Trade Friction in Two-Country HANK with Financial Friction

Abstract

This article develops the Two-Country HANK model(Heterogeneous Agents New Keynesian) in the context of trade frictions between China and the United States. Diverging from the conventional RANK (Representative Agent New Keynesian) model, our approach more accurately mirrors real-world economic dynamics by incorporating household heterogeneity and financial market frictions. Our investigation into the effects of reciprocal tariff impositions by these nations has revealed several critical insights. First, tariffs induce a measure of economic downturn in both countries. While a larger country can inflict more significant harm on a smaller one at a lower cost, it is not immune to output reductions. Second, the imposition of tariffs amid trade disputes alters the investment landscape of the smaller country, exacerbating the disparity in foreign investment holdings between wealthier and poorer households and thereby affecting wealth inequality. Lastly, although monetary policy can invigorate investment via financial leverage in the face of trade-induced recessions, its short-term benefits on output are constrained due to its limited capacity to boost labor effectively.

Appendix

1.1 Algorithm to Solve Two-Country HANK model

This section presents the algorithm employed to solve the Two-Country HANK model. Technically, our primary contributions within the framework of the two countries involve formulating and solving the Hamilton-Jacobi-Bellman (HJB) equations incorporating three heterogeneous assets, following the approach outlined in Kaplan et al. (2018).

In the HANK model, optimization outcomes in the firm sector manifest as the wage and return rates of illiquid assets, which are subsequently incorporated into households’ balance sheets. Consequently, the optimization process for both sectors-households and firms-is effectively encapsulated by the optimization undertaken for households.

1.1.1 The Derivation for Total Consumption

In this subsection, we introduce the simplification of the utility function and assume the existence of a total consumption $c_t$ for heterogeneous households, satisfying:

$$\begin{aligned} c_t P_t = P_{ht} c_{ht} + P_{ft} e_t (1+\tau _{im}) c_{ft} \end{aligned}$$

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The total price level $P_t$ in the home country comprises the domestic goods price $P_{ht}$ and the price of imported goods $P_{ft}e_t(1+\tau _{im})$:

$$\begin{aligned} P_t = \left[ \rho _p (P_{ht})^{\varepsilon _{p}} + (1-\rho _p)(P_{ft} e_t(1+\tau _{im}))^{\varepsilon _{p}} \right] ^\frac{1}{\varepsilon _{p}}, \end{aligned}$$

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In the equation, $\rho _p$ represents the proportion of domestic goods price, while $\varepsilon _p$ indicates the elasticity between domestic and foreign goods prices.

For households’ optimization problems, the Hamilton-Jacobi-Bellman equation is:

$$\begin{aligned} V(b_t,a_{ht},a_{ft},z_t) = \max _{c_{ht},c_{ft},\ell _t} u(c_{ht},c_{ft},l_t) + V_b \dot{b_t} + V_{ah} \dot{a_{ht}} + V_{af} \dot{a_{ft}} + g_z V_z + \frac{\sigma _z^2}{2} V_{zz} \end{aligned}$$

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where $V(b_t,a_{ht},a_{ft},z_t)$ is the value function of households, $V_b,V_{ah},V_{af},V_{z},V_{zz}$ are the partial derivatives of the value function.

The partial derivatives of the HJB equation over $c_{ht},c_{ft}$ are

$$\begin{aligned} \begin{aligned}&0 = c_{ht}^{-\gamma _c} - \frac{P_{ht}}{P_t} V_b\\&0 = \varphi _{c} c_{ft}^{-\gamma _c} - \frac{P_{ft} e_t (1+\tau _{im})}{P_t} V_b \end{aligned} \end{aligned}$$

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This implies that $c_{ht}$ and $c_{ft}$ are linked through their respective prices:

$$\begin{aligned} c_{ft} = c_{ht} \left[ \frac{P_{ft} e_t (1+\tau _{im})}{\varphi _{c} P_{ht}}\right] ^{-\frac{1}{\gamma _c}} \end{aligned}$$

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By substituting the equation mentioned above into the total consumption $c_t$, we subsequently obtain:

$$\begin{aligned} \begin{aligned}&c_{ht} = \frac{c_t P_t}{P_{ht}+\left[ P_{ft} e_t (1+\tau _{im})\right] ^{1-\frac{1}{\gamma _c}} \left( \varphi _{c} P_{ht}\right) ^{\frac{1}{\gamma _c}}}\\&c_{ft} = \frac{c_t P_t}{\left[ P_{ft} e_t (1+\tau _{im})\right] ^{\frac{1}{\gamma _c}} \left( \varphi _{c} P_{ht}\right) ^{-\frac{1}{\gamma _c}} P_{ht} + P_{ft} e_t (1+\tau _{im})} \end{aligned} \end{aligned}$$

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Finally, the updated utility function is a function of total consumption $c_t$ and labor supply $\ell _t$:

$$\begin{aligned} \begin{aligned}&u(c_{ht},c_{ft},\ell _t) = \alpha _c \frac{\left( c_t\right) ^{1-\gamma _c}}{1-\gamma _c} - \varphi _{\ell } \frac{(\ell _t)^{1+\gamma _\ell }}{1+\gamma _\ell } \triangleq u(c_{t},\ell _t),\\&\alpha _c \equiv \left[ \frac{ P_t}{P_{ht}+ P_{ft} e_t (1+\tau _{im}) \left[ \frac{P_{ft} e_t (1+\tau _{im})}{\varphi _{c} P_{ht}}\right] ^{-\frac{1}{\gamma _c}} }\right] ^ {1-\gamma _c} \left( 1+\varphi _{c} \left[ \frac{P_{ft} e_t (1+\tau _{im})}{\varphi _{c} P_{ht}}\right] ^{\frac{\gamma _c-1}{\gamma _c}} \right) \end{aligned} \end{aligned}$$

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1.1.2 Endogenous Labor Supply

According to Moll’s website, this section delves into the intricacies of deriving the endogenous labor supply equation. For more detailed information and derivations, kindly refer to the provided link: https://benjaminmoll.com/codes/.

$$\begin{aligned} \begin{aligned} V(b_t,a_{ht},a_{ft},z_t)&= \max _{c_{t},\ell _t} u(c_{t},l_t) + V_b \dot{b_t} + V_{ah} \dot{a_{ht}} + V_{af} \dot{a_{ft}} + g_z V_z + \frac{\sigma _z^2}{2} V_{zz} \\ s.t. \quad \dot{b_t}&= (1-\tau _w)w_t \ell _t z_t + r_b b_t + T_t - d_{ht} - \chi _h(d_{ht}, a_{ht}) \\&- d_{ft} - \chi _f(d_{ft}, a_{ft}) - c_t, \\ u(c_{t},l_t)&= \alpha _c \frac{\left( c_t\right) ^{1-\gamma _c}}{1-\gamma _c} - \varphi _{\ell } \frac{(\ell _t)^{1+\gamma _\ell }}{1+\gamma _\ell }. \end{aligned} \end{aligned}$$

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The variables $V_b,V_{ah},V_{af},V_z$ represent the first-order partial derivatives of the value function V, while $V_{zz}$ denotes the second-order partial derivative. The first-order conditions of the HJB equations are defined as follows:

$$\begin{aligned} \begin{aligned} 0&= \alpha _c c_t^{-\gamma _c} - V_b \\ 0&= -\varphi _{\ell } \ell _t^\frac{1}{\varphi _\ell } + V_b (1-\tau _w)w_t z_t \end{aligned} \end{aligned}$$

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The labor supply, denoted by $\ell _t$, is indeed a function of total consumption $c_t$:

$$\begin{aligned} \ell _t = \left( c_t\right) ^{-\gamma _c \varphi _\ell } \left[ \frac{\alpha _c (1-\tau _w)w_t z_t}{\varphi _{\ell }}\right] ^{\varphi _\ell } \end{aligned}$$

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Given the budget constraint equation $\dot{b}$, in the equilibrium state, the changing rates of all assets are nil, and deposits are also stationary. Consequently, the consumption $c_0$ and labor supply $\ell _0$ are determined by

$$\begin{aligned} c_0 = (1-\tau _w)w_t \ell _0 z_t + r_b b_t + T_t - fixedcost_f \end{aligned}$$

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By substituting $c_0$ into the previous equation, we obtain

$$\begin{aligned} \ell _0 = \left( (1-\tau _w)w_t \ell _0 z_t + r_b b_t + T_t - fixedcost_f\right) ^{-\gamma _c \varphi _\ell } \left[ \frac{\alpha _c (1-\tau _w)w_t z_t}{\varphi _{\ell }}\right] ^{\varphi _\ell } \end{aligned}$$

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The equation provided earlier outlines how the labor supply $\ell _0$ is determined in the equilibrium state.

1.1.3 Algorithm for HJB

Let’s return to the updated version of HJB equation:

$$\begin{aligned} \begin{aligned} V&(b_t,a_{ht},a_{ft},z_t) = \max _{c_{t},\ell _t} u(c_{t},l_t) + V_b \dot{b_t} + V_{ah} \dot{a_{ht}} + V_{af} \dot{a_{ft}} + g_z V_z + \frac{\sigma _z^2}{2} V_{zz} \\ s.t.&\ \dot{b_t} = (1-\tau _w)w_t \ell _t z_t + r_b b_t + T_t - d_{ht} - \chi _h(d_{ht}, a_{ht}) - d_{ft} - \chi _f(d_{ft}, a_{ft}) - c_t, \\&\dot{a}_{ht}= r_{ah} a_{ht} + d_{ht},\quad \dot{a}_{ft}= r_{af} a_{ft} + d_{ft}\frac{P_{t}}{P^{*} _{t} e_t}. \end{aligned} \end{aligned}$$

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The income flow of liquid assets $\dot{b}_t$ can be partitioned into three distinct components:

$$\begin{aligned} \begin{aligned}&S^c = (1-\tau _w)w_t l_t z_t + r_b b_t + T_t - c_{t}\\&S^{dh} = - d_{ht} - \chi _h(d_{ht}, a_{ht}), \qquad S^{df} = - d_{ft} - \chi _f(d_{ft}, a_{ft})\\&\dot{b_t} = S^c + S^{dh} + S^{df} \end{aligned} \end{aligned}$$

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Let ${x}^{F+} \triangleq max (x,0)$ and ${x}^{B-} \triangleq min (x,0)$. Then we assume $x = {x}^{F+} + {x}^{B-}$. Let $b_t,a_{ht},a_{ft},z_t$ be situated in distinct grids, and we can label a household as (i, j, k, nz). Subsequently, we define the approximate forward derivative $V^{F}_b$ and backward derivative $V^B_b$ as follows:

$$\begin{aligned} \begin{aligned}&V^{n+1}_b \approx \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b} \approx \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \\&V^{F}_b = \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b}, \qquad V^B_b = \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \end{aligned} \end{aligned}$$

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Given that $V_b \dot{b_t} = V^B_b (S^{c,B-} + S^{dh,B-} + S^{df,B-}) + V^{F}_b (S^{c,F+} + S^{dh,F+} + S^{df,F+})$, we can determine the appropriate approximation for the derivative. This approximation approach enables the program to solve HJB equations efficiently.

We can then obtain the approximate solution for the household sector by solving the discrete HJB equation. As defined earlier, the discrete HJB equation is formulated as follows:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta } + \rho V^{n+1}_{i,j,k,nz} =&\max _{c_{ht},c_{ft},l_t} U + V_b \dot{b_t} + V_{ah} \dot{a_{ht}} + V_{af} \dot{a_{ft}} + V_z g_z + V_{zz} \frac{\sigma _z^2}{2} \\ =&U + V^B_b (S^{c,B-} + S^{dh,B-} + S^{df,B-}) \\&+ V^{F}_b (S^{c,F+} + S^{dh,F+} + S^{df,F+}) \\&+ V^B_{ah} d^{h,-}_t + V^{F}_{ah} (d^{h,+}_t + r_{aht} a_{ht}) + V^B_{af} d^{f,-}_t \frac{P_t^* e_t}{P_t} \\&+ V^{F}_{af} (d^{f,+}_t \frac{P_t^* e_t}{P_t}+ r_{af} a_{ft}) \\&+ V_z g_z + V_{zz} \frac{\sigma _z^2}{2} \end{aligned} \end{aligned}$$

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It is worth noting that the approximate derivatives, specifically $V^{F}_b = \frac{V_{i+1,j,k,nz} - V_{i,j,k,nz}}{\Delta b}$ and $V^B_b = \frac{V_{i,j,k,nz} - V_{i-1,j,k,nz}}{\Delta b}$, can be factorized. Subsequently, we can rephrase the discrete HJB equation in the following manner:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta } + \rho V^{n+1}_{i,j,k,nz} =&U + \frac{V_{i-1,j,k,nz}}{\Delta b}(-S^{c,B-} - S^{dh,B-} - S^{df,B-}) \\&+ \frac{V_{i+1,j,k,nz}}{\Delta b}(S^{c,F+} + S^{dh,F+} + S^{df,F+})\\&+ \frac{V_{i,j,k,nz}}{\Delta b}(S^{c,B-} -S^{c,F+} + S^{dh,B-} - S^{dh,F+}\\&+ S^{df,B-} - S^{df,F+}) \\&+ \frac{V_{i,j-1,k,nz}}{\Delta a}( - d^{h-}_t) + \frac{V_{i,j+1,k,nz}}{\Delta a}(d^{h+}_t+r_{aht} a_{ht}) \\&+ \frac{V_{i,j,k,nz}}{\Delta a}(d^{h-}_t - d^{h+}_t - r_{aht} a_{ht}) \\&+ \frac{V_{i,j,k-1,nz}}{\Delta a^{*}}( - d^{f,-}_t \frac{P_t^* e_t}{P_t}) \\&+ \frac{V_{i,j,k+1,nz}}{\Delta a^{*}}(d^{f,+}_t \frac{P_t^* e_t}{P_t}+ r_{aft} a_{ft}) \\&+ \frac{V_{i,j,k,nz}}{\Delta a^{*}}(d^{f,-}_t \frac{P_t^* e_t}{P_t} - d^{f,+}_t \frac{P_t^* e_t}{P_t} - r_{aft} a_{ft}) \\&+ \sum _{nz' \ne nz}\lambda _{nz}(V_{i,j,k,nz'} - V_{i,j,k,nz}), \end{aligned} \end{aligned}$$

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This indicates that the items $V_{i-1,j,k,nz}$ and $V_{i+1,j,k,nz}$ reside on distinct subdiagonals within matrix V, while the items $V_{i,j,k,nz}$ occupy the main diagonal of V. Referring to the work of Achdou et al. (2022), similar observations hold true for other derivatives as well. Consequently, the aforementioned discrete HJB equation can be concisely represented as:

$$\begin{aligned} \frac{1}{\Delta }\left( V^{n+1}_{i,j,k,nz}-V^{n}_{i,j,k,nz}\right) +\rho V^{n+1}_{i,j,k,nz}=U_{i,j,k,nz}^{n}+\left( \textbf{A}^{n}+\mathbf {\Lambda }\right) V^{n+1}_{i,j,k,nz}, \end{aligned}$$

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This implies that given the value at period n, we can obtain the result for period $n+1$.

By repeatedly applying this equation until the value function V reaches convergence, we can solve the distribution $g(b_t,a_{ht},a_{ft},z_t) \triangleq g_{i,j,k,nz}$, for households using the Kolmogorov Forward equation:

$$\begin{aligned} \left( \textbf{A}^{n}+\mathbf {\Lambda }\right) ^T g = 0 \end{aligned}$$

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According to Achdou et al. (2022), the Pollution Method can be utilized to compute the distribution matrix g, ultimately leading to the solution of the HJB function for households.