## _wp09275

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---

### I. Introduction — valuation channel and research objective
- Focus: the “valuation channel” — capital gains and losses on the stock of gross foreign assets and liabilities due to expected or unexpected asset price changes.
- Context: empirical literature finds the financial channel quantitatively important for the United States; Gourinchas and Rey (2007b) estimate the financial channel contributed on average about 30 percent of the (cyclical) external adjustment of the United States since the 1950s.
- Objective: study the valuation channel theoretically in a two-country, dynamic, stochastic, general equilibrium (DSGE) portfolio model with international trade in equity, encompassing complete and incomplete asset market scenarios.
- Key modeling features:
  - International equity trading in a two-country DSGE model with production under monopolistic competition.
  - Households supply labor, consume a CES aggregated basket of differentiated domestic and foreign goods, and hold shares in domestic and foreign firms.
  - Production uses only labor and is subject to country-wide productivity shocks.
  - Uncertainty arises from productivity and government spending shocks; asset markets are incomplete when both shocks are present.
- Analytical contribution: separating asset prices and asset quantities in the definition of net foreign assets enables characterization of first-order dynamics of valuation effects and portfolio adjustment and their relative contributions to net foreign asset and macroeconomic dynamics.

### II. Methodology and solution technique
- Approximation approach:
  - Combine a second-order approximation of the portfolio optimality conditions with a first-order approximation of the rest of the model.
- Key modeling implications:
  - The excess return on foreign assets is a fully unanticipated, i.i.d. variable in the model.
  - Capital gains or losses on equity positions (valuation channel) are generally not i.i.d.; the initial response of valuation to a shock at time t is unanticipated as of t−1, but dynamics in following periods are fully anticipated as of the time of the shock.
  - Valuation response to relative productivity shocks is described by an ARMA(1,1) process.
  - Valuation response to relative government spending shocks is i.i.d., stemming from the model’s random-walk property of the consumption differential.

### III. Main analytical results on valuation vs. portfolio adjustment
- Cross-country dividend differential determines relative equity values and is proportional to contemporaneous productivity and consumption differentials, producing ARMA dynamics for valuation under productivity shocks.
- Separation of prices and quantities is necessary to obtain first-order decomposition of valuation and portfolio adjustment and to characterize anticipated capital gain effects.
- Comparative roles by shock type:
  - Productivity shocks:
    - The share of valuation is positive and constant in all periods after the impact of a productivity shock.
    - Valuation can represent a significantly larger share of net foreign asset movements than portfolio adjustment in an incomplete markets scenario.
    - Numerical illustrations show a non-negligible difference between the valuation measure used here and the excess-return-based measure in Devereux and Sutherland (2009b) for productivity shocks.
  - Government spending shocks:
    - The share of valuation in adjustment is zero in all periods but the impact one; portfolio adjustment is responsible for all changes in net foreign assets in subsequent periods.
    - Valuation in response to government spending shocks is i.i.d. because the consumption differential follows a random walk in the model.
- Methodological note: when asset positions do not treat prices and quantities separately, obtaining the first-order decomposition and anticipated valuation effects requires higher-order approximations.

### IV. Implications for macroeconomic dynamics and risk sharing
- The paper disentangles excess returns, capital gains, and portfolio adjustment for consumption risk sharing when financial markets are incomplete.
- These channels can dampen or amplify the impact response of the cross-country consumption differential to shocks and can keep it constant in subsequent periods.
- Insurance and specialization results:
  - Households achieve perfect insurance against country-specific productivity shocks when there is no government spending uncertainty or when steady-state government spending is zero.
  - With inelastic labor supply (φ = 0), output and firm profits are determined independently of government spending; international trade in equities provides no hedge against government spending shocks. Optimal portfolio strategy: fully insure against idiosyncratic productivity shocks while remaining exposed to government spending shocks.
- Quantitative remarks and limitations:
  - The model focuses on impulse responses for numerical illustration and omits adjustment costs or other frictions that would ensure stationarity of responses to non-permanent shocks.
  - The implied random-walk property of relative consumption affects analytical solutions; introducing stationarity-inducing devices would alter long-horizon stationarity but likely yield small quantitative departures from the core results for plausible calibrations.
  - Extensions to richer, quantitative models (including home bias or additional assets) could incorporate the methodology without changing the core mechanism.

### V. Model structure — households, consumption, and budget constraints (selected)
- World and agents:
  - Two countries, home and foreign; world population continuum [0,1]; home households in [0,a), foreign in [a,1].
  - Continuum of monopolistically competitive firms on [0,1], home firms z ∈ [0,a), foreign firms z* ∈ [a,1].
- Preferences (home representative household):
  - Expected utility: E_t ∑_{s=t}^∞ β^{s−t} [ C_s^{1−1/σ} / (1−1/σ) − χ L_s^{1+1/φ} / (1+1/φ) ], with 1>β>0 and σ, χ, φ >0.
- Consumption aggregation:
  - C_t = [ a^{1/ω} C_{Ht}^{(ω−1)/ω} + (1−a)^{1/ω} C_{Ft}^{(ω−1)/ω} ]^{ω/(ω−1)}, with ω>0.
  - Sub-baskets and price indexes as specified (θ>1).
- Asset holdings notation:
  - x_{t+1}, x^*_{t+1}, x^{**}_{t+1} defined as aggregate per capita holdings entering period t+1.
- Government spending:
  - G_t and G^*_t denote exogenous government spending (following AR(1) processes in logs); balanced budgets with lump-sum taxation.
- Household budget constraints (equilibrium versions) and gross returns:
  - v_t x_{t+1} + v^*_t x^*_{t+1} + C_t + G_t = (v_t + d_t) x_t + (v^*_t + d^*_t) x^*_t + w_t L_t.
  - R_t ≡ (v_t + d_t) / v_{t−1}, R^*_t ≡ (v^*_t + d^*_t) / v^*_{t−1}.
  - Euler: C_t^{−1/σ} = β E_t ( C_{t+1}^{−1/σ} R_{t+1} ).
  - Indifference: E_t ( C_{t+1}^{−1/σ} R_{t+1} ) = E_t ( C_{t+1}^{−1/σ} R^*_{t+1} ).
  - Optimal labor supply: C_t^{−1/σ} w_t = χ L_t^{1/φ}.
- Law of one price and cashless specification noted; focus on real variables.

### VI. From the budget constraints to the law of motion for net foreign assets — key equations
- Home aggregate per capita net foreign assets entering period t+1:
  - nfa_{t+1} ≡ v^*_{t} x^*_{t+1} − (1−a)/a v_{t} x^*_{t+1}. (7)
- Law of motion for net foreign assets:
  - nfa_{t+1} = R^D_t α_t + R_t nfa_t + y_t − C_t − G_t. (8)
  - Foreign analogue: nfa^*_{t+1} = R^D_t α^*_t + R_t nfa^*_t + y^*_t − C^*_t − G^*_t. (9)
  - With asset market clearing and subtraction:
    - nfa_{t+1} = R^D_t α_t + R_t nfa_t + (1−a)(y^D_t − C^D_t − G^D_t). (10)
  - Notation: R^D_t ≡ R^*_t − R_t; α_t ≡ v^*_{t−1} x^*_{t}; y^D_t ≡ y_t − y^*_t; C^D_t ≡ C_t − C^*_t; G^D_t ≡ G_t − G^*_t.

### VII. Firms, production, and useful properties
- Real prices (constant markup θ/(θ−1)):
  - RP_t = θ/(θ−1) w_t / Z_t and RP^*_t = θ/(θ−1) w^*_t / Z^*_t. (11)
- GDP:
  - y_t = RP_t Z_t L_t and y^*_t = RP^*_t Z^*_t L^*_t. (12)
- Income distribution:
  - w_t L_t = (θ−1) y_t / θ and d_t = y_t / θ. (20)
- Selected identities:
  - y_t / y^*_t = (Z_t / Z^*_t)^{(1+φ)(ω−1)/(ω+φ)} (C_t / C^*_t)^{−φ(ω−1)/(σ(ω+φ))}. (14)
  - TOT_t ≡ RP_t / RP^*_t = (Z_t / Z^*_t)^{−(1+φ)/(ω+φ)} (C_t / C^*_t)^{φ/(σ(ω+φ))}. (17)
  - w_t / w^*_t and L_t / L^*_t formulas (18)-(19).

### VIII. Steady-state portfolio and comparative statics
- Steady-state portfolio α given by equation (21); key special cases preserved:
  - If G = 0 then α = β(1−a)/(1−β).
  - α = β(1−a)/(1−β) also arises when φ = 0 (inelastic labor supply), regardless of G.
- Interpretation: when G = 0 (or φ = 0), households choose the portfolio that fully insures against idiosyncratic productivity shocks; government spending risk is not hedged by equity when φ = 0 because G does not affect profits.
- Comparative static signs (assuming ω > 1):
  - ∂α/∂a < 0, ∂α/∂σ > 0, ∂α/∂σ^2_{εGD} < 0, ∂α/∂φ_G < 0, ∂α/∂σ^2_{εZD} > 0, ∂α/∂φ_Z > 0; ∂α/∂G < 0.
  - For plausible parameters: ∂α/∂β > 0, ∂α/∂φ < 0, ∂α/∂ω > 0.

### IX. Log-linear decomposition: portfolio adjustment vs valuation
- Normalized nfa (equation (22)):
  - n̂fa_{t+1} = α/(1−G) (x̂^D_{t+1} − v̂^D_t).
- First-difference decomposition (equation (23)):
  - Δ n̂fa_{t+1} = α/(1−G) Δ x̂^D_{t+1} − α/(1−G) Δ v̂^D_t = ĉa_t + v̂al_t,
    - ĉa_t ≡ α Δ x̂^D_{t+1}/(1−G) (portfolio adjustment/current account),
    - v̂al_t ≡ −α Δ v̂^D_t/(1−G) (valuation: asset price effect).
- Log-linearized law of motion (equation (24)):
  - n̂fa_{t+1} = (1/β) n̂fa_t + α/[β(1−G)] R̂^D_t + (1−a)/(1−G) ŷ^D_t − (1−a) Ĉ^D_t − (1−a)G/(1−G) Ĝ^D_t.
- Relation between excess returns and v̂^D_t:
  - R̂^D_t = −β v̂^D_t − (1−β) d̂^D_t + v̂^D_{t−1}.
- First-order portfolio adjustment identity (equation (25)):
  - Δ x̂^D_{t+1} (scaled) = (1−β)/β (x̂^D_t − d̂^D_t) + (1−a)/α [ŷ^D_t − (1−G) Ĉ^D_t − G Ĝ^D_t].
  - Interpretation: Δ x̂^D_{t+1} equals the income balance from the pre-existing net equity position plus the trade balance.

### X. Solution structure for portfolio adjustment and valuation
- Relative productivity and government spending processes (AR(1)):
  - Ẑ^D_t = φ_Z Ẑ^D_{t−1} + ε^Z_D_t, 1 ≥ φ_Z ≥ 0. (26)
  - Ĝ^D_t = φ_G Ĝ^D_{t−1} + ε^G_D_t, 1 ≥ φ_G ≥ 0. (27)
- First-order portfolio adjustment solution (equation (28)):
  - Δ x̂^D_{t+1} = η_{Δx^D,Z^D} ε^Z_D_t + η_{Δx^D,G^D} ε^G_D_t.
- Condition for positive elasticity of net equity adjustment to relative productivity (equation (29)):
  - σ(ω−1) (1 + φ)^2 (ω + φ) φ > G^2 (1 − β φ_Z)^2 σ^2_{εGD} (1 − G) (1 − β φ_G)^2 σ^2_{εZD}.
  - This condition also ensures α > 0.
- Relative equity valuation solution (equation (30)):
  - v̂^D_t = η^vD_a n̂fa_t + η^vD_ZD Ẑ^D_t + η^vD_GD Ĝ^D_t + η^vD_ξ ξ̂_t.
  - With ω > 1: η^vD_a < 0, η^vD_GD > 0, η^vD_ξ < 0. η^vD_ZD > 0 under a given condition.

### XI. Valuation dynamics and persistence
- Valuation growth to productivity innovations (equation (31)-(32)):
  - Δ v̂^D_t = η^{ΔvD}_{εZD} ε^Z_D_t − (1−β) (1+φ) (ω−1) φ_Z (1−φ_Z)^{-1} (ω+φ)^{-1} (1−β φ_Z)^{-1} Ẑ^D_{t−1} + η^{ΔvD}_{εGD} ε^G_D_t.
  - For productivity innovations, valuation effects follow an ARMA(1,1):
    - Δ v̂^D_t = φ_Z Δ v̂^D_{t−1} + η^{ΔvD}_{εZD} ε^Z_D_t + η^{ΔvD}_{εZD,−1} ε^Z_D_{t−1}.
- For government spending innovations valuation effects are limited to the i.i.d. innovation (no persistence).
- Interpretation: valuation can be persistent and anticipated after productivity shocks but is an i.i.d., impact-only effect after government spending shocks.

### XII. Shares of valuation and current account in net foreign asset changes
- Definitions (equation (34)):
  - v̂al^S_t ≡ v̂al_t / Δ n̂fa_{t+1} = (1 − Δ x̂^D_{t+1} / Δ v̂^D_t)^{−1},
  - ĉa^S_t ≡ ĉa_t / Δ n̂fa_{t+1} = [1 − (Δ x̂^D_{t+1} / Δ v̂^D_t)^{−1}]^{−1},
  - with v̂al^S_t + ĉa^S_t = 1.
- Analytical results for shocks:
  - Productivity innovation ε^Z_D_0 = 1, none thereafter:
    - v̂al^S_0 = (1 − η_{Δx^D,Z^D} / η^{ΔvD}_{εZD})^{−1}.
    - For t ≥ 1: v̂al^S_{t} = [1 + η_{Δx^D,Z^D} (ω+φ) (1 − β φ_Z) / ((1 − β) (1 + φ) (ω−1) (1 − φ_Z))]^{−1} (constant in t ≥ 1).
    - Shares of valuation are constant and fully anticipated after impact.
  - Government spending innovation ε^G_D_0 = 1, none thereafter:
    - Δ v̂^D_t = η^{ΔvD}_{εGD} ε^G_D_t implies valuation is impact-only.
    - v̂al^S_0 = (1 − η_{Δx^D,G^D} / η^{ΔvD}_{εGD})^{−1}; for t ≥ 1: v̂al^S_{t} = 0 and ĉa^S_{t} = 1.
    - Portfolio adjustment performs all net foreign asset adjustment for t ≥ 1.

### XIII. Cross-country consumption differential and role of valuation/current account
- Cross-country consumption differential solution (equation (35)):
  - Ĉ^D_t = η^C_D_a n̂fa_t + η^C_D_ZD Ẑ^D_t + η^C_D_GD Ĝ^D_t + η^C_D_ξ ξ̂_t.
  - With ω > 1: η^C_D_a > 0, η^C_D_GD < 0, η^C_D_ξ > 0; η^C_D_ZD > 0 under condition in the Technical Appendix.
- First-difference decomposition:
  - Δ Ĉ^D_t = η^C_D_a ĉa_{t−1} + η^C_D_a v̂al_{t−1} + η^C_D_ZD Δ Ẑ^D_t + η^C_D_GD Δ Ĝ^D_t + η^C_D_ξ Δ ξ̂_t.
  - Interpretation: consumption growth between t−1 and t depends on prior period current account and valuation, plus contemporaneous shocks.

### XIV. Calibration used for numerical illustrations
- a = .5
- β = .99
- σ = .5
- φ = 4
- θ = 6
- ω = 2
- G = .2
- φ_Z = .95
- σ^2_{εZD} = 1.08 (computed from σ^2_{εZ} = .73 and σ_{εZ εZ^*} = .19)
- φ_G = .87
- σ^2_{εGD} = 5.12

### XV. Numerical impulse-response highlights (exact figures)
- Productivity shock (one-percent relative productivity innovation at t = 0):
  - On impact:
    - Capital gains/losses share: v̂al^S_0 = 1.0056.
    - Current-account share: ĉa^S_0 = −.0056.
  - For t ≥ 1:
    - v̂al^S_{t} = .9162.
    - ĉa^S_{t} = .0838.
  - Interpretation: valuation represents the main portion of net foreign asset movements after a productivity shock; current account contributes positively but is smaller.
  - The difference between valuation measure v̂al_t and excess-return-based measure ξ̂_t (valDiff_t = v̂al_t − ξ̂_t) is non-negligible for several years after the shock.
- Government spending shock (one-percent relative government spending innovation at t = 0):
  - On impact:
    - v̂al^S_0 = .9116.
    - ĉa^S_0 = .0884.
  - For t ≥ 1:
    - v̂al^S_{t} = 0.
    - ĉa^S_{t} = 1.
  - Interpretation: valuation accounts for most of the initial movement at impact, but portfolio adjustment is responsible for all net foreign asset movements in all periods t ≥ 1.

### XVI. Main conceptual conclusions
- Separating asset prices and quantities in the definition of net foreign assets permits first-order characterization of valuation and portfolio adjustment using first-order approximation for non-portfolio conditions and second-order approximation for portfolio optimality conditions.
- Excess returns (R̂^D_t) are i.i.d. and unanticipated at first order, but capital gains/losses can be persistent and anticipated after productivity shocks (ARMA(1,1) dynamics).
- For productivity shocks: valuation is the dominant driver of net foreign asset changes; near-complete-markets allocation implies small consumption differentials and large price-driven adjustments.
- For government spending shocks: valuation matters only on impact; thereafter portfolio adjustment (current account) carries net foreign asset dynamics.
- Valuation and portfolio adjustment jointly influence consumption risk sharing by damping or amplifying impact responses and maintaining consumption differentials in subsequent periods.

*Source: _wp09275 - 1. From the Budget Constraints to the Law of Motion for Net Foreign Assets (IMF Working Paper content as provided).*

### References.................................................................................................37

### _wp09275 - References.................................................................................................37

### I. Introduction — valuation channel and research objective
- Focus: a specific component of the financial channel of external adjustment that works through valuation effects only, termed the “valuation channel” (capital gains and losses on the stock of gross foreign assets and liabilities due to expected or unexpected asset price changes).
- Context: empirical literature finds the financial channel quantitatively important for the United States; Gourinchas and Rey (2007b) estimate the financial channel contributed on average about 30 percent of the (cyclical) external adjustment of the United States since the 1950s.
- Objective: study the valuation channel theoretically in a two-country, dynamic, stochastic, general equilibrium (DSGE) portfolio model with international trade in equity, encompassing complete and incomplete asset market scenarios.
- Key modeling features:
  - International equity trading in a two-country DSGE model with production under monopolistic competition.
  - Households supply labor, consume a CES aggregated basket of differentiated domestic and foreign goods, and hold shares in domestic and foreign firms.
  - Production uses only labor and is subject to country-wide productivity shocks.
  - Uncertainty arises from productivity and government spending shocks; asset markets are incomplete when both shocks are present.
- Analytical contribution: separation of asset prices and asset quantities in the definition of net foreign assets enables characterization of first-order dynamics of valuation effects (changes in relative, cross-country equity prices) and portfolio adjustment (changes in quantities of net foreign equity holdings, i.e., the current account in this model), and their relative contributions to net foreign asset and macroeconomic dynamics.

### II. Methodology and solution technique
- Approximation approach:
  - Combine a second-order approximation of the portfolio optimality conditions with a first-order approximation of the rest of the model, following Devereux and Sutherland (2009a) and Tille and van Wincoop (2008).
- Key modeling implications:
  - The excess return on foreign assets is a fully unanticipated, i.i.d. variable in the model.
  - Capital gains or losses on equity positions (valuation channel) are generally not i.i.d.; the initial response of valuation to a shock at time t is unanticipated as of t−1, but dynamics in following periods are fully anticipated as of the time of the shock.
  - Response of valuation to relative productivity shocks is generally described by an ARMA(1,1) process.
  - Response of valuation to relative government spending shocks is i.i.d., stemming from the model’s random-walk property of the consumption differential.

### III. Main analytical results on valuation vs. portfolio adjustment
- General results:
  - The cross-country dividend differential, which determines relative equity values, is proportional to contemporaneous productivity and consumption differentials.
  - The proportionality to productivity (and consumption) produces richer ARMA dynamics of valuation in response to productivity shocks.
  - Separation of prices and quantities is necessary to obtain first-order decomposition of valuation and portfolio adjustment and to characterize anticipated capital gain effects.
- Comparative roles by shock type:
  - Productivity shocks:
    - The share of valuation is positive and constant in all periods after the impact of a productivity shock.
    - Valuation can represent a significantly larger share of net foreign asset movements than portfolio adjustment in an incomplete markets scenario, consistent with an equilibrium allocation close to the complete markets outcome.
    - Numerical illustrations show a non-negligible difference between the valuation measure used here and the excess-return-based measure in Devereux and Sutherland (2009b) for productivity shocks (i.e., predictable valuation effects along the dynamics following these shocks).
  - Government spending shocks:
    - The share of valuation in adjustment is zero in all periods but the impact one; portfolio adjustment is responsible for all changes in net foreign assets in subsequent periods.
    - Valuation in response to government spending shocks is i.i.d. because the consumption differential follows a random walk in the model.
- Methodological note:
  - When asset positions do not treat prices and quantities separately, obtaining the first-order decomposition and anticipated valuation effects requires higher-order approximations (e.g., third-order approximation of portfolio optimality conditions with second-order approximation of the rest of the model).

### IV. Implications for macroeconomic dynamics and risk sharing
- Role of valuation, excess returns, and portfolio adjustment:
  - The paper disentangles excess returns, capital gains, and portfolio adjustment for consumption risk sharing when financial markets are incomplete.
  - These channels can dampen or amplify the impact response of the cross-country consumption differential to shocks and can keep it constant in subsequent periods.
- Insurance and specialization results:
  - Regardless of labor supply elasticity, households achieve perfect insurance against country-specific productivity shocks when there is no government spending uncertainty or when steady-state government spending is zero.
  - With inelastic labor supply (regardless of government spending), output and firm profits are determined independently of government spending; international trade in equities provides no hedge against government spending shocks. Optimal portfolio strategy: fully insure against idiosyncratic productivity shocks while remaining exposed to government spending shocks.
- Quantitative remarks and limitations:
  - The model focuses on impulse responses for numerical illustration and omits adjustment costs or other frictions that would ensure stationarity of responses to non-permanent shocks.
  - The implied random-walk property of relative consumption affects analytical solutions; introducing stationarity-inducing devices would alter long-horizon stationarity but likely yield small quantitative departures from the core results for plausible calibrations.
  - Extensions to richer, quantitative models (including home bias or additional assets) could incorporate the methodology of separating asset prices and quantities without changing the core mechanism.

### V. Relationship to the literature
- Connections:
  - Builds on and contrasts with Gourinchas and Rey, Lane and Milesi-Ferretti, Devereux and Sutherland, Tille and van Wincoop, Pavlova and Rigobon, Kollmann, Coeurdacier, Kollmann, and Martin, Benigno, and others.
- Distinctive aspects:
  - Emphasis on separating asset prices and quantities to isolate capital gains (valuation) from the income balance and from portfolio (quantity) adjustment.
  - Analytical characterization of anticipated capital gain effects and their macroeconomic consequences, particularly in production economies with monopolistic competition rather than endowment economies.

### VI. Model: households, consumption, and budget constraints (selected structure and equations)
- World and agents:
  - Two countries, home and foreign, world population continuum [0,1], home households in [0,a), foreign in [a,1].
  - Continuum of monopolistically competitive firms on [0,1], home firms z ∈ [0,a), foreign firms z* ∈ [a,1].
- Preferences (home representative household):
  - Expected utility: E_t ∑_{s=t}^∞ β^{s−t} [ C_s^{1−1/σ} / (1−1/σ) − χ L_s^{1+1/φ} / (1+1/φ) ], with 1>β>0 and σ, χ, φ >0.
- Consumption aggregation:
  - C_t = [ a^{1/ω} C_{Ht}^{(ω−1)/ω} + (1−a)^{1/ω} C_{Ft}^{(ω−1)/ω} ]^{ω/(ω−1)}, with ω>0.
  - Home and foreign sub-baskets:
    - C_{Ht} = [ (1/a)^{1/θ} ∫_0^a c_t(z)^{(θ−1)/θ} dz ]^{θ/(θ−1)}
    - C_{Ft} = [ (1/(1−a))^{1/θ} ∫_a^1 c_t(z*)^{(θ−1)/θ} dz* ]^{θ/(θ−1)}, with θ>1.
  - CPI and price sub-indexes:
    - P_t = [ a P_{Ht}^{1−ω} + (1−a) P_{Ft}^{1−ω} ]^{1/(1−ω)}
    - P_{Ht} = ( (1/a) ∫_0^a p_t(z)^{1−θ} dz )^{1/(1−θ)}
    - P_{Ft} = ( (1/(1−a)) ∫_a^1 p_t(z*)^{1−θ} dz* )^{1/(1−θ)}
- Asset holdings notation:
  - x_{t+1} ≡ aggregate per capita home holdings of shares in home firms entering period t+1.
  - x^*_{t+1} ≡ aggregate per capita home holdings of shares in foreign firms.
  - x^*_{t+1} (foreign holdings of home shares) and x^{**}_{t+1} (foreign holdings of foreign shares) defined analogously.
- Government spending:
  - G_t and G^*_t denote exogenous government spending (following AR(1) processes in logs); governments run balanced budgets with lump-sum taxation equal to spending.
- Household budget constraints (equilibrium versions):
  - v_t x_{t+1} + v^*_t x^*_{t+1} + C_t + G_t = (v_t + d_t) x_t + (v^*_t + d^*_t) x^*_t + w_t L_t
  - v_t x^*_{t+1} + v^*_t x^{**}_{t+1} + C^*_t + G^*_t = (v_t + d_t) x^*_t + (v^*_t + d^*_t) x^{**}_t + w^*_t L^*_t
  - Where v_t, v^*_t are share prices; d_t, d^*_t are dividends; w_t, w^*_t are real wages — all in units of the common consumption basket.
- Gross returns and optimality:
  - Gross returns: R_t ≡ (v_t + d_t) / v_{t−1}, R^*_t ≡ (v^*_t + d^*_t) / v^*_{t−1}.
  - Euler equation for home equity holdings: C_t^{−1/σ} = β E_t ( C_{t+1}^{−1/σ} R_{t+1} ).
  - Indifference between home and foreign equity: E_t ( C_{t+1}^{−1/σ} R_{t+1} ) = E_t ( C_{t+1}^{−1/σ} R^*_{t+1} ).
  - Optimal labor supply: C_t^{−1/σ} w_t = χ L_t^{1/φ}.
- Notes:
  - All prices in each country are denominated in units of the country’s currency; cashless specification follows Woodford (2003); nominal prices and wages are perfectly flexible so focus is on real variables.
  - Law of one price holds for each individual good; consumption-based PPP holds when consumption preferences are identical across countries.

*Source: _wp09275 - References.................................................................................................37*

### 1.    From the Budget Constraints to the Law of Motion for Net Foreign Assets

### From the Budget Constraints to the Law of Motion for Net Foreign Assets

### Key equations and definitions
- Equity market clearing conditions: equations (5)-(6) as stated in the source.
- Home aggregate per capita net foreign assets entering period t+1:
  - nfa_{t+1} ≡ v^*_{t} x^*_{t+1} − (1−a)/a v_{t} x^*_{t+1}. (7)
- Home budget constraint (rewritten) and law of motion for net foreign assets:
  - nfa_{t+1} = R^D_t α_t + R_t nfa_t + y_t − C_t − G_t. (8)
  - Foreign analogue: nfa^*_{t+1} = R^D_t α^*_t + R_t nfa^*_t + y^*_t − C^*_t − G^*_t. (9)
  - Imposing asset market clearing and subtracting yields:
    - nfa_{t+1} = R^D_t α_t + R_t nfa_t + (1−a)(y^D_t − C^D_t − G^D_t). (10)
  - Notation: R^D_t ≡ R^*_t − R_t; α_t ≡ v^*_{t−1} x^*_{t}; y^D_t ≡ y_t − y^*_t; C^D_t ≡ C_t − C^*_t; G^D_t ≡ G_t − G^*_t.

### Firms, production, and income distribution
- Real prices (constant markup θ/(θ−1)) in consumption units:
  - RP_t = θ/(θ−1) w_t / Z_t and RP^*_t = θ/(θ−1) w^*_t / Z^*_t. (11)
- GDP (production side):
  - y_t = RP_t Z_t L_t and y^*_t = RP^*_t Z^*_t L^*_t. (12)
- Income distribution (constant proportions):
  - w_t L_t = (θ−1) y_t / θ and d_t = y_t / θ. (20)

### Useful properties (selected)
- Cross-country GDP ratio:
  - y_t / y^*_t = (Z_t / Z^*_t)^{(1+φ)(ω−1)/(ω+φ)} (C_t / C^*_t)^{−φ(ω−1)/(σ(ω+φ))}. (14)
- Terms of trade:
  - TOT_t ≡ RP_t / RP^*_t = (Z_t / Z^*_t)^{−(1+φ)/(ω+φ)} (C_t / C^*_t)^{φ/(σ(ω+φ))}. (17)
- Wage and labor effort differentials:
  - w_t / w^*_t = (Z_t / Z^*_t)^{(ω−1)/(ω+φ)} (C_t / C^*_t)^{φ/(σ(ω+φ))}. (18)
  - L_t / L^*_t = (Z_t / Z^*_t)^{φ(ω−1)/(ω+φ)} (C_t / C^*_t)^{−φω/(σ(ω+φ))}. (19)

### Steady-state portfolio and role of labor supply elasticity
- Steady-state portfolio α (equation (21); exact form preserved in source). Key special cases:
  - If G = 0 then α = β(1−a)/(1−β).
  - α = β(1−a)/(1−β) also arises when φ = 0 (inelastic labor supply), regardless of G.
- Interpretation:
  - When G = 0 (or φ = 0), households choose the portfolio that fully insures against idiosyncratic productivity shocks; government spending risk is not hedged by equity when φ = 0 because G does not affect profits.
- Comparative static signs (from inspection of equation (21) and text):
  - Assuming ω > 1: ∂α/∂a < 0, ∂α/∂σ > 0, ∂α/∂σ^2_{εGD} < 0, ∂α/∂φ_G < 0, ∂α/∂σ^2_{εZD} > 0, ∂α/∂φ_Z > 0; ∂α/∂G < 0. For plausible parameters, ∂α/∂β > 0, ∂α/∂φ < 0, ∂α/∂ω > 0.

### Log-linear decomposition: portfolio adjustment vs valuation
- Log-linear normalized definition of nfa (equation (22)):
  - n̂fa_{t+1} = α/(1−G) (x̂^D_{t+1} − v̂^D_t).
- First-difference decomposition (equation (23)):
  - Δ n̂fa_{t+1} = α/(1−G) Δ x̂^D_{t+1} − α/(1−G) Δ v̂^D_t = ĉa_t + v̂al_t,
    - where ĉa_t ≡ α Δ x̂^D_{t+1}/(1−G) (portfolio adjustment/current account),
    - v̂al_t ≡ −α Δ v̂^D_t/(1−G) (valuation: asset price effect).
- Log-linearized law of motion for net foreign assets (equation (24)):
  - n̂fa_{t+1} = (1/β) n̂fa_t + α/[β(1−G)] R̂^D_t + (1−a)/(1−G) ŷ^D_t − (1−a) Ĉ^D_t − (1−a)G/(1−G) Ĝ^D_t.
- Relation between excess returns and v̂^D_t:
  - R̂^D_t = −β v̂^D_t − (1−β) d̂^D_t + v̂^D_{t−1}.
- First-order portfolio adjustment identity (equation (25)):
  - Δ x̂^D_{t+1} (scaled) = (1−β)/β (x̂^D_t − d̂^D_t) + (1−a)/α [ŷ^D_t − (1−G) Ĉ^D_t − G Ĝ^D_t].
  - Interpretation: Δ x̂^D_{t+1} equals the income balance from the pre-existing net equity position plus the trade balance.

### Solution structure for portfolio adjustment and valuation
- Relative productivity and government spending processes (AR(1)):
  - Ẑ^D_t = φ_Z Ẑ^D_{t−1} + ε^Z_D_t, 1 ≥ φ_Z ≥ 0. (26)
  - Ĝ^D_t = φ_G Ĝ^D_{t−1} + ε^G_D_t, 1 ≥ φ_G ≥ 0. (27)
- First-order portfolio adjustment solution (equation (28)):
  - Δ x̂^D_{t+1} = η_{Δx^D,Z^D} ε^Z_D_t + η_{Δx^D,G^D} ε^G_D_t,
    - with elasticities given in text (preserved exactly).
- Condition for positive elasticity of net equity adjustment to relative productivity (equation (29)):
  - σ(ω−1) (1 + φ)^2 (ω + φ) φ > G^2 (1 − β φ_Z)^2 σ^2_{εGD} (1 − G) (1 − β φ_G)^2 σ^2_{εZD}.
  - This is the same condition that ensures α > 0 (home agents hold positive foreign equity).
- Relative equity valuation solution (equation (30)):
  - v̂^D_t = η^vD_a n̂fa_t + η^vD_ZD Ẑ^D_t + η^vD_GD Ĝ^D_t + η^vD_ξ ξ̂_t.
  - With ω > 1: η^vD_a < 0, η^vD_GD > 0, η^vD_ξ < 0. η^vD_ZD > 0 under a given condition in text.

### Valuation dynamics and persistence
- Valuation growth dynamics to productivity innovations (equation (31) and (32)):
  - Δ v̂^D_t = η^{ΔvD}_{εZD} ε^Z_D_t − (1−β) (1+φ) (ω−1) φ_Z (1−φ_Z)^{-1} (ω+φ)^{-1} (1−β φ_Z)^{-1} Ẑ^D_{t−1} + η^{ΔvD}_{εGD} ε^G_D_t.
  - For productivity innovations, valuation effects follow an ARMA(1,1) — see (32):
    - Δ v̂^D_t = φ_Z Δ v̂^D_{t−1} + η^{ΔvD}_{εZD} ε^Z_D_t + η^{ΔvD}_{εZD,−1} ε^Z_D_{t−1}.
- For government spending innovations valuation effects are limited to the i.i.d. innovation (no persistence).
- Interpretation:
  - Valuation (capital gains/losses) can be persistent and anticipated after productivity shocks (ARMA dynamics) but is an i.i.d., impact-only effect after government spending shocks.

### Shares of valuation and current account in net foreign asset changes
- Definition (equation (34)):
  - v̂al^S_t ≡ v̂al_t / Δ n̂fa_{t+1} = (1 − Δ x̂^D_{t+1} / Δ v̂^D_t)^{−1},
  - ĉa^S_t ≡ ĉa_t / Δ n̂fa_{t+1} = [1 − (Δ x̂^D_{t+1} / Δ v̂^D_t)^{−1}]^{−1},
  - with v̂al^S_t + ĉa^S_t = 1 (but not constrained to [0,1] individually).
- Analytical results for shocks:
  - Productivity innovation ε^Z_D_0 = 1, none thereafter:
    - v̂al^S_0 = (1 − η_{Δx^D,Z^D} / η^{ΔvD}_{εZD})^{−1}.
    - For t ≥ 1: v̂al^S_{t} = [1 + η_{Δx^D,Z^D} (ω+φ) (1 − β φ_Z) / ((1 − β) (1 + φ) (ω−1) (1 − φ_Z))]^{−1} (constant in t ≥ 1).
    - Shares of valuation are constant and fully anticipated after impact.
  - Government spending innovation ε^G_D_0 = 1, none thereafter:
    - Δ v̂^D_t = η^{ΔvD}_{εGD} ε^G_D_t implies valuation is impact-only.
    - v̂al^S_0 = (1 − η_{Δx^D,G^D} / η^{ΔvD}_{εGD})^{−1}; for t ≥ 1: v̂al^S_{t} = 0 and ĉa^S_{t} = 1.
    - Portfolio adjustment performs all net foreign asset adjustment for t ≥ 1.

### Role of valuation and portfolio adjustment in macro dynamics
- Cross-country consumption differential solution (equation (35)):
  - Ĉ^D_t = η^C_D_a n̂fa_t + η^C_D_ZD Ẑ^D_t + η^C_D_GD Ĝ^D_t + η^C_D_ξ ξ̂_t.
  - ω > 1 implies η^C_D_a > 0, η^C_D_GD < 0, η^C_D_ξ > 0; η^C_D_ZD > 0 iff condition in the Technical Appendix.
- First-difference decomposition:
  - Δ Ĉ^D_t = η^C_D_a ĉa_{t−1} + η^C_D_a v̂al_{t−1} + η^C_D_ZD Δ Ẑ^D_t + η^C_D_GD Δ Ĝ^D_t + η^C_D_ξ Δ ξ̂_t.
  - Interpretation: consumption growth between t−1 and t depends on prior period current account and valuation, plus contemporaneous shocks.

### Calibration used for numerical illustrations (values preserved exactly)
- a = .5
- β = .99
- σ = .5
- φ = 4
- θ = 6
- ω = 2
- G = .2
- φ_Z = .95
- σ^2_{εZD} = 1.08 (computed from the stated standard values: σ^2_{εZ} = .73 and σ_{εZ εZ^*} = .19)
- φ_G = .87
- σ^2_{εGD} = 5.12

### Numerical impulse-response highlights (exact figures from the source)
- Productivity shock (one-percent relative productivity innovation at t = 0):
  - On impact:
    - Capital gains/losses share: v̂al^S_0 = 1.0056.
    - Current-account share: ĉa^S_0 = −.0056.
  - For t ≥ 1:
    - v̂al^S_{t} = .9162.
    - ĉa^S_{t} = .0838.
  - Interpretation: valuation (capital gains/losses) represents the main portion of net foreign asset movements after a productivity shock; current account contributes positively but is smaller.
  - The difference between valuation measure v̂al_t and excess-return-based measure ξ̂_t (valDiff_t = v̂al_t − ξ̂_t) is non-negligible for several years after the shock (persistent relative equity price dynamics not captured by ξ̂_t).
- Government spending shock (one-percent relative government spending innovation at t = 0):
  - On impact:
    - v̂al^S_0 = .9116.
    - ĉa^S_0 = .0884.
  - For t ≥ 1:
    - v̂al^S_{t} = 0.
    - ĉa^S_{t} = 1.
  - Interpretation: valuation accounts for the majority of the initial movement in net foreign assets at impact, but portfolio adjustment is responsible for all net foreign asset movements in all periods t ≥ 1 after a government spending shock.

### Main conceptual conclusions (as presented in the source)
- Separating asset prices and quantities in the definition of net foreign assets permits characterization of first-order dynamics of both valuation and portfolio adjustment using:
  - First-order approximation for non-portfolio equilibrium conditions, and
  - Second-order approximation for portfolio optimality conditions.
- Excess returns (R̂^D_t) are i.i.d. and unanticipated at first order, but capital gains/losses (valuation) can be persistent and anticipated after productivity shocks (ARMA(1,1) dynamics).
- For productivity shocks: valuation is the dominant driver of net foreign asset changes; the economy’s near-complete-markets allocation implies small consumption differentials and thus large price-driven adjustments.
- For government spending shocks: valuation matters only on impact (i.i.d. innovation); thereafter portfolio adjustment (current account) carries net foreign asset dynamics.
- Valuation and portfolio adjustment jointly influence consumption risk sharing: they dampen or amplify the impact response of the cross-country consumption differential and maintain consumption differentials constant in subsequent periods.

*Source: _wp09275 - 1. From the Budget Constraints to the Law of Motion for Net Foreign Assets (IMF Working Paper content as provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09275.pdf_