The experiments begin in Chapter 3. The experiments in Chapter 3 are descriptive, and these can be very useful for introductory students. The experiments give students a feel for the data, especially for how variables change over time. All of Chapter 3 is introductory. Chapter 4 is also introductory; it contains descriptive experiments about the National Income and Product Accounts and the Flow of Funds Accounts.

The analysis begins in Chapter 5, where fiscal-policy effects are examined. This material is accessible to introductory students who have had the equivalent of the IS-LM model. (Although most introductory texts do not refer to the model that they are teaching as the "IS-LM" model, this is in fact the model that they are using.) The first four experiments pertain to changing government spending under different assumptions about monetary policy, which are standard IS-LM questions. The rest of the experiments in this chapter pertain to changing other fiscal-policy variables, and this can be skipped by introductory users if desired.

Chapter 6 discusses monetary-policy effects, and again this material is accessible to those who have had the equivalent of the IS-LM model. Going through the experiments in this chapter is a good way of getting introductory students to understand the tools that are available to the Federal Reserve to carry out monetary policy.

The first part of Chapter 7 discusses price shocks, and this material is accessible to those who have had the equivalent of the AS-AD model. The first two experiments concern changing the import price index, PIM, and these changes can be looked upon as shifts of the AS curve. Both of these experiments are good ways of showing how stagflation can arise. The second part of Chapter 7 examines wealth effects through stock market shocks, and this is intermediate material.

Chapter 8 is probably the limit for most introductory students. It discusses the macroeconomic consequences of deficit targeting. Going through the deficit-targeting experiments in this chapter brings together a lot of macroeconomics and is very useful review.

Chapters 9-13 are intermediate to advanced. Many of the experiments require changing coefficients in the equations and examining how the properties of the model differ for these changes. At the intermediate level this is an excellent way of getting students to have a deeper understanding of macroeconomic issues. The suggestions in Chapter 13 about imposing rational expectations on the model and making major changes to the model are for advanced users.

To summarize, the following is recommended for introductory users:

1. Chapter 1 - read carefully
2. Chapter 2 - look over and then use for reference purposes
3. Chapter 3 - do all the experiments
4. Chapter 4 - do all the experiments
5. Chapter 5 - do at least the first four experiments
6. Chapter 6 - do all the experiments
7. Chapter 7 - do the first two experiments
8. Chapter 8 - do all the experiments (good review material)

Everything in the workbook is accessible to intermediate users except perhaps for Sections 13.3 and 13.4.

The complete description of the US model is in Estimating How the Macroeconomy Works -- Fair (2004). The latest version of the model has been estimated through 2008:3. These estimates and the complete specification of the model are presented in The US Model Appendix A: October 30, 2008 , which is an update of Appendix A in Fair (2004). See the Forecast Memo for the specification changes that have been made to the model from the version in Fair (2004).

The reference Fair (2004) should be read by those contemplating using the model for research. This reference may also be of interest to teachers who would like a deeper understanding of the model than can be obtained by reading Chapter 2 of this workbook. There are a number of experiments in Fair (2004) that may be of interest to discuss with advanced students.

1.1 Macroeconometric Models
1.2 Data

There are two types of variables in macroeconometric models: endogenous and exogenous. Endogenous variables are explained by the equations, either the stochastic equations or the identities. Exogenous variables are not explained within the model. They are taken as given from the point of view of the model. For example, suppose you are trying to explain consumption of individuals in the United States. Consumption would be an endogenous variable-a variable you are trying to explain. One possible exogenous variable is the income tax rate. The income tax rate is set by the government, and if you are not interested in explaining government behavior, you would take the tax rate as exogenous.

Specification

It is easiest to consider what a macroeconometric model is like by considering a simple example. The following is a simple multiplier model. C_t is consumption, I_t is investment, Y_t is total income or GDP, G_t is government spending, and r_t is the interest rate. The t subscripts refer to period t.

(1) C_t = a₁ + a₂Y_t + e_t

(2) I_t = b₁ + b₂r_t + u_t

(3) Y_t = C_t + I_t + G_t

Equation (1) is the consumption function, equation (2) is the investment function, and equation (3) is the income identity. Equations (1) and (2) are stochastic equations, and equation (3) is an identity. The endogenous variables are C_t, I_t, and Y_t; they are explained by the model. r_t and G_t are exogenous variables; they are not explained.

The specification of stochastic equations is based on theory. Before we write down equations (1) and (2), we need to specify what factors we think affect consumption and investment in the economy. We decide these factors by using theories of consumption and investment. The theory behind equation (1) is simply that households decide how much to consume on the basis of their current income. The theory behind equation (2) is that firms decide how much to invest on the basis of the current interest rate. In equation (1) consumption is a function of income, and in equation (2) investment is a function of the interest rate. The theories behind these equations are obviously much too simple to be of much practical use, but they are useful for illustration. In practice it is important that we specify our equations on the basis of a plausible theory. For example, we could certainly specify that consumption was a function of the number of sunny days in period t, but this would not be sensible. There is no serious theory of household behavior behind this specification.

e_t and u_t are error terms. The error term in an equation encompasses all the other variables that have not been accounted for that help explain the endogenous variable. For example, in equation (1) the only variable that we have explicitly stated affects consumption is income. There are, of course, many other factors that are likely to affect consumption, such as the interest rate and wealth. There are many reasons that not all variables can be included in an equation. In some cases data on a relevant variable may not exist, and in other cases a relevant variable may not be known to the investigator. We summarize the effects of all of the left out variables by adding an error term to the equation. Thus, the error term e_t in equation (1) captures all the factors that affect consumption other than current income. Likewise, the error term u_t in equation (2) captures all the factors that affect investment other than the interest rate.

Now, suppose that we were perfectly correct in specifying that consumption is solely a function of income. That is, contrary to above discussion, suppose there were no other factors that have any influence on consumption except income. Then the error term, e_t, would equal zero. Although this is unrealistic, it is clear that one hopes that consumption in each period is mostly explained by income. This would mean that the other factors explaining consumption do not have a large effect, and so the error term for each period would be small. This means that the variance of the error term would be small. The smaller the variance, the more has been explained by the explanatory variables in the equation. The variance of an error term is an estimate of how much of the left hand side variable has not been explained. In macroeconomics, the variances are never zero; there are always factors that affect variables that are not captured by the stochastic equations.

Equation (3), the income identity, is true regardless of the theories one has for consumption and investment. Income is always equal to consumption plus investment plus government spending (we are ignoring exports and imports here).

Estimation

Once stochastic equations have been specified (written down), they must be estimated if they are to be used in a model. Theories do not tell us the size of coefficients like a₁, a₂, b₁, and b₂. These coefficients must be estimated using historical data. Given the data and the specification of the equations, the estimation techniques choose the values of the coefficients that best "fit" the data in some sense.

Consider the consumption equation above. One way to think of the best fit of this equation is to graph the observations on consumption and income, with consumption on the vertical axis and income on the horizontal axis. You can then think of the best fit as trying to find the equation of the line that is "closest to" the data points, where a₂ would be the slope of the line and a₁ would be the intercept. The ordinary least squares technique picks the line that minimizes the sum of the squared deviations of each observation to the line.

A common estimation technique in macroeconometrics is two-stage least squares, which is the technique used to estimate the US model. This technique is similar to the ordinary least squares technique except that it adjusts for certain statistical problems that arise when there are endogenous variables among the explanatory (right hand side) variables. In our current example, one would estimate the four coefficients a₁, a₂, b₁, and b₂.

Solution

Once a model has been specified and estimated, it is ready to be solved. By "solving" a model, we mean solving for the values of the endogenous variables given values for the exogenous variables. Remember that exogenous variables are not explained within the model. Say that we are in period t-1 and we want to use the model to forecast consumption, investment, and income for period t. We must first choose values of government spending and the interest rate for period t. Since t is in the future, we do not know for certain what government spending and the interest rate will be in period t, and so we must make our best guesses based on available information. (For example, we might use projected government budgets.) In other words, our model forecasts what consumption, investment, and income will be in period t if the values we chose for government spending and the interest rate in period t are correct. We must also choose values for the error terms for period t, which in most cases are taken to be zero. Given the exogenous variable values, the values of the error terms, and the coefficient estimates, equations (1), (2), and (3) are three equations in three unknowns, the three unknowns being the three endogenous variables, C_t, I_t, and Y_t. Thus, the model can be solved to find the three unknowns.

A model like equations (1)-(3) is called simultaneous. Income is an explanatory variable in the consumption function, and consumption is a variable in the income identity. One cannot calculate consumption from equation (1) unless income is known, and one cannot calculate income from equation (3) unless consumption is known. We thus say that consumption and income are "simultaneously" determined. (Investment in this model is not simultaneously determined because it can be calculated once the value for the exogenous variable r_t has been chosen.)

The above model, even though it is simultaneous, is easy to solve by simply substituting equations (1) and (2) into (3) and solving the resulting equation for Y_t. Once Y_t is solved, C_t can then be solved. In general, however, models are not this simple, and in practice models are usually solved numerically using the Gauss-Seidel technique. The steps of the Gauss-Seidel technique are as follows:

1. Guess a set of values for the endogenous variables.
2. Using this set of values for the right hand side variables, solve all the equations for the left hand side variables.
3. Step 2 yields a new set of values of the endogenous variables. Replace the initial set with this new set, and solve for the left hand side variables again.
4. Keep replacing the previous set of values with the new set until the differences between the new set and the previous set are within the required degree of accuracy. When the required accuracy has been reached, "convergence" has been attained, and the model is solved. The right hand side values are consistent with the computed left hand side values.

If a variable computed by an equation is used on the right hand side of an equation that follows, usually the newly computed value is used rather than the value from the previous iteration. It is not necessary to do this, but it usually speeds convergence. The usual procedure for the above model would be to guess a value for Y_t, compute C_t from equation (1) and I_t from equation (2), and use the computed values to solve for Y_t in equation (3). This new value of Y_t would then be used for the next pass through the model.

Testing

Once a model has been specified and estimated, it is ready to be tested. Testing alternative models is not easy, and this is one of the reasons there is so much disagreement in macroeconomics. The testing of models is discussed extensively in Fair (2004), Fair (1994), and Fair (1984), and the interested reader is referred to this material.

One obvious and popular way to test a model is to see how close its predicted values are to the actual values. Say that you want to know how well the model explained output and inflation in the 1970s. Given the actual values of the exogenous variables over this period, the model can be solved for the endogenous variables. The solution values of the endogenous variables are the predicted values. If the predicted values of output and inflation are close to the actual values, then we can say that the model did a good job in explaining output and inflation in the 1970s; otherwise not.

The solution of a model over a historical period, where the actual values of the exogenous variables are known, is called an ex post simulation. In this case, we do not have to guess values of the exogenous variables because all of these variables are known. One can thus use ex post simulations to test a model in the sense of examining how well it predicts historical episodes.

Forecasting

Once a model has been specified and estimated, it can be used to forecast the future. Forecasts into the future require that one first choose future values of the exogenous variables, as we described in the Solution section above. Given values of the exogenous variables, we can solve for the values of the endogenous variables. The solution of a model for a future period, where "guessed" values of the exogenous variables are used, is called an ex ante simulation.

Analyzing Properties of Models

Perhaps the most important use of a model is to try to learn about the properties of the economy by examining the properties of the model. If a model is an adequate representation of the economy, then its properties should be a good approximation to the actual properties of the economy. One may thus be able to use a model to get a good idea of the likely effects on the economy of various monetary and fiscal policy changes.

In the simple model above there are two basic questions that can be asked about its properties. One is how income changes when government spending changes, and the other is how income changes when the interest rate changes. In general one asks the question of how the endogenous variables change when one or more exogenous variables change. Remember, in our simple model above the only exogenous variables are government spending and the interest rate.

The Gauss-Seidel technique can be used to analyze a model's properties. Consider the question of how Y_t changes when G_t changes in the above model. In other words, one would like to know how income in the economy is affected when the government changes the amount that it spends. One first solves the model for a particular value of G_t (and r_t), perhaps the historical value of G_t if the value for period t is known. Let Y_t^* be the solution value of Y_t. Now change the value of G_t (but not r_t) and solve the model for this new value. Let Y_t^** be this new solution value. Then Y_t^** - Y_t^* is the change in income that has resulted from the change in government spending. (The change in Y divided by the change in G is sometimes called the "multiplier," hence the name of the model.) Similarly, one can examine how income changes when the interest rate changes by 1) solving the model for a given value of r_t, 2) solving the model for a new value of r_t, and 3) comparing the predicted values from the two solutions. You can begin to see how all sorts of proposed policies can be analyzed as to their likely effects if you have a good macroeconometric model.

Most of the experiments in this workbook are concerned with examining the properties of the US model. You will be comparing one set of solution values with another. If you understand these properties and if the model is an adequate approximation of the economy, then you will have a good understanding of how the economy works.

Further Complications

Actual models are obviously more complicated than equations (1)--(3) above. For one thing, lagged endogenous variables usually appear as explanatory variables in a model. The value of consumption in period t-1, denoted C_t-1, is a lagged endogenous variable since it is the lagged value of C_t, which is an endogenous variable. If C_t-1 appeared as an explanatory variable in equation (1), then the model would include a lagged endogenous variable.

When lagged endogenous variables are included in a model, the model is said to be dynamic. An important feature of a dynamic model is that the predicted values in one period affect the predicted values in future periods. What happens today affects what happens tomorrow, and the model is dynamic in this sense. Again, in the case of consumption, the idea is that how much you decide to consume this year will affect how much you decide to consume next year.

Also, most models in practice are nonlinear, contrary to the above model, where, for example, consumption is a linear function of income in equation (1). In particular, most models include ratios of variables and logarithms of variables. Equation (1), for example, might be specified in log terms:

(1)' log C_t = a₁ + a₂log Y_t + e_t

Nonlinear models are difficult or impossible to solve analytically, but they can usually be solved numerically using the Gauss-Seidel technique. The same kinds of experiments can thus be performed for nonlinear models as for linear models. As long as a model can be solved numerically, it does not really matter whether it is nonlinear or not for purposes of forecasting and policy analysis.

The error terms in many stochastic equations in macroeconomics appear to be correlated with their past values. In particular, many error terms appear to be first order serially correlated. If e_t is first order serially correlated, this means that:

(4) e_t = p e_t-1 + v_t

where p is the first order serial correlation coefficient and v_t is an error term that is not serially correlated. In the estimation of an equation one can treat p as a coefficient to be estimated and estimate it along with the other coefficients in the equation. This is done for a number of the stochastic equations in the US model.

The US model is a quarterly model; all the variables are quarterly. An important point should be kept in mind when dealing with quarterly variables. In most cases quarterly variables are quoted seasonally adjusted at annual rates. For example, in the National Income and Product Accounts real GDP for the fourth quarter of 1994 is listed as $7461.1 billion, but this does not mean that the U.S. economy produced $7461.1 billion worth of output in the fourth quarter. First, the figure is seasonally adjusted, which means that it is adjusted to account for the fact that on average more output is produced in the fourth quarter than it is in the other three quarters. The number before seasonal adjustment is higher than the seasonally adjusted number. Seasonally adjusting the data smooths out the ups and downs that occur because of seasonal factors.

Second, the figure of $7461.1 billion is also quoted at an annual rate, which means that it is four times larger than the amount of output actually produced (ignoring seasonal adjustment). Being quoted at an annual rate means that if the rate of output continued at the rate produced in the quarter for the whole year, the amount of output produced would be $7461.1 billion. For variables that are quoted at annual rates, it is not the case that the yearly amount is the sum of the four published quarterly amounts. The yearly amount is one fourth of this sum, since all the quarterly amounts are multiplied by four.

It is also important to understand how growth rates are computed. Consider a variable Y_t. The change in Y from period t-1 to period t is Y_t - Y_t-1. The percentage change in Y from t-1 to t is (Y_t - Y_t-1)/Y_t-1, which is the change in Y divided by Y_t-1. If, for example, Y_t-1 is 100 and Y_t is 101, the change is 1 and the percentage change is .01. The percentage change is usually quoted in percentage points, which in the present example means that .01 would be multiplied by 100 to make it 1.0 percent.

The percentage change in a variable is also called the growth rate of the variable, except that in most cases growth rates are given at annual rates. In the present example, the growth rate in Y at an annual rate in percentage points is

Growth Rate (annual rate) = 100[(Y_t/Y_t-1)⁴ - 1] .

If Y_t-1 is 100 and Y_t is 101, the growth rate is 4.06 percent. Note that this growth rate is slightly larger than four times the quarterly growth rate of 1.0 percent. The above formula "compounds" the growth rate, which makes it slightly larger than 4 percent. The growth rate at an annual rate is the rate that the economy would grow in a year if it continued to grow at the same rate in the next three quarters as it did in the current quarter.

2.1 Introduction and History
2.2 Tables of Variables and Equations
2.3 The Structure of the Model
2.4 Properties of the Model
2.5 Alternative Monetary Policy Assumptions
2.6 Important Things to Know About the Program

Work began on the theoretical basis of the model in 1972. The theoretical work stressed three ideas: 1) basing macroeconomics on solid microeconomic foundations, 2) allowing for the possibility of disequilibrium in some markets, and 3) accounting for all balance sheet and flow of funds constraints. The stress on microeconomic foundations for macroeconomics has come to dominate macro theory, and this work in the early 1970s is consistent with the current emphasis. The introduction of disequilibrium possibilities in some markets provides an explanation of business cycles that is consistent with maximizing behavior. The model explains disequilibrium on the basis of non rational expectations. Agents must form expectations of the future values of various variables before solving their multiperiod maximization problems. It is assumed that no agent knows the complete model, and so expectations are not rational. Firms, for example, make their price and wage decisions based on expectations that are not rational, which can cause disequilibrium in the goods and labor markets.

The theoretical model was used to guide the specification of the econometric model. This work was done in 1974 and 1975, and by 1976 the model was essentially in the form that it is in today. The explanatory variables in the econometric model were chosen to be consistent with the assumption of maximizing behavior, and an attempt was made to model the effects of disequilibrium. Balance sheet and flow of funds constraints were accounted for: the National Income and Product Accounts (NIPA) data and the Flow of Funds Accounts (FFA) data are completely integrated in the model. This latter feature greatly helps in considering alternative monetary policies, and it allows one to consider carefully "crowding out" questions.

Table A.1 presents the six sectors in the US model: household (h), firm (f), financial (b), foreign (r), federal government (g), and state and local government (s). In order to account for the flow of funds among these sectors and for their balance-sheet constraints, the U.S. Flow of Funds Accounts (FFA) and the U.S. National Income and Product Accounts (NIPA) must be linked. Many of the identities in the US model are concerned with this linkage. Table A.1 shows how the six sectors in the US model are related to the sectors in the FFA. The notation on the right side of this table (H1, FA, etc.) is used in Table A.5 in the description of the FFA data.

Table A.2 lists all the variables in the US model in alphabetical order, and Table A.3 lists all the stochastic equations and identities. The functional forms of the stochastic equations are given in Table A.3, but not the coefficient estimates. The coefficient estimates are presented in Table A.4, where within this table the coefficient estimates and tests for equation 1 are presented in Table A1, for equation 2 in Table A2, and so on. The tests for the equations, which are reported in Table A.4, are explained in Fair (2004), and this discussion is not presented in this workbook.

The remaining tables provide more detailed information about the model. Tables A.5-A.7 show how the variables were constructed from the raw data. Table A.8 shows how the model is solved under various assumptions about monetary policy. Table A.9 lists the first stage regressors per equation that were used for the two-stage least squares estimates. Finally, Table A.10 shows which variables appear in which equations. This table can be useful for tracing through how one variable affects other variables in the model.

1. Household sector (h)
2. Firm sector (f)
3. Financial sector (b)
4. Federal government sector (g)
5. State and local government sector (s)
6. Foreign sector (r)

Each of these sectors will be discussed in turn.

The Household Sector

In the multiplier model in Chapter 1, consumption is simply a function of current income, but this is obviously much too simple as a description of reality. As noted above, the stress in the model is on microeconomic foundations and possible disequilibrium effects. In the microeconomic story households maximize a multiperiod utility function. Households make two decisions each period. They decide how much to consume and how many hours to work. If households can work as many hours as they wish (no disequilibrium), then income, which is the wage rate times the number of hours worked, is not an appropriate explanatory variable in a consumption equation, because part of it (the number of hours worked) is itself a decision variable. If there is no disequilibrium, decisions about consumption and hours worked are made at the same time. Households do not earn income and then decide how much to consume. Consumption and hours worked are instead determined jointly, and hours worked should not be considered as helping to "explain" consumption if there is no disequilibrium. Both variables are "explained" by other variables.

The main variables that explain consumption and hours worked in the microeconomic story are the wage rate, the price level, the interest rate, tax rates, the initial value of wealth, and nonlabor income. The interest rate affects consumption because of the multiperiod nature of the maximization problem.

This microeconomic story has to be modified if households are not allowed to work as many hours as they would like. If households want to work more hours than firms want to employ and if firms employ only the amount they want (which seems reasonable), then households are "constrained" from working their desired number of hours. These periods correspond to periods of "unemployment." The existence of binding labor constraints is likely to lead households to consume less than they otherwise would. Also, a binding labor constraint on a household means that income is a legitimate explanatory variable for consumption, since the number hours worked is no longer a decision variable. It is imposed from the outside by firms. As discussed below, an attempt has been made in the econometric work to handle possible disequilibrium effects within the context of the microeconomic story.

In the empirical work the expenditures of the household sector are disaggregated into four types: consumption of services (CS), consumption of nondurable goods (CN), consumption of durable goods (CD), and investment in housing (IHH). Four labor supply variables are used: the labor force of men 25--54 (L1), the labor force of women 25--54 (L2), the labor force of all others 16+ (L3), and the number of people holding more than one job, called "moonlighters" (LM). These eight variables are determined by eight estimated equations.

There are two main empirical approaches that can be taken regarding the use of wage, price, and income variables in the consumption equations. The first is to add the wage, price, nonlabor income, and labor constraint variables separately to the equations. These variables in the model are as follows. The after tax nominal wage rate is WA, the price deflator for total household expenditures is PH, the after tax nonlabor income variable is YNL, and the labor constraint variable is Z. The price deflators for the four expenditure categories are PCS, PCN, PCD, and PIH. Z takes on a value of zero when there is no binding constraint (periods of full employment), and it gets more and more negative as the economy gets further and further from full employment. Consider as an example the CS equation. Under the first approach one might add WA/PH, PCS/PH, YNL/PH, and Z to the equation. The justification for including Z is the following. By construction, Z is zero or nearly zero in tight labor markets. In this case the labor constraint is not binding and Z has no effect or only a small effect in the equation. This is the "classical" case. As labor markets get looser, on the other hand, Z falls and begins to have an effect in the equation. Loose labor markets, where Z is large in absolute value, correspond to the "Keynesian" case. Since Z is highly correlated with hours paid for in loose labor markets, having both WA and Z in the equation is similar to having a labor income variable in the equation in loose labor markets.

The second, more traditional, empirical approach is to replace the above four variables with real disposable personal income, YD/PH. This approach in effect assumes that labor markets are always loose and that the responses to changes in labor and nonlabor income are the same. One can test whether the data support YD/PH over the other variables by including all the variables in the equation and examining their significance. The results of doing this in the four expenditure equations supported the use of YD/PH over the other variables, and so the equations that were chosen for the model use YD/PH. This is a change from the version of the model in Fair (1984), where the first approach was used.

The dominance of YD/PH does not necessarily mean that the classical case never holds in practice. What it does suggest is that trying to capture the classical case through the use of Z does not work. An interesting question for future work is whether the classical case can be captured in some other way.

The first nine equations in Table A.4 in Appendix A are for the household sector. As noted above, there are four expenditure equations and four labor supply equations. These are the first eight equations in Table A.4. The ninth equation explains the demand for money of the household sector, and it will be discussed later in this chapter.

The Firm Sector

There are twelve stochastic equations for the firm sector (equations 10 through 21 in Table A.4). The firm sector determines production given sales (i.e., inventory investment), nonresidential fixed investment, employment demand, the price level, and the wage rate, among other things.

In the multiplier model in Chapter 1 investment is only a function of the interest rate. There is no labor market, and so employment demand is not determined. Also, no distinction is made between production and sales, and so there is no inventory investment. (Inventory investment in a period is the difference between what firms produce and what they sell.) Finally, no mention is made as to how the price level is determined. A realistic model of the economy must obviously take into account more features of firm behavior.

Production in the model is a function of sales and of the lagged stock of inventories (equation 11). Production is assumed to be "smoothed" relative to sales. The capital stock of the firm sector depends on the amount of excess capital on hand and on current and lagged values of output (equation 12). It also depends on two cost of capital variables: a real interest rate variable and a stock market variable. Nonresidential fixed investment is determined by an identity (equation 92). It is equal to the change in the capital stock plus depreciation. The demand for workers and hours depends on output and the amount of excess labor on hand (equations 13 and 14). (Excess labor is labor that the firm holds (pays for) that is not needed to produce the current level of output.)

The price level of the firm sector is determined by equation 10. It is a function of the lagged price level, the wage rate, the price of imports, the unemployment rate, and a time trend. The lagged price level is meant to pick up expectational effects, the unemployment rate is meant to pick up demand pressure effects, and the wage rate and import price variables are meant to pick up cost effects.

The nominal wage of the firm sector is determined by equation 16. The nominal wage rate is a function of the current and lagged value of the price level and a time trend. The equation is best thought of as a real wage equation, where the nominal wage rate adjusts to the price level with a lag.

Equation 17 determines the demand for money of the firm sector. It is discussed later. Equation 19 explains the level of interest payments of the firm sector.

The other stochastic equations for the firm sector are fairly minor. The level of overtime hours is a nonlinear function of total hours (equation 15). The level of dividends paid is a function of after tax profits (equation 18). Inventory valuation adjustment is a function of the change in the price level times the lagged stock of inventories (equation 20), and capital consumption is a function of nonresidential investment (equation 21).

The Financial Sector

The multiplier model in Chapter 1 is the IS part of the IS-LM model. The LM part of this model is as follows:

(4) M_t^d/P_t = d₁ + d₂Y_t + d₃r_t + w_t

(5) M_t^s = M_t

(6) M_t^d = M_t^s

where M_t^d is the quantity of money demanded, M_t^s is the quantity of money supplied, and P_t is the price level. In equation (4) the demand for real money balances is a function of income and the interest rate. Equation (5) states that the supply of money is equal to M_t, which is assumed to be the policy variable of the monetary authority. Equation (6) is the equilibrium condition in the money market; the money market is assumed to clear---the quantity of money demanded equals the quantity of money supplied. The three endogenous variables in equations (4)-(6) are M_t^d, M_t^s, and r_t. The exogenous variables to this block of equations are M_t, Y_t, and P_t.

When equations (4)-(6) are added to equations (1)-(3) in Chapter 1, Y_t, which is exogenous in the LM model, becomes endogenous and r_t, which is exogenous in the IS model, becomes endogenous. The exogenous variables in this expanded model, the overall IS-LM model, are G_t, M_t, and P_t.

The demand for money equations in the model are consistent with equation (4) of the LM model. There are two main demand for money equations, one for the household sector---equation 9---and one for the firm sector-equation 17. In these equations the demand for money is a function of the interest rate and a transactions variable. There is also a separate demand for currency equation---equation 26---which is similar to the other two.

An important difference between the present model and the LM model is that the present model accounts for all the flows of funds among the sectors and all balance sheet constraints. This allows the "tools" of the monetary authority in the model to be the actual tools used in practice: the discount rate, the reserve requirement rate, and open market operations. The money supply in this setup is an intermediate target rather than a direct control variable as in the LM model. More will be said about this below.

The other equations of the financial sector consist of an equation explaining bank borrowing from the Fed, two term structure equations, and an equation explaining the change in stock prices. Bank borrowing from the Fed is a function of the three month Treasury bill rate and the discount rate (equation 22). The bond rate in the first term structure equation is a function of current and lagged values of the bill rate (equation 23). The same is true for the mortgage rate in the second term structure equation (equation 24). In the stock price equation, the change in stock prices is a function of the change in the bond rate and the change in after tax profits (equation 25).

The Foreign Sector

There is one stochastic equation in the foreign sector, which explains the level of imports (equation 27). The level of imports depends on consumption plus fixed investment and on the domestic price level relative to the price of imports. If the price of imports rises relative to the domestic price level, imports are predicted to fall, other things being equal, as people substitute domestic goods for imported goods. Otherwise, the level of imports is just run off of total consumption and fixed investment.

The State and Local Government Sector

There is one stochastic equation in the state and local government sector, an equation explaining unemployment insurance benefits (equation 28). The level of unemployment insurance benefits is a function of the level of unemployment and the nominal wage rate. The inclusion of the nominal wage rate is designed to pick up effects of increases in wages and prices on legislated benefits per unemployed worker.

The Federal Government Sector

There are two stochastic equations in the federal government sector. The first explains the interest payments of the federal government (equation 29), and the second explains the three month Treasury bill rate (equation 30). The federal government sector is meant to include the Federal Reserve as well as the fiscal branch of the government.

Equation 29 for the federal government sector is similar to equation 19 for the firm sector. It explains the level of interest payments of the government sector.

The bill rate is determined by an "interest rate reaction function," where the Fed is assumed to "lean against the wind" in setting its interest rate targets. That is, the Fed is assumed to allow the bill rate to rise in response to increases in inflation and lagged money supply growth and to decreases in the unemployment rate. There is a dummy variable multiplying the money supply variable in the equation. This variable takes on a value of one between 1979:4 and 1982:3 and zero otherwise. It is designed to pick up the change in Fed operating policy between October 1979 and October 1982 when the Fed switched from targeting interest rates to targeting the money supply.

When the interest rate reaction function (equation 30) is included in the model, monetary policy is endogenous. In other words, Fed behavior is explained within the model. How the Fed behaves is determined by what is going on in the economy. The program that you will be using, however, allows four other assumptions to be made about monetary policy. These are 1) the bill rate is exogenous, 2) the money supply is exogenous, 3) the level of nonborrowed reserves is exogenous, and 4) the amount of government securities outstanding is exogenous. If one of these four assumptions is made, then monetary policy is exogenous and equation 30 is dropped.

You may need to use Tables A.2-A.4 in Appendix A when you are puzzled about some aspect of the results. As noted above, these tables provide a complete listing of the variables and equations of the model.

Interest Rate Effects

There are many channels through which interest rates affect the economy. It will first help to consider the various ways that an increase in interest rates affects consumption and housing investment. 1) The short term after tax interest rate RSA is an explanatory variable in equation 1 explaining service consumption (CS), and the long term after tax interest rate RMA is an explanatory variable in equation 2 explaining nondurable consumption (CN), in equation 3 explaining durable consumption (CD), and in equation 4 explaining housing investment (IHH). The interest rate variables have a negative effect in these equations. In addition the long term bond rate is an explanatory variable in the investment equation 12, where a change in the real bond rate has a negative effect on plant and equipment investment.

Interest rates also affect stock prices in the model. The change in the bond rate RB is an explanatory variable in equation 25 determining capital gains or losses on corporate stocks held by the household sector (CG). An increase in RB has a negative effect on CG (i.e., an increase in the bond rate has a negative effect on stock prices). When CG decreases, the net financial assets of the household sector (AH) decrease---equation 66---and thus total net wealth (AA) decreases---equation 89. AA is an explanatory variable in the consumption equations (wealth has a positive effect on spending). Therefore, through the wealth channel, an increase in interest rates has a negative effect on consumption. When RB rises, AA falls and thus spending falls. (It should also be the case that an increase in interest rates lowers wealth in the model through a fall in long term bond prices, but the data are not good enough to pick up this effect.)

It is also the case that an increase in interest rates increases the interest income of the household sector because the household sector is a net creditor, i.e. the household sector lends more than it borrows. Interest income is part of personal income, which has a positive effect on consumption and housing investment, and so on this score an increase in interest rates has a positive effect on consumption and housing investment. This "income effect" of a change in interest rates on household expenditures is now quite large because of the large federal government debt holdings of the household sector. The negative income effect from a fall in interest rates now offsets more of the positive substitution effect than it did earlier.

A change in interest rates thus affects GDP through a number of channels. The size of the net effect on GDP of a change in interest rates is an empirical question, which the model can be used to answer. The final answer obviously depends on the specification of the stochastic equations, and you may want to experiment with alternative specifications to see how the final answer is affected. The size of the net effect is, of course, of critical importance for policy purposes.

Tax Rate Effects

An increase in personal income tax rates and/or social security tax rates (D1GM, D1SM, D4G) lowers the after tax wage (WA)---equation 126---and disposable income (YD)---equation 115. Disposable income is an explanatory variable in the consumption and housing investment equations---an increase in YD increases spending. Therefore, an increase in tax rates lowers consumption and housing investment by lowering disposable income. An increase in personal income tax rates also lowers the after tax interest rates RSA and RMA, which on this score has a positive effect on consumption and housing investment because the after tax interest rates have a negative effect.

One obvious exercise with the model is to change the corporate profit tax rate D2G and see how this affects the economy. You will find, for example, that an increase in D2G has a fairly small effect on GDP in the model. It appears from an exercise like this that the government can raise a lot of tax revenue (and thus lower the government deficit) by raising D2G with only a small negative effect on the economy. The way an increase in D2G affects the economy is as follows. An increase in D2G increases corporate profits taxes (TFG)---equation 49---which lowers after tax profits . The decrease in after tax profits results in a capital loss on stocks---equation 25---which lowers household wealth, which has a negative effect on consumption and housing investment and thus on sales and production. Also, the decrease in after tax profits results in a decrease in dividends---equation 18---which lowers disposable income, which has a negative effect on consumption and housing investment.

Both of these effects of a change in D2G on GDP are initially quite small. It takes time for households to respond to changes in wealth, and it takes time for dividends to respond to changes in after tax profits. Whether this specification is realistic is not clear. Changes in D2G may affect the behavior of the firm sector in ways that are not captured in the model, and you should thus proceed cautiously in changing D2G (or D2S for the state and local government sector). This may not be as easy a revenue raiser as the model implies.

Labor Supply and the Unemployment Rate

The unemployment rate UR is determined by equation 87. UR is equal to the number of people unemployed divided by the civilian labor force. The number of people unemployed is equal to the labor force minus the number of people employed. The labor force is made up of three groups---prime age men (L1), prime age women (L2), and all others (L3). The three labor force variables along with the number of moonlighters (LM) are the labor supply variables in the model. Three of the four labor supply variables depend positively on the real after tax wage rate (WA/PH), which means that the substitution effect is estimated to dominate the income effect on labor supply.

It is important to note that anything that, say, increases the labor force will, other things being equal, increase the number of people unemployed and the unemployment rate. (Other things equal here includes no change in employment.) For example, suppose the personal income tax rate is lowered, thereby raising the after tax wage rate WA. Then the labor force variables L2 and L3 will rise---equations 6 and 7. This, other things being equal, leads the unemployment rate to rise. Other things are not, of course, equal because the decrease in the tax rate also leads, for example, to a increase in consumption and housing investment, which leads to an increase in production and then to employment. Employment thus rises also. Whether the net effect is an increase or a decrease in the unemployment rate depends on the relative sizes of the increased labor force and the increased employment, which you can see when you run the experiments. The main point to remember is that the labor force responses can be important in determining the final outcome. There is, for example, no simple relationship between the unemployment rate and output (i.e., there is no Okun's "law") because of the many factors that affect the labor force.

L1, L3, and LM depend negatively on the unemployment rate, and for L1 and L3 this is the discouraged worker effect at work. In bad times (i.e., when the unemployment rate is high) some people get discouraged from ever finding a job and drop out of the labor force. (When people drop out of the labor force, they are no longer counted as unemployed, and so this lowers the measured unemployment rate.) When things improve, they reenter the labor force. This "discouraged worker effect" is captured by the unemployment rate in the L1 and L3 equations. Be aware when you run experiments that part of any change in the labor force is due to the discouraged worker effect in operation. This effect can be quantitatively very important in slack periods. The number of people holding two jobs (LM) also decreases in slack periods, and this is captured by the unemployment rate in the LM equation.

Note finally that L2 and L3 depending positively on the after tax wage rate (substitution effect dominating) is consistent with the theory behind the Laffer curve. Labor supply does respond to tax rates in the model. That is, when taxes decrease, the after tax wage rate increases, leading to an increase in the labor force. If you run various experiments, however, you will see that the quantitative responses are fairly small.

"Productivity" Movements

The productivity variable PROD in the model is defined in equation 118. PROD is equal to Y/(JF*HF), where Y is output, JF is the number of jobs, and HF is the number of hours paid for per job. PROD is thus output per paid for worker hour. Although this variable is usually called "productivity," it is important to realize that it is not a good measure of true productivity. In slack periods firms appear to pay for more worker hours than they actually need to produce the output; they hold what is called in the model "excess labor." This means that JF*HF is not a good measure of actual hours worked, and so Y/(JF*HF) is an imperfect measure of the true ability of the economy to produce per hour worked.

PROD is a procyclical variable. It falls in output contractions as excess labor is built up (output falls faster than hours paid for), and it rises in output expansions as excess labor is eliminated (output rises faster than hours paid for). The amount of excess labor on hand appears as an explanatory variable in the employment and hours equations---equations 13 and 14.

Output and the Unemployment Rate

It was mentioned above that there is no simple relationship between output and the unemployment rate in the model (no Okun's law) because of the many factors that affect the labor force. Even though the relationship is not stable, one can say that changes in output are likely to correspond to less than proportional changes in the unemployment rate. That is, when output increases (decreases), the unemployment rate will decrease (increase) but by proportionally less than output. There are three main reasons for this in the model. First, when output decreases by a certain percentage, the number of jobs falls by less than this percentage because firms cut hours worked per job as well as jobs and also build up some excess labor. Second, the number of people employed falls by less than the number of jobs because some of the jobs that are cut are held by people holding two jobs. These people are still employed; they just hold one job now rather than two. Third, as the economy contracts, the discouraged worker effect leads some people to drop out of the measured labor force and thus the measured labor force falls, which ceteris paribus decreases the unemployment rate. These three effects show why the unemployment rate tends to change by proportionately less than output does. You should examine these effects when you do the various experiments with the model.

Price Responses to Output Changes

One of the most difficult issues in macroeconomics is trying to determine how fast inflation increases as the economy approaches full capacity. The data are not good at discriminating among alternative specifications because there are so few observations at very high levels of capacity or low unemployment rates. The demand pressure variable used in the price equation (equation 10) is simply the level of the unemployment rate. No nonlinear transformation of the unemployment rate or output gap measure has been used. When other functional forms were tried, the fit of the equation was not quite as good as the fit using the level of the unemployment rate.

Because of the uncertainty of how the aggregate price level behaves as unemployment approaches very low levels, you should be cautious about pushing the model to extremely low unemployment rates. The price response that the model predicts for very low unemployment rates may be much less than would actually exist in practice. Again, the data are not good in telling us what this response is. You should probably not push the economy much below an unemployment rate of about 3.5 percent if you want to trust the estimated price responses.

You will notice if you run an experiment that increases output that the estimated size of the price response is modest, especially in the short run. This is a common feature of econometric models of price behavior. The estimated effects of demand pressure variables on prices are usually modest. This is simply what the data show, although many people are of the view that the effects should be larger. If you would like a larger response in the model, simply make the coefficient of the unemployment rate in equation 10 larger in absolute value (i.e., make it more negative). Advanced users may want to reestimate the equation using various nonlinear transformations of the unemployment rate. Be warned, however, that this is unlikely to make much difference to the fit of the price equation.

The key price variable in the model is PF, which is determined by equation 10, and this is the variable you should focus on. For most experiments PF and the GDP price deflator GDPD, respond almost identically. If, however, you, say, increase government purchases of goods, COG, which is a common experiment to perform, this will initially have a negative effect on the GDP price deflator even though it has a positive effect on PF. One would expect a positive effect because the increase in COG increases Y, which lowers the gap. The problem is that the GDP price deflator is a weighted average of other price deflators, and when you change COG you are changing the weights. It so happens that the weights change in such a way when you increase COG as to have a negative effect on the GDP price deflator. This is not an interesting result, and in these cases you should focus on PF, which is not affected by the change in weights.

Although demand pressure effects on prices are modest in the model, the effects of changing the price of imports (PIM) on domestic prices are fairly large, as you can see if you change PIM. In fact, much of the inflation of the 1970s is attributed by the model to the increases in PIM in this period. The model also attributes much of the drop in output in the 1970s to the rise in import prices. The reason for this is as follows. When PIM rises, domestic prices initially rise faster than nominal wages (because wages lag prices in the model). Higher prices relative to wages have a negative effect on real disposable income (YD/PH) and thus on consumption and housing investment, which leads to a drop in sales and production. In addition, if the money supply is held unchanged, the rise in prices leads to an increase in interest rates (through the standard LM story), which has a negative effect on consumption and housing investment. Interest rates also rise if the Fed instead behaves according to the interest rate reaction function---equation 30---because the Fed is estimated in this equation to let interest rates rise when inflation increases. One of the experiments in Chapter 8 is to examine what the 1970s might have been like had PIM not risen in this period.

Response Lags and Magnitudes

You will soon see as you begin the experiments that the effects of any change on the economy take time. There are significant response lags estimated in the model; it is by no means the case that firms and households respond quickly to policy changes.

You should also be aware regarding the magnitudes of the responses that they depend on the sizes of the estimated coefficients in the stochastic equations. If, say, one category of consumption responds more to a particular change than does another category, this reflects the different coefficient estimates in the two relevant stochastic equations. Also, some potentially relevant explanatory variables have been dropped from one equation and not from another (variables are generally dropped if their coefficient estimates have the wrong sign), which can account for the differences in responses. Another way of putting this is that no prior constraints on, say, the consumption equations have been imposed in order to have the responses of the different categories of consumption be the same. The data are allowed to determine these differences.

The notation f( ) means that the equation is stochastic. The variables inside the parentheses are explanatory variables. For the sake of the present discussion, only the explanatory variables that are needed for the analysis are listed in the parentheses.

The relevant notation is:

AG     net financial assets of the federal government
BO     bank borrowing from the Fed
BR     total bank reserves
CUR    currency held outside banks
DISG   discrepancy for the federal government
G1     reserve requirement ratio
M1     M1 money supply
MB     demand deposits and currency in banks
MDIF   discrepancy between M1 and other variables
MF     demand deposits and currency of firms
MG     demand deposits and currency of the federal government
MH     demand deposits and currency of households
MR     demand deposits and currency of the foreign sector
MS     demand deposits and currency of the state and local governments
Q      gold and foreign exchange of the federal government
RD     discount rate
RS     three month Treasury bill rate
SG     saving of the federal government

Equations 9, 17, and 26 are the demand for money equations. Equation 22 explains bank borrowing from the Fed. The ratio of borrowed to total reserves is a function of the difference between the bill rate and the discount rate. Equation 57 determines total reserves. G1 is the reserve requirement ratio and MB is the amount of demand deposits in commercial banks, and so -G1*MB is the amount of required reserves. (MB is negative because demand deposits are liabilities of banks, and so -G1*MB is positive, which is as desired.) Since -G1*MB is the amount of required reserves and BR is the amount of total reserves, equation 57, by setting the two equal, reflects the assumption that banks hold no excess reserves. Banks hold almost no excess reserves in practice, and so this assumption is a good approximation.

Equation 71 is a balance sheet constraint. -MB is the total amount of demand deposits in banks, and CUR is the total amount of currency outstanding. Therefore, -MB + CUR is the total amount of demand deposits and currency. This amount is held by the five other sectors (household, firm, foreign, federal government, and state and local government). Equation 71 simply states that the change in the amount of demand deposits and currency held by the five sectors is equal to the change in demand deposits in banks plus the change in currency---a balance sheet constraint.

Equation 77 is the budget constraint of the federal government. SG is the saving of the federal government. SG is almost always negative because the federal government almost always runs a deficit. If the government runs a deficit, it can finance it in a number of ways. Two minor ways are that it can decrease its holdings of demand deposits in banks (MG) and it can decrease its holdings of gold and foreign exchange (Q). More importantly, it can increase the amount of high powered money (currency plus non borrowed reserves) in the system, which is CUR + (BR - BO). Finally, it can increase the amount of government securities in the hands of the public, meaning the government borrows from the public. -AG in the model is the amount of government securities outstanding. (AG is negative because government securities are a liability of the government, and so -AG is the (positive) amount of government securities outstanding). The variable DISG in equation 77 is a discrepancy term that is needed to make the NIPA data and the FFA data match. It reflects errors of measurement in the data. Ignoring MG, Q, and DISG in equation 77, the equation simply says that when, say, SG is negative, either government securities outstanding must increase or the amount of high powered money must increase. This is the budget constraint that the government faces, the government being defined here to be the federal government inclusive of the monetary authority.

Equation 81 is the definition of M1. The change in M1 is equal to the change in MH + MF + MR + MS plus some minor terms that are included in the MDIF variable.

The importance of the above equations for understanding the tools of monetary policy and the relationship between monetary policy and fiscal policy cannot be overstated. We are now ready to consider various cases. First, it is important to consider what is exogenous in the above equations, what is determined elsewhere in the model, and what is endogenous. The following variables are all minor and taken to be exogenous in the model: MR, MG, MS, Q, DISG, and MDIF. In addition, there are three monetary policy tools, RD, G1, and AG, which for now can be assumed to be exogenous. The variable SG is determined elsewhere in the model (by equation 76 in Table A.3 in Appendix A). This leaves eight endogenous variables for the eight equations (we are not using equation 30 yet): MH, MF, CUR, BO, BR, MB, RS, and M1. Given the other values, the eight equations can be used to solve for the eight unknowns. (It is usually the case in macroeconometric models that counting equations and unknowns is sufficient for solution purposes.)

In terms of solving the model by the Gauss-Seidel technique, which needs one left hand side variable per equation, the eight equations have to be rearranged somewhat. The details of the various arrangements that are discussed in this section are presented in Table A.8 in Appendix A, although these details are of more numerical than economic interest. In the present case equation 9 would be used to solve for RS, equation 17 for MF, equation 26 for CUR, equation 22 for BO, equation 57 for MB, equation 71 for MH, equation 77 for BR, and equation 81 for M1.

Remember in this setup that the three tools of the Fed, RD, G1, and AG, are exogenous. You can play the Fed in the model and change any of the three tools to see the effects on the economy. Experiments of this type are examined in Chapter 6.

If AG is exogenous and if fiscal policy is changed in such a way that the deficit is made larger (SG larger in absolute value), then the increase in the deficit must be financed by an increase in high powered money. With AG exogenous, there is nothing else to give in equation 77. If you increase government spending with AG exogenous, you will see that this leads to a large fall in the interest rate (RS). Here is where the insights from the IS-LM model might help (although be careful of oversimplification). When high powered money increases, households and firms must be induced to hold more currency and demand deposits, which is done through a lower interest rate.

In practice, RD and G1 are not tools that are used very often by the Fed. G1 is seldom changed, and RD is usually changed after other interest rates have changed (i.e., RD usually follows rather than leads the market). Therefore, although you can use RD and G1 as tools in the model (when AG is exogenous), this is not an accurate representation of the way that monetary policy is carried out.

Although AG is the main tool used by the Fed in its conduct of monetary policy, it is actually not realistic to take AG to be exogenous. The Fed is much more concerned about RS and M1 than it is about AG. A better way to think about this setup is that the Fed uses AG to achieve a target value of RS or M1.

Consider an M1 target first. If the Fed picks a target value for M1, then M1 is now exogenous (set by the Fed). This means that we now have an extra equation---81---with no unknown matched to it. The obvious choice for the unknown is AG. In other words, AG is now endogenous; its value is whatever is needed to have the target value of M1 be met. A similar argument holds for an RS target. If the Fed picks a target value for RS, then RS is exogenous and AG is endogenous. AG is whatever is needed to have the RS target be met. This is also where equation 30 can come in. Instead of setting a target value of RS exogenously, the Fed can use an equation like 30 to set the value. If this is done, then RS is endogenous---it is determined by equation 30. AG, of course, is also endogenous, because its value must be whatever is needed to have the value of RS determined by equation 30 be met. Finally, one can have the Fed target the level of non borrowed reserves, BR - BO, which makes BR - BO exogenous and AG endogenous. The program you will be using allows the various cases just discussed to be considered. Remember, however, that taking AG to be exogenous is not likely to be a sensible assumption about the way that the Fed actually behaves.

You will see if you change RD or G1 when AG is endogenous that nothing much happens. If, say, AG is endogenous and RS is exogenous, then whatever changes are made in RD or G1 are offset by changes in AG. The target is RS, and AG adjusts to achieve this target regardless of the values of RD and G1. A similar argument holds if M1 is exogenous or equation 30 is used. Changing RD or G1 when AG is endogenous is thus not an interesting exercise.

As a final point about the above equations, note the relationship between fiscal policy and monetary policy. Fiscal policy changes SG, which from equation 77 must be financed in some way. If the Fed keeps AG unchanged, an unrealistic assumption, the financing will all be through high powered money. If the Fed keeps M1 unchanged, then AG will adjust to offset any effects of the fiscal policy change that would have otherwise changed M1. If the Fed keeps RS unchanged, then AG will adjust to offset any fiscal policy effects on RS. It should be clear from this that the size of fiscal policy effects on the economy are likely to be sensitive to what is assumed about monetary policy.

Table A.8 shows how the model is solved for each of the monetary policy assumptions. When either the money supply, the level of non borrowed reserves, or the amount of government securities outstanding is taken to be exogenous, the left hand side variables for some of the equations have to be rearranged. This is shown in the table.

The datasets are discussed in About US User Datasets, and you should read this now if you have not already done so. It is important to note that the data in BASE up to the beginning of the forecast period are actual data. From the beginning of the forecast period on, the data are forecasted data. Each quarter the model is used to make a forecast of the future, and the new forecast is added to the Web site as soon as it is done. The exogenous variable values for the forecasted quarters are the values chosen before the model is solved. The endogenous variable values for the forecasted quarters are the solution values (given the exogenous variable values).

BASE also contains the coefficient estimates that existed at the time of the forecast (the model is reestimated quarterly). All the other information that is needed to solve the model is also in BASE, and this is also true of the datasets that you create.

It is important that you understand the following. If you ask the program to solve the model for the forecast period (or any subperiod within the forecast period) and you make no changes to the coefficients and exogenous variables, the solution values for the endogenous variables will simply be the values that are already in BASE. If, on the other hand, you ask the program to solve the model for a period prior to the forecast period, where actual data exist, the solution values will not be the same as the values in BASE because the model does not predict perfectly (the solution values of the endogenous variables are not in general equal to the actual values). It is thus very important to realize that the only time the solution values will be the same as the values in BASE when you make no changes to the exogenous variables and coefficients is when you are solving within the forecast period.

Using the Forecast Period to Examine the Model's Properties

The easiest thing to do when running experiments with the model is to run them over the forecast period. Say that you want to examine what happens to the economy when government purchases of goods (COG) is decreased by $10 billion. You make this change in the program and ask for the model to be solved, creating, say, dataset NEW. If you do this over the forecast period, the solution values in NEW can be compared directly to the values in BASE. If you had not changed COG, the solution values NEW would have been the same as those in BASE, and so any differences between the solution values in NEW and BASE can be attributed solely to the change in COG.

Many of the experiments in this workbook are thus concerned with making changes in the exogenous variables, solving the model over the forecast period, and comparing the new solution values to the solution values in BASE. The difference between the two solution values for a particular endogenous variable each quarter is the estimated effect of the changes on that variable.

You should be aware that most of the time one is comparing the solution value in one dataset with the solution value in another dataset. One is not generally examining solution values over time in a single dataset. Students often confuse changes over time in a given dataset with changes between two datasets. It is critical that you understand this distinction, and so make sure that you do before going on.

You should also be clear on what is meant when a variable is said to be "held constant" for an experiment, such as the money supply or the interest rate being held constant. This means that the values of the variable for the experiment are the same as the values in the base dataset. They are not changed from the base values during the experiment. Being held constant does not mean that the variable is necessarily unchanged over time. If the values of the variable in the base dataset change over time (which most values do), then the variable values in the new data set will change over time; they just won't be changed from the values in the base dataset.

Using Non Forecast Periods to Examine the Model's Properties

In some cases one must use non forecast periods to carry out the experiment. If, for example, you are concerned with examining how the economy would have behaved in the 1970s had the price of imports (PIM) not increased so dramatically, you must deal with the period of the 1970s. Say you are interested in the 1973:1-1979:4 period, and you want to know what would have happened in this period had PIM not changed at all. The first thing you must do is solve the model for this period making no changes to PIM (or any other exogenous variable). Call the dataset of these solution values BASEA. At this point you can compare the solution values in BASEA with the values in BASE, which are the actual values. The differences in these values are the prediction errors, and you can examine these errors to see how well the model predicted the period. Remember that this prediction uses the actual values of all the exogenous variables---it is an ex post prediction.

Once you have created BASEA, this dataset is your "base" dataset for v any further experiments. Using BASEA as your base, you can then change PIM for the 1973:1-1979:4 period, where the new values of PIM are unchanged over time, and solve the model again, creating, say, NEWA. You can then compare the values in NEWA with those in BASEA to examine the effects of the PIM change. You do not compare the values in NEWA with those in BASE. The differences between these values are a combination of the prediction errors of the model and the effects of the changes in PIM, and you cannot separate the differences into the two parts. Again, it is extremely important that you understand what is going on here.

Changing Coefficients

Some of the experiments call for changing one or more coefficients in the model. Even if you are dealing with the forecast period, once you make a change in a coefficient, the solution values are not the same as the values in BASE even if no exogenous variables have been changed. When you change one or more coefficients, you must first solve the model with only these coefficients changed. The dataset created by this solution, say BASEA, is your base dataset. You can then make changes in the exogenous variables and solve the model for these changes. The dataset created from this solution, say NEWA, can then be compared to BASEA. It is not correct to compare NEWA to BASE, because the differences between these two datasets are due both to the coefficient changes and to the exogenous variable changes.

Selecting Variables to Display

There are 127 endogenous and slightly over 100 exogenous variables in the model. It is clear that you do not want to display results for each of these variables, or even for each of the endogenous variables, for each experiment. The program allows you to display a subset of these variables, and you should in general use this option. A few key variables should always be displayed, and other variables should be displayed as the experiment warrants.

The following list includes most of the variables that you will ever want to examine. You can select from this group as the experiment warrants. The variables are in alphabetical order.

AA      Total net wealth of the household sector
AG      Net financial assets of the federal government
        (-AG is roughly the federal government debt)
BO      Bank borrowing from the Fed
BR      Total bank reserves
CD      Consumer expenditures for durables
CG      Capital gains (+) or losses (-) on stocks held by households
CN      Consumer expenditures for nondurables
COG     Federal government purchases of goods
COS     State and Local government purchases of goods
CS      Consumer expenditures for services
CUR     Currency held outside banks
D1G     Personal income tax parameter for federal taxes
E       Total employment
EX      Exports
GDPD    GDP price deflator
GDPR    Real GDP
IHH     Residential investment, household sector
IKF     Nonresidential fixed investment of the firm sector
IM      Imports
INTF    Interest payments of the firm sector
INTG    Interest payments of the federal government
IVF     Inventory investment of the firm sector
JF      Number of jobs in the firm sector
KK      Capital stock of the firm sector
L1      Labor force, men 25-54
L2      Labor force, women 25-54
L3      Labor force, all others 16+
LM      Number of moonlighters
MB      Net demand deposits and currency of banks
MF      Demand deposits and currency of the firm sector
MH      Demand deposits and currency of the household sector
M1      Money supply
PCGDPD  Percentage change in the GDP price deflator (annual rate)
PCGDPR  Percentage change in real GDP (annual rate)
PCM1    Percentage change in the money supply (annual rate)
PF      Price deflator for the firm sector
PG      Price deflator for COG
PIEF    Before tax profits of the firm sector
PIM     Import price deflator
POP     Population, 16+
POP1    Population, men 25-54
POP2    Population, women 25-54
POP3    Population, all others 16+
PROD    Output per paid for worker hour ("productivity")
RB      Bond rate
RM      Mortgage rate
RS      Three month Treasury bill rate
SG      Savings of the federal government
SGP     Federal government surplus (+) or deficit (-)
SHRPIE  Ratio of after tax profits to the wage bill
SRZ     Household saving rate
SR      Savings of the foreign sector (-SR is the U.S. current account)
THG     Personal income taxes to the federal government
TRGH    Transfer payments from the federal government to households
U       Total number of people unemployed
UR      Unemployment rate
WA      After tax nominal wage rate
WF      Nominal wage rate of the firm sector
WR      Real wage rate
X       Total sales of the firm sector
Y       Total production of the firm sector

The Possible Transformations for Modifying the Stochastic Equations

The program allows you to modify the stochastic equations in the US model subject to two restrictions. The first restriction is that the left hand side variables of the stochastic equations cannot be changed. The second restriction is that only certain variables can be used as right hand side variables. The right hand side variables can be any endogenous or exogenous variable in the model, that is, any variable listed in Table A.2 in Appendix A. In addition, certain transformations of the endogenous and exogenous variables are allowed. These are as follows:

Name     Transformation
 
LCSZ      log(CS/POP)
LCNZ      log(CN/POP)
CDZ       CD/POP
IHHZ      IHH/POP
L1Z       log(L1/POP1)
L2Z       log(L2/POP2)
L3Z       log(L3/POP3)
LMZ       log(LM/POP)
LMHZ      log[MH/(POP*PH)]
LPF       logPF
IKFZL     IKF-1-DELK*KK-1
IKF1      <>IKF
LJF1      <>logJF
LHF1      <>logHF
LHO       logHO
LWF       logWF
LMFZ      log(MF/PF)
LY1       <>logY
LHF       logHF
PCPD      100*[(PD/PD-1)4-1]
LPX       logPX
BOZBR     BO/BR
LXMFAZ    log[(X-FA)/POP]
LPIEFAZ   log[(PIEF-TFG-TFS)/DF-1]
RB1       <>RB
LCURZ     log[CUR/(POP*PF)]
LIMZ      log(IM/POP)
LUB       logUB
LDF1      <>logDF
CFM1      <>(CF-TFG-TFS)
TXCRA     TXCR*IKFA
LPIKIKF   log(PIK*IKF)
LCCF      logCCF
LXMFA     log(X-FA)
LU        logU
LWFD5     log[WF*(1+D5G)]
RMACDZ    RMA*CDA
RMALIHHZ  RMA-1*IHHA
PCM1L1A   D794823*PCM1-1
LMHL1Q    log[MH-1/(POP-1*PH)]
LMFL1Q    log(MF-1/PF)
LCURL1Q   log[CUR-1/(POP-1*PF)]
KDZ       KD/POP
LGAPA     log[(YS-Y)/YS + .04]
LEXL      log(JF/JHMIN)
LY        logY
GDPRZ     GDPR/POP
YDZ       YD/(POP*PH)
LWAZPH    log(WA/PH)
LYNLZ     log[YNL/(POP*PH)]
LAAZ      log(AA/POP)
AAZ       AA/POP
PX1VL     (PX-PX-1)*V-1
LEXLL1D   DD772*log(JF/JHMIN)-1
LL1Z      log(L1/POP1)
LJF1L1D   DD772*Ch logJF-1
RS1       <>RS
LPFZPIM   log(PF/PIM)
YNLZ      YNL/(POP*PH)
KHZ       KH/POP
LL2Z      log(L2/POP2)
LL3Z      log(L3/POP3)
TDD       DD772*T
LCNZ1     <>log(CN/POP)
LPIM      logPIM
LYDZ      log[YD/(POP*PH)]
RSB       RS*(1-D2G-D2S)
LLMZ      log(LM/POP)
RBA4L3A   IKFA*(real, after tax RB-3)
EXKK      KK-KKMIN
Y1        <>Y

If you study Table A.3 carefully, you will see that many of the above transformations are used in the 30 stochastic equations. To give an example of how these transformations can be used, say that one wanted to add the log of the lagged value of assets to the housing investment equation, equation 4. This addition could be made using LAAZ above. When you are modifying the equations, different lag lengths can be used for any variable. Likewise, different values of the order of the autoregressive process for the error term can be used.

Annual Rates versus Quarterly Rates

As noted in About US User Datasets, the flow variables are displayed at annual rates even though the variables are at quarterly rates in the model. This generally poses no problem in performing the experiments in this workbook except for examining a budget constraint equation like 77. (See the discussion of equation 77 in Section 2.5.) Equation 77 is:

77 0 = SG - <>AG - <>MG + <>CUR + <>(BR - BO) - <>Q - DISG

SG is a flow variable, and so it is displayed in at an annual rate. A variable like AG, on the other hand, is a stock variable, where the quarterly rate/annual rate distinction is not relevant. The change in AG (<>AG), however, is a flow variable. (<>AG is the value of AG this quarter minus the value of AG last quarter.) Say that you display all the variables in equation 77 and want to see if equation 77 is met in the data (which it will be). When you compute a change like <>AG, it will be at a quarterly rate unless you multiply it by four. If you don't multiply all computed changes by four, equation 77 will not be met because SG and DISG are at annual rates. Therefore, remember to multiply all computed changes by four when examining an equation like 77.

In this chapter you are asked to table and graph the variables listed in Section 2.6 for the period since 1970:1 and examine how they behave during the recessionary and inflationary subperiods. You should save these tables and graphs for future reference. You may also want to table and graph the variables for the entire period since 1952:1. If you do so, how many other recessionary and inflationary subperiods can you pick out?

In doing the work in this chapter (and the others) you should always use Table A.2 in Appendix A as the reference for the variable names. The definition of a variable is not always repeated when the notation for the variable is used in describing the experiments.

Experiment 3.1: Variables for the 1970:1-2008:3 Period

• Table and graph the variables listed in Section 2.6 for the 1970:1-2008:3 period.

Some of the following questions may require more knowledge of macroeconomics than you currently have. Do your best for now to get a feel for the data, and then come back to the tables and graphs to review the data from time to time as you gain more knowledge.

Recessionary Periods

1. From the table and graph for real GDP, GDPR, can you pick out the four recessionary periods mentioned above? Which of the other variables tend to follow the business cycle, and which do not? Can you think of reasons why some do and some don't? Focus in particular on the components of GDPR and on the saving rate, SRZ.

Inflationary Periods

1. From the table and graph for the percentage change in the GDP deflator, PCGDPD, can you pick out the two inflationary periods? How does the import price index, PIM, behave during these periods? How about the nominal wage rate, WA, and the real wage rate, WR?
2. How much overlap is there between the recessionary and inflationary periods? (These overlap periods are called periods of "stagflation.") Can you speculate on what might have caused the stagflation? Can you pick out any variables that you have graphed as possible culprits?

Labor Market Variables

1. How did the unemployment rate, UR, behave during the recessionary and inflationary periods?
2. Is there a tight relationship between UR and the rate of inflation, PCGDPD?
3. How did productivity, PROD, behave during the recessionary periods? Why might PROD be procyclical?
4. Calculate the values of the labor force participation rates L1/POP1, L2/POP2, and L3/POP3 for a few of the quarters within the overall period. Note how they have changed over time, and note the remarkable rise in L2/POP2, the participation rate for women 25-54, since 1970. What might have caused this rise?
5. Are the labor force participation rates procyclical or countercyclical or neither? Do they seem to vary with UR? Why might they vary with UR?

Fiscal Policy Variables and Other Federal Government Variables

1. Examine how the fiscal policy variables COG, D1G, D2G, D3G, D4G, D5G, and TRGH have changed over time.
2. Examine how the government deficit SGP and the government debt -AG have changed over time. What are some of the factors that led to the large rise in the deficit?
3. Why are interest payments of the government, INTG, growing so rapidly? What consequences does this have for SGP and AG?

Monetary Policy Variables and Other Financial Variables

1. How did the bill rate, RS, behave during the recessionary periods? During the inflationary periods? What does this say about Fed behavior?
2. What is the relationship between the long term bond rate, RB, and RS? Why might RS fluctuate more than RB?
3. Examine carefully the behavior of RS during the period 1979:4-1982:3. Does this behavior seem different than at other times? How did the money supply, M1, behave during this period versus at other times? Does it seem to you that the Fed behaved differently during this period than otherwise?
4. Compute velocity, GDP/M1, for a few quarters and examine how it has changed over time. How stable over time does it seem?
5. What happened to GDPR and GDPD in 1974? What was the monetary response to this in 1975? (Look at RS and M1.)
6. What was the monetary response following the stock market crash in 1987? Can you pick out the effects of the stock market crash on household wealth, AA?

Foreign Sector Variables

1. Examine how PIM, IM, PEX, and EX have changed over time.
2. How has the saving of the foreign sector, SR, changed over time? What are the main factors that have led to this change? (-SR is roughly the U.S. current account.)
3. SGP and -SG are sometimes call the "twin deficits." Examine their relationship over time. Can you think of reasons why the two deficits might be positively correlated?

Extra Credit

1. If you have tabled or graphed the variables from 1952:1 on, answer the relevant questions above for this longer period. For example, how many recessions can you pick out since 1952? How many high inflation periods? Are some of the relationships that were not stable after 1970 closer to being stable before 1970?

4.1 National Income and Product Accounts
4.2 The Flow of Funds Accounts

The National Income and Product Accounts (NIPA) and the Flow of Funds Accounts (FFA) are more than just the places where much macroeconomic data come from. They help us organize our thoughts about the structure of the economy, and they provide the framework for constructing models of the economy. The exercises in this chapter are designed to get you acquainted with the two sets of accounts.

By definition, GDP is equal to consumption plus investment plus government spending plus exports minus imports. In the US model there are six sectors and a number of categories of consumption, investment, and government spending, which make the GDP definition and other definitions somewhat more involved. It will be useful to begin with the definition of total sales of the firm sector, denoted X, which is defined in equation 60. Equation 60 is:

60. X= CS + CN + CD + IHH + IKF + EX - IM + COG + COS + IKH + IKB + IKG + IHF + IHB - PIEB - CCB

Experiment 4.1: The Components of X

• Table variable X and its components (from equation 60) for 2008:3.

1. Note that some components are quite small. You should use this experiment to get a feel for the size of the various components. Note what a large fraction consumption is of total sales.

By definition production minus sales is the change in inventories. This definition is equation 63:

63. V = V_-1 + Y - X

where V is the stock of inventories at the end of the quarter, Y is production, and X is sales. Also, by definition, the change in inventories is inventory investment, which is equation 117:

117. IVF = V - V_-1

where IVF is inventory investment of the firm sector. In the model Y is determined by equation 11, which reflects production smoothing behavior, X is determined by the identity 60, V is determined by the identity 63, and IVF is determined by the identity 117.

Y is not total GDP; it is only the part of GDP produced by the firm sector. Some production also takes place in the financial and government sectors. Equation 83 defines real GDP as production in the firm, financial, and government sectors:

83. GDPR = Y + PIEB + CCB + PSI13*(JG*HG + JM*HM + JS*HS) + STATP

where PIEB + CCB is production in the financial sector and the term after that is production in the government sector. STATP is a statistical discrepancy pertaining to the use of the chain weighted data, which is discussed in Fair (1994).

Experiment 4.2: Going from X to GDPR

• Table X, Y, IVF, and GDPR for 2008:3.

1. Note that IVF is small relative to Y and that most of GDPR is Y (i.e., most of GDPR is produced by the firm sector).

4.1.2 Nominal versus Real GDP

As any introductory economics textbook discusses, it is important to distinguish between nominal and real GDP. By definition, nominal GDP is equal to real GDP times the GDP price index. In the model, this relationship is equation 84, which is used to determine the GDP price index:

84. GDPD = GDP/GDPR

1. Use the tables and graphs of GDP, GDPR, and GDPD from the previous chapter to examine the relationship between GDP and GDPR, i.e., to examine GDPD. During which periods would it have been particularly misleading to have focused on GDP instead of GDPR as a measure of output?

4.1.3 Federal Government Variables

The NIPA are useful for examining the role that the government plays in the economy. Total expenditures of the federal government (EXPG) are defined in equation 106, and total receipts (RECG) are defined in equation 105:

106. EXPG = PUG + TRGH + TRGR + TRGS + INTG + SUBG - WLDG - IGZ

105. RECG = TPG + TCG + IBTG + SIG

The federal government surplus (+) or deficit (-) (SGP) is the difference between receipts and expenditures, which is defined in equation 107:

107. SGP = RECG - EXPG

Experiment 4.3: The Federal Government Budget

• Table the variables in equations 105, 106, and 107 for 2008:3.

1. What are the largest components of government expenditures? What are the largest sources of government receipts?

There is also an equation for each sector that defines its budget constraint. If, for example, a sector's financial saving is positive, this must result in an increase in at least one of its assets or a decrease in at least one of its liabilities. The budget constraint of the household sector is equation 66 (remember that <> means "change in"):

66. 0 = SH - <>AH - <>MH + CG - DISH

AH is the household sector's net financial assets except for its holding of demand deposits and currency (MH). CG is the capital gains variable, and DISH is a discrepancy term. If CG is positive, then AH increases because corporate stocks held by the household sector are included in AH. Taking CG as given, equation 66 shows that if SH is positive and MH is unchanged, then AH must increase. In other words, a positive level of saving must result in an increase in net financial assets unless it all goes into demand deposits and currency.

The same considerations apply to the other sectors of the model. The five other saving equations are 69 (firm sector), 72 (financial sector), 74 (foreign sector), 76 (federal government sector), and 78 (state and local government sector). Note that federal government saving (SG) is almost always negative because the federal government almost always runs a deficit. (The federal government surplus or deficit variable in the model is actually SGP, not SG, but for all intents and purposes SG and SGP are the same. There are minor accounting differences between the two variables.) Note also that the saving of the foreign sector (SR) is the negative of the U.S. balance of payments on current account.

The five other budget constraint equations are 70 (firm sector), 73 (financial sector), 75 (foreign sector), 77 (federal government sector), and 79 (state and local government sector). Equation 77, the federal government budget constraint, was discussed in Section 2.5.

An important constraint in the FFA is that the sum of the financial saving across sectors is zero. Someone's expenditure is someone else's receipt, which is what this constraint says. In the notation in the model the constraint is:

0 = SH + SF + SB + SG + SS + SR

If you combine this equation with the six budget constraints, you get equation 80 in the model:

80 0 = <>AH + <>AF + <>AB + <>AG + <>AS + <>AR - CG + DISH + DISF + DISB + DISG + DISS + DISR + STAT + WLDF - WLDG - WLDS - DISBA

Equation 80 is another way of saying that the sum of the saving across sectors is zero, in this case using the net financial asset variables rather than the saving variables. Because of the budget constraints, these two ways are equivalent. Equation 80 is redundant in the model because it is implied by the six budget constraints plus the fact that the saving variables sum to zero. It is, of course, a good way of checking that the data are correct.

Experiment 4.4: Saving Equations and Budget Constraints

• Table the variables in equations 65 and 66 for 2008:2 and 2008:3. Do the same for the variables in equations 69 and 70, in equations 72 and 73, in equations 74 and 75, in equations 76 and 77, and in equations 78 and 79.

1. Note that the budget constraints are met in the data aside from rounding errors. (Remember when you check the budget constraints, you must multiply the computed changes in the stock variables by four, since the saving variables are at annual rates. See the discussion in Section 2.6. The changes in this case are 2008:3 values minus 2008:2 values.)
2. Observe that the six saving variables, SH, SF, SB, SG, SS, and SR, sum to zero. Who are the big savers and who are the big dissavers?
3. Check equation 77 carefully, and review the discussion of this equation in Section 2.5.
4. Check the budget constraint for the foreign sector carefully. Why has AR been increasing so rapidly recently?

The final constraint that will be discussed is the demand deposit identity, equation 71:

71. 0 = <>MB + <>MH + <>MF + <>MR + <>MG + <>MS - <>CUR

MH, MF, MR, MG, and MS are demand deposit and currency holdings of the various sectors. CUR is the amount of currency in the hands of the public. MB is the total amount of demand deposits held; it is negative because demand deposits are a liability of banks.

Experiment 4.5: The Demand Deposit Identity

• Table the variables in equation 71 for 2008:2 and 2008:3.

1. Check that equation 71 is met for 2008:3. Which sector holds the largest amount of demand deposits and currency? Which sector holds the smallest?

5.1 Changes in Government Purchases of Goods
5.2 Other Fiscal Policy Variables
5.3 Balanced Budget Multiplier

A good way to learn about the properties of a model (and if the model is any good about the properties of the economy) is to consider the effects of changing various fiscal policy variables. We know from Chapter 2 that fiscal policy effects depend on what is assumed about monetary policy, and so we must also consider monetary policy in this chapter. We begin with a very straightforward experiment: a decrease in government purchases of goods with the money supply remaining unchanged. Federal government purchases of goods in the model is denoted COG. COG is to be decreased with M1 remaining unchanged.

Experiment 5.1: Decrease in Government Spending, Money Supply Unchanged

• Change COG by -20 in each quarter of the forecast period. Take M1 to be exogenous.

The following are some of the questions you should consider about this experiment, but they are by no means exhaustive. You should add to the list. For most questions, you should focus on the results about four quarters out. After four quarters the economy has adjusted enough to the government spending change for the effects to be noticeable.

1. How are the following variables affected over time and why: real GDP (GDPR), the price level (PF), the unemployment rate (UR), the short term interest rate (RS), and the federal government deficit (SGP)? Note that the change in GDPR divided by the change in COG is the multiplier. How does the multiplier change over time? Why?
2. Examine how the main components of GDP are affected and try to explain why they move the way they do: CS, CN, CD, IHH, IKF, IVF, and IM.
3. Note that in the first quarter of the change the decrease in X is greater than the decrease in Y. Why? Relate this to the theory behind equation 11.
4. Examine the employment and labor force responses in the model, and again try to explain the reasons behind the movements: JF, E, L1, L2, L3, and LM. What happened to productivity, PROD?
5. With M1 fixed, why did RS fall? (For this answer you can appeal to the ideas behind the LM curve.) Why did RB and RM fall?
6. What happened to the change in stock prices, CG, initially and then over time? Why?
7. How did AG change? (-AG is the amount of federal government securities outstanding.) Why? What happened to nonborrowed reserves, BR - BO, and why?
8. How was the level of profits, PIEF, affected? How was the saving rate of the household sector, SRZ, affected?
9. What happened to the percentage change in the real wage, WR, and why?
10. Use the discussion in Section 2.4 to help you in understanding the reasons for the results.

The second experiment is the same as the first except that the interest rate, RS, rather than the money supply, M1, is kept unchanged in response to the decrease in COG.

Experiment 5.2: Decrease in Government Spending, Interest Rate Unchanged

• Change COG by -20 in each quarter of the forecast period. Take RS to be exogenous. Solve the model.

1. The key feature of this experiment is that it is more contractionary than Experiment 5.1. Explain carefully why.
2. How did the Fed's behavior differ in this experiment from that in the first one? In which experiment was the change in AG larger? Why?

The third experiment is the same as the first except that the Fed is taken to behave according to the interest rate reaction function-equation 30.

Experiment 5.3: Decrease in Government Spending, Interest Rate Reaction Function

• Change COG by -20 in each quarter of the forecast period. . The default option in the program is to use the interest rate reaction function, and so no changes here are needed regarding the assumptions about monetary policy. Solve the model.

1. An interesting question regarding this experiment is whether the interest rate falls more or less here than it does in Experiment 5.1. In other words, does the Fed ease up more or less here than it does when it keeps M1 unchanged. This is an empirical question, since in theory the result could go either way. Given whatever the interest rate differences are, explain how and why the rest of the economy is different here after the COG change than it is in Experiment 5.1.
2. How did the change in AG here compare to the changes in Experiments 5.1 and 5.2?

The experiments so far have been for a permanent change in government purchases of goods. The next experiment examines the effects of a temporary change. This experiment is the same as Experiment 5.1 except that the change in COG is only for the first quarter.

Experiment 5.4: Temporary Decrease in Government Spending, Money Supply Unchanged

• Change COG by -20 in the first quarter of the forecast period. Take M1 to be exogenous. Solve the model.

1. Compare the results of this experiment with those of Experiment 5.1. (Note that the results for the first quarter are the same.)
2. How long does it take for the effects of the COG change on the economy to be essentially negligible?

Say that instead of cutting government spending COG by $20 billion, you wanted to raise personal income taxes by approximately the equivalent amount. How do you do this? The federal personal income tax rate in the model is D1G, and so D1G needs to be raised. Say that we want D1G to be raised so that the initial impact of the tax increase takes about the same amount away from the economy as the decrease in COG did. COG is in real terms and tax payments are in nominal terms, and so the first thing we need to do is to convert the $20 billion decrease in COG into nominal terms. PG in the model is the price index for COG, and its value in 2008:3 was 1.264. The nominal change in government spending corresponding to a $20 billion real change is thus $20 billion times 1.264 = $25 billion. We thus need to raise taxes by $25 billion.

The level of federal personal income taxes in the model (THG) is determined by equation 47:

47. THG = [D1G + (TAUG*YT)/POP]*YT

where YT is taxable income. The federal tax system is estimated in the model to be slightly progressive, and so the tax rate increases as YT increases. TAUG is the estimated progressivity parameter. Now, the question is, in order to increase THG by $25 billion, how much do we have to raise D1G? To answer this, we need to know the level of YT. YT in 2008:3 was $10,655.8 billion, and $25 billion is .235 percent of this. Therefore, we need to raise D1G by .00235. We will thus raise D1G by .00235 in the following experiment.

You should be aware that calculations like we have just done are rough. You cannot change D1G to hit a particular change in THG exactly because YT is endogenous. As D1G is changed, the economy changes, including YT, and so THG will also change for this reason as well as from the initial change in D1G. Calculations like the above give one a fairly good idea where to start, but it may be after the first run that you want to adjust D1G slightly to meet more accurately the THG target.

Experiment 5.5: Increase in the Personal Income Tax Rate, Interest Rate Reaction Function

• Change D1G by .00235 in each quarter of the forecast period. Solve the model.

1. Compare the results of this experiment with those of Experiment 5.3. Why is the economy. slower to respond in this case? Which policy change is best for decreasing the federal government deficit and why?
2. What does the Fed do in response to the increase in taxes and why?
3. How has the tax rate increase affected the labor force? How has it affected the personal saving rate, SRZ?

Another important fiscal policy variable