V“‘ .‘, -INFLATIO’N. EXPECTATIONS, AND WEALTH REDESTRIBUTION Thesis for the Degree of Ph. D. M!CH!GAN STATE UNIVERSITY WOODWARD CLARK PRWHARD * 19.70 This is to certify that the thesis entitled INFLATION , EXPECTATIONS , AND WEALTH REDISTRIBUTION presented by WOODWARD CLARK PRICHARD has been accepted towards fulfillment of the requirements for flk 0. degree in .égdflflajf 5 Major professor Date 6‘ [5/7&) / / 0-169 W¢umx¢ whammy, :J n LIu.';..4R Y Mich: 331 State University A 4Md#_’— ABSTRACT INFLATION, EXPECTATIONS, AND WEALTH REDISTRIBUTION By' Woodward Clark Prichard This work is an attempt to determine the nature and extent of wealth redistribution from inflation in the market for debt instruments. A rigorous analysis of the bond market is presented, showing the influence of infla- tion on eXpectations in this market. Inflation is pre- sented in the context of a general correctness of expecta- tions framework. A precise analysis of what constitutes wealth redistribution is then presented. The incorporation of a model of expectations into the theory of wealth redistribution is then undertaken. Empirical testing of the model of wealth redistribu- tion constitutes the next portion of the work. Various types of testing methods are applied to the collected data on wealth and debt. Finally, the conclusions of the author regarding wealth redistribution are presented. lNFLATlON, EXPECTATIONS, AND NEALTH REDISTRIBUTION By Woodward Clark Prichard A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1970 ACKNOWLEDGMENTS I would like to acknowledge the assistance of Professors Byron Brown, Myles Delano, and especially Professor Bruce Allen. I would also like to acknowledge the assistance of Miss Elaine Otto in the editorial preparation of this work. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . ii LIST OF TABLES . . . . . . . . . . . . iv Chapter I. INTRODUCTION . . . . . . . . . . 1 11. REVIEW OF PREVIOUS STUDIES . . . . . 3 III. THE THEORY OF WEALTH REDISTRIBUTION . . . 22 IV. EMPIRICAL TESTING . . . . . . . . . 68 V. CONCLUSION . . . . . . . . . . . 93 REFERENCES . . . . . . . . . . . . . 96 iii LIST OF TABLES Table Page 1. Weights given to periods in formulating Kt+l o o o o o o o o o o o o o 65 2. Annual rate of inflation: 19A9-1966 . . . 71 3. Actual and eXpected rates of inflation: 19"‘9—1967 o o o o o o o o o o o 87 A. Predicted values of (Kt - KS): 19u9-1967 . 88 iv CHAPTER I INTRODUCTION The makers of economic policy are rarely afforded the luxury of selecting Pareto optimal paths of action. Various alternatives among which they have the liberty to choose almost always involve not only gains for some set of members of society, but also losses for some set of members of society. In making their choice, then, policymakers should make some attempt to weigh the rela- tive gains and losses accruing to various segments of society. Their choice should be the wiser the more information economists can provide about the scope and extent of these gains and losses from the effects of various possible policy actions. One of the most significant of these tradeoffs is the one between unemployment and inflation, which has recently come to occupy an important place, not only in the literature of economics, but also in the established body of issues concerning the informed general citizenry. This paper is concerned with inflation. Inflation is important as a public issue partly because the public views it as a vehicle of wealth redistribution. A crucial task is thus handed to the economist, that of determining 1 the validity of this View by examining the actual condi- tions under which wealth redistribution from inflation occurs, and the possible extent of any wealth redistribu- tion which has occurred, is occurring, or might occur in the future. This work is restricted to inflation-caused wealth redistribution in the market for debt instruments. Any effects resulting in other markets from any wealth redistribution in this particular market are not con- sidered. The reason for limiting the study in this way is the convenience of focusing the attention of the study solely on one problem. Developing a rigorous definition of wealth redistribution in this market occupies a major part of this work. Therefore, in certain preliminary sections and in reviewing some of the literature in this field, the term wealth redistribution will be used in a more general sense. It will refer to changes in the distribution of wealth which occur from the existence of debt obligations and changes in price levels. CHAPTER II REVIEW OF PREVIOUS STUDIES This review will include not only studies of inflation—caused wealth redistribution in the market for debt instruments, but also inflation-induced wealth redistribution in other markets. The latter is included to outline the method of previous studies involving wealth redistribution from inflation, regardless of the market involved. Actually, studies dealing with this subject are so few as to render a complete treatment of all of them both convenient and instructive. Previous study of the effects of inflation on the distribution of wealth has proceeded along seVeral distinct lines. The earliest and crudest work consists of historical studies examining the possible existence of a lag of wages behind prices in inflationary periods, this lag supposedly signifying a redistribution away from labor and to capital or entrepreneurship. These efforts have utilized time Series data on real wages and price index levels for a variety of countries and a variety of inflationary periods. While some have found a lag,l Alchian and Kessel2 and Felix 3 have cast doubt upon the validity of the results of these studies. With the appearance of comprehensive national income data came numerous investigations into the effect of inflation on relative shares. The wage lag hypothesis seems to be consistent with the data during some inflations in the U.S., particularly during the period 1945-5“, but in another inflation (1956-57), the data do not seem to support it.“ A problem with most of these studies is that the real effects of productivity changes occurring with changes in levels of output over the business cycle are difficult to separate from any effects due to the presence of wage lags. This difficulty has been partially overcome by the inclusion of productivity as a factor in 5 some studies. Phelps-Brown and Browne surveyed a whole lEarl J. Hamilton, "Profit Inflation and the Indus- trial Revolution," gig, S6 (19A2); and Wesley C. Mitchell, Gold, Prices, and Wages under the Greenback Standard, University of California Press, 1958, reprinted Augustus Kelley, Chicago, 1966. 2Armen Alchian and Reuben Kessel, "The Meaning and Validity of the Inflation-Induced Lag of Real Wages Behind Prices," AER, 50 (March 1960). 3David Felix, "Profit Inflation and Industrial Growth," QJE, 70 (August 1956). “See G. L. Bach and Albert Ando, "The Redistribu- tional Effects of Inflation," Review of Economics and Statistics, 37 (February 1957), pp. 1:13. 5E. H. Phelps-Brown and M.-H. Browne, "Distribution and Productivity under Inflation, l9A7-57," Economic Journal, 70 (December 1960). \JW range of countries during a period of world-wide inflation, while Schultze6 studied a range of different industries in the U.S. Both studies fail to reveal that any significant redistribution away from labor has occurred. Alchian and Kessel7 used an entirely different technique which also failed to reveal any gain to business firms from a wage lag. However, work on income distribution using national income figures has been rather conclusive on the proposi- tion of redistribution away from certain "passive" classes of income recipients (rentiers and lenders, mainly) to "active" classes (entrepreneurship and labor, mainly) during inflationary periods in the U.S. This type of redistribution is that of the oft-cited "erosion" of purchasing power of claims on income which are fixed in nominal terms belonging to pensioners, bondholders, lease- holders, and others. A particular form of this theory, labeled the "debtor—creditor” hypothesis,8 concerns the market for debt instruments, bonds, mortgages, and other forms of debt, which are fixed in nominal money terms. This hypothesis states that during periods of rising prices, 6Charles Schultze, "Recent Inflation in the United States," Study Paper #1, Employment, Growth, and Price Levels, Joint Economic Committee, 1960. 7 8 That this is a special case of incorrect expecta- tions will be discussed in Chapter III. Alchian and Kessel, op. cit. purchasing power is redistributed from creditors to debtors because the repayment of debt involves a payment of less purchasing power than would have been made if prices had been stable for the period for which the debt had been contracted. This hypothesis clearly involves an assumption about eXpectations; the assumption being that creditors do not anticipate correctly the rate of inflation, under- estimating it, and/or do not incorporate this knowledge into their market behavior. It is clear that wealth redistribution can occur if creditors anticipate inflation in part of the period covered by a debt contract, but because a contractual agreement has already been reached are unable to alter their behavior. However, even in this case wealth redistribution occurs because anticipa- tions about future price levels were not correct at the time of the contractual agreement. The agreement is the manifestation of market behavior at the time of its making, and this behavior preSumably was influenced by expectations of the future values of economic variables, including price levels. The integration of price level variations, expecta- tions, and the market for debt instruments can be shown as follows: Assume riskless debt (riskless in the sense that default is ruled out) is in the form of obligations to pay a fixed nominal amount (F) at time (t + I). These are sold at time (t) at market price (V) to lenders. ‘m-In Then , (r is not necessarily used here on an annual basis, but is the rate of interest over any single period, so that J“: there are no compounding problems.) The payment of V occurs at the time t, whereas the payment of F occurs at period t + I. If the general price index (I) in period t, I(t), does not equal I(t + l), then the "real" interest rate (r') does not equal the nominal interest rate (r). (The distinction between these two rates was developed by Fisher.)9 Summarizing the above, F(t + l) r,,=I(t+15_1=F(t+l). I(t) -1 V t V( 5 I(t + l) I t With both V(t) and F(t + l) contractually fixed I t + l at time t, then r' = r if and only if _£I7E7—l'= 1. Suppose inflation (or deflation) is anticipated at time t, and both demand and supplies of debt are functions of r', not r. Then if anticipated ;£%T%7ll > 1, equi— librium r must be greater than if anticipated ££%T%7ll had equaled one, in order that r' remain unchanged. 9Irving Fisher, The Theory of Interest (New York: Macmillan, 1930), p. A2. Thus the basis of any wealth redistribution occur- ring from the debtor—creditor hypothesis in this example must be that r does not rise sufficiently or at all, and that as a result r' 3 lower in inflationary periods than in non—inflationary periods. The redistribution effect operates through a price, r', the real cost of borrowing, I“-_‘L .1 which lags during inflation. Because this cost is reduced during inflation, wealth is redistributed from creditors to debtors. It is clear that if we postulate that demand and supply functions for debt instruments are based on the real cost of borrowing (r'), wealth redistribution occurs only if individuals do not correctly anticipate price level changes over time and r' over time is thus altered. if the foregoing conditions hold, the extent of wealth redistribution from the debtor-creditor hypothesis will depend upon the degree of accuracy of anticipations 10 and the extent of debt instruments outstanding. Fisher, in The Theory of Interest, clearly was of the opinion that wealth redistribution did ordinarily occur from the debtor-creditor mechanismz. When prices are rising, the rate of interest tends to be high but not so high as it should be to com- pensate for the rise; and when prices are falling, the rate of interest tends to be low, but not so low as it should be to compensate for the fall.11 lOThis will be shown riEOPOUS1y in Chapter 111' lllbid., p. U3. He went UN to observe that the real rate of interest in the United States from March to April, 1917 (a period of rising prices), fell below minus seventy per cent.12 DC Alessil3 has constructed a model which is designed to measure the degree of accuracy of anticipations of inflation. He defines the net monetary position of an .fim individual (M) as the current market value of his stock .91‘ _i.. 9. of monetary liabilities (ML) (accounts payable, bonds, loans, and other obligations to pay fixed nominal amounts) less his stock of monetary assets (MA) (accounts receivable, cash, and other claims to fixed dollar amounts of income). Each variable is a function of time so that: (2.1) M(t) E ML(t) — MA(t) M(t) > 0 denotes net debtor status. Non—monetary position (R) is defined as the current market value of the stock of non-monetary assets (NMA) held by an individual (land, buildings, inventory) less the current market value of the stock of non-monetary liabilities (NML) (depreciation, maintenance), so that: (2.2) R(t) 2 NMA(t) — NML(t) 12Ibid., p. “U. 13Louis de Alessi, "The Redistribution of Wealth by Inflation: An Empirical Test with United Kingdom Data," Southern Economic Journal, 30 (October 1963). 10 Note here that R(t) and W(t) are not symmetrically defined in terms of assets and liabilities. Nominal wealth (W) is then defined as: (2.3) w(t) E R(t) - M(t) De Alessi excludes all other wealth-affecting phenomena except "normal income under conditions of static equilibrium," so that net non-monetary assets (R) grow at some rate (r), normal income rate, and this is termed the real rate of interest stated in terms of constant pur— chasing power. Net monetary liabilities (M) grow at a "money" rate of interest (m) which is contractually specified. Under de Alessi's assumptions, m = r. This gives: (2.“) W(t + l) - W(t) = r[R(t)] - m[M(t)], or since r = m, r[R(t) - M(t)], which is specified as continuous.lu De Alessi's definition of wealth differs from that which is generally used in economic theory. The usual definition of wealth [W'(t)] at any time is: ' Y t 1 Y t+2 y t+ W) = m . +17% rim): . (Hg; lulbid., p. 11h. ll where (Y) equals an economic unit's income in period (t) and where the economic unit's time horizon is assumed to be a constant number of periods from the period for which its wealth position is defined. If we define its wealth position at t, and its time horizon is then t + n, then if we define its wealth position at t + 1, its time L'M'h‘.“ horizon is t + n + l. a‘1Y5l Now W'(t + l) > W'(t) if and only if H, [Y(t+l) + Y(t+2) + Y(t+3) + ... + Y(t+n+l) ] (1+r) (l+r)2 (1+r)n - > [Y(t) + —-———-Y(t+l) + ———Y(t+2% + + Y—(tmr), l (1+r) (1+r) (1+r) We can reconcile the two definitions to the extent of making one a monotonic transformation of the other by making the following assumptions: In the de Alessi definition if M is assumed zero in t, a) W(t + l) — W(t) = Y(t + l) and if Y(t + l) > O and any Y(t + n), n > 1, is a positive function of W(t + 1), then if b) W(t + l) > W(t) then, c) W'(t + l) > W'(t). If all prices increase at some rate (K) per period, and only normal income is assumed to affect wealth, then the income stream produced by net non-monetary assets R 12 must also increase at the same rate of inflation K. Specified in current prices, B would grow at a rate r + K. Net monetary liabilities (M) would grow at a rate r + Ka in current prices, where K8 is the anticipated rate of inflation at the time of debt contraction and is the supplement to r required if behavior in debt markets is ru- a function of real variables. This gives: (2.5) W(t + 1) - W(t) = (r + K)R(t) — (r + Ka)M(t) Substituting for R(t) from (2.3) gives: W(t+1) — W(t) = (r+K)W(t) + (r+K)M(t) - (r+Ka)M(t) 01" (2.6) W(t + 1) - W(t) = (r + K)W(t) + (K - Ka)M(t) Since K[W(t)] is the change in nominal wealth necessary to maintain constant purchasing power during an inflation of rate K, real wealth will be redistributed between debtors and creditors if Ka # K, assuming M(t) # 0. Having assumed only one r for all W, we can say that no redistribution occurs from a rate of inflation equal to K only if all wealth grows at a rate r + K. From (2.6) it can be seen that this will hold only if K is correctly anticipated (Ka = K) and/or net debtor position (M) is equal to zero during any period of inflation. 13 When inflation is not anticipated and Ka < K, then during inflation, an economic unit "gains nominal wealth on its monetary liabilities at a rate K - Ka’ and loses wealth on its monetary assets at the same rate."15 In de Alessi's terms: (a) gain of net debtor (M > O) = (K - Ka)[M(t)], F.“ ,1 (b) loss of net creditor (M < O) = - (K - Ka)[M(t)]. De Alessi then introduces a variable denoting the degree of anticipation of inflation which is labeled (B), where: and is restricted by de Alessi to be 0 < B :_1. Substituting B in equation (2.6) gives: (2.7) W(t + l) — W(t) = (P + K)W(t) + BEKM(t)] De Alessi then makes the crucial assumption that all other phenomena which may change W over time are independent of net monetary debtor status (M) so that his model becomes stochastic with these phenomena allowed 16 for by the error term u(t). (2.8) W(t + 1) — W(t) = (r + K)W(t) + BEKM(t)] + u(t) lSIbid., p. 115. 16Ibid. 1A Variables are now converted to units relative to their values at t = 0 (they are converted to indices with base year t = O), with a time interval from zero to t considered, and rates of return not compounded, giving: (2.9) %%g—)y = r(t) + HH ‘3? + Btl‘t}(g)l(0) <§é>1 + u(t) where l is used as an index of prices, so that l%%% is H K over period 0 to t. (%;) is an estimate of M/W over period 0 to t. The formal model is now completed. De Alessi has formulated it so that the debtor-creditor hypothesis is reduced to a hypothesis about the value of B, the degree of anticipation of inflation. If the debtor-creditor hypothesis does in fact hold, then the value of B should be significantly different from zero. His empirical test consisted of using "business firms whose common stocks are quoted on an exchange"17 in the United Kingdom during a period of inflation. The necessary data for approximating M and W are available for these economic units. M' is derived by observing the firm's balance sheet, and W' is computed by taking the number of shares outstanding multiplied by their market price. As an estimate of %%%%, relative change in wealth over time, relative changes in the market price of firm 17Ibid. 1‘ IA'!‘ 15 shares, g%%%, is used with appropriate adjustments made for changes in the number of shares outstanding and cash dividends paid over time from O to t. De Alessi's final regression equation is derived by manipulating terms in (2.9) and by his assumption that r is the normal rate of return, replacing it by a constant (a), and replacing B by b. (2.10) §§%} — 113793 = a + b [I(t)1?oi(0) (9%)] + u(t) 'fhe nmxiel rubw inas: . (a) an independent variable [I(t)I(0)(O) (%%)] whose economic meaning is the net debtor status of a unit divided by its total wealth, all weighted by the rate of inflation, over time from O to t. (b) one constant, a, whose economic meaning is that all firms experience some normal income from R, their non-monetary wealth, and that this income is constant between firms, so that no change in wealth between firms arises from R. (c) a dependent variable, E%%%»- %%%%, whose economic meaning is the relative change in adjusted share prices of firms less the rate of inflation. This is an indicator of whether firms have gained or lost in real I(t) terms, and since TIE? is assumed the common denominator of real wealth for all firms (the usual price index 16 axtsumption that for all individuals, all weights are ewqual), then relative %%%% among firms is dependent on t;he values taken by the independent variable for each i‘irm. A brief summary of de Alessi's method of testing sand results follows. The sample was drawn from firms in ‘the United Kingdom, the period covered was from December 31, 19A8 to December 31, 1957, during which time the annual rate of inflation (retail price index) varied between 1.2% and 12.5% and the price index climbed 52% 18 over the entire time period covered. De Alessi's tests for the levels of significance at which the null hypothesis that b = 0 could be rejected, yielded indecisive results. In only three out of a possible set of 36 possibilities was this level less than the .05 level. Although his other tests did not yield much more ciecisive results either, de Alessi concludes that, "The I“esults observed therefore are not inconsistent with t;he results predicted by the debtor-creditor hypothesis; t>hey suggest that individuals failed to anticipate infla- tLion correctly over the period from 19148-1957."19 De Alessi's model has been implicitly used in several C)ther studies which have proceeded along quite similar \ lBlbid., p. 117. lglbid., p. 123. l7 l:ines, using firm balance sheet information and changes it: stock prices. These studies have used U.S. data, llcnveVEH‘. Ando and Bach20 used a random sample of 52 firms £1rfl.investigated the performance of these economic units CDVGP the period 1939-1952, an inflationary one. The t3esting period was divided into three subperiods due to tzhe prevalence of switching from debtor to creditor status among the firms over the entire period. Apparently 11et debtor status was not determined for each year, but classification proceeded from only three samples in the period. Rank correlation testing was used, with the variables being as follows: Rank Correlation Coefficient Case 1939- 1939- 19135- 1919-- 1952 1956 191‘9 1952 Case A LInd.-creditor-debtor rank .04 —.Ol -.02 -.01 [)ep.-increase in net return C3ase B 12nd.-creditor-debtor rank .26 .23 .09 .18 De~p.-increase in stock prices Source: Ando and Bach, p. 12. aOAndo and Bach, op. cit. 18 In no case did the rank correlation coefficient inke on values greater than .26, leading Ando and Bach t») conclude that "These results do not confirm the pre— ciiction that debtor companies will gain more during fignflation than will creditors, for any of the three gaeriods shown, the results are mixed, and over all show r10 very significant differences."21 Kessel22 used three samples to test the debtor— <2reditor hypothesis. A sample of 16 firms in the banking industry was submitted to a rank correlation test, the rankings based (an relative net creditor status (all 16 firms were found to be net creditors) and relative per cent increase in stock prices from 19U2 to l9u8, an inflationary period. "Roughly 23% (R2 = .A8) of the observed variation was (explained by the debtor-creditor hypothesis."23 Industrial firms were tested, one test covering a 53eriod of inflation and one test covering a period of cieflation. During the interval 1942-1948, a random Szimple of 30 industrials drawn from the population of fVirms listed on the New York Stock Exchange was submitted tco the Mann and Whitney test for differences in random \ 211bid., p. 10. ‘22Reuben A. Kessel, "Inflation-Caused Wealth Rsadistribution: A Test of a Hypothesis," AER, 66 (March 1f956), pp. l28-1H1. 23Ibid., p. 132. l9 vzariables, the variables being the changes in share gxrices of the 15 creditor firms and the changes in share pxrices of the 15 debtor firms in the sample over the pneriod. The test indicated a difference at a signifi- czance level less than .0025?!4 Rank correlation testing was also used on the same :Sample, which yielded results significant at the .002 Zlevel (R2 = .A7), indicating correlation, the rankings tuased again on relative net debtor status, and per cent czhange in stock prices. Another sample using a different (iebt classification and industrials was also tried with r zan R2 equal to .63.2) As a further test, 31 industrial firms were randomly sielected from the N.Y.S.E. listings during a deflation, 1929-1933, with testing supporting the debtor—creditor liypothesis (this time creditors would be expected to ggain) at significance levels of .05“ and .03.26 27 Alchian and Kessel used similar methods, but irnproved upon the other studies by enlarging the sample Siize, using a more comprehensive testing period, using fflirms divided into four distinct industry groups, and teesting each individually. \ 2“Ibid., p. 135. 251b1d. 26 Ibid., p. 137. 27Armen Alchian and Reuben Kessel, "The Redistri— bLition of Wealth through Inflation," Science, 130 (ESeptember A, 1959), pp. 535-537. 20 The test used was the t test for differences toeatween means, the means being the relative change in astock prices times number of shares, adjusted for divi- Ciends paid out of debtor firms and creditor firms, respectively.28 Their tests, for various samples, yielded good ‘results in support of the debtor-creditor hypothesis in the context of the Alchian-Kessel definitions of debtor and creditor. Since the testing was based on differences in means of data on two classes of firms, the method of classifica- tion is crucial. In this respect, the Alchian—Kessel study falls short. Firms were classified as net debtors or creditors for the whole period studied if they were net debtors or creditors during at least two-thirds of this period. This method permitted a firm which was a net creditor of five dollars in each of two-thirds of the years, and a net debtor of $5,000,000 in each of the remaining one-third years, to be classified as a net creditor over this period. A general view of previous studies seems to indicate ‘vast room for further studies of wealth redistribution from inflation, and particularly so with respect to this type of redistribution in the debt instrument market. .Although each previous study seems to have made a valuable 28Ibid., p. 537. 21 ccantribution, there are areas apparent in each of them 111 which further work should be done. Since the most recent study devoted to the debt 111strument market, de Alessi's, was done prior to 1963, 14:3ing more recent data would be a significant contribution film itself. However, much more than simply this will be Clone here. This study is an attempt to fill in at least ssome of the gaps left in our information about inflation- craused wealth redistribution. Theoretical gaps will be t;reated in Chapter III and empirical gaps will be treated .in.Chapter IV. CHAPTER III THE THEORY OF WEALTH REDISTRIBUTION We have seen that the debtor—creditor hypothesis i.s simply a particular case of a general correctness of eeXpectations problem. All wealth, not just monetary vvealth, may increase, decrease, or remain constant as texpectations about the future are correct or not. Once “time is introduced into economic analysis in the form of courses of action open to economic units which are non— instantaneous, i.e., some positive units of time elapse between the decision-making process and when the results of the decision are known, uncertainty is also introduced since the multitude of endogenous and exogenous variables Eiffecting the economic system is constantly changing. TPhe problem for the decision-making economic unit is tzhus adopting the proper strategy, given its utility or ssatisfaction function. This necessarily involves some €3stimation procedure about future values of variables I‘elevant to the particular economic unit. While the EStrategy adopted by any unit may be the outcome of a Cauite complex process, it is reasonable to say that in Egeneral, the better the unit's ability to estimate future 22 23 values of variables correctly, the more successful that unit will be, and the greater its wealth will be.. Alchian and Kessel, Fisher, de Alessi, and others have concentrated on one aspect of the expectations problem. They have focused on expectations about one variable, the price level, and on one market, that for debt instruments. The debtor-creditor hypothesis is the result, a hypothesis concerning how behavior in the debt market is influenced by expectations and how wealth changes in accordance with the correctness of those expectations. Inflation is a special factor only in that it describes the behavior of the variable, the general price index. While the studies surveyed here have almost exclu- sively dealt with only an asymmetrical aspect of the correctness of expectations problem, a generalized study of this problem must necessarily admit of all aspects of the correctness of expectations in the debt market. The debtor—creditor hypothesis deals with an underestimate of future levels of prices, but there is at least an equal theoretical, if not actual, problem in the possibility of overestimation of future price levels and its effect on wealth through its effect on interest rates in the debt market. 2“ ,(I) in this section demand and supply functions for bonds will be developed from a theory of individual behavior. The assumptions used are as follows: (a) There are only two periods considered, t and t+l. (b) Each individual has a fixed income in period t, Y and a fixed income in period t+l, Y t+l° (c) Individuals always buy the same mix of goods t’ in each period. All prices in each period are fixed and the only decision which individuals make is how much to consume in period t, X and how much to consume in period t, t+l, xt+1. (d) Individuals' utility (U) is an additive function of Xt and Xt+l' tion of goods in one period depends solely on the amount (The utility derived from the consump- of goods consumed in this period and not on the amount of goods consumed in any other period. This assumption greatly simplifies the analysis, yet does not seem to be an outrageous affront to reality) where BU/BX > 0 and i 32U/8X12 < o. (e) At the end of period t+l, there are no savings. All wealth is spent on goods in periods t and t+l. (f) Individuals maximize utility subject to a wealth constraint. 25 (g) All relevant variables, W, r, I are known t, with certainty except It+l' More definitions of relevant items are now given. (a) It and It+l represent the price indices in periods t and t+l. Define It as E 1. Let It+1/It = 1 + K. (K is the per cent change in the price index.) (b) C is total expenditures on goods in periods t and t+l. I. L R.‘ n..-Ill. I, ‘. . , = (3'1) C Itxt + It+1Xt+1' (c) C' is the present value of total expenditures in t and t+l. (3.2) c' = I x + I (1 + r)‘1 t t t+lXt+l (r is the money rate of interest.) (d) Wealth (W) is the present value of income streams in t and t+l. (3.3) w = Yt + Yt+l(l + r)‘1 Ifrom (d): U = f(xt) + f(Xt+l) Frwm1(e), W = C', so w — 1 x — I x (1 + r)“1 = o t t t+l t+1 ()l" W - X (l + K 1 + r )X t+1 ' 0 However, the maximization problem would contain expected wealth (We) as the budget constraint because it is the expected price in period t+l that matters. represent the expected decimal change in prices from period t to t+l. From (f), individuals maximize: (3.“) U' = f(Xt) + f(X '! %.[..J_..:f!(t)_)\ t ! aiu = f'(t+1) ‘t+1 BU' _ e §—— - W — Xt - t+l) -A[We —x ’ t+l t Assuming second order conditions hold, llibrium position requires that f'(t) = f'(t+1) Iflanipulating terms Let K8 (l + K 1 + r the equi- 27 If K = 0 (there is no expected inflation or deflation) then f'(t+1) _ l + (ii-6) WET" 1 r F?“ The equilibrium position (the optimum ratio of quantities of Xt and X consumed) depends on the price t+l ratio between Xt and Xt+l and the nominal rate of interest. The decision rule involves r (the nominal rate) explicitly—— not the real rate r'. However, once we allow K8 to differ from zero, then this term (1 + Ke) cannot be dropped from the equilibrium equation. The equilibrium equation is, after manipulation, f'(t+l) _ (1 + K9) + (5'7) “ITIET’ ‘ 1 r The optimum allocation of consumption of Xt and 1 + Ke l + r ) is the real e ;Xt:+1 depends on K and r. NOW; ( FYPice ratio between goods Xt and X and r', the real t+1’ l_i_§_) _ 1. Twate of interest is ( e l + K When price levels are assumed to vary, it is the TWeal rate of interest which determines the equilibrium (lonsumption in t and t+l. The nominal rate of interest 28 is the determining price only when price levels are assumed constant. Since incomes are fixed in each period, individuals must enter the bond market to finance expenditures when- ever their income in any period is not equal to their expenditures. They are demanders or suppliers of bonds according to whether their period t incomes are greater than or less than their expenditures in period t. Bond demand and supply functions are thus functions of the allocation of expenditures between t and t+l. Since this optimum allocation has been established as depending on the real rate of interest, bond demand and supply functions depend on the real rate of interest, except in the special case where It = It+1' Graphically, this can be shown with indifference curves representing time preference and the budget 1 + r ). constraint slope representing ( e l + K m .’..£ :1 '27. 29 E: (D r t+l The equilibrium quantity of X is E and of X t t+l If the price ratio changes, because of a change :in inflationary expectations (say K'8 > Ke), then {l—i—Eg)we > (l—:—£—§)We, so the new budget constraint 1 + K 1 + K' (Ban be represented by the dotted line. 30 Xt E F .’/’ we .\ E ,1 L i ,l‘ \ /’ J x' i\ U1 0 \ D F ( l+re)we (1+re)we Xt+l 1+K' l+K The change in relative prices has resulted in a :shift to an equilibrium position where L of Xt and D of ixt+1 are consumed. Since L/D is greater than E/F, Icelatively more of Xt is now being consumed because the Iaeal rate of interest is now viewed as lower than before. (11) The market for debt instruments can be summarized as follows: 31 We have one behavioral function for potential buyers of debt instruments (lenders or creditors). This can be represented as: l + r (3.8) qD = f'( e 1 + K ,X) where qD is demand for debt instruments (the units on qD l + r are dollar amounts), e l + K is real rate of interest, and A is a vector of other variables affecting qD which for simplicity can be assumed to be exogenous to the debt instrument market. (All variables are assumed to be flow variables, defined as values per period.) We also have one behavioral function for potential sellers of debt instruments (borrowers or debtors). This can be represented as a l + (3.9) qo = g(—-——-£-é-, y) l + K 1 + r Vvhere qS is supply of debt instruments, —————5' is real 1 + K Iwate of interest, and y is a vector of other variables Eiffecting qS which are assumed exogenous. Assuming that debt instruments are not a Giffen Esood, i.e., that demand functions for debt instruments Eire a positive function of the real rate of interest, (Zeteris paribus, we establish the following qualitative PEIation: 3 (gh) 3(l_i_E_Q 1 + K6 (3.10) > 0 On the supply side, if the good, debt instruments, is "non-Giffen" in the sense that supply functions for debt instruments are a positive function of the real rate of interest, ceteris paribus, we also establish the following qualitative relation: :8 (3.11) —-—i%-§-- < O a( ) 1 + x9 And, finally, we postulate equilibrium in the debt instrument market be setting demand and supply equal. (3.12) dB = qS = q 1 + r , 1 + Ke Now, as has been demonstrated heretofore, ‘the real rate of interest, is the nominal rate weighted by ishe relevant price index change over the period covered lfiy the debt contract. Under most contractual agreements for the purchase ainl sale of debt instruments, the nominal rate of interest, r‘, is specified in the contract and thus known with (Zertainty. However, the degree of price change, if any, 141 future time periods is not known with certainty. We can assume, however, that buyers and sellers of debt '“:1 1"” . ..‘. _.a"~ ante-A 33 instruments possess some expectations of values for future prices. Although different individuals will attach different weights to various future prices, this problem does not concern us here and can be resolved if we use some single price index, I(t), as that value, the expected value of which is an argument in the demand and supply functions for debt instruments. These are r, the nominal interest rate, and Ke, the percentage eXpected change in the value of I. The relevant behavioral functions in the market for debt instruments are, after this transformation: e (3.13) qD = f(r, KD, x) where K5 is demanders' expectations, and (3.1“) q8 = g(r, Kg, Y) \Nhere K; is supplierr' expectations. Now, given the qualitative relations that have been (Established in (3.10) and (3.11), namely that demand and tEUpply functions for debt instruments are "non—Giffen" l + r 1+Ke “With respect to , the real rate of interest, we can live the information we have on the relationship between 1’ .. ) _.li_36 , KL, and r to compute partial derivatives of (3.13) l + K' and (3.1M) with respect to r and Ke. 3“ We have assumed that 3(+D) > 0. We have also 8( I’> 1+xe sliown in Chapter II that the definitional formula for 1" is P, = b‘(t+l_)_. I(t) _ 1 W W In order to generalize, yet avoid interest com- pxaunding problems, assume that both interest rates and iriflation or deflation rates are defined over some period ‘t to t+n, and not on a yearly basis. Also, it is the erXpectational value of l%%%§l which is relevant. The symbols in the formula for r' represent the fa) llowi ng: F(t+n) = face value of promise to pay amount at t+n V(t) = market price of debt instrument at t. [_£(£)_Je = ——l——— = expected relative change in i(t+n) l + Ke price index. We now have qualitative relations for the relevant Vat'iables in (3.13) and (3.1“), and the complete beflasdcuel system is: .e f(r, h (3-13) (11) D, A) (3-ilU) qS .e 3.);(113 Rs: Y) 35 WHUIN‘ a£%%l > O, 3:921 < 0 3 (KD) 21nd giggl < 0, éiagl > O. 3(Ks) Since A and y, the vectors of exogenous variables, cic3rnu.directly concern this particular analysis, they veill be assumed constant and drOpped from the system. Initially, assume that both demanders and suppliers fkormulate expectations so as to arrive at similar expected veilues of price changes over any given period. In our tearminology, assume K: equals KS. This assumption will bee relaxed later. This gives a complete system, adding the market 6‘QUilibrium equations of: e \ = ( qD f,r, KD,S) (3-1:? 0 = (1' K9 J) (~10 g 3 13,8) SL1butituting q for qD and q8 yields I k’v- V v 36 q = f(r, KS,S) (3.16) e g(r, KD,S) .0 II We have two equations and two unknowns (q, r) with e D,S (but a comparative statics analysis on this system to find I< assumed to be formulated exogenously. We can work t;he direction of change of equilibrium values of endogenous \rariables with respect to changes in values of exogenous \Jariables, which in this system consist only of one \Jariable, K; S’ the expected price level change. 3 Writing (3.16) implicitly and taking the total 1 ciifferential of the system gives: e D,S e )dKD,S f'(r)dr - ldq = -f'(K (53.17) _ , e e g (KD,S)dKD,S g'(r)dr - ldq Now, what is the sign of the change in the equili- t"I”"ium value of r when K; S changes? Writing (3.17) in 3 (jesterminant form and using Cramer's rule, we have: 1 - The analysis used here is based on the framework tC) be found in Don Patinkin, Money, Interest, and Prices (22nd ed.; New York: Harper and Row, 1965), Appendix 1. 37 e e dr D1 (13 l”) - = —— M"? D “U,S f'(r) -l g'(r) -l h works out to equal [g'(r) - f'(r)] and since we klave established that g'(r) < O and f'(r) > O, D < O. u 'u u * f) ' e - ' e [)1 works out to equal [f (KD,S) g (KD,S)] and since vve have also established that g'(Kg S) > O and f'(KS S) < O f , ’ I l I) C 3.19) , if — -l > O d(K[.)’S) D 'Fliis completes iini‘formal proof cd‘iflue effect of an firuérease in the expected degree of price increase on the 6“~lllilibrium nominal rate of interest. Nominal rates will r'iize when the price level is expected to rise. Now the same type of analysis can be used if the e e aizszumption that KS = KD now have two exogenous variables and we can observe is dropped and we let Kg # Kg. Wee ‘thee sign of the effect on equilibrium r When one of these OXOgenous variables changes, and the other is held C 0h stant . 38 Suppose We hold K: constant and desire the direction «>f change in r when K5 changes. We want to know the sign ()f' (12:. dkh , .e q - f.r, KD) ’ 3 20) e q - g(r, Ks) From writing the new system (3.20) implicitly and Viriting the determinant of its total differential, teecalling that since we assume ng = O, we have only CIKS on the right hand side of the system. We get the determinant: e I -f(KD) -l 0 -l . , D we D D works out to equal [f'(Kg)], which is negative, J L. 13 (g) £2-93. e _ D de w! . . . . d? aCare D” < O and D < O, yielding > O. “ dK D 39 With suppliers' expectations constant, an increase in: the expected price level on the part of demanders of (iebt instruments will, therefore, result in an increase itithe equilibrium nominal rate of interest. Next, a change in suppliers' expectations is con- e D (on the system as before yields: :aidered, holding K constant. The same process performed dr=_.l (1K3 U l dr smiere D < 0, D < 0, giving > O. 3 dKe D Hence, an increase in suppliers' expectations of tile value of K, holding demanders' expectations constant, alxso results in an increase in the equilibrium nominal 1%1te of interest. What is the effect on q when these variables change? I he? can use the same method. For instance, HO e 0/ _v f .r) f (KD,S) g'(r’) -g' E II II OI LP _ (1“ d K D . D “A works out to equal -g'(Kg’s)[f'(r)] — g'(r)[—f‘(KS,S)]. The first term is positive and the second term is negative so no qualitative prediction is given. Specific values of the variables must be known. This turns out to be the result for derivatives of q with respect to the possible variables also. This analysis, simplified though it may be, is basically the eXplanation for a tendency toward high interest rates during inflationary periods, abstracting from any governmental policy-making actions (which might also push interest rates upward during inflationary periods). As stated previously, this process can be entirely symmetrical, and when deflation is expected, the equili— brium interest rate will be lower than if no price level (Wrange s eXpected. With this in mind, it is, of course, no surprise to see interest rates in the immediate past and the present (1968-1969) at near record highs. If one surveys the 1"Lnancial literature of the period, one finds a per- aistent belief in the probability of future inflation. “l ESomc examples are given below. In well known business ixvriodicnls the following appeared: Today, inflation is once again a major problem and it may get worse.2 Why should a company wait to invest later-—in the insatiable seventies-~if it can do it now? In two or three years, we'll have to pay at least 10% to 15% more for the equipment we're buying now.3 Speaking of a recent rise in the prime lending rate, Secretary of the Treasury Kennedy called the increase "Another indication of the strength and pervasiveness of o \ — u inflationary pressures." (III) In this section we shall define what is termed wealth redistribution in the bond market due to inflation. This definition can be best arrived at by means of graphical analysis of various possible situations relevant to this zztluiy. For simplicity and clarity of exposition some assump- tions will be stated at the outset: (a) Bond markets are competitive and are described by the behavioral functions and equilibrium conditions {Elven in the section in Chapter III. 2Fortune, October, 1966, p. 120. 3Business Week, April 26, 1969. “Wall Street Journal, June 10, 1969. U2 (b) While various degrees of inflation and deflation {ire possible, only two rates of price level change are (zonsidered, a zero rate or no price level change, and a xeate of price level increase of E, where K is greater ‘tlian zero and is a constant.‘ This E is hereafter referred t<3 in this section as inflation. The period for which t,hesc are defined is t to t+l. (c) All bonds are of one period, and all are lJcDught and sold at the beginning of the period at price X7(t.), and the face value of the bond is paid at the end ()f the period. This latter amount is F(t+l). We have arlready shown that r can be determined from these two veilues and r' can be determined from these two values L)lus price level changes from t to t+l. (d) Suppliers and demanders always have identical 9; KO (0 - E) bonds were sold at ré. Had Kg equaled K, the sUpply curve would have been 8'. More bonds would have been sold at a higher interest rate (r;). Demanders have received a payment of ré(0 - E), exactly what they eXpected to receive. However, suppliers have made a payment of ré(0 — E), less than had their expectations been correct. The shaded area ET_’ represents a "gain" to suppliers but not a loss to ; demanders. In cases (E) and (G), there is clearly a loss involved for the group having incorrect eXpectations. ,;__. However, unlike cases (A) through (D), neither case (E) nor (G) involves direct wealth redistribution from debtors to creditors or from creditors to debtors. In cases (F) and (H), there appears to be some "gain" to the group having incorrect expectations. However, this "gain" should be considered a sort of added payment occurring in spite of, not because of, incorrect expectations. Take, for example, (F). Also, consider the demand and supply curves as boundaries between attain- able and unattainable points. Now, the point representing the ordered pair r? and E lies within the set of attain- able points, i.e., the area to the left and above the demand curve, D. (0 - E) bonds could have been demanded at either ré or r%. The payment r%(0 - E) would have been acceptable to bond demanders for the purchase of (O - E) bonds. 5“ in case (H), the point representing the ordered pair rs and E lies with the set of attainable points for suppliers; it is to the left and below the supply curve, S. This payment would have been acceptable to suppliers for the sale of (0 - E) bonds. In cases (E) and (G), there is a clear loss since the points representing E and the ex post real rate of interest paid do not lie with the set of attainable points for demanders and suppliers, respectively. In case (E), the point ré - E is to the right of the demand curve, D, in the unattainable area. In case (F), the point r? - E is to the right of the supply curve, S, in the unattainable area. In situations where Ke and K8 D S a loss occurs only in the following situations: are not equal, then, (1) When bond demanders (creditors) under- estimate inflation. (2) When bond suppliers (debtors) over- estimate inflation. A sort of gain, in the sense that some extra consumers or producers surplus is reaped, occurs in the following situations: (1) When bond demanders (debtors) over- estimate inflation. (2) When bond suppliers (debtors) under- estimate inflation. 55 (IV) Emphasis has been placed here on the generality of -the possibility of wealth redistribution from imperfect eXpectations in the bond market. De Alessi has restricted his model to only one aSpect of this by limit- ing the values which B may take.5 His model deals with a narrow interpretation of the debtor-creditor hypothesis where wealth redistribution can only occur under special conditions of incorrect expectations. This is readily seen from re-examining his basic model (2.7), which is: ._i_ (2.7) W(t+l) - W(t) = (r + K)(W(t) + B[KM(t)] Now it is entirely possible that incorrect eXpecta- tions in the debt market can result in wealth redistribu— tion even if there is no inflation. If inflation is expected, and the equilibrium r increases but no inflation actually occurs, wealth redistribution will occur from debtors to creditors in the form of the high price debtors must pay for borrowing. However, this cannot occur in (2.7) since when K = 0, the second term drops out and the wealth change of an economic unit does not depend on M, its net debt position. In addition, some forms of wealth redistribution possible under the more general view of incorrect L 5B is the degree of anticipation of price level changes. 56 expectations in the debt instrument market, are denied by the restrictions on B. De Alessi restricts B to 0 < B i l, but according to the more general View, B can take on values outside this range. The values of B can be positive, negative, or zero, according to various combinations of values of K8 and K (it is assumed below that Ke = Ke = Ke) with the economic S D meaning of the various values of B given as follows: K‘- K8 e (a) B = 0, so ——E—-— = 0 if and only if K = K . Inflation or deflation is correctly anticipated. ' e (o) B = l, o 5-” K = 1 if and only if K8 = o. Inflation or deflation is wholly unanticipated. K - K8 K < IKI. Inflation or deflation is partially, (c) 0 < B < 1, so 0 < 8l < 1 if and only if 0 < IK but not completely, correctly anticipated. K - K9 K and K > 0, or xe > o and K < 0. Individuals anticipate (d) a > 1, so > 1 if and only if Ke < l deflation when inflation actually occurs, or vice versa. x - Ke K negative values of K, deflation and Ke < K, or positive (e) B < 0, so < 0 which could occur with values of K with Ke > K. Individuals overestimate the rate of deflation or inflation. Possibilities (d) and (e) are not included in the de Alessi view of the debtor-creditor hypothesis, but 57 are included in a more general view. (a), (b), and (c) are possible under both views. It can be seen from the above that wealth redistri- bution can occur for any case whean # 0, (a), and that for each other case, (b, c, d, e), wealth redistribution may occur from creditors to debtors, 93 from debtors to creditors, depending upon whether K is negative or u'.’ f- dd “In? ! positive (deflation or inflation occurs). EXpectations are a crucial part of the debtor- creditor hypothesis (either View) of wealth redistribution. .4. .'.‘ , Although expectations have, in previous studies, been treated as a sort of residual, emerging out of various values of the "degree of anticipation" coefficient obtained from various studies, it may be worthwhile at this juncture to depart from this path and venture some sort of hypothesis concerning eXpectations in the debt instrument market. The question of how eXpectations are formed has been a question which several economists have attempted to answer with some type of plausible hypothesis. Any number of hypotheses could be offered as to how eXpectations of future rates of inflation or deflation are formed. Ideally, it might be assumed that expectations of price level changes are derived from expectations of variables that influence price level changes: Federal Reserve actions, fiscal policy, productivity, the demand for money, etc. Shackle has suggested the concept of an "inflative" or "deflative" index of government actions.6 One hypothesis which has gained support through its seeming consistency with observed data is the "adaptive eXpectations" hypothesis.7 This hypothesis has been used by Nerlove8 to study cobweb phenomena, Meiselman9 to study the term structure of interest rates, and used in studies of the effect of inflation on the velocity of money by Cagan.lO The adaptive expectations model is basically this: (3.2“) ——— = u(K - K Expectations of any variable, here Ke, the expected rate of inflation at time t, change directly with errors in eXpectations (Kt - Ki). 6G. L. S. Shackle, Uncertainty in Economics (Cambridge: Cambridge University Press, 1955), pp. 194—21“. 7David Meiselman, The Term Structure of Interest Rates (Englewood Cliffs, New Jersey: Prentice-Hall, 1952), pp. 18-19. Marc Nerlove, "Adaptive Expectations and Cobweb Phenomena," QJE, 72 (May l958), pp. 227—2uo. 9 10Phillip Cagan, "The Monetary Dynamics of Hyper- inflation," in Studies in the Quantity Theory of Money, ed. by Milton Friedman (Chicago: University of Chicago Press, 1956). David Meiselman, op. cit. 59 e ax The time path of change, 533-, is determined by (K - KS) and an adjustment coefficient, a. t To give an example of its applicability to the debtor-creditor hypothesis, suppose that the rate of inflation were a constant (Kt = a). We can solve the differential equation as follows: . .F 3': -5 ail-r . D (3.25) ax: = u(a - K:)3t or, after separating the variables e aKt: (3.26) ——————-= oat. (a-KE) Both sides of (3.26) can be integrated to give: (3.27) In Ia - K: = at + c where c is the constant of integration. Multiplying both sides by -1 yields -ln Ia - KEI = -ot - c which is equivalent to -(a - KS) = e—O‘t-C which is equivalent to e , _ —dt Kt ’ d "’ L? where L = e_c, and limKE-a=lim—%~E- t+oo t-Mao e L n For any value of a > 0, lim ——— = 0 and so lim Ke - a = o. 4 at t t+0° e t+00 With the rate of inflation or deflation constant, the adaptive expectations model predicts that in the limit, the eXpectational value of K will approach the actual value of K. This means that in the limit, no wealth redistribution will occur from the debtor-creditor hypothesis, since its existence depends on incorrect eXpectations of K. The foregoing is included only as an example, since a constant rate of inflation is, in fact, most unlikely. In difference equation form, the adaptive eXpecta— tions model is: _ _ e - u(Kt Kt) e e: e .- reotricted here by 0 < a < 1. Since AKt K(t+l) Kt’ e e _ 61 This difference equation may be manipulated so as to show how it implies that the anticipated rate of inflation at time t, (K3), is a function of weighted past rates of inflation. 7 e = e _ e (J03O) K(t+l) Kt + u(Kt Kt) Replacing a by l - A gives: f’) e = e _ — e \3031) K - (1-A)Kt + A[K(t_1) + (1'*)(K[t-1] - Kft_l])1 which, after substituting for successive K?t—n) in the same manner, reduces to mm“. - e. _ 2 + . . . + (1-A)A“K(t_n). The eXpected rate of inflation in period t + l is a weighted average of past rates of inflation, the weights declining exponentially for successive periods so that Kitil) is more heavily influenced by rates of inflation in more recent periods. The actual values of the weights (A's) depend on the adjustment coefficient in (3.2“), d, since a = l - A. It is helpful at this point to interject a note on the relationship of the adaptive expectations model to the most generally used frame of reference in the field of eXpectations, the concept of the elasticity of eXpectations. A brief definition of this concept is given by Ozga. It is a measure of the responsiveness of prOSpects to changes in results, and has been defined as the ratio of the proportional change in the former to that of the latter. If, for instance, eXpectations are sure prOSpects of prices, the elasticity of eXpectations is the ratio of the proportional change 63 in the eXpected future price to the proportional change in the price which has been observed in the past. If the elasticity of expectations is equal to unity, a 10% increase in the actual price leads to a 10% increase in expected price. Thus, if formerly the price was expected to remain unchanged, and then it increased by 10%, it is now expected to remain unchanged at the new, higher level. If the elasticity of eXpecta— tions is greater than unity, an increase in the actual price gives rise to a prOSpect that it will increase still further.1 Since we are dealing in the rate of change of prices, not absolute prices, this definition must be modified a bit. This can be done by defining the object of expecta- tions to be the rate of change in the price level. e .__._ (K equals actual rate of change, and Kt equals expected t rate of change.) Now, if the elasticity of eXpectations is equal to unity, a 10% increase in the actual rate of price change leads to a 10% increase in the expected rate of price change. With the elasticity of expectations equal to unity, the rate of price change eXpected is exactly the rate of price change actually prevailing at the time the expectations are formed. This is equivalent to the Special case of the adaptive expectations model with the adjustment coefficient equal to one. From equation (3.34), which is 11S. A. Ozga, EXpectations in Economic Theory (Chicago: Aldine Publishing Company, 1965), p. Ifi§. 6N . e _ e e (3.5M) K(t+1) — (l—A)Kt4-A[K(t_l)+-(1-A)(K[t_lJ-K[t_l])] it can be seen that when a = l, and remembering that d = l - A, so A = 0, the second term in (3.3M) drops out and the weight attached to K would equal 1, and, in fact, t e K(t+l) would equal Kt' .' More formally, according to the definition given by Ozga, the elasticity of expectations (08) is given by the ) formula: ,e afi(t+l) K9 axe K (3 36) are = -——————-—(t+1) = ———)-(t+1 --———-——t ' ext BKt Ke T (t+1) t Differentiating (3.34) with respect to Kt gives e 3K(t+1) _ 1 A A T‘ - .0 t e Kt (3.37) o = (l - A) e K(t+1) ' e e When a = l, we know that a = l - A = 1, so 0 = Kt/K(t+l)’ . _ =6 and from (3.3“) we know that when a - l, Kt K(t+l)’ so 6 = e: Kt/K(t+l) l and o 1. However, when a is not equal to unity, we cannot be as specific about the value of the elasticity of expecta— tions in the adaptive model. This is because we were 65 .. . e able to identify the value of Kt/K(t+l) since in the case _ e . where a- l, Kt equaled K(t+l)’ but when a is not equal to one we cannot identify the value of Kt/K?t+l) without a knowledge of other variables, specifically all past rates of price change, since when a is not equal to one, these e (t+1)' case when a equals one that they do not contribute to rates contribute to K It is only in the special e K (t+1) €t+l) and we can therefore establish the value of Kt/K without them. Some numerical examples of various weights attached to selected periods for different values of a are given below. TABLE l.--Weights given to periods in formulating K t+1' 2 V: ilu: Of K K K 2 Kt—n t t-l t-2 n=0 3/U 3/A 3/16 3/6A 63/6“ 1/2 1/2 1/u 3/16 15/16 1/4 1/4 3/16 9/6“ 37/6“ As a increases from its lower restriction [see (3.27)] of zero to its upper restriction of one, the relative weights placed on the three preceding periods N E z Kt-n] increases, and for all three examples shown n O 66 the combined weights of the three preceding periods comprise well over one-half of the sum of all weights (the sum being equal to one). Since, by assuming this expectations model we have obtained a solution for K: in terms of past Kt's, we can make the apprOpriate substitution in the de Alessi model. Substituting (3.35) in (2.6) yields: ' (3.38) W(t+1) — W(t) = (r + K)w(t) n . n + [Kt - (1-A); x Kt_n]M(t) I..—-'\r‘ ._ — 1M“ ‘0 9‘3“1 What this means for_the focus of this study, wealth redistribution due to changes in the price level, is that if the adaptive eXpectations hypothesis holds, with values of a substantially above zero, say over l/U, then expectations about the rate of inflation or deflation in any period will be mostly influenced by what the actual rate of inflation was in periods immediately preceding. If this is indeed the case, then wealth redistribution, which according to the propositions advanced here results from incorrect anticipations (K: # Kt)’ will be most acute when there are sharp changes in the rate of price level increase or decrease. For, when expectations about K:+l are largely based on rate of inflation or deflation in is significantly different e t+1 the immediate past, and Kt+l from these past rates, then K will differ significantly 67 from Kt+l’ and massive wealth redistribution will occur. It is interesting to note that this leads to the policy recommendation of avoidance of these sharp changes in the rate of inflation, rather than a policy recommendation of the avoidance of inflation, if the goal is to minimize wealth redistribution from this source. PM CHAPTER IV EMPIRICAL TESTING A precise theoretical analysis of wealth redistri- bution due to incorrect expectations in the bond market has been presented in the foregoing chapters. Further work in this essay will be directed toward an examination of the empirical evidence relating to this topic. This empirical work will follow two lines of approach. First, extensive testing will be conducted along the same lines used by de Alessi, Ando and Bach, Kessel, and Alchian and Kessel. This work will attempt to improve upon these studies by using more comprehensive data, more recent data, and somewhat different and hopefully more powerful testing methods. The second method of approach assumes the adaptive expectations model outlined in Chapter III, and using this, together with the de Alessi model, tests for wealth redistribution from incorrect expectations. The final model which de Alessi used as his regres- sion equation, equation (2.10), is ,, F(t) I(t) _ I(t) - I(O) M' (1.010) W— I O -a +b[ I(O) (WI—7)] +U(t) 68 69 This model includes one dependent variable and one independent variable. However, the independent variable is, itself, a multiple of two variables--[I(t)IEO§(O)], which is a time series variable, and (%%), which is a cross—section variable. De Alessi's model actually incorporates a pooled independent variable. The use of pooled data ordinarily F“ is accompanied by special econometric problems which necessitate appropriately specialized techniques. However, de Alessi's tests cover only one period, so they are actually only cross-section studies. The rate I(t) - 1(0) 1(0) 0 regression; only (%T) varies. It is quite possible that of inflation [ ] does not vary in each de Aless 's time interval for stock prices changes, one year, was too short for the effects of inflation on wealth to appear in the data. An interval longer than one year may be a better one for testing purposes. Ando and Bach, Alchian and Kessel, and Kessel all used longer periods, but avoided the pooled data problem by the simplification eXplained below. The researcher is confronted with a three dimen- sional array of data. Time is represented by different periodic rates of inflation. Differences at any time in (%%), which are the cross-section dimension, are scattered among various economic units. The remaining dimension is the dependent variable. Note in (2.10) that it is only when I(t)1%01(0) is not equal to zero that the model 70 predicts any differences in changes in wealth caused by net debtor position. Excluding the use of pooled data, two approaches can be used in testing the model: (a) Inflation can be held constant among all economic units while %% varies, or (b) %% can be held constant while inflation varies. The latter could be done by picking economic units whose %% position remained relatively constant, and testing for different degrees of changes in wealth during various periods of varying price level changes. The ilimnm‘mt tau 1...“. .._...i ._P 2101.. \n I ‘ | former approach is the one used by the studies cited. It is also the one which will be used here. It seems to offer the more incisive testing method because it avoids some of the econometric problems of (b)--chiefly auto— correlation and other time series related problems--and it offers a stronger possibility of wider variation in the independent variable. Once this method of testing is adopted, it is then necessary to select the time periods for which data is to be tested. A period of at least several years may be necessary before any wealth redistribution from the debtor- creditor hypothesis shows up in the data. This is due to the possibility of lags between changes in wealth and the time that the changes are recognized by the market, since market price data will be used in the study (see section 71 on data used). The periods used here vary somewhat, but are all at least three year periods. Which periods to use is the next problem. It is desirable to use not only periods of inflation, but also periods of at least relatively stable prices. This is due to the possibility of some excluded variable, highly correlated with net indebtedness, being a prime cause of changes in wealth. If the model were tested in periods of stable prices, and if we obtained results indicating net indebtedness significantly influencing changes in wealth, suspicion would be cast on the model. TABLE 2.--Annual rate of inflation: 1949-1966. Year Kt“ Year Kt 19U9 1.00 1958 .79 1950 8.00 1959 1.58 1951 2.21 1960 1.07 1952 1.08 1961 1.15 1953 .93 1962 1.23 195“ - .32 1963 1.31 1955 1.50 1964 1.67 1956 3.98 1965 2.91 1957 2.76 1966 2.83 *K = annual rate of inflation (% increase in CPI). t 72 The rates of inflation or deflation occurring in each year, JUNO-66, are given in Table 2. These are percentage changes in the Consumer Price Index of the U.S. Bureau of Labor Statistics, I