Simple Analytics of the Government Expenditure Multiplier

Woodford states at the outset that “the answer does not depend solely on the assumed structure of the economy… If prices or wages are sticky, monetary policy affects real activity, and so the consequences of an increase in government purchases depend on the monetary policy response” (Sec. 2, p. 8). He stresses that even the seemingly natural benchmark of “leaving monetary policy unchanged” is ambiguous – “it is not the same thing to assume that the path of the money supply is unchanged as to assume that the path of interest rates is unchanged, or that the central bank’s inflation target is unchanged, or that the central bank continues to adhere to a ‘Taylor rule’” (p. 8) – so the paper works through several alternative, precisely specified monetary policies in turn rather than positing a single number.

With flexible wages and prices, the equilibrium condition u’(Yt − Gt) = μṽ’(Yt) implies a multiplier dY/dG = ηu/(ηu + ηv), which “is positive, but necessarily less than 1,” so that “private expenditure… is necessarily crowded out, at least partially, by government purchases” (Sec. 1.1, pp. 4-5). The multiplier can be “only a small fraction of 1” when the intertemporal elasticity of substitution of private expenditure is high and the marginal cost of extra output rises steeply. Critically, this multiplier is unaffected by the introduction of Dixit-Stiglitz monopolistic competition with a constant markup μ (Sec. 1.2): the markup changes the level of output but not the multiplier, since “the key to obtaining a larger multiplier is an endogenous decline in the labor-efficiency wedge,” which requires some form of nominal stickiness, not market power per se (p. 6).

If monetary policy maintains a constant real rate rt = r̄ regardless of the fiscal shock, then households’ Euler equation implies constant consumption Ct = C̄, so output moves one-for-one with government purchases: Yt = C̄ + Gt, “and the multiplier (dYt/dGt) is equal to 1… There is no crowding out of private expenditure by government purchases, though no stimulus of additional private expenditure, either” (Sec. 2, p. 9). Woodford calls this “essentially the standard ‘multiplier’ calculation in undergraduate textbooks, where the question asked is how much the ‘IS curve’ shifts to the right” (p. 8), and notes it holds “under a wide range of alternative assumptions about the nature of price or wage stickiness” (p. 9) – it is a property of the demand side alone, given that monetary policy is assumed capable of holding the real rate fixed.

Under a Taylor rule responding to the flexible-price output gap, the multiplier γy = (1−ρ+ψΓ)/(1−ρ+ψ) satisfies Γ < γy < 1 – strictly between the neoclassical multiplier Γ and the constant-real-rate value of 1 (Sec. 3.3, eq. 3.16) – because inflation is allowed to rise somewhat (unlike under strict inflation targeting) but real rates also rise in response (unlike under the constant-real-rate policy), partially crowding out private spending. If instead the Taylor rule responds to output relative to its trend level rather than the flexible-price gap (the way “most central banks” actually specify potential output, not adjusted for government purchases), the multiplier can fall “even smaller than the one predicted by the neoclassical model” for a sufficiently strong output-response coefficient (Sec. 3.3, eq. 3.17, p. 18) – price stickiness can thus produce a smaller multiplier than full flexibility once the central bank’s reaction function is specified realistically.

Modeling a financial disturbance as driving the natural real rate rnet_t to a negative level rL for a random duration governed by persistence parameter µ (following Eggertsson 2009), Woodford shows the multiplier while the ZLB binds is ϑG = [(1−µ)(1−βµ) − κσµΓ] / [(1−µ)(1−βµ) − κσµ], which is “necessarily greater than 1 (for any µ > 0)” and “monotonically increasing in µ,” becoming “unboundedly large as µ approaches [its upper bound]” (Sec. 4.1, pp. 22-24). Using Eggertsson’s Great-Depression-calibrated parameters (µ = 0.903, rL = −4% annualized), the implied multiplier is 2.3 – comparable to the “multiplier above 2” that Christiano et al. (2009) find in a larger estimated New Keynesian model, and broadly consistent with defense-spending-based multiplier estimates of 2.1-2.5 from Depression-era cross-country panel evidence (Almunia et al. 2010) that Woodford cites as corroboration.

Woodford attributes the discrepancy not mainly to model complexity but to a difference in the assumed fiscal-policy design: his calculation assumes spending increases “lasts precisely as long as credit spreads are elevated… following which period Gt = Ḡ again,” whereas “Cogan et al. instead consider increases in government purchases that… extend much longer than the period over which the interest rate is assumed to remain at zero” (Sec. 4.2, p. 27). He shows this distinction is quantitatively decisive: decomposing Eggertsson’s multiplier of 2.3, “1.0 of this is due to the increase in government purchases during the current quarter, while the other 1.3 is the effect of higher anticipated government purchases in the future” (p. 29) being expected to persist only within the crisis window – and when a probability λ of post-crisis persistence is introduced, the multiplier falls below 1 for λ ≥ 0.8 (roughly 4+ quarters of expected post-crisis duration) and turns negative for λ ≥ 0.91 (10+ quarters), because anticipated future crowding-out and disinflation depress current private spending even while the ZLB still binds (Sec. 4.2, pp. 28-29).

No – Woodford is explicit that “this does not follow trivially from the existence of a positive multiplier (or even a multiplier greater than 1); one must consider the value of the use to which the resources consumed by the government would otherwise be put” (Sec. 5, pp. 30-31). Away from the ZLB, when monetary policy can freely stabilize the output gap, he argues government purchases should be set purely by the “principle of efficient composition of expenditure” – comparing the marginal utility of public spending g’(Gt) to that of private spending u’(Yt − Gt) – because “to the extent that the problem [of insufficient demand] can be solved using monetary policy, it is costless to do so… whereas… government spending… has a cost, since it requires the diversion of real resources to alternative uses” (Sec. 5.2, p. 33).

No: solving for the fiscal policy that minimizes a quadratic loss in inflation, the output gap, and the distortion from deviating spending away from its efficient composition, Woodford finds that “it is not optimal to fully stabilize inflation and the output gap, despite the feasibility of doing so, because of the inefficient composition of expenditure that this would involve” (Sec. 5.3, p. 37). For a financial disturbance of only moderate expected persistence (µ ≤ 0.5 in his calibration), “the optimal increase in government purchases is only a small fraction of the increase that would be required to eliminate the output gap”; only when the disturbance is expected to be very persistent does welfare-maximizing stimulus approach (without reaching) the amount needed for full stabilization (p. 37).

Woodford shows that financing spending with a proportional tax on sales revenue changes the flexible-price multiplier calculation to (1−τt)u’(Yt−Gt) = ṽ’(Yt), and that with logarithmic utility and a balanced budget this multiplier can fall to exactly zero, or even negative if the intertemporal elasticity of substitution is low enough (Sec. 6, eq. 6.1, p. 38). But he argues this matters less than it might seem for his main results: “in the benchmark case considered in section 2, where monetary policy is assumed to maintain a constant path for the real interest rate, taking account of tax distortions would not change the conclusion that the government expenditure multiplier is equal to 1,” as long as the fiscal change does not alter the long-run level of tax distortions (p. 38) – because that result rests on the demand side alone, not on the supply-side tax wedge.

In the conclusion, Woodford states that “under circumstances like those of a Great Depression… standard models… imply that the government expenditure multiplier should be larger than one, and may be well above one,” and that fiscal stimulus can raise welfare up to nearly the amount needed to fully close the output gap when the constraining disturbance is expected to be very persistent (Sec. 7, p. 40). But “under less extreme circumstances, the case for using variations in government purchases for stabilization purposes is much weaker”: away from the ZLB, “there is a good case for leaving output-gap stabilization largely to monetary policy”; and even at the ZLB, the case for stimulus “applies only… in which the increased government purchases will be terminated as soon as the constraint ceases to bind,” since spending or debt-financed tax increases that are correctly anticipated to continue afterward “significantly reduce the stimulative effects… and a fortiori… the net welfare gains” (p. 40).