The Inflation Tax in a Real Business Cycle Model

Questions & answers

Q1. How is money introduced into the real business cycle model, and which activities are “cash goods” versus “credit goods”?

Money enters via a cash-in-advance constraint applied only to consumption – “consumption is a ‘cash good’ while leisure and investment are ‘credit goods,’ in the terminology of Lucas and Stokey (1983, 1987)” – so that when households want to reduce cash holdings in response to higher anticipated inflation, “they can only do so by reducing consumption.” The underlying real economy is otherwise identical to Hansen’s (1985) one-sector stochastic growth model with indivisible labor: households can only work a fixed positive number of hours or not at all, and this nonconvexity is “convexified” by having households trade employment lotteries (following Rogerson 1988) so that a fraction of households work full hours each period and the rest are unemployed.

Q2. Why can’t the paper use the usual real-business-cycle solution shortcut (the second welfare theorem), and how does it solve for equilibrium instead?

Because the cash-in-advance constraint introduces a “wedge of inefficiency” (the paper’s phrase, borrowed from Lucas 1987), the competitive equilibrium is not the solution to a planner’s problem, so the paper cannot simply compute the Pareto-optimal allocation and invoke the second welfare theorem the way Hansen (1985) did; instead the authors apply a method due to Kydland (1987) that solves for the competitive equilibrium directly, via a linear-quadratic approximation to the household’s dynamic programming problem solved by iterating on a version of Bellman’s equation subject to a rational-expectations consistency condition at each iteration. The paper notes that when money is supplied “optimally” (at a growth rate equal to the discount factor β, so the cash constraint never binds), the resulting real economy and its cyclical and steady-state properties are identical to the moneyless Hansen (1985) economy – a useful internal check on the solution method.

Q3. What is the paper’s central finding about constant versus erratic (variable) money growth and the business cycle?

When money is supplied according to a constant growth-rate rule – even one implying a high average inflation rate – “the features of the business cycle are unaffected”: the standard deviations and output-correlations of the real variables are statistically indistinguishable from the moneyless model, because a constant anticipated inflation rate only shifts the steady state (lowering output, consumption, investment, capital, and hours) without altering how the economy responds to technology shocks around that steady state. When money instead grows erratically – calibrated via an AR(1) process fit to postwar quarterly M1 growth (autoregressive coefficient of 0.48) – “the cyclical behavior of real variables are altered slightly: consumption becomes more variable relative to income and the price level becomes quite volatile,” and the correlations of consumption and prices with output become smaller in absolute value, bringing the simulated moments closer to the actual U.S. 1955:3-1984:1 statistics reported in the paper’s Table 1.

Q4. How does the paper measure the welfare cost of anticipated inflation, and what magnitudes does it find?

Following the tradition of Bailey (1956), Fischer (1981), and Lucas (1981), the paper measures the welfare cost of inflation as the percentage increase in consumption (ΔC) that would make a household in a given inflationary steady state as well off as under the Pareto-optimal allocation (money growing at rate β, where the cash constraint never binds); for a quarterly cash-in-advance constraint (consistent with treating M1 as the relevant money stock), a sustained 10% annual inflation costs about 0.387% of steady-state GNP – “very close to estimates that have been suggested by others” (Fischer’s ~0.3%, Lucas’s ~0.45%) – while under a monthly constraint (closer to the monetary base), the same 10% inflation costs only about 0.1% of GNP, and even a 400% annual inflation costs only “slightly more than 1.5 percent” under the monthly definition. The paper is explicit that the large gap between the quarterly and monthly estimates reflects which monetary aggregate – and hence which fraction of households’ liquid balances – is assumed to bear the inflation tax, and cautions that because institutions to shield balances from inflation plausibly evolve in very-high-inflation economies, “our model may not be very reliable for analyzing hyperinflation.”

Q5. What happens to steady-state hours, output, and the capital stock as the anticipated inflation rate rises, and why?

Anticipated inflation taxes the “cash good” (consumption) relative to the untaxed “credit goods” (leisure and investment), so as the money growth rate rises above its optimal level (g = β), agents substitute toward leisure – “the consumption of leisure increased because agents substitute this ‘credit good’ for the consumption good in the face of a positive inflation tax” – which lowers hours worked and hence lowers steady-state output, consumption, investment, and the capital stock, while the share of output allocated to investment does not change with inflation, because in steady state investment finances future consumption that will itself face exactly the same inflation tax as current consumption.

Q6. What does the paper claim about the long-run relationship between inflation and employment, and what empirical evidence does it offer?

The model implies that steady-state employment rates are lower in higher-inflation economies – “the operational Phillips curve is upward sloping” in the long run, a possibility the paper attributes to Friedman’s (1977) Nobel lecture – and the authors report a cross-section regression of average employment rates on average inflation rates for 23 developed countries over 1976-85, finding a coefficient of -0.5 (standard error 0.17), which becomes -0.44 (standard error 0.22) when the most extreme observation (Chile) is dropped, which the paper reads as a “statistically significant negative correlation between inflation rates and employment rates” consistent with the model’s implication. The same qualitative upward-sloping long-run Phillips curve result is noted as also arising in the related model of Greenwood and Huffman (1987).

Q7. What channels does the paper explicitly say it is – and is not – capturing, and what does it speculate about the sources of money’s real effects it has not modeled?

The paper is explicit that its results reflect “only the costs and consequences of money that are due to the inflation tax: there are no informational problems created by the money supply process” in the model, and that unanticipated inflation and “sticky price” mechanisms – channels many believe are the most important ways money affects the real economy – are not present here at all. The authors conclude that “if money does have a major effect on the cyclical properties of the real economy it must be through channels that we have not explored here,” speculating that the money supply process may matter chiefly through its informational consequences for private agents’ expectations of relative prices, “as in the natural rate literature,” rather than through the anticipated-inflation-tax channel this paper isolates.