Model | RDP 2025-04: HANK and the Transmission of Shocks to Demand and Supply

RDP 2025-04: HANK and the Transmission of Shocks to Demand and Supply 3. Model

Karsten Chipeniuk, Gulnara Nolan and Matt Nolan

June 2025

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Broadly speaking, a HANK model is an economic model in which agents differ in some way (heterogeneous agents) and firms face frictions when they change prices (New Keynesian). The latter feature leads to a role for monetary policy. While a number of such models currently exist in the literature, in this paper we consider the simplest extension of a classical representative agent specification that is capable of generating a distribution of net worth similar to that of the Australian economy. We do this as a starting point to demonstrate the value of these models. In particular, we assume that households have access to a single savings vehicle, government bonds, with which to smooth consumption in response to aggregate shocks. This is the same assumption underlying the broader dynamic stochastic general equilibrium (DSGE) literature.

Time is continuous and infinite, and all agents are infinitely lived. The economy is populated by a unit measure of households who consume and work, and a unit measure of monopolistic goods firms who set prices to maximise profits. There is also a representative final good firm which combines output from the monopolistic firms into a final consumption bundle. The government adjusts lump sum transfers to stabilise debt, and the central bank sets the nominal interest rate in response to inflation. In the HANK model, however, households are unable to insure against their own individual productivity shocks, resulting in differences in income, consumption, and wealth.

In the remainder of this section, the precise optimisation problems solved by the individuals in the economy are described in greater detail. Specifically, we describe the objectives that they try to maximise, what constraints they are subject to, and the resulting optimal behaviour rules. These rules, taken together, characterise equilibrium in the economy.

3.1 Households

There is a unit measure of ex ante identical households who value consumption and dislike working. Given streams for consumption c_t and hours worked n_t, a household's expected lifetime utility is given by

(1)

𝔼_{0} \int_{0}^{\infty} e^{- ρ^{t}} u (c_{t}, n_{t}) d t

where $ρ$ is the time rate of preference. Flow utility u takes the form

(2)

u (c_{t}, n_{t}) = {\begin{matrix} \frac{c_{t}^{1 - σ}}{1 - σ} - Ψ \frac{n_{t}^{1 + φ}}{1 + φ} & σ \neq 1 \\ \log (c_{t}) - ψ \frac{n_{t}^{1 + φ}}{1 + φ} & σ = 1 \end{matrix}}

in which $σ$ denotes constant relative risk aversion (CRRA), $φ$ is the reciprocal of the Frisch elasticity of labour supply, and $ψ$ is the weight on disutility from working.

Households have access to a single asset, government bonds, with which to self-insure and hence smooth consumption and working hours. Denoting household bond holdings by b_t, the time rate of change db_t/dt of these holdings is given by the household's income minus its expenditures on consumption. Income is derived from interest payments on current bond holdings, wages for hours worked, dividend payments derived from ownership of monopolistic firms, and lump sum government transfers. Interest income is given by I_tb_t, where I_t denotes the nominal interest rate. Wages are given by $(1 - τ) W_{t} z_{t} n_{t}$ , where $τ$ is the labour income tax rate, $W_{t}$ is the nominal wage rate, and z_t is individual labour productivity. Denoting the real value of firm dividends by D_t and real lump sum transfers by T_t, then government bond holdings evolve according to

(3)

\frac{d b_{t}}{d t} = I_{t} b_{t} + (1 - τ) W_{t} z_{t} n_{t} + P_{t} D_{t} + P_{t} T_{t} - P_{t} c_{t}

where P_t is the price of the consumption good and dividends are distributed independently of individual productivity or wealth. Defining the inflation rate as

(4)

Π_{t} = \frac{1}{P_{t}} \frac{d P_{t}}{d t}

it can be shown that this can be rewritten in real terms as

(5)

\frac{d a}{d t} = R_{t} a_{t} + (1 - τ) W_{t} z_{t} n_{t} + D_{t} + T_{t} - c_{t}

where $R_{t} = I_{t} - Π_{t}$ is the real interest rate, W_t is real wages, and a_t = b_t/P_t is the real value of government bond holdings. Households can borrow, but only up to some limit, so that bond holdings must satisfy

(6)

a_{t} \geq \underline{a}

In the bond evolution equation (Equation (5)), households take the inflation rate, wage rate, dividends and transfers as given, while the labour income tax rate is fixed throughout time. Individual productivity, on the other hand, evolves according to an exogenous stochastic process. Specifically, we assume that productivity can take on J different states, with jumps between states j and j′ arriving according to a Poisson process with rate $λ_{j j^{'}}$ . Critically, due to the lack of a market for assets apart from government bonds, and in particular the lack of contingencies for productivity outcomes, households are unable to insure against their individual productivity risk. As a result, households in the model vary in income reflecting their productivity level, and consequently asset accumulation, resulting in a non-trivial equilibrium distribution of wealth.

The solution to the household problem is a process for consumption and hours which maximises Equation (1) subject to the Constraints (5) and (6). It can be shown that this problem can be rewritten recursively by defining the value function v_t (a, z) as the maximum value of Equation (1) for a given level of bond holdings a and productivity state z. The solution is then given by behaviour rules c_t (a, z) and n_t (a, z) which decide consumption and hours for these levels of bond holdings and productivity, which in turn satisfy the Hamilton-Jacobi-Bellman equation

(7)

\begin{array}{l} ρ v_{t} (a, z) = \max_{c_{t}, n_{t}} [u (c_{t}, n_{t}) + ∑_{z^{'} \neq z} λ_{z z^{'}} (v_{t} (a, z^{'}) - v_{t} (a, z)) \\ + (R_{t} a + (1 - τ) W_{t} z n_{t} + D_{t} + T_{t} - c_{t}) \frac{\partial v_{t}}{\partial a} + \frac{1}{d t} 𝔼_{t} d v] \end{array}

The first order conditions for the maximisation in this equation give the intertemporal smoothing motive,

(8)

\frac{\partial u}{\partial c_{t}} = \frac{\partial v_{t}}{\partial a}

expressing the trade-off between consumption now versus in the future, and the consumption leisure trade-off,

(9)

\frac{\partial u}{\partial n_{t}} = (1 - τ) W_{t} z \frac{\partial v_{t}}{\partial c_{t}}

which reflects the disutility of working. Given the solution of the above equations for the consumption and labour rules, the distribution of wealth evolves from its initial state according to the associated optimal savings behaviour, in a manner dictated by the Kolmogorov forward equation

(10)

\frac{\partial F_{t}}{\partial t} = - \frac{\partial}{\partial a} [s_{t} (a, z) F_{t} (a, z)] - \underset{}{\sum_{z^{'} \neq z} λ_{z z^{'}} F_{t} (a, z^{'}) + λ_{z z} F_{t} (a, z)}

Here F_t denotes the probability density function of households over assets a and productivity states z, while s_t denotes the optimal time rate of savings of a household with asset holdings a in productivity state z. Thus the above equation expresses the change in the share of households in each state as the net flow of agents out of the wealth state while remaining in the same productivity state, combined with the flow of agents to new productivity states.

3.2 Firms

The supply side of the economy consists of monopolistic firms which produce individualised goods and a competitive sector which bundles these goods into a final consumption product. All firms are profit maximising. Workers are employed by monopolistic firms, who adjust their prices while incurring a cost for doing so via a Rotemberg rigidity. All profits from the sale of intermediate goods are paid as dividends to the households.

Specifically, the final goods firms choose how much of each intermediate product to include in the consumption basket, in order to minimise their costs while meeting demand, which they take as given. Given inputs $y_{j, t}, j \in [0, 1]$ , the amount of the final good produced is given by

(11)

{(\int_{0}^{1} y_{j, t}^{\frac{ε - 1}{ε}} d j)}^{\frac{ε}{ε - 1}}

where $ε$ denotes the elasticity of substitution between inputs. Denoting the price of intermediate good j by p_j,t, then these firms seek to minimise

(12)

\int p_{j, t} y_{j, t} d j

subject to the constraint that the quantities of intermediate goods demanded sum to meet demand Y_t :

(13)

{(\int_{0}^{1} y_{j, t}^{\frac{ε - 1}{ε}} d j)}^{\frac{ε}{ε - 1}} = Y_{t}

It can be shown that the solution to this problem yields the demand schedule for intermediate goods,

(14)

y_{j, t} = {(\frac{p_{j, t}}{P_{t}})}^{- ε} Y_{t}

and the zero profit condition then delivers the price level,

(15)

P_{t} = {(\int_{0}^{1} p_{j, t}^{1 - ε} d j)}^{\frac{1}{1 - ε}}

Monopolistic intermediate goods firms, on the other hand, take the demand schedule (Equation (14)) as given and hire labour n_j,t to produce their individual products according to the technology

(16)

y_{j, t} = Z_{t} n_{j, t}

where Z_t is total productivity common across all firms. These firms hire sufficiently to meet their demand, while minimising wage costs; however, due to there only being one factor of production, cost minimisation is trivial and the firms simply hire according to

(17)

n_{j, t} = \frac{1}{Z_{t}} {(\frac{p_{j, t}}{P_{t}})}^{- ε} Y_{t}

The firm then chooses the price p_j,t of its output to maximise expected discount future real profits, computed as revenue from sales minus wage and price adjustment costs

(18)

𝔼_{0} \int_{0}^{\infty} e^{- ρ t} (\frac{p_{j, t}}{P_{t}} y_{j, t} - W_{t} n_{j, t} - \frac{θ}{2} {(π_{j, t})}^{2} y_{j, t}) d t

where $π_{j, t} = \frac{1}{p_{j, t}} d p_{j, t} / d t$ and $θ$ controls the magnitude of the cost to adjust prices. We consider a symmetric equilibrium in which all firms choose the same price every period. In this case, it can be shown that the solution of the firm's problems delivers a Phillips curve,

(19)

(ρ - \frac{1}{Y_{t}} \frac{1}{d t} 𝔼_{t} d Y) Π_{t} = \frac{ε}{θ} (\frac{W_{t}}{Z_{t}} - \frac{ε - 1}{ε}) + \frac{1}{d t} 𝔼_{t} d Π

This is a similar condition to that in representative agent models, namely it relates inflation to firm marginal costs (wages) and expected future inflation.

3.3 Government

The government in the model economy taxes labour income and supplies bonds in order to finance its own expenditures, lump sum transfers to households, and interest payments on existing debt. In real terms, its budget identity is given by

(20)

R_{t} A_{t} + G_{t} + T_{t} = τ W_{t} N_{t} + \frac{d A}{d t}

where A_t is the real value of aggregate bonds outstanding, G_t is government expenditures, and N_t is aggregate efficiency hours. In order to stabilise the level of debt, transfers adjust to deviations of the debt away from its steady-state value according to

(21)

T_{t} - T = ξ (A_{t} - A) + ω_{t}^{F}

where variables without time subscripts denote steady-state values, $ξ$ controls the strength of the fiscal response to deviations of debt from target, and $ω_{t}^{F}$ represents a fiscal shock.

3.4 Monetary policy

Monetary policy follows a Taylor rule in which the nominal interest rate responds to inflation according to

(22)

I_{t} = R + μ Π_{t} + ω_{t}^{M}

in which $μ$ controls the central bank response to deviations of inflation from the steady-state value of zero, and $ω_{t}^{M}$ represents a monetary policy shock.

3.5 Representative agent model

To draw out differences in shock transmission between HANK and RANK we consider an equivalent RANK model. The representative agent analogue of the model described in the previous sections is taken to be equivalent apart from the individual productivity process for households. In this version of the model, we assume that households have access to rich financial markets which allow them to insure against all idiosyncratic risk and hence receive the wages of an average worker every period. In this case, it can be shown that the household optimisation delivers as an equilibrium condition the Euler equation

(23)

\frac{1}{C_{t}} \frac{1}{d t} 𝔼_{t} d C_{t} = R_{t} - ρ

along with the leisure/consumption trade-off

(24)

Ψ N_{t}^{φ} = (1 - τ) W_{t} z C_{t}^{- σ}

in which C_t is household consumption, N_t is household labour supply, and z is average individual labour productivity. In contrast with the heterogeneous agent specification, the distribution of wealth consists of all households simply holding the aggregate level of bonds A_t.