The welfare costs of inflation reconsidered
Abstract
EconStor is a publication server for scholarly economic literature, provided as a non-commercial public service by the ZBW.
Full text
Benati, Luca; Nicolini, Juan Pablo Working Paper The welfare costs of inflation reconsidered Discussion Papers, No. 24-08 Provided in Cooperation with: Department of Economics, University of Bern Suggested Citation: Benati, Luca; Nicolini, Juan Pablo (2024) : The welfare costs of inflation reconsidered, Discussion Papers, No. 24-08, University of Bern, Department of Economics, Bern This Version is available at: https://hdl.handle.net/10419/304423 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/
Faculty of Business, Economics and Social Sciences Department of Economics The Welfare Costs of Inflation Reconsidered Luca Benati, Juan-Pablo Nicolini 24-08 September, 2024 Schanzeneckstrasse 1 CH-3012 Bern, Switzerland http://www.vwi.unibe.ch DISCUSSION PAPERS
The Welfare Costs of Inflation Reconsidered∗ Luca Benati University of Bern† Juan-Pablo Nicolini Federal Reserve Bank of Minneapolis and Universidad Di Tella‡ Abstract We revisit the estimation of the welfare costs of inflation originating from lack of liquidity satiation for 11 low-inflation and 5 high-inflation countries, and for Weimar Republic’s hyperinflation. Our evidence suggests that, contrary to the implicit assumption in much of the literature, these costs are far from negligible. For the U.S. our point estimates are equal to about one-third of those computed by Lucas (2000), and an order of magnitude larger than those obtained by Ireland (2009). Crucially, the most empirically plausible moneydemand functional form points towards sizeable ‘upward risks’ for these costs, with the 90% confidence interval associated with a 4% nominal interest rate stretching beyond 0.5 per cent of GDP. The welfare costs of inflation in the Euro area are about twice as large as in the U.S., thus suggesting that, ceteris paribus,theinflation target should be materially lower. At the peak of the inflation episodes, welfare costs had ranged between 0.3 and 1.9 per cent of GDP for low-inflation countries; between 4 and nearly 7 per cent for highinflation ones; and between 26 and 36 per cent for Weimar’s hyperinflation. ∗We wish to thank participants to the EEA 2024 meeting in Rotterdam for useful comments and discussions. The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of Minneapolis, or of the Federal Reserve System. †Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001, Bern, Switzerland. Email: luca.b[email protected]e.ch ‡Federal Reserve Bank of Minneapolis, 90 Hennepin Avenue, Minneapolis, MN 55401, United States. Email: [email protected] 1
1 Introduction The welfare costs of inflation originating from lack of liquidity satiation–as discussed in the classic work of Bailey (1956), Friedman (1969), Lucas (2000), and Ireland (2009)–tend to be consistently disregarded in the current policy debate. A case in point is the proposal to increase inflation targets in order to decrease the probability that the Zero Lower Bound (ZLB) on monetary policy rates may become binding. Following Blanchard, Dell’Ariccia and Mauro (2010), a vast literature has explored the costs of a binding ZLB on monetary policy rates within cashless economies. Although this literature has documented the benefits that increasing the inflation target delivers to society, such increase also comes at a cost, originating from moving the economy further away from liquidity satiation. A likely reason for the literature’s disregard of these costs is the widespread belief in the absence of stable money-demand relationships, which are a necessary condition for the computation of such costs. Following Goldfeld (1973, 1976), who first documented the (alleged) instability of the U.S. demand for M1, a large literature has confirmed his findings, and it has produced qualitatively the same evidence for different countries. The recent work of Lucas and Nicolini (2015) for the U.S., and Benati, Lucas, Nicolini, and Weber (2021) for 38 countries since World War I, has shown however that the notion of instability of the long-run demand for M1 is, in fact, incorrect. On the contrary, the existence of a stable long-run demand for M1 appears to be one of the most robust stylized facts in the entire field of macroeconomics. As discussed by Benati et al. (2021), and as we discuss below in detail, crucial issues in being able to identify a stable long-run demand for M1 are ()imposingunitary income elasticity, which is implied by theory and it is in fact supported by the data; and () adopting the correct functional form for the demand for real money balances. A second reason for the literature’s consistent disregard of the welfare costs of inflation originating from lack of liquidity satiation is the (by now) long-standing tradition of evaluating monetary policies within cashless economies, with monetary aggregates playing no role whatsoever. As we show below, however, there is no sense in which modern advanced economies are becoming cashless. For countries such as the U.S., the U.K., Sweden, and Switzerland, or for the Euro area, the ratio of M1 to GDP has fluctuated in recent years between 40 and 100 per cent. As we show, this implies that for empirically plausible parameterizations of the demand for money balances, the point estimates of the welfare costs of inflation are in general nonnegligible. Further, we show that once taking into account of statistical uncertainty, in several cases (notably, the U.S.) the most plausible functional form for the demand for real M1 balances points towards non-negligible ‘upward risks’ for the welfare costs of inflation, in the sense that (e.g.) 90 per cent confidence intervals for these costs include values that are definitely non-negligible, and in fact often sizeable. Afinal likely reason for the literature’s disregard for these costs is that they are often thought to be negligible. For example, Feldstein (1997, p. 145), in his review of 2
the costs of inflation, characterized the welfare costs originating from lack of liquidity satiation as ‘small relative to the other effects that have been discussed in this paper’. As we show in this work, this presumption is in general incorrect. The number of studies presenting empirical estimates of the welfare costs of inflation originating from lack of liquidity satiation is quite surprisingly limited. Further, previous studies are sub-optimal along several dimensions. First, several of them (first and foremost Lucas, 2000) are based on calibrated, as opposed to estimated models. Second, previous estimates are uniformly based on functional forms for the demand for real money balances that Benati et al. (2021) have shown to be empirically highly implausible.1As we show, in general the money-demand specification does matter for the computations of welfare costs along several dimensions. Third, to the best of our knowledge the only previous analysis of the welfare costs for hyperinflations is Barro (1972). Last, but not least, the period following the outbreak of the financial crisis has provided previously unavailable information on the behavior of money demand at very low interest rates, which as discussed by Lucas (2000) should play a crucial role in the determination of the welfare costs of inflation. In principle this should allow us to obtain better and more precise estimates of these costs. In this paper we revisit the estimation of the welfare costs of inflation for eleven low-inflation and five high-inflation countries, and for Weimar Republic’s hyperinflation. In our analysis we follow the tradition of considering the most liquid monetary assets, which include cash and transactional deposits. We abstract from a detailed discussion of the demand for each of the components, an issue recently addressed by Kurlat (2019).2 Our evidence suggests that, contrary to the explicit or implicit assumption in much of the literature, these costs are often far from negligible. For the U.S. our point estimates are equal to about one-third of those computed by Lucas (2000), and an order of magnitude larger than those obtained by Ireland (2009).3Crucially, the most empirically plausible money-demand functional form points towards sizeable ‘upward risks’ for these costs, with the 90% confidence interval associated with a 4% nominal interest rate stretching beyond half a percentage point of GDP. At the peak of the inflation episodes, welfare costs had ranged between 0.3 and 1.9 per cent of GDP for low-inflation countries; between 4 and nearly 7 per cent for high-inflation ones; and between 26 and 36 per cent for Weimar’s hyperinflation. Our evidence suggests that ignoring money in analyzing optimal monetary policies can be seriously misleading. For instance, Coibion, Gorodnichenko and Wieland 1Notable cases in point are Lucas (2000) and Ireland (2009). 2He shows that addressing these considerations in a model with imperfect competition substantially increases the estimates of the welfare cost, relative to models that ignore the creation of inside money. 3For a steady-state interest rate of 5 per cent, Lucas (2000) computed the cost to be around 1.1 per cent of lifetime consumption. Ireland (2009) challenged Lucas’ interpretation of the data, and estimated a mere 0.04 per cent of lifetime consumption. 3
(2012)4make a compelling argument against increasing the inflation target in countries like the U.S., based on a model with frictions in price-setting and with recurrent, though not very frequent, episodes with the nominal interest rate at the ZLB. Based on their preferred specification they compute the welfare effect of an interest rate of 5 per cent to be close to 0.6 per cent of lifetime consumption. Such an estimate, combining the cost created by price frictions and the probability to be at the ZLB, is of the same order of magnitude of the estimates we obtain for the U.S., and it is in fact slightly smaller than the upper bound of our 90 per cent confidence interval. In any of the low-inflation countries the demand for M1 as a fraction of GDP at very low, or even negative interest rates has (so far) exhibited no obvious difference compared to its behavior at higher interest rates. These results contrast with those of Mulligan and Sala-i-Martin (2000), who, based on U.S. households micro data, provided evidence that money demand becomes comparatively flatter at low interest rates. We provide a straightforward explanation for Mulligan and Sala-iMartin’s (2000) finding, by showing mathematically that if the true money demand specification is the one that, as shown by Benati et al. (2021), is the most empirically plausible at low inflation rates, Mulligan and Sala-i-Martin’s (2000) approach automatically produces spurious evidence of a flatter demand curve at low interest rates. From a theoretical standpoint, following Alvarez, Lippi and Robatto (2019) we construct upper and lower bounds for the welfare costs of inflation. As they show, the areaunderthemoneydemandcurveisanalmostexactmeasureofthewelfarecost for a very general class of monetary models in the neighborhood of zero. We extend their results for a quite general sub-class of the models they analyze and compute exact lower and upper bounds for the costs, using the area under the money demand curve, for any value of the interest rate. As we show, the difference between the upper and the lower bound is extremely small for the range of interest rates ever observed in low-inflation countries such as the U.S. For policy purposes, our main finding is that the welfare costs inflation in the Euro area are about twice as large as in the U.S.. This suggests that, ceteris paribus, the inflation target should be materially lower. The paper proceeds as follows. In Section 2 we discuss a family of monetary models for which we derive very tight lower and upper bounds for the welfare cost of inflation using the area under the real money demand curve. In Section 3 we discuss the data and several figures that, in our view, present very solid evidence in favor of stable money demand relationships for the countries we analyze. Section 4 makes formally this statement by analyzing unit root and cointegration properties of the series. Section 5 presents our computations for the welfare cost functions. For each interest rate level, we compute the bootstrapped distribution of the welfare costs (and therefore median estimates, and confidence intervals) expressed in percentage points 4Coibion et al. (2012) explicitly acknowledge that they do not take into account the costs derived from lack of money satiation. 4
of GDP. Section 6 concludes. 2TheModel We study a labor-only economy with uncertainty in which making transactions is costly.5The economy is inhabited by a unit mass of identical agents with preferences given by 0 ∞ X =0 ()(1) where is differentiable, increasing and concave. Every period, the representative agent chooses a number of portfolio transactions that allow her to exchange interest-bearing illiquid assets for money, that is needed to buy the consumption good. The total cost of those transactions, measured in units of times, is given by a function ()where is an exogenous stochastic process. This formulation generalizes the linear function assumed by Baumol (1952) and Tobin (1956). The production technology for the consumption good is given by == where is time devoted to the production of the final consumption good and is an exogenous stochastic process. The representative agent is endowed, in each period, with a unit of time that is used to produce goods and to make transactions. Thus, equilibrium in the labor market implies that 1=+() and feasibility is given by =(1 −()) It follows that the real wage is equal to . Purchases are subject to a cash in advance constraint ≤(2) where are average money balances and is the number of portfolio adjustments within each period. The variable is the only economically relevant decision to be made by the representative agent. We allow for money to pay a nominal return that we denominate which in what follows, in line with the literature, we will set to zero. 5The baseline model is discussed at length in Benati et. al. (2020). 5
At the beginning of each period, the agent starts with nominal wealth that can be allocated to money or interest bearing bonds, soarestrictiontotheoptimal problem of the agent is +≤(3) Nominal wealth at the beginning of next period, in state +1will then be given by +1 ≤(1 + )+(1 + )+(4) +[1−()] − where is the return on government bonds and is a transfer made by the monetary authority. Notice that the unconstrained efficient outcome is to allocate all the labor input to the production of the consumption good so as to set = Thus, a measure of the welfare cost of making transactions, as a fraction of consumption, is given by the value of ()in equilibrium. In the Online Appendix 1, we show that as long as the cost function ()is differentiable, an interior solution for must satisfy 2 () (1 −()) = − (5) We also show that as long as − 0the cash in advance is binding, which implies that =1 (6) so real money demand, as a proportion of output, is equal to the inverse of Note that equation (5) is independent of . Thus, secular increases in productivity do not affect the optimal solution for so the theory implies a unit income elasticity of real money demand. Note that the solution for and therefore the solution for real money demand, depends on the interest rate differential between bonds and money. As mentioned above, since we assume that =0, real money demand only depends on the interest rate on bonds. For further references, we let the interest rate differential between bonds and money to be ≡ − For the maximum problem of the agent to be well defined, it has to be the case that = − ≥0(7a) which is the well-known lower bound on the interest rates in bonds.6The popular zero-bound restriction on policy rates is obtained from (7)plus the standard assumption in the literature that =0The analyses of both Lucas (2000) and Ireland (2009) are done under this standard assumption. 6Intuitively, where ()−()to be negative, the representative agent would have incentives to borrow from the government unbounded quantities and hold money. 6
2.1 The functional form for the demand for real money balances The functional form of the real money demand function depends on the functional form of the transactions technology (), and at this level of generality the model is consistent with many different possibilities. In what follows, and to clarify the main difference between Lucas (2000) and Ireland (2009), we consider three wellknownfunctionalformsthathavebeenusedinpreviousempiricalwork. Allofthe three functional forms exhibit a unit income elasticity, as implied by the model. The first specification is the log-log one, ln =1−ln +1 (8) that exhibits a constant interest rate elasticity equal to .Noticethatas→0real money demand goes to infinity. It is this asymptote at zero that Lucas used to argue that the welfare cost of inflation is sizeable, even at low values for the interest rate. The other two formulations that we explore, the semi-log ln =2−+2 (9) that exhibits a constant semi-elasticity, , and the Selden-Latané =1 3++3 (10) both imply a finite level of the demand for real money balances when the interest rate becomes zero. This feature is emphasized by Ireland, who uses (9) in his revision of Lucas’s estimate. By exploiting recent data that include, for a few countries, several years of very low (or even negative) interest rates, we can provide a sharper comparison of the empirical performance of the three alternative functional forms. As we show below, the welfare costs implications of the last two functional forms are similar. We do however choose to include the Selden-Latané specification, together with the others since it does have an overall better performance than the other two, as our econometric analysis shows.7 In the next Section we show how to build tight upper and lower bounds for the welfare cost of inflation, using the area under the estimated real money demand function. 2.2 The welfare costs of inflation and the area under the money demand curve In this section we apply the techniques developed in Alvarez, Lippi and Robatto (2019) to a class of models that is more restrictive than the ones they used. Specifi7This is in line with the evidence in Benati, Lucas, Nicolini, and Weber (2021). 7
Table 1 Results from Wright’s tests: 90% bootstrapped confidence interval for the second element of the normalized cointegration vector, based on systems for (log) M1 velocity and (the log of) a short-term rate Money demand specification: SeldenCountry Period Latané Semi-log Log-log Low-inflation countries: United States 1959Q1-2019Q4 [-0.587 -0.343] [-0.148 -0.068] [-0.347 -0.067] 1959Q1-2023Q2 [-0.694 -0.438] [-0.175 -0.103] [-0.529 -0.181] United Kingdom 1955Q1-2023Q2 [-0.564 -0.371] [-0.115 -0.079] [-0.393 -0.196] Canada 1947Q3-2006Q4 [-0.713 -0.543] NCD [-0.512 -0.440] 1967Q1-2023Q1 [-0.634 -0.397] [-0.119 -0.035] [-0.409 -0.245] Australia 1969Q3-2023Q1 [-0.858 -0.764] [-0.181 -0.045] [-1.275 -0.974] Switzerland 1972Q1-2023Q1 [-0.457 -0.336] [-0.233 -0.128] — Sweden 1998Q1-2023Q1 [-0.614 -0.299] [-0.153 -0.121] — Euro area 1999Q1-2023Q1 [-0.586 -0.409] [-0.217 -0.165] — Denmark 1991Q1-2023Q1 [-0.373 -0.187] [-0.139 -0.043] — South Korea 1964Q1-2023Q1 [-0.579 -0.513] [-0.148 0.027] NCD Japan 1960Q1-2023Q2 [-0.452 -0.366] [-0.317 -0.033] [-0.610 -0.122] Hong Kong 1985Q1-2023Q2 [-1.128 -0.769] [-0.257 -0.100] [-0.495 -0.091] High-inflation countries: Bolivia 1980-2019 — — [-0.696 -0.260] Chile 1946-2019 — — [-0.443 -0.095] Ecuador 1980-2019 — — [-1.217 -1.021] Israel 1982Q1-2019Q4 — — [-0.428 -0.392] Mexico 1982Q1-2019Q4 — — [-0.621 -0.325] Hyperinflations: Weimar Republic Sep. 1920-Oct. 1923 — — [-0.526 -0.349] Based on 10,000 bootstrap replications. NCD = No cointegration detected. The last observations for the interest rate are either zero or negative.
Benati (2024) shows that for 20 hyperinflations evidence in favor of the log-log is overwhelming. As for low-inflation countries, the evidence in Benati et al. (2021) suggests that the Selden-Latané specification is the most plausible one, but the evidence is less clear-cut. For this group of countries we therefore now turn to a systematic model comparison exercise. 4.2 Which specification do the data prefer? Since it is not possible to nest the three money demand specifications into a single encompassing one, we proceed as follows. We start from the comparison between the semi-log and the log-log. Intuitively, the comparison between (9)and(8)boils down to whether the dynamics of log M1 balances as a fraction of GDP (i.e., minus log velocity) is better explained by the level of the short rate, or by its logarithm. For low-inflation each country we therefore regress ln ()on a constant, lags of itself, and lags of either the level of the short rate or its logarithm. A natural way of interpreting these regressions is the following. Under the assumption that cointegration is indeed there for all countries,15 and based on either specification, both =[ln()]0and =[ln()ln()]0have a cointegrated VECM(- 1) representation, which maps into a restricted VAR() representation in levels (where the restrictions originate from the cointegration relationship). The equations we are estimating can therefore be thought of as the corresponding unrestricted form of the equations for ln ()in the VAR() representation in levels for either or . It is important to stress that the two specifications we are estimating are in fact nested: the easiest way of seeing this is to think of them as two polar cases– corresponding to either =1or =0–in the following representation based on the Box-Cox transformation of : ln µ ¶=+ X =1 ln µ− −¶+ X =1 Ã −−1 !+(14) We estimate (14) via maximum likelihood, stochastically mapping the likelihood surface via Random-Walk Metropolis (RWM). The only difference between the ‘standard’ RWM algorithm which is routinely used for Bayesian estimation and what we are doing here is that the jump to the new position in the Markov chain is accepted or rejected based on a rule which does not involve any Bayesian priors, as it uniquely involves the likelihood of the data.16 So one way of thinking of this is as Bayesian estimation via RWM with completely uninformative priors, so that the log-posterior collapses to the log-likelihood of the data. All of the other estimation details are identical to Benati (2008), to which the reader is referred to. 15If this assumption did not hold, the entire model comparison exercise would obviously be meaningless. 16So, to be clear, the proposal draw for the parameter vector ,˜ , is accepted with probability min[1, (−1,˜ |,)], and rejected otherwise, where −1is the current position in the Markov 12
Table 2aModel comparison exercise, semi-log versus log-log: mode of the log-likelihood in regressions of log velocity on lags of itself and either the short rate or its logarithm p=2 p=4 p=8 SemiLog SemiLogSemiLogCountry Period log log log log log log United States 1959Q1-2023Q2 766.1394 756.6280 763.2818 751.3266 765.1439 740.3543 United Kingdom 1955Q1-2023Q2 879.6821 877.9350 898.6224 893.7504 892.1970 887.1920 Canada 1947Q3-2006Q4 820.2401 807.8379 813.8001 804.9218 802.7001 794.7403 1967Q1-2023Q1 775.0890 767.0845 775.9595 766.4531 771.9264 766.1943 Australia 1969Q3-2023Q1 650.7331 656.0624 649.9510 655.1057 642.6046 650.3903 South Korea 1964Q1-2023Q1 630.9515 633.8825 628.2222 634.6372 623.8333 628.0991 Japan 1960Q1-2023Q2 845.3632 850.7677 841.5156 848.6520 832.2577 840.2434 Hong Kong 1985Q1-2023Q2 328.0148 325.5701 326.1339 324.9236 319.8478 325.2641 For Switzerland, Sweden, Euro area, and Denmark there is no comparison because the last observations for the short rate are negative. Table 2bModel comparison exercise, Selden-Latané versus semi-log: mode of the log-likelihood in regressions of the short rate on lags of itself and either velocity or its logarithm p=2 p=4 p=8 SeldenSemiSeldenSemiSeldenSemiCountry Period Latané log Latané log Latané log United States 1959Q1-2023Q2 -22.9102 -24.0809 -5.8335 -7.3440 12.9347 10.3522 United Kingdom 1955Q1-2023Q2 -85.7350 -84.1422 -85.4391 -83.9044 -83.6446 -82.1970 Canada 1947Q3-2006Q4 -72.0532 -71.7812 -64.4770 -66.2576 -62.2194 -64.2760 1967Q1-2023Q1 -65.0057 -65.9778 -56.1253 -59.0260 -50.7916 -53.9112 Australia 1969Q3-2019Q4 -136.4591 -137.1389 -132.5116 -133.5144 -116.7487 -118.2407 Switzerland 1972Q1-2023Q1 -45.6989 -45.8396 -39.9744 -40.8984 -20.5636 -22.4888 Sweden 1998Q1-2023Q1 65.5876 65.4372 66.9821 66.9083 70.5126 68.9721 Euro area 1999Q1-2023Q1 63.8008 64.3157 64.5967 65.3372 74.7778 75.5777 Denmark 1991Q1-2023Q1 50.9544 50.7969 60.7088 60.1085 65.6600 64.4409 South Korea 1964Q1-2023Q1 -131.5950 -135.8924 -118.2770 -131.1253 -86.1317 -93.5032 Japan 1960Q1-2023Q2 -141.5147 -141.6026 -140.6631 -140.7865 -129.5219 -130.1270 Hong Kong 1985Q1-2023Q2 -65.6601 -65.7389 -60.9537 -61.4880 -50.8999 -51.6665 For Switzerland, Sweden, Euro area, and Denmark there is no comparison because the last observations for the short rate are negative.
Table 2reports, for either specification, and for ∈{248}themodeofthe log-likelihood. The main result in the table is that whereas the semi-log appears as the preferred functional form for the U.S. the U.K., Canada, and Hong Kong, the log-log produces a larger value of the likelihood for Australia, South Korea, and Japan, so that neither of the two specifications clearly dominates the other one.17 Turningtothecomparisonbetweenthesemi-log and the Selden-Latané, we adopt the same logic as before, but this time we ‘flip’ the specifications for velocity on their head, by regressing the interest rate on lags of itself and of either the level or the logarithm of velocity. Once again, these two regressions can be thought of as particular cases of the nested regression =+ X =1 −+ X =1 ⎡ ⎢ ⎣³− −´ −1 ⎤ ⎥ ⎦+(15) with either =1(corresponding to Selden-Latané) or =0(corresponding to the semi-log). At first sight this approach might appear as questionable: since we are here dealing with the demand for real M1 balances for a given level of the short-term nominal interest rate, why would it make sense to regress the short rate on M1 velocity? In fact, this approach is perfectly legitimate, for the following reason. As shown by Benati (2020), M1 velocity is, to a first approximation (and up to a scale factor), the permanent component of the short-term rate,18 so that focusing (e.g.) on the SeldenLatané specification, =+ ,whereis velocity, 0are coefficients, and is the unit-root component of the short rate (), with = + ,and being the transitory component.19 This can be seen quite clearly in Figure 2 for Australia, Canada, the Euro area, Hong Kong, Sweden, Switzerland, and the U.K.. chain, and (−1˜ |)= (˜ |) (−1|) which uniquely involves the likelihood. With Bayesian priors it would be (−1˜ |)= (˜ |)(˜ ) (−1|)(−1) where (·)would encodes the priors about . 17This crucially hinges on the fact that we are here exclusively focusing on low-inflation countries. As we discuss in Section ??, for high-inflation countries, and especially hyperinflationary episodes, the data’s preference for the log-log is overwhelming. 18This expresses in the language of time-series analysis Lucas’ (1988) point that real M1 balances are very smooth compared to the short rate. 19A simple rationalization of this fact is provided by a ‘preferred habitat’ model (see Modigliani and Sutch, 1966, and Vayanos and Vila, 2021) in which ‘long’ investors such as pension funds play an important role in money demand. The intuition is that whereas permanent shocks to the short rate shift the entire term structure of interest rates, and therefore affect the demand for M1 coming 13
Regressing on therefore amounts to regressing the short rate on its (rescaled) stochastic trend, i.e. the dominant driver of its long-horizon variation, and it is therefore conceptually akin to (e.g.) regressing GDP on consumption.20 TheresultsarereportedinTable2. The evidence is much sharper than for the comparison between the semi-log and the log-log: in particular, for equal to either 4 or 8 the Selden-Latané specification is preferred to the semi log for all countries except the United Kingdom and the Euro area. Summing up, whereas the Selden-Latané functional form appears to be quite clearly preferred to the semi-log, the semi-log and the log-log seem to be, from an empirical standpoint, on a roughly equal footing. We draw two main conclusions from the evidence so far. First, in line with the evidence in Figures 1 and 2, the data provide substantial support to the existence of a stable long-run demand for M1, as predicted by the theory. Second, for lowinflation countries the Selden-Latané specification appears to exhibit the best overall performance among the three. Based on this, for low-inflation countries we choose to use the Selden-Latane as our benchmark functional form. But we will also provide estimates for the other two specifications. 4.3 Exploring stability and non-linearities A main concern in working with estimated money demand curves pertains to the stability of the long-run relationship over time. As previously mentioned, even without the econometric evidence produced (e.g.) by Friedman and Kuttner (1992), the simple visual evidence had been sufficient to discredit, long ago, any notion of stability of the U.S. demand for real M1 balances. As our results make clear, the solution proposed by Lucas and Nicolini (2015) has re-established stability of the U.S. demand for M1. However, since for all of the other countries in our dataset we work with the ‘standard’ M1 aggregate, it is a legitimate question whether for (some of) these countries, too, some adjustment to the standard aggregate might be required in order to obtain stability of the long-run demand for M1. 4.3.1 Testing for stability in the cointegration vector Table 3 reports evidence from Hansen and Johansen’s (1999) tests for stability in the cointegration vector21 for our dataset, based on any of the three money demand from all investors, transitory shocks only impact the short end of the yield curve, and therefore have a much smaller (and in the limit negligible) effect. 20See Cochrane (1994) on consumption being the permanent component of GDP. 21On the other hand, we do not test for stability of the loading coefficients, since they pertain to the short-term adjustment dynamics of the system towards its long-run equilibrium, and they are therefore irrelevant for the purpose of computing the welfare costs of inflationinthesteady-state. Finally, we eschew Hansen and Johansen’s (1999) fluctuation tests because, as shown by Benati et 14
specifications. Only in two instances, Denmark, and Japan based on the SeldenLatané specification, the tests detect evidence of instability.22 Table 3 Bootstrapped p-valuesfor Hansen and Johansen’s (1999) tests for stability in the cointegration vector for (log) M1 velocity and (the log of) a short-term rate Money demand specification: SeldenSemiLogCountry Period Latané log log Low-inflation countries: United States 1959Q1-2023Q2 0.5875 0.8030 0.9940 United Kingdom 1955Q1-2023Q2 0.5905 0.5480 0.9365 Canada 1947Q3-2006Q4 0.3535 0.6710 0.6910 1967Q1-2023Q1 0.6900 0.7945 0.6070 Australia 1969Q3-2023Q1 0.7835 0.7880 0.6950 Switzerland 1980Q1-2023Q1 0.6378 0.8102 — Sweden 1998Q1-2023Q1 0.2335 0.1690 — Euro area 1999Q1-2023Q1 0.4880 0.2915 — Denmark 1991Q1-2023Q1 0.0085 0.2605 — South Korea 1964Q1-2023Q1 0.1460 0.5835 0.4485 Japan 1960Q1-2023Q2 0.0030 0.2600 0.4030 Hong Kong 1985Q1-2023Q2 0.5280 0.4510 0.8465 High-inflation countries: Bolivia 1980-2019 — — 0.1020 Chile 1946-2019 — — 0.3740 Ecuador 1980-2019 — — 0.1335 Israel 1982Q1-2019Q4 — — 0.5330 Mexico 1982Q1-2019Q4 — — 0.3940 Hyperinflations: Weimar Republic Sep. 1920-Oct. 1923 — — 0.2105 Based on 10,000 bootstrap replications. Null of 0 versus 1 cointegration vectors. The last observations for the interest rate are either zero or negative. Overall, there is very little evidence of a break in the real money demand relaal. (2021) via Monte Carlo, they exhibit, overall, a significantly inferior performance compared to the tests for stability in the cointegration vector and loading coefficients. 22This is in line with the evidence in Benati et al.’s (2021) Section 6.2. The main finding there was that evidence of breaks in either the cointegration vector or the loading coefficients vector is weak to non-existent. The estimated break dates for the cointegration vector are 2008Q1 for Denmark and 1979Q4 for Japan. The second element of the normalized cointegration vector for the first and second sub-periods is equal to -0.37 and -0.66 for Denmark, and to -0.41 and -0.74 for Japan. 15
tionship derived from the theory. This is reassuring in itself, but also in reference to the issue raised by Ireland and that has prevailed the discussion in the United States, related to a structural break in this relationship somewhere between the late 70s and the early 80s. It is the assumption of such a break that justifies focusing the analysis using only the recent data. These tests show, on the one hand, that once we take into account United States specific regulatory changes, there is no break in the money demand relationship over the post-WWII period. On the other hand, they show that in other similar developed countries that did not experience regulatory changes, the high inflation episode of the late 70s and early 80s is consistent with a stable demand for real M1 balances, just based on the standard M1 monetary aggregate. 4.3.2 Are there non-linearities in money demand at low interest rates? A conceptually related issue pertains to the possibility that, at low interest rates, money demand might exhibit sizeable non-linearities, due to the presence of fixed costs associated with the decision to participate, or not to participate, in financial markets (see e.g. Mulligan and Sala-i-Martin, 2000).23 Based on this argument, at sufficiently low interest rates money demand (and therefore money velocity) should be largely unresponsive to changes in interest rates, since most (or all) households simply do not participate in financial markets. The implication is that it should not be possible to reliably estimate money demand functions (and therefore the welfare costs of inflation) based on aggregate time series data, as only the use of micro data allows to meaningfully capture the non-linearities associated with the cost of participating in financial markets. Although Hansen and Johansen’s (1999) tests detect little evidence of instability in the cointegration vector, for the specific purpose of testing whether money demand curves might be flatter at low interest rates these results should be discounted for (at least) two reasons. First, as discussed by Bai and Perron (1998, 2003), when a coefficient experiences twobreaksinoppositedirections(e.g.,first an increase, and then a decrease), break tests which have not been explicitly designed to search for multiple breaks may have ahardtimeindetectingthefirst break to begin with. Within the present context this could be relevant for three countries, the U.S., the U.K., and Canada. In any of these cases the short rate had been below 5% (which, following Mulligan and Sala-iMartin, 2000, we take as the relevant threshold) at the beginning of the sample; it then 23The intuition is straightforward. Suppose that the interest rate, , is initially equal to zero, and consider a household with nominal assets , which are entirely held in either cash or non-interestbearing deposits. Crucially, suppose that if the household wants to switch a fraction of its assets into bonds ,ithastopayafixed cost .Asincreases from zero to 0, unless the household will keep all of its wealth in either cash or deposits form, and only when the inequality is satisfied it will have an incentive to buy bonds. This implies that, under the plausible assumption that is heterogenous across the population, money demand should exhibit sizeable non-linearities (rather than a strict discountinuity) at low interest rates. 16
24 Figure 3 Informal evidence on the possible presence of non-linearities at low interest rates
25 Figure 4 M1 velocity and short-term nominal interest rates: observations with the short rate above and below 5 per cent (quarterly data)
significantly increased above 5% during the Great Inflation; and it has progressively decreased since the early 1980s. Under the assumption that money demand curves are comparatively flatter at low rates, this implies that the slope of the curve should have first increased, and then decreased, which is precisely the kind of circumstance in which these tests may have problems in detecting a break. Second, Hansen and Johansen’s (1999) are tests for breaks at unknown points in the sample. In principle, it should be possible to perform more powerful tests if we had strong reasons for choosing a specific threshold for the short rate, which, as mentioned, we take it to be 5%. Before delving into the econometric evidence, however, it is of interest to see what a simple visual inspection of the data suggests. Figure 3 shows informal evidence on the possible presence of nonlinearities for five countries for which both sub-samples with the short rate above, and respectively below 5% are sufficiently long. In order to provide sharper evidence, for four countries (the U.S., the U.K., Canada, and Australia) we consider long samples of annual data that we do not further analyze.24 The figure shows the raw data for M1 velocity and a short rate. The evidence speaks for itself, and it provides no support to the notion that velocity–and therefore money demand–may be less responsive to interest rate changes at low interest rates. The only possible exception is the U.S. until WWII. Overall, the ‘big picture’ emerging from Figure 3 suggests that the relationship between M1 velocity and the short rate is virtually the same at all interest rate levels. Although we will shortly discuss the econometric results, in fact we regard this evidence, because of its simplicity, as the strongest argument against the notion that money demand curves may be flatter at low interest rates.25 Figure 4 shows evidence based on quarterly data for the four countries with sufficiently long continuous samples with the short rate both above and below the 5% threshold. The top row shows scatterplots of M1 velocity and the short rate, with the observations with the short rate above and below the threshold being shown in black and red, respectively.26 (The sub-samples with the short rate below and above 5% are reported in Table ??.) The panels also show an horizontal red line corresponding to an extreme version of the non-linearity hypothesis, in which when the short rate falls below 5% by an arbitrarily small quantity 0, velocity becomes completely insensitive to interest rate fluctuations (and therefore perfectly flat). The reason for 24This is because, these being annual series, for all of them at least one of the sub-samples with the short rate either above or below 5% features too few observations to produce reliable results. 25This is in line with Summers’ (1991) point that the most convincing type of evidence, and the one that, historically, had the most impact in terms of changing the profession’s views, is simple evidence based on either raw data, or data that have been subjected to very simple manipulations. 26For Canada (1947Q3-2006Q4) it would seem that there is a discontinuity in the relationship between velocity and the short rate. In fact, this is not the case: rather, in order to obtain ‘clean’ samples with the short rate almost entirely below or above 5% we had to eliminate the period 1967Q41973Q1, during which the short rate fluctuated around 5%. By the same token, for the U.S. we exclude the period 1991Q4-2000Q4. 17
29 Figure 7b Comparing the estimated welfare cost functions produced by alternative specifications: point estimates of the lower and upper bounds, 5th and 16th percentiles of the lower bounds, and 84th and 95th percentiles of the upper bounds of the bootstrapped distributions
In what follows, we discuss in detail our results using the Stock and Watson DOLS estimates, and leave for the appendix the analysis with Wright’s (2000) tests, where, as ‘point estimates’, we pick the value that is most difficult to reject at the 10% level. Results based on Wright’s (2000) approach are very similar to those produced by Stock and Watson’s estimator, except for a few cases in which the estimated welfare cost are slightly higher.32 The methodology we use in order to estimate the demand for real M1 balances follows Luetkepohl (1991, pp. 370-371). Specifically, we start by estimating via OLS the cointegrating regression corresponding to any of the three specifications, i.e. to either (8), (9), or the inverse of (10).33 This gives us the point estimates of the parameters we need in order to compute the point estimates of the welfare cost functions. We then estimate the relevant VECM via OLS by imposing in estimation the previously estimated cointegration vector, and we characterize uncertainty about the point estimates of the welfare cost function by bootstrapping the VECM as in Cavaliere et al. (2012). In line with the previous discussion, this procedure is valid if the series contain exact unit roots. Under the alternative possible interpretation of the results from unit root tests, i.e. that the series are local-to-unity, we proceed as in Benati et al. (2021, Section 4.2.1). Specifically, we compute, based on the just-mentioned VECM, the corresponding VAR in levels, which by construction features one, and only one exact unit root, and we turn it into its corresponding near unit root VAR by shrinking theunitrootto=1-0.5·(1/), where isthesamplelength. 34 The bootstrapping procedure we implement for the second possible case, in which the processes feature near unit roots, is based on bootstrapping such a near unit root VAR. In short, the two bootstrapping procedures produce numerically near-identical results, and in what follows we will therefore exclusively report and discuss those based on bootstrapping the VECM (the alternative set of results is however available upon request). Figure 5 shows, for any of the eleven countries, the estimated welfare cost functions basedontheSelden-Latanéspecification, which based on the previous discussion we take as our benchmark functional form for this group of countries. We plot the point estimates of the lower and upper bounds–which, in line with the discussion in Section 2, are nearly indistinguishable–the 5th and 16th percentiles of the lower bounds, and the 84th and 95th percentiles of the upper bounds of the bootstrapped distributions. Assuming that the inflation target and the natural rate of interest are both equal to 2 per cent, the monetary policy rate will be equal, on average, to 4 per cent. In turn, by arbitrage the same will approximately hold for short-term nominal rate. On the other hand, the thought experiment considered by both Lucas (2000) and Ireland (2009) involved a steady-state nominal rate of 5 per cent. In order to be able to draw 32These cases are Japan and the U.S. for the semi-log, the Euro area for the Selden-Latané and Switzerland for the log-log. 33So, to be clear, when we work with the Selden-Latane specification we run the cointegrating regression for M1 velocity, rather than its inverse, M1 as fraction of GDP. 34For details see Benati et al.’s (2021) footnote 24. 21
a comparison with their results we therefore plot the welfare cost functions over the domain from 0 to 5 per cent for the short rate. Starting from the U.S., which has been the focus of most previous research, based on a calibrated log-log Lucas (2000) computed a welfare loss of a 5 per cent nominal rate equal to 1.1% of permanent consumption. Based on a semi-log estimated over the period 1980Q1-2006Q4, on the otehr hand, Ireland (2009) estimated a welfare loss of just 0.04 per cent of lifetime consumption. Our point estimate in Figure 5 is equal to 0.3 per cent of GDP, about one-third of the loss computed by Lucas (2000), and an order of magnitude larger than that estimated by Ireland (2009). So, even just focusing on point estimates, our evidence for the U.S. suggests that these losses are not negligible, since–it is important to recall–these losses are suffered every single year.Further,andcrucially,the90percentconfidence interval in Figure 5 stretches between 0.19 and 0.71 per cent. This points towards the existence of sizeable ‘upward risks’ potentially associated with the welfare costs of inflation, which policymakers obviously should take into account when choosing inflation objectives. Sure enough, this is not the case for all countries: e.g., Denmark, the Euro area, and especially Sweden have quite tight confidence bands. For all of the remaining countries, however, these bands are uniformly wide, or very wide: this is the case, in particular, for the U.K., Australia, and Hong Kong. A policymaker aiming at implementing robust policies would take such ‘upward risks’ into account, and ceteris paribus would set the inflation target lower than the value just implied by the point estimate. A second robust finding emerging from Figure 5 is a non-negligible extent of heterogeneity across countries. For example, focusing on point estimates, and our benchmark thought experiment of a natural rate of interest and an inflation target both equal to 2 per cent, welfare costs in the Euro area, equal to 0.4 per cent, are about twice as large as those in the U.S., thus suggesting that, ceteris paribus,the inflation target should be materially lower. More generally, point estimates range betweeen 0.07 per cent for Japan and 0.45 per cent for Australia, whereas the 95th percentiles of the bootstrapped distribution ranges betweeen 0.13 per cent and 1.3 per cent for the same two countries. Again, these are sizeable figures. Figure 6 brings this home in the starkest possible way. The figure reports the point estimates of the lower and upper bounds of these costs at each point in time along the sample, together with the 5th percentiles of the lower bounds and 95th percentiles of the upper bounds of the bootstrapped distributions. Once again, the point estimates of the lower and upper bounds are nearly indistinguishable. A first finding emerging from the figure is a dramatic extent of variation in these costs, due to a corresponding large variation in the level of short-term nominal interest rates over the sample periods. The main finding in Figure 7, however, is that if we uniquely focus on countries whose samples include the Great Inflation, at the peak of the inflation episodes point estimates of the welfare costs had ranged between 1.2 per cent of GDP for the U.S. and 1.9 per cent for Australia, whereas the 95th percentiles of the upper bounds of the bootstrapped distributions had ranged between 2.1 per 22
cent for Canada and 3.5 per cent for Australia. These numbers are very far from negligble, and in fact they are uniformly sizeable, reiterating onec again one of our main points: the assumption, explicit or implicit in the previous literature, that these costs can be ignored is unwarrated. A key implication of our results is that the common practice in the literature of ignoring money in the analyses of optimal monetary policies is equally unwarranted, and it can be seriously misleading. For Example, Coibion, Gorodnichenko and Wieland (2012)35 advance a compelling argument against increasing the inflation target in countries like the U.S., based on a model with frictions in price-setting and with recurrent, although not very frequent episodes with the nominal interest rate constrained by the ZLB. Based on their preferred specification they compute the welfare effect of an interest rate of 5 per cent to be close to 0.6 per cent of lifetime consumption. Such an estimate, combining the cost created by price frictions and the probability to be at the ZLB, is of the same order of magnitude of the estimates we obtain for the U.S., and it is in fact slightly smaller than the upper bound of our 90 per cent confidence interval in Figure 5. Finally, Figures 7-7explore whether the three functional forms for the demand for real money balances do, or do not produce materially different welfare cost functions. Overall, evidence is mixed. Starting from the U.S., at a 4 per cent short rate point estimates are equal to about 0.2 per cent for all functional forms. This shows that the conventional-wisdom notion that, ceteris paribus (e.g., for given observations for velocity and the interest rate), a log-log specification should be expected to produce comparatively higher welfare costs is, in general, incorrect. The reason for this is straightforward: Although the log-log specification does not have a finite satiation level of money balances at =0 , empirical estimates of the intercepts and the coefficient on the (logarithm of the) interest rate do in fact matter. Therefore, focusing e.g. on the semi-log and the log-log, it is perfectly possible that the parameters estimates are such that −log()=2 µ1−1+ ¶≥1 1−1−=log −log() In several instances, however, the standard assumption appears to be validated: this is the case for Canada, Japan, and South Korea, whereas evidence for Hong Kong runs, once again, against conventional wisdom.We now turn to the welfare costs of high inflations and Weimar Republic’s hyperinflation 35Coibion et al. (2012) explicitly acknowledge that they do not take into account the costs derived from lack of money satiation. 23
30 Figure 8 Estimated welfare cost functions and welfare losses at each point in time for high-inflation countries and Weimar: Republic’s hyperinflation: point estimates of the lower and upper bounds, 5th and 16th percentiles of the lower bounds, and 84th and 95th percentiles of the upper bounds of the bootstrapped distributions
5.2 High inflation countries and Weimar Republic’s hyperinflation As it has been extensively documented, very high inflations have uniformly been associated with macroeconomic mayhem and the destruction of wealth held in nominal assets. Evidence is especially stark for hyperinflations. For Weimar Republic’s episode, for example, the data reported in Table XL of Graham (1930, p. 317) show that the unemployment rate among trade union members, which in 1922 had oscillated between 0.6 and 3.3 per cent, increased rapidly following the invasion of the Ruhr on the part of France in January 1923, which as pointed out by Bresciani-Turroni (1937) ‘gavethecoupdegrâcetothenationalfinances and the German mark’, thus inaugurating the most extreme phase of the hyperinflation. Unemployment reached 6.2 per cent in May, 9.9 in September, and it further increased to a remarkable 28.2 per cent in December, the last month of the hyperinflation. In this section we explore the welfare costs of these episodes originating from lack of liquidity satiation, which, quite surprisingly, have been near-uniformly overlooked by the previous literature. The only exception we are aware of is Barro (1972), which reports evidence for the Weimar Republic’s episode and four other hyperinflations that had been studied by Cagan (1956). For all countries we estimate Meltzer’s (1963) log-log specification. Our main finding is that for very high inflations and hyperinflations these costs are very far from negligible, as at the inflation peaks of the respective episodes they range from about 4 per cent of output for Ecuador to between 26 and 36 per cent for Weimar’s hyperinflation. Figure 8 reports the evidence. The top row shows the estimated welfare cost functions (in percentage points of GDP) for values of the opportunity cost of money fromzerotothemaximumvaluethatithadtakenoverthesampleperiod. The bottom row shows the estimated welfare losses at each point in time. The thick black lines are the point estimates of the lower and upper bounds, whereas the red lines are the 84th and 95th percentiles of the bootstrapped distribution of the upper bound, and the 5th and 16th percentiles of the corresponding distribution of the lower bound. Several facts clearly emerge from the figure. In particular, first, in line with the previous evidence for low-inflation countries, the point estimates of the upper and lower bounds of both the welfare cost functions, and the welfarelossesateachpointintimeareverytight,tothepointthatinafewinstances (in particular, the welfare losses in the second row) they are nearly indistinguishable. The only exception is Weimar’s hyperinflation, for which for high values of the opportunity cost they can be clearly distinguished. Second, focusing again on the point estimates, the evidence in Figure 8 uniformly suggests that for all of these episodes the welfare costs had been sizeable-to-large. In particular, at the peaks of the inflation episodes these costs had been equal to about 4 per cent of GDP for Ecuador; between 4 and 5 per cent for Bolivia, Chile, and Israel; nearly 7 per cent for Mexico; and between 26 and 36 per cent of income for 24
the Weimar Republic. This provides a stark illustration of how, beyond the already well known and widely documented costs of very high inflations and hyperinflations in terms of economic mayhem and the destruction of wealth held in nominal assets, these episodes have consistently imposed non-negligible, and sometimes large costs uniquely in terms of lack of liquidity satiation, by compelling agents to hold comparatively low levels of real money balances. Third, uncertainty is near-uniformly substantial, sometimes remarkably so. The only exceptions are the historical welfare losses for Chile, Israel, and Mexico: for any of these countries, the inflation peaks pertained to comparatively small fractions of the respective samples, with the result that for most of the sample the opportunity cost had been comparatively low, which, as the top row of Figure 8 shows, is associated with comparatively tighter confidence intervals. Statistical uncertainty is exceptionally large for Weimar’s hyperinflation, for which starting from the second half of 1922 nearly any value of the welfare losses is in principle plausible. 6Conclusions In this paper we revisit the estimation of the welfare costs of inflation for eleven lowinflation and five high-inflation countries, and for Weimar Republic’s hyperinflation. In our analysis we follow the tradition of considering the most liquid monetary assets, which include cash and transactional deposits. We abstract from a detailed discussion of the demand for each of the components, an issue recently addressed by Kurlat (2019).36 Our evidence suggests that, contrary to the explicit or implicit assumption in much of the literature, these costs are often far from negligible. For the U.S. our point estimates are equal to about one-third of those computed by Lucas (2000), and an order of magnitude larger than those obtained by Ireland (2009).37 Crucially, the most empirically plausible money-demand functional form points towards sizeable ‘upward risks’ for these costs, with the 90% confidence interval associated with a 4% nominal interest rate stretching beyond half a percentage point of GDP. At the peak of the inflation episodes, welfare costs had ranged between 0.3 and 1.9 per cent of GDP for low-inflation countries; between 4 and nearly 7 per cent for high-inflation ones; and between 26 and 36 per cent for Weimar’s hyperinflation. Our evidence suggests that ignoring money in analyzing optimal monetary policies can be seriously misleading. For instance, Coibion, Gorodnichenko and Wieland 36He shows that addressing these considerations in a model with imperfect competition substantially increases the estimates of the welfare cost, relative to models that ignore the creation of inside money. 37For a steady-state interest rate of 5 per cent, Lucas (2000) computed the cost to be around 1.1 per cent of lifetime consumption. Ireland (2009) challenged Lucas’ interpretation of the data, and estimated a mere 0.04 per cent of lifetime consumption. 25
(2012)38 make a compelling argument against increasing the inflation target in countries like the U.S., based on a model with frictions in price-setting and with recurrent, though not very frequent, episodes with the nominal interest rate at the ZLB. Based on their preferred specification they compute the welfare effect of an interest rate of 5 per cent to be close to 0.6 per cent of lifetime consumption. Such an estimate, combining the cost created by price frictions and the probability to be at the ZLB, is of the same order of magnitude of the estimates we obtain for the U.S., and it is in fact slightly smaller than the upper bound of our 90 per cent confidence interval. In any of the low-inflation countries the demand for M1 as a fraction of GDP at very low, or even negative interest rates has (so far) exhibited no obvious difference compared to its behavior at higher interest rates. These results contrast with those of Mulligan and Sala-i-Martin (2000), who, based on U.S. households micro data, provided evidence that money demand becomes comparatively flatter at low interest rates. We provide a straightforward explanation for Mulligan and Sala-iMartin’s (2000) finding, by showing mathematically that if the true money demand specification is the one that, as shown by Benati et al. (2021), is the most empirically plausible at low inflation rates, Mulligan and Sala-i-Martin’s (2000) approach automatically produces spurious evidence of a flatter demand curve at low interest rates. From a theoretical standpoint, following Alvarez, Lippi and Robatto (2019) we construct upper and lower bounds for the welfare costs of inflation. As they show, the areaunderthemoneydemandcurveisanalmostexactmeasureofthewelfarecost for a very general class of monetary models in the neighborhood of zero. We extend their results for a quite general sub-class of the models they analyze and compute exact lower and upper bounds for the costs, using the area under the money demand curve, for any value of the interest rate. As we show, the difference between the upper and the lower bound is extremely small for the range of interest rates ever observed in low-inflation countries such as the U.S. For policy purposes, our main finding is that the welfare costs inflation in the Euro area are about twice as large as in the U.S.. This suggests that, ceteris paribus, the inflation target should be materially lower. 38Coibion et al. (2012) explicitly acknowledge that they do not take into account the costs derived from lack of money satiation. 26
7 References Alvarez, F., and F. Lippi (2009): “Financial Innovation and the Transactions Demand for Cash”, Econometrica, 77(2), 363-402. Alvarez, F, F. Lippi, and R. Robatto ( 2019): "Cost of Inflation in Inventory Theoretical Models," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 32, pages 206-226, April. Attanasio, O. P., L. Guiso, and T. Jappelli (2002): “The Demand for Money, Financial Innovation, and the Welfare Cost of Inflation: An Analysis with Household Data”, Journal of Political Economy, 110(2(April)), 317-351. Bai, J. and P. Perron (1998): “Estimating and Testing Linear Models with Multiple Structural Changes”, Econometrica,66(1),47-78 Bai, J. and P. Perron (2003): “Computation and Analysis of Multiple Structural Change Models”, Journal of Applied Econometrics,18(1),1-22 Bailey, M. J. (1956): “The Welfare Cost of Inflationary Finance”, Journal of Political Economy, 64(2), 93-110. Barro, R.J. (1972): “Inflationary Finance and the Welfare Cost of Inflation”, Journal of Political Economy, 80(5), 978-1001. Baumol, W. J. (1952): “The Transactions Demand for Cash: An Inventory Theoretic Model”, Quarterly Journal of Economics, 66, 545-372. Belongia, M.T., and P.N. Ireland (2019): “The Demand for Divisia Money: Theory and Evidence”, mimeo, May 2019 Benati, L. (2008): “Investigating Inflation Persistence Across Monetary Regimes”, Quarterly Journal of Economics, 123(3), 1005-1060. Benati, L. (2015): “The Long-Run Phillips Curve: A Structural VAR Investigation”, Journal of Monetary Economics, 76(November), 15-28. Benati, L.(2020): “Money Velocity and the Natural Rate of Interest”, Journal of Monetary Economics, 116, 117-134 Benati, L.(2024): “The Monetary Dynamics of Hyperinflation Reconsidered”, mimeo Benati, L., R. E. Lucas Jr., J.P. Nicolini, and W. Weber (2021): “International Evidence on Long-Run Money Demand”, Journal of Monetary Economics, 117, 43-63 Blanchard, O.J., G. Dell’Ariccia, and P. Mauro (2010): “Rethinking Macroeconomic Policy”, IMF StaffPosition Note SPN/10/03 Bresciani-Turroni, C. (1937): The Economics of Inflation, John Dickens and Co. Northampton. Cavaliere, G., A. Rahbek, and A. M. R. Taylor (2012): “Bootstrap Determination of the Cointegration Rank in Vector Autoregressive Models”, Econometrica,80(4), 1721-1740. Christiano, L. J., and T. J. Fitzgerald (2003): “The Bandpass Filter”, International Economic Review,44(2),435-465. Coibion, Olivier. “The Optimal Inflation Rate in New Keynesian Models: Should 27
Central Banks Raise their Inflation Targets in Light of the ZLB?”, with Yuriy Gorodnichenko and Johannes Wieland, 2012, Review of Economic Studies 79, 1371-1406. Diebold, F. X., and C. Chen (1996): “Testing Structural Stability with Endogenous Breakpoint: A Size Comparison of Analytic and Bootstrap Procedures”, Journal of Econometrics,70(1),221-241. Dotsey, M., and P. Ireland (1996): “The Welfare Cost of InflationinGeneral Equilibrium”, Journal of Monetary Economics, 37(29), 29-47. Elliot, G., T. J. Rothenberg, and J. H. Stock (1996): “Efficient Tests for an Autoregressive Unit Root”, Econometrica,64(4),813-836. Engle, R. F., and C. W. Granger (1987): “Cointegration and Error Correction: Representation, Estimation, and Testing”, Econometrica, 55(2), 251-276. Feldstein, Martin S. (1997): “The Costs and Benefits of Going from Low Inflation to Price Stability”, in Christina D. Romer and David H. Romer, editors, Reducing Inflation: Motivation and Strategy, University of Chicago Press, 123—166 Friedman, B.M., and K.N. Kuttner (1992): “Money, Income, Prices, and Interest Rates”, American Economic Review, 82(3), 472-492. Goldfeld, S.M. (1973): “The Demand for Money Revisited”, Brookings Papers on Economic Activity,3,577-646. Goldfeld, S.M. (1976): “The Case of the Missing Money”, Brookings Papers on Economic Activity,3,683-730. Graham, F. D. (1930): Exchange, Prices, and Production in Hyperinflation,Princeton University Press. Hamburger, M.J. (1977): “Behavior of the Money Stock: Is there a puzzle?”, Journal of Monetary Economics,3,265-288. Hansen, B. E. (1999): “The Grid Bootstrap and the Autoregressive Model”, Review of Economics and Statistics, 81(4), 594-607. Holtfrerich, C.L. (1980): Die Deutsche Inflation 1914-1923: Ursachen und Folgen in Internationaler Perspektive. Berlin/New York: Walter de Gruyter, S. 70. Primärquelle: Die Wirtschaftskurve 1922, 1923 passim. Ireland, P. (2009): “On the Welfare Cost of Inflation and the Recent Behavior of Money Demand”, American Economic Review, 99(3), 1040-1052. Kurlat, P (2019). "Deposit Spreads and The Welfare Cost of Inflation", Journal of Monetary Economics, October. Latané, H. A. (1960): “Income Velocity and Interest Rates: A Pragmatic Approach”, Review of Economics and Statistics, 42(4), 445-449. Lucas Jr., R.E. (1988): “Money Demand in the United States: A Quantitative Review”, Carnegie-Rochester Conference Series on Public Policy, 29, 137-168. Lucas Jr., R.E. (2000): “Inflation and Welfare”, Econometrica,68(2),247-274. Lucas Jr., R. E., and J.P. Nicolini (2015): “On the Stability ofMoney Demand”, Journal of Monetary Economics, 73, 48-65. Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis, 2nd edition. Springer-Verlag. 28
for nominal GDP in U.S. dollars (“Producto interno bruto (PIB), Miles de dólares”), available for the period 1965-2011, is from Chapter 4 of “85 Años”. An important point to stress is that since we are working with M1 velocity–defined as the ratio between nominal GDP and nominal M1–the specificunitinwhichthetwoseriesare expressed (US dollars, or Ecuadorian sucres) is irrelevant. B.2.4 Israel Quarterly seasonally adjusted data on nominal GDP and the CPI are from the Central Bureau of Statistics, whereas a series for M1 is from Israel’s central bank. A series for the Treasury bill rate is from the International Monetary Fund’s International Financial Statistics. B.2.5 Mexico Quarterly seasonally adjusted data on nominal GDP are from Mexico’s statistical agency, INEGI. Quarterly seasonally adjusted data for the CPI, M1, and a 3-month government bond yield are all from the Banco de México. For all countries, in what follows we work with money velocity (i.e., the inverse of money balances as a fraction of GDP), and a series for the opportunity cost of money, which we compute as the maximum, at each point in time, between inflation and the series for the nominal short-term interest rate (for Argentina we were not able to find an interest rate series, and we therefore work with inflation). B.3 Weimar’s Republic Monthly data on the velocity of circulation of money based on wholesale prices are from Table XXII of Bresciani-Turroni (1937). The series had been normalized by 1913 (i.e. for the year 1913 it took a value of one). Based on Benati, Lucas, Nicolini and Weber’s (2021) data, however, in 1913 German money velocity had been equal to 7.49. Consequently, we have rescaled Bresciani-Turroni’s money velocity series by multiplying it by 7.49.39 A series for the inflation rate is from Cagan (1956). A series for the money market rate (‘Tägliches Geld’) is from Table 23 of Holtfrerich (1980). The sample period is September 1920-October 1923. C Why We Do Not Use Divisia Aggregates Throughout the entire paper we work with ‘simple-sum’ M1 aggregates. In this appendix we briefly discuss why we have chosen to ignore Divisia indices. A first problem is that, to the very best of our knowledge, such indices are only available 39In fact, working with Bresciani-Turroni’s original series produces manifestly absurd results, with the welfare costs of inflation even taking values in excess of 100 per cent of GDP. 35
for the United States (from the Center for Financial Stability, henceforth CFS)and for the United Kingdom (from the Bank of England). A second problem is that, for the United States, the Divisia M1 series constructed by the CFS does not feature MMDAs (which are instead included in Divisia M2). This means that although the resulting index of monetary services has been constructed by optimally weighting the underlying individual assets, it suffers from the crucial shortcoming that it is not including a key component of the transaction technology. As a result, although Divisia M1 is in principle superior to the standard simple-sum M1 aggregate, it ultimately suffers from the same shortcoming of not including MMDAs. So the key question is: What is more important? Including MMDAs, or optimally weighting the underlying assets? Figure C.1 provides evidence on this, by showing the same evidence shown in Figure 2 in the main text of the paper, but this time with velocity being computed based on Divisia aggregates. The figure speaks for itself, and provides no evidence of a stable relationship between the velocity of any Divisia aggregate and its opportunity cost (computed based on the user cost series from the CFS). In particular, a comparison between the first panel of Figure C.1, and the second panel in Figure 2, clearly shows that, for the purpose of detecting a stable long-run demand for M1 in the United States, the crucial issue is including MMDAs in the definition of M1, rather than computing the aggregate by optimally weighting the underlying assets. So although, in theory, Divisia M1 possesses optimal properties, because of the specific way in which is has been constructed, within the present context such optimal properties are trumped by the fact that, exactly as its simple-sum counterpart, it does not include MMDAs. 36
Table A.1aBootstrapped p-values for Elliot, Rothenberg, and Stock unit root tests M1 velocity short rate p=2 p=4 p=6 p=8 p=2 p=4 p=6 p=8 Low-inflation countries: United States 1959Q1-2023Q2 0.9809 0.9323 0.9236 0.9207 0.3122 0.1794 0.1053 0.2918 1959Q1-2019Q4 0.8633 0.8362 0.9048 0.8764 0.4382 0.2861 0.1903 0.4334 1959Q1-2001Q4 0.3529 0.2989 0.4112 0.3768 0.3238 0.2756 0.1428 0.2979 United Kingdom 1955Q1-2023Q2 0.9257 0.8719 0.8042 0.8490 0.2896 0.3225 0.4118 0.4673 1955Q1-2008Q3 0.8187 0.8012 0.7262 0.7740 0.1416 0.1640 0.2386 0.2600 Canada 1947Q3-2006Q4 0.4641 0.6307 0.3987 0.5405 0.2298 0.2466 0.2224 0.3600 1967Q1-2023Q1 0.9772 0.9682 0.9557 0.9554 0.3996 0.3831 0.3605 0.6097 Australia 1969Q3-2023Q1 0.9665 0.9600 0.9580 0.9213 0.4143 0.3218 0.5442 0.7099 1969Q3-2008Q4 0.9883 0.9823 0.9834 0.9679 0.3078 0.2719 0.4116 0.5810 Switzerland 1980Q1-2023Q1 0.8459 0.7684 0.7566 0.6917 0.3639 0.4453 0.1953 0.2024 Sweden 1998Q1-2023Q1 0.5217 0.4343 0.5006 0.6716 0.3254 0.4637 0.5665 0.5411 Euro area 1999Q1-2023Q1 0.1096 0.0847 0.0515 0.0075 0.4403 0.2511 0.2468 0.3625 Denmark 1991Q1-2023Q1 0.1027 0.2238 0.2289 0.1194 0.0999 0.0433 0.0212 0.0090 South Korea 1964Q1-2023Q1 0.0195 0.0000 0.0150 0.0535 0.5079 0.4863 0.3828 0.0867 Japan 1960Q1-2023Q2 0.7305 0.7323 0.8163 0.5827 0.2394 0.3962 0.4375 0.4192 Hong Kong 1985Q1-2023Q2 0.6293 0.6881 0.5964 0.6065 0.3103 0.1820 0.1338 0.3085 Based on 10,000 bootstrap replications of estimated ARIMA processes. Tests are with an intercept and no time trend.
Table A.1bBootstrapped p-values for Elliot, Rothenberg, and Stock unit root tests Logarithm of: M1 velocity short rate p=2 p=4 p=6 p=8 p=2 p=4 p=6 p=8 Low-inflation countries: United States 1959Q1-2023Q2 0.9850 0.9688 0.9712 0.9759 0.1579 0.0883 0.2274 0.2411 1959Q1-2019Q4 0.9749 0.9550 0.9784 0.9611 0.4835 0.3839 0.4577 0.2198 1959Q1-2001Q4 0.3054 0.2694 0.3626 0.3169 0.4250 0.3985 0.3023 0.3665 United Kingdom 1955Q1-2023Q2 0.9793 0.9496 0.9284 0.9224 0.2758 0.2210 0.2902 0.7804 1955Q1-2008Q3 0.8090 0.8838 0.8299 0.8706 0.1484 0.2162 0.3100 0.4094 Canada 1947Q3-2006Q4 0.1103 0.2339 0.1159 0.2931 0.0590 0.0474 0.0229 0.0275 1967Q1-2023Q1 0.9951 0.9898 0.9803 0.9902 0.1001 0.1006 0.3021 0.7613 Australia 1969Q3-2023Q1 0.9898 0.9863 0.9770 0.9609 0.1160 0.0509 0.4413 0.8191 1969Q3-2008Q4 0.9978 0.9946 0.9953 0.9923 0.3915 0.3312 0.5893 0.6785 Switzerland 1980Q1-2023Q1 0.8647 0.8136 0.7686 0.7438 ———— Sweden 1998Q1-2023Q1 0.7177 0.5494 0.5106 0.6041 0.2756 0.2584 0.3197 0.3893 Euro area 1999Q1-2023Q1 0.4266 0.3080 0.2638 0.0954 0.4084 0.6636 0.6686 0.6779 Denmark 1991Q1-2023Q1 0.2648 0.3825 0.4415 0.2066 ———— South Korea 1964Q1-2023Q1 0.3815 0.1906 0.4337 0.4606 0.6510 0.6106 0.5806 0.3996 Japan 1960Q1-2023Q2 0.9694 0.9607 0.9528 0.8539 0.5903 0.6472 0.6553 0.6345 Hong Kong 1985Q1-2023Q2 0.7613 0.7888 0.7563 0.7412 0.1532 0.1217 0.1317 0.2013 High-inflation countries: Israel 1982Q1-2019Q4 0.8997 0.8346 0.7721 0.5814 0.5826 0.5698 0.5049 0.4309 Mexico 1982Q1-2019Q4 0.4687 0.4159 0.2106 0.1188 0.2463 0.1085 0.0400 0.0147 p=1 p=2 p=1 p=2 Bolivia 1980-2019 0.7971 0.8018 0.7588 0.7822 Chile 1946-2019 0.6862 0.5399 0.7214 0.6946 Ecuador 1980-2019 0.9309 0.8715 0.8018 0.8928 Hyperinflations: p=1 p=2 p=3 p=4 p=1 p=2 p=3 p=4 Weimar Republic Sep. 1920-Oct. 1923 0.9309 0.9601 0.8592 0.8569 0.5006 0.5459 0.7271 0.5317 Based on 10,000 bootstrap replications of estimated ARIMA processes. Tests are with an intercept and no time trend. The short rate has a few negative observations at the end of the sample.
32 Figure C.1 United States: money velocity based on Divisia aggregates, and the corresponding opportunity costs