New Keynesian Economics with Household and Firm Heterogeneity

The paper asks whether the lessons of a decade of Heterogeneous-Agent New Keynesian research on households – that heterogeneity and financial constraints change how monetary and fiscal policy transmit to spending – “also apply to investment using a model where heterogeneous firms face idiosyncratic productivity risk and financial frictions,” arguing this question is important because “investment accounts for a large share of changes in output, both over the business cycle and in response to monetary shocks,” yet “comparatively less work has been done to incorporate heterogeneous firms” relative to heterogeneous households (Abstract; Introduction, pp. 1-2).

The representative-agent benchmark, which extends the textbook three-equation New Keynesian model with sticky wages and capital investment, shows that both consumption (following the permanent income hypothesis) and investment (following Q theory) are “dominated by direct effects” of interest rates, and that deficit-financed transfers have no effect on spending because of Ricardian equivalence on the household side and a zero MPI on the firm side (Section 2, pp. 3-4, Section 2.4). Reviewing the canonical HANK literature, the paper reiterates that once households are heterogeneous and face borrowing constraints, “monetary transmission propagates through indirect effects because high-MPC households are not very sensitive to interest rates but are quite sensitive to changes in income,” and that high-MPC households “break Ricardian equivalence,” so deficit-financed household transfers do stimulate consumption (Section 3, pp. 4-5, Section 3.3).

Firms face a collateral constraint limiting new debt to a fraction chi of new capital and a non-negativity constraint on dividends; Proposition 1 characterizes three regions of firm behavior – “unconstrained” firms (MPI = 0, investment follows the Q-theory first-order condition, independent of debt), “actively constrained” firms (both constraints bind, so investment levers up current cash on hand, with MPI = 1/(1 - chi/(1+r)), which is “generally larger than 1 because firms can lever up”), and “potentially constrained” firms (currently unconstrained but with positive probability of future binding constraints, giving an MPI strictly between 0 and the actively-constrained level, reflecting a form of “effective risk aversion”) (Section 4.2, pp. 28-31, Proposition 1). In the calibrated steady state, 66% of firms are actively constrained, 21% are potentially constrained, and 13% are unconstrained, producing an unweighted average MPI of 0.50 (Section 5.3, pp. 37-38, Figure 6).

Because cash flows are proportional to output under the model’s Cobb-Douglas production function, a given proportional change in aggregate demand changes every firm’s cash flow by the same proportion of its own size, so the aggregate effect of general-equilibrium cash-flow changes on investment is governed by the “size-weighted MPI” (the MPI weighted by each firm’s share of aggregate cash flow), which the authors calibrate at just 0.05 – an order of magnitude below the unweighted average MPI of 0.50 – because in their model high-MPI (constrained) firms “tend to be small” (Section 6.1, pp. 39-40). The authors are explicit that this “extreme negative correlation between size and MPI” is “the result of our particular calibration, not a general property of these models,” drawing an analogy to how early heterogeneous-household models similarly overstated the income-MPC correlation before direct MPC estimates revealed a weaker relationship (Introduction, p. 5; Section 6.1, pp. 40-41).

In response to a monetary easing shock, investment responds through both a “direct effect” (the traditional Q-theory channel, in which lower interest rates lower firms’ discount rate) and a new “indirect” cash-flow effect (lower real wages under sticky wages raise firms’ cash flows, which increases investment among constrained, positive-MPI firms), but the authors find the indirect channel “relatively modest, accounting for 13% of the total response,” with “the majority of the response of investment continu[ing] to be driven by the direct effects of monetary policy” (Section 6.1, pp. 39-40, Figure 7). This modest size directly reflects the small size-weighted MPI (0.05) that governs the indirect channel, as discussed in Q4.

A one-time, deficit-financed transfer to firms equal to 1% of steady-state GDP raises investment by about 3% on impact (over 0.5% of GDP), with “most of this increase… driven by the direct effect of the transfer” governed by the large unweighted MPI (0.50) – larger than the analogous household transfer’s effect on consumption, since the unweighted MPI exceeds the unweighted MPC – while the indirect, cash-flow-driven response is “an order of magnitude smaller” than the direct effect, again reflecting the small size-weighted MPI (Section 6.2, pp. 41-42, Figure 8). The authors also note the firm transfer’s effects “are not persistent,” with investment reverting close to steady state after impact, in contrast to the household model where intertemporal MPCs generate sustained persistence – a difference they attribute to the firm model not featuring similarly persistent “iMPIs” and flag as an open question for future research (Section 6.2, p. 42).

Because a spending shock raises aggregate demand, which stimulates both high-MPC household consumption and high-MPI firm investment simultaneously, the two heterogeneity channels reinforce each other, producing an output multiplier of the form dY0 = [MPC_uw . dT^h_0 + MPI_uw . dT^f_0] / (1 - MPC_weighted - MPI_weighted), in which the unweighted MPC/MPI govern the direct effect and the income-/size-weighted MPC/MPI enter the denominator nonlinearly (Section 7, pp. 42-44, equation 26). Quantitatively, combining heterogeneous households with heterogeneous firms raises the total response to a firm transfer by about 20% relative to a model with heterogeneous firms and a representative household (from a multiplier of MPI/(1-MPI) to MPI/(1-MPI-MPC)), while the analogous complementarity for a household transfer is “modest,” since the firm side’s small size-weighted MPI contributes little to the denominator (Section 7, pp. 44-45, Table 4).

Table 5 (Section 8) maps target buffer-stock assets to target capital stock, the household’s elasticity of intertemporal substitution to the firm’s adjustment-cost parameter, the household borrowing constraint to the firm’s collateral and dividend constraints, precautionary saving to firms’ “concern about hitting constraints,” concave consumption to concave investment, and MPCs/iMPCs to MPIs/iMPIs, concluding that “the presence of high-MPI firms creates new transmission channels for investment, just like the presence of constrained, high-MPC households did for consumption” (Section 8, pp. 45-46, Table 5). The authors call the firm side of this table “as underexplored as their cousins were on the household side ten years ago” and highlight the absence of direct quasi-experimental estimates of firm-level MPIs – unlike the well-developed empirical MPC literature – as the paper’s central limitation and the field’s most pressing open question (Abstract; Introduction, p. 5; Section 8, p. 46).

The authors report developing code, built on the sequence-space Jacobian methodology of Auclert, Bardóczy, Rognlie and Straub (2021), for solving models combining heterogeneous firms and heterogeneous households together, noting that “the full model, with both sources of heterogeneity and three state variables for firms, can be solved in a matter of minutes” (Introduction, p. 6), which they present as lowering the barrier to entry for future work extending their canonical heterogeneous-firm model.