Phillips Curves, Expectations of Inflation and Optimal Unemployment over Time

The paper is “a study of the ‘optimal’ fiscal control of aggregate demand,” presenting a dynamic macroeconomic model from which the optimal time-path of aggregate employment, and the resulting time-path of the actual rate of inflation given an initially expected inflation rate, are derived (Introduction, p. 254). To keep the dynamic optimization problem tractable, Phelps adopts a deliberately simplified setting: “a closed, non-stochastic economy… in which exogenous monetary policy immunizes investment against variations in capacity utilization” so as to keep potential capital intensity constant over time (p. 254) – what he later calls “virtual golden-age growth” (Part I.A, pp. 257-259).

The conventional approach, as Phelps describes it, superimposes social indifference curves over a single Phillips curve (drawn for zero expected inflation) and picks the point of tangency, at some unemployment rate u below the “equilibrium” rate u at which actual and expected inflation coincide* (Introduction, pp. 254-255). Phelps’s objection is dynamic: if that statically optimal rate is chosen and held, “it is reasonable to suppose that the participants in product and labour markets will learn to expect inflation… [and] the Phillips Curve will gradually shift upward… by the full amount of the newly expected and previously actual rate of inflation” (p. 255); repeating this, “the rate of inflation will continue to increase as long as the unemployment ratio is smaller than u*,” with utilization eventually forced back toward u* but only after “a very high rate of inflation – much higher than the policy-makers myopically bargained for” is reached. “Thus the conventional approach goes wrong in implicitly discounting future utilities infinitely heavily” (p. 256).

A Quasi-Phillips Curve is the relation p/p = f(y) - x between the actual inflation rate and the utilization ratio y, for a given expected rate of inflation, -x (Part I.B, eq. 3, pp. 261-262). Unlike a single fixed empirical Phillips curve, “every increase of the expected rate of inflation by one point will increase by one point the actual rate of inflation associated with any given utilization ratio” (p. 261) – so there is a whole family of parallel curves, one for each value of expected inflation (illustrated for -x = -0.01, 0, 0.01 in Figure 1), and f(y) itself is increasing and strictly convex in y, equal to zero at the equilibrium ratio y*, positive above it, and approaching infinity as y approaches its upper feasible bound (p. 261).

y is the utilization ratio at which f(y) = 0, so that the actual rate of inflation exactly equals the expected rate and expectations are not disturbed** (Part I.B, p. 262). Phelps labels y greater than y* as “over-utilization” and y less than y* as “under-utilization,” stressing that this is “merely from the point of view of equilibrium” and “without intending normative significance” (p. 262) – y* is a property of the model’s dynamics, not by itself a claim about the socially best employment level (that role is instead played by a separate constant y-hat in the utility function, which the paper allows to differ from y*, reflecting involuntary unemployment in some sectors even at y = y*; Part I.C, pp. 265-266).

Phelps adopts “the mechanism of ‘adaptive expectations’ first used in this context by Phillip Cagan”: the rate of change of the expected inflation rate is an increasing function of the gap between actual and expected inflation, equal to zero when the gap is zero (Part I.B, pp. 262-263, eq. 4). Substituting the Quasi-Phillips Curve into this adjustment rule yields a single law of motion for the expected deflation rate x as a function of utilization alone, x-dot = G(y), with G(y*) = 0 and G strictly decreasing (eq. 5, p. 263) – so once a utilization path y(t) is chosen, the entire time path of expected (and hence actual) inflation is pinned down given the initial condition x(0) = x0 (p. 263).

The government (the “Fisc”) is assumed to maximize the integral over time of a possibly discounted “rate of utility” U(x, y) that depends on utilization directly (through consumption and leisure) and on the money interest rate i = r(y) - x indirectly (through a demand to hold real money balances) (Part I.C, pp. 264-269, eq. 7-8). Utility from utilization alone is dome-shaped, peaking at a constant y-hat between y* and the upper feasible bound, reflecting the trade-off between more consumption/less involuntary unemployment (from higher y) and less leisure, with a discrepancy between y and y* also representing a welfare cost from disappointed expectations (pp. 265-266). Utility is flat in the money interest rate over a “full liquidity” range (i less than or equal to a threshold i-bar) but falls, at an increasing rate, as the interest rate rises further, going to minus infinity as i approaches a “barter point” i-super-b at which the monetary system breaks down (pp. 266-267) – capturing the real cost of economizing on money balances when expected inflation, and hence the money interest rate, is high.

Because the ordinary discounted-utility integral can diverge to infinity along many feasible paths when there is no discounting, Phelps adopts the “over-taking principle”: a path is preferred to another if, for some sufficiently large T0, its cumulative utility through every T greater than T0 exceeds the other path’s (Part II, pp. 273-274). Under this criterion, if the initially inherited expected deflation rate x0 is below the rate needed for “full liquidity” at equilibrium utilization, x-bar(y*), then the optimal policy requires under-utilization (y less than y*) for all time, with utilization approaching y* and expected deflation approaching x-bar(y*) only asymptotically (Part II, pp. 274-275); if x0 already equals x-bar(y*), equilibrium utilization with full liquidity is immediately and permanently optimal; and the paper shows that sustained over-utilization is never optimal in the no-discount case, since any such “binge” is dominated by a slower one, with the striking implication that “over-utilization is not optimal whether or not [the current static optimum] y(x) is larger than y*” (pp. 275-277).

With positive discounting, the optimal path of expected deflation x(t) converges to a long-run value x (rather than to the full-liquidity level x-bar(y)), and if the inherited x0 exceeds x*, over-utilization (y greater than y*) becomes optimal, with utilization approaching y* only asymptotically from above** (Part III, pp. 278-279, eq. 14-17). “The greater is the utility discount rate, the smaller algebraically will be the equilibrium deflation rate” (p. 279): a higher discount rate pushes x* further from full liquidity and makes over-utilization more likely to be optimal for a given x0, though x* can never be pushed below the level that would make the static, myopic policy optimal even as the discount rate goes to infinity (p. 279).

“A tight fiscal policy producing ‘under-utilization’… is optimal if and only if the currently expected inflation rate exceeds the asymptotically optimal inflation rate. The latter is determined by liquidity considerations and by social time preference (the utility discount rate), not by the strength of preferences for high or low utilization (at a given rate of interest)” (Part IV, “Concluding Remarks,” p. 281). Phelps draws out the implication directly: “what characterizes the advocates of a ‘high-pressure’ policy of over-utilization is their implicit adoption of a large utility discount… they reveal high ’time preference,’” reframing the inflationist/deflationist policy debate as a disagreement about how heavily to discount the future rather than a disagreement about the employment-inflation trade-off itself.

Phelps explicitly lists several simplifications he regards as provisional (Part IV, p. 281): inflation in the model depends only on the level of the utilization ratio, not (as he suggests would be more realistic) on its rate of change as well; investment is exogenously fixed on a “virtual golden-age” path rather than being made endogenous and jointly optimized with aggregate demand; and the model is closed, with no balance-of-payments considerations, even though Phelps suggests “the model’s greatest relevance may be for a nation’s optimal objectives in the international co-ordination of aggregate demand and price trends among countries.” Earlier in the paper he also flags the stationarity of the utility-rate function as his “greatest reservation” when labour-augmenting technical progress is present (Part I.C, p. 273), and notes that the non-negativity constraint on the money interest rate is not taken fully seriously in the analysis (Part I.B, pp. 263-264).

Although the paper does not use the term “natural rate,” its equilibrium utilization ratio y plays the same structural role: it is the ratio at which expectations are not being revised, and the paper’s central dynamic argument – that holding utilization away from y forever requires ever-adjusting, ultimately unsustainable expectations – is the same logic later associated with the natural-rate hypothesis.** Phelps frames his critique explicitly against economists who treat the Phillips curve as a stable, exploitable menu (citing Lipsey and Okun in his opening footnotes, p. 255), and against econometric work that “probably estimate[s] some average of different Phillips Curves, corresponding to different expected rates of inflation… which have varied only over a small range” (p. 256, fn. 3) – an early statement of the idea that an empirically estimated Phillips curve conflates multiple expectations-conditioned relationships rather than representing one stable, policy-exploitable curve.