Partial differential equation models in macroeconomics

It is a review written with an explicit recruitment purpose: to interest mathematicians in the PDE systems that arise naturally in macroeconomics. The opening sentence states the goal directly — “The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics.” The authors’ pitch has two halves: these PDEs “come from models designed to study some of the most important questions in economics,” and at the same time “they are highly interesting for mathematicians because their structure is often quite difficult.” The article is labelled a Review and is “One contribution of 13 to a Theme Issue ‘Partial differential equation models in the socio-economic sciences’.” Its recurring refrain — “that interesting economics and challenging mathematics go hand in hand is one of the main themes of this paper” — is stated in §2a.

A system of two coupled nonlinear PDEs: a Hamilton-Jacobi-Bellman equation for the optimal control problem of one atomistic individual, and an equation for the evolution of the distribution of individual state variables in the population. The distribution equation takes the form “a Fokker-Planck equation, Fisher-KPP equation or Boltzmann equation” depending on the model. The authors note this pairing is what Lasry and Lions termed a “mean field game,” and that Lasry, Lions have “obtained some theoretical characterizations for special cases, but many open questions remain.” A footnote carries an important scope condition on the claim that two equations suffice: heterogeneous-agent macro models “typically make the assumption that individuals have identical preferences (even though they are heterogeneous in other dimensions). It is for this reason that only two equations are sufficient… Models with heterogeneous preferences can be considered but they involve more equations.”

The authors’ stated reason is pragmatic: the discrete-time versions are workhorses but poorly understood theoretically and hard to compute, so recent work has moved to continuous time “to make progress.” Their words in §1: “Heterogeneous agent models are usually set in discrete time. While they are workhorses of modern macroeconomics, relatively little is known about their theoretical properties and they often prove difficult to compute. To make progress, some recent papers have therefore studied continuous time versions of such models. Our paper reviews this literature.” The claim being made is about tractability and mathematical structure, not about continuous time being a better description of the world.

The system is a stationary HJB equation for the value function v coupled with a Fokker-Planck equation for the density g, closed by a market-clearing condition that bonds are in zero net supply, with existence of a stationary equilibrium proved and uniqueness open. Households maximize discounted utility over consumption with wealth evolving as da = (z + r(t)a − c)dt, income following a reflected diffusion dz = μ(z)dt + σ(z)dW on a closed interval, and the state constraint a ≥ a_underbar. Market clearing is ∫ag(a,z,t)dadz = 0 — “for every dollar borrowed, there is someone else who saves a dollar.” The paper reports that Achdou, Lasry, Lions and Moll “prove the existence of a solution to (2.1)–(2.4), i.e. of a stationary equilibrium,” with the key step being to show the first moment m(r) of g goes to a_underbar as r → −∞ and becomes unbounded as r → ρ⁻, so that some r satisfies market clearing by continuity. Three questions are then listed as open: uniqueness of the stationary solution, and existence and uniqueness of the time-dependent equilibrium. The obstacle to uniqueness is identified precisely: “showing that (or finding conditions under which) the first moment of g, m(r), is monotone as a function of r.”

Treated as a state constraint, it makes the optimal saving policy’s derivative blow up at the floor and puts a Dirac mass in the stationary wealth density — a genuine mathematical complication that the authors also call one of the model’s economically most interesting predictions. Under the additional assumption a_underbar > −z_underbar/r, the state constraint binds for low enough income, and the stationary saving policy s(a,z) = z + ra + ∂_pH(∂_a v) satisfies the expansion s(a,z) ~ −s̄_z·√(a − a_underbar) with s̄_z > 0 for z_underbar ≤ z ≤ z*, “meaning that in particular the derivative ∂_a s becomes unbounded when we let a go to a_underbar.” It follows “that the stationary distribution g is unbounded and has a Dirac mass at a = a_underbar for z ≤ z*.” The authors immediately note why economists care: “What fraction of individuals in an economy such as that of the USA are borrowing constrained and how we would expect this to change when various features of the environment… change is an important question with wide-reaching policy implications.”

No, and the paper says so plainly: the wealth inequality it generates is substantially smaller than observed. “In the data, wealth is extremely unequally distributed. For example, in the USA, the top 1% richest individuals own around 35% of aggregate wealth. In contrast, it turns out that the degree of wealth inequality generated by this model is substantially smaller than the one observed in the data. This observation was first made by Aiyagari.” The paper reports that this gap “has motivated the study of richer models of individual heterogeneity and wealth accumulation” — heterogeneous returns to saving, private enterprise with imperfect capital markets, and heterogeneous time preference. This is a failure of the calibrated benchmark reported as such, not a result the paper claims.

Geometric Brownian motion with a negative drift plus a “small friction” such as a reflecting barrier delivers a stationary density that is exactly a power law with exponent ζ = 1 − 2μ̄/σ̄²; adding an optimal-stopping exit decision makes the lower barrier endogenous and turns the value-function problem into a variational inequality of the obstacle type. In §3 the process is dz/z = μ̄dt + σ̄dW with μ̄ < 0 and σ̄ > 0, with a minimum size z_min imposed as a reflecting barrier; solving the Fokker-Planck equation gives f(z) = ζ z_min^ζ z^(−ζ−1), “that is, a power law with exponent ζ > 1.” In §3a the paper reviews a simplified Luttmer model — itself “a continuous time formulation of that originally studied by Hopenhayn” — in which firms choose only whether to remain active or exit for scrap value ψ, with exiting firms mechanically replaced by an equal mass of entrants at productivity z_0. The variational inequality “now determines an endogenous threshold z_min at which firms exit,” and Luttmer shows that under geometric Brownian motion plus appropriately chosen assumptions the system solves explicitly, with a right tail f(z) = cz^(−ζ−1) and the same exponent formula. The scope condition is explicit: “While the case in which z_t follows a geometric Brownian motion is very well understood, a natural question is what the exit decision and the firm size distribution look like for more general stochastic processes” — existence, uniqueness and numerical methods for that general case are all listed open.

They replace the local Fokker-Planck law of motion with non-local Fisher-KPP or Boltzmann-type equations that generate sustained growth rather than a stationary distribution, and in the simplest case deliver growth = σ√(2α) alongside tail inequality 1/ζ = σ/√(2α) — so experimentation trades growth off against inequality but diffusion does not. In §4a agents meet at Poisson rate α, keep the better of the two productivities, and individually experiment with d log z = σ dW; the distribution of log productivity satisfies the Fisher-KPP equation ∂_t F − (σ²/2)∂_xx F = −αF(1 − F). Travelling-wave solutions F(x,t) = Φ(x − γt) exist, and from a Dirac initial mass the limiting wave has γ = σ√(2α), with the wave’s tail an asymptotic power law of exponent ζ = √(2α)/σ. The authors read the growth formula as economically natural: “it is the combination of ’experimentation’ parametrized by σ and ‘diffusion’ parametrized by α that is the engine of growth in this economy. Either force in isolation would lead to stagnation, but the two together create sustained growth.” The comparative statics are then stated carefully — a rise in σ “leads not only to higher growth, but also higher inequality,” whereas “a rise in α leads to both higher growth and lower inequality,” so that policies increasing knowledge diffusion “have the twin benefits of stimulating growth while at the same time reducing inequality.” This last sentence is framed as an invitation to imagine an extension where α and σ are choices, not as a result of the model as stated.

The system becomes a pair of integro-PDEs, with the optimal search share generally varying across productivity types, and the paper offers its balanced-growth formula there only as an explicit conjecture. Individuals split one unit of time between producing with knowledge they have and searching for ideas, with Poisson meeting rate α(s) increasing and concave and output (1 − s)e^x. Lucas and Moll study the special case σ = 0, show travelling-wave solutions v(x,t) = w(x − γt), f(x,t) = φ(x − γt) exist, and develop numerical methods for computing them. Three questions are open: existence and uniqueness of a solution, the asymptotic behaviour of f from a Dirac initial condition, and numerical methods for the time-dependent problem. On the second, the paper writes “a natural conjecture would be that the limiting distribution is a travelling wave with growth rate and tail inequality” given by integral generalizations of the σ = 0 formulae, adding “If this conjecture turns out to be correct, one prediction of the model would be that policies that increase s* for part of the population have the benefit of simultaneously stimulating growth and reducing inequality.” The hedge is the paper’s own.

Because the cross-sectional distribution becomes a random variable and must therefore enter each individual’s own state space, the Bellman equation becomes an equation over an infinite-dimensional space; the workaround is to allow aggregate shocks only at finitely many dates so that each path is finite-dimensional. With income z_t A_t where A_t is a two-state Poisson process, “the introduction of aggregate shocks creates a major difficulty: in contrast to the case without aggregate uncertainty studied in §2, it becomes necessary to include the entire distribution of income and wealth g as a state variable in the optimal control problem of individuals… calendar time t is no longer a sufficient statistic to describe the behaviour of the system.” The resulting equation (5.1) contains the functional derivative δv/δg and “is not an ordinary HJB equation because of the presence of g in the state space. The difficulty, of course, is that g is an infinite-dimensional object.” The paper’s stated reason for the workaround is directly that “its numerical approximation is very difficult.” In the workaround, shocks occur at times τ_n = nΔ for n = 1,…,N with N = 1/Δ, so there are finitely many paths (v_t, g_t), each solving forward-backward PDEs in the finite-dimensional variables (t, a, z) with transmission conditions at the shocks; “a situation with, for example, 10 shocks leads to 2¹⁰ = 1024 paths and can be simulated numerically.” The convergence claim is a hope, not a theorem: “The hope is that the model with a finite number of shocks approximates the model when A_t is a two-state Poisson process, as Δ → 0.”

Because the object policy most needs — the marginal propensity to consume out of income — depends on where a household sits in the wealth distribution, and that is exactly what a representative-agent model cannot say. The paper motivates §5 through fiscal stimulus: “A policy that is often advocated is fiscal stimulus, that is a one-time transfer from government to households with the aim of increasing their disposable income… The crucial question is usually whether such fiscal stimulus will be effective and in particular whether households will actually increase their spending.” The critical object is MPC_i(a,z,g) = ∂_z c_i(a,z,g), and the state constraint matters: households at a_underbar with low enough income consume their entire income and hence have a high MPC. The paper then reports the calibrated model’s second quantitative failure: “calibrated versions of the model do not generate high enough average MPCs when compared with the data, mainly because not enough individuals are borrowing constrained for reasonable parameter values,” which “has motivated the development of alternative models, for example models with more than one asset (e.g. Kaplan & Violante, who argue for the importance of distinguishing between liquid and illiquid assets).”

It acknowledges that the financial-frictions and finite-agent literatures obtain tractability by shrinking the heterogeneity to one or two dimensions, grants the advantage, and states its own bet that richer heterogeneity will be necessary anyway. The paper cites Brunnermeier and Sannikov, He and Krishnamurthy, Adrian and Boyarchenko and Di Tella as studying business cycles in models of financial intermediation with frictions, noting these frictions “give rise to interesting nonlinear behaviour of macroeconomic aggregates. For example, GDP may have a bimodal stationary distribution even if the driving stochastic process is unimodal” — but that “these papers all make the assumption that there are only two (or three) types of agents, so that the wealth distribution can be summarized by the share of wealth of one of the two types. The big advantage of these two approaches is that this is a one-dimensional rather than an infinite-dimensional object.” It also credits Scheinkman and Weiss with the early demonstration that in a model with only two agents, idiosyncratic shocks plus missing insurance markets can generate aggregate fluctuations. The authors’ position is stated as a judgement, not a demonstration: “However, for many interesting economic questions, it may be necessary to consider richer forms of heterogeneity. Our hope is therefore that some progress can be made on infinite dimensional problems such as (5.1).”

A continuous-time version of the Ericson-Pakes model of firm dynamics in an oligopolistic industry, which is a differential game rather than a mean field game because the number of agents is finite and each acts strategically. Two firms with profits π(z_i, q_i, q_j) — increasing in own productivity and own quantity, decreasing in the rival’s quantity — face productivity that evolves with learning-by-doing, so the drift μ is increasing in own output. In a symmetric Nash equilibrium the value function v(z,x) satisfies equation (6.1), whose Hamiltonian and optimal choice are defined jointly by a fixed point in best responses. The paper notes the model generalizes to n > 2 and to entry and exit, and observes that the literature typically assumes a finite set of firm states instead, whereas “the first route leads naturally to PDE methods. We are not aware of a general characterization of these problems.” Existence and uniqueness for (6.1) and numerical methods for a continuous state variable are both listed open.

Optimal dynamic contracts and policies, labour-market models, and normative analysis of optimal allocations — all named as omissions forced by space, not as areas where PDE methods fail. §1 lists “the large literature studying the design of optimal dynamic contracts and policies” (citing Sannikov, Williams, Farhi and Werning) and “models of the labour market” (Alvarez and Shimer; the Lentz and Mortensen review). The paper also flags a normative scope condition on everything it does cover: “throughout this paper, we focus on equilibrium allocations in which individuals take as given the actions of others rather than coordinating with them. As a result, these equilibrium allocations are in general suboptimal from the point of view of society as a whole,” with optimal allocations in heterogeneous-agent models to be analysed instead along the lines of Nuño and of Lucas and Moll.

That the literature it surveys shares one mathematical structure, that some of its problems are well-understood PDEs while others are new and hard, and that developing numerical methods for them is as important as proving theorems. The conclusion (§7) restates the unifying claim — these theories “share a common mathematical structure which can be summarize[d] by a system of coupled nonlinear PDEs or mean field game” — and closes on the recruitment note the paper opened with: “We view this to be a very promising area for future research, or, as economists like to say, we see large ‘gains from trade’ between macroeconomists and mathematicians working on PDEs.” Two further remarks recur and are worth carrying: existence results are not always the interesting ones, since “non-uniqueness is a very real possibility in many equilibrium models arising in economics, and… a better understanding of the conditions under which non-uniqueness can arise is equally interesting to economists as proving uniqueness”; and numerical method development appears in the open-question list of every single section.