Workers, capitalists, and the government: fiscal policy and income (re)distribution

Highlights • Propose a tractable capitalist-worker New Keynesian model to study the interaction of household heterogeneity and fiscal policy.• Modeling limited asset market participation through portfolio adjustment costs generates a realistic pattern of intertemporal marginal propensities to consume.• The capitalist-worker structures ensures that the transmission of demand shocks under sticky prices does not rely on income effects on labor supply induced by countercyclical markups.• Relative to the predictions of the traditional two-agent model, fiscal multipliers are smaller; and the sensitivity of the output path to public deficits is dampened.• Overall, the model matches the implications of richer Heterogeneous-Agent New Keynesian (HANK) models in key respects, while remaining analytically tractable.


E Estimation through IRF matching 42
Appendix A Partial equilibrium consumption models
Since the log-linearization is done around a steady-state with x W = 1, the results correspond to iMPCs computed from a linearized version of the model. A.1

A.2 Comparison to alternative frameworks A.2.1 Bonds in utility
Section 2.2 of the main text showed, using a simple partial equilibrium consumption-savings setup, that the introduction of bond portfolio adjustment costs (PACs) into the budget constraint, as in Schmitt-Grohe and Uribe (2003) and Neumeyer and Perri (2005), can give rise to intertemporal marginal propensities to consume (iMPCs) that are consistent with empirical data and match the A.1 Log-linearizing the model, albeit perhaps less common in this context than plain linearization, is convenient as it ensures consistency with the remainder of the paper. predictions of multi-asset, heterogeneous-agent models. Dropping indices for the household type to ease notation, the relevant Euler equation could be written in the more general form u (c t ) + u (c t )ρ (b t ) = βE t u (c t+1 )(1 + r t ). (A.8) where ρ (·) is the first derivative of a convex function. Impose that ρ(b t ) = ψ 2x (b t − b) 2 and that u(c t ) = log(c t ). Then log-linearizing around a steady state with consumption normalized to unity, 1 + r = β −1 and zero net assets (as in the steady state of the general equilibrium version of the model), we arrive at the log-linearized Euler equation (3): An alternative specification relies on the introduction of bonds into the utility function (BU) and can, under suitable functional form assumptions, be shown to be first-order equivalent, so that once linearized it also implies equation (A.9). In particular, suppose we set ψ = 0 but introduce an additional bond term into the household's objective function: Thus, in general, equivalence between the two approaches requires that v (b t ) = −u (c t )ρ (b t ), where the multiplicative term in the case of PACs arises because the costs are measured in financial rather than utility terms. Even thus, supposing that v(b t ) = − ψ 2x (b t − b) 2 and linearizing around the same steady state, the multiplicative term disappears and, therefore, the PAC and BU approaches deliver exactly the same predictions up to first order.
This analysis demonstrates, thus, that there are multiple, equivalent, tractable ways of modeling household behavior consistent with the micro data on intertemporal marginal propensities to consume reviewed in Section 2.2. Intuitively, it is not surprising that the introduction of a convex cost to savings and a concave benefit, respectively, can give rise to similar behavior. Both specifications imply that the marginal net benefit from saving is decreasing. Consequently, the household wants to consume more out of an income windfall than if ψ = 0, and the pattern of iMPCs displays a gradual decay as the household dis-saves. As described in the main text, the particular functional form assumption adopted here can be seen as formalizing the idea of households as exhibiting "target saving" behavior, that is, they aim to return, in the absence of shocks, to some long-run level, and are penalized the farther away from that target they move in an attempt to smooth consumption when hit by transitory shocks. The equivalence result here clarifies that this penalty could take either a financial or a psychological form.
We conclude with two practical observations. First, both the PAC'and the BU specifications can be introduced into a two-agent framework in different ways. For instance, Auclert et al. (2018) propose'a two-agent model that combines one type of agent with bonds in the utility with a fraction of hand-to-mouth agents, thus modifying the consumption behavior of the unconstrained households.
By contrast, in our model we primarily tackled the consumption behavior of the constrained type of household. We adopted the latter approach in an effort to generalize the notion of "limited asset market participation" that is often viewed as underpinning hand-to-mouth consumption as incorporated into the traditional two-agent models of Campbell and Mankiw (1989), Galí et al. (2007) and Bilbiie (2008).
Second, it is quite common (e.g., Hagedorn (2018); Michaillat and Saez (2019)  Notes: The two panels show the dynamic consumption response to a period s = 0 unanticipated income shock under alternative specifications of the consumption/savings problem: the benchmark permanent income hypothesis (PIH), which corresponds to the "unconstrained" household in Figure 2 in the main text; an economy with habit formation (with ξ = 0.9); and a household subject to portfolio adjustment costs (PACs), denoted "worker" in the main text. The horizontal axis shows time measured in quarters. The vertical axis displays the marginal propensity to consume ∂c t /∂ x 0 .

A.2.2 Habits
The paper proposes limited asset market participation modeled through convex portfolio adjustment costs as a tractable way of describing household consumption behavior in line with both micro and macro data. This approach contrasts with the extensively used assumption of 'habit formation' in the utility function, which we briefly consider here. A.3 According to a common model of habit formation, the period utility function of a representative where c t is aggregate consumption, σ is the coefficient of relative risk aversion, and ξ ∈ (0, 1) is a habit formation coefficient. As is well known, the resulting model can give rise to the kind of persistent and hump-shaped response of aggregate consumption to shocks commonly found in empirical macro studies.
Habit formation implies consumption dynamics that are inconsistent with the micro evidence, A.3 We consider external habit formation as in Smets and Wouters (2007). Internal habits give rise to similar properties. however, as discussed in greater detail by Auclert et al. (2020a) (also see Carroll et al. (2018) Havranek et al.'s (2017) survey of the micro literature furthermore finds little evidence for the existence of consumption habits sufficiently strong to explain the persistence or 'excess smoothness' of aggregate consumption. a one-off change in the real interest rate s ≥ 0 periods later is The subsequent expected path of consumption, for t ≥ 1 obeys for t > s. (with log utility) predicts consumption to respond one-for-one after a one-off interest rate cut, the effect is smaller when adjustment costs are positive (ψ W > 0). In the following period, consumption mildly declines, in fact, as the household replenishes her savings. Furthermore, a worker household discounts news about future interest rate changes, that is, consumption responds less for greater values of s. This is illustrated in the right-hand panel A.2d.

Appendix B TANK models
In this section we summarize the equilibrium as well as steady-state conditions for the simple TANK models discussed in Section 2. Notation is as in the main text. We also show the sensitivity of each model to the two key parameters, bond adjustment costs ψ W and population share λ.  Table B.1 summarizes the non-linear equilibrium conditions of a simple two-agent New Keynesian model with unconstrained and hand-to-mouth households (TANK-UH).

Description
Equation In Section 2.1.1 we consider a log-linearized version of the model. The equilibrium conditions are approximated around the zero-inflation steady state (Π = 1), in which hours worked are normalized to unity (n = y = 1). Then from the Euler condition of unconstrained households we have that R = 1 β . An optimal production subsidy τ S = (η − 1) −1 ensures zero profits in equilibrium, so that w = mc = (1 + τ S ) η−1 η = 1 and d = 0. For simplicity, here we assume zero government spending and debt in steady state (b = b U = g = t = 0). Given equal hours, we have an equal-consumption result; c = c U = c H = 1. Finally ν = w cn ϕ = 1.
The steady-state conditions are as in the UH model, with the additional stipulation that b W = 0. As regards the steady state, workers' labor is now pinned down by n w = n λ = 1 λ . To ensure equal consumption in steady state, we introduce a lump-sum transfer from workers to capitalists

B.1.1.3 TANK-CW
The labor dis-utility weight is now ν = w c w n W ϕ = λ ϕ . Then the remaining conditions are unchanged.

B.2.1 Calibration with equal population shares
We start this section by presenting an alternative calibration of λ and ψ W . While in 3.1 they were set to target both quarterly and annual impact iMPC evidence, here we compare models where each agents represent half of the population in the economy, that is, λ = 0.5, and then set PACs to match the quarterly impact MPC (ψ W = 0.2265). This calibration generates more heterogeneous average iMPCs across models and helps highlighting the impact of our modeling choices on the aggregate labor supply and fiscal multiplier. Figure B.1 shows that, keeping population proportions constant, the introduction of workers reduces by half the output multiplier for a deficit-financed government spending shock. This result occurs due to the lower consumption propensity of workers compared to hand-to-mouth households. Introducing, on top of that, capitalists instead of the usual unconstrained households drives the impulse responses of hours and, hence, output to almost zero in the resulting TANK-CW model. This is because the labor supply curve of households suffering a loss in profit income (due to markups moving countercyclically) does not shift outwards. Hence this calibration implies virtually no aggregate expansionary effects of fiscal stimulus, while preserving the same redistributive movements in real wages and profits as observed in the TANK-UW model with unconstrained households and workers. Notes: The figure shows the impulse responses of selected variables to a government spending shock according to different, simple TANK models. All series are in proportional deviations from their steady state (in %) except for the fiscal variables (government spending, bonds and taxes) and profits, which are measured in percentage of steady-state output. Consumption components are weighted by population shares. Explanations for the acronyms are as follows: UH -unconstrained and hand-to-mouth households; UW -unconstrained and worker households; CW -capitalist and worker households.

B.2.2 Variations in λ
Next, consider variations in the population share parameter λ, holding all other parameter values constant. Figures B.2-B.4 reveal that a higher value of λ generates a more positive consumption and output response in all three model variants. This result follows directly from household heterogeneity and the fact that both hand-to-mouth and workers' (impact) MPCs (and for workers also MPCs in subsequent periods) are higher than those of permanent-income consumers. Hence, raising the value of λ raises the higher average marginal propensity to consume. B.1 B.1 Notice that we cannot simulate the TANK-UH model for a high value of λ such as λ = 0.9 due to indeterminacy issues, as discussed in Section C.

B.2.3 Variations in ψ W
Turning to variations in the strength of portfolio adjustment costs, as indexed by ψ W , we now show what happens in the TANK-UW and TANK-CW models when we raise ψ W above the baseline level of ψ W = 0.0742. This exercise serves to illustrate that the two differences between benchmark TANK model and the proposed alternative TANK-CW model interact. For recall that only in the UW model households with unconstrained access to financial markets (elastically) supply labor.
In particular, Figure B.5 demonstrates that whereas in the UW model increasing the value of ψ W reinforces both the aggregate and redistributive effect of fiscal policy, this result does not carry over to the CW model ( Figure B.6). Instead, increasing the strength of PACs reinforces the redistributive effect of fiscal policy but reduces its impact on aggregate output. Intuitively, the reason is a higher value of ψ W pushes up workers' consumption on impact, reducing their desire to work, ceteris paribus. Other things are not equal, however, because the expansion in demand also pushes up (down) wages (profits). Unlike in the UW model, however, this fails to trigger an increase in labor supply through a profit income effect. As a result, there is no compensating rise in total hours worked and, hence capitalists' income and aggregate output.

B.2.4 Variations in fiscal rule parameters
In Footnote 30 we remarked upon the fact that in Figure 5 the path of workers' consumption is lightly hump-shaped, a pattern that is inherited also by aggregate consumption. Since private consumption alongside public spending pins down labor demand in this model, the same shape is observed also for the labor share. This result is, however, contingent upon the particular fiscal rule we consider rather than being a direct consequence of workers' iMPCs; after all, the latter peak on impact. Figure B.7 illustrates this point by considering an altered parameterization of the fiscal rule, whereby the coefficient on debt is one and that to government spending is zero. Consequently, bonds peak on impact. It can be seen that the response of consumption and the labor share is no longer hump-shaped. Thus, the presence of portfolio adjustment costs generates a hump-shaped path for workers' consumption provided the path for the supply of liquidity in form of government bonds is hump-shaped as well. Intuitively, the greater the excess of bonds issued by the government to finance its additional spending relative to workers' target level, the greater is workers' desire to consume so as to avoid a penalty in the form of adjustment costs. Notes: The figure shows the impulse responses of selected variables to a government spending shock equal to one percent of steady-state output in different, simple TANK models when in the tax rule the coefficient on debt is one and that to government spending is zero. All series are in percent deviations from their steady state except for the fiscal variables (government spending, bonds and taxes) and profits, which are measured in percentage of steady-state output. Consumption components are weighted by population shares. Explanations for the acronyms: UH -unconstrained and hand-to-mouth households; UW -unconstrained and worker households; CW -capitalist and worker households.

B.3 Medium-scale models
We concluded our analysis in the main text, Section 3.3, by considering medium-scale variants of the different TANK models that allow for physical investment, nominal wage stickiness and other frictions commonly incorporated into medium-scale DSGE. Here we provide more details on these models. Importantly, capitalists receive income not only from holding firm equity -as in our simple variants, following Bilbiie (2008) -but they also invest in physical capital -as in Galí et al. (2007).
Intermediate firms rent capital and use it alongside labor according to a Cobb-Douglas production function. Rigidity in nominal wages is modeled as for prices using Rotemberg adjustment costs. As such, hours worked are largely demand-determined, worker households are not always on their labor supply curve, and the average marginal propensity to earn is low (on the importance of which for consistency with empirical data, see Auclert et al. (2020b)). In addition, we allow for fixed costs in production, investment adjustment costs, variable capital utilization, a more general Taylor rule featuring interest rate smoothing and, finally, positive government spending as well as debt in steady state. Table B.3 summarizes the non-linear equilbrium conditions of the medium-scale TANK-CW model. B.2 Relative to the simple model described in Appendix Section B, new endogenous variables are as follows: y m gross output (includes fix costs of production); capital, k; investment, i; the real rental rate r K ; investment adjustment costs s, which imply that Tobin's q, denoted q, is no longer necessarily equal to unity; utilization of capital, u, and the associated cost Ψ; workers' marginal rate of substitution, mr s. Given nominal wage stickiness, in the form of Rotemberg adjustment costs, we also have gross wage inflation, Π w . Notice, furthermore, that now variables in the fiscal rule are expressed in deviations from their own (positive) steady-state. It would be straightforward to add various other common frictions such as habits or firm-specific capital.
As far as additional parameter values are concerned, notice that α denotes the capital share, η W B.2 The UW variant can be obtained by replacing capitalist-type with unconstrained-type households and adding back the aggregate labor supply equation. The benchmark UH model is instead obtained by replacing worker-type with hand-to-mouth households who do not participate in financial markets at all. the elasticity of substitution between differentiated labor, ξ w Rotemberg wage adjustment costs, υ = γ 1 γ 2 is the utilization elasticity, φ r is interest rate inertia while φ y is the interest rate response to output. Steady-state relationships involve calibrating the steady state government spending and debt to output ratios (respectively g y and b y and the parameter ν scaling the dis-utility of working: ν = w η w −1 η w n W ϕ . Additionally, we impose fixed costs in production to ensure zero monopoly profits in steady-state, F = n k n α − w + r K k n . As before to ensure equal consumption in steady-state, we introduce a lump-sum transfer from workers to capitalists tr W = ((wn W ) − c W − t)λ = 1 − λ. generates impulse responses for consumption and investment that are consistent with evidence from macro time series, as considered, for instance, in Section D. In particular, aggregate consumption exhibits a positive but relatively short-lived response to an expansionary, deficit-financed increase in public spending. In addition to elastic investment, which makes capitalists' consumption response less negative, allowing for (empirically plausible) real interest rate smoothing also plays an important role. In its absence, consumption by workers, in particular, is more negative. This result, too, can be directly tied to the partial equilibrium model analyzed in Section 2.2. There we noted that, unlike for hand-to-mouth households, workers' consumption is somewhat responsive to real interest rate movements, especially at a near horizon. Consequently, if the central bank raises real interest rates more slowly following an expansion in public spending that pushes up inflation, workers will consume more.

Description
Equation Rental rate of capital

Appendix C Determinacy in TANK models
In the main text, we focused on consumption dynamics and the determinants of labor supply in different TANK models. Here we draw attention to a third dimension along which the traditional characterization of household heterogeneity in a two-agent setting and our proposed amendment To add intuition, Bilbiie (2008) shows that it is the interplay between labor markets and asset markets described in Section 2.3 above that underpins these determinacy properties of the TANK-UH model.
This point is most easily seen in a special case of the model that imposes budget balance and zero persistence government spending, and assumes that the central bank responds to expected C.1 For surveys of labor supply elasticity estimates, see, e.g., Chetty et al. (2013) and Attanasio et al. (2018). C.2 On this point, also see Gali et al. (2004), Galí et al. (2007) and Bilbiie (2008), but also note Maliar and Naubert (2019).
next-period inflation. Then the model can be reduced to a two-equation system: where ζ = 1−λ 1−λ χ and κ = χ+ϕ −1 ϕ −1 +1 , with χ = ϕ + 2 denoting the elasticity of hand-to-mouth households' consumption to aggregate income. C.3 The crucial parameters pinning down the stability properties of the system are those appearing in ζ, which is interpretable as the elasticity of aggregate demand with respect to the real interest rate. These parameters are the share of hand-to-mouth households λ and the inverse Frisch elasticity ϕ. When ζ is strictly positive, the standard Taylor principle applies.
Aggregate demand becomes a negative function of the real interest rate, however, when λ > 1 2+ϕ .
Under this "inverted aggregate demand logic," the central bank needs to obey an inverted Taylor principle (φ π < 1) for stability to obtain. If asset market participation is sufficiently limited (λ is high) and/or labor supply inelastic enough (ϕ is high), a fall in the real interest rate can become contractionary due to the negative demand effect arising from the strong fall in profit income - (1 − λ) −1 > 1 units per type-U household for any unit drop in total profits d t -that occurs due to hand-to-mouth households' high marginal propensity to consume, which amplifies the rise and wages (fall in profits) relative to the case where all households smooth consumption by participating in financial markets. Under the standard Taylor principle, a non-fundamental increase in inflation expectations can then be self-fulling: it triggers a rise in the real interest rate which pushes up demand, output, and inflation, thus validating expectations.
Introducing even moderate portfolio adjustment costs gives rise to more plausible determinacy properties. Paralleling the above analysis, but now for the model with workers rather than hand-tomouth households, Figure C.2 plots the stability properties of the rational expectations equilibrium (in a neighborhood of the steady-state) as a function of the population share parameter λ, the inverse C.3 Differences from the value of the same elasticity reported in Bilbiie (2019) are due to different assumptions on the determination of labor supply -other things equal, the value of χ is higher in the present setup -and the fact that we abstracted from fiscal redistribution of profits.  Table 2.
Frisch elasticity ϕ, as well as the strength of PACs as indexed by ψ W (Panel C.2a) or the Taylor rule coefficient on inflation φ π (Panel C.2b). For higher values of ψ W the behavior of workers approximates that of hand-to-mouth households and, consequently, the indeterminacy problems for parameter combinations other than low λ and low ϕ reappear under a conventional value of φ π = 1.5. By contrast, for lower values the standard Taylor principle is restored for any combination of λ and ϕ. The reason is that the introduction of a partially constrained worker household type implies a less stringent interdependence of labor and financial markets. In particular, for values of the adjustment cost parameter ψ W below infinity, workers' consumption propensity lies below that of hand-to-mouth households. The elasticity of real wages and, hence, profits, to output is consequently lower for smaller values of ψ W (also see Figure B.5). Hence, the "inverted aggregate demand logic" region shrinks as ψ W declines (Panel C.2a). In particular, when λ and ψ W are calibrated to match micro consumption data as in our baseline calibration, the standard Taylor principle is restored (Panel C.2b).
Lastly, we observe that in the TANK framework introducing capitalists is feasible without compromising on the stability properties only in the presence of intermediate portfolio adjustment  Table 2. Notably, panel C.2a assumes φ π > 1.
costs. Suppose, for the sake of argument, that we are back in the setting with hand-to-mouth households (that is, PACs are infinitely large). Then after replacing aggregate labor supply with an equation for hand-to-mouth households only, a few substitutions show thatn t =g t 1+ϕ . Thus, hours worked are independent of profits. Unfortunately, however, solving the problem of profit income effects in this way exacerbates the determinacy problem that already afflicted the benchmark TANK-UH specification: irrespective of the value of λ a unique equilibrium is now unattainable given φ π > 1. C.4 Panel C.3a indicates that there are no combinations of λ and ϕ for which the rational expectations equilibrium is locally unique given an active Taylor principle φ π > 1. under an active Taylor principle.
Panel C.3b instead shows that the introduction of an intermediate level of portfolio adjustment costs, with ψ W = 0.0742, restores this possibility. With the exception of the corner case λ → 0 that is irrelevant for all practical purposes, there is a unique saddle-path stable solution irrespective of the value of λ and ϕ. C.5 C.4 More formally, if we write the system as E t z t+1 = Γ 0 z t + Γ 1 ν t , where z t (c t , Π t ) and ν t = ( m t , g t ) , then for any φ π > 1 it is never the case that both eigenvalues of Γ 0 are outside the unit circle, as required for determinacy (since both consumption and inflation are forward-looking variables).
C.5 It can be shown that the model with capitalists and hand-to-mouth households can be represented by the same system (C.1)-(C.2), except that the slope of the demand curve now is ζ C H = 1−λχ C H 1−λ and the coefficient on consumption in (C.2) curve becomes κ C H = χ C H (λ ϕ+1) . While ζ C H < 0 once the profit income effect on labor supply is removed, we now have that κ C H < 0. Consider then once more a non-fundamental increase in inflation expectations. Under an active Taylor principle the real interest rises, pushing down consumption. But in the CH model this triggers a rise in  Table 2.
inflation that makes the initial sunspot shock self-fulfilling.

D.1 Baseline specification and results
Our baseline econometric tool to document the empirical effects of an unanticipated increase in government purchases is the structural vector autoregression (SVAR) approach devised by Forni and Gambetti (2016). It combines the recursive identification strategy of Blanchard and Perotti (2002) with a news variable constructed based on data from the Survey of Professional Forecasters (SPF). Jointly, these two components allow extracting "surprise" government spending shocks from the data. We find this approach appealing for two reasons.  (2016)). Such fiscal foresight means that recursive identification, by itself, may not be sufficient to clearly distinguish between unanticipated and anticipated shocks, because some changes in fiscal expenditures are anticipated by agents even though they are unpredictable based on the variables in the econometrician's information set. Including the SPF news variable serves to enrich this information set and thus helps identify spending shocks "purified" of the anticipated component. Specifically, define the implied cumulated forecasts for government spending growth between t = s and t = h, s < h as F t (s, h) = h j=s E P t g t+ j , D.1 The empirical literature on fiscal policy offers a range of other approaches to identifying government spending shocks but is too extensive to summarize comprehensively. Relevant contributions include Caldara and Kamps (2008); Mountford and Uhlig (2009);Ramey (2011);Ben Zeev and Pappa (2015); Caldara and Kamps (2017); Ramey and Zubairy (2018). Several contributions have highlighted the challenge arising from fiscal foresight, including Yang (2005); Ramey (2011); Leeper et al. (2013); Forni and Gambetti (2016). Our identification approach is also related to that of Ricco (2015).
where E P t denotes the median expectation in the SPF in period t and g t+ j denotes the realized growth rate of government spending at t + j. In practice, we follow Forni and Gambetti (2016) and place F t (1, 4) as the second variable in the SVAR after government spending.
In terms of practical implementation, the benchmark specification is a ten-variable VAR estimated for the U.S. relying on quarterly data spanning from 1981:Q3 to 2007:Q4 and using standard Bayesian methods. The data comprise: log real government spending (consumption plus gross investment); the cumulated forecast of government spending growth over the next four quarters, F t (1, 4); D.2 log real net taxes; log real GDP; log real consumption (durables and non-durables); log real investment; log labor share; log real corporate profits; the GDP deflator; and the 10-year real interest rate. The labor share deserves particular attention. Theoretically, it is defined as the share of total compensation of the labor force in the aggregate output of the economy. The empirical counterpart to this theoretical construct is ambiguous, however. As our baseline measure we use the labor share in the domestic corporate non-financial business sector, constructed in line with the methodology of Gomme and Rupert (2004). D.3 All data sources are described in further detail below.
The starting date is dictated by the availability of SPF data for fiscal variables and coincides approximately with the beginning of the Great Moderation. The end date is prior to the start of the Great Recession to avoid potential structural breaks, but below we also report results obtained ending the sample in 2016, and using rolling windows. The lag length is chosen based on information criteria, which suggest the use of two lags for the baseline SVAR. The equations are estimated in D.2 Including the one-step-ahead forecast (h = 1) as the second variable in the SVAR, instead, and identifying the "purified" surprise spending shock as the first Cholesky shock would essentially be equivalent to the strategy followed by Auerbach and Gorodnichenko (2012) as well as Born et al. (2013). However, if the number of periods of anticipation exceeds one, then this variable will not include the news shock. By contrast, using F t (1, 4) as the news variable in the VAR increases the chances of capturing all relevant anticipation effects. We have also experimented with a news variable capturing expectations revisions (N t (1, 3) in the notation of Forni and Gambetti (2016) and the results are very similar to the F t (1, 4) approach. D.3 This approach is consistent with recent studies underscoring the importance of adjusting labor share measures for income from self-employment and housing in the context of lower-frequency movements in the labor share, an example being Gutierrez and Piton (2019). Also seem among others, Rognlie (2018). In the present context it is, furthermore, relevant that our benchmark measure excludes the public sector, alleviating any concerns that increased government spending on employment might mechanically increase the labor share of the economy as a whole. Notes: This figure shows empirical impulse responses for an unanticipated government spending shock. Impulse responses are scaled such that the log change of government spending is unity at its peak. Solid lines indicate the median posterior density of impulse responses, while the shaded area represents the 16th to 84th percentiles. All series except interest rate and inflation rate shown in proportional deviations from baseline. levels to preserve potential cointegrating relationships among the variables. We include a quadratic time trend as in Ramey (2016)   spending. Notice that in the main text we considered a five times larger shock (approximately one percent of GDP) in order to facilitate a more direct comparison with theoretical results coming from models with zero and positive government spending in steady-state, respectively. Three results stand out. Firstly, the response of real output is positive, but for only about one and a half years. The impact multiplier is 0.8, but reflecting the short-lived nature of the expansion, the present-value cumulative multiplier (Mountford and Uhlig (2009)) after two years is only 0.48; it falls to 0.16 after three years. D.4 Secondly, and considering the components of national income, aggregate consumption is crowded in following an expansionary unanticipated government spending shock, whereas investment falls sharply (cf. Ramey (2016)).
Finally, and focusing on the novel aspect of our empirical exercise, the labor share exhibits a positive, persistent and hump-shaped response. The peak effect in percentage deviations from baseline is comparable to that of GDP, but the rise is gradual. In addition, the response is statistically significant for several quarters around its peak. This results suggests that the expansionary government spending shock induces a redistribution of national income away from recipients of capital income towards workers. D.5

D.2 Results for Australia, Canada, and the UK
The scope of the new stylized fact about the response of the labor share to government spending shocks is not restricted to the US, as this section demonstrates by examining the cases of Canada, Australia and the UK. In all three cases, we limit ourselves to recursive identification given limited data availability in terms of proxies for news shocks, and we study the sample 1970:I-2007:IV for which high-quality data is available for all three countries. We use two lags for Canada and Australia and three for the UK. Figure D.2 shows that in all three countries, in response to a surprise government spending shock, the labor share initially increases in a statistically significant manner D.4 The magnitude and (relatively low) persistence of the output response is consistent with other studies that noted the decline of the output effect after 1980 in U.S. data (see, e.g., Perotti (2005) and Caldara and Kamps (2008)). D.5 These cyclical shifts in the functional income distribution -with government spending shocks inducing redistribution from firm owners to workers -may help further motivate the use of a two-agent model that distinguishes, in a stylized fashion, between households primarily relying on labor income (workers) and those owning the economy's dividendsdistributing firms (capitalists). Notes: This figure shows impulse responses for a recursively identified government spending shock, estimated separately for Australia, Canada, andthe UK (1970:I-2007:IV). The U.S. case is shown for comparison purposes. It is identified and estimated using the same method over the same sample. The median posterior density of impulse responses is displayed in form of a solid line while the 16th and the 84th percentiles are shown as dotted lines. Impulse responses are scaled such the log change of government spending is unity at its peak. All series except interest rate shown in %. before reverting back to the mean, potentially with a degree of undershooting after several years.
Qualitatively, these dynamics are remarkably close to those reported earlier for the US. D.6

D.3 Further robustness checks
In the working paper version of this article (link), we report extensive robustness checks, e.g., Jordà's (2005) local projection method and varying sub-samples. D.6 The magnitude of the labor share increase for the Canada and Australia is notably larger than observed for the US, but it is significantly smaller for the UK where, in addition, the multiplier is negative (consistent with Afonso and Sousa (2012)).  D.7 The SPF provides separate forecasts for state, local and federal government spending, whereas our variable of interest is total government spending. We aggregate the individual components to obtain a forecast for the latter, and constructed news variables on this basis. This procedure may introduce bias in our estimates, because in 1996, the U.S. Bureau of Economic Analysis (BEA) switched its method for aggregating the headline components of real GDP and the associated price indexes from the fixed-weight aggregation method to the chain-weight aggregation method. Under the latter ("Fisher ideal"), additivity of real levels does not hold (Whelan 2002  We use a Gamma distribution for the standard deviation of the government spending shock and a Beta distribution for the autoregressive parameter. For the percentage of hand-to-mouth/workers in the economy we use a Normal distribution centered around 0.5. The prior distribution for ψ W is a normal centered around 0.2265, a value chosen following the same rationale applied in calibrating the simple model and that matches both the impact quarterly and annual iMPCs. E.2 Furthermore, to estimate the Rotemberg price and wage rigidity parameters we use use the following approach.

D.4 Data Sources and Transformation
We estimate directly the slope of the price and wage Phillips curves, respectively κ = η−1 ξ and κ w = η w −1 ξ w , and then use the calibration of η and η w to back out the price/wage adjustment costs. For κ and κ w we use a Beta centered around 0.1. A Gamma distribution centered around usual values found in the literature is also used for investment adjustment costs, and the response to inflation and output in the Taylor rule. Lastly, we use a Beta distribution for interest rate smoothing.

R
. Figure E.1 and Table E.1 document the results. Three observations stand out. First, all three model variants can fit the aggregate data reasonably well in sample, with the exception of sluggish inflation dynamics and, for the TANK-UH model perhaps the persistence of the negative response of investment. Second, while the posterior means for common parameters between the three models are estimated to be very close, the results for the population share λ are notable. In E.2 Note however that this matching to micro evidence is only valid at the prior mean and that both parameters will be estimated separately. particular, to fit the macro data the TANK-UH model requires 60% of the population not to have any access to financial markets and behave in a hand-to-mouth way. As such, while the model can, in principle, replicate empirical impulse responses, it does so at the expenses of implying a fraction of hand to mouth agents in the economy that is very far from empirical estimates (see Kaplan and Violante (2014)).
Finally, and concentrating on the models with workers, consider the estimated values of λ and ψ W . Our baseline calibration approach targeted evidence on intertemporal marginal propensities to consume from micro, cross-sectional data. Matching median moments for the fraction consumed out of an unanticipated income windfall after one quarter and one year, we set λ equal to 0.8 and ψ W equal to 0.074. In the estimation on macro, time-series data, even when imposing a prior on λ that is centered around 0.5, the share of workers in the UW and CW models is likewise found to be 0.8 and 0.75, respectively. At 0.21 (in UW) and 0.15 (in CW), the portfolio adjustment cost parameter, ψ W is estimated to be slightly higher than implied by our partial equilibrium matching exercise on micro data, but given a prior centered on 0.0227 the posterior mean is likewise pushed in the 'right' direction and the 95% HPD interval for TANK-CW include the value 0.074. As such, estimation results from both macro and micro data point towards similar values of the key parameters λ and ψ W in two-agent models with workers.