HANK: Heterogeneous Agent New Keynesian Models

• This paper: Kaplan, Moll, and Violante (2018)

• Framework for quantitative analysis of the transmission mechanism of monetary policy

• Three Building Blocks:

  1. Uninsurable idiosyncratic income risk
  2. Nominal price rigidities
  3. Assets with different degrees of liquidity

How Monetary Policy Works in RANK

• Total consumption response to a drop in real rates:

\[ C \text{ response}=\underbrace{ \text{direct response to } r}_{>95\%} + \underbrace{\text{indirect effects due to } Y}_{<5\%} \]

• Direct response is everything: pure intertemporal substitution

However, data suggest:

1. Low sensitivity of \(C\) to \(r\)
2. Sizable sensitivity of \(C\) to \(Y\)
3. Micro sensitivity is vastly heterogeneous and depends crucially on household balance sheets

How Monetary Policy Works in HANK

• Once matched to micro data, HANK delivers realistic:
  • Wealth distribution: small direct effect
  • MPC distribution: large indirect effect (depending on \(\Delta Y\))

\[ C \ \text{response} = \underbrace{\text{direct response to } r}_{\text{RANK:}~>95\%~|~\text{HANK:}~<1/3} + \underbrace{\text{indirect effects due to } Y}_{\text{RANK:}~<5\%~|~\text{HANK:}~>2/3} \]

• Overall effect depends crucially on fiscal response, unlike in RANK where Ricardian equivalence holds.

Households

• A continuum of households is indexed by liquid assets \(b\), illiquid assets \(a\), and idiosyncratic productivity \(z\)
• Time is continuous and the aggregate state is the cross-sectional distribution \(\mu_t(da,db,dz)\)
• Productivity \(z\) follows an exogenous Markov process
• Households die at Poisson rate \(\zeta\); newborns enter with zero wealth and \(z\) drawn from the ergodic distribution

Households choose consumption \(c_t\), labor supply \(\ell_t\), and deposits \(d_t\) into the illiquid account:

\[ \max \; E_0 \int_0^\infty e^{-(\rho+\zeta)t} u(c_t,\ell_t)\,dt \]

where \(u\) is increasing and concave in consumption, and decreasing and convex in labor.

Household Balance Sheets

Liquid and illiquid assets evolve according to

\[ \dot b_t = (1-\tau_t) w_t z_t \ell_t + r_t^b(b_t)b_t + T_t - d_t - \chi(d_t,a_t) - c_t \]

\[ \dot a_t = r_t^a a_t + d_t \]

subject to

\[ b_t \geq -\underline b, \qquad a_t \geq 0. \]

• Liquid borrowing pays the higher rate \(r_t^{b-} = r_t^b + \kappa\)
• Illiquid assets earn a higher return in equilibrium: \(r_t^a > r_t^b\)
• \(d_t < 0\) denotes withdrawals from the illiquid account

Why Two Assets Matter

The transaction cost for moving resources across accounts is

\[ \chi(d,a) = \chi_0 |d| + \chi_1 \left|\frac{d}{a}\right|^{\chi_2} a \]

• The linear term \(\chi_0 |d|\) creates an inaction region
• The convex term ensures finite deposit and withdrawal rates
• This wedge generates households with very different liquid positions and MPCs
• Monetary policy therefore redistributes both through interest rates and through income changes

Production Side

A representative final-good producer aggregates differentiated intermediate goods:

\[ Y_t = \left( \int_0^1 y_{j,t}^{\frac{\varepsilon-1}{\varepsilon}} dj \right)^{\frac{\varepsilon}{\varepsilon-1}} \]

which implies demand for variety \(j\):

\[ y_{j,t} = \left(\frac{p_{j,t}}{P_t}\right)^{-\varepsilon} Y_t, \qquad P_t = \left( \int_0^1 p_{j,t}^{1-\varepsilon} dj \right)^{\frac{1}{1-\varepsilon}}. \]

Each intermediate producer has technology

\[ y_{j,t} = k_{j,t}^{\alpha} n_{j,t}^{1-\alpha}, \]

so real marginal cost is

\[ m_t = \left(\frac{r_t^k}{\alpha}\right)^{\alpha} \left(\frac{w_t}{1-\alpha}\right)^{1-\alpha}. \]

Price Setting

Intermediate firms face Rotemberg price adjustment costs

\[ \Theta_t(\pi_t) = \frac{\theta}{2}\pi_t^2 Y_t, \qquad \pi_t = \frac{\dot P_t}{P_t}. \]

This yields a continuous-time New Keynesian Phillips curve:

\[ \left(r_t^a - \frac{\dot Y_t}{Y_t}\right)\pi_t = \frac{\varepsilon}{\theta}(m_t - m^*) + \dot \pi_t, \qquad m^* = \frac{\varepsilon-1}{\varepsilon}. \]

• Inflation rises when markups are too low, i.e. when \(m_t > m^*\)
• Future profits are discounted at the illiquid return \(r_t^a\)

Illiquid Asset Composition

• The illiquid account contains physical capital and equity claims on intermediate-firm profits
• Let \(q_t\) be the share price and \(s_t\) the household’s equity holdings, so

\[ a_t = k_t + q_t s_t \]

• Since resources move freely within the illiquid account, no arbitrage implies

\[ \frac{\Pi_t + \dot q_t}{q_t} = r_t^k - \delta =: r_t^a \]

• This lets the model collapse capital and equity into one composite illiquid asset with return \(r_t^a\)

Monetary And Fiscal Policy

The central bank sets the nominal liquid return via a Taylor rule:

\[ i_t = \bar r^b + \phi \pi_t + \epsilon_t, \qquad \phi > 1, \]

and the Fisher equation pins down the real liquid return:

\[ r_t^b = i_t - \pi_t. \]

The government budget constraint is

\[ \dot B_t^g + G_t + T_t = \tau_t \int w_t z \ell_t(a,b,z)\, d\mu_t + r_t^b B_t^g. \]

• Fiscal closure can come through taxes \(\tau_t\), transfers \(T_t\), or spending \(G_t\)
• Because Ricardian equivalence fails, the fiscal response is quantitatively central in HANK

Equilibrium

An equilibrium consists of household choices, firm choices, prices, policy paths, and a distribution \(\mu_t\) such that agents optimize and markets clear.

Key market-clearing conditions are:

\[ B_t^h + B_t^g = 0, \qquad B_t^h = \int b \, d\mu_t \]

\[ K_t + q_t = A_t, \qquad A_t = \int a \, d\mu_t \]

\[ N_t = \int z \ell_t(a,b,z)\, d\mu_t \]

\[ Y_t = C_t + I_t + G_t + \Theta_t + \chi_t + \kappa \int \max\{-b,0\}\, d\mu_t \]

• The liquid-rate shock affects consumption directly through \(r_t^b\)
• In HANK, the larger amplification comes indirectly through wages, illiquid returns, and fiscal adjustment

Continuous-Time HA Models

• Following Achdou-Han-Lasry-Lions-Moll, solving a heterogeneous-agent model means solving a system of PDEs

• There are two core objects:

  1. A Hamilton-Jacobi-Bellman (HJB) equation for individual optimal choices
  2. A Kolmogorov Forward (KF) equation for the evolution of the cross-sectional distribution

• This apparatus is extremely general:

  1. Heterogeneous-household models: Bewley, Huggett, Aiyagari, HANK
  2. Heterogeneous-producer models: e.g. Hopenhayn-style environments

• A simple but powerful numerical method is the finite difference method

• See Benjamin Moll’s continuous-time HA codes: http://www.princeton.edu/~moll/HACTproject_htm

HJB: Individual Optimization

Let \(V(a,b,z)\) denote the value of a household with illiquid assets \(a\), liquid assets \(b\), and productivity \(z\).

In steady state, the HJB takes the form

\[ (\rho + \zeta) V(a,b,z) = \max_{c,\ell,d} \left\{ u(c,\ell) + V_a(a,b,z)\big(r^a a + d\big) \\ + V_b(a,b,z)\big((1-\tau)w z \ell + r^b(b)b \\ + T - d - \chi(d,a) - c\big)\\ + \mathcal{L}^z V + \zeta V_0(z) \right\}, \]

where \(\mathcal{L}^z\) is the generator for idiosyncratic income risk.

• The HJB pins down policy rules for \(c(a,b,z)\), \(\ell(a,b,z)\), and \(d(a,b,z)\)
• Once we know these policies, we also know the drift of assets at each point in the state space

FOCs And “Tomorrow Is Today”

Because time is continuous, the first-order conditions are local and look static.

• For liquid assets, the relevant shadow value is \(V_b(a,b,z)\)
• With separable CRRA utility, the consumption FOC is

\[ u_c(c,\ell) = V_b(a,b,z) \]

and if \(u(c,\ell)\) is CRRA in consumption, this becomes

\[ c^{-\gamma} = V_b(a,b,z). \]

• This is what Moll means by “tomorrow is today”
• Instead of an expectation over a discrete next period, the HJB works with infinitesimal changes in the value function
• As a result, one can often compute the optimal controls directly from local derivatives of \(V\)

KF: Evolution Of The Distribution

Given optimal household policies, the cross-sectional distribution \(g(a,b,z,t)\) evolves according to the Kolmogorov Forward equation

\[ \partial_t g = - \partial_a \big( \dot a(a,b,z) g \big) - \partial_b \big( \dot b(a,b,z) g \big) + \mathcal{L}^{z*} g, \]

where \(\mathcal{L}^{z*}\) is the adjoint of the income-shock generator.

• The KF equation is the law of motion for the distribution

• In steady state, it sets net probability flows equal to zero

• Solving the model therefore means finding:

  1. Policies that satisfy the HJB
  2. A distribution that is stationary under those policies

Finite Difference Method

• Put a grid on the state space for \((a,b,z)\)
• Approximate derivatives like \(V_a\) and \(V_b\) with one-sided finite differences
• Use an upwind scheme so that the numerical derivative follows the direction of the endogenous drift
• This turns the continuous PDE problem into a sparse linear algebra problem

Schematic steady-state HJB system:

\[ \rho v = u(v) + A(v) v \]

where \(A(v)\) is the discretized generator implied by policies and drifts.

• Solving the Bellman equation becomes solving or iterating on a sparse matrix system
• After that, the distribution is recovered from the corresponding discretized KF equation

Why Continuous Time Is Computationally Convenient

1. Borrowing constraints are boundary conditions

• In continuous time, inequality constraints show up at the boundary of the state space
• Interior FOCs still hold with equality
• This is cleaner than discrete time, where occasionally binding constraints directly distort Euler equations

2. Sparsity

• Over an infinitesimal interval, a state can only move to nearby grid points
• In one dimension this often gives tridiagonal matrices
• In multiple dimensions the matrices are not literally tridiagonal, but they remain very sparse

3. Tight link between economics and numerics

• The drift of assets determines both the HJB discretization and the KF discretization
• The numerical objects have a direct economic interpretation as local probability flows

Two Birds With One Stone

The HJB and KF problems are tightly linked.

• Discretize the HJB and obtain a generator matrix \(A\)
• Then the discretized KF equation uses the transpose of that same matrix:

\[ \partial_t g = A^\top g \]

and in steady state

\[ A^\top g = 0, \qquad \mathbf{1}^\top g = 1. \]

• Reason: the KF operator is the adjoint of the HJB operator
• So once the HJB is coded correctly, much of the distributional machinery comes “for free”
• This is one of the main computational insights in the Achdou et al. approach

Aggregate Shocks

• Achdou et al. and the baseline HANK model here focus on idiosyncratic shocks only

• Aggregate shocks make the problem much harder because the entire distribution becomes a state variable

• A companion line of work develops efficient methods for this case:

  • “When Inequality Matters for Macro and Macro Matters for Inequality”
  • Open-source Matlab toolbox: https://github.com/gregkaplan/phact

• Conceptually, this extends linearization methods such as Campbell (1998) and Reiter (2009)

• Intuition: instead of a single global slope, the method tracks different local slopes at different points in the state space

• This is the bridge from continuous-time HA methods to Krusell-Smith-style environments with aggregate risk