Incomplete Markets and Aggregate Demand

The paper asks: “What are the effects of market incompleteness on aggregate demand?” (Introduction, p. 2), framing this as the underdeveloped counterpart to the extensive scrutiny the “supply block” (the Phillips curve) has received in the New Keynesian literature. Werning notes that behind the standard representative-agent Euler equation “lies an assumption of complete markets or the adoption of a representative agent,” an assumption “easily rejected at face value” given well-documented idiosyncratic income risk and incomplete insurance, even though its tractability has made it hard to displace (Introduction, pp. 1-2).

Werning shows that naively aggregating individual Euler equations fails: with idiosyncratic uncertainty and no binding constraints, Jensen’s inequality implies aggregate consumption growth strictly exceeds the representative-agent prediction, while with binding borrowing constraints (and no uncertainty) it falls strictly short – so combining both, “there is no telling in which direction the departure…goes” (Section 2.3, pp. 12-13). His approach instead exploits an additional equilibrium condition not used by simple aggregation exercises: the general-equilibrium requirement that aggregate consumption equal aggregate income, contrasting this with earlier aggregation work by Huggett (1993) and Aiyagari (1994), who “aggregate consumption and savings for given interest rates…taking the income process as given” (Section 2.2, footnote 4, p. 10).

With no borrowing and no outside assets, no intertemporal trade is possible in equilibrium, so the allocation coincides exactly with autarky (each household simply consumes its own income each period), and Proposition 1 shows the equilibrium interest rate path is pinned down by a single generalized Euler relation, g_t(R_t, C_t, C_t+1) = 0 (Section 3.1, pp. 13-15). Because wealth is zero for every household at all times under this assumption, “there is no wealth distribution to keep track of,” so unlike a Krusell-Smith-type numerical solution, computing the aggregate path “does not require carrying a large endogenous state and confronting the curse of dimensionality” (Section 3.1, p. 15).

Proposition 2 shows this holds when utility is power (CRRA) with multiplicative taste shocks and individual income is proportional to aggregate income; the aggregate relation becomes U’(C_t) = beta_t R_t U’(C_t+1), with the discount factor beta_t an explicit function (equation 16) of idiosyncratic uncertainty in relative income growth (Section 3.2, pp. 15-16). Critically, “in periods with greater uncertainty or downward tail risk…we can expect the discount factor to be higher,” which lowers current consumption for given future consumption and the interest rate – so, as the paper stresses under the heading “Are Incomplete Markets Irrelevant? No, not at all,” market incompleteness still has a first-order level effect on demand even where it leaves the elasticity to interest rates unchanged (Section 3.2, p. 16).

Wherever the representative-agent representation holds, “aggregate consumption reacts to changes in the path of interest rates in the same manner as in representative agent models,” so “forward guidance is exactly as effective as in representative agent models” (Introduction, p. 4; Section 3.2, p. 16), a result the paper explicitly contrasts with McKay, Nakamura, and Steinsson (2015), who find forward guidance is muted under incomplete markets. Werning is careful to frame this as one specific, if important, benchmark: departing from the conditions of Proposition 2 – for instance if the elasticity of individual to aggregate income is not exactly one across all types – can make current consumption react more or less than one-for-one to future consumption depending on model specifics (Section 3.3, pp. 17-19).

In the intensive/extensive-margin employment example, when gamma < 1 (some of the income adjustment occurs via a fluctuating probability of underemployment, lambda(Y)) the paper derives (Proposition 4) a modified Euler relation in which the discount factor is strictly decreasing in future aggregate consumption – unlike the gamma = 1 case, where it is constant (Section 3.4, pp. 19-21). Intuitively, “when gamma < 1 we are departing from the neutrality result by assuming that lower aggregate income increases individual income risk,” so anticipated higher future income lowers future uncertainty and the precautionary-saving motive, amplifying the usual consumption-smoothing response to expected future rate cuts (Section 3.4, p. 21). The paper connects this countercyclical-risk assumption to the equity-premium asset-pricing literature (citing Constantinides and Duffie 1996; Alvarez and Jermann 2001) and notes it has empirical support (Storesletten et al. 2004; Guvenen et al. 2014).

With logarithmic utility, income proportional to aggregate income, and borrowing constraints proportional to aggregate income, Proposition 5 (zero initial bond holdings) and Proposition 6 (arbitrary initial bond holdings) show a standard Euler equation, U’(C_t) = beta_t R_t U’(C_t+1), continues to hold even though the underlying household allocation is nontrivial and generally has no closed form (Section 4.1, pp. 22-26). The key technical device is that, given one equilibrium for a normalized economy with constant output, any other equilibrium path for {C_t, R_t} satisfying the aggregate Euler equation can be constructed by proportionally rescaling household consumption and wealth by C_t, since the budget and borrowing constraints are homogeneous of degree one in aggregate income under these assumptions (Section 4.1.1, pp. 24-25).

“Yes and no”: Proposition 5 shows the amount of liquidity has no effect on the response of aggregate consumption to changes in current and future interest rates – that response is identical to a representative agent’s – but liquidity still matters at the microeconomic level for consumption smoothing, and this shows up in the level of the discount factor beta_t, so liquidity has a genuine macroeconomic effect on the level of consumption and interest rates, just not on their comovement (Section 4.1.1, “Does the Level of Liquidity Matter?”, p. 24). The paper also connects the discount factor’s dependence on idiosyncratic uncertainty to New Keynesian “natural interest rate” logic: periods of heightened uncertainty or deleveraging (as in Eggertsson and Krugman 2012; Guerrieri and Lorenzoni 2011) raise beta_t and so lower the natural rate, potentially pushing the economy against the zero lower bound (Section 4.1.1, “Discounting and Natural Interest Rates,” p. 25).

In a three-period example with uncertainty only at the middle date, Proposition 7 (slack borrowing constraints) shows that when the elasticity of intertemporal substitution 1/sigma is below one (sigma > 1), a drop in the future interest rate R1 raises consumption today, C0, by more than the standard representative-agent effect, because the market value of the outside asset rises faster than output when sigma > 1, expanding liquidity and easing consumption smoothing; the effect reverses when sigma < 1 (Section 4.2, pp. 27-28). Proposition 8 shows this amplification is stronger still once some households are liquidity-constrained, because a rising asset value has a direct, one-for-one marginal effect on the consumption of constrained households (Section 4.2, p. 28).

Proposition 9 shows that in a real business cycle model with capital, idiosyncratic productivity risk, and incomplete markets, if utility is logarithmic and capital fully depreciates each period (the Brock-Mirman case), then aggregate capital, labor, and consumption dynamics are exactly identical to the corresponding complete-markets representative-agent economy – individual consumption and labor simply scale with the aggregate, ci(z^t,s^t) = ci(z^t) C(s^t) (Section 6.2, pp. 32-33). The paper is explicit that “this occurs despite potentially arbitrarily large departures at the microeconomic household level in the allocation,” and frames the result as complementing, with an exact analytical counterpart under a special (full-depreciation) case, the approximate aggregation that Krusell and Smith (1998) found numerically in a more general version of the same class of model (Section 6, pp. 31-32).

Werning summarizes that idiosyncratic uncertainty, incomplete markets, and borrowing constraints affect both the level of aggregate demand (greater uncertainty or tighter borrowing constraints “typically depress consumption”) and, outside the benchmark cases identified in the paper, its sensitivity to interest rates – with the paper’s examples suggesting that in “plausible cases,” and especially for future interest rate changes, “consumption becomes more sensitive” than in the representative-agent benchmark (Conclusion, p. 33). He is careful throughout not to claim generality beyond the specific benchmark conditions identified (zero liquidity with proportional income and multiplicative taste shocks; positive liquidity with logarithmic utility and proportional income/borrowing limits), describing departures from these conditions as “suggestive” and dependent “ultimately…on the particular assumptions one makes” (Section 2.3, footnote 7, p. 12; Conclusion, p. 33).