Inflation and Welfare

Questions & answers

Q1. What is the paper’s headline quantitative estimate, and how is it obtained?

“It is estimated that the gain from reducing the annual inflation rate from 10 percent to zero is equivalent to an increase in real income of slightly less than one percent” (Abstract). This is obtained by fitting a money-demand curve – real balances (M1) as a ratio to nominal income, as a function of the short-term nominal interest rate – to U.S. annual data for 1900-1994, then applying Bailey’s (1956) method of computing the area under the corresponding inverse demand curve (the consumers’ surplus recoverable by moving from a given interest rate down to zero) as the welfare-cost measure (Sections 1-2). Lucas describes the underlying motivation plainly: because holding non-interest-bearing cash is individually costly (its opportunity cost is the nominal interest rate) but privately wasteful in aggregate (everyone spends resources trying to get someone else to hold it), “these person-hours are simply thrown away, wasted on a task that should not have to be performed at all,” and reducing money growth (and hence inflation and interest rates) can make everyone better off as long as rates are positive (Section 1).

Q2. What two functional forms does Lucas fit to the money-demand data, and what do they imply?

A “log-log” demand curve, m(r) = A r^(-eta), with the interest elasticity eta = 0.5 giving the best fit, and a “semi-log” curve, m(r) = B e^(-epsilon r), with epsilon = 7 fitting best (Section 2). Lucas states plainly that “the semi-log function…provides a description of the data that is much inferior to the log-log curve.” The two forms diverge sharply in their welfare implications away from the fitted range: “at a six percent interest rate, for example, the log-log curve implies a welfare cost of about one percent of income, while the semi-log curve implies a cost of less than 0.3 percent,” but because the two curves are “nearly parallel between interest rates of 3 and 10 percent,” they give very similar estimates of the cost of exceeding zero inflation by moderate amounts (Section 2). The key remaining divergence is at very low rates: log-log demand implies a substantial further welfare gain from moving from zero inflation all the way to the interest-rate-minimizing deflation rate, while under semi-log demand “this gain is trivial.”

Q3. How does the Sidrauski framework provide a theoretical rationale for the Bailey consumers’-surplus calculation?

Lucas builds a simplified, deterministic version of Sidrauski’s (1967) representative-household model in which households derive utility from consumption and from real money balances under homothetic, constant-relative-risk-aversion preferences, and shows that solving the household’s balanced-growth-path optimization yields a steady-state money-demand relation m(r) that can be identified directly with the empirically fitted curves of Section 2 (Section 3). He shows the resulting theoretical welfare-cost function, obtained by solving an exact differential equation implied by the model, is numerically almost indistinguishable from the Bailey consumers’-surplus approximation at realistic interest rates – “on a plot such as Figure 5 the exact and the approximate solutions cannot be distinguished by the eye” – so “giving colorful names to statistical relationships is not a substitute for economic theory,” but in this case the atheoretical curve-fitting turns out to have been measuring the right object all along. Within this framework the Friedman (1969) rule – a deflation equal to the real interest rate, driving the nominal rate to zero – is confirmed as the utility-maximizing policy whenever money demand is downward-sloping.

Q4. Does allowing for realistic fiscal financing (income taxation instead of lump-sum transfers) change the welfare estimates?

Only marginally, and only at extremely low interest rates. Lucas shows that because U.S. money demand implies real balances grow without bound as the nominal rate approaches zero (m(r) -> infinity as r -> 0 under the fitted log-log form), literally implementing r = 0 would require an infinite lump-sum transfer of resources to the public, which is infeasible if only distortionary income taxation is available – placing “a positive lower bound on r.” But he calculates that with realistic calibration, “an income tax rate of 0.03 would implement an interest rate of 0.001 (that is, one-tenth of one percent),” so “the Friedman rule requires qualification in this case, but the qualification is of no quantitative interest” (Section 4). Extending the model further to include elastic labor supply and government consumption financed by a mix of income and inflation taxation, Lucas reports the minimized welfare cost across several assumed labor-supply elasticities differs from the frictionless benchmark by “0.0015 times income” at most, concluding “above about half a percent [interest], estimated welfare costs are the same as in the inelastic labor supply, lump sum tax case” (Section 4).

Q5. What is the McCallum-Goodfriend transactions-technology model, and how does it relate to the Baumol inventory model?

In this alternative framework (Section 5), household utility depends only on goods consumption, not directly on real balances; instead, households must devote time to “transacting” according to an explicit technology relating time spent, cash held, and the flow of spending achieved, so that the time cost of economizing on cash, s(r), is itself “a direct measure of the welfare cost of inflation, interpreted as wasted time.” Lucas shows that working backward from the empirically estimated money-demand function to the implied transactions technology, the resulting welfare-cost function s(r) is again numerically almost identical to the Bailey consumers’-surplus approximation across a wide range of interest rates. In the specific case that best fits the U.S. data – a log-log demand curve with elasticity 0.5 – the implied transactions technology reduces to a straight line through the origin, which Lucas notes “corresponds to the celebrated inventory-theoretic model introduced by Baumol (1952),” in which halving average cash holdings and doubling the number of bank trips sustains the same spending pattern – yielding the classic “square root rule” for cash management, s(r) proportional to the square root of r.

Q6. What numerical checks does Lucas perform to confirm that the McCallum-Goodfriend model’s balanced-path behavior is actually utility-maximizing?

Solving the household’s dynamic program numerically over a grid of possible money holdings (rather than relying only on first-order conditions), Lucas confirms that for realistic values of the risk-aversion parameter, “the cash holdings of a single consumer with arbitrary initial balances converges to the steady state value” implied by the model, validating the balanced-path analysis (Section 5, Figures 9-10) But he also documents a genuine failure mode: for near-linear utility (risk aversion at or very close to zero), the optimal policy is discontinuous – the household sets transacting time to zero for a while, consuming nothing while accumulating cash, and then “enjoy[s] a consumption orgy in which all cash is spent at once,” a cycle that repeats. Lucas concludes this pathology “arise[s] only under near-linear utility, with values of [risk aversion] far below any available estimates,” so for realistic parameter values the model’s balanced-path money-demand relation remains a valid implication of optimizing behavior.

Q7. Why does Lucas caution that the welfare gains from approaching zero interest rates specifically cannot be reliably estimated from aggregate time series?

He points to Mulligan and Sala-i-Martin’s finding, based on the Survey of Consumer Finances, that “about 59% of American households in 1989 hold no financial assets beside cash and their checking account,” which Lucas interprets as evidence for a fixed cost of holding interest-bearing assets at all – a cost invisible in aggregate time series but potentially large enough to “completely negate any welfare gain from reducing interest rates from, say, 1.5 percent to zero,” since households with low asset holdings would never find it worth paying the fixed cost regardless of how low rates fall (Section 6). He states directly: “there is good reason to doubt that accurate estimates of cash holding at very low interest rates can be obtained from aggregate U.S. time series evidence alone,” and separately flags concerns about M1’s adequacy as a monetary aggregate in the 1990s (his own fitted demand functions “do very badly” in that decade), pointing to Divisia monetary aggregates as offering “much the best prospects for resolving the difficulty.”

Q8. What is Lucas’s final summary assessment of what has and has not been reliably established?

“In all of the models I have reviewed, the estimated gains of reducing inflation and interest rates are positive, starting from any interest rate above, say, one tenth of one percent,” and “reducing interest rates from 14 percent to 3 percent would yield a benefit equivalent to an increase in real income of about 0.008, eight tenths of one percent” – an estimate Lucas says “is about the same whether one uses the fitted log-log demand curve for money or the semi-log version,” is not sensitive to the fiscal-financing assumptions of Section 4, and would not be much altered by adding realistic productivity or money-supply shocks to the model (Section 6, citing Cooley and Hansen 1989). By contrast, the further gain from reducing interest rates from 3 percent (roughly the U.S. rate under zero inflation) all the way to zero is far less certain: “about 0.009” (0.9% of GDP) under log-log demand – larger than the entire 14-to-3-percent gain – but “less than 0.001” under semi-log demand, and Lucas notes that “insofar as the fixed costs postulated by Mulligan and Sala-i-Martin are important, even this figure may be an overstatement.” He closes by framing the paper’s contribution as a rare case of a normative economic theory that is “quantitatively reliable over a wide range of interest rates” despite monetary theory’s persistent lack of a single, universally accepted microfoundation.