5 and section 6. However, for now, it is important to note that the models we discuss in this study are one-sector models.

The data series of this study is from 1960(1) to 2012(2) and is chosen based on data availability.^{4} All the data, except for total hours worked, is obtained from the Federal Reserve Bank of St. Louis. For more details, see Appendix D at the end.

Gross Domestic Product (GDP) is the market value of all final goods and services produced in an economy during a given period of time and thus represents output in our models. GDP is generally calculated as the sum of consumption, investment, government expenditure and net exports. However, the model economies to be discussed are very abstract as it contains no government sector, no household production sector, no foreign sector and no explicit treatment of inventories. Accordingly, we need to calculate our consumption and investment so that their sum is equal to GDP.

In our models, there is only one consumption good available and therefore following Farmer & Guo (1995) the final consumption expenditure from all sectors in the economy represents the consumption variable which also includes government expenditure.

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Figure 1: Output Construction (Y vs. C+I)

Cooley (1997) stated that investment for a one sector economy should correspond to the sum of gross fixed capital formation from all sectors, consumption of consumer durables, changes in inventories and net exports. Consumption of durable goods are included in investment rather than in consumption expenditure because they are seen as additions to the household’s stock of capital. Net exports are also included because there is no foreign sector. However, due to the unavailability of suitable data, our investment measure does not include consumer durables and the resulting addition to output. Con- sumption of durable goods is a part of our measure for consumption expenditure. Figure 1 above shows the sum of consumption and investment against output which shows that the two measures are very close to each other.

Labor input is a multi-faceted concept and can cover broad definitions. Our interest is in the intensive margin of labor input and hence it is represented by total hours worked. Since there is no household production sector or farm sector, this is a measure of hours worked by all labor engaged in the production of goods and services in the non-farm business sector. An intensive margin such as total hours worked is presumed to be a better measure of labor input than an extensive margin such as civilian employment because it captures changes in weekly hours, changes in the proportion of part-time workers, overtime hours and annual leave.

Consumer Price Index (CPI) is the price measure used to represent the price level.

Inflation is then calculated from the data for CPI as the change in the natural logarithm of CPI, i.e., as ΔLN(CPI).

M1 and M2 are the two monetary aggregates used as a measure to represent money supply in our models. In the model in section 5, money stock is simply cash balances held for the purchase of consumption goods and so we use M1 as its empirical counterpart. However, in section 6, money represents both cash and bank deposits^{5} and accordingly we use M2 which is a broader concept of money.

Finally, since the nominal interest rate in our models are either return on government bond holdings or return on risk-free deposits at the bank, 3-month treasury bill rate in the secondary market is used as its empirical counterpart.

Having constructed the data, we now turn to present some stylized facts of the US business cycle.

## 4 Stylized Facts of U.S. Business Cycles

In order to evaluate the performance of business cycle models we need to extract the cyclical component from the actual data. In this section we want to accomplish two things. First we want to discuss how to extract business cycle component from the data. Second, having de-trended the data, we want to take the business cycle component and document the main stylized facts of the U.S. business cycle.

### 4.1 The Hodrick-Prescott Filter

Most economic aggregates grow over time while exhibiting transitory fluctuations around the growth trend. The idea is to characterize an observed time series, *y t*, as the sum of a cyclical component, *y c t*,andagrowthcomponent, *y t*.Thestatisticalmeasurement of business cycles involve making the series stationary by removing the secular trend. We intend to employ a widely used technique for representing growth and business cy- cle components, known as the Hodrick-Prescott (HP) filter, introduced by Hodrick & Prescott (1997 ). The HP filter is derived by solving the following minimization problem:

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The first term is the sum of the squared deviations from trend and therefore measures the degree of fit between *y t* and *y g t*.Thesecondtermmeasuresthedegreeofsmoothness in *y g t*.Forquarterlydata,thestandardvaluechosenforthesmoothingparameter *λ* is 1600. When *λ* = 0 the trend coincides with the original series, while with *λ* = *∞* the solution to this problem is a linear trend.

Figure 2 in the next page shows how cyclical component of output is constructed. In panel (a), the logarithm of current output is the series exhibiting more variability whereas trend output is the smoother one. The HP filtered cyclical component of output is the series in panel (b) defined by the difference from the series in the first panel, i.e.

illustration not visible in this excerpt ^{6}

It is important to note that there has been controversies regarding the appropriateness of the use of HP filter for business cycle research. Prescott (1986) points out that the HP filter is a high-pass filter as it is designed to eliminate stochastic components with periodicities greater than 32 quarters. This implies that in using the HP filter we are necessarily defining business cycles as fluctuations in economic time-series with periodicity of 8 years or less. Low frequency movements of the data, which are omitted by this subjective definition, may have important implications for business cycle research.

### 4.2 Features of U.S. business cycles

In this section the business cycle facts of the U.S. economy are represented by calculating several statistics and displaying the cyclical components from the HP filtered time series data. We report the amplitude of the fluctuations in aggregate variables in order to assess their relative magnitudes, measure the correlation of aggregate variables with real output to capture the extent to which variables display co-movement, and finally measure the cross-correlation over time to indicate whether there is any evidence that variables lead or lag one another. Table 1 shows the summary statistics whereas Table 8 in Appendix A shows a more elaborate table with up to 5 leads and lags. Entries in column x are the contemporaneous cross-correlation coefficients between the cyclical component of the series and the cyclical component of output. Entries in columns x(-1) and x(+1) are the non-contemporaneous cross-correlation coefficients at one lag and one lead respectively.

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From observing the relationship between output (GDP) and other data sets, the following characteristics are regarded as the most significant features of the US business cycle for the period 1960(1) *−* 2012(2):

*-* Consumption is less volatile than output and is pro-cyclical.

*-* Investment is more than five times as volatile as output and is pro-cyclical.

*-* Total hours worked is slightly more volatile than output and is pro-cyclical.

*-* Pro-cyclical and leading money: There is a slight contemporaneous positive corre- lation between the nominal money stock (measured as either M1 or M2) and real output. More importantly, there is also a pronounced phase shift in the correlation between output and money stock. The cross-correlation of output with the monetary aggregates show that output is more highly correlated with lagged values of the aggregates, implying that money peaks before output.

*-* Counter-cyclical prices: Price level, measured as Consumer Price Index (CPI), shows that prices are counter-cyclical.

*-* Inflation is positively correlated with output. It also tends to lag the GDP cycle by three quarters.

*-* There is a positive correlation between output and nominal interest rates (3-month T-bill rate).

These are the primary facts that characterize the business cycle of the US economy. Now we proceed to describe the models used in this study.

## 5 Baseline Cash-in-Advance Model

This section describes our first model where, following Cooley & Hansen (1989), we introduce the cash-in-advance motive for holding money into the basic indivisible labor real business cycle model. A specific characteristic of the cash-in-advance model is that it requires agents to have carried over money from the previous period which they use in the current period to make purchases. In our model, the use of money carried over from the previous period is restricted to the purchase of consumption goods. However, investment goods do not require the use of money and can be purchased using current period’s income. We are interested in exploring both the qualitative and quantitative effects of monetary shocks for fluctuations of economic aggregates over the business cycle.

### 5.1 The Structure

The economy is assumed to be populated by a continuum of agents indexed by *i* of unit mass so that per capita variables are equal to the aggregate of the same variable. An infinitely lived agent maximizes the discounted expected utility function,

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where *β* is the discount factor, *c i t* istime *t* consumption, *h t* istime *t* laborsupplyand *B* is the marginal disutility attached to an extra unit of work.

Following Hansen (1985), the utility function has indivisible labor where each family signs a contract to provide a fixed amount of labor with a certain probability. Hansen (1985) shows that introducing labor contracts in which wages are paid to all agents but only a fraction of the agents end up working smooths out the goods consumption set and makes it convex over goods consumption and expected hours worked. Adding such an unemployment insurance allows the optimization problem to have a well-defined solution.

Production is assumed to be characterized by constant returns to scale and occurs through a Cobb-Douglas aggregate production function

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where *Y t* is output, *H t* is aggregate labor, *K t* is capital, 0 *< θ <* 1 is the capital share parameter and *λ t* is total factor productivity. A constant capital share (and thus labor share) is supported by empirical studies.^{7} Technology evolves exogenously according to

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where the error term is independently and identically distributed as *ϵ λ t ∼ N* (0 *, σ* 2 ), *ϵ λ*

0 *< γ <* 1 and the stationary state value of the level of technology is *λ* = 1.

An agent *i* carries over an amount of money from the previous period, *m i t −* 1,and receives a nominal transfer from or pays a tax to the government equal to *T t*. The cash-in-advance constraint on consumption purchases implies that

illustration not visible in this excerpt

where *p t c i t* isconsumptionexpenditureinperiodt,(1 + *i t −* 1) *b t −* 1 istheprincipalplus interest on government bond holdings and *b t* is bond acquired in current period and carried into the next period.

In addition to the cash-in-advance constraint, individual *i* faces the flow budget constraint,

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where the right-hand side shows the sum of labor income, capital income, the amount of undepreciated capital at the end of the period and the real value of money held at the beginning of the period (including the transfer/tax from the government and the return on bond holdings) while the left-hand side shows the sum of consumption, capital to be taken to the next period and the real value of money and bonds to be held at the beginning of next period.

Real government spending, *G t*, and the per capita money stock, *M t*, are assumed to follow a stochastic process. In addition, the government must satisfy a budget constraint in each period which is given as follows,

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where *B t* is the nominal government debt and the initial stock of government debt, *B* 0, is given^{8}.

Since we are not studying the impact of government spending shocks here, we set *G t* =

0. * Ricardian equivalence theorem* states that a change in the timing of taxes/transfers by the government is neutral. This means that in equilibrium a change in current taxes/transfers, which is exactly offset in present value terms by an equal and oppo- site change in future taxes/transfers, has no effect on the real interest rate or on the

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consumption of individual consumers. By the virtue of this theorem, we assume that *B t* = 0⋁ *t*: 0 *→ ∞*. This assumption implies that no bonds are held in this economy in equilibrium and that *T t* = *M t* +1 *− M t*. The money stock is assumed to grow at the rate *g t* where *g t* evolves stochastically according to the AR(1) process,

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The inclusion of the term (1 *−π*) *lng* causes this money growth process to have a stationary state value of *g* and the error term is independently and identically distributed as *ϵ g t ∼ N* (0 *, σ* 2 *ϵ g*).

Note that in order to solve this model, the cash-in-advance constraint must be binding in every period. Cooley & Hansen (1989 ) have shown that this condition is met as long as the expected gross growth rate of money, *g t*, is greater than the discount factor, *β*.

### 5.2 The Full Model

When the stationary state gross growth rate of money is anything but 1, money stock will be either growing or shrinking over time and we will not be able to find stationary states for the nominal variables. Since we are developing a model similar to Cooley & Hansen (1989), we will follow their procedure and normalize the nominal variables in each period by dividing them by *M t*, and define

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The presence of money creates a friction in the economy so that the competitive equilibrium is no longer Pareto optimal and therefore we cannot simply solve a social planner’s problem to find the equilibrium allocations. Instead, we will use the recursive competitive equilibrium concept. All individuals are identical by assumption and all will end up doing the same thing. However, if each individual chooses to do something else it will not have an effect on aggregate outcome. The equilibrium is found by solving the following maximization problem,

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To find the first order conditions of optimality we form the Lagrangian,

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where *φ t* is the Lagrange multiplier on the cash-in-advance constraint and *ψ t* is the multiplier on the flow budget constraint.

The first order conditions are:

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The first order conditions with respect to *c i t* and *h t* givesusexpressionsforthe two Lagrangian multipliers,

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We use these expressions to remove the multipliers from the other two first order conditions which then together with our two budget constraints give the following four optimality conditions:

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Moreover, assuming perfect competition in factor markets gives us the following two conditions,

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### 5.3 The Stationary State

Here we obtain the stationary states of the model.^{9} The model works in such a way that in the absence of technology and money growth shocks, the optimal choice of the real variables and the nominal variables (normalized by dividing them by *M t*) will converge to steady states or constant values. Taking aggregation into account, the equations of the model in a stationary state are

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We use the parameter values as discussed in the calibration section in 5 *.* 4. Using these parameter values, the stationary states for the variables are shown in Table 2.

Table 2: Stationary state values of variables for the baseline cash-in-advance model

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### 5.4 Calibration

The term calibration can take two distinct definitions. One definition, known as strict calibration, refers to the use of economic theory as the basis of restricting our model’s structure and at the same time using the implications of that structure in the mea- surement of data. The other definition, known as classic calibration, refers to choosing parameters such that they are consistent with empirical studies on micro level data about the economy.^{10} In this paper, we will use both strict and classic calibration and our mea- surement of data in Section 3 contains techniques of calibration procedure according to these definitions. A list of values for the parameters are given in Table 3.

Table 3: Parameter Values

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The discount factor (*β*) and the depreciation rate (*δ*), following King & Rebelo (1999), are chosen to be 0 *.* 98 and 0 *.* 025 respectively. Quarterly depreciation rate on capital, *δ*, is derived in King & Rebelo (1999) from a conventional depreciation rate of 10% per annum. Next, following Gollin (2002), we use a capital share (*θ*) of 0 *.* 336 which implies a labor share (1 *− θ*) of 0 *.* 664.

We justify the use of steady state value of labor as follows. Federal Reserve Bank of St. Louis reports *average hours of work per week* between 1947 and 1969 while it reports *average annual hours worked per employed person* between 1971 and 2011. Using these data, average weekly hours between 1960 and 1969 is 40 *.* 5 hours while between 1970 and 2011 is 35 hours. Burnside & Eichenbaum (1996) used 15 hours as the daily time endowment as sleep was not included. Multiplying 15 by 7 gives the weekly time endowment of 105 hours. To find *H*, we divide average weekly hours by the weekly time endowment and this gives a steady state value which is approximately equal to 1 */* 3.

In order to simulate our models so that we can compare them to the U.S. economy, we need to obtain measures for the technology shocks and money growth shocks that feed into our model. Once the shocks are obtained, we feed them in the recursive equilibrium law of motion. We set all the variables to zero for 1959(4) and feed our first shock in the next quarter, 1960(1). The simulated series are then de-trended by the HP filter to remove any growing trend in the data before it is compared to actual data. Following King & Rebelo (1999), the *AR* (1) coefficient for technology shock, *γ*, is chosen to be 0 *.* 979 and the standard deviation, *σ λ ϵ*,issetto 0 *.* 0072.

**[...]**

^{1} For instance, see Christiano et al. (2005).

^{2} Fisher (1930).

^{3} See Christiano (1991).

^{4} Number in parenthesis refers to the quarter.

^{5} In this model deposits should not be interpreted as checking deposits but rather as less liquid deposits that earn higher interest rates.

^{6} This difference is multiplied by 100 so that cyclical output is a percentage. 9

^{7} See Gollin (2002).

^{8} Note that there is a unit mass of individuals so the per capita variables are equal to the aggregate of the same variable.

^{9} Variables with a bar on top represent the steady states of the corresponding variables.

^{10} See Cooley (1997) for more detail on calibration.

- Quote paper
- Qazi Haque (Author), 2013, The Role of Monetary Shocks in the U.S. Business Cycle, Munich, GRIN Verlag, https://www.grin.com/document/293495

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