Rational Expectations and the Theory of Price Movements

Muth opens (Sec. 1, pp. 316-317) by citing two conclusions from studies of expectations data: (1) “averages of expectations in an industry are more accurate than naive models and as accurate as elaborate equation systems, although there are considerable cross-sectional differences of opinion,” and (2) “reported expectations generally underestimate the extent of changes that actually take place.” He argues dynamic economic models up to that point had used ad hoc expectations formulas with “little evidence… that the presumed relations bear a resemblance to the way the economy works” (p. 316), and frames his hypothesis as the opposite of the common view that assuming rationality leads to theories inconsistent with observed dynamics – “our hypothesis is based on exactly the opposite point of view: that dynamic economic models do not assume enough rationality” (p. 317).

The hypothesis: “expectations of firms… tend to be distributed, for the same information set, about the prediction of the theory (or the ‘objective’ probability distributions of outcomes)” (Sec. 2, p. 317). Muth specifies it asserts three things: "(1) Information is scarce, and the economic system generally does not waste it. (2) The way expectations are formed depends specifically on the structure of the relevant system describing the economy. (3) A ‘public prediction’… will have no substantial effect on the operation of the economic system (unless it is based on inside information)" (p. 317). He is careful to note the hypothesis does not claim entrepreneurs’ scratch work literally resembles a system of equations, nor that predictions are perfect or identical across agents (p. 317). For tractable analysis he further assumes normally distributed disturbances, the existence of certainty equivalents, and linearity of the system including the expectations formulas (Sec. 2, p. 318), noting these assumptions are mutually near-implying.

In a market with quantity produced C_t = -βp_t^e (demand-side notation aside, production decided on expected price) and market equilibrium P_t = C_t, with an additive supply shock u_t, the reduced form gives the market price as a linear function of the expected price and the shock (Sec. 3, eq. 3.1-3.3, pp. 317-318). Imposing rationality – that the aggregate expectation of firms equals the theory’s own prediction – implies, when shocks are serially uncorrelated, that the expected price equals the equilibrium price (p_t^e = 0 in deviation-from-equilibrium terms), so that all price and quantity movement traces along the demand curve and is driven entirely by the (unpredictable) supply shock (p. 318). Muth notes this simplest case is “of little empirical interest, because the shocks were assumed to be completely unpredictable,” motivating the extension to serially correlated and partly observable shocks (pp. 318-319).

Writing the shock u_t as a linear combination of past independent innovations with weights w_j (eq. 3.6), Muth derives the rational multi-period-ahead price expectation p_{t,L}^e as a function of these weights (eqs. 3.9-3.11), then converts this into a forecasting rule V_j expressed in terms of observable past prices via a triangular system of equations (eqs. 3.12-3.14) (Sec. 3, pp. 318-320). Two illustrative special cases: if shocks are purely transitory (serially independent), the rational forecast collapses to the simple equilibrium-price prediction of Q3 above (eq. 3.15); if instead a shock permanently shifts all future supply conditions, the rational expected price becomes a geometrically weighted moving average of past prices (eq. 3.17), matching the exponential-smoothing forecasting formula Nerlove had used empirically to estimate agricultural supply elasticities – with Muth noting his derivation shows the correct smoothing coefficient should depend on the demand and supply parameters, not be fitted independently (pp. 320-321).

Muth shows the framework can represent systematic over- or under-discounting of recent information by multiplying the most recent shock’s weight by a factor λ in the expectations formula (eq. 3.18); with permanent-type disturbances, under-discounting (λ<1) produces expectations that overweight the latest observed price while over-discounting (λ>2, say) underweights it (Sec. 3, pp. 321-322, Fig. 3.1). He offers a second interpretation of this same modification: if a fraction λ of firms are “insiders” with a one-period information lag while the rest lag two periods, the same biased aggregate formula results (p. 322). Muth stresses that cross-sectional heterogeneity in individual expectations has a negligible aggregate effect as long as individual deviations from the rational forecast are not strongly, systematically correlated with each other or with other explanatory variables (p. 321).

Muth derives an individual’s demand for speculative inventories from expected-utility maximization over uncertain future profit, using a second-order Taylor approximation of the utility function; under normality of prices and a small expected-price-change assumption, the optimal inventory position I_t is approximately proportional to the expected price change and inversely proportional to the conditional variance of next period’s price (eq. 4.9, “a” being this proportionality factor) (Sec. 4, pp. 322-324). Embedding this speculative demand into the market-clearing conditions (eqs. 4.10-4.14) yields a second-order difference equation whose dynamic stability depends on the parameter combination α/(β+γ) (eq. 4.16-4.17); for serially uncorrelated shocks the solution reduces to an AR(1) rational expected-price rule, p_t^e = λ₁ p_{t-1} (eq. 4.20), with λ₁ close to unity when speculative inventories are an important factor, giving prices a high positive serial correlation, and close to zero when they are negligible.

Table 4.1 and the accompanying discussion (Sec. 4, pp. 327-329) show that speculation reduces the variance of prices by “spreading the effect of a market disturbance over several time periods,” that this effect is negligible when the speculative-demand parameter α is small relative to the sum of demand and supply slopes (β+γ), and that mean income to speculators is always positive even though (per rational expectations) no speculative profit opportunity remains unexploited in the aggregate. Producers’ revenue and consumers’ expenditure both rise with more active speculation, with consumer expenditure initially rising slightly faster than producer revenue, so that “the effect of speculation on welfare is therefore not obvious” (p. 328) – Muth does not claim speculation is unambiguously welfare-improving, only that it stabilizes prices while remaining individually rational and profitable.

Muth catalogs three earlier expectations formulas used in cobweb theory (Sec. 5, pp. 329-331): the classical assumption that expected price equals the latest observed price (p_t^e = p_{t-1}); Goodwin’s extrapolative rule adding a fraction of the latest price change (eq. 5.3); and Nerlove’s adaptive-expectations rule, which revises the forecast by a fraction of the latest forecast error (eq. 5.4). All three, Muth notes, imply that the cobweb-model’s prediction and farmers’ actual expectations move in opposite directions from each other – “the prediction of the cobweb theory would ordinarily have the sign opposite to that of the firms” (p. 330) – a point noted decades earlier by Schultz, who called it “not… extremely improbable” that farmers simply fail to learn from experience.

Several studies had estimated the regression of actual on expected price/output changes (eq. 5.6) and found the coefficient b positive but less than one (Sec. 5, pp. 331-332) – a finding Muth calls “clearly inconsistent with the cobweb theory, which ordinarily requires a negative coefficient.” Citing Bossons and Modigliani, Muth shows this positive-but-attenuated coefficient is exactly what rational expectations predicts via a regression/measurement-error argument: since the rational expected price is, in the aggregate, an unbiased predictor of the actual price, the probability limit of the least-squares estimate of b is Var(expected price)/Var(actual price), which is necessarily less than one (eq. 5.8, p. 333). This is presented as a genuinely discriminating empirical test between the two classes of theories, one that the data favor for rational expectations.

Comparing predicted mean cycle length (measured either by intervals between successive trend-line “upcrosses” or between peaks/troughs) across the four theories (Table 5.2, Sec. 5, pp. 333-334), the rational model without storage implies a cycle length around 3-4 production periods, while allowing for storage/speculation lengthens it further (roughly 3 to more than 4 periods). Muth notes that the observed length of real cycles, such as the hog cycle – shown too long for the classical cobweb theorem as early as 1935 by Coase and Fowler – and Ezekiel’s cattle-price evidence (implying an implausible 5-7 year production period under the strict cobweb reading) tend to exceed three production periods, which is more consistent with the rational-expectations-with-speculation account than with the classical cobweb theorem (p. 334), though Muth cautions such comparisons must be interpreted carefully given possible serial correlation in the underlying exogenous disturbances.