Monetary Policy and Labor Market Friction in a HANK Model

Zelin Deng

doi:10.1515/bejm-2024-0148

Article

Monetary Policy and Labor Market Friction in a HANK Model

Zelin Deng

Published/Copyright: June 18, 2025

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From the journal The B.E. Journal of Macroeconomics Volume 25 Issue 2

Abstract

I develop a two-asset heterogeneous-agent New Keynesian model with search and matching frictions in the labor market, which extends the transmission mechanism of monetary policy to household consumption. Uninsurable countercyclical unemployment risk plays a crucial role in the transmission of monetary shocks to consumption through a novel channel driven by countercyclical precautionary saving motives. Following an increase in the real interest rate, unconstrained households raise their liquid savings and reduce current consumption to insure against the risk of lower future individual labor income, resulting from longer expected unemployment durations. This mechanism accounts for 16 % of the total decline in consumption in a model calibrated to a realistic wealth distribution. The strength of the countercyclical precautionary saving motive depends on the degree of wage rigidity and the fiscal policy rule in general equilibrium. Additionally, I extend the sequence-space Jacobian algorithm to a continuous-time framework, where the efficiency of constructing partial equilibrium Jacobians is enhanced by a generalized approach to handling a large number of income grid points in the heterogeneous-agent block.

Keywords: New Keynesian; unemployment risk; sequence-space-jacobians; monetary policy

JEL Classification: C61; E42; E24; E12

Corresponding author: Zelin Deng, School of Economics, Shanghai University of Finance and Economics, Guoding Road 777, Yangpu District, Shanghai, China, E-mail: dengzelin@163.sufe.edu.cn

I would like to thank Youzhi Yang and Shengliang Ou for valuable advice.

Appendix A: Equations of RANK Models

A.1 The Simple RANK Model in Section 2

The model is standard and shares many similarities with the production side of the HANK model in the main text. I summarize the equations as follows:^[22]

(A.1) C ̇ t / C t = γ r t b − ρ ( Euler equation ) ,

(A.2) r t b W t = x − w − δ W t + ∂ t W t Job creation ,

(A.3) κ = f t W t Free entry ,

(A.4) U ̇ t = δ N t − θ t f t U t , N t = 1 − U t LoM of unemployment

(A.5) f t = μ θ t − ς Matching rule ,

(A.6) Y t = A N t Production function ,

(A.7) x = 1 / A , Labor price ,

(A.8) D t = Y t − w N t − κ v t Profit ,

(A.9) r t b B = − S t = − T t − T 0 U t Fiscal budget ,

(A.10) C t = Y ̃ t , Y ̃ t = Y t − κ v t Market clear .

The correspondingly linearized equations are

(A.11) C ̂ ̇ t / C = γ r ̂ t b ,

(A.12) f ̂ ̇ t = k f ̂ t − f r ̂ t b ,

(A.13) U ̂ ̇ t = − δ + θ f U ̂ t − θ ̂ t f + θ f ̂ t U ,

(A.14) f ̂ t = − ς f θ ̂ t θ ,

(A.15) Y ̂ t = A N ̂ t ,

(A.16) D ̂ t = Y ̂ t − w N ̂ t − κ v ̂ t ,

(A.17) r ̂ t b B = − S ̂ t = − T ̂ t − T 0 U ̂ t ,

(A.18) C ̂ t = Y ̂ t − κ v ̂ t ,

(A.19) r ̂ ̇ t b = − ρ ζ r ̂ t b .

Note that (A.12) follows by the combination of linearization to (A.2) and (A.3) with form r t b + δ κ / f t = x − w − κ f ̇ t / f t 2 .

The calibration basically comes from the baseline model with several differences. First, since there is no physical capital in this simple RANK-SAM model, I normalize the output as 1 and choose steady-state TFP as A = Y/N. Second, discount rate is equal to r ^b required by Euler equation. Third, there is no government purchase, and the unemployment insurance and transfer are calibrated by steady-state wage and labor share: w = 0.6Y/N, T ⁰ = 0.4w and T = −r ^b B − T ⁰ U.

A.2 The Two-Asset RANK Model

The difference with the HANK model in main text is that there is no heterogeneity in households. The representative household problem is

(A.20) max C t , D t , B t , A t t ≥ 0 ∫ 0 ∞ e − ρ t u ( C t ) d t

subject to

(A.21) B ̇ t = ( 1 − τ t ) Z t + r t b B t + T t − D t − χ ( D t , A t ) − C t ,

(A.22) A ̇ t = r t a A t + D t ,

where labor earning is Z t = w t N t + T t 0 ( 1 − N t ) . The Lagrangian is

(A.23) L = ∫ 0 ∞ e − ρ t u ( C t ) + λ 1 , t ( 1 − τ t ) Z t + r t b B t + T t − D t − χ ( D t , A t ) − C t − B ̇ t + λ 2 , t r t a A t + D t − A ̇ t d t

Integrating by parts yields

(A.24) L = ∫ 0 ∞ e − ρ t u ( C t ) + λ 1 , t ( 1 − τ t ) Z t + r t b B t + T t − D t − χ ( D t , A t ) − C t + λ 2 , t r t a A t + D t − ρ λ 1 , t B t + λ ̇ 1 , t B t − ρ λ 2 , t A t + λ ̇ 2 , t A t d t + λ 1,0 B 0 + λ 2,0 A 0

First order conditions follow as

(A.25) u ′ ( C t ) = λ 1 , t ,

(A.26) − λ 1 , t 1 + ∂ d χ + λ 2 , t = 0 ,

(A.27) r t b + ξ λ 1 , t − ρ λ 1 , t + λ ̇ 1 , t = 0 ,

(A.28) − λ 1 , t ∂ a χ + r t a λ 2 , t − ρ λ 2 , t + λ ̇ 2 , t = 0 ,

(A.29) λ 1,0 = 0 , λ 2,0 = 0 .

Substituting (A.25) into (A.27) and (A.26) into (A.28), the optimal path of C _t, D _t should satisfy

(A.30) C ̇ t C t = − u ′ ( C t ) u ′ ′ ( C t ) C t r t b − ρ = γ r t b − ρ ,

(A.31) r t a − r t b 1 + ∂ d χ − ∂ a χ − ∂ d d χ D ̇ t − ∂ d a χ r t a A t + D t = 0

Setups in other sectors remains unchanged deliberately. I choose ρ and χ ₀ to match the liquid and illiquid wealth implied by main text. First-order conditions combined with adjustment cost (20) implies ρ = r ^b = 0.005 and χ 0 = 1 − χ 1 ( 1 − χ 2 ) ( r a ) χ 2 / ( r a − r b ) − χ 1 χ 2 ( r a ) χ 2 − 1 = 0.99 . The recalibrated adjustment cost gives a residual consumption C = 0.51.

Appendix B: Proofs

B.1 Proof of Proposition 1

Proof.

Substituting the linearized budget constraint B ̂ ̇ t = r ̂ t b B + ρ B ̂ t + Y ̂ t + S ̂ t − C ̂ t into the linearized Euler equation C ̂ ̇ t / C = γ r ̂ t b gives a bond demand equation which satisfies a second-order differential system

(B.1) B ̂ ̈ t − ρ B ̂ ̇ t = F t ,

where F t = r ̂ ̇ t b B + Y ̂ ̇ t + S ̂ ̇ t − γ C r ̂ t b .

It implies a zero eigenvalue and a positive eigenvalue equal to ρ associated with forward-looking saving B ̂ ̇ t . The solutions are then given by

(B.2) B ̂ ̇ t = − ∫ t ∞ e − ρ ( τ − t ) F τ d τ B ̂ t = ρ − 1 ∫ 0 ∞ e − ρ τ F τ d τ − ∫ 0 t F τ d τ − ∫ t ∞ e − ρ ( τ − t ) F τ d τ

Plugging them into the budget constraint, it follows

(B.3) C ̂ t = r ̂ t b B + ρ B ̂ t + Y ̂ t + S ̂ t − B ̂ ̇ t = r ̂ t b B + ρ B ̂ t + Y ̂ t + S ̂ t + ∫ t ∞ e − ρ ( τ − t ) r ̂ ̇ τ b B + Y ̂ ̇ τ + S ̂ ̇ τ − γ C r ̂ τ b d τ = − γ C ∫ t ∞ e − ρ ( τ − t ) r ̂ τ d τ + ρ ∫ t ∞ e − ρ ( τ − t ) r ̂ τ b B + Y ̂ τ + S ̂ τ d τ + ρ B ̂ t .

Denote F t d = r ̂ ̇ t b B + Y ̂ ̇ t + S ̂ ̇ t and represent F _t as F t = F t d − γ C r ̂ t b , then the explicit expression of wealth follows

(B.4) B ̂ t = ρ − 1 ∫ 0 ∞ e − ρ τ F τ d τ − ∫ 0 t F τ d τ − ∫ t ∞ e − ρ ( τ − t ) F τ d τ = ρ − 1 ρ ∫ 0 ∞ e − ρ τ r ̂ τ b B + Y ̂ τ + S ̂ τ d τ − ρ ∫ t ∞ e − ρ ( τ − t ) r ̂ τ b B + Y ̂ τ + S ̂ τ d τ − F 0 d + F 0 d − F t d + F t d − γ C ∫ 0 ∞ e − ρ τ r ̂ τ b d τ − ∫ 0 t r ̂ τ b d τ − ∫ t ∞ e − ρ ( τ − t ) r ̂ τ b d τ = ∫ 0 ∞ e − ρ τ r ̂ τ b B + Y ̂ τ + S ̂ τ d τ − ∫ t ∞ e − ρ ( τ − t ) r ̂ τ b B + Y ̂ τ + S ̂ τ d τ − ρ − 1 γ C ∫ 0 ∞ e − ρ τ r ̂ τ b d τ − ∫ 0 t r ̂ τ b d τ − ∫ t ∞ e − ρ ( τ − t ) r ̂ τ b d τ .

Plugging this bond demand function into (B.3) then gives the aggregate consumption function that only depends on interest and income:

(B.5) C ̂ t = ∫ 0 ∞ ∂ C ̂ t ∂ r ̂ τ b r ̂ τ b d τ + ∫ 0 ∞ ∂ C ̂ t ∂ Y ̂ τ Y ̂ τ d τ + ∫ 0 ∞ ∂ C ̂ t ∂ S ̂ τ S ̂ τ d τ ,

where

(B.6) ∂ C ̂ t ∂ r ̂ τ b = B ρ e − ρ τ − γ C e − ρ τ + γ C I { τ < t } , ∂ C ̂ t ∂ Y ̂ τ = ∂ C ̂ t ∂ S ̂ τ = ρ e − ρ τ .

□

B.2 Proof of Proposition 2

Proof.

Using (A.14), the corresponding solution to f ̂ t and θ ̂ t are

(B.7) f ̂ t = ∫ t ∞ f r ̂ τ e − k ( τ − t ) d τ = f r ̂ 0 b e k t ∫ t ∞ e − ( k + ρ ζ ) τ d τ = K f r ̂ 0 b e − ρ ζ t , θ ̂ t = K θ r ̂ 0 b e − ρ ζ t ,

where K f = f ρ ζ + k , K θ = − K f θ ς f and k = (x − w)f/κ is assumed greater than 0. Substituting these results into the evolution of unemployment (A.13) yields

(B.8) U ̂ t = − ∫ 0 t e − ( δ + θ f ) ( t − τ ) θ ̂ τ f + θ f ̂ τ U d τ = 1 − ς ς θ K f U r ̂ 0 b e − ( δ + θ f ) t ∫ 0 t e − ρ ζ − δ − θ f τ d τ = K u e − ( δ + θ f ) t − e − ρ ζ t r ̂ 0 b ,

where K u = 1 − ς ς ρ ζ − δ − θ f θ K f U . Note that d U ̂ t d r ̂ 0 b is always positive, i.e. unemployment is always countercyclical. Therefore,

(B.9) Ω t u = ∫ 0 ∞ ∂ C ̂ t ∂ U ̂ τ U ̂ τ d τ = ( T 0 − w ) K u ∫ 0 ∞ ρ e − ρ τ e − ( δ + θ f ) τ − e − ρ ζ τ d τ = ρ ( T 0 − w ) K u ρ ζ − δ − θ f ( ρ + δ + θ f ) ( ρ + ρ ζ ) .

□

B.3 Proof of Proposition 3

Proof.

In the symmetric equilibrium, the total profit of intermediate firms is described as (49) with equity price q t I = J t ( P t ) , thus r t q t I = Π t I + q ̇ t I . For the capital producer, total equity is q t K ≡ q t K t and note that

(B.10) q ̇ t K ≡ d ( q t K t ) d t = q ̇ t K t + q t K ̇ t

With (51) and (39), it implies

(B.11) r t q t K = Π t K + q ̇ t K

Since I define the value per filled vacancy as W _t, the value or equity for all jobs is q t W ≡ W t ( 1 − U t ) . Multiply (32) both side by 1 − U _t, and combine it with free entry condition (35),

(B.12) ( r t + δ ) q t W = ( x t − w t ) L t − κ V t + θ t f t U t W t + ( 1 − U t ) W ̇ t

Note that

q ̇ t W ≡ d [ ( 1 − U t ) W t ] d t = − U ̇ t W t + ( 1 − U t ) W ̇ t = − ( δ ( 1 − U t ) − θ t f t U t ) W t + ( 1 − U t ) W ̇ t

where the second line uses the motion of the unemployment rate. It then simplified to be

(B.13) r t q t W = Π t W + q ̇ t W

□

B.4 Proof of Lemma 4

Proof.

(B.14) Δ f ≡ f ( s i t + Δ s i t ) − f ( s i t )

(B.15) = f ( s i t − β i s i t Δ t + ( η i − s i t ) Δ P i t ) − f ( s i t )

(B.16) = f ( s i t − β i s i t Δ t ) ( 1 − Δ P i t ) + f ( η i − β i s i t Δ t ) Δ P i t − f ( s i t )

Notice that

(B.17) f ( η i − β i s i t Δ t ) Δ P i t = f ( η i ) − f ′ ( η i ) β i s i t Δ t Δ P i t + o ( Δ t )

(B.18) = f ( η i ) Δ P i t + o ( Δ t )

Similarly,

(B.19) f ( s i t − β i s i t Δ t ) ( 1 − Δ P i t ) = f ( s i t ) − f ′ ( s ) β i s Δ t − f ( s ) Δ P i + o ( Δ t )

In summary,

(B.20) Δ f = − f ′ ( s i t ) β i s i t Δ t + f ( η i ) − f ( s i t ) Δ P i t + o ( Δ t )

(B.21) ( Δ f ) 2 = f ( η i ) − f ( s i t ) 2 ( Δ P i t ) 2 − 2 f ′ ( s i t ) β i s i t Δ t Δ P i t + o ( Δ t )

Take the conditional expected value with respect to Δf and (Δf)² given s _it = s and divide by Δt to obtain

(B.22) 1 Δ t E Δ f | s i t = s = − f ′ ( s ) β i s + λ i ∫ − ∞ + ∞ f ( x ) − f ( s ) ϕ i x d x + o ( Δ t ) Δ t

(B.23) 1 Δ t E ( Δ f ) 2 | s i t = s = λ i ∫ − ∞ + ∞ f ( x ) − f ( s ) 2 ϕ i x d x + o ( Δ t ) Δ t

Let Δt → 0 to get

(B.24) μ i ( f ( s ) ) ≡ lim Δ t ↓ 0 1 Δ t E Δ f | s i t = s = − f ′ ( s ) β i s + λ i ∫ − ∞ + ∞ f ( x ) − f ( s ) ϕ i x d x

(B.25) σ i 2 ( f ( s ) ) ≡ lim Δ t ↓ 0 1 Δ t E ( Δ f ) 2 | s i t = s = λ i ∫ − ∞ + ∞ f ( x ) − f ( s ) 2 ϕ i x d x

To derive stationary expectation and variance, note that

(B.26) d E s i t = − β i E s i t d t + E ( η i − s i t ) E d P i t

(B.27) = − β i E s i t d t − λ i d t E s i t

(B.28) d E s i t 2 = − 2 β i E s i t 2 d t + E η i 2 − s i t 2 E d P i t

(B.29) = − 2 β i E s i t 2 d t + λ i d t σ i 2 − E s i t 2

Assume Es _i0 = s and E s i 0 2 = 0 , then these differential equations have solutions

(B.30) E s i t = s e − ( β i + λ i ) t

(B.31) E s i t 2 = σ i 2 λ i 2 β i + λ i 1 − e − ( 2 β i + λ i ) t

When t → +∞,

(B.32) E s i t → 0 E s i t 2 → σ i 2 λ i 2 β i + λ i

The half life of s _it is the time t when Es _it = Es _i0/2. By (B.30), we get t = log 2/(β _i + λ _i). □

Appendix C: Numerical Algorithm

C.1 Solve HJB and KFE

I solve the HJB equation by using the upwind finite difference method (FD) following Achdou et al. (2022) and KMV.

Denote grid points by a _i, i = 1, …, I, b _j, j = 1, …, J, n _e, e = 0, 1 and s _k, k = 1, …, K and discretized value function by

V i , j , e , k = V ( a i , b j , n e , s k )

To approximate the derivatives, I use non-equispaced grids and denote Δ a i + = a i + 1 − a i and Δ a i − = a i − a i − 1 . ∂ _a V(a _i, b _j, n _e, s _k) ≡ V _a,i,j,e,k is either a forward or backward difference approximation

V a , i , j , e , k ≈ V b , i , j , e , k F = V i + 1 , j , e , k − V i , j , e , k Δ a i + or V a , i , j , e , k ≈ V b , i , j , e , k B = V i , j , e , k − V i − 1 , j , e , k Δ a i +

Similarly for the b dimension. Given a guess for value function V i , j , e , k n , the FD approximation to HJB equations is

(C.1) V i , j , e , k n + 1 − V i , j , e , k n Δ + ( ρ + ξ ) V i , j , e , k n + 1 = u c i , j , e , k n + V b , i , j , e , k n + 1 s i , j , e , k b , n + V a , i , j , e , k n + 1 s i , j , e , k a , n + ∑ k ′ ≠ k Λ k , k ′ s V i , j , e , k ′ n − V i , j , e , k ′ n + 1 + Λ e , e ′ n V i , j , e ′ , k n − V i , j , e , k n + 1

(C.2) V b , i , j , e , k n = u ′ c i , j , e , k n

(C.3) V a , i , j , e , k n = V b , i , j , e , k n ( 1 + χ d d i , j , e , k n , a i )

(C.4) s i , j , e , k a , n = ( r a + ξ ) a i + d i , j , e , k n

(C.5) s i , j , e , k b , n = ( 1 − τ ) z ( n e , s k ) + ( r b ( b j ) + ξ ) b j + T − c i , j , e , k n

(C.6) − d i , j , e , k n − χ d i , j , e , k n , a i

In the a dimension, use a forward difference approximation whenever the drift s ^a is positive and a backward approximation otherwise. In the b dimension, additionally split the drift s b = ( 1 − τ ) z ( n , s ) + ( r t b ( b ) + ξ ) b + T − d − χ ( d , a ) − c into s ^c = (1 − τ)z(n, s) + (r ^b(b) + ξ)b + T − c and s ^d = −d − χ(d, a). Denote c i , j , e , k B as the optimal consumption calculated by backward approximation V b , i , j , e , k B , c i , j , e , k F by forward approximation V b , i , j , e , k F , and c i , j , e , k 0 = ( 1 − τ ) z ( n e , s k ) + ( r b ( b j ) + ξ ) b j + T . Define corresponding drifts and Hamiltonians for b dimension

s i , j , e , k c , B = ( 1 − τ ) z ( n e , s k ) + ( r b ( b ) + ξ ) b j + T − c i , j , e , k B s i , j , e , k c , F = ( 1 − τ ) z ( n e , s k ) + ( r b ( b ) + ξ ) b j + T − c i , j , e , k F H i , j , e , k c , B = u c i , j , e , k B + V b , i , j , e , k B s i , j , e , k c , B H i , j , e , k c , F = u c i , j , e , k F + V b , i , j , e , k F s i , j , e , k c , F H i , j , e , k c , 0 = u c i , j , e , k 0

For boundary at j = 1 and j = J, define H ̲ to be a sufficient small negative integer, and

H i , 1 , e , k c , B = H ̲ H i , J , e , k c , F = H ̲

To determine the final s ^c and c, use selecting rule

I i , j , e , k c , F = I s i , j , e , k c , F > 0 ⋂ I s i , j , e , k c , B ≥ 0 ⋃ I H i , j , e , k c , F > H i , j , e , k c , B × ⋂ I H i , j , e , k c , F > H i , j , e , k c , 0 I i , j , e , k c , B = I s i , j , e , k c , B < 0 ⋂ I s i , j , e , k c , F ≤ 0 ⋃ I H i , j , e , k c , B > H i , j , e , k c , F × ⋂ I H i , j , e , k c , B > H i , j , e , k c , 0 I i , j , e , k c , 0 = I ̃ i , j , e , k c , F ⋂ I ̃ i , j , e , k c , B

The upwind rule is then

s i , j , e , k c = I i , j , e , k c , F s i , j , e , k c , F + I i , j , e , k c , B s i , j , e , k c , B c i , j , e , k = I i , j , e , k c , F c i , j , e , k F + I i , j , e , k c , B c i , j , e , k B + I i , j , e , k c , 0 c i , j , e , k 0

Denote d i , j , e , k F B , d i , j , e , k B B , d i , j , e , k F F , d i , j , e , k B F in a fashion of

V a , i , j , e , k F = V b , i , j , e , k B ( 1 + χ d d i , j , e , k F B , a i )

The corresponding drifts and Hamiltonians for intersectional dimension are

s i , j , e , k d , F B = − d i , j , e , k F B − χ d i , j , e , k F B , a i s i , j , e , k d , B F = − d i , j , e , k B F − χ d i , j , e , k B F , a i s i , j , e , k d , B B = − d i , j , e , k B B − χ d i , j , e , k B B , a i H i , j , e , k d , F B = V a , i , j , e , k F d i , j , e , k F B + V b , i , j , e , k B s i , j , e , k d , F B H i , j , e , k d , B F = V a , i , j , e , k B d i , j , e , k B F + V b , i , j , e , k F s i , j , e , k d , B F H i , j , e , k d , B B = V a , i , j , e , k B d i , j , e , k B B + V b , i , j , e , k B s i , j , e , k d , B B

For boundary at i = 1, i = I, j = 1 and j = J, let

H I , j , e , k d , F B = H i , 1 , e , k d , F B = H ̲ H 1 , j , e , k d , B F = H i , J , e , k d , B F = H ̲ H 1 , j , e , k d , B B = H i , 1 , e , k d , B B = H ̲

Denote the valid indicators for d as

I i , j , e , k F B = I d i , j , e , k F B > 0 ⋂ I H i , j , e , k d , F B > 0 I i , j , e , k B F = I s i , j , e , k d , B F ≥ 0 ⋂ I H i , j , e , k d , B F > 0 I i , j , e , k B B = I s i , j , e , k d , B F < 0 ⋂ I H i , j , e , k d , B F > 0 ⋂ I d i , j , e , k B B ≤ 0

The selection rule is

I i , j , e , k d , F B = I i , j , e , k F B ⋂ I ̃ i , j , e , k B F ⋃ I H i , j , e , k d , F B ≥ H i , j , e , k d , B F × ⋂ I ̃ i , j , e , k B B ⋃ I H i , j , e , k d , F B ≥ H i , j , e , k d , B B I i , j , e , k d , B F = I i , j , e , k B F ⋂ I ̃ i , j , e , k F B ⋃ I H i , j , e , k d , B F ≥ H i , j , e , k d , F B × ⋂ I ̃ i , j , e , k B B ⋃ I H i , j , e , k d , B F ≥ H i , j , e , k d , B B I i , j , e , k d , B B = I i , j , e , k B B ⋂ I ̃ i , j , e , k B F ⋃ I H i , j , e , k d , B B ≥ H i , j , e , k d , B F × ⋂ I ̃ i , j , e , k F B ⋃ I H i , j , e , k d , B B ≥ H i , j , e , k d , F B

The upwinding rule for policy d is then

d i , j , e , k = I i , j , e , k d , F B d i , j , e , k F B + I i , j , e , k d , B F d i , j , e , k B F + I i , j , e , k d , B B d i , j , e , k B B s i , j , e , k d = − d i , j , e , k − χ ( d i , j , e , k , a i )

With the selected policies c _i,j,e,k, s i , j , e , k c , d _i,j,e,k and s i , j , e , k d at hand and denoting x ⁺ = max{x, 0} and x ⁻ = min{x, 0}, the upwind FD approximation to the HJB equation is given by

(C.7) V i , j , e , k n + 1 − V i , j , e , k n Δ + ( ρ + ξ ) V i , j , e , k n + 1 = u c i , j , e , k n + V b , i , j , e , k B , n + 1 s i , j , e , k c , n − + V b , i , j , e , k F , n + 1 s i , j , e , k c , n + + V b , i , j , e , k B , n + 1 s i , j , e , k d , n − + V b , i , j , e , k F , n + 1 s i , j , e , k d , n + + V a , i , j , e , k B , n + 1 d i , j , e , k n − + V a , i , j , e , k F , n + 1 × r a a i + d i , j , e , k n + + ∑ k ′ ≠ k Λ k , k ′ s V i , j , e , k ′ n − V i , j , e , k ′ n + 1 + Λ e , e ′ n V i , j , e ′ , k n − V i , j , e , k n + 1

In matrix notation, represent V as a matrix with I × J rows and 2K columns

(C.8) 1 Δ ( V n + 1 − V n ) + ( ρ + ξ ) V n + 1 = u n + A n V n + 1 + V n ( Λ ′ − diag ( Λ ) ) + V n + 1 diag ( Λ )

where

A n = A 1 n A 2 n … A 2 K n V n + 1 = V 1 n + 1 V 2 n + 1 ⋱ V 2 K n + 1

and Λ = Λ^s ⊗ Λⁿ. A ⁿ has I × J rows and I × J × 2K columns and is and Λ is a square with 2K rows. In particular, A ⁿ summarize the information of asset transition, and have a sparsity structure sharing the same structure as Achdou et al. (2022). I use a matrix notation with subscript k means the k column in this matrix.

This problem can be broken up to K smaller problems thus the routine can take advantage of parallelization to speed up matrix calculation. In particular, for k = 1…2K,

(C.9) 1 Δ V k n + 1 − V k n + ( ρ + ξ ) V k n + 1 = u k n + A k n V k n + 1 + V n ( Λ ′ − diag ( Λ ) ) k + V n + 1 diag ( Λ ) k

Since the KFE equation is a transposed version of the HJB equation, the matrix form of the KFE equation is then

(C.10) 1 Δ ψ k n + 1 − ψ k n + ξ ψ k n + 1 = D ψ k n + ( A k ) ′ ψ k n + 1 + ψ n ( Λ − diag ( Λ ) ) k + ψ n + 1 diag ( Λ ) k

Here, D is the matrix where all entries are zero except for the row that describes those newborns with zero assets, which is filled by death rate ξ. A _k is the stationary generator matrix implied by (C.9).^[23] Due to the normalization of the distribution matrix, ψ should be interpreted as the mass of each state rather than the density function.

C.2 Solved Blocks

See Figures 7 and 8.

Figure 7:

(Solved) Labor blocks given x , r ^a, L ^d.

Figure 8:

(Solved) Capital blocks given Z , Y , x , r ^a.

Appendix D: The Decomp. of Impact Response of Cons. on Liquid Wealth

See Figures 9–12.

Figure 9:

Elasticity on liquid asset. (a) Averaged. (b) Employed. (c) Unemployed.

Figure 10:

Decomposition of total elasticity on liquid asset. (a) Average household. (b) Employed household. (c) Unemployed household.

Figure 11:

Decomposition of direct elasticity on liquid asset. (a) Averaged. (b) Employed. (c) Unemployed.

Figure 12:

Decomposition of indirect elasticity on liquid asset. (a) Averaged. (b) Employed. (c) Unemployed.

Appendix E: The Impulse Responses of Other Models

See Figures 13–15.

Figure 13:

IRF: competitive labor market. (a) Interest rates and inflation. (b) Aggregate quantities. (c) Aggregate price. (d) Profits.

Figure 14:

IRF: flexible wage. (a) Interest rates and inflation. (b) Aggregate quantities. (c) Aggregate price. (d) Profits.

Figure 15:

IRF: budget balanced. (a) Interest rates and inflation. (b) Aggregate quantities. (c) Aggregate price. (d) Profits.

Appendix F: Robustness Check of Fiscal Policy Parameters

In this section, I implement the robustness check of three parameters in fiscal policy: ρ ^B, ρ ^E and ρ ^R. Values of these parameters are controversial and matters on debt cyclicality. The plausible values of these parameters should be around values of baseline and balanced-budget fiscal policy. I consider possible values as follows: ρ ^B = 0.03, 0.1, 1; ρ ^E = 0, 0.15, 0.5 and ρ ^R = 0.5, 1, 1.5. Figure 16 shows the decomposition of consumption response in different ρ ^B, ρ ^E and ρ ^R. Since it does not change the hiring behavior of wholesale firms, a remarkable share of unemployment risk in consumption response to monetary policy shock still appears insofar as slightly different values. Figures 17–19 illustrate the impulse response in different ρ ^B, ρ ^E and ρ ^R respectively. These parameters deliver distinct debt path and surplus responses but little differences in other variables especially like output, investment and consumption.

Figure 16:

Consumption decomposition in different ρ ^B, ρ ^E and ρ ^R.

Figure 17:

Response to a monetary policy shock in different ρ ^B.

Figure 18:

Response to a monetary policy shock in different ρ ^E.

Figure 19:

Response to a monetary policy shock in different ρ ^R.

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Received: 2024-08-27

Accepted: 2025-05-15

Published Online: 2025-06-18

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Articles in the same Issue

https://doi.org/10.1515/bejm-2024-0148

Keywords for this article

New Keynesian; unemployment risk; sequence-space-jacobians; monetary policy