m.” M‘W'Tw~ Al II-' 7" ' ' .1 INN? +1’:5..”°¢ ABSTRACT THE SHORT-RUN IMPACT OF AN AD'VALOREM EXCISE TAX ON THE AUTOMOBILE MARKET: A DYNAMIC APPROACH By Zane A. Spindler This study seeks to determine the short-run impact of manufacturers' excise taxes on the market for new automobiles. Normally, potential output and incidence effects of any proposed change in excise taxes are analyzed via comparative statics analysis, which compares the long- run equilibrium solution for the model before and after the change. Prediction of effects based on such analysis can be misleading since: (1) it may take a long time before the long-run effects are realized, and (2) the interim effects may be quite different than those predicted for the long run. Therefore, in order to get a good estimate of the immediate and intermediate effects of a change in a policy parameter, a method of comparative dynamics must be employed. A dynamic model has been constructed for the new car market which formulates structural relations for new car buyers, new car dealers, and manufacturers. The parameters of the model are estimated simultaneously by the full information maximum likelihood method using monthly data for the years 1960 to 1965. The fundamental dynamic equation, the Zane A. Spindler characteristic equation, and the equilibrium solution are then deve10ped for the model. ‘The characteristic equation is analyzed to determine the stability properties of the model. ‘The fundamental dynamic equation is used to develop the dynamic multipliers for the excise tax variable. The dynamic multipliers and the cumulative dynamic multipliers are then compared to the equilibrium or long-run multipliers which are_given by the equilibrium solution to the model. The new car market as represented by the model is found to be stable, with a period of oscillation of approximately fourteen months. The largest of the roots of the characteristic equation is given by a complex conjugate pair whose modulus is close to but less than one. As a result, the model will have a time path that is oscillatory and stable but which converges very slowly. Consequently, the equilibrium values for the model are not approached rapidly. Therefore, for short and intermediate term policy purposes, the equilibrium solution is not relevant. Instead, the dynamic multipliers and cumulative dynamic multipliers must be considered. These will show that the net effects of an excise tax change depend on the period considered. The direction of the effects is reversed approximately every six months. The size of the short and intermediate term effects varies from zero to several times the long-run effect, depending on the period considered. Zane A. Spindler It can be concluded that the main effect of a change in excise tax rates will be to introduce a new cyclical component into the determination of the endogenous variables in this market. No clear statement about the beneficial aspects for consumers of such a change can be made without Specification of the appropriate time period. THE SHORT-RUN IMPACT OF AN AD VALOREM EXCISE TAX ON THE AUTOMOBILE MARKET: A DYNAMIC APPROACH BY. Zane A: Spindler A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1968 -- '3" r. E’ (2 VI) ._~-’ v 6 f Copyright by ZANE ANDRE SPINDLER 1968 Copyright by ZANE ANDRE SPINDLER 1968 ACKNOWLEDGEMENTS Special mention must be made at this time of the assistance that I have received from Dr. Paul E. Smith and Dr. Jan Kmenta. Dr. Smith has suggested many elements of the approach taken on this tOpic, and Dr. Kmenta has_given valuable advice on the econometric problems involved. Both have read the earlier drafts and have been instrumental in the correction of initial errors 0 I would like to thank the Economics Department at Michigan State University for underwriting the many hours of computer time that I have used on the University's CD 3600 computer. My Special thanks are due to Roy Gilbert for his programming assistance. The National Science Foundation must be acknowledged for the financial support given to me during the summer of 1966 when the initial research on this project was being performed. I would also like to thank Dr. R. A. Holmes, iii who has helped me with various econometric problems while I was finishing the dissertation at Simon Fraser University. iv TABLE OF CONTENTS TITLE ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES CHAPTER I: INTRODUCTION iii vii ix CHAPTER II: PREVIOUS ECONOMETRIC STUDIES OF AUTOMOBILE DEMAND USING TIME SERIES DATA Section 1: R005 and von Szeleski section 2: Chow Section 3: Nerlove Section 4: Suits Section 5: Turnovsky Section 6: Ueno and Tsurumi Section 7: Conclusion CHAPTER III: CONSTRUCTION OF THE MODEL Section 1: The Demand for New Cars Section 2: Production of New Cars Section 3: Total Inventory Demand Section 4: Inventory Supply Identity Section 5: Stock Identity Section 6: The Complete Model APPENDIX TO CHAPTER III 5 5 9 12 16 21 25 30 CHAPTER IV: ESTIMATION OF THE MODEL 50 Section 1: Single Equation Estimation 50 Section 2: Determining Identification 54 Section 3: Complete System Estimation 59 CHAPTER V: DYNAMIC ANALYSIS 69 Section 1: Derivation of the Fundamental Dynamic Equations for the Endogenous Variables 69 Section 2: The Characteristic Component, the Characteristic Equation, and the Characteristic Roots 73 Section 3: The Particular Component and the Equilibrium Solution 81 Section 4: The Full Complete Solution 82 CHAPTER VI: DERIVATION OF THE FUNDAMENTAL DYNAMIC EQUATIONS, THE PARTICULAR SOLUTION, AND THE DYNAMIC MULTIPLIERS FOR.EXCISE TAXES 91 CHAPTER VII: INCIDENCE AND OUTPUT EFFECTS 113 Conclusions 124 BIBLIOGRAPHY 125 vi 6.11 6.12 6.13 6.14 6.15 6.2 6.3 6.4 LIST OF TABLES Summary of Elasticities Demand for New Cars Production of New Cars Inventory Demand Fundamental Dynamic Equations: Sales Production Inventory Price Stock Equilibrium Solutions Fundamental Dynamic Equations With Tax Variable: Sales Production Inventory Price Stock Equilibrium Solutions with Tax Variable Dynamic Multipliers for Tx u Dynamic Multipliers for Yt vii PAGE 33 66 67 68 84 85 86 87 88 89 98 98 99 100 100 109 110 List of Tables, cont. .6.S Cumulative Dynamichultipliers for Tx_ 111 6.6 Cumulative Dynamic Multipliers for.Yt 112 7.1 Long-Run Incidence and Output Effects 115 .7.2 Maximum Changes in Endogenous Variables Induced By A Change in the Excise Tax Variable 120 viii LIST OF FIGURES FIGURE PAGE 5.1 78 5.2 80 ix LIST OF APPENDICES PAGE APPENDIX TO CHAPTER III: SYMBOL INDEX 47 CHAPTER I INTRODUCTION At mid-year in 1965, the federal excise tax on automobiles was reduced from ten percent to seven percent and was made retroactive to May 15, 1965. The benefits of such a tax reduction might have gone solely to the consumer, solely to the suppliers, or might have been shared by both. Using the "comparative statics" analysis traditional in the public finance sphere of economics, one would predict that such a reduction of an ad valorem tax would result in a reduction in price and an increase in sales.1 The actual size of these changes would depend on the elasticities of the supply and demand curves for the good involved. Given the estimated equations for supply and demand, the size of the effect of the tax change could be predicted. 1R. A. Musgrave, The Theory of Public Finance (New York: McGraw Hill Book C0., 1959), ch. 13. One question that arises at this point is whether the predicted effect will be immediate or whether it will only occur after a period of time. If the latter is the case, two more questions arise: (1) how long will it take before this effect is realized? and, (2) what will be the nature of the effects in the intermediate period, before the predicted effect is realized? The methods of comparative statics are not designed to answer such questions. Yet these are the very answers that are required for analysis of the real world in the short run. In comparative statics, two equilibrium positions resulting from different levels of an exogenous variable are compared to determine the effect of changing the exogenous variable from one level to another. The system is only analyzed when it is at its equilibrium position after all adjustments within the given system have been completed. However, the adjustment process and the time path of the endogenous variables are of crucial interest for short-run analysis. Therefore, it is necessary to introduce the method of comparative dynamics. This method involves the comparison of the time paths of endogenous variables generated by a system when it is subject to different levels of a given exogenous variable. There are two basic procedures for implementing this method. The first, a simulation method, is actually to calculate the time paths of the endogenous variables when the exogenous variable of interest is at one level and subtract from them the time paths of the endogenous variables when the exogenous variable is at another level - the difference showing the effect in each period from the change in the exogenous variable. The second procedure is to calculate the "dynamic multipliers" for the exogenous variable, which give the period-byéperiod impact of a change in that variable. The period-by-period effect shown by each procedure will be the same; however, the dynamic multiplier procedure will be more general. In this investigation, the dynamic multiplier procedure of comparative dynamics will be used to determine the short-run incidence and output effects on the auto- mobile market of a change in the excise tax rate. Only the internal dynamics of the automobile market will be considered in this study. The interaction between the automobile market and other markets will not be considered. The relationship between activity in the automobile market and national income will be assumed to be uni-directional; income will be an exogenous variable. Thus, the scope of this investigation will be narrowed to the internal incidence and output effects of an excise tax reduction. In Chapter II previous econometric studies of the automobile market will be reviewed. The models used for these studies are not appropriate for a short-run dynamic analysis with monthly data. Therefore a short— run dynamic model will be developed in Chapter III and the parameters will be estimated in Chapter IV. Chapter V deve10ps the dynamic aSpects of the model. In Chapter VI the dynamic multipliers for the tax variable will be derived and calculated. The conclusions about the incidence and output effects of excise tax changes will be given in Chapter VII. CHAPTER II PREVIOUS ECONOMETRIC STUDIES OF AUTOMOBILE DEMAND USING TIME SERIES DATA In this chapter, some of the previous works on automobile demand will be discussed briefly. This discussion will be limited to those studies which used time series data because there is a similarity between the approaches taken in those and the present study. The studies specifically treated will be those of R005 and von Szeleski, Chow, Nerlove, Suits, Turnovsky, and Ueno and Tsurumi. The symbols used in the discussion will be those of the original authors, so that transference between the explanation given here and the original work will be facilitated. Section 1: R005 and von Szeleski The first significant econometric study of automobile demand was presented by C. F. R005 and V. von Szeleski in 1938.1 In this study, the demand by new owners and the replacement demand are treated separately at first. Sales to new owners Snt are made a function of the difference between the maximum ownership level and the existing stock Ct‘ Thus, Snt = Ao(ftmt - Ct)’ (2.11) where A0 is an adjustment coefficient; ft is the number of families at time £3 and mt represents the price of cars pt, the durability of cars dt' and the per capita money income of the population It' The factors ft and mt together determine the maximum ownership level. Since the adjustment coefficient A0 is a function of the price pt, the trade-in price ratio Tt’ the level of income It' and the stock of cars Ct' . W“ O A3tpt TtI C A tt’ (2.12) (where A includes all neglected factors) can be used in J. equation (2.11). Also in equation (2.11),: is replaced it 1"Factors Governing Changes in Domestic Automobile Demand," The Dynamics of Automobile Demand (New York: General Motors Corp., 1939), pp.21-95. -6 e 6 Btpt Itdt (2.13) (where Bt includes all neglected factorsz), since mt depends on income It, price pt’ and durability dt of new cars. Thus, equation (2.11) is rewritten as: - B Y '5 E g a 2- - Snt A3tpt TtItCt{ftBt(d)t It Ct}. (2.14) Replacement sales are made a function of theoretical scrapping X. The adjustment coefficient here is also made a function of the price level pt, income It’ trade- in price ratio Tt' and per capita income I t; that is, -a BY SR = A4tpt T tI:xt' (2.15) (where A4 includes the neglected factors). t Sales to new owners and replacement sales are 3 then combined to get total sales. The formulation now I _ k 2AlthoughA and B are both said to include the neglected factors, tHéy are not necessarily the same since they measure the effect of these factors on different variables. 3This combination is performed by simply adding (2.14) and (2.15) and factoring out the common terms p;fiJ IB, and IY. Since the coefficients of p, T, and I are the same in bETh the new owner and replacement sales adjustment is: ‘GBY P-58 St AStpt TtIt{Ct(ftBt(a)t It - Ct) + ASXt}, (2.16) where A5 replaces A4 since the exponents for pt, Tt’ and It for new owner and replacement sales are combined. The formulation that the authors actually choose . 4 . . . . . . to fit to the data 15 a modification of th1s form. The1r fitted equation is: 1.20 -.65 s = j p {.0254c (M - c ) + .6SX } t t t 3t t 2t 1 (2.17) where j_represents supernumerary money income (which is defined as the excess of disposable income over subsistence expenditures), M3t represents the maximum ownership level, and th represefifg replacement pressure (which is a function of tHe rate of theoretical scrappings). The price coefficients, it is obvious that the elasticities of the respective adjustment coefficient with respect to these variables are assumed to be the same. 4R005 and von Szeleski, pp. 60-62. STheoretical scrapping is estimated by using a shifting mortality table to adjust the age distribution of cars. elasticity is -.65 and the income elasticity is +1.20. R005 and von Szeleski have a complete discussion on the impact of other variables that might be relevant. They also include a section on the month-to-month fluctuation in sales. Under the latter heading they discuss such short-term factors as seasonal variation, new model stimulus, price change anticipation, inventory and production of new cars, and the trend of business activity throughout the year. In some cases, they regress new car sales against these variables individually but do not deve10p a complete monthly model to compare with their yearly model. Through variation of the years covered in the analysis they find that the price elasticity varies between -.65 and -2.5. That is, by fitting the equation to data from different time periods, different values for price elasticity were obtained. Specifically, if observations from the years 1919, 1920, and 1921 were omitted, price elasticity would be at a high of -2.5. The elasticity also varied depending on the measure used for price. Section 2: Chow A significant contribution was made by Gregory 6 C. Chow in 1957, using data from 1921 to 1953. In his 6Demand for Automobiles in the United States (Amsterdam: North Holland PfiinShing C6., 1957). 10 study, Chow tests his alternative theory of demand for durable goods against the existing theory. The existing theory is interpreted as setting the stock demand of a durable good as a function of income of the consuming unit and the relative price of the good. This is stated as a b X = J I 2.21 t pt t. ( ) where XL is stock, 1 is a constant, pt is the relative price, It is the level of income, a is the price elasticity, and b'is the income elasticity. The alternative theory is that the individual has some desired asset structure (the main assets being the stock of money, consumer durables, and interest- bearing assets), and that the durable goods consumed may depend as much on that desired asset structure as on the individual's consumption plan.» Under this hypothesis, the stock demand is stated as: ,'a, It' and b have the same meaning as where Xt’ g, L33 before, but 1 represents the real per capita stock of it money and glis the elasticity of demand with respect to money. 11 The best regression equation found for the existing theory uses Friedman's concept of permanent income as the measure of income. This is found to be superior to using regular income or regular income and a time variable. This fitted equation is: -.950 2.027 X = “6.276p I t t 2 et , R = .948, (2.23) where I6 is permanent income. t With the alternative formulation, the regression coefficient for regular income is not significant while that for the money stock is, and the multiple correlation coefficient is higher than for existing theory formulation. But when permanent income is used, the existing theory formulation is better in that its multiple correlation coefficient is higher. Also, the money variable becomes insignificant in the alternative formulation. Under both theoretical formulations, the own-price elasticity of demand for the stock of automobiles in the United States is approximately -1, while the income elasticity tends to be twice as high for permanent income as for real disposable income per period: +1.745 and +.774 respectively. The best fitted equation under the alternative hypothesis is: -1.026 1.745 .243 2 X1: = ‘50562pt Iet Mat ’ R = .951, (2.24) 12 where Ma is the per capita stock of currency, demand deposi??? and time deposits. Chow also tests three basic purchase models. The desired purchase per capita is a linear function of relative price of cars in different age groups, "expected income", and old stock in the first model; of relative price of cars in different age groups and "expected income" in the second; and new car price, income, and old stock in the third. It is found that the third model (the model treating new and used cars as imperfect substitutes) is the best fitting. Another purchase model tested includes saving as an independent variable by including both diSposable income and a fraction (the marginal prOpensity to consume) times the level of "expected income". This model is significantly better in explaining purchase than models without saving. Section 3: Nerlove Nerlove's main contribution is the derivation of the long-run demand for cars from the short-run demand for new cars.7 The long-run demand for automobiles is the level of the stock of automobiles that consumers desire to hold, given certain economic conditions. It is, in fact, 7"A Note on Long-run Automobile Demand," Journal of Marketing, XXII (July, 1957), pp. 57-64. 13 the level of the stock of automobiles that would prevail if the relevant economic conditions remain unchanged over time. Actually, these conditions are continually changing and, hence, so is the desired stock. However, due to lags in the adjustment process, the actual stock may not be changed in one period by the same amount as the desired stock; thus, actual stock which is observable will generally not be equal to desired stock which is not observable. Consequently, the relationship between the desired stock and the relevant economic variables can not be straight- forwardly determined from the observable data. This complication is avoided by Nerlove in the following way. First, he formulates a linear demand function for stock: 5: = a0 + alpt + azyt + a3zt, (2.31) where s: is the long-run equilibrium stock per capita, pt is the relative price of automobiles, yt is real disposable income per capita, and 2t represents all other variables. It is assumed that an adjustment toward the desired level is actually made in period t and that the adjustment is equal to a prOportion b'of the difference between desired stock in period t and actual stock in period t-l; that is, st - s = b(s: - ), (2.32) t-1 St—l 14 where st is the actual stock in period t and s is the t-l actual stock in period t-l. Also, the actual stock in period E'is equal to the new car sales in that period plus the depreciated stock from the last period; that is, st = xt + (l-d)st_1, (2.33) where xt is the new car sales and 9.15 the rate of depreciation. From equations (2.31), (2.32), and (2.33), Nerlove derives the following demand function for new cars: x = aobd + albpt - a1b(1-d)pt_.1 + azbyt -a2b(1-d)yt + a3bzt - a3b(1-d)zt_ 1 +(1-d)xt (2.34) -1’ This formulation is then fitted by least squares to data from the 1922-41 and 1948-53 periods with the exception of variables zt and zt_1. The result is: 15 x = 0.0046 - 0.013 p + 0.006 t (.006) t (.006)pt'1 +0.013 y - 0.007 y t + 0.268 x (.002) (.003) t'1 (.211) t'l' R = .91. (2.35) The price elasticity of new car sales is -.9 and the income elasticity of new car sales is +2.8. By comparing the numerical coefficients of equation (2.35) with the coefficients of equation (2.34), the numerical value of the coefficients for the long-run demand function for stock can be derived.8 It turns out that the long-run price elasticity of automobiles is higher than the short-run price elasticity. The long-run price elasticity of automobile stock is -1.2. The long-run income elasticity is also higher, being +3.8. The rate of depreciation d_is found to be 45 percent while the adjust- ment coefficient is about .73. It must be noted that Nerlove's "long-run" elasticities are elasticities of equilibrium stock and not elasticities of long-run flow. The former are found by relating the percentage change in 8Casual observation reveals that these coefficients are overidentified. There are, in fact, two ways to find the value of d, neither of which gives the same value. Nerlove notes-this and chooses to determine d_by dividing the coefficient of y - by the coefficient for y and subtracting the resu from one. He prefers this method since he expects lesser errors in the income series than in the price series. 16 equilibrium stock to a change in price or income, while the latter are found by relating the percentage change in equilibrium flow (sales) to a change in price or income. Section 4: Suits A particularly interesting formulation of automobile demand was undertaken by Daniel B. Suits in 1958.10 This formulation differs from previous ones in several respects. First, some allowance is made for the effect of credit conditions on demand by including a measure for credit tightness (the number of months the average automobile installment contract runs = M). The difference between the real retail price of new cars 2 9Equilibrium flow will not be achieved until equilibrium stock is achieved. At that time equilibrium flow would just equal the replacement investment for equilibrium stock; that is (in Nerlove's terms), x = d - 5* Thus, long-run flow elasticity with respect to price would simply be: nx = A5*d . = n d = -1.2 ' .45 = -.54 LR Ap 5* 5*LR 10"The Demand for New Automobiles in the United States, 1929-1956," The Review of Economics and Statistics, XL (August, 1958), pp. 273-80. 17 and the average retail price of used cars U is divided by the average credit terms M’in order to arrive at a measure for the average monthly payment. Second, Suits sets up a simultaneous equation model consisting of four equations: two for demand (2.41) and supply (2.42) in the retail market for new cars and two for supply (2.43) and demand (2.44) in the used car market. Specifically, the equations are: R = al(P-U)/M + azY + aSAY + a0 + 111 (2.41) R = blP + bzw + b3T + b0 + u2 (2.42) R' = CIR + co + u3 (2'43) R' = dIU/M + dZY + dsAY + dds + d0 + ud, (2-44) where R is retail sales of new cars; Y is real diSposable income; W is the wholesale price of new cars; T is dealers' costs of operations; 3; is used car sales; ui's represent all other neglected variables (error terms); a0, b0, c0, and d0 is the stock of used cars in existence on January 1 of represent the constant terms; and S the year. Since data was not available for several of these variables, Suits does not estimate these equations simultaneously. Instead, he solves the supply and demand equations for used cars to obtain an expression for U/M 18 in terms of R, Y, AY, and S and eliminates R'. He then replaces ELM in the demand equation for new cars with this expression. Thus, he obtains a demand equation for new cars which contains the following independent variables: ELM, X, Al, and S. From this point on, the retail supply equation for new cars is not used, although it remains part of the system. To estimate the demand equation, Suits expresses all variables in terms of first differences, in order to avoid the problem of autocorrelation of the residuals.11 11This procedure will not necessarily eliminate autocorrelation. For example, consider a simple linear system with autocorrelation, one independent variable X, and one dependent variable Y. Thus, where Then, taking first differences yields 0.. g + - . This will eliminate autocorrelation only if 1V. at - Et-l = ut' which will be the case from (ii) only if §_is equal to one. This, however, is an unusual case. In fact, if there is no autocorrelation this procedure may introduce it. 19 The resulting fitted equation is: AR = .106 AYg- .234 A(P/M) - .507 AS (.011) (.088) (.086) - .827 AX + .115, (.261) (2.45) where 3.3nd.§ are in millions of cars, PAM is an index, Y_is in billions of dollars, and AX is the first difference in the dummy shift variable (AX = 0 for all years except 1941 when AX’= +1 and 1952 when AX a -l). The adjusted coefficient of correlation is .93. By testing alterations of this formulation, Suits deriVes two important consequences: (1) when the credit variable is omitted, the regression coefficient for price not only has the wrong sign but also is not signficant; and (2) the rate of increase in income is not significant and, when included in the formulation, lowers the adjusted coefficient of correlation to .86.12 The elasticity of sales was found to be -.59 with respect to price per month, +4.16 with respect to income, and -3.65 with respect to the stock of cars. When the stock of cars was dropped from the formulation, the price elasticity was near unity and the income elasticity dr0pped 12The rate of increase in income in this 2 formulation (due to the first differences method) is A Yt’ not AYt. .___. 20 to about 3. In a later paper Suits tests the desirability of including other factors in the demand equation.13 Those included are supernumerary income, age distribution of stock, and separated price and credit terms. The function is also tested on prewar and postwar data separately. The formulations are made in logarithmic form for all analyses. It was found that supernumerary income (1.6., the level of income over a certain subsistence level) improved the explanation of new car demand when included in the formulation. This effect was the greatest when the subsistence level was taken to be $1500 per household (1947-49 dollars). The elasticity of new cars to supernumerary income was +2.88. The age distribution of stock of used cars was taken into account by including lagged new car sales divided by stock. The regression coefficient for this variable turned out to have the expected sign L: but was not significant. Including this variable did improve the overall explanation by a small amount. The separation of Suits' price-credit variable 13"Exploring Alternative Formulations of Automobile Demand," 'Review of Economics and Statistics, XLIII (February, 1961): pp. 66-69. 21 into component terms yields the interesting result that theregression coefficients for each.term separately are about the same as for the combined term although less significant; the values and.standard errors are -.657 (.195), >-.656,(.28), and +.636 (.28) for ELM, P, and M respectively. Thus, the elasticities of retail price and credit are about the same when separated since the formulation is in terms of logarithms. When real wholesale price was substituted for retail price in the separated formulation, it was found to have a higher elasticity (-1.12) than that of retail price. Analysis of the prewar versus postwar regressions shows the difference is significant. In the postwar period, the elasticity of income is higher, price elasticity is lower, and stock elasticity is lower. Section 5: Turnovsky A more recent work on automobile demand is an analysis of the New Zealand automobile market by S. J. Turnovsky.14 The case of New Zealand is unusual in that no cars are produced there; hence, all new cars must be imported. Imports of new cars are limited by the country's foreign exchange earnings. Thus, the supply 14"The New Zealand Automobile Market, 1948-63: An Econometric Case-Study of Disequilibrium," The Economic Record, XLIII (June, 1966), pp. 256-71. 22 of new cars is more or less determined by factors exogenous to the automobile market. In light of this fact, Turnovsky decides to formulate a short-run price adjustment equation. To derive this equation, he first formulates a behavioral equation which expresses the desired level of stocks at time t: 8* = A + aYt + 8Pt + gt, (2.51) where S: is the desired level of car stock per 1,000 persons at time t, A.is a constant, Yt is real disposable income per 1,000 persons at time 3, Pt is an index of relatively new second-hand cars divided by the consumers' price index at time t, and £_is time in years. New car demand is given as: * g * - ' Dt y(St St), (2.52) where D; is the rate of demand for new car purchases per 1,000 pgrsons at time E, St is the actual level of car stock per 1,000 persons af—time t, and I_is the stock adjustment coefficient. The level of stock will be changing since new cars are being imported and old cars are depreciating. This is expressed as: 23 st =0t - ds , (2.53) where‘st is the derivative of §_with respect to time, Dt is tEe rate of supply of new cars per 1,000 persons at time 3, and glis the rate of depreciation. The short- run price adjustment mechanism is given as: o = * - .- P r(Dt Dt)’ (2.54) where 2.15 the price adjustment coefficient. To take account of the unique supply situation in New Zealand, Turnovsky Specifies the following equation: (2.55) where the supply of stock St 1 is a linear function of lagged stock and lagged export receipts. Xt-1 represents the exports per 1,000 persons divided by import price at time t-l. From these equations an estimating equation is derived and fitted to the data. Two estimating methods are used: (1) ordinary least squares, and (2) three- pass least squares. The results are as follows: 24 OLS p = 317.8 - 0.626 p 1 + 0.00652 Y t (525.5) (0.246) t“ (0.00194) P -28.01 s + 16.65 s + 33.65 t (4.61) t (3.72) t‘1 (14.66) R2 = 0.87, D.W. = 2.46 (2.56) 3PLS P = 563.4 - 0.680 Pt 1 + 0.00532 Y t (329.5) (0.122) ' (0.00089) -24.34 s + 13.26 s + 34.99 t (2.16) t (2.34) t‘1 (9.08) - 0.1025 a (0.1215) t’1 2 R = 0.96, D.W. = 2.21, (2.57) where the standard errors are given under the regression _ coefficients, the R2 's are the reSpective multiple correlation coefficients, the D.W.‘s are the respective Durbin Watson statistics, and Gt_l is the three-pass variable. From the regression coefficients of the estimating equation, Turnovsky derives the estimates for the structural parameters and the estimates for the various 25 elasticities. The income elasticity of demand for new cars is 7.20 for OLS and 9.94 for 3PLS. The price elasticity of demand for new cars is -0.57 for OLS and -9.74 for 3PLS. The elasticity of demand for new cars with respect to present stock is -2.65 for OLS and -4.04 for 3PLS. Turnovsky also derives the long-run income elasticity of demand for stock (1.97 for both OLS and 3PLS), the short-run income elasticity of demand for stock (0.91 for OLS, 0.95 for 3PLS), the long-run price elasticity of demand for stock (-l.53 for OLS, -1.93 for 3PLS), and the short-run price elasticity of demand for stock (-0.70 for OLS, -0.93 for 3PLS). The supply function for stock was also tested by statistical methods. In this case, the OLS estimate turned out to be the best. The elasticities of supply of stock and supply of new cars with respect to lagged exports turned out to be 0.13 and 0.82 respectively for OLS. Section 6: Ueno and Tsurumi The most recent study of the automobile market is that of H. Ueno and H. Tsurumi.15 Their paper presents a fairly complete dynamic annual model of the American automobile industry. The model that they present 15"A Dynamic Supply and Demand Model of the United States Automobile Industry 1921-1945" (Economic Research Services Unit Discussion Paper No. 58, University of Pennsylvania, July, 1967).- (Mimeographed.) 26 contains twenty-one distinct endogenous variables, , several endogenous variables that are composites of other endogenous variables and exogenous variables, and ten exogenous variables.17 With these, the authors developed a system of thirty-one equations, and, using annual data from the years 1921-42 and 1942-65, estimated the values of the parameters by using the two-stage least squares, the ordinary least squares, and the scanning estimating procedures. The estimated equations in conjunction with projected values of the exogenous variables were used to predict the future values of selected endogenous variables. The Ueno-Tsurumi model is a more complete specification of the automobile market than any given in previous models. It considers explicitly the firms' behavior and supply relationships and the interdependent relationships 6Desired stock of passenger cars, desired stock of buses and trucks, actual stock of passenger cars, actual stock of buses and trucks, desired domestic demand for passenger cars, desired domestic demand for buses and trucks, actual production of vehicles by selected companies, actual production of passenger cars, actual production of buses and trucks, inventory of passenger cars, inventory of buses and trucks, capital stock, retained earnings, net investment, factory retail price of passenger cars, wholesale price of buses and trucks, wage rate, total employment, dividend payments, profits after taxes, and value of sales. 17Desired foreign demand for passenger cars, desired foreign demand for buses and trucks, actual foreign demand for passenger cars, actual foreign demand for buses and trucks, accumulated depreciation of selected companies, disposable income, gross national product, the gross national product deflator, total pOpulation of the United States, and the unemployment rate. 27 between supply and demand. For both supply and demand, the stock adjustment method is the deus ex machina of their system. There is not enough space in this summary to sketch the deve10pment of the complete model. Instead, a few points about the demand functions for passenger cars will be mentioned since similarities exist between their formulation and previous formulations. In fact, their formulation is derived from Nerlove's, and it differs only by the inclusion of p0pulation and the rate of population growth.18 The per capita demand for passenger cars for the Ueno-Tsurumi model is exactly the same as equation (2.34) in the Nerlove model except that the depreciation coefficient [d is replaced by 2;, where d' = (l-d)/(1+nt), (2.61) and nt equals the rate of population growth. The coefficients of t Z demand formulation were estimated with a scanning method to avoid a problem with the nonlinear parameters contained in the formulation. The equation with estimated coefficients, standard errors, and multiple correlation coefficient is: 18"A Note on Long-Run Automobile Demand." 28 x =,+ 6.198 - .0235 (P - . .55 p t (.0067) t (.1106) t'1 + .0525 (Y - .55 Y ) + .2056 x (.0069) t t‘1 (.1101) t'l’ R2 = .8170, (2.62) where the variables have the same meaning as those in . 1 . . . Nerlove's formulation."9 The deprec1ation-p0pulat1on growth coefficient d;_is equal to .55. Using a trend estimate of pOpulation growth of one percent per year, the adjusted depreciation rate would be approximately .44. The long-run stock elasticity coefficients for price and income, using the above formulation, would be -l.6 and +2.9 respectively. The coefficients of the regression equation, the depreciation rate, and the elasticites are, for the most part, close to those obtained by Nerlove. On the supply side, production is a linear function of desired demand and lagged inventory. Price is determined (they claim) by the full cost principle, but actually it is made simply a linear function of average 19See pages 13 through 15. 29 labor cost and lagged price.20 When tested, both of these functions have high multiple correlation coefficients 2 = .8978 and R2 = .9916 respectively), and satisfactory (R Durbin Watson statistics (D.W. a 1.9 and D.W. = 1.2 respectively). The model also includes formulations for determining capacity production, investment, capital stock, wages, labor requirements, dividends, profits after taxes, retained earnings, and market shares. Consequently, the entire nature of the model from the supply side (as well as from the demand side) is one of long-run adjustment. In fact, much of the data used is not available on a regular basis for periods shorter than one year. The model does not include a decision function for inventories (which are held for the most part by the dealers), nor did inventories enter explicitly into the price determination function. These facts tend to make the model inappropriate for short-run analysis - i.e., analysis of the market within one year. 20Capital enters this formulation only implicitly through its effect on labor requirements for any output and its effect on wage rates through its effect on capacity output. The wage rate times the labor requirement divided by output gives average labor cost. Capital costs are ex- plicitly excluded from the price determination equation, as are material costs. Thus, it seems that, in their equation, price is not determined by the full cost principle as commonly understood. 30 Section 7: Conclusion In each of these studies,only yearly data were used. All of the studies except those of Turnovsky and Ueno and Tsurumi used the least squares method of estimation. As is well known, when the variables within a system are determined simultaneously by a number of structural relationships (as in the case of the automobile market), the estimation of one of these equations by the least squares method will yield inconsistent estimates of the parameters. None of the past studies deal with this problem by its prOper solution: that is, the simultaneous estimation of a complete model. Ueno and Tsurumi at times use the two-stage least squares method (which should give 21). better estimates than OLS but at other times use ordinary least squares and a scanning procedure. Another problem of estimation arises when there is a lagged dependent variable in the specification (as in Nerlove's, Turnovsky's, or Ueno and Tsurumi's demand equations): the error term in the estimating equation will be autocorrelated, and, as a result, the estimate for the coefficient of the lagged dependent variable 21J. Johnston, Econometric Methods (New York: McGraw Hill Book Co., 1963), pp.293-95. 31 . . . 22 will be inconSistent. Turnovsky attempts to avoid this problem by using the three-pass least squares method which will reduce autocorrelation but will not necessarily give 23 consistent estimates for all parameters. Of these studies, Suits', Turnovsky's, and Ueno and Tsurumi's are the only ones which provide a complete model for the market they are studying (although they do not estimate it as a complete model). Only Nerlove, Turnovsky, and Ueno and Tsurumi formulate models that are essentially dynamic in nature. The Ueno-Tsurumi model is perhaps the best and most complete of those studied, but it is essentially long-run in nature and not suited for short-run analysis. The demand formulations of R005 and von Szeleski and Chow are very similar. The same can be said of the demand formulations of Nerlove, Turnovsky, and Ueno and Tsurumi. Suits' demand formulation is essentially different from the rest. So, of the models reviewed, there are three basic types. In the initial stage of the preparation of the present work, all three types of formulation were tested with monthly data for the years 1960 to 1965. None of the 22D. Cockrane and G. H. Orcutt, "Applications of Least Squares Regression to Relationships Containing Autocorrelated Error Terms," Journal of the American Statistical Association, XLIV (March, 1949777pp. 32-61. 23L. D. Taylor and T. A. Wilson, "Three-Pass Least Squares: A Method for Estimating Models with a Lagged Dependent Variable," The Review of Economics and Statistics, XLVI (November, 1964), pp. 329-42. 32 formulations performed very well in that signs were Opposite those expected, key variables did not have significant coefficients, and the correlation coefficients were extremely low compared to those obtained in the original studies. As a result, it appeared that it was necessary to construct a fairly complete model that would be dynamic in nature, suitable for the use of monthly data, 24 and capable of being estimated simultaneously. The formulations of such a model are the object of the next chapter. 24 The last two requirements were most restrictive and eliminated a number of possible formulations since monthly data on many crucial variables were not available, and the storage capacity of the MSU CD 3600 computer was not sufficient to handle models with a large number of variables by the full information maximum likelihood estimation ‘ procedure. 33 Table 2.1 Summary of Elasticities ........................ INCOME PRICE ELASTICITY ELASTICITY AUTHOR, . _ OF DEMAND OF DEMAND R005 and von Szeleski +1.20 - .65 Chow Existing Hypothesis +2.027 - .950 Alternative Hypothesis +1.74S -l.026 Nerlove New Car Sales +2.8 -0.9 Equilibrium Stock +3.8 -l.2 Suits +4.16 - .59* Turnovsky OLS 3PLS OLS 3PLS Sales +7.20 +9.94 -O.S7 -9.74 Stock (Long-Run) +1.97** +1.97** -l.53** -l.93** Stock (Short-Run) +0.91 +0.95 -0.70 -0.93 Ueno and Tsurumi +2.9** -1.6** * Price per month (price-credit variable) ** Elasticities of equilibrium stock CHAPTER III CONSTRUCTION OF THE MODEL It has been decided that the model used for this study should describe the short-run behavior of the new car market. The month-to-month adjustment in price and quantity will be of specific interest. The model consists of three structural equations and two identities. The structural equations give the relationships for demand for new cars, for production of new cars, and for inventory adjustment. The first identity gives the relationship between inventories, sales, and production. The second identity gives the relationship between present stock, past sales, and past stock. The structural equations are ex ante in nature, giving the plans of consumer, manufacturer, and dealer as determined by current and past economic data. The identities impose the ex post relationship. Linear equations are used to simplify the formulation. A listing and eXplanation of the variables introduced in this chapter is given in the 34 35 Appendix to Chapter III. Section 1: The Demand for'New cars The demand for any good by consumers will depend upon the price of the good, the prices of other goods and services, the level of income of consuming units, and other less quantifiable variables such as tastes, consumer Spending attitudes, eXpectations, and so forth.1 The demand for new units of a durable good, such as automobiles, will also be influenced by the Size of the stock of used units left over from past periods.2 The reason for the inclusion of the stock variable is that some types of the services yielded by an automobile are not used up in the first period of use but are also available for use in subsequent periods. Thus, these services could be obtained either from new units or used units. The more used units available to supply these services, the lower will be the need to obtain the services from new cars. Also, since the size of the stock of used cars will have an effect on the price relationship between new and used cars, a larger stock might mean that it would be less costly to obtain certain services by buying a used car rather than a new car. 1J. M. Henderson and R. E. Quandt, Microeconomic Theory: ’A Mathematical Approach (New York: MCGraw Hill Book Co., 1958), pp.87-8. 2A. C. Harberger (ed.), The Demand for Durable Goods (Chicago: University Of Chicago Press, 1960), pp. 316. cu fl! (’7‘ f" I! 36 The stock adjustment approach has been used by others3 to relate the level of stocks to the level of sales; however, it will not be used in this model. The reason for not formulating the demand for new cars as some preportion of the difference between the desired stock demand and the actual stock is that there is reason to believe that there are certain services rendered by a completely new car that are not rendered by cars that have been used. That is, used cars (even those only one month old) may not be perfect substitutes for new cars.4 If this is the case (as is assumed), it would be inapprOpriate to consider the flow demand of one good solely in terms of the difference between the desired and actual stock of its imperfect substitute (as would the stock adjustment approach). Other than the stock adjustment approach, there are three basic ways of including the effect of the stock of used cars: (l) by including it directly in the formulation; (2) by including the price of used cars (which is a function of the demand for used cars as well as stock) in the formulation; (3) by using both used car price and stock. The first approach is used and justified a posteriori. 3See Nerlove, Turnovsky, and Ueno and Tsurumi in Chapter II. 4 Some evidence of this was found by Chow, as discussed on page 12 of Chapter 11. 5Preliminary tests using the ordinary least squares method Showed that the latter two methods yield insignificant coefficients and Signs contrary to those expected. 37 Taking all these things into consideration, the demand for new cars will be stated, in general, as: Dt = D(PNt/P0t' Yt’ St' Ot); (3.11) that is, the demand for new cars is a function of the price of new cars PNt relative to the price of all other goods and services Pot' the level of income of all consuming units Yt' the level of the stock of cars inherited from the last period St' and other factors Ot‘ The influence of these other faztors must be ignored-in this study since there is no reliable monthly data available (although certain expectational and demographic data are available from the Survey Research Center at the University of Michigan on a quarterly basis). In addition, variables must be added to take into account the regular seasonal changes in consumer buying patterns and the effects of shocks peculiar to the automobile market. Thus, the following linear form will be used as the demand equation: 12 a + P '+"S+ , .4- N a a1 + a It as t i§1a1+3A1 ult’ (3.12) where Nt represents sales, Pt represents real new car price, Yt represents real disposable income, St represents stock, the Ai's are adjustment variables, the ai's are regression 38 coefficients, and “It gives the influence of omitted factors. Further explanation of these variables is _given in the Appendix to this chapter. Section 2: Production of NeW'Cars The automobile industry can be described as being oligopolistic in nature: that is, a few large companies produce virtually all the units sold in the United States. If these companies do seek to maximize profits, they will set their levels of production where marginal cost equals marginal revenue.6 Assuming that the total cost function for the industry could be represented as 2 TCt = 80 + Bth + Btha (3021) where the 81's are parameters and Qt represents the quantity of new cars produced per period, we can derive the marginal cost function for the industry by taking the first derivative of the total cost function. We have: Met = 81 + zBZQt' (3.22) 6Henderson and Quandt, ch. 6. 39 Marginal revenue for the industry could be represented as: MRt = Pt(l-Txt)(l-Mt)(1-l/nt), . (3,23) where Pt is the retail market.price in period t, Txt is the manufacturer's excise tax rate in period t, Mt is the dealer's margin in period 3 (i.e., the difference between retail and wholesale price), and ”t is the elasticity of demand in period 3, Let (l-Txt)(1-Ht)(1-1/nt) be represented by Yt' Setting marginal cost equal to marginal revenue, we have: = + 3 = MCt 31 ZBZQt MRt YtPt’ (3.24) or 81 + ZBZQt = Ytpt' (3-25) Solving for quantity, we have: Qt = -81/282 + (yt/2823Pt- (3.26) Letting b0 represent “81/282 and b1 represent (Yt/ZBZ), (3.26) can be simplified to:7 7The simplifying assumption iS that MRt is eXpected marginal revenue and then Txt M , and n .——— are replaced by their ———- —£ t average values Tx, M, and 3 respectively, so that 1.15 not a function of time. 40 Qt é b0 + blPt. (3.27) This is the form of the linear short-run supply function for the industry inthe normal stationary state context. However, the supply function needed for this study must be dynamic Since the conditions of the market are constantly changing when the market is in disequilibrium. The manufacturers must take recent information about the market into consideration. At the beginning of each period E, the manufacturers would have information on the level of sales and the level of inventories in period Ell' These would be relevant Since what the industry produces must be taken up either by retail new car sales or by new car inventories. High levels of retail new car sales in period 3;; would induce the manufacturers to increase production since more cars could be expected to be taken up by sales. Higher levels of inventory in period £;£_might induce manufacturers to decrease production if they desired to keep inventories at some pre-set level. This latter effect might not be so clear due to changes in inventory requirements throughout the season and/or to changes in inventory requirements because of changes in expected sales levels. In any case, lagged sales and lagged inventory should be included in the dynamic supply function to compensate for the month-to-month changes in the market. Also, over the period of the data 41 used, about five years, some changes in the technology and the cost situation could be expected. This would cause the supply function to shift. To take account of this, a time variable will be added as a dummy variable for these changes. Taking the previous analysis into consideration, the linear form of the equation for the production of new cars will be Specified as: = b + b P + b N + b I + b T + Qt 0 1 t 2 t-l 3 t-l 4 t ”Zt’ (3.28) where Qt represents production, Pt represents real new car price, Nt-l represents lagged sales, I represents lagged t-l inventories, Tt represents time, and the bi's are regression coefficients; p2 gives the influence of omitted factors. Further explanation of these variables is given in the Appendix of this chapter. Section 3: Total Inventory Demand The net demand for new cars to be held in inventory by dealers will depend on the dealers' desired equilibrium level of inventory It for any given period as well as the actual level of inventory It-l left over from 42 the past period and the rate at which dealers want to adjust 8 their stock to the desired level. This can be stated as: It - It_1 = 6(I: - It_1). (3.31) Given the level of past inventory, if the "desired" equilibrium level of inventory is different, the dealers will want to make some adjustment in the actual level of inventory during the current period. The desired adjustment may be complete or partial. The basic assumption is that the rate of adjustment of actual inventory is proportional to the difference between the desired equilibrium inventory and the actual inventory. The coefficient of proportionality is given by g) which could take on values between 0 and 1. This inventory adjustment coefficient is assumed to be a constant for all periods and all dealers. Desired inventory is assumed to be a function of present and past prices and past levels of sales. If past prices were high, the assumption is that the market is "strong" and more cars should be held in inventory to meet anticipated demand. If current price is low, the dealers would want to sell fewer cars in the current period, holding them in inventory until the price is higher. The higher the 8For similar models see R. G. D. Allen, Mathematical Economics (London: MacMillan Co., 1956). 43 level of sales in the most recent period, the higher would be the level of inventory the dealers would desire to hold in order to be able to meet the demand for their product. Thus, the equation for desired inventory can be Specified as: 1* = 0 + e 1Pt + e P N (3.32) 2 t-l + 63 t-l’ where the 61's are the parameters, and Pt' Pt-l’ and N ‘__ ___ _____ t-l have the same meaning as previously. 02 and 63 are expected to be positive, and 61 is expected to be negative. Sub- stituting (3.32) into (3.31), we have: .- = + I I 6(00 0 P + P + N - I t t-l t 6 0 1 Z t-l 3 t-l t-l). (3.33) Multiplying through by g, tranSposing It-l from left to right, and collecting terms, we have an equation for the actual level of inventory demand: I = 60 + 50 P + 50 + 60 t 0 1 t 2 t-l 3 I (1'5)It-1° N t-l (3.34) By replacing 860, 661, 662, 663, and 1-6 with :2, c1, c2, c3, 44 and c4respectively and adding a random term "3’ we obtain the following estimating equation: It ‘ C0 * C1Pt * Czpt-i + C3Nt-1 * C4It-1 I “St' (3.35) The variables and symbols used have been eXplained previously except for the ci's and.p3, which are the regression coefficients and the error term, respectively, for this equation. Section 4: Inventory Supply Identity The number of new cars available to be held as inventory by dealers and manufacturers is the sum of inventory held over from the past period plus current production less current sales. This can be stated symbolically as: = - 3.41 where the symbols have been previously defined. Section 5: Stock Identity The stock of cars on the road at the beginning of 45 each period will depend on the sales of new cars in all previous periods and the depreciation rate Q (assumed constant) applicable to cars. Thus, 2 3 s =_(l-d)Nt_1 + (l-d) Nt_2 + (l-d) Nt_ t 3 + 000 l'd)nN 3. c M. c 51) where 2_equals the number of years automobiles have been produced. Lagging (3.51) and multiplying by (l-d) gives: 2 3 (1-d)st,1 - (l-d) Nt_2 + (l-d) Nt_3 + ... (l-d)nNt_n, (3.52) since Nt- 1 will equal zero. n- Equation (3.51) and (3.52) can be combined to yield the following equivalent form for (3.51): s = (l-d)(Nt_1 + s (3.53) t t-1)' That is, stock in the current period will equal one minus the depreciation rate times the last period's stock and the additions to stock in the last period. 46 Section 6: ’Th ‘Complete Model As developed in the first five sections of this chapter, the complete model is as follows: 12 N =_a + a P + azYt + aSSt + 2 ai+3Ai + "1t i=1 (3.12) Qt 7 b0 I blpt + bZNt-l I b3It-1 I b4Tt I u2t (3.28) = + + It c0 + Clpt + CZPt-l + C3Nt-l C4It-l ”3t (3.35) It = It“1 + Qt - Nt (3.41) st = (l-d)(Nt_1 + st_1), (3.53) where the variables are explained in the Appendix to this chapter. CHAPTER III APPENDIX SYMBOL INDEX Adjustment variables. A1 through A11 represent the months -—- -——- from January to November and compensate for seasonal variation in demand. A represents the months in which ._l£ there were strikes during the 1960-65 period; it compensates for the impact strikes might have on consumer expectations. Structural and/or regression coefficients of the variables in equation (3.12). Structural and/or regression coefficients of the variables in equation (3.28). Structural and/or regression coefficients of the variables in equation (3.35). The constant percentage depreciation rate applicable to the stock of cars in general. For the purpose of computation, the depreciation rate was assumed to be three percent per month. This gives a yearly depreciation rate of approximately thirty- one percent. The value of stock was computed using other depreciation rates, but this rate seemed to give the best performance in terms of 33 and significance levels of 47 t-1 it 48 coefficients. 'The inventory adjustment coefficient which gives the pr0portion of the difference 'between actual inventories and desired inventories that will be reduced or increased in one period. Inventory for period t of new cars held by the dealers and thE'manufacturers (in thousands). The total number of cars produced but not yet sold in the retail market. Inventory of new cars lagged one period (in thousands). ' The error terms for equations (3.12), (3.28), and (3.35). Can be regarded as the influence of the omitted factors in each equation which is assumed to be random. The number of new domestic cars sold in the retail domestic market in each period 3 (in thousands)2. Sales of new cars lagged one period (in thousands). The new car price index for period 3 divided by the consumer price index for period t, both of which were compiled by the Bureau of Labor Statistics. Real new car price index lagged one period. 1 Ward's Autpmotive Yeargpok, 1960-65. 2Survey of Current Business, 1960-65. 49 The total number of new cars produced for the domestic market in period t by the U. S. producers (in thousandS). Derived by subtracting exports per period from total production per period. The stock of cars on the road at the beginning of each period, adjusted for age composition of stock (in thousands). Stock lagged one period (in thousands). The level of real disposable income seasonally4adjusted at annual rates (in billions). Although real disposable income is computed by the Commerce Department only on a quarterly basis, real personal income is computed both on monthly and quarterly bases. Therefore, real diSposable income can be derived on a monthly basis under the assumption that the ratio of monthly real diSposable income to quarterly real disposable income is the same as the ratio of monthly real personal income to quarterly personal income during the same period of time. Thus, for example, real disposable income for December equals real disposable income for the fourth quarter times real personal income for December divided by real personal income for the fourth quarter. That is, RDI = RDI ° RPI /RPI Dec. IV Dec. IV Time measured in single units. Ranges in value from 1 to 60. Used as a dummy variable to take account of changes in cost and technology. 3 Ward's Automotive Yearbook. 4Survey of Current Business. CHAPTER IV ESTIMATION OF THE MODEL In this chapter, the model deve10ped in the last chapter will be estimated by two methods using monthly data for the months from May, 1960, to April, 1965 (60 observations). The results from these two methods will be presented, discussed, and compared. Section 1: Single Equation Estimation Equations (3.12), (3.28), and (3.35) given in the last chapter were each fitted to the data separately with the ordinary least squares estimating procedure. Each equation with its estimated parameters and its coefficient of determination isgiven below in equations (4.11), (4.12), and (4.13) reSpectively. 1For an explanation of this procedure, see J. Johnston, Econometric Methods (New York: McGraw Hill Book Co., 1963), ch. 4. 50 51 Nt = 1691.48 16.69 P «(1427.39) ~ (10.97) +. 3.64 Y - 0.0477 8 t (1.21) t (0.0269) t - 72.74 A1 92.25 A2 + 20.57 A3 + 31.13 A4 (37.15) (34.71) (34.23) (33.76) + 34.71 A (33.93) 32.19 A - 90.56 A - 130.47 A 5 (34.95) 6 (31.62) 7 (31.72) 8 ”215.64 A + 54.26 A + 33.35 A ’112071 A (35.26) 9 (33.25) 10 (37.38) 11 (33.41) 12 2 R = .868 (4.11) Q = -3830.66 + 40.10 Pt + 0.4868 Nt_1 - 0.0863 1t_1 t (1393.44) (14.19) (0.2012) (0.1245) + 9.36 Tt (2.41) 2 R = .462 (4.12) I a -528.56 - 8.96 P + 11.77 P + 0.3965 N 1 (540.17) (13.18) t (12,763't-1 (0.1312) t-l + 0.7085 I (0.0655) t'1 2 R = .7155 (4.13) 52 Consider equation (4.11). The signs of the important variables (price, income, and stock) are in the expected direction. ‘Income and stock are highly significant, while price is significant at the .24 level. Using the average quantities for price, income, and stock, representative elasticities can be derived for these variables. The elasticity of sales with respect to price is -2.68, the elasticity of sales with respect to income is +2.35, and the elasticity of sales with respect to stock is -1.45. In equation (4.12), the signs are also in the expected direction for all variables. Price, lagged sales, and time are highly Significant. Lagged inventory is significant only above the .49 level. The F test on the entire equation is significant at the .0005 level. The elasticity of production with respect to price (using average values for price, lagged sales, lagged inventory, and time) was +6.37. The signs are in the expected direction in equation (4.13) as well. Lagged sales and lagged inventories are highly significant, while lagged and present prices are not very significant. The inventory adjustment coefficient g ’5 is equal to +.2915.a This means that approximately thirty percent of a necessary inventory adjustment is undertaken 2The inventory adjustment can be found directly (See equation (3.34) in Chapter III) by subtracting the coefficient of It-l from 1.000. 53 every month. A significant proportion of the variation in the three dependent endogenous variables is explained by the three equations. However, the pr0portion of variation explained by equation (4.12) is low and would indicate that some important variables might have been left out of this formulation. Alternative tests of other possible formulations for this relationship with the variables for which monthly data were available failed to produce a superior formulation (superior in the sense of having a higher 3: while at the same time not having adverse Signs or less Significant coefficients). One might suspect that equation (4.11) made a better Showing as far as the coefficient of determination was concerned, Simply because of the greater number of variables included in the formulation. This is certainly true in that the fines higher for (4.11) when the twelve Ai terms were included than when they were not included: However, this was not the reason for their inclusion. The main ex posteriori reason was that the signs of the other variables Pt’ I and St were all in t, the expected direction and more significant when these terms were included. In the other equations, it was not necessary to include these variables to obtain the prOper signs and so they were not included.3 3In addition, it was felt that more than one set of these seasonal and shock variables would be redundant in the context of simultaneous estimation, and, in one program where they were included, the memory capacity of the computer was exceeded. S4 The ordinary least squares method was used in this investigation as a first test of the apprOpriateness of the equations in each model considered. Its advantage in this capacity is its cheapness in terms of computing time. It does, however, have some major disadvantages. When used on separate equations of a simultaneous model, it will not only lead to biased estimates of the parameters of the equation, but also, the estimates will be inconsistent.4 If the simultaneous system is solved to find its reduced form relations, and ordinary least squares applied to these reduced form relations, then, from the estimates of the reduced form parameters, estimates of structural parameters can be derived. This method is called "indirect least squares;" it will give consistent estimates for the structural parameters.5 However, it is feasible only if the structural parameters in the simultanous system are exactly identified. Thus, to find which estimating technique is appropriate, if any, it is necessary to find whether the model as postulated is just identified, overidentified, or underidentified. This is the subject of the next section. Section 2: Determining Identification Whenever a system that contains Simultaneous relationships is studied, the so-called "identification 4J. Johnston, p. 253. SIbid. 55 problem" arises. Briefly, the identification problem may be stated as that of "deducing the values of the parameters of the structural relations from a knowledge of the reduced- form parameters."6 If the equations in a simultaneous system are too alike in form, it may not be possible to find the values of the structural parameters. In this case, the structural parameters are said to be unidentifiable or underidentified. Consequently, it may not be worthwhile to conduct a statistical investigation for the purpose of determining those parameters. Thus, at this point, it will be shown that the parameters of the model are not under- identified. It will be convenient to state the model given by: a + + + N a + alPt aZYt aSSt 2 a = b + b P + b N + b I + b T + u Qt 0 1 t 2 t-l 3 t-l 4 t 2t (4.22) I - c + c P + c N + c I + c P + u 56 It = It-1 + Qt - Nt (4.24) st . (l-d)Nt_1 + (1-d)st_1 (4.25) in the following matrix notation: Nt Pt Yt St A1 B1 1 B1 2 "' B1 23 ' 111 + a0 B2 1 32 2 "' B2 23 ‘ 112 I b0 BS 1 B3 2 "' B3 23 A12 = '13 I C0 B4 1 B4 2 ... B4 23 Qt 0 BS 1 B5 2 "° B5 23 Nt-l ° It-1 Tt It Pt-1 St-l (4.26) where the Bij's give the structural coefficient for the 1th variable in the ith equation. The matrix containing the Bi.'s 57 can be represented by £1. The nature of the structural equations was such that not all of the variables in the system entered into each equation. Consequently, some of the_Bii'S in the matrix Q: will be zero. Taking this into consideration, matrix M: can be specified as: 1 -a1 -a2 -a3 -a4 ... -a15 0 0 0 0 0 0 0 0 -b1 0 0 0 ... 0 ‘ 1 -b2 -b3 -b4 0 0 0 B* - 0 -c1 0 0 0 ... 0 0 -c2 -c3 0 1 -c4 0 l 0 0 0 0 ... 0 -l 0 -l 0 1 0 0 0 0 0 l 0 ... 0 0 (d-l) 0 0 0 O (d-l)/ (4.27) It will be important to Show the possibility of identification for equations (4.21), (4.22), and (4.23) only since (4.24) and (4.25) are identities. Hence, we are concerned only with rows one to three in matrix 3:. A necessary condition for the identification of the ith equation in a system of M_linear equations is that at least m-l of the parameters in equation 1 must be zero.7 Since the number of structural equations in the model is three, there Should be at least two (m-lf 3 - l . 2) Bij's 7W. C. Hood and T. C. Koopmans (eds.), Studies in Econometric Method, Cowles Commission Monograpfi No. 10 (New York: John Wiley and Sons, 1953), pp. 135-42. 58 in each row equal to zero in order for the model to be identified. As is apparent from (4.27), this condition is more than satisfied - i.e., each equation is overidentified. Another necessary condition for the identification of the ith equation is that for every other equation §_there exists one variable that appears in equation §_but not in equation ‘i.8 Again referring to (4.27), this condition is met for every equation Since each of the top three rows of B:_ contains a Eli.n0t in the other rows. ‘The sufficient condition for the identification of the ith equation is that the matrix formed by the exclusion of row i must be of rank 2:1. That is, from the remaining rows and columns, there must be at least Ell rows and columns that form a non-zero determinant.9 By the manipulation of the rows and columns, (4.27) can be shown to satisfy this criterion for every structural equation. It is apparent that the equations are not only identified but are overidentified. There are more than enough restrictions to determine the parameters of each structural equation. Fortunately, overidentification does not pose the same type of problem as underidentification. It merely means that certain estimation procedures will give inconsistent estimates when used, such as the ordinary or limited least squares methods. However, there are estimation procedures that can be used to give consistent estimates of the parameters 8Ibid., pp. 135-42. 9Ibid. 59 of overidentified equations, some of which are Single equation methods and some of which are complete system methods.10 Section 3: ‘Complete SyStem'EStimation A complete system estimation procedure was chosen which would Simultaneously determine the structural coefficients of all structural equations in the model. The estimation procedure used was the full information maximum likelihood method. With this method, the likelihood function is maximized subject to all the a priori restrictions (i.e., requirements that certain Bij's are equal to zero) for every equation in the system. TH; advantage of this method from the standpoint of statistical theory is that it will generally give estimates that are asymptotically consistent and efficient and less 11 The disadvantage is that it involves solving biased. systems of non-linear equations, which is very time consuming. Fortunately, a computer program and computer time were available. The results of the full information estimation of the equations in the model are given for the structural equations by equations (4:3111.(4fl32)’ and (4.33) below. 10Johnston, pp. 254-68. 11J. G. Cragg, "On the Relative Small Sample Properties of Several Structural-Equation Estimators," Econometrica, XXXV (Januarv. 1967). on. 91 8 100. 60 N . + 4425.87 - 37.75 P + 2.856 Y (2372.90) (18.33) t (1.083) t - .07378 St + 3.939 A1 - 23.91 AZ .(.03416) (34.22) t (30.50) t + 53.36 A3 + 59.51 A4 + 65.76 A5 (28.84) t (27.40) t .(28.03) t .+ 13.86 A6 - 65.87 A - 78.62 A (33.13) t (30.84) 7t (52.03) 8t - 232.52 A + 88.83 A + 88.31 A (42.77) 9t (30.62) 10t (38.16) 11t (31.32) t (4.31) (1347.50) (13.21) (.2193) - .01544 1t_1 + 8.580 T, (.11413) (1.721) (4.32) 1 a - 433.12 - 39.47 pt +- .3070 Nt_1 t (653.92) (12.52) (.1645) + .6887 I + 41.75 Pt_ (.0872) t‘1 (9.43) 1 (4.33) 61 The reduced-form equations, along with the respective correlation coefficients and Durbin Watson statistics, are given by equations (4.34), (4.35), (4.36), and (4.37) which follow. N = 1920.99 + 1.941 Yt - .05015 S - 13.37 Pt- t l + .04527 Nt_ + .09472 It_ + 2.747 rt 1 1 + 2.678 A - 16.25 A2 + 36.27 A3 + 40.45 A 1t t t 4t + 44.70 A5t + 9.424 A6 " 44.78 A7 "' 53.45 A8 t t t - 158.07 A + 60.39 A10 + 60.03 A " 38004 A t t 12 9 t t 11 2 R = .815, D.W. = 1.72 (4.34) p s + 66.35 + .02422 Yt - .0006257 st + .3541 Pt_1 "' " o " o 2 .001199 Nt- 002509 It- 07 76 Tt 1 1 + .0334 A1 " .2027 A2 + .4525 A3 + .5047 A4 t t t t + .5577 A5 + .1175 A6t - .5587 A7t - .6668 A8 t t 9 t t R2 = .967, D.W. = 1.29 (4.35) 62 D II -- 1130.81 + . - . , » 98S4_Yt , 02546 St + 14.40 pt_1 + .3996 Nt 1 - .1175 It-l + 5.620 Tt + 1.359 A1t - 8.248 A + 18.41 A + 20.53 A + 22.69 A 2t 3t 4t 51'. + 4.782 A - 22.73 A - 27.13 A - 80.23 A 6t 7t 8t 9t R2 = .522, D.W. - 1.68 (4.36) I = - 3051.81 - .9561 Yt + .02469 St + 27.77 Pt t 1 + .3543 Nt-l + .7878 It-l + 2.872 Tt - 1.329 Alt + 8.002 A2t - 17.86 A3t - 19.92 A4t - 22.01 A5t - 4.641 A6t + 22.05 A7t + 26.32 A8t - 77.84 A9t - 29.74 Alot - 29.56 Allt + 18.73 Alzt R2 = .676, D.W. . 1.66 (4.37) Tables 4.1, 4.2, and 4.3 give a comparison of estimates, errors, and elasticities for the least squares and full information estimates. Comparison of rows 5 and 6 of Table 4.1 shows that the ratio of the regression coefficient 63 to the standard error for each important variable is greater.with the full information method except for the income variable. 'The price and stock elasticities of demand are greater under full information, while income elasticity of demand is somewhat less. 'The most significant difference is in the estimate of the regression coefficient for price and its standard error - the impact of price on sales is estimated to be considerably greater under full information estimation. Table 4.2 reveals that there is not a great deal of difference in the estimates, standard errors, or elasticities given by the two methods with the exception of the lagged inventory variable. From Table 4.3, it is apparent that the full information method results in an increase in the size and significance of price and lagged price. However, the significance of the other variables deteriorates somewhat since the ratios of the regression coefficient to standard error are lower under full information. This should not be taken to mean that the estimates for those variables obtained by the least squares method are superior since they will, in general, be inconsistent and more biased. In reference to the correlation coefficients given after the reduced-form equations, it appears that 12Ibid. 64 these are reasonable values for such statistics, although the lower Bffs for the production and inventory relation- ships might indicate that there are some important variables missing from their structural equations. But a Significant prOportion of the variation in the endogenous variables is eXplained by the existing equations. The Durbin Watson statistic is difficult to inter- pret for two reasons: (1) if the dependent variable is a function of the lagged dependent variable in the equation to be estimated, as is the case in (4.34), (4.35), and (4.37), the Durbin Watson statistic is biased against discovering either positive or negative autocorrelation;13 and, (2) the 14 only give values Durbin Watson statistic tables available for D. W. for a maximum of five (R - 5) independent variables, while the reduced-form equations here have eighteen pre- determined variables. Since the band of indeterminacy gets wider as the number of independent variables 3 increases, and Since the D. W. statistics are barely within that band at R - 5, it is reasonable to assume that the hypothesis of no positive autocorrelation would not be definitely rejected or 13C. F. Christ, Econometric Models and Methods (New York: John Wiley 8 Sons, 1966), p. 527. 14J. Durbin and G. S. Watson, "Testing for Serial Correlation in Least Squares Regressions, I and II," Biometrika, XXXVII (December, 1951), pp. 409-28, and XXXVIII (June, 1952), pp. 159-78. See also J. Durbin, "Testing for Serial Correlation in Systems of Simultaneous Regression Equations," Biometrika, XLIV (December, 1957), pp. 370-77. 65 accepted. It is more unlikely that the hypothesis of no negative autocorrelation would be rejected. The reduced-form equations are obtained by solving the model foreach endogenous variable in turn, ending up with each endogenous variable being Specified in terms of the exogenous and lagged endogenous variables. The coefficient of these variables in each equation are referred to as "impact multipliers" since the change in any pre- determined variable multiplied by its reduced-form coefficient will give the total change of the respective endogenous variable in the current period. For example, a one billion dollar change in diSposable income Yt will result in 1.941 thousand increase in car sales, a 2.42—percent increase in price, a .985 thousand increase in production, and a .956 thousand reduction in inventory held during the current period (the month of the change).15 In the next chapter, the regression coefficients estimated here will be used to determine the dynamic characteristics of the specified model for the automobile market. 15Note that the reduction in inventories plus the increase in production exactly equal the increase in car sales (.985 + .956 - 1.941). 66 Section 4:"Summary of Data Table 4.11 Demand.for New Cars .Sales. > -. ................. t...... t t l OLS Reg. Coefficient 1691.48 -l6.69 3.647 -0.04774 2 PI Reg. Coefficient 4425.87 -37.75 2.856 -0.03416 3 OLS Std. Error 1427.39 10.97 1.210 0.0269 4 Fl Std. Error 2372.90 18.33 1.083 0.03416 5 OLS RC/SE C01. 1/3 1.185 -1052]. 3.013 -10773 6 Fl RC/SE Col. 2/4 1.865 -2.060 2.638 -2.160 7 OLS Elast. Sales to: - -2.68 2.35 -l.45 8 PI Elast. Sales to: - -6.93 2.105 -2.56 67 Table 4.2 Production of New Cars Produc ion = C n t ’ S 1, . a - t . o 5 ant IIIFIF...a est_1 Inv t-l Timet ........................................................ l OLS Reg. . Coefficient -3830.66 +40.10 +.4868 -.08631 +9.359 ........................................ 2 Fl Reg. Coefficient -3829.90 40.68 .4484 -.01544 8.580 3 OLS Std. Error 1393.44 14.19 .2012 .1245 2.410 4 FI Std. Error 1347.50 13.21 .2193 .1141 1.721 ........ 5 OLS RC/SE C01. 1/3 -2.749 2.826 2.419 -.6933 3.883 6 PI RC/SE ' Col. 2/4 -2.842 3.0788 2.045 -.1353 4.986 7 OLS Elast. Prod. To: - 6.37 .5008 -.l407 - 8 PI Blast. Prod. To: - 6.40 .4562 -.0251 - *Inv. = Inventory 68 Table 4.3' Inventory Demand Inventory =_ Constant Pricet Salest_1 Inv.t_1* Timet l OLS Reg. Coefficient 222.64 -8.96 .3965 .7085 11.77 2 PI Reg. Coefficient -106.47 -39.47 .3070 .6887 41.75 3 OLS Std. Error 540.17 13.18 .1312 .0655 12.76 4 PI Std. Error 653.92 12.52 .1645 .0872 9.43 5 OLS RC/SE Col. 1/3 -.9785 -.680 3.022 8.284 .9638 6 PI RC/SE Col. 2/4 -.6623 -3.153 1.865 7.896 4.427 7 OLS Elast. Invent. to: - -.832 .230 .659 1.096 8 PI Elast. Invent. to: - -3.663 .178 .640 3.8869 *Inv. = Inventory CHAPTER V DYNAMIC ANALYSIS In this chapter, the fundamental dynamic equation, the characteristic equation, and the equilibrium equation will be deve10ped for the model. The approach of this chapter follows that of Chapter VI of Impact Multipliers and Dynamic Properties of the Klein - Goldberger Model by Arthur S. Goldberger.1 Section 1: Derivation of the Fundamental Dynamic Equations For the Endogenous Variables The terms of the structural equations can be arranged as follows: Nt - alpt - aZYt - a3st - 2 a A - a0 = 0 (5.11) l . . (Amsterdam: North Holland Publishing Co., 1959). 69 70 Qt ' blpt ' bZNt-l ' bSIt-l ' b4Tt ' b0 7 0 (5.12) I "' P " N " I " - = t C1 C2 t-l C3 t-l C4Pt-1 Co 0 (5.13) Nt + It - Qt - It_1 =_0 (5.14) - - c - _ = St (1 d)9t_1 (1 d)Nt_1 0 (5.15) Separating the endogenous, lagged endogenous, and exogenous terms and putting them in matrix form gives: 71 100-a1-a3 Nt\ 0 1 0.-b1 0 Qt 0 0 1 -c1 0 ' It 1-110 0 0 0 0 0 0 Nt_1 -b2 0 -b3 0 0 Qt-l + -c2 0 -c3 -c4 0 ° It-l o 0-1 0 0 / pt_1 -(l-d) 0 0 0 -(l-d) st_1/ -a2 0 -a4 ... -a15 -l 0 0\ Yt 0 -b4 0 ... 0 0 -1 0 Tt + 0 0 0 0 0 0 -1 - A1 - 0 0 0 o 0 0 0 I o 0 0 o 0 0 0/ A12 a0 (5.16) This expression can be written in matrix notation as: + F x + Gz = 0, (5.17) 72 whereFO and F1 are, respectively, the matrices of the coefficients of current endogenous variables x and lagged t endogenous variables xt_1.“G is a matrix of coefficients of the exogenous variables zt (including the constants which can be regarded as fixed exogenous variables). To simplify (5.17), the lag operator E.is defined as Ext 5 x 1 and introduced into (5.17) to give: 1:. + J 3 0 Ext Czt 0, (5 18) where F = (F0 + FIE) is the coefficient matrix with its elements being linear functions in E. The solution of (5.18) for xt in terms of the ex- ogenous variables zt is: Xt a 'F G2 (5.19) t, -1 . . . where F is the inverse of matrix E. Separating the inverse of matrix E into its componentSIgives: . . (5.20) F - IFI t' where‘lFl is the determinant of matrix £_and (Adj F) is the adjoint of E. Multiplying both sides by [Fl results in: Ilet.= Hz (5.21) t! where H = -(Adj F)G. M_is a matrix whose elements are polynomials in E, The determinant of E will also be a polynomial in E. Equation (5.21) gives the fundamental dynamic equation for each endogenous variable. The equations for this model are given in Tables 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6 at the end of this chapter. These will be used later to derive the dynamic multipliers for tax Txt and income Yt' Section 2: The Characteristic Component, the Characteristic _Equation, and_the Characteristic_Rpots The determinant'|F| can be written as: 1 '-b2E "CzE 1 (d-l)E 74 0 0 "' u .a .33 1 -b3E -b1 0 (l-c3E) (-c1-c4E) 0 0 -1 l-E 0 0 0 0 0 1-(1-d)E ( Substituting into (5.22) the values of the 5.22) coefficients as obtained from the full information estimates we have: 1 -.4484E |F| - -.30705 1 -.97E 0 0 +37.75 + 1 +.01544 -40.68 0 1-.6887E 39.47-41.75E -1 1-E 0 0 0 0 ( It can be shown that this determinant is given by: 3 |F| = 37.1690 E + 248.5942 E - 184.1913 E2 - 117.90 ( .07378 0 0 0 1-.97E 5.23) 5.24) 75 Thus, the left Side of equation (5.21) for any particular endogenous variable can be written as: 37.1690 xi.“3 - 184.1913 xi,t-2 + 248.5942 Xi,t-1 - 117.90 Xi,t’ (5.25) where xi t 'S represent the'i}h.variable lagged 2 time .__L_LE periods back from the present period 3, The coefficients of (5.25) are exactly the same regardless of which endogenous variable is used in place of xi. The left Side of (5.21) is often called the basic characteristic component of the system; it gives the intrinsic response characteristics of the entire system of equations. By putting (5.25) in homogeneous form, substituting l for xi, and raising l_to the power indicated by its nearness to period E (i.e., the nearer to time p the higher the power), we can form: 3 2 117.90 A - 248.5942 4 + 184.1913 4 - 37.1690 - 0 (5.26) or, dividing through by the first coefficient: ,3 - 2.1085 ,2 + 1.5622 1 = .3152 = 0, (5.27) 76 which is the characteristic equation of the system. roots of this equation are: 11 = +.3170 12 - +.8957 + 43811 = +.8957 - .438li These roots can be put into a column vector denoted {h A‘ A2 .3) t and 5_ can be defined as: A; that is, where 3.15 the current time period. The as 77 Since the basic characteristic component it can be expressed as a function of_A:,,i.e., x =IKA , _ (5.28) where the elements of §_(where E is a five by three matrix) are arbitrary constantsgiven by the initial conditions, the significance of the roots of the characteristic is clear. The roots raised to a power given by £_(the current time period) will determine the size and rate of change (+ or -) of the characteristic component in the current period 3, Thus, the roots determine the stability, periodicity, and damping characteristics of the intrinsic response.3 These characteristics can be noted from 11, 12, and 13 for the present model.4 Consider IE as'p increases from zero to infinity. Since the absolute value of 11 is i will approach zero. Thus, 11 will contribute a component that directly less than one, as t increases, the value of I converges in zero, as shown by the dashed line in Figure 5.1 on the next page. 2Ibid., p. 109. 3161a. 4This analysis follows Baumol, Economic Dynamics (New York: MacMillan Co., 1959), ch. 11. 78 _xtfi 1.0000 .j___] I I I I l I l I .3170 - '---| I I .1005 , ._____. '0101 A . a :———5"""’r . a an- 0 1 2 3 4 5 6 7 t Figure 5.1 Since roots 02 and 13 are complex roots, they must be analyzed together. The value 2: that the complex conjugate pair (c + dip c - d1) will add to the characteristic component can be expressed as: ;: 8 Dt{e cos (tR) + f sin (tR)}, (5.29) where D31] c2 + d2 79 and is the absolute value of the complex conjugate pair, “3 is the angle found by letting cos R = ..... 6.. c2 + d2 sin R = "d‘ V 2 2 c + d and g_and f are constants which are determined by the initial conditions. By substituting the values for D and B for this model, we can rewrite (5.29) as: A xc = (.9971)t{e cos t (26°3') + f sin t (26°3')}. t (5.30) Consideration of (5.30) will yield the relevant points. First, it is apparent that the bracketed expression takes on identical values every 360°. Thus, the length of the cycle will be equal to t = 360° =360° = 13.82 periods. —_ R 110°30' That is, a cycle for the part of the characteristic 80 component contributed by the complex conjugate pair is completed approximately every fourteen months. Second, since the value of 2_is leSs than one, as £_increases the value of;2:'will approach zero, and resultingly, expression (5.30) will approach zero. These two points Show that the influence of roots 12 and).3 in this model is that of dampened oscillation, as shown by the dashed line in Figure 5.2. SEC ti F'-_-I .1 I F" I I I | l I , I 0 . 4 T i . I . n» 3.5 7.0 10.5 14.0 17.5 2110 24.5 t I I ' I I L._._T I I._..__J Figure 5.2 Since the absolute value of E_is larger than the absolute value of 11, the amplitude of the oscillation introduced by the complex conjugate pair will decrease more slowly than the amplitude of the oscillation introduced by 81 The preCeding considerations have shown that the system is inherently stable, as would be expected by observation of the market that this model represents. Section 3: The Particular Component and the Equilibrium Solution ' The effect of the exogenous variables upon the endogenous variables is given by the so-called "particular component" of the full solution and can be found by taking a particular solution to (5.21).6 That is, x = Hzt, (5.31) where each particular solution depends on the time paths of the exogenous variables. If the assumption is made that the exogenous variables are constant over time, that is, (5.32) 5The value of It converges to zero rapidly: within 8 periods it will be zero- to four decimal places. Dt, on the other hand, converges to zero very slowly: it tEkes 120 periods (or 10 years) for the value of 2: to go from .9971 to .7036. Thus, it will take about ten years for the amplitude of the cycle added by (5.30) to be reduced only 30 percent. 6Goldberger, p. 110. 82 then the particular solution is also a constant - that is, 3? = 36. (5.33) This solution is given by: _." 1 X = —:- 6.2., (5.34) f whereZE is the determinant of E with the value of E=l and . 7 . the elements of §:are obtained by also setting E=1. This solution is called an "equilibrium solution" and is given for this model in Table 5.2. gpgtion 4: The Full Complete Solution N The "full complete solution" xt is defined as the sum of the basic characteristic component xt and the particular component i}. That is,8 =x+3€=KA+5€ (5.35) where E'is a matrix the elements of which are determined by the initial conditions. In the model under consideration, E 7Goldberger, p. 111. 8Ibid., p. 112. 83 would be a five by three matrix. The values of each of the endogenous variables in each of the first three months would provide a sufficient set of initial conditions necessary to determine the fifteen elements of 5. These are easily obtained from the data. Since it will not be used later, the full complete solution is not derived for this model. However, it is useful to consider the importance of the particular and characteristic components in the context of it. The full complete solution will give the time path of the endogenous variables given the initial conditions and the time paths of the exogenous variables. If none of the roots).i in A: are greater than one, then the vector A: will approazH the zero vector in value as 3 goes to infinity. The system is then said to be stable and the values of kt will tend toward the values of if; that is, the values for the full complete solution will tend to approach the values of the particular solution; thus, the particular solution is denoted as the "equilibrium solution." The values of the roots in the characteristic component are of interest since they indicate whether the system is stable or unstable, converging or diverging. The particular component is of interest for this investigation since the matrix EZZ:will give us the equilibrium multipliers for the exogenous variables. AS was stated earlier, the fundamental dynamic equations are of interest since they can be used to find the dynamic multipliers for the exogenous variables. 84 Section 5:' Fundamental Dynamic Equations ‘Table‘5.l"'Sa1eS Nt -.2.1085Nt_1 +1.5622Nt_2 - .3152Nt_3 é 1.9415Yt - 4.5104)!t +-3.5448Yt_z - '9644Yt-3 + 2.7470Tt - 4.552.2Tt“1 + 1.8334Tt_2 + 2.6780A1 - 6.2250A1 + 4.8925A1 - 1.3318A1 t t-l t-Z t-3 - 16.2464A2 + 37.7788A2 - 29.6916A2 + 8.0824Az t t-l t-2 t-3 + 36.2679A3 - 84.3361A3 + 66.2825A3 - 18.0429A3 t t-l t-z t-3 + 40.4517A4 - 94.0650A4 + 73.9288A4 - 20.1243A4 t t-l t-Z t-3 + 44.6997A5 - 103.9431AS + 81.6923A, - 22.2376A5 t t-l ‘t-z t-3 + 9.4176A - 21.8994A6 + 17.2115A6 - 4.6852A6 6t t-l t-2 t-3 - 44.7718A7 + 104.1108A - 81.8241A7 + 22.2735A7 t 7t-1 t-2 t-3 - 53.4440A8 + 124.2769A8 - 97.6733A8 + 26.5878A8 t t-l t-Z t-3 - 158.0638A9 + 367.5563A9 - 288.8746A9 + 78.6351A9 t t-l H t-2 t-3 + 60.3850A - 140.4173A + 110.3586A - 30.0409A lot 10t_1 10t_2 10t_3 + 60.0324A - 186.3781A + 109.7140A - 29.8655A 11t 11t-1 11t-2 11t-3 - 38.0392A + 88.4550A - 69.5197A + 18.9241A 12t 12t-1 lzt-z 12t-3 + 1.9150 1 85 Table 5.2 Production Qt -,2.1085Qt_1 + 1.5622Qt_2h¢ '3152Qt-3 =_.9854Yt - .8869Yt_1 -l.2658Yt_2 +1.6169Yt_3 - .4394Yt_4 + 5.6200Tt - 12°1923Tt-1 + 8.5136Tt_2,- 1.9108Tt_3 +.1.3587A1 - 1.2241A1 - 1.7470A1 t t-l t-2 + 2.2315111 - .6065A1 - 8.2480A2 + 7.4290A2 + 10.6024A2 t-3 t-4 t t-l t-Z - 13.5430A2 + 3.6807A2 + 18.4077A - 16.5843A3 t-3 t-4 3t t-l -23.6684A3 + 30.2329A - 8.2167A3 + 20.5312A4 t-z 3t-3 t-4 t - 18.4974A4 - 26.3988A4 + 33.7205A4 - 9.1645A t-l t-2 t-3 4t-4 + 22.6872A5 - 20.4349A - 29°1710A5 + 37°2616A5 t ~ 5t-1 t-Z t-3 - 10.1269A5 + 4.7799A6 - 4.3064A6 - 6.]459A6 t-4 t t-l t-Z + 7.8505A6 - 2.1336A6 - 22.7238A7 + 20.4729A7 t-3 t-4 t t-l + 29.2181A7 - 37.3217117 + 10.1433117 - 27.1254A8 t-2 t-3 't-4 t t-l t-2 t-3 te4 - 80.2250A9 + 72.2781A9 + 103.1525A9 - 131.7619A9 t t-1 t-2 t-3 + 35.8102A + 30.6483A - 27.6123A - 39.4073A ~9t-4 lot 10t-l 1Ot-Z + 50.3369A - 13.6805A + 30.4693A - 27.4511A lot-3 10t-4 11t llt-l - 39.1771A + 50.0429A - 13.6006A - 19.3067A 11t_2 11t_3 11t_4 121 + 17.3942A + 24.8244A - 31.7094A + 8.6180A 12t-1 12t-z 12t-3 12t-4 + 5.2246 86 Table 5.3 ' ‘I‘n‘v‘en‘t‘opy 'r - 2.10851 + 1.56221 ._ .3 21. a . ' , t t_1 . A t_2., 15 't-3 _ 9561Yt 4 2 6681Yt_ —- 2.14 7 + .439 I . - . T 3 1 Yt_2 , 4Yt-3 + 2 87201t .5 1815 t_1 + 2. 442'rt_2 - 1.3290A1 + 3.6825A1 - 2.9559A1 + .6065A1 + 8.0020A. t t—l t-z t-3 ”t -— 22.3484A2 + 17.9387A2 - 3.6807A2 - 17.8602A3 t-l t-Z t-3 't -+- 49.8898A3 - 40.0458A3 + 8.216743 - 19.9205A4 t-l t-z t-3 t -+- 55.6405A ~ 44.6654A4 + 9.16454.4 - 22.0124A5 4t-1 t-Z t-3 t —+- 61.4885A5 - 49.3559AS + 10. 1269AS - 4.6377A6 t-l t-2 t-3 t t-l t-2 t-3 t -— 61.5877A7 + 49.4355117 - 10.1433A7 + 26.3186A t-l t-Z t-3 8t - 73.5172A8 + 59.0111118 - 12.1080A8 + 77.8388A9 tr]. t-2 1.1-3 t - 217.4313A9 + 174.5289A9 - 35.8102A9 - 29.7367A10 t-l t-2 t-3 ‘t 4+ 83.0651A - 66.6571A + 13.6805A - 29.5630A 10t_1 10t_2 10t_3 111: + 82.5800A - 66.2858A. + 13.6006A + 18.7325A 11t_1 11t_2 11t_3 12t - 52.3264A + 42.0016A - 8.6180A - 15.2190 12t-1 lzt-Z 12t'3 87 Table'554 Price I’t - 2.1085Pt_1 r 1.5622Pt-2 - .31529t_3,-It.0242Yt _. .043_6Yt__l r .0196Yt_2 - .00021t_3 5.07271t t .1154Tt_1 - .0450Tt_2 + .0334A1 - .0601A1 + .0271A1 - .0002A1 - ' . - t *t-z t-Z t-3 - .2027A2 + .3649A2 - .1645A2 + .0013A2 t t-l t-Z t-3 ‘*' .4525A3 - .8145A3 + .3672A3 - .0029A3 t t-l t-Z t-3 '*’ .5047A4 ' .9085A4 + .4096A4 - .0032A4 + .5577A t t-l t-2 t-3 5t ‘- 1.0039AS + .4526A: - .0035AS + .1175A6 ’ .2115A t-l “t-Z t-3 t 6t-. ** .0954A6 - .0007A6 - .5586A7 + 1.0055A7 - .4534A7 t-2 t-3 t t-l t-' 4* .0035A7 ' .6668A8 + 1.2003A8 - .5412A8 + .0042A8 t-3 t t-l t-2 t-; ‘* 1.9721A + 3.5500A9 - 1.6006A9 + .0124A9 9t t-l t-2 t-3 '* .7534A - 1.3562A + .6115A - .0047A 10t 10t_1 10t_2 10t_3 '* .7490A - 1.3483A + .6079A .0047A llt 11t_1 11t_2 11t_3 ‘ .4746A + .8543A .3852A . + .0030A 12t 121:.1 12t_2 121:.3 + 1.2158 88 st -2.10855t_1 + 1.56225t_-.-..31525t_ 2 . 3 f1.8814Yt -1 - 2.5499Yt_2 t -9649Yt_3 + 2.6621Tt_1 - 1.8334Tt_2 + 2.5967A - 3.5193A + 1.3318A 1t-1 1t-z - 15.7590A 1 2 t-3 t-l + 21.3584A2 - 8.0824A t- + 35.1798A3 2 - 47.6797A.5 t-3 t-l ‘ 2 t-2 + 18.0429A3t- 4 + 39.2382A4 - 53.1800A4 + 20.1243A t t- 3 -1 2 t-3 + 22.2376A + 9.1351A + 43.3587A5 - 58.7646A t t-3 -1 5 5 6 t-2 t-l + 58.8595A 7 -12.3809A6 + 4.6842A - 43.4286A 7 t- 2 6t-3 t-l t-Z - 22.2735A - 51.8407A + 70.2604A - 26.5878A 7 8t-1 8t-2 8t-3 t-3 - 153.3219A + 207.7994A + 58.5735A t-l - 78.6351A t-2 9 9 9 10 t-3 t-l - 79.3855A + 30.0409A. + 58.2314A - 78.9218A 10t-z 10t-3 11t-1 11t-z - 18.9241A + 29.8655A - 36.8980A + 50.0084A 11t_ 121:,2 12t_3 3 lzt-l + 91.6238 89 Table 5.6 ‘ ‘Eq‘u‘ill'i'b'ri'um Solutions '+..0794Ye + .2036T t .0989A - .5545A 1 + 1.2375A 2 3 + 1.3805A + 1.5256A + .3212A - 1.5277A - 1.8238A , 4 5 6 7 3 - 5.3935A + 2.0606A + 2.0158A - 1.2981A12 9 10 11 + 1.9150 (5.61) ‘+ .0736Ye + .2202T + .0916A1 - .5696A2 + 1.2361A3 4 6 . - . + 2. 8 + 2.0462 1 8216A8 5 387A9 05 4A10 A11 1.2960A12 + 5.2246 (5.62) .0700Ye + .2505T + .0296A1 - .6382A2 + 1.4476A3 + 1.6144A + 1.7833A + .3716A - 1.7877A7 - 2.1335A8 4 5 6 - , + 2.3956 - 1.5184A 6.3090A9 + 2 4101A10 A11 12 - 15.2190 (5.63) Tablej5.6p;cont. + .0189A4 + - + .0739A9 + 40.0216A3 - 49.4050A7 + 66.2462A ,,.0009Ye e .0166T + .0013A - .0209A + .0282A + 2.1400Ye + 5.9833T + 2.9545A + 44.6389A - 58.9754A8 - 174.4231A 11 90 1 , .0076A2 f .0171A 3 5 7 8 .0281A 12 + 1.2158 (5.64) 1 2 + 49.3263A 4 5 + 10.3855A 6 9 + 66.6346A10 - 41.9761A + 91.6238 (5.65) 12 CHAPTER VI DERIVATION OF THE FUNDAMENTAL DYNAMIC EQUATIONS, THE PARTICULAR SOLUTION, AND THE DYNAMIC MULTIPLIERS FOR EXCISE TAXES The previous chapters have developed the model and its dynamic prOperties. In this chapter, the effect of excise taxes will be analyzed by the use of dynamic multipliers. AS was illustrated in Section 2 of Chapter III, excise taxes enter the model as a wedge between retail price and the net price received by the suppliers. The supply equation could, in fact, be Specified as a function of net price (price after taxes) rather than retail price. In equation (3.28) we could replace Pt with P: and fit this relationship to the data. If the-Excise €31 rate was constant during the estimating period, this estimation would give us the same parameters for all variables except price. Since the net price P2 is a constant proportion of retail price (if the excise—zax rate is constant), the new regression coefficient for the net price variable will be a constant proportion of the regression coefficient for the retail price variable. That is, if 91 92 n Pt =‘(l-Tx)Pt’ . (6.11) where_P: is net price, IE is the excise tax rate, andPt is the retail price, rearranging terms, ...1 n. P = P (6.12) t l-Tx t, and then substituting Pt into (3.28) will give: + + + t 0 l'TX t 2 t-l b3It-1 b4Tt I'12. (6013) Since 15 is assumed constant during the estimating period, b1 l-Tx will also be constant and would be equal to the regression coefficient obtained for the price variable if net price had been used in (3.28) rather than retail price. That is, n l b = , (6.14) 1 l-Tx n . . . . . . where b1 is the regre551on coeff1c1ent for net price in the supply equation. If the value of b1 is known and if the excise tax rate was constant during the estimation period, it is not necessary to estimate b“: it can be derived from b1 and 1;. _l.’ — 93 In the Specification of the supply function, quantity is related to the actual retail price Since a constant relationship is assumed between the retail price and the net price. If the tax rate were variable, the relationship between retail price and net price would not be constant, and therefore, the relationship between quantity and retail price would not be constant either. Thus, b1 would not be constant as the tax variable varied. However, b? would be constant as the tax variable varied Since the tax rate does not enter into the relationship between net (after tax) price and quantity Supplied. Combining (6.11), (6.13), and (6.14), the following supply relationship is obtained: n — n - . Qt - b0 + blPt blTth + bZNt-l + b31t-1 + b4Tt + Hz- (6.15) Equation (6.15) Specifies the form of the relationship between quantity, retail price, the excise tax rate, and other variables. However, this relationship is not linear since the third term on the right hand side of (6.15) is non- linear. In order to estimate the separate effect of the tax variable, it is desirable that it enter linearly in equation (6.15). Fortunately, there is a procedure available that gives a linear approximation to a non-linear 94 term. The term n blrxpt (6.16) is replaced by the following linear approximation: bnTP bnTP bnT P‘ + x - - 1 t 1 X t 1 xt t' (6‘17) where the variables with bars are the means of those variables during the estimating period, and therefore are constants. Replacing (6.16) by (6.17) in (6.15) yields: n- ._ I) __ n_ 0 = b - b Tx P + b l-Tx P + b P Tx ‘11 0 1tt 1( )t 1tt a + bZNt-l + bslb1 + b4Tt + ”2t, (6.18) where the variables are defined in the same way as for (3.28). The error term p* may be different since (6.17) 2t is only an approximation. By substituting into (6.18) the values of the coefficients obtained by the full information estimation 1L..R. Klein, A Textbook of Econometrics (Evanston, Illinois: Row Peterson 6 Co., 1953), p. 121. 95 procedure and the means, the following is obtained:2 Q = -3400.22 + 40.68P - 42.99Tx + .4484N t t t t where Txt is measured in percentage terms. This regression equation differs from (3.52), the full information estimation procedure estimated relation- ship for supply, in that the value of the constant term is different, and it contains a tax variable. Equation (6.19) can be rearranged and represented as: -bP-bN -b - - - = Qt 1 t 2 t-l 3It-l b4Tt bsTxt b5 0’ (6.20) where the bi's have the same meaning as before except for b1 which is the new constant term, and b5 which is the 0 ~ EEefficient of the exogenous tax variable. Now equation (6.20) can be used to replace (5.12) in the dynamic analysis of the model. The only difference between these two equations is in the constant term and the number of exogenous variables. _, 2Average retail price 8'Pt = 95.116. Constant tax rate =15 = ten percent. —— 96 The endogenous variables and the values.of their coefficients are exactly the same. 'Thus, the coefficients of the characteristic equation (which depend only on the values of the coefficients of the endogenous variables) will be exactly the same. Consequently, the characteristic roots are the same so that the dynamic preperties of the model are unaffected by the change in the form of the supply equation. In equation (5.17) the matrix §_will be expanded by one column (the element in the second row of the column will be b5, the other elements will be zeros) and the column vector of—exogenous variables will be expanded by one row (having Txt as its only element), and b: will be sub- .___ 0 stituted for b0. The new matrix will be rEEresented by g; and the new column vector of exogenous variables will be represented by zé. Equation (5.21) can now be respecified as: Ile = H'z' (6.21) where H' -- - (Adj F)G'. (6.22) Equation (6.21) gives the fundamental dynamic equation for each endogenous variable. These equations are 97 _ given in Tables 6.11, 6.12, 6.13, 6.14, and 6.15, which follow. In each case, the coefficient of Txt (Txt._1 for stock) gives the impact multipliers of the excise tax variable. The new equilibrium solution is given in Table 6.2, which also follows. In these equations, the coefficient of the Txe gives the equilibrium multiplier for the excise tax variEEIe. The interpretation of these coefficients is as follows: for each one percentage point increase (decrease) in the excise tax rate, the new equilibrium monthly sales level will be approximately 928 units below (above) the old equilibrium; the new equilibrium monthly production level will be approximately 928 units below (above)the old equilibrium; the new equilibrium level of inventories will be approximately 1,165 units below (above) the old equilibrium; the new equilibrium price index will be .083 above (below) the old equilibrium; and the new equilibrium level of stock will be approximately 30,008 units below the old equilibrium. 98 Table 6.11 Fundamental Dynamic Equations - Sales With , .. . 'Tax Variable .. .. w " 2.1 .1” ' o = o " ..- Nt OBSNt-l + 1 3622Nt 3152Nt-3 _1 9415Yt 4 3104Yt-1 t 2 t‘ t t' -3 1 + 1.8334Tt_2 - 13.7637Txt + 22.8298TX - 9.1946Tx + 3.0903* t-l t-Z .0. * The dots indicate that the variables and coefficients (A1 to A12 ) are the same as those found in Table 5.1, p. 84. t t-3 . Table 6.12 Fundamental Dynamic Equations - Production With Tax Variable Qt - Z‘IOSSQt-l + 1.5622Qt_2 - .3152Qt_3 = .98S4Yt - .8869Yt_ l - 1.2658Yt__2 + 1.6169Yt_3 - '4394Yt- + 5.6200Tt - 12.1923Tt-1 + 8'5136Tt-2 - 1.9108Tt_3 - 28.1544Txt + 61.1458Txt_1 - 42'6967Txt-2 + 9.5768Txt_3 ... + 6.5092** ** The dots indicate that the variables and coefficients (A1 to A12 ) are the same as those found in Table 5.2, p. 85. t t-4 99 Table 6.13 Fundamental Dynamic Equations - Inventory ‘ " " With Tax Variable '_pp‘* .. - 7 g- It 2.10851t_l r 1.56221t_2 .315..1,C_3 .9561Yt + 2.6681Yt-1 - 2.1417Y + .4394Y t- t 3 + 2.8720Tt - 5.1815Tt_ + 2.3442Tt_ 2 1 2 - 11.7566TX coo " 13.6064* - 14.3908Txt + 25.9860Txt_1 t_2 * The dots indicate that the variables and coefficients (A to A ) are the same as those found in Table 5.3 p. 86. 1t 12t-3 ’ Table 6.14 Fundamental Dynamic Equations - Price With Tax Variable P - 2.1085Pt_ t + 1.5622Pt_ - .3152Pt_ = .0242Yt - .0436Y 1 2 3 t-l + '0196Yt-2 - .0002Yt_ - .0727Tt + '1154Tt- - '0450Tt- 3 1 2 + .3646Tx - .5787Tx + .2256Tx ...+ 1.1007** 1: t- 112 1 ** The dots indicate that the variables and coefficients (A1 to A12 ) are the same as those found in Table 5.4, p.87. t t-3 100 Table 6.15 Fundamental Dynamic Equations - Stock With 5 - 2.5499Y t - 13.3507Txt * ** t - 2.10858t_1 + 1.56228t_ Tax Variable """ —'_ - .31528 = t 3 1.8814Y t 2 -l + .9649Y + 2.6621T - 1.8334T 2 t t 1 t- 2 1 + 9.1946Txt_ + 133.1671* 2 0.. The dots indicate that the variables and coefficients (A1 to A ) are the same as those found in Table t-l 12t-3 5.5, p.88. Table 6.2 Eqpilibrium Solutions with Tax Variable +.0794Ye + .2036T - .9277Txe ... + 3.0903** (6.61) - .0736Ye + .2202T - .9277Txe ... + 6.5092** (6.62) + .0700Ye + .2505T - 1.1653Txe ... - 13.6064** (6.63) + .0009Ye - .0166T + .0830Txe ... + 1.1007** (6.64) + 2.1400Ye + 5.9833T - 30.0079Txe ... + 133.167l** (6.65) The dots indicate that the variables and coefficients (A1 to A12) are the same as those found in the equations __ ___ in Table 5.6, pp. 89-90. 101 The equations given in Tables 6.11, 6.12, 6.13, 6.14, and 6.15 can be used to find the dynamic multipliers for each endogenous variable with respect to given endogenous variables. The derivation of dynamic multipliers can best be explained by reference to a simple example.3 Suppose that one of our fundamental dynamic equations was given by: Pt + Ylpt-l - 10 + A Txt + AZTx (6.23) 1 t-l’ where Pt represents an endogenous variable (say price), Y1 is the coefficient of the lagged endogenous variables, Txt represents an exogenous variable (say tax rate), and the 11's are the coefficients of the exogenous variables. By subtracting YlPt-l from both Sides of equation (6.23), we obtain: P i A + AlTxt + AZTXt- t 0 (6.24) 1 ' YlPt-l' Now lag all the terms in (6.24) one period to obtain: 3For alternative methods, see Goldberger, pp. 80-83, and J. Kmenta and P.E. Smith, "Autonomous Expenditure versus Money Supply: An Application of Dynamic Multipliers," (Michigan State University Econometric Workshop Paper #6604, February, 1967). (Mimeographed.) 102 Pt-l _‘A0 + AlTxt-l + Aszt'o - Ylpt 9. (6.25) Substitute (6.25) into (6.24) in place of Pt 1 to get: Pt =»A0 - 7110 +AlTxt + (AZ-ylkl)Txt_1 - 1 T + 2P (6 26 Y1 2 Xt-2 Y1 t-2' ° ) Lagging (6.25) another period and substituting the result into (6.26) gives: _ 2 Pt -_40(1-Y1+Y1) + 111xt + (12-y111)1xt_1 + (-y A + yzk ) Tx + y21-Tx 1 2 1 1 t-2 1 4 t-3 3 - Y1P+_7’ (6‘27) This successive substitution process can be continued indefinitely. Each time it is performed, it will bring in an exogenous term lagged one period more than the other exogenous terms and, while eliminating one endogenous term, will bring in another endogenous term lagged one period more than the previous lagged endogenous variable. The coefficient of the next to last exogenous variable will also be changed. 5...: (3 LR The coefficient of the.current exogenous variable .gives the immediate effect of any changein the exogenous variable on the endogenous variable in the current period. That is, it is the impact multiplier. The coefficient of the exogenous variable lagged one period gives the multiplier effect of a change in the exogenous variable on the endogenous variable one period later. If the above substitution process was performed p_times, the coefficient for the exogenous variable lagged 3:3 periods would be found. It would give the effect of a change in the exogenous variable on the endogenous variable 3 periods later. Thus, these coefficients are called "dynamic multipliers." If the system converges to an equilibrium, the value of the pth dynamic multiplier for an exogenous variable Should approach zero as p_goes to infinity. That is, as‘p gets larger, the value of the 2th dynamic multiplier Should get smaller. The total of all of the dynamic multipliers for a given exogenous variable should be equal to the total multiplier for that variable given by the coefficient for that variable in this equilibrium solution, given by equation (5.34) or in Table 6.2. By applying this successive substitution process to the fundamental dynamic equations given in Tables 6.11 to 6.15, the dynamic multipliers for the tax and income variables were derived with respect to each of the 104 endogenous variables. These are presented in Tables 6.3 and 6.4 at the end of this chapter. The value of the dynamic multiplier for the t+n period and the ith endogenous variable gives the effect of a one unit change in the exogenous variable in period 3 on the endogenous variable in period 313, For example, in Table 6.3, the value of the dynamic multiplier for the excise tax rate with respect to sales in period pill is -7.2916. This indicates that a one percentage point increase (decrease) in the excise tax rate will induce a decrease (increase) in sales of approximately 7,292 units.4 Fer another example, in Table 6.4 the value of the dynamic multipliers for the exogenous variable income with respect to price in period 3:32 is -.0119. This indicates that a one billion dollar increase (decrease) at a yearly rate (or a 83.3 million dollar increase (decrease) at a monthly rate) in income will induce a decrease (increase) in the price index of .0119. Notice that for each endogenous variable the tax multiplier for each exogenous variable does not retain the same Sign throughout all periods. The values of these 4The word "approximately" is used here not only because of the rounding off of the number, but also (and chiefly) because a linear approximation was employed in working toward these dynamic multipliers. These dynamic multipliers will be reasonably close approximations only when Txt and Pt are reasonably close to the mean values used . "" -- earlier in this chapter. 105 multipliers, in fact, move in a cyclical fashion. This cycle is a damped cycle since previous analysis showed the model to be stable.5 The central value around which this cycle takes place is, of course, the equilibrium multiplier for the given exogenous variable with respect to a particular endogenous variable, as given in Table 6.2 by the coefficients of the exOgenous variables. The period of this cycle is approximately 14 months.6 Since the dynamic multipliers vary in value in a regular Cyclical way, the values of the endogenous variables can also beexpected to vary in a regular cyclical way as a result of a change in the exogenous variables. Consequently, the endogenous variables will at times be above their old equilibrium values and at times be below their old equilibrium values. Such behavior can be traced by finding the values of the cumulative dynamic multipliers. These are found by successively adding up the values of the dynamic multipliers; that is, the cumulative dynamic multiplier in period 212 equals the sum of the dynamic multipliers from period EDup to and including period pig. The cumulative dynamic multipliers for excise taxes and income with respect to the endogenous variables are given in Tables 6.5 and 6.6 at the end of this chapter. The values in these tables give the cumulative effect of a one unit change in the main SSee,Chapter V. 6See Chapter V, p.80. 106 exogenous variables on the values of the endogenous variables. 'For example, a one percentage point increase (decrease) in the excise tax rate will cauSe the price index to beabove (below) its old equilibrium value7 by approximately .3646 in period 3 (the period of the tax change), to be below (above) its equilibrium value by .4401 in period £19 (nine periods after the period in which the tax changed), and to be above (below) its old equilibrium value by .5935 in period Eilé (sixteen periods after the period in which the tax changed). The tables for the multipliers for income presented here are of interest since they give an idea of the size of the impact of income on the variables in the automobile market. In the first few periods, these values could be expected to give a reasonable first approximation to actual effects. However, not only is the antomobile market functionally dependent upon income, but income is also functionally dependent on the automobile market since the auto industry is such a large part of the national economy in the United States. This second relationship is not Shown in the present model and therefore the feedback effects 7This is true only if the change is initiated when price was at equilibrium. Otherwise, read "normal value for that period" in place of "equilibrium value" where "normal value for that period" is understood to mean the value that would-have prevailed in that period if the tax change had not taken place (as given by the fundamental dynamic equations in Tables 5.1 to 5.6). 107 of sales, production, prices, and inventory on income are not taken into account. The tables simply give the effect of a "once and for all" change in income. Hence, these tables and the fundamental dynamic equations are not of .great value for forecasting for longer periods, unless fiscal and monetary policy authorities decide to keep the level of real disposable income constant for 3:2 periods. To make them more useful for forecasting, they would have to be derived after the model given here was respecified in the context of a simultaneous macro model. The same type of criticism could apply to the fundamental dynamic equations and the tables_giving the multipliers for the excise tax variable. However, the purpose of these equations and tables was to give the ceteris paribus effect of an excise tax change. The ceteris paribus assumption in the context of traditional analysis is applied to those variables normally included in the demand equation for a particular good with the exception of the price of that good (such things as income, income distribution, population, and other prices are assumed constant). This assumption fixes the analysis in that it is undertaken for a given demand curve and the conclusions necessarily hold only if that demand curve remains unchanged. In the present analysis, one other main variable in the demand function is allowed to vary: stock, being an endogenous variable, will vary with sales and depreciation. 108 This is roughly equivalent to allowing the price of the main .substitute,commodity to vary in the traditional analysis. The effect of changing this assumption for comparative statics analysis is to reduce the size of the effect of a change in excise taxes but not a change in direction of the effect on price and quantity.8 The difference in the conclusions of the present analysis will not be due to the assumption of the variability of stock, but rather to the internal dynamics of the automobile market. The same type of phenomena could undoubtedly have been shown had the present model been Specified in the context of a macro model, since adjustments in this model would also be of a cyclical nature. Instead, the present study has been designed to Show that the internal dynamics of the automobile market alone are enough to warrant different, more complicated predictions of the incidence and output effects. These incidence and output effects will be discuSsed in the next chapter. 8If the ad valorem excise tax is increased on good x, the net demand curVE and nef marginal revenue curve will — rotate downward. The equality of net marginal revenue and marginal cost will occur at a lower level of output and hence, price, as read off the demand curve (which at this point has not shifted), will be higher (by less than the tax if marginal cost is constant or increases with output, but by more than the tax is marginal cost decreases with output at a sufficiently rapid rate {see Musgrave, chapter l3}). Since the price of x has increased, the demand for its substitute will increase_- which, in the short run at least, will cause an increase in the price of y, The increase in the price of‘y 'will in turn cause an increase in the demand for 5, and, as a result, an increase in the price and output of x. After all such interaction has wrought itself out, price and output will be higher than that predicted by a partial analysis holding the price of X constant. Table 6.3 ...... Dynamic Multipliers for Tx 00000 TIME N ‘Q I P s PERIOD t ”t. t t t t -l3.7637 -28.1544 -l4.3908 +.3646 0 t+1 - 6.1910 + 1.7822 - 4.3570 +.1901 -13.3507 t+2 - 0.7470 + 4.9715 + 1.5376 +.0568 -18.1554 t+3 + 3.7585 + 8.5527 + 5.5128 -.0622 -17.4245 t+4 + 7.1412 +ll.408l + 7.8495 -.1602 -12.5855 t+5- + 8.9485 +10.8238 + 8.4210 -.2224 - 5.0376 t+6 + 8.8973 + 8.7764 + 7.2318 ‘ -.2384 + 3.5459 t+7 + 7.0318 + 4.9749 + 4.5676 -.2058 +11.3792 t+8 + 3.7486 + .1919 + .9882 -.1316 +16.8658 t+9 - 0.2783 - 4.6049 - 2.7741 -.0310 +18.9035 t+10 - 4.2247 - 8.4359 - 5.9513 +.0753 +17.0963 t+11 - 7.2916 -10.5351 - 7.9036 +.l657 +11.8336 t+12 - 8.8646 -10.4907 - 8.2444 +.2220 + 4.2024 t+13 - 8.6297 - 8.3142 - 6.9100 - +.2329 - 4.2384 t+14 - 6.6470 - 4.4671 - 4.1826 +.1966 -11.7713 t+15 - 3.3278 + .2642 - .6230 +.1205 -16.8746 t+16 + 0.6491 + 4.9193 + 3.0449 +.0206 -18.5275 t+l7 + 4.4708 + 8.5463 + 6.0732 -.0830 -16.4126 t+18 + 7.3624 +10.4172 + 7.8509 -.l693 -10.9812 t+19 + 8.7451 +10.l676 + 8.1417 -.2205 - 3.3553 t+20 + 8.3481 + 7.8605 + 6.5764 -.2269 + 4.9068 t+21 + 6.2593 + 3.9683 + 3.7990 -.1872 +12.1276 t+22 + 2.9133 - .7048 + .2670 -.1097 +16.8477 t+23 - 1.0037 - 5.2071 - 3.2980 -.0105 +18.1241 t+24 - 4.6958 - 8.6301 - 6.1749 +.0903 +15.7184 t+n 0 0 0 0 0 110 ....................... ' Table 6.4 Dynamic; Mu'l't‘ip‘li‘ers for Yt- .......... TIME N ‘ Q 1 P s PERIOD t t t t t t +1.9415 + .9854 -.9561 +.0242 0 t+l -0.4167 +1.1908 +.6522 +.0074 +1.8814 t+2 -0.3669 -0.2943 +.7270 -.0025 +1.4170 t+3 -0.4755 -0.5534 +.6521 -.0095 +1.0136 t+4 -O.5610 -o.7712 +.4449 -.0138 + .5166 t+5 -0.5553 -0.8542 +.l483 -.0150 - .0478 t+6 -0.4445 -0.7706 -.l766 -.0130 - .5880 t+7 -0.2467 -0.5338 -.4639 -.0084 -1.0024 t+8 -0.0009 -0.1908 -.6554 -.0023 -1.2101 t+9 +0.2439 +0.1889 -.7129 +.0045 -1.1710 t+10 +0.4373 +0.5277 -.6256 +.0103 - .8944 t+ll +0.5410 +0.7576 -.4120 +.0138 - .4381 t+12 +0.5347 +0.8328 -.ll6l +.0147 + .1044 t+l3 +0.4196 +0.7386 +.2018 +.0125 + .6228 t+14 +0.2204 +0.4952 +.4769 +.0078 +1.0118 t+lS -o.0225 +0.1527 +.6537 +.0015 +1.1935 t+l6 -0.2596 -0.2190 +.6970 -.0050 +1.1322 t+l7 -0.4426 -0.5439 +.5986 -.0104 + .8415 t+18 -0.5346 -0.7567 +.3794 -.0138 + .3818 t+19 -0.5178 -0.8l47 +.0846 -.0144 - .1525 t+20 -0.3964 -0.7074 -.2257 -.0119 - .6528 t+21 -0.1949 -0.4572 -.4884 -.0071 -l.0182 t+22 +0.0448 -0.1158 -.6507 -.0009 -1.1747 t+23 +0.2740 +0.2742 -.6800 +.0056 -1.0922 t+24 +0.4465 +0.5581 -.5713 +.0106 - .7887 t+~ 0 o 0 0 0 111 Table 6.5 ‘ Cumulat‘ive' Multipliers for Tx ...... TIME N Q 1 P s PERIOD 5 t t t t t -13.7637 -28.lS44 -l4.3908 +.3646 + .0000 t+1 -19.9547 -26.3722 -18.7478 +.5547 -13.3507 t+2 -20.7017 -21.4007 -17.2102 +.6115 -31.5061 t+3 -16.9432 -12.8480 -ll.6974 +.5493 -48.9306 t+4 - 9.8020 - 1.4399 - 3.8479 +.3891 -6l.5161 t+5 - .8535 + 9.3839 + 4.5731 +.l667 -66.5537 t+6 + 8.0438- +18.l603 +11.8049 -.0717 -63.0078 t+7 +15.0756 +23.1352 +16.3725 -.2775 -Sl.6286 t+8 +18.8242 +23.3271 +17.3607 -.4091 -34.7628 t+9 +18.5459 +18.7222 +14.5866 -.4401 -lS.8593 t+10 +14.3212 +10.2863 + 8.6353 -.3648 + 1.2370 t+11 + 7.0296- - 0.2488 + .7317 -.1991 +13.0706 t+12 - 1.8350 -10.7395 - 7.5127 +.0229 +17.2730 t+13 -10.4647 -19.0537 -14.4227 +.2558 +13.0346 t+l4 -17.1117 -23.5208 -l8.6053 +.4524 + 1.2633 t+15 -20.4395 -23.2566 -19.2283 +.5729 -15.6113 t+l6 -19.7904 -18.3373 -16.1834 +.5935 -34.l388 t+l7 -15.3196 - 9.7910 -10.1102 +.5105 -50.5514 t+18 - 7.9572 +-0.6262 - 2.2593 +.3412 -6l.5326 t+19 + 0.7879 +10.7938 + 1.8824 +.1207 -64.8879 t+20 + 9.1360 +18.6543 + 8.4588 -.1062 -59.9811 t+21 +15.3953 +22.6226 +12.2578 -.2934 -47.8535 t+22 +18.3086 +21.9178 +12.5248 -.4031 -3l.0058 t+23 +17.3049 +16.7107 + 9.2268 -.4l36 -12.88l7 t+24 +12.6091 + 8.0806 + 3.0519 -.3233 + 2.8367 t+m - .9277 - .9277 - 1.1653 +.0830 -30.0079 Table 6.6 ' Cumulative Dy‘n‘am'i‘c Multipliers for 112 (.1 TIME N Q ' I P s PERIOD t . t - t -t t t +1.9415 + .9854 - .9561 +.0242 0.0000 t+l +1.5248 +2.1762 - .3039 +.03l6 +1.88l4 t+2 +1.1579 +1.8819 + .4231 +.0291 +3.2984 t+3 + .6824 +1.3285 +1.0752 +.0196 +4.3120 t+4 + .1214 + .5573 +1.5201 +.0058 +4.8286 t+5 - .4339 - .2969 +1.6684 -.0092 +4.7808 t+6 - .8784 -l.0675 +1.4918 -.0222 +4.1928 t+7 -1.1251 -1.6013 +1.0279 -.0306 +3.1904 t+8 -l.1260 -1.7921 + .3725 -.0329 +1.9803 t+9 - .8821 -1.6032 - .3404 -.0284 + .8093 t+10 - .4448 -1.0755 - .9660 -.0181 - .0851 t+ll + .0962 - .3179 -1.3780 -.0043 - .5232 t+12 + .6309 + .5149 -1.4941 +.0104 - .4188 t+13 +1.0505 +1.2535 -1.2923 +.0229 + .2040 t+14 +1.2709 +1.7487 - .8154 +.0307 +1.2158 t+15 +1.2484 +1.9014 - .1617 +.0322 +2.4093 t+l6 + .9888 +1.6824 + .5353 +.0272 +3.5415 t+l7 + .5462 +1.1385 +1.1339 +.0168 +4.3830 t+18 + .0116 + .3818 +1.5133 +.0030 +4.7648 t+19 - .5062 - .4329 +1.5979 -.0114 +4.6123 t+20 - .9026 -1.1403 +1.3722 -.0233 +3.9595 t+21 -1.0975 -1.5975 + .8838 -.0304 +2.9413 t+22 -1.0527 -1.7133 + .2331 -.0313 +1.7666 t+23 - .7787 -1.4661 - .4469 -.0258 + .6744 t+24 - .3322 - .9080 -l.0182 -.0152 - .1143 t+~ + .0794 - .0736 + .0700 +.0009 +2.1400 CHAPTER VII INCIDENCE AND OUTPUT EFFECTS In this chapter, the incidence and output effects of an excise tax on automobiles will be analyzed. The basic assumptions are that the automobile market is dynamic in nature with the particular Specification and parameters of the model presented in the previous chapters, and that all variables assumed to be exogenous to this market are held constant. Consider, first, the estimated coefficients for the excise tax variable in the equilibrium or particular solutions given in Table 6.2. These coefficients are -.9277 with reSpect to sales, -.9277 with respect to production, -l.l653 with reSpect to inventories, +.0830 with respect to price, and -30.0079 with respect to stock. As stated previously, these values give the equilibrium multipliers for the tax variable with regard to the respective endogenous variables. That is, given the change in the tax rate, the eventual change in the equilibrium values of the endogenous variables can be found directly by multiplying the 113 114 change in.tax times the value of.the respective equilibrium multiplier. 'This has been done in Table 7.1, which follows, for these eXcise tax reductions: a one percent reduction, a three percent reduction correSponding to the actual change in the excise tax rate on automobiles in May, 1965, and a ten percent reduction corresponding to a complete removal of the excise tax. For each endogenous variable, the eventual absolute change is given along with the percent of the average value of the endogenous variable from 1960 to 1965 that each change represents. 115 Table 7.1 Long-Run Incidence and Output Effects .................................................. RESULTING EVENTUAL CHANGE IN:" REDUCTION SALES PRODUC-. INVEN- PRICE STOCKS IN EXCISE (Units/ TION TORY. INDEX (N0. of TAXES Month) (Units/ .(No. of Units) Month) Units) 1 PERCENTAGE POINT Absolute Change +928 +928 +l,l65 -.083 +30,008 Percentage Change*‘ .14% .15% .12% .09% .16% 3 PERCENTAGE POINTS Absolute Change +2,783 +2,783 +3,496 -.249 +90,024 Percentage Change* .44% .474 .36% .26% .504 10 PERCENTAGE POINTS Absolute Change +9,277 +9,277 +11,653 +.830 ‘ +300,080 Percentage Change* 1.49% 1.56% 1.20% .87% 1.66% * Percentage of average values during the period of estimation: P’- 95.116 '6 a 591,000 5 a 17,979,000 T' 'N = 619,000 964,000 116 These values can be interpreted as the incidence and output effects that would be derived frOm the traditional, comparative Statics analysis. 'As can be seen from the table, the long-run effect of excise tax changes is quite small for the automobile market. The 1965 excise tax change of three percentage points (a 30 percent decrease) would only cause an eventual .26 percent decrease in prices and an eventual .44 percent increase in sales. Over ninety percent of the change in excise tax rates would eventually be absorbed by the suppliers and less than ten percent would be passed on to the consumer. Even a complete removal of the excise tax would have a very small effect in the long run on the automobile market and, as a result, on the economy as a whole. The conclusion of this analysis would be that consumers would benefit very little in the long run from a reduction in Agovernment revenues that was effected by a change in the excise tax rate on automobiles. However, when the incidence and output effects are considered in the context of dynamic analysis, the conclusions become less clear and depend upon the particular period under investigation. Consider the estimated values for the dynamic multipliers and the cumulative dynamic multipliers. The values of the former give the period-by- period impact of a one percentage point excise tax change on the respective endogenous variables. The latter give the cumulative effect of a one percentage point excise tax change on the respective endogenous variables. As can be 117 observed, the values of the dynamic multipliers vary in sign as £:p_increases;‘ that is, these values go through a cycle. The nature of this cycle is dependent upon the nature of the characteristic equation (derived in Chapter V); the duration of the cycle will be equal to the duration of the cycle introduced by the characteristic component (although the two cycles are not necessarily in phase); and the amplitude of this cycle will be a multiple (possibly a fractional multiple) of the amplitude of the cycle intro- duced by the characteristic component. If the characteristic component does not introduce a cyclical variation (i.e., all roots of the characteristic equation are positive), then the values of the multipliers will not vary either, but will converge or diverge directly (converge if all roots are less than one in value, diverge otherwise). Even if there is a converging fluctuation introduced by the characteristic equation, it may converge rapidly or very Slowly, depending on the absolute value of the largest root(s) of that equation. The size and Sign of the roots of the characteristic equation of a model will depend on the way that the model is specified and on the values of the estimated parameters for that model. For the automobile market, as Specified and estimated here, the roots are such that the component added by the characteristic equation, although stable, converges very slowly. Consequently, the dynamic multipliers will 118 fluctuate in value and converge toward zero very slowly. Likewise, the cumulative dynamic multipliers will fluctuate in value and converge toward the value of the equilibrium multiplier very slowly.1 Thus, for predictive or analytical purposes, the equilibrium multipliers and the results of comparative statics analysis would not be of much short- run interest. Dynamic analysis Shows the effects in each period of a change in excise taxes and is therefore more valuable for predictive or analytical purposes. Table 6.2 .gives the ineremental monthly incidence and output effects, while Table 6.3 gives the cumulative effects. The price index column in these tables gives the incidence effects of the tax change: if the sign of the amount in the column is negative, part of the change is being passed on to the consumer; if it is positive, the suppliers are absorbing the change (or more than the change). The "Sales" and "Production" columns give the output effects: if the sign of the amount in these columns is positive, more cars are produced and sold; if it is negative, sales and production are decreased. As these tables show, the incidence and output effects will depend upon the period considered. In each period 1Since the "half-life" of I; and I; is about twenty years, "very Slowly" can be taken to —— .—_ mean that it will take more than twenty years to converge reasonably close to zero or the equilibrium value. 119 these effects will be different.. In some periods, the direction of theSe effects is in the direction predicted by comparative statics analysis and in other periods the direction of these effects is opposite that predicted. The size of these effects will also vary with the period considered. The maximum and minimum values of the reSpective cumulative dynamic multipliers for the first year after the tax change are taken from Table 6.3 and presented in Table 7.2 for excise tax reductions of l, 3, and 10 percent. The absolute change and the percentage that change is of average value for this endogenous variable from 1960 to 1965 are presented. Although in some months the incidence and output effects will be very low - lower than the equilibrium effects - other months Show these effects being much higher than might be predicted on the basis of equilibrium analysis. Thus, the 1965 excise tax reduction of three percentage points would result in a maximum increase in sales of about ten percent in one month and a decrease in sales of about nine percent in another month, all within the first year after the tax reduction. In one month, about sixty percent of the tax change is passed on to the consumer, while in another month suppliers not only absorb the change completely but also extract an extra forty percent of the change from consumers. 120 ...pozcflpcoo HH+H _meu m+p w+u w+u eofinom name who. see. som.fl seo.m neo.m .omcmeu ommucoOHom oso.ms- Hose.+ oom.sa- st.mN- eNm.mH- owcocu ousaomn< m+o N+o H+o o N4o eoenoa onus sum. see. sea.a nos.e sem.m «omsecu ammucoouom mmm.oo+ mHHc.- ses.ma+ emH.mN+ Hos.ON+ omsesu ousflomn< ezHoo mosezmummo H Amends.. moose: fiasco: fiasco: mmx xme omfloxm on» :w owwmsu <.Nm wousch moanoflpm> ozocowopcm a“ mowcmnu ESwaoz ~.s canoe 121 ...poscwucou HH+D w+o w+o w+o m+o eoasoa asap «HM» swm.H use.m new.HH sNo.o .omcesu omnpcouwom eH~.am mo~m.a+ omo.Nm- Hwa.oo- Nae.om omsesu OHSHOmn< m+o ~+e H.“ o ~+o eoesoa ages EHH.H amo.s smm.m sa~.eH ”mo.os «owsesu ommueoouom ‘mno.oma+ memm.H- Hem.om+ Noe.em+ moa.~o+ omcmcu oHSHomn< mezHoo muzH oneuacoaa mmszH oneusoomo mms ommuo>m mo owmucoonom e .ocoo .~.a oases 124 ConclusionS' The reduction of the excise tax on automobiles will introduce a new cyclical component into the determination of the endogenous variables in this market. 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