The Intertemporal Keynesian Cross

The static consumption function Ct = C(Yt − Tt) underlying the textbook Keynesian cross “does not respect intertemporal budget constraints and therefore cannot be microfounded in a dynamic model” (Sec. 1, p. 2), since it ignores both the drag from anticipated future taxes on deficit-financed spending and the boost from past saved income. The authors note that “following the pioneering work of Modigliani and Brumberg (1954) and Friedman (1957), a large body of empirical work has shown that past and expected future income affect current consumption” (p. 2), which is precisely what a static function cannot capture. Their fix is to derive, from an explicit class of microfounded consumption-saving models (Sec. 2.2), an intertemporal consumption function Ct({Zs}) that depends on the entire path of aggregate post-tax income {Zs}, not just current income – allowing saving of unspent income and borrowing against anticipated future income.

The IKC is equation (13), dY = dG − M·dT + M·dY, the first-order (linearized) impulse-response analog of the static equation dYt = dGt − mpc·dTt + mpc·dYt, with the scalar mpc replaced by the infinite matrix of iMPCs M (Proposition 1, p. 12). Because M alone determines how any given path of government spending dG and taxes dT translates into an output path dY, “in the intertemporal Keynesian cross, the matrix of iMPCs M is the sufficient statistic for this response” (p. 12) – the same sufficient-statistic role mpc plays in the static cross, but now capturing the entire time-profile of consumption responses rather than a single number. A key mathematical restriction is that the present value of each column of M must equal 1 (eq. 14): whatever is not spent on impact is necessarily saved and spent later, at some date.

Using Norwegian administrative data on lottery winnings analyzed by Fagereng, Holm and Natvik (2021), the authors find a contemporaneous annual MPC of about 0.51, but also a sizable follow-on iMPC of about 0.18 in the year after the shock, with the response decaying slowly and becoming statistically insignificant only around year four (Sec. 3.1, pp. 16-17). They corroborate this with a second, independent source – a lower-bound estimate constructed from the 2016 Italian Survey of Household Income and Wealth (SHIW) – which gives a weighted contemporaneous MPC of 0.44 and a year-one lower bound of 0.14, “closely aligned with those obtained from the Norwegian data” (p. 17). The authors are explicit about the limits of this evidence: “existing data are currently too limited to allow us to fully construct M without imposing more structure from a model” (p. 18), since there is essentially no reliable evidence yet on consumption responses to anticipated (rather than surprise) income changes.

Representative-agent models “fail immediately on the grounds that they cannot match the high static MPC M00” (p. 5, Introduction), while two-agent models with hand-to-mouth and saving households can match M00 but “predict a very low subsequent iMPC M10, almost an order of magnitude below our estimate” (p. 5). The model that fits is a heterogeneous-agent model with limited liquidity, in which “many households have short but nonzero effective planning horizons,” so they save part of an unanticipated income shock and spend it down gradually over the next few years, matching the observed M10 (p. 5). The paper studies two variants: HA-one, a single-liquid-asset (Bewley-style) incomplete-markets model, and HA-two, a two-account model with both a liquid and an illiquid asset (following Kaplan and Violante 2014 and Bayer et al. 2019).

Although “both models have cumulative multipliers on deficit spending that are well above one,” HA-one’s iMPCs “continue to decline rapidly after several years, as any remaining above-normal liquidity is spent down,” producing a cumulative MPC over several years near one and correspondingly “very large Keynesian multipliers,” whereas HA-two’s iMPCs “decline more slowly after several years, because at that point, unspent savings have mostly migrated to the illiquid account” (p. 6). While the authors say “our evidence on iMPCs is not precise enough to select between these models directly,” they argue HA-two “is more consistent with the Norwegian evidence on asset accumulation after lottery earnings” (p. 6), and later show it is the only model among those considered that also matches the empirical evidence on spending out of capital gains (Sec. 6, discussed further in Q8).

Proposition 3 shows that “if there is a unique equilibrium… the fiscal multiplier is 1 at every date, dY = dG” whenever the budget is balanced (dG = dT), irrespective of the iMPC matrix M (p. 26). The logic is that output and pre-tax income rise exactly with spending, taxes rise by the same amount, so after-tax income – and therefore consumption in every model – is unchanged. This holds for RA, TA, and HA economies alike: “HA and RA economies, despite very different iMPCs, have the same balanced-budget multiplier” (p. 27). The result depends on income and taxes having the same incidence across households; the paper notes in an appendix that lump-sum taxation instead produces a multiplier below 1, because taxpayers then have higher short-run iMPCs than income earners.

Under deficit financing, Proposition 4 shows the output response is dY = dG + M·(dG − dT), so the consumption response “only depends on the path of primary deficits dG − dT” via the iMPC matrix (p. 27), implying that, holding the deficit path fixed, the spending multiplier equals the transfer multiplier plus one. In the paper’s stylized IKC environment (Table 1/Table 4), the RA and TA models still deliver a unit or near-unit cumulative multiplier (Ricardian equivalence effectively holds for RA; TA’s initial boom is exactly offset by a later contraction from future taxes), while TABU, HA-one, and HA-two all generate cumulative multipliers strictly above one – as high as roughly 15-17 for the stylized TABU/HA-one calibrations, reflecting that “deficit-financed spending leads to an increase in income without an immediately offsetting increase in taxes… leading to an output boom in the future that triggers its own intertemporal consumption feedback” (p. 4).

Once the model is extended beyond the IKC’s assumptions of a constant real rate and no capital, “iMPCs out of income are no longer sufficient statistics for the general equilibrium effect of fiscal policy: now, the consumption response to real interest rates and the iMPCs out of a surprise capital gain also matter” (p. 4). The authors prove that under two assumptions on preferences and initial portfolios there remains an analytical link between these objects, but they show empirically that TABU-style analytical models, when calibrated to match income iMPCs, imply MPCs out of capital gains “that are far too large” (p. 4) relative to the data – whereas HA-two is “unique, among the models we consider, in its ability to fit empirical evidence on spending out of capital gains in addition to income” (p. 5). This distinction turns out to reverse the ranking of multipliers once real rates and investment are allowed to move (Q9).

In the full quantitative HANK model of Section 7, an active Taylor rule pushes up real interest rates in response to expansionary fiscal policy, crowding out investment in every model, and crowding out consumption too in the RA and (eventually) TA models – but “HA-two stands out in that, despite the active Taylor rule and rising real rates, consumption contributes positively to output for about two years, offsetting investment crowd-out” (Sec. 7.2, p. 39). Table 4 reports that under substantial deficit financing (ρB = 0.93), HA-two’s impact and cumulative multipliers are both around 1.3 – “the only model that generates deficit-financed multipliers above 1” in the quantitative environment – while TABU’s multipliers, despite matching income iMPCs well, are pulled down by its unrealistically large capital-gains MPC, which makes the fall in the stock market (as investment is crowded out) contract consumption much more than in HA-two, where “the MPC out of capital gains is very low,” consistent with the data (Sec. 7.2, p. 40).

Comparing an active Taylor rule to a zero lower bound (ZLB) episode of three years, the paper finds that “irrespective of the degree of deficit financing, the ZLB dramatically increases the output response,” since unresponsive nominal rates mean the inflation generated by fiscal expansion pushes real rates down, crowding in both consumption and investment (Sec. 7.3, p. 41). A third scenario – holding the real interest rate constant, as in the paper’s baseline IKC environment – produces outcomes “somewhere in between the ZLB and the Taylor rule” (p. 41), which the authors offer as direct confirmation that their constant-real-rate assumption in Sections 2-6 is an intermediate case between very loose (ZLB) and very tight (active Taylor rule) monetary policy, not an extreme case in itself.

They flag three explicit limitations (Sec. 7.4, pp. 42-43): the model economy “was assumed to be entirely closed; openness should reduce multipliers somewhat as it dampens the feedback between consumption and income”; the empirical literature “typically characterizes a single type of ‘multiplier’” whereas the model implies multipliers depend heavily on the degree of deficit financing, which they describe as “a new testable prediction for the empirical literature”; and the analysis “mostly restricted attention to fiscal policies which adjust income taxes… without altering tax progressivity,” even though less progressive taxation is shown (Sec. 5.1) to reduce multipliers. They also note that their quantitative HA-two multipliers (roughly 0.3 to 1.3 depending on horizon and financing) fall within the range Ramey’s (2019) survey reports as plausible (“probably between 0.8 and 1.5,” with 0.5 or 2 not rejected by the data), though they caution that a genuine comparison would require matching the exact degree of deficit financing in each historical episode studied empirically.