- Email: [email protected]
Comments on “Developments in Monetary Aggregation Theory”
Comments on "Developments in Monetary Aggregation Theory" J. Huston McCulloch, The Ohio State University
If all the components of " money" were perfect substitutes, or if they all had a uniform opportunity cost relative to nonmonetary assets, simple sum aggregation of the components of the money stock would provide a useful and meaningful macroeconomic variable. As Bill Barnett has often reminded us, in any other case simple sum monetary aggregates are not economically meaningful. Instead, index number theory points us toward a Divisia index of the various com ponents, computed by weighting the rate of change of each component by its share in the total "expense" on monetary services, as measured by its opportunity cost relative to the return on nonmonetary assets . Prior to the DIDMCA deregulatory reforms that were begun in 1981, Ml consisted primarily of currency that paid zero interest , plus demand deposits that perhaps paid some implicit interest through the back door, but zero explicit interest. If we ignore the fractional "moneyness" of savings accounts and other near moneys, M I may perhaps have been a fair approximation to a meaningful monetary aggregate. Since 1981, however, this has clearly not been the case. Currency still pays zero interest, while nationally available NOW accounts pay substantial ex plicit interest. As Barnett himself points out in Developments in Mon etary Aggregation Theory (this issue, Section 15B) the need for such an index is even greater in the 1980s than it was previously. If we are ever to find a stable money demand function , we would expect it to be a demand for Divisia money, expressed as a function of the dual Divisia opportunity cost index. Barnett is therefore to be congratulated for persevering in his study of Divisia monetary index theory and for his investigations into its empirical performance. The Barnett article is primarily a review of previous research into monetary aggregation theory, the lion's share of which has been done by Barnett himself and his collaborators. Address correspondence to Prof. J. Huston McCulloch, Department of Economics, The Ohio State University, 1945 N. High Street . Columbus, OH 43210--1172 . Journal of Policy Modeling 12(2):259-263 (1990) © Society for Policy Modeling . 1990
259 0161·8938/90/$3.50
260
I.H. McCulloch
One part of the article that I found particularly interesting was Sec tion 7 and 8 on the implications of relaxing the critical homotheticity assumption. In order for a Divisia index to behave exactly like an aggregate commodity, it is well known that certain integrability con ditions must be met in order to guarantee that the line integral that defines the log of the Divisia index be path-independent. These con ditions imply basically that money demand is homothetic: An equi proportionate increase in all opportunity costs, holding the volume of transactions constant, will reduce the demand for all components equi proportionately, and also that an increase in the volume of transactions to be conducted with money, holding opportunity costs constant, will increase the demand for all components equiproportionately. (Chow, 1985). Samuelson and Swamy (1974) disparagingly referred to these conditions as "Santa Claus assumptions." Yet in the present article Barnett argues that aggregation theory is not completely dependent on these homotheticity assumptions. He points out that a "distance function" can be defined that has many of the desirable properties of a Divisia index, even when preferences do not exhibit linear homogeneity. The reader will want to consider the implications of this alternative approach. Another particularly interesting part of the article is Barnett's dis cussion, in Section l ID, of the velocity-weighted monetary index proposed by Paul Spindt. Barnett argues that in order for such an index to be meaningful, it must have arisen from one of two optimization problems, neither of which, he maintains, is realistic. His argument in this section sounds cogent, and deserves particularly careful study. Elsewhere in Section 11 he also considers other weighted monetary indices that have been proposed. In Section 13, Barnett mentions areas in which the existing Divisia monetary indices might be refined and touches briefly on the issue of the benchmark return used to compute the opportunity cost on each monetary component. Barnett in the past has used a Baa corporate bond yield as this benchmark. I find this rate to be questionable for two primary reasons: First, it contains a substantial default premium that ideally should not be present in the benchmark return. And second, it does not reflect an expected return over a short money-hold ing horizon, but rather is a long-term rate that reflects expected short term returns averaged over many years in the future. If short-term rates are expected to drift up or down over the future, the Baa yield to maturity can be quite different from the expected short-term holding period return on Baa bonds, even abstracting from the de fault issue. Furthermore, the actual holding period return on Baa bonds
COMMENTS
261
is only a very noisy observation on their expected short-term holding period return. U.S . Treasury securities are free from default risk , and have no more inflation risk than any other dollar-denominated asset. It would therefore make more sense to base the benchmark return on their yields than on those of Baa corporates . Long-term Trea sury bonds have the same problem as Baa corporates , however, in that their yields to maturity do not reflect current expected holding period returns , and their actual short-term returns are very noisy. Empirically, short-term U.S . Treasury bills exhibit a strong term premium , which gives them a lower expected return on average than longer-term Treasury securities and sug gests that they themselves possess some of the " rnoneyness" the Divisia index attempts to measure. The 1- or 3-month T-bill yield to maturity would therefore be inappropriate as a benchmark yield. However, the marginal empirical term premium on U.S. Treasury bills is essentially zero after about 6 months (McCulloch , 1987). This would suggest that something like the instantaneous forward interest rate implicit in the U.S . Treasury yield curve would be a nearly ideal benchmark . It is close enough to a zero maturity to not be far different from current expected short-term holding period returns , yet far enough out that it is yet not contaminated by the erm premium. Nan-Ting Chow (1975) has computed and investi gated an alternative set of Divisia monetary indices based on this improved benchmark. One issue that Barnett does not address in the present article, but that I hope will be investigated in future research , is the sensitivity of Divisia indices to monetary disequilibrium . Divisia index theory is based on the assumption that consumers continuously and precisely maximize strictly concave utility subject to given prices , so that at any moment they are on their demand curves. Yet as James Davidson and Jonathan Ireland point out in their contribution to this volume , plausible inventory models of money demand predict that most of the time most money holders are equally content with a broad range of money hold ings. Their behavior implies a certain long-run average level of money demand, but at any moment they feel themselves under no compulsion to be on their "demand schedule." How does index number theory apply when money is not in the utility function , but rather is merely a convenience inventory designed to minimize lumpy transactions costs, and when optimizing behavior does not require one to move instantly to one's " optimal" position? Another issue that I hope will be examined in future research is the
262
J.H . McCulloch
sensitivity of Divisia indices to the stochastic behavior of prices and quantities. In Francois Divisia' s original exposition 0925/1926), he made the then standard assumption that the price Pi(t) and hence the quantity q,{t) demanded of the r commodity would be differentiable functions of time. Under this assumption, d q/ql
= - d (I /q,)/(I /q,)
= d log q..
(I)
This implied that the linked Laspeyres quantity index (f and the linked Paasche quantity index QP had a common limit, as the observation interval went to zero, equal to the Divisia index QD, defined as the line integral of the share-weighted d log p/s. Today, however, it is common, particularly in finance theory, to assume that prices (and therefore the equilibrium quantities demanded) will follow a diffusion process or Brownian motion if actually observed in continuous time. If qi is governed by
at + odz.,
d log q, = .....
where dz, is a standard (unit volatility per unit time) Wiener process, then Equation I above , on which the Divisia index is based , no longer holds, contrary to Barnett's statement in Sections 6A and 6B that in continuous time the Divisia index is perfect and unambiguous. Instead, Ito's lemma enables us to write d q/q, = (Ili
+
cr~l2) dt
+ odz;
and -d(l /q)/(l lqJ = (111 - cr~/2)dt
+ o dz;
Over a finite interval of length T, this effect implies that even with literally continuous revision of the Laspeyres and Paasche weights, we will find that
and
0: ~ (ZO e- vlrt2 , where v2 is derived volatility of the log of the Divisia index. Put differently, the familiar theorem concerning the path indepen dence of a line integral when the integrability conditions are met as sumes that the path leading from the initial quantities at time 0 to the final quantities at time T has been parameterized in such a way that prices and quantities are functions of finite variation of the parame terizing variable. However, if time t is the variable that has been chosen
COMMENTS
263
to parameterize the path, and if quantities follow a diffusion process, then quantity will be a function of infinite variation of the parameter izing variable. That is to say, it would take a bottomless bottle of ink to draw the complete path of a diffusion process over a single unit interval of time . We may still be able to place some interpretation on the line integral, but the usual theorems may no longer apply. Under such conditions, then, the theoretical issue is still open of which , if any, index is a meaningful aggregate . And empirically , how big is the gap between the continuously linked Laspeyres and Paasche indices? It is quite conceivable that this problem is not a big one, but this remains to be demonstrated. A related problem, which has been recognized (see Barnett's Section 15A) but in practice is often swept under the rug, is that of true discontinuities in price and/or quantity data, such as occur at the mo ment when regulations are lifted or imposed. A Gaussian diffusion process at least is continuous , if not differentiable , which enables us to quantify the spread between the Laspeyres and Paasche linked in dices by means of It6's lemma. The situation is much more serious, however, with discontinuities, since then even It6' s lemma is of no avail. The primary such events recently were the introduction of NOW accounts nationwide in January of 1981, which would be expected to have a major impact on Divisia Ml , and the introduction of Money Market Deposit Accounts in January 1983, which could have a com parable impact on Divisia M2. These episode s should be carefully scrutinized to see how robust Divisia-like aggregates are to variations in the details of computation that under ordinary circumstances would be inconsequential.
REFERENCES Barnett, W.A. (1990) Developments in Monetary Aggregation Theory, Journal of Policy Mod eling 12: 00-00. Chow, N.-T. (1985) The Demand for Divisia Money. Unpublished Ph.D. dissertation, Ohio State University. Davidson, James, and Jonathan Ireland (1990) Buffer Stocks, Credit, and Aggregation Effects in the Demand for Broad Money: Theory and an Application to the U. K. Personal Sector. Journal of Policy Modeling 12: 00-00. Divisia, F. (1925/1926) L' indice monetaire et la theorie de la monnaie, Revue d' Economie Politique 39: 842-861, 980-1008,11 21-1151 (1925); 40: 49-81 (1926). McCulloch, I .H . (1987) The Monotonicity of the Term Premium: A Closer Look, Journal of Financial Econom ics 18(March): 185-191. Samuelson, P.A. , and Swamy, S. (1974) Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis, American Economic Review (September): 567-593.
If all the components of " money" were perfect substitutes, or if they all had a uniform opportunity cost relative to nonmonetary assets, simple sum aggregation of the components of the money stock would provide a useful and meaningful macroeconomic variable. As Bill Barnett has often reminded us, in any other case simple sum monetary aggregates are not economically meaningful. Instead, index number theory points us toward a Divisia index of the various com ponents, computed by weighting the rate of change of each component by its share in the total "expense" on monetary services, as measured by its opportunity cost relative to the return on nonmonetary assets . Prior to the DIDMCA deregulatory reforms that were begun in 1981, Ml consisted primarily of currency that paid zero interest , plus demand deposits that perhaps paid some implicit interest through the back door, but zero explicit interest. If we ignore the fractional "moneyness" of savings accounts and other near moneys, M I may perhaps have been a fair approximation to a meaningful monetary aggregate. Since 1981, however, this has clearly not been the case. Currency still pays zero interest, while nationally available NOW accounts pay substantial ex plicit interest. As Barnett himself points out in Developments in Mon etary Aggregation Theory (this issue, Section 15B) the need for such an index is even greater in the 1980s than it was previously. If we are ever to find a stable money demand function , we would expect it to be a demand for Divisia money, expressed as a function of the dual Divisia opportunity cost index. Barnett is therefore to be congratulated for persevering in his study of Divisia monetary index theory and for his investigations into its empirical performance. The Barnett article is primarily a review of previous research into monetary aggregation theory, the lion's share of which has been done by Barnett himself and his collaborators. Address correspondence to Prof. J. Huston McCulloch, Department of Economics, The Ohio State University, 1945 N. High Street . Columbus, OH 43210--1172 . Journal of Policy Modeling 12(2):259-263 (1990) © Society for Policy Modeling . 1990
259 0161·8938/90/$3.50
260
I.H. McCulloch
One part of the article that I found particularly interesting was Sec tion 7 and 8 on the implications of relaxing the critical homotheticity assumption. In order for a Divisia index to behave exactly like an aggregate commodity, it is well known that certain integrability con ditions must be met in order to guarantee that the line integral that defines the log of the Divisia index be path-independent. These con ditions imply basically that money demand is homothetic: An equi proportionate increase in all opportunity costs, holding the volume of transactions constant, will reduce the demand for all components equi proportionately, and also that an increase in the volume of transactions to be conducted with money, holding opportunity costs constant, will increase the demand for all components equiproportionately. (Chow, 1985). Samuelson and Swamy (1974) disparagingly referred to these conditions as "Santa Claus assumptions." Yet in the present article Barnett argues that aggregation theory is not completely dependent on these homotheticity assumptions. He points out that a "distance function" can be defined that has many of the desirable properties of a Divisia index, even when preferences do not exhibit linear homogeneity. The reader will want to consider the implications of this alternative approach. Another particularly interesting part of the article is Barnett's dis cussion, in Section l ID, of the velocity-weighted monetary index proposed by Paul Spindt. Barnett argues that in order for such an index to be meaningful, it must have arisen from one of two optimization problems, neither of which, he maintains, is realistic. His argument in this section sounds cogent, and deserves particularly careful study. Elsewhere in Section 11 he also considers other weighted monetary indices that have been proposed. In Section 13, Barnett mentions areas in which the existing Divisia monetary indices might be refined and touches briefly on the issue of the benchmark return used to compute the opportunity cost on each monetary component. Barnett in the past has used a Baa corporate bond yield as this benchmark. I find this rate to be questionable for two primary reasons: First, it contains a substantial default premium that ideally should not be present in the benchmark return. And second, it does not reflect an expected return over a short money-hold ing horizon, but rather is a long-term rate that reflects expected short term returns averaged over many years in the future. If short-term rates are expected to drift up or down over the future, the Baa yield to maturity can be quite different from the expected short-term holding period return on Baa bonds, even abstracting from the de fault issue. Furthermore, the actual holding period return on Baa bonds
COMMENTS
261
is only a very noisy observation on their expected short-term holding period return. U.S . Treasury securities are free from default risk , and have no more inflation risk than any other dollar-denominated asset. It would therefore make more sense to base the benchmark return on their yields than on those of Baa corporates . Long-term Trea sury bonds have the same problem as Baa corporates , however, in that their yields to maturity do not reflect current expected holding period returns , and their actual short-term returns are very noisy. Empirically, short-term U.S . Treasury bills exhibit a strong term premium , which gives them a lower expected return on average than longer-term Treasury securities and sug gests that they themselves possess some of the " rnoneyness" the Divisia index attempts to measure. The 1- or 3-month T-bill yield to maturity would therefore be inappropriate as a benchmark yield. However, the marginal empirical term premium on U.S. Treasury bills is essentially zero after about 6 months (McCulloch , 1987). This would suggest that something like the instantaneous forward interest rate implicit in the U.S . Treasury yield curve would be a nearly ideal benchmark . It is close enough to a zero maturity to not be far different from current expected short-term holding period returns , yet far enough out that it is yet not contaminated by the erm premium. Nan-Ting Chow (1975) has computed and investi gated an alternative set of Divisia monetary indices based on this improved benchmark. One issue that Barnett does not address in the present article, but that I hope will be investigated in future research , is the sensitivity of Divisia indices to monetary disequilibrium . Divisia index theory is based on the assumption that consumers continuously and precisely maximize strictly concave utility subject to given prices , so that at any moment they are on their demand curves. Yet as James Davidson and Jonathan Ireland point out in their contribution to this volume , plausible inventory models of money demand predict that most of the time most money holders are equally content with a broad range of money hold ings. Their behavior implies a certain long-run average level of money demand, but at any moment they feel themselves under no compulsion to be on their "demand schedule." How does index number theory apply when money is not in the utility function , but rather is merely a convenience inventory designed to minimize lumpy transactions costs, and when optimizing behavior does not require one to move instantly to one's " optimal" position? Another issue that I hope will be examined in future research is the
262
J.H . McCulloch
sensitivity of Divisia indices to the stochastic behavior of prices and quantities. In Francois Divisia' s original exposition 0925/1926), he made the then standard assumption that the price Pi(t) and hence the quantity q,{t) demanded of the r commodity would be differentiable functions of time. Under this assumption, d q/ql
= - d (I /q,)/(I /q,)
= d log q..
(I)
This implied that the linked Laspeyres quantity index (f and the linked Paasche quantity index QP had a common limit, as the observation interval went to zero, equal to the Divisia index QD, defined as the line integral of the share-weighted d log p/s. Today, however, it is common, particularly in finance theory, to assume that prices (and therefore the equilibrium quantities demanded) will follow a diffusion process or Brownian motion if actually observed in continuous time. If qi is governed by
at + odz.,
d log q, = .....
where dz, is a standard (unit volatility per unit time) Wiener process, then Equation I above , on which the Divisia index is based , no longer holds, contrary to Barnett's statement in Sections 6A and 6B that in continuous time the Divisia index is perfect and unambiguous. Instead, Ito's lemma enables us to write d q/q, = (Ili
+
cr~l2) dt
+ odz;
and -d(l /q)/(l lqJ = (111 - cr~/2)dt
+ o dz;
Over a finite interval of length T, this effect implies that even with literally continuous revision of the Laspeyres and Paasche weights, we will find that
and
0: ~ (ZO e- vlrt2 , where v2 is derived volatility of the log of the Divisia index. Put differently, the familiar theorem concerning the path indepen dence of a line integral when the integrability conditions are met as sumes that the path leading from the initial quantities at time 0 to the final quantities at time T has been parameterized in such a way that prices and quantities are functions of finite variation of the parame terizing variable. However, if time t is the variable that has been chosen
COMMENTS
263
to parameterize the path, and if quantities follow a diffusion process, then quantity will be a function of infinite variation of the parameter izing variable. That is to say, it would take a bottomless bottle of ink to draw the complete path of a diffusion process over a single unit interval of time . We may still be able to place some interpretation on the line integral, but the usual theorems may no longer apply. Under such conditions, then, the theoretical issue is still open of which , if any, index is a meaningful aggregate . And empirically , how big is the gap between the continuously linked Laspeyres and Paasche indices? It is quite conceivable that this problem is not a big one, but this remains to be demonstrated. A related problem, which has been recognized (see Barnett's Section 15A) but in practice is often swept under the rug, is that of true discontinuities in price and/or quantity data, such as occur at the mo ment when regulations are lifted or imposed. A Gaussian diffusion process at least is continuous , if not differentiable , which enables us to quantify the spread between the Laspeyres and Paasche linked in dices by means of It6's lemma. The situation is much more serious, however, with discontinuities, since then even It6' s lemma is of no avail. The primary such events recently were the introduction of NOW accounts nationwide in January of 1981, which would be expected to have a major impact on Divisia Ml , and the introduction of Money Market Deposit Accounts in January 1983, which could have a com parable impact on Divisia M2. These episode s should be carefully scrutinized to see how robust Divisia-like aggregates are to variations in the details of computation that under ordinary circumstances would be inconsequential.
REFERENCES Barnett, W.A. (1990) Developments in Monetary Aggregation Theory, Journal of Policy Mod eling 12: 00-00. Chow, N.-T. (1985) The Demand for Divisia Money. Unpublished Ph.D. dissertation, Ohio State University. Davidson, James, and Jonathan Ireland (1990) Buffer Stocks, Credit, and Aggregation Effects in the Demand for Broad Money: Theory and an Application to the U. K. Personal Sector. Journal of Policy Modeling 12: 00-00. Divisia, F. (1925/1926) L' indice monetaire et la theorie de la monnaie, Revue d' Economie Politique 39: 842-861, 980-1008,11 21-1151 (1925); 40: 49-81 (1926). McCulloch, I .H . (1987) The Monotonicity of the Term Premium: A Closer Look, Journal of Financial Econom ics 18(March): 185-191. Samuelson, P.A. , and Swamy, S. (1974) Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis, American Economic Review (September): 567-593.