A re-consideration of Money Demand Theory

Kapur, Basant Article A re-consideration of Money Demand Theory German Economic Review (GER) Provided in Cooperation with: Verein für Socialpolitik / German Economic Association Suggested Citation: Kapur, Basant (2025) : A re-consideration of Money Demand Theory, German Economic Review (GER), ISSN 1468-0475, De Gruyter, Berlin, Vol. 26, Iss. 2, pp. 71-92, https://doi.org/10.1515/ger-2024-0055 This Version is available at: https://hdl.handle.net/10419/331951 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ ger 2025; 26(2): 71–92 Basant K. Kapur* A Re-Consideration of Money Demand Theory https://doi.org/10.1515/ger-2024-0055 Received May 23, 2024; accepted December 19, 2024; published online January 29, 2025 Abstract:Portfolio models typically ignore precautionary transactions demands for liquid assets, and models of precautionary demands typically ignore asset rateof-return risk. If asset-holders are risk-averse, however, both transactions risk and rate-of-return risk affect demands for both liquid and illiquid assets, even when the two risks are independent of each other. We demonstrate this in a four-asset framework, and show that our integrated treatment produces unexpected and instructive results and insights. For example, (a) an increase in the expected return to risky securities increases the demand for M1, even when M1 is used entirely for transactions purposes, (b) an increase in the variance of securities returns reduces the demand for M1, and (c) an increase in the asset-holders’ wealth reduces her demand for M1. A broader framework for the study of money demand is thus called for. Keywords: money demand; asset demands; non-separability; expected utility JEL Classification: E4; G1; G2 1 Introduction Consider a risk-averse wealth-holder who can allocate her wealth across four assets – a risky long-term bond or equity, a riskless long-term asset, a lower-yielding riskless short-term asset, and cash. Her investment horizon is shorter than the maturity period of the risky long-term asset, and she wishes to maximize the expected utility of her terminal wealth. However, ‘half-way’ through the period she faces an *Corresponding author: Basant K. Kapur, Emeritus Professor, Department of Economics, National University of Singapore, Kent Ridge, Singapore 117570, Singapore, E-mail: [email protected]. https://orcid.org/0000-0002-0685-8425 Open Access. ©2025 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 72 —B. K. Kapur uncertain transactions requirement, which could be xwith a specified probability, and 0 otherwise. Her long-term assets are too costly to liquidate to meet this requirement, which can be met either from cash or from liquidating part or all of her short-term asset, for which she has to incur a transactions cost of, say, tper dollar liquidated. As we discuss below, the above setting is, in a stylized way, characteristic of many real-world asset allocation problems. We argue in this article that this fairly elaborate framework, with four assets and two qualitatively different sources of risk, leads to striking new insights, and provides the basis for a re-consideration of money demand theory. The riskless long-term asset could be fixed deposits or CD’s (certificates of deposit), the interest-bearing short-term asset could be savings deposits or other instruments, and ‘cash’ could represent demand deposits (possibly paying minimal interest) and cash-on-hand. Intuition might suggest that one could partition the asset-holder’s decision problem into two components: the choice among the long-term assets on the one hand, and that among the short-term assets on the other, with the obvious restriction being that the sum of her short-term assetholdings not exceed x plus the amount of transactions costs incurred in the event of a transactions shock. With this restriction holding, intuition might further suggest that, for example, a mean-preserving change in the variance of the return on the risky long-term asset (we refer to this as a bond henceforth) would affect the choice between bonds and CD’s, but would have no bearing on the choice between cash and short-term savings instruments (we refer to these instruments as savings deposits henceforth). Vice versa for a change in, say, the interest rate on savings deposits. In this article, we show that this latter intuition is incorrect. A mean-preserving change in the variance of bond returns will affect the choice between cash and savings deposits, notwithstanding that the sum of holdings of cash and savings deposits prior to the transactions shock remains at xplus required transactions costs, and that the distribution of the transactions shock is independent of that of the bond return. Moreover, the direction of change is quite unexpected. A mean-preserving increase in bond return variance decreases the demand for cash, and increases the demand for savings deposits, as seen in our simulation analysis below. Perhaps even more surprisingly, an increase in the mean return to bond-holding induces a substitution towards cash and away from savings deposits. Risk aversion is a necessary condition for these results. There is a positive probability that the transactions shock xwill not materialize, and her terminal wealth will include her holdings of cash and savings deposits. Randomness of her terminal wealth is thus due to both the randomness of the bond rate of return, and the randomness of her transactions requirement. Being risk-averse, she will thus need A Re-Consideration of Money Demand Theory —73 to consider both sources of randomness in her expected-utility-of-wealth calculus, notwithstanding that they are independent of each other.1For conciseness, we refer to this as the ‘non-separability property’ of risk-averse expected utility. Our four-asset framework thus generates interesting patterns of substitutability and complementarity across these assets, which are not induced by patterns of correlation across asset returns as in other models. Our analysis should also provide cautionary advice to asset-holders, including chief financial officers of corporations, against engaging in the partitioning of the decision problem described above, however convenient or logical it might appear to be. 2 Literature Review We seek here to synthesize two hitherto separate strands of literature, albeit in a simplified setting. The first is a discussion originating from James Tobin’s classic 1958 article, ‘Liquidity Preference as Behavior towards Risk,’ which was followed by a critical article by Chang, Hamberg, and Hirata (1983) with the self-explanatory title ‘Liquidity Preference as Behavior toward Risk is a Demand for Short-Term Securities – not Money.’ Chang et al. correctly point out that other ‘safe’ assets, with fixed capital and interest rate values, dominate noninterest-bearing cash for portfolio diversification purposes alone. In terms of our four-asset framework, longterm assets such as fixed deposits would dominate short-term assets such as savings deposits for portfolio diversification purposes. This leaves unanswered two questions, of which the second follows from the first. First, how then do we account for people’s significant holdings of M1 (cash outside banks plus demand deposits) observed all over the world? The obvious answer is that M1 enables transactions demands to be met more cheaply than through the drawing down of other short-term, fixed capital-value assets such as savings deposits. There has developed an extensive literature on this, starting from the classic articles of Baumol (1952) and Tobin (1956), and proceeding through a sequence of complex models allowing for deterministic and stochastic transactions demands of varying forms, as well as various forms of transactions costs. Alvarez, Lippi and Robatto (2019, Section 5) provide a comprehensive overview of this literature (see also Alvarez and Lippi 2017). This literature has indeed assumed that the (single) short-term asset that is alternative to M1 is of fixed capital value and offers a fixed interest rate, while its liquidation to meet transactions demands incurs transactions costs. The asset-holder’s problem is cast as one of expected-cost-minimization over the asset-holder’s infinite lifetime, subject to meeting the transactions demands, 1I owe this interpretation to John Quah. 74 —B. K. Kapur with exogenous income inflows being represented as negative transactions outlays.2 This formulation effectively implies risk-neutrality on the part of asset-holders. However, the assumption of risk-neutrality raises further complications, which do not appear to have been adequately appreciated. In reality, of course, an assetholder has access to multiple assets, various of which offer stochastic returns. Under risk-neutrality she would, cet. par., channel all her asset holdings in excess of the short-term assets required to meet transactions needs to the asset with the highest expected return: her long-term portfolio holdings would become degenerate, which is clearly counterfactual. Second, suppose that asset-holders are not risk-neutral, but instead risk-averse. As mentioned in the Introduction, and shown below, it is then no longer the case that the precautionary transactions demands (adapting the terminology of Frenkel and Jovanovic (1980), and others) for individual short-term assets is independent of, say, the risk of and return on long-term bonds, which implies the necessity for a more inclusive approach to the determination of optimal asset allocations. This is the synthesis that we seek to effect in this study, in that we allow for both rate-ofreturn risk and transactions risk, whereas earlier studies have abstracted from one or the other of these (and, when considering transactions risks, have assumed the special case of risk-neutrality). Our analysis also differs from three other strands of the literature. The first is that of ‘background risk’ (see in particular Fagereng, Guiso, and Pistaferri 2018) – if, for example, an asset holder is also confronted with labour income risk. By definition, this refers to uninsurable risk – risk that ‘cannot be diversified or avoided’ (ibid., p. 437). In our model, however, the asset-holder can adjust the amount of liquidity risk she effectively bears by changing the ratio of money to savings deposits in her portfolio. We show this more precisely below. The second is the issue of optimal portfolio allocation across two or more risky assets (see, for example, Hadar and Seo 1990), wherein a major concern is establishing the conditions under which a stochastically dominating shift in the returns to one of the risky assets unambiguously increases the amount invested in that asset. In our model, we also have multiple (two) risks, but one of them is a liquidity risk, which is qualitatively different from rate-of-return risk. This is intuitively evident, and we explicitly discuss the differences below. Thirdly, and overlapping somewhat with the Baumol-Tobin-type transactions models discussed earlier, there have been studies seeking to distinguish between 2For example, Alvarez, Lippi, and Robatto (2019): ‘The problem for the agent is to minimize the (expected lifetime) cost incurred to finance an exogenous consumption stream’ (p. 211), and ‘The agent withdraws from and deposits to an asset account with real rate of return r’(p.212). A Re-Consideration of Money Demand Theory —75 the transactions roles of currency outside banks, bank deposits, and possibly other short-term monetary instruments such as MMDA’s (money market deposit accounts). Freeman and Kydland (2000) assume a single kind of interest-bearing bank deposit (this is characterized as a demand deposit, and included in their definition of M1), but there is a fixed cost of using these for payments purposes (which ‘may be thought of as a check-clearing cost or a cost of verifying the identity of the person writing a check or making a withdrawal’ (ibid., p. 1126)). By contrast, currency outside banks pays no interest, but incurs no fixed cost when used for payments. They then show that it is optimal to use currency for small purchases, and checks drawn on demand deposits for larger ones,3with the purchase threshold between the two determined endogenously. A puzzling, and possibly inconsistent, feature of their study is that they assume infinitely-lived individuals with strictly concave per-period utility functions, as well as technology and money supply shocks (which may be auto-correlated), but do not incorporate any risk premia into their analyses.4 Belongia and Ireland (2019) postulate a linear homogeneous monetary services aggregator of currency and a single type of interest-earning bank deposit, and adopt a ‘simplified, perfect foresight partial equilibrium framework’ (p.3).Lucas (2000) also adopts a deterministic framework in analyzing money demand, with money either entering the utility function or being explicitly used for transactions purposes. Lucas (1980) allows for random transactions demands, and, working with general utility and transactions risk functions, focuses on characterizing the general equilibrium of the economy without devoting much attention to studying the properties of individual money demand, other than showing that individual Engel curves for real money balances are upward-sloping. He abstracts from asset rateof-return risk. Finally, Lucas and Nicolini (2015) construct a deterministic model with three monetary assets – currency outside banks, demand deposits, and moneymarket deposit accounts – with varying transactions costs, reserve requirements, and interest rates, and show that these individual assets are used for transactions of differing sizes. Plotting their NewM1 measure, including money-market deposit account holdings, and ratios of its individual components against short-term Treasury Bill rates, they find that ‘while the ability of the model to match the ratios 3It is assumed that there are n, apparently exogenously determined, sub-periods within each period, and household stocks of currency and deposits are replenished after each sub-period, so that currencyand depositholdings are constant over the period as a whole. 4They refer to earlier works by Lacker (1988) and Freeman and Huffman (1991), but these models have overlapping generations of two-period-lived individuals, with risk-neutral second-period utility functions. The latter explicitly note (p. 650) that risk premia become relevant if secondperiod utility is not linear in consumption. There is a similar puzzle in the infinite-horizon model of Benati et al. (2021), which has productivity, interest rate, and intermediation technology shocks. 76 —B. K. Kapur between the components of NewM1 is mixed, the behavior of the aggregate is remarkably close to the data’ (p.60). 5 There is abundant empirical evidence that the four-asset classification we propose corresponds well with reality. In fact, the plausibility of such a correspondence can be established deductively, as well as shown empirically. Since savings deposits offer a higher interest rate than demand deposits, asset-holders would opt entirely for the former unless there is some offsetting disadvantage. Indeed, in the US there is typically a limit of six cheque or debit-card withdrawals per month from savings deposits,6any further withdrawals requiring a visit to the bank or to an ATM machine, and minimum-balance requirements are often imposed as well. Banks are obliged to impose such restrictions given that they face lower reserve requirements against savings deposits than against demand deposits: the lower reserve holdings are precisely what enables them to offer higher returns on savings deposits. Similarly, a penalty is often imposed on premature cash withdrawals from fixed deposits and CD’s,7and again the lower reserve requirements are a major factor, along with investments in longer-term assets, enabling banks to offer higher returns on these than on savings deposits. Lastly, risk-averse individuals will require a risk premium to compensate them for holding bonds or equities, rather than fixed-capital-value fixed deposits and CD’s. There are also restrictions or fees of varying kinds on ‘linked savings accounts’ (Kagan 2020a), ‘high-yield savings accounts’ offered by both online and brick-andmortar institutions (Karl 2020c), and money market deposit accounts and money market mutual funds (Barba 2023). Bankrate (June 21 2020) lists a total of 121 of the ‘best available rates across different account types,’ and the APY (Annual Percentage Yield), across savings accounts and money market accounts, of the top 15 of these range from 1.15 % to 1.36 %. In contrast, the best CD rates at that time ranged from 1.50 % APY upwards for deposits of 6 months or longer (Karl 2020d). 3 The Model Our analysis is explicitly partial-equilibrium in nature, as in our view various (though not all) general-equilibrium analyses adopt, for tractability reasons, 5New-Monetarist, search-theoretic approaches to the study of money demand include Kim and Marchesiani (2024) and Berentsen, Huber, and Marchesiani (2015,2018), and abstract entirely from rate-of-return risk. 6Fontinelle (2020). We abstract from NOW accounts, which typically have minimum balance requirements and other restrictions. Edmondson (2021) observes, ‘Today, there are very few NOW accounts used anymore as many types of checking accounts can bear (low) interest.’ 7Karl (2020b). A Re-Consideration of Money Demand Theory —77 simplifying assumptions that do not do full justice to the richness of the determinants of individual asset demands. A general-equilibrium extension of our elaborately-specified partial-equilibrium analysis is postponed to future research. We consider a risk-averse asset-holder who has an initial wealth endowment of W0 at time 0, and seeks to maximize the expected utility of her terminal wealth, W2,at the end of period 2.8She receives no income in period 1. At the beginning of period 1 she faces an uncertain liquidity demand, which is x>0 with probability ptand 0 with probability 1 −p𝓁. This is an essential ‘maintenance’ expense (e.g. a medical expenditure), but does not otherwise enhance her utility: it could simply be viewed as forestalling a sharp decline in her utility that would otherwise occur. She can invest her initial wealth across four assets: (1) cash or demand deposits M0(assumed noninterest bearing for convenience9), which can be liquidated (utilized to pay for the expense) without any transactions cost at time 1, (2) a short-term savings deposit S0, on which, following the discussion earlier, an inconvenience or transactions cost of tper dollar withdrawn at time 1 is incurred, (3) a long-term fixed deposit or CD F0, which matures at time 2, and (4) a risky security B0, which yields an interest payment at time 2, as well as any capital gain or loss then.10 For simplicity, we assume that the penalty for a premature withdrawal from the fixed deposit at time 1, inclusive of the foregone interest, is sufficiently high that the asset holder does not entertain this possibility. Similarly, we assume that foregoing any interest payment if the risky security is liquidated at time 1, together with the brokerage or transactions cost incurred, is sufficiently costly that this too is not a worthwhile option for her.11 We believe these last two specifications accord well with reality. In principle, should the liquidity demand not transpire, the asset-holder may wish to convert her Mand Sholdings into holdings of Fand Bat time 1. However, the returns from converting into Fwould be sharply reduced on account of her reduced holding period, as well as the brokerage cost incurred. This is particularly 8This is specified as the terminal period for notational convenience. Below, we point out that our analysis is equally applicable in a suitably-specified infinite-horizon optimization framework. 9As will be evident from the subsequent analysis, introducing a further distinction between cash holdings outside banks – which pay no interest and incur no transactions costs – and demand deposits – which pay lower interest than savings deposits, and incur positive but smaller transactions costs than the latter – will not affect our qualitative insights. 10 As mentioned we assume that the bond’s maturity date is at some date beyond time 2, so that it is not redeemable at par at time 2. 11 We also assume a prohibitively high cost of short sales of any asset (borrowing), so as not to unduly complicate the discussion. Our assumption that the liquidity demand can only occur once, at time 1, is simply designed as an approximation to the fact that in reality there may be myriad small liquidity demands between times 0 and 2, and the asset holder may well have to incur fixed ‘attention costs’, as well as brokerage costs, each time she has to liquidate small amounts of bonds. As such, she does not entertain this option. 78 —B. K. Kapur so if, following the argument in fn. 11 above, she is uncertain as to precisely when all liquidity demands might be incurred. Regarding B,theclosersheistotime2at the time she converts from Mand Sinto this asset, the lower will be her gain as the price of the security would, with adjustment for risk, tend to converge towards the expected gross return, inclusive of interest payment, at time 2 so as to rule out ‘abnormal’ expected capital gains or losses.12 In addition she would still have to incur the brokerage cost of such conversion. Thus, we rule out the possibility of re-optimization of asset holdings at time 1. Allowing for re-optimization would significantly complicate the analysis without generating new insights, since Mwould still command a liquidity advantage over Sgiven that she would first have to incur a transactions cost to convert from Sto Mbefore investing in either Bor F.Ourspecification is equivalent to assuming, as in infinite-horizon models such as the one we outline briefly below, that asset-holdings are only optimized at the beginning of each period (where the ‘period’ corresponds to the dual periods in our current model). Her wealth constraint at time 0 is W0=M0+S0+F0+B0(1) Since Mpays no interest, its gross return (1 plus interest) is simply 1, and the gross returns on S,F,andB, net of any brokerage costs incurred in investing in them at time 0 (assumed proportional to the amounts invested), are, respectively, rs,rf, and, for B,r1>1 with probability pband r2<1 with probability 1 −pb. For obvious reasons we assume: 1<rs<rf<pbr1+(1−pb)r2.(2) Given these inequalities, it would clearly not be optimal for her to hold any Mor Sabove the sums required to meet the liquidity demand at time 1. This does not imply the restriction M0+S0=x, however, as we also have to account for the transactions costs incurred if any Sis utilized for meeting the liquidity demand at that time. Interestingly, it can be shown that, provided that t<1, it is a matter of indifference whether these transactions costs are incurred out of Mor out of further drawdown of S,13 and for concreteness we assume the former. Let M1denote the 12 This ‘no-arbitrage’ argument would also ensure that bond prices could not be overly high at time 1 (as there would then be high expected capital losses going forward), again serving to dissuade the use of bonds to meet liquidity demands at that time. 13 It can be shown that for any given M0the amount of S0held is the same in either case. Formally, if Sis used to defray the transactions costs, then we have (1−t)S0=x−M0,andifMis used to defray the transactions costs then we have S0=x−M1=x−(M0−tS0),whichgenerates the same equation for S0. The restriction t<1 ensures that holdings of Sdo not ‘explode’ if S A Re-Consideration of Money Demand Theory —85 Differentiating (18) with respect to r1and setting dr2∕dr1=−1itisreadily shown that dB∗ 0∕dr1<0, as expected, and hence dF∗ 0∕dr1>0. Although there is separability in that r1and r2do not affect M∗ 0and S∗ 0 under CARA, the same is not true of rf.From(18) we have that dB∗ 0∕drf<0, and from (17) it is readily shown that dM∗ 0∕drf>0. The rationale for this latter result is novel. Manipulating (3) and (4) we easily obtain that S∗ 0+M∗ 0= x−tM∗ 0 1−t. Thus, when M∗ 0goes up S∗ 0+M∗ 0goes down, and so some further funds are released for investment in F0. This is the main source of non-separability in the CARA case.21 It should be noted that in the present experiment of a mean-preserving increase in bond returns, there are no wealth effects. Thus, the key difference between CARA and CRRA is not the DARA (Decreasing Absolute Risk Aversion) entailed by CRRA utility (which plays some part below), but the much more limited degree of non-separability under CARA utility. (B) Returning to the CRRA case, the next comparative-static experiment is perhaps even more stark. Let there instead be a marginal increase in the expected bond return. We raise r1to 1.1102, and r2to 0.9502. As expected, B∗ 0rises, to 0.7829, and F∗ 0falls, to 0.0167. What is entirely unexpected is that M∗ 0increases, to 0.0795, and S∗ 0decreases, to 0.1208. D∗is again extremely low, at 1.3204e-14. In this case, the realized variance of total bond returns has gone up, to 0.0039, and heuristically the asset-holder mitigates the increase in her overall risk exposure by substituting away from S0and towards M0in meeting precautionary demands.22 Remarkably, a positive relationship between US money (M1) demand and the 10-year Government Bond Rate was indeed observed in the 1990–2019 period for interest rates above 4 % p.a.,asKim and Marchesiani (2024) document. Not only does our result further exemplify the nonseparability property of risk-averse expected utility, it also shows that the conventional wisdom that expected bond returns should enter negatively in the demand function for M1 need not always hold, and the effect is instead model-dependent. Moreover, we obtain the same qualitative outcomes when 𝜎=2. Henceforth, we maintain 𝜎at unity.23 21 Another source, which results rather mechanically from the wealth constraint, has to do with the effect of rson F∗ 0, the analysis of which is very similar to the effect in the CRRA case (fn.25 below). 22 There is an offsetting wealth effect: evaluated at her initial level of bond-holding, an increase in the mean bond return increases the asset-holder’s wealth, and with DARA she would wish to increase her holding of Sand reduce her holding of M. It turns out that this only partially offsets the effect described above. 23 A seemingly puzzling feature of our result is the very high interest-elasticity of money demand that is generated, of the order, in fact, of 155. Berentsen, Huber, and Marchesiani (2015) obtain an 86 —B. K. Kapur A third experiment yields somewhat less clear-cut but nonetheless highly useful additional insights. We return to our original calibration, except that we increase tmarginally to 0.01152: at the same time, in order to keep M∗ 0+S∗ 0unchanged, at 0.2004, we increase x, and to 4 decimal places the required value of xis also 0.2004. Now M∗ 0rises drastically, to 0.198, and S∗ 0falls drastically, to practically 0 (actual value is 0.0024). Following our earlier argument, one might expect B∗ 0to increase and F∗ 0to decrease, since the asset-holder can now ‘afford’ to take on additional risk. Instead, however, B∗ 0decreases marginally, to 0.7540, and F∗ 0increases marginally, to 0.0456. These very small effects are explained by the wealth effect: evaluated at her initial levels of M0and S0, increases in tand xreduce the asset-holder’s expected terminal wealth net of expected precautionary costs, and with DARA she chooses to marginally decrease her holdings of Band increase her holdings of F. Any further increase in twould result in a corner solution for S0, which would make comparative-static experiments difficult to interpret. Evidently, the optimal level of S0is highly sensitive to t.24 One may examine the effects of varying other parameters, such as the rates of return on Sand F, but the results are similar.25 Our final thought-provoking experiment is to examine the effects of changes in W0relative to changes in x.Thereare two noteworthy cases here: (a) Suppose we increase only W0in our baseline calibration, to 1.005. The results are, quite remarkably, that M∗ 0decreases quite significantly to 0.0345, and S∗ 0 increases to 0.1664. B∗ 0and F∗ 0increase slightly, to 0.7587 and 0.0454 respectively. We note that M∗ 0has decreased, by over 50 %, notwithstanding that estimate of this elasticity in the upward-sloping region of money demand of 0.16. However, in our model the increase in money demand is more than offset by a concomitant decrease in demand for S(from (3) and (4) above we have that (M0+S0)=(x−tM0)∕(1−t)), while the resultant decrease in interest earnings from Sis ‘compensated for’ by the increased bond expected return. As further discussed below, this result underscores the necessity of a ‘systems’ or structural approach to the study of money demand. 24 In order to generate more variation in twe changed the initial calibration to x=0.38874 and t =0.00878, resulting in an initial value of S∗ 0of 0.3526 (we also increased rfto 1.0265 to generate an interior solution for F0). However, even here we could only increase tand xvery slightly before S∗ 0 approached the corner at 0, and the results are qualitatively unchanged. 25 We increased rsmarginally to 1.0151, keeping all other parameters at their baseline values, and there occurs a large increase in S∗ 0, a virtually zero (0.0023) solution for M∗ 0, and very small decreases in B∗ 0and F∗ 0. These small declines are mainly due to the larger value of M∗ 0+S∗ 0owing to the higher transactions costs incurred when S∗ 0goes up. Similarly, a small increase in rf,to 1.025, generates as expected an increase in F∗ 0and decrease in B∗ 0, both of which are fairly large (to 0.0948 and 0.7050 respectively), as well as a moderate decrease in S∗ 0(to 0.1085) and increase in M∗ 0(to 0.0917). (With F∗ 0+B∗ 0increasing, M∗ 0+S∗ 0has to decrease, which is effected by reducing S∗ 0 and increasing M∗ 0((3) and (4) above).) A Re-Consideration of Money Demand Theory —87 the initial overall portfolio size W0has increased. There are a number of noteworthy implications of these results: (i) With DARA the demand for the risky assets has gone up. However, strikingly, the demand for Srises much more than the demand for B, in both absolute and relative terms. This is due to a feature of the model that has been alluded to earlier: investing in Sreduces the demand for Mfor precautionary purposes almost pari passu, and the funds thus released can be utilized for other investments. No such offsetting release occurs upon increasing holdings of B. (ii) The fact that the demand for Mhas gone down marks a notable departure from the well-known result expressed in Theorem 1 of Cass and Stiglitz (1972) and in earlier literature, in which, in a pure ‘portfolio’ analysis of the demand for one risky asset and money, the wealth elasticity of the demand for money is unity under constant relative risk aversion. With our more elaborate asset structure, the use of Mfor portfolio diversification purposes is dominated by F, and the demand for M,vis-à-vis S, is governed by precautionary requirements and expected-utility maximization. We thus support the opinion of Cass and Stiglitz (ibid. p. 331): ‘without stringent conditions it does not appear to be possible to derive a simple theory of the demand for money from portfolio analysis’ (p. 331). A Small Digression, Continued: W0does not appear in (17) and (18), and hence M∗ 0(and S∗ 0)aswellasB∗ 0are invariant to any changes in W0,whichsimplychange F∗ 0pari passu. This result clearly exhibits the role played by DARA in the analysis of wealth effects above. In view of the well-known analytical limitations of CARA utility (exemplified, in fact, by the result here), we will continue to work with CRRA utility.26 (b) Suppose that the increase in W0is accompanied by an increase in xsuch that the initial portfolio size, net of the expected size of the precautionary shock, is unchanged. In our example, with p𝓁=0.5, this requires that xincreases by twice the increase in W0, to 0.209. The asset-holder’s expected initial net wealth has not changed, but she confronts higher risk. In this case, we find that, notwithstanding the small absolute size of these changes, S∗ 0decreases, to such an extent that the zero lower bound on it becomes binding. The final solution then is that M∗ 0increases substantially, also to 0.209 (since with S∗ 0=0 26 Kim and Marchesiani (p. 1223) observe that assuming a CRRA specification for the utility function ‘is a standard practice in the literature.’ Obiter dictum our analysis should provide a cautionary note against uncritical adoption of specifications (such as CARA utility) which give rise to ‘neat’ closed-form solutions, in preference to CRRA utility, the analysis of which is more ‘messy’ but also more realistic. 88 —B. K. Kapur there are no transactions costs to be incurred), and with increased provision for the possible precautionary shock owing to the increase in xboth B∗ 0and F∗ 0decrease marginally. We discuss the implications of these results below. The analysis thus far has been conducted within a single- (or dual-) period optimization framework. Under simplifying assumptions, the results also hold in an infinite-horizon framework, noting that as mentioned we confine ourselves in this study to a partial-equilibrium analysis of asset demands. We assume that the assetholder receives an exogenous endowment each period, which, consistent with the data, may be growing over time, and is sufficiently impatient that she wishes to consume her entire end-of-period wealth each period. (From the definition of the stochastic discount factor, such impatience is even more pronounced if consumption is generally rising over time.) We further assume that she faces a borrowing constraint that does not permit any borrowing in any period. Finally, we assume that the rate-of-return risk and the precautionary risk are independent over time, and independent of each other. Under these specifications, it is easy to see that our one-period analysis is replicated every period, with possibly different values of beginning-of-period wealth, and of tand xeach period. If desired, the various asset rates of return can also be treated as exogenously varying over time. Replication, mutatis mutandis, of our one-period analysis may help to explain certain noteworthy broad trends over time.27 Federal Reserve data (Federal Reserve 2020) show a pronounced decline in the M1/GDP ratio in the US from about 0.275 in 1959 to about 0.09 in 2007, prior to the onset of the Global Financial Crisis. (There were upward movements in the late-1980s and mid-1990s.) Absolute levels of M1 rose throughout this period (ibid.). Basing on our various results above, our model is in principle capable of accounting for these, and concurrent developments in other asset-holdings, by suitable combinations of rises in GDP (a proxy for yearly endowments), smaller rises in x, some declines in tin accordance with the data, and, although this is not essential, some adjustments in asset returns. The first two of these changes could produce a falling M1/GDP ratio and a rising M1,28 and the 27 We are abstracting here from savings over time, implicitly assuming, as do Lucas (2000) and Lucas and Nicolini (2015), that all income is consumed. The US national savings rate has been below 10 % from the mid-1960’s till 2007, except for one year in the early 1970’s (Peterson 2020). The personal savings rate has been below 11.4 % (8.8 %) since 1980 (1990) up to 2007 (Statistica Research Department 2020), and part of these savings, e.g. those placed in Individual Retirement Accounts, are for longer-term retirement purposes. 28 There is a possible additional reason for these first two developments. In 2012 the Federal Reserve introduced the annual Diary of Consumer Payment Choice, one of the findings of which has been, ‘Historically, cash has been the most used payment instrument for small-value payments (payments under $25)’ (Cubides and O’Brien (2023),p. 11). One might in fact extend the argument to M1 as a whole, since small-value payments can also be defrayed through debit cards and ATM A Re-Consideration of Money Demand Theory —89 third of these could help to better match the generally rising behaviour of Sover time. By contrast, the framework of Belongia and Ireland (op. cit.), building on the earlier work by Lucas (2000,1980), abstracts completely from rate of return and precautionary risks, and imposes an unchanging equilibrium value of M1/GDP over time if user costs of holding money are constant. This does not appear to do justice to the complexity of the determinants and behaviour of money demand.29 Beyond the foregoing observations, formal econometric modelling of money demand over time is outside the scope of this essentially conceptual study. Such modelling would have to take into consideration changes in financial regulations, ‘the introduction of more innovative financial products’ (Kim and Marchesiani, op cit.), measurement issues, mechanism design issues (ibid.), and other factors. It is unclear as to what extent various of these considerations can simply be subsumed under the rubric of changes in our tparameter. Changes in both tand xover time would have to be identified and estimated, as also would possibly time-varying patterns of correlation, if any, between differing sources of risk. There is also a further consideration, which none of the studies cited above have taken into account: money demand is affected, not only by the average level of bond returns, but also, we have shown, by a mean-preserving increase in the variance of bond returns. Relatedly, one must be cognizant of the fact that in our setup it is not possible to vary M0independently of S0, and, again, a structural approach to financial asset demands (viewing an asset-holder’s portfolio in its entirety) is called for. Lastly, to the extent that parsimony is an advantage, the fact that our model can parsimoniously match the observed positive relationship, over a certain range, between money demand and the mean bond return gives it an edge over the extraordinarily complicated (and highly stylized) New-Monetarist-Mechanism-Design article by Kim and Marchesiani (op cit.), the only other study among those cited that can generate such a relationship.30 Our model thus, we believe, provide a valuable ‘springboard’ for further studies, both theoretical and empirical. cards. One could for convenience then model the use of M1 for this purpose as a concave deterministic function of GDP, which could easily be introduced into our model as an additional source of demand for M1, thus helping to account for the first two developments mentioned above. Unfortunately, data limitations preclude an empirical examination of this hypothesis over the 1959–2007 period. 29 Lucas (2000) himself notes (looking at the 1900–94 period), ‘The money-income ratio is essentially trendless over the entire century, although there has been a strong downward trend since World War II’ (p.249). 30 The following comment by Kim and Marchesiani (p. 1232) should, however, be acknowledged: ‘Of course, we do not claim that the mechanism is the only determinant of the observed money demand behavior after the 1990s. We only argue that the mechanism, in the form of MI (Market Intelligence), could have, together with other factors studied in the literature ... explained part of the behavior of the money demand. Our paper complements this literature.’ 90 —B. K. Kapur 5 Conclusions Why is a four-asset framework more suitable for the study of money demand, and demand for other assets, than a twoor threeasset framework? There are two independent sources of risk in the model – a longer-term portfolio or rate-of-return risk, and a shorter-term transactions risk. With four assets, two of them can be ‘assigned’ to deal with each source of risk, in the specific senses that Band Fare not deployed to handle transactions risk, while there would be no demand for M and Sif xwere 0. By contrast, if there were for example only three assets, one of them, namely S, would have to help satisfy both portfolio diversification and transactions motives. As such, one would expect the demand for Sto go down by less (and hence the demand for Mto go up by less) when tgoes up, since Sis also held for portfolio diversification purposes. Asset-holders are in reality aware of the benefits of optimizing across four assets, as evidenced by positive real-world holdings of all these assets, and if one instead works with a threeor twoasset framework, one is likely to obtain an inaccurate characterization of asset demands. Quite remarkably, even though the demands for Mand Sarise only when x is positive, the optimal allocation between M and Sdoes depend on characteristics of long-term assets, such as the rate of return on bonds and the variance of bond returns, as we have demonstrated. Similarly characteristics of short-term assets such as the unit transactions cost tdo affect the demands for long-term assets. Changes in the asset-holder’s wealth also affects the demands for various assets in unexpected ways. These results are due to the non-separability property of risk-averse expected utility of terminal wealth under DARA, explained earlier. The juxtaposition of this property with a four-asset framework thus yields novel and important results and insights. In an intertemporal context, our model is potentially capable of matching the pronounced decline in the M1/GDP ratio over the 1959–2007 period, which various other works cited above do not. There is, finally, a need to extend the analysis to a dynamic, stochastic general-equilibrium framework, without imposing simplifying assumptions that yield, often counterfactually, an unchanging equilibrium M1/GDP ratio over time. Acknowledgments: I would like to thank two anonymous referees for their invaluable comments, John K H Quah and Denis Tkachenko for enlightening discussions, Ren Jie, Lee Ming Xuan, and Wang Yu Han for superb research assistance, and the Department of Economics, National University of Singapore, for financial support. The usual disclaimer applies. Competing interests: None declared. A Re-Consideration of Money Demand Theory —91 References Alvarez, F., and F. Lippi. 2017. “Cash Burns: An Inventory Model with a Cash-Credit Choice.” Journal of Monetary Economics 90: 99−112.. Alvarez, F., F. Lippi, and R. Robatto. 2019. “Cost of Inflation in Inventory Theoretical Models.” Review of Economic Dynamics 32: 206−26.. Bankrate. 2020. “Best Money Market Account.” https://www.bankrate.com/banking/money-market/ rates/. June 21. Barba, M. 2023. “Money Market Funds vs. Money Market Accounts.” https://www.money-rates.com/ basicguides/money-market-account-vs-money-market-fund.htm. Aug 16. Baumol, W. J. 1952. “The Transactions Demand for Cash: An Inventory Theoretic Approach.” Quarterly Journal of Economics LXVI: 545−56.. Belongia, M. T., and P. N. Ireland. 2019. “The Demand for Divisia Money: Theory and Evidence.” Journal of Macroeconomics 61: 1−24.. Benati, L., R. E. Lucas, J. P. Nicolini, and W. Weber. 2021. “International Evidence on Long-Run Money Demand.” Journal of Monetary Economics 117: 43−63.. Berentsen, A., S. Huber, and A. Marchesiani. 2015. “Financial Innovations, Money Demand, and the Welfare Cost of Inflation.” Journal of Money, Credit, and Banking 47: 223−61.. Berentsen, A., S. Huber, and A. Marchesiani. 2018. “Limited Commitment and the Demand for Money.” Economic Journal 128: 1128−56.. Cass, D., and J. E. Stiglitz. 1972. “Risk Aversion and Wealth Effects on Portfolios with Many Assets.” The Review of Economic Studies 39: 331−54.. Chang, W. W., D. Hamberg, and J. Hirata. 1983. “Liquidity Preference as Behaviour toward Risk Is a Demand for Short-Term Securities −Not Money.” The American Economic Review 73: 420−7. Christiano, L., M. Eichenbaum, and S. Rebelo. 2011. “When Is the Government Spending Multiplier Large?” Journal of Political Economy 119: 78−121.. Cubides, E., and S. O’Brien. 2023. 2023 Findings from the Diary of Consumer Payment Choice.San Francisco: FedCash Services, Federal Reserve System, May. Edmondson, B. 2021. “What is a NOW Account?” The Balance.https://www.thebalance.com/what-is-anow-account-4582920.Jan31. Fagereng, A., L. Guiso, and L. Pistaferri. 2018. “Portfolio Choices, Firm Shocks, and Uninsurable Wage Risk.” The Review of Economic Studies 85: 437−74. Federal Reserve. 2020. Diary of Consumer Payment Choice, Annual Issues Since 2012.StLouis:Federal Reserve. Federal Reserve Bank of St Louis. M1/GDP Time Series 1959-2020. https://fred.stlouisfed.org/graph/?g=dQQ. Federal Reserve Bank of St Louis. M1 Time Series 1959-2020.https://fred.stlouisfed.org/series/M1SL. Fontinelle, A. 2020. “What Are the Withdrawal Limits for Savings Accounts?” https://www .investopedia.com/faq-what-are-the-withdrawal-limits-for-savings-accounts-4589981.Sep30 Freeman, S., and G. Huffman. 1991. “Inside Money, Output, and Causality.” International Economic Review 32: 645−67.. Freeman, S., and F. E. Kydland. 2000. “Monetary Aggregates and Output.” The American Economic Review 90: 1125−35.. Frenkel, J. A., and B. Jovanovic. 1980. “On Transactions and Precautionary Demand for Money.” Quarterly Journal of Economics 95: 25−43.. 92 —B. K. Kapur Gandelman, N., and R. Hernandez-Murillo. 2014. “Risk Aversion at the Country Level.” Federal Reserve Bank of St Louis Working Paper 2014-005B. October. Hadar, J., and T. K. Seo. 1990. “The Effects of Shifts in a Return Distribution on Optimal Portfolios.” International Economic Review 31: 721−36.. Kagan, J. 2020a. “Linked Savings Account.” https://www.investopedia.com/terms/l/linked-savingsaccount.asp. Feb 11. Karl, S. 2020b. “Certificate of Deposit.” https://www.investopedia.com/terms/c/certificateofdeposit .asp. April 30. Karl, S. 2020c. “How to Open a High-Yield Savings Account.” https://www.investopedia.com/how-toopen-a-high-yield-savings-account-4770631.Nov3. Karl, S. 2020d. “Best CD Rates.” (Updated Periodically), https://www.investopedia.com/best-cd-rates4770214. June 30. Kim, S. T., and A. Marchesiani. 2024. “Market Intelligence Gathering, Asymmetric Information, and the Instability of Money Demand.” Economic Inquiry 62: 1216−45.. Lacker, J. 1988. “Inside Money and Real Output.” Economics Letters 28: 9−14.. Lucas, R. E. 1980. “Equilibrium in a Pure Currency Economy.” In Models of Monetary Economies, edited by J. H. Kareken, and N. Wallace, 131−45. Minneapolis: Federal Reserve Bank of Minneapolis. Lucas, R. E. 2000. “Inflation and Welfare.” Econometrica 68: 247−74.. Lucas, R. E., and J. P. Nicolini. 2015. “On the Stability of Money Demand.” Journal of Monetary Economics 73: 48−65.. MathWorks. 2024a. “MultiStart.” https://www.mathworks.com/help/gads/multistart.html. MathWorks. 2024b. “Portfolio Selection and Risk Aversion.” https://www.mathworks.com/help/ finance/portfolio-selection-and-risk-aversion.html. Panis, C. W. A., and M. J. Brien. 2016. Financial Asset Holdings of Households in the United States in 2013. Advanced Analytical Consulting Group and Deloitte. Unpublished Report, Jan 21. Parkhomenko, A. 2017. Grid Search in MATLAB. Spring. https://www.andrii-parkhomenko.net/files/Grid %20Search%20in%20Matlab.pdf. Peterson, M. 2020. The US National Saving Rate in Historical Perspective. Peter G Peterson Foundation. https://www.pgpf.org/chart-archive/0078_Savings-Rate.March9. Roulleau-Pasdeloup, J. 2018. “The Government Spending Multiplier in a (Mis-)Managed Liquidity Trap.” Journal of Money, Credit, and Banking 50: 293−315.. Srinivas, V. 2019. How have US Household Balance Sheets Changed since the Financial Crisis? Deloitte. https://www2.deloitte.com/us/en/pages/financial-services/articles/how-us-householdbalance-sheets-changed-since-the-financial-crises.html, Nov 11. Statistica Research Department. 2020. Personal Saving Rate in the United States from 1960 to 2019. https://www.statista.com/statistics/246234/personal-savings-rate-in-the-united-states/. Nov 23. Tobin, J. 1956. “The Interest-Elasticity of Transactions Demand for Cash.” The Review of Economics and Statistics 38: 241−7.. Tobin, J. 1958. “Liquidity Preference as Behaviour towards Risk.” The Review of Economic Studies 25: 65−86..