## Appendix I

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---

### Introduction and research questions
- Investigates portfolio choice in an open macroeconomy with asset trade and its role in macro fluctuations.
- Key questions:
  - Does the international transmission mechanism depend on who owns firms?
  - Do changes in valuations of internationally traded assets play a role in macroeconomic adjustment to shocks?
  - Is there an interaction between the stock market and exchange rates?2
- Emphasizes need for a model consistent with empirical evidence that equity prices depend on the market’s perception of monetary policy.3

### Model setup and core assumptions
- Symmetric, two-country general-equilibrium model:
  - Agents have identical preferences in each country; firms use identical technologies; market structure identical across countries.
  - Stochastic processes for driving variables (productivity and monetary) are identical.
- Crucial assumption: claims to human capital are not traded.
- Implication: despite symmetry, equilibrium home and foreign portfolios can differ and may exhibit home bias in equities.

### Mechanism: sticky nominal prices and portfolio implications
- Monetary policy affects real returns through sticky nominal prices: some firms set nominal prices without full information about the state.
- If all nominal prices are sticky, short-run output becomes demand determined.
- Under a positive productivity shock to home firms:
  - Demand for labor declines; employment and wages fall; firm profits increase.
  - Returns to labor (human capital) and returns to firm ownership are negatively correlated in the short run—opposite the typical neoclassical presumption.
  - This negative correlation makes ownership of domestic firms an effective hedge against employment and wage risk.

### Connection to the home bias puzzle
- The model may help explain the empirical “home bias” puzzle: foreign equities comprise a small proportion of investors' portfolios.4
- In a framework with sticky nominal prices, optimal portfolios may be strongly biased toward home equities as a hedge against labor income fluctuations (nontradable risk).
- Home-biased optimal portfolios can arise endogenously; the model is not constructed solely to explain home bias.

### Static model — setup and key results
- Economy features:
  - Two countries (Home and Foreign), world population normalized to unity, half in Home and half in Foreign.
  - All goods tradable and perishable; only firms can export; firms monopolistically produce differentiated varieties using labor only; prices preset in consumers' currency (local currency pricing).
  - Assets traded before shocks: forward contracts in foreign exchange and equities.
  - Shocks: monetary and technology shocks, identical distributions across Home and Foreign.
- Household portfolio choices made ex ante: shares of Home equities (γh), shares of Foreign equities (γf), forward position in foreign exchange (δ%); constraint γh + γf = 1 (equation (2.7)).
- Preferences use CES aggregation with parameters including ω and λ; key parameter conditions: 1ρ>, 0χ>, 0ψ>, and 0η> (from equation (2.1)).
- Main analytical results:
  - Equity portfolio determined by covariances and variances of shocks to profits and labor income orthogonal to exchange rate risk (equation (2.26)).
  - Complete home bias result in symmetric sticky-price static model: 0γ= (Home households hold 100 percent of Home equities) (derived from (2.26) and (2.27)–(2.31)).
  - Forward position δ solved from (2.31) hedges exchange-rate-related portions of monetary shocks.
  - World-consumption identity in linearized model: *tt    t cs  cρρ=+  (equation (2.33)).
  - Money-holding first-order condition and exchange rate relation: tt mcρ= . With symmetry: *ttt smm=− (equation (2.35)).
- Income and hedging distinction:
  - Monetary shocks create real income effects hedged via bond portfolios or forward positions; productivity shocks affect distribution between labor income and profits and are hedged by equity portfolios.
  - Predetermined nominal prices imply monetary shocks have real effects and incomplete consumption correlation across countries.

### Intuition (static model)
- Equity holdings hedge only the component of income orthogonal to exchange rates; exchange-rate-correlated risk is hedged by forward contracts.
- If the component of labor income orthogonal to exchange rates is uncorrelated with relative profits, portfolio is balanced (γ = 1/2).
- The covariance between wage income and Home profits relative to Foreign profits determines home vs. anti-home bias:
  - Positive covariance → anti-home bias (1/2γ>), as in Baxter and Jermann (1997).
  - Negative covariance → home bias (γ<1/2).
- In the symmetric sticky-price static model the term leads to complete home bias (0γ=) because Home equity returns perfectly hedge labor income shocks once projected orthogonally to the exchange rate.

### Dynamic model — setup and distinguishing features
- Infinite-horizon model to study persistent technology shocks and degrees of price stickiness.
- Price-setting: fraction τ of firms set prices in advance (preset), remaining firms set prices after shocks (flexible); allows comparison across degrees of stickiness.
- Equities of all firms in a country are bundled.
- Households maximize expected discounted utility with discount factor β and choose portfolio positions and consumption; first-order conditions summarized in (3.2)–(3.6).
- Financial wealth, human capital, and return definitions introduced (3.7)–(3.12), including foreign equity share γ (3.12).
- Firms: flexible-price firms’ optimal prices (3.16); preset-price firms’ optimal prices (3.17)–(3.18). Aggregate profit expressions (3.22)–(3.23).
- Stochastic processes and persistence:
  - Money: 11 m ttt mmv ++ =+ and symmetric assumptions (3.24).
  - World and relative technology: 11WWW tWtt aavθ ++ =+ , 11RRR tRtt aavθ ++ =+ (3.25) with θW, θR in [0,1) indicating persistence; v shocks i.i.d.
  - Variances: 2var()W W vσ=, 2var()R R vσ=, cov(,) 0 WR vv =; symmetric initial conditions 0 0 R a=, 0 0 R m=.

### Dynamic model — qualitative implications
- Allowing for price flexibility (lower τ) and persistent productivity shocks reduces home bias compared with static sticky-price case.
- When 1ω> (elasticity of substitution between Home and Foreign goods more than unity), the optimal Home portfolio is less home biased than in the static model because households consider future adjusted prices.
- Degree of home bias depends on τ, θW, θR, and β: “The degree of home bias depends on the persistence of price stickiness, the persistence of productivity shocks, and the weight that households assign to future consumption.”

### Salient features of the solution (linearized dynamic model)
- Linearly approximated dynamic model replicates allocation achievable with a full set of nominal contingent bonds. Key replicated condition:
  - **()tt    t  t  t cc  s p pρ−=+−**. (3.26)
- Purchasing power parity holds in expectation because prices are sticky at most one period:
  - ***11 ()   () tttt EcEc ++ =.** (3.27)
- Stationarity property: forward-looking expected consumption is equalized due to equilibrium asset and human-capital wealth equalization.
  - Relative total wealth equality (with ζ defined in Appendix):
    - **(1)0 RR ttt vhsζζ+−   −= .** (3.28)
  - Interpretation: tV is the value of equities Home household carries into t+1 and tH is expected value at time t of returns to work from t+1 onward; (3.28) implies Home and Foreign total wealth at end of period t are equal.
- Certain coefficients are constant over time:
  - **11 (1) 2 t δδτ ρ ≡=  −,** (3.29)
  - ** *1 2(1) tt γγ γ ζζ Α ≡== Β+  −  Α ,** (3.30) with Α and Β defined in Appendix.
- Comparative statics for γ:
  - γ increases in Α and decreases in Β.
  - Demand is decreasing in ζ when Β>Α; to have home bias (1 2 γ<) generally requires Β>Α, implying condition (3.31):
    - **1(1)1 0 1(1) 1    1 R R βθ ωτω ωτψ  ωψ βθ −−    − − > +−   + − .** (3.31)
  - Condition (3.31) does not depend on ρ or ζ; ζ determines level of home bias.

### Role of parameters and special cases
- Parameter interpretations:
  - Rβθ: weight on the future / persistence of shocks.
  - τ: fraction of firms setting price in advance (price stickiness).
  - ω: elasticity of substitution between Home and Foreign aggregates.
  - ζ: labor’s share in national income.
  - ψ: elasticity of labor supply.
- Extremes and limits:
  - If 1τ= and 0R βθ= → complete home bias (0γ=) — static model result.
  - If 0τ= → optimal portfolio approaches **11   1 212 γ ζ => −** (matches Baxter and Jermann (1997)).
  - Increasing τ tends to make home bias more likely; γ is decreasing in τ when 1ω>.
  - As Rβθ increases, Α increases and share of Foreign equities increases; in limit 1 R βθ→ portfolio approaches flexible-price value **11 21 γ ζ = −**.
  - When 1ω= and price stickiness absent, 100 percent home bias arises: **0γ=**; if 1ω= and prices flexible, γ indeterminate (Cole and Obstfeld (1991); Obstfeld and Rogoff (2002) analog).

### Calibration and quantitative results
- Calibration choices:
  - Set 1τ= and calibrate period length by half-life of price adjustment; choose half-life of 1 year → one period = two years.
- Baseline parameter values:
  - 23 ζ = .
  - 8 (0.855)0.286 R θ = ≈ .
  - 8 (0.99)0.923 β = ≈ .
  - 1.5 ω = .
  - 1 ψ = .
- Baseline numerical result:
  - With baseline parameters, foreign equity share γ ≈ 0.052. (Symmetric unbiased portfolio would be 0.5 γ = .)
- Correlations and interpretation:
  - Negative conditional correlation between labor hours and productivity (conditioning on productivity shock) is key driver of home bias; unconditional correlation can be positive due to hedging via forward contracts.
  - Empirical literature: Gali (1999) finds negative conditional correlation between labor hours and productivity for technology shocks; literature contested by Christiano, Eichenbaum and Vigfusson (2003), Francis and Ramey (2003, 2004), Gali, Lopez-Salido and Valles (2003).
- Numerical illustration (correlation between return on domestic equities and human capital under alternative technology shock assumptions):
  - Table (rows cor(a t , a t *) and columns Standard Deviation of Home Productivity Shock relative to Home Monetary Shock = 0.01, 0.5, 1, 2, 4, 8, 100):
    - For cor(a t , a t *) = 0.00: 0.990 0.959 0.875 0.614 0.117 -0.314 -0.569
    - For cor(a t , a t *) = 0.25: 0.990 0.960 0.878 0.625 0.137 -0.296 -0.560
    - For cor(a t , a t *) = 0.50: 0.990 0.961 0.882 0.637 0.159 -0.287 -0.579
    - For cor(a t , a t *) = 0.75: 0.990 0.962 0.886 0.651 0.186 -0.290 -0.656
    - For cor(a t , a t *) = 0.99: 0.990 0.963 0.889 0.666 0.219 -0.317 -0.960
  - Interpretation: with parameter set (including ρ = 5, W θ = 0.75, and time period = two years), unconditional correlation between returns to human capital and Home equities is not necessarily negative; it becomes negative only when productivity shock standard deviation is large relative to monetary shock standard deviation.
- Sensitivity to speed of price adjustment (Foreign equity share under alternative half-life values; comparison for 1τ= and 0.8τ= with implied aggregate half-life in parentheses):
  - Half-life (quarters) / Foreign equity share (1τ=) / Foreign equity share (0.8τ=; implied aggregate half-life)
    - 0.5  (0.375) : 0.533 / 0.734
    - 1    (0.75)  : 0.303 / 0.482
    - 2    (1.5)   : 0.143 / 0.284
    - 4    (3)     : 0.052 / 0.162
    - 6    (4.5)   : 0.023 / 0.121
    - 8    (6)     : 0.011 / 0.104
  - Interpretation: faster price adjustment (shorter half-life) raises foreign equity share; half-life of 2 quarters → foreign share 0.143 (1τ=); half-life 1 quarter → 0.303; half-life 0.5 quarter → anti-home bias consistent with Baxter and Jermann (1997).

### Main conclusions and implications
- The model provides a general-equilibrium theory where home bias emerges naturally in a symmetric dynamic economy with partial price stickiness and nontradable claims to labor income.
- Even though the linear approximation replicates complete-market allocations, the model does not resolve other international-finance puzzles (e.g., high exchange-rate volatility, consumption-real exchange-rate anomaly described in Chari et al. (2002)).
- Potential extensions:
  - Embedding credit constraints (short-sale constraints) as in Julliard (2004) could produce stronger home bias and relax the tight link between real exchange rate and relative consumption.
  - Compatible with other explanations for home bias (information costs, institutional frictions); identified forces do not exclude those channels.
- Model strengths: analytically tractable, yields closed-form expressions (e.g., equation (3.30) for foreign equity share) and offers a tractable framework to study valuation effects, portfolio adjustment, stock prices and exchange rate interactions.

### Formal definition and linearized equilibrium (Appendix summary)
- Formal Definition A: equilibrium is a sequence of variables solving full system of (50) equations (equations (2.5), (2.8), (2.9), (3.3), (3.4), (3.7)-(3.22), their foreign counterparts, plus 3 asset-market clearing conditions), given stochastic sequences { A t t , M t t , A t * , M t * } and specified initial conditions.
- Approximated system: log-linearization around unconditional means (real variables stationary under stationary productivity processes; nominal variables have unit roots).
- Key log-linearized first-order conditions: (A.1)–(A.6), (A.9)–(A.17).
- Conjectured equilibrium allocation in closed-form log-linear solutions (A.18)–(A.37), including:
  - Stationary constant: **11 (1) 2 t δδτ ρ ≡=  −** (A.32)
  - Foreign equity share: ** *1 2(1) tt γγ γ ζζ Α ≡== Β+  −  Α** (A.33)
  - Replication of full contingent-claims allocation: ** **()tt    t  t  t cc  s p pρ−=+−.** (A.38)
- Verification: proofs that first-order conditions, market clearing, returns on assets, asset allocation conditions (A.3)–(A.5), human wealth (A.15), and budget constraints are satisfied by conjectured allocation (summarized in Appendix D).

*Source: _wp05165 - Appendix I (Appendix and solution summary)*

### Appendix I........... ..................................................................................................

### Appendix I

### Introduction and research questions
- Investigates portfolio choice in an open macroeconomy with asset trade and its role in macro fluctuations.
- Key questions highlighted:
  - Does the international transmission mechanism depend on who owns firms?
  - Do changes in valuations of internationally traded assets play a role in macroeconomic adjustment to shocks?
  - Is there an interaction between the stock market and exchange rates?2
- Emphasizes the need for a model consistent with empirical evidence that equity prices depend on the market’s perception of monetary policy.3

### Model setup and core assumptions
- A symmetric, two-country general-equilibrium model:
  - Agents have identical preferences in each country.
  - Firms use identical technologies.
  - Market structure is identical across countries.
  - Stochastic processes for driving variables (productivity and monetary) are identical.
- Crucial assumption: claims to human capital are not traded.
- Implication: despite symmetry, equilibrium home and foreign portfolios can differ and may exhibit home bias in equities.

### Mechanism: sticky nominal prices and portfolio implications
- Monetary policy affects real returns through sticky nominal prices: some firms set nominal prices without full information about the state.
- If all nominal prices are sticky, short-run output becomes demand determined.
- Under a positive productivity shock to home firms:
  - Demand for labor declines.
  - Employment and wages fall.
  - Firm profits increase.
- Consequently, returns to labor (human capital) and returns to firm ownership are negatively correlated in the short run—opposite the typical neoclassical presumption.
- This negative correlation makes ownership of domestic firms an effective hedge against employment and wage risk.

### Connection to the home bias puzzle
- The model may help explain the empirical “home bias” puzzle: foreign equities comprise a small proportion of investors' portfolios.4
- Argument: in a framework with sticky nominal prices, optimal portfolios may be strongly biased toward home equities as a hedge against labor income fluctuations (nontradable risk).
- The paper does not construct the model solely to explain home bias, but finds home-biased optimal portfolios can arise endogenously.

### Empirical relevance and related findings
- Cites empirical findings that support model implications:
  - Equity prices respond to perceptions of monetary policy.3
  - Gali (1999) shows that sticky-price closed-economy models can generate falls in labor hours after positive technology shocks; labor hours decline in response to positive technology shocks in most G-7 countries.6
  - Bottazzi, Pesenti and van Wincoop (1996), and Julliard (2002) find returns to human capital and equities are negatively correlated in most OECD countries.7
- Relates to literature explaining home bias as a hedge against non-tradable risks; here the nontradable risk is labor income.8
- Contrasts with neoclassical models where, because labor income is more correlated with domestic firms' profits, optimal portfolios may be more foreign-weighted than classical endowment models predict (citing Baxter and Jermann (1997)).

*Source: _wp05165 - Appendix I........... ..................................................................................................*

### introduction of nontradable risk generally has not been helpful in explaining home bias.

### introduction of nontradable risk generally has not been helpful in explaining home bias.

### Purpose and overview
- Chief aim: provide a model of portfolio choice under sticky nominal prices in the open economy (not necessarily a model of home bias in equities).
- Key insight previewed in the text: home bias arises endogenously under nominal price stickiness because monetary shocks can be hedged with bonds/forward positions while equity portfolios primarily hedge productivity (real) risks. Home bias depends on the persistence of price stickiness, persistence of productivity shocks, and the discounting of future consumption.

### Static model — setup and assumptions
- Economy: two-country general-equilibrium model with sticky prices; countries called Home and Foreign; world population normalized to unity; half in Home and half in Foreign; identical preferences.
- Goods: all goods are tradable and perishable; markets segmented so that only firms can export goods; firms monopolistically produce differentiated varieties using labor only; prices are preset in consumers' currency (local currency pricing).
- Assets traded before shocks: forward contracts in foreign exchange and equities.
- Shocks considered: monetary shocks and technology shocks, with identical distributions across Home and Foreign.
- Household portfolio choices made ex ante: shares of Home equities (γh), shares of Foreign equities (γf), and forward position in foreign exchange (δ%); constraint γh + γf = 1 (equation (2.7)).
- Preferences and consumption structure use CES aggregation with parameters including ω (elasticity between Home and Foreign goods) and λ (elasticity among varieties, with 1λ>).
- Key parameter conditions stated in utility specification: 1ρ>, 0χ>, 0ψ>, and 0η> (from equation (2.1)).

### Static model — main analytical results
- Equity portfolio determined by covariances and variances of shocks to profits and labor income orthogonal to exchange rate risk (equation (2.26) and discussion).
- Complete home bias result: equilibrium implies 0γ= (i.e., Home households hold 100 percent of Home equities) under the model’s symmetry and sticky-price assumptions (conclusion following substitution into (2.26) and supporting equations (2.27)–(2.31)).
- Forward position δ: solved from equation (2.31) under symmetry; an equilibrium forward position hedges exchange-rate-related portions of monetary shocks.
- World-consumption identity in linearized model: *
tt    t
cs  cρρ=+  (equation (2.33)), implying the linearized model replicates allocation achievable with a full set of nominal contingent bonds.
- Money-holding first-order condition and exchange rate relation: tt
mcρ=. Combining with symmetry yields *ttt smm=− (equation (2.35)); exchange rates determined by relative money supplies.
- Income and hedging distinction:
  - Monetary shocks create real income effects that can be hedged via bond portfolios or forward positions; monetary shocks induce nominal exchange rate changes that alter bond returns.
  - Productivity (technology) shocks affect distribution between labor income and profits; equity portfolios hedge productivity risk. Because firm revenue depends only on monetary shocks in the sticky-price environment, holding Home firms’ equities perfectly hedges human-capital exposure, producing complete home bias.
- Role of predetermined nominal prices: preset nominal prices imply that monetary shocks have real effects and that goods prices in consumers’ currencies do not arbitrage away differences across countries when markets are segmented, producing incomplete consumption correlation across countries.

### Intuition and mechanism (static model)
- Equity holdings hedge only the component of income orthogonal to exchange rates; exchange-rate-correlated risk is hedged by forward contracts (equation (2.26) discussion).
- If the component of labor income orthogonal to exchange rates is uncorrelated with relative profits, the portfolio would be balanced (γ = 1/2). The second term in (2.26) — covariance between wage income and Home profits relative to Foreign profits — determines home vs. anti-home bias:
  - Positive covariance → anti-home bias (1/2γ>) as in Baxter and Jermann (1997).
  - Negative covariance → home bias (γ<1/2).
- In the symmetric sticky-price static model, that term leads to complete home bias (0γ=) because Home equity returns perfectly hedge labor income shocks once projected orthogonally to the exchange rate (equations (2.27)–(2.30) logic).

### Dynamic model — setup and distinguishing features
- Infinite-horizon model to study persistent technology shocks and degrees of price stickiness.
- Price-setting: fraction τ of firms in each country set prices in advance (preset), remaining firms set prices after shocks (flexible). This allows comparison across degrees of stickiness.
- Equity bundling: equities of all firms in a country bundled together.
- Households maximize expected discounted utility with discount factor β and choose sequences of portfolio positions and consumption; first-order conditions summarized in (3.2)–(3.6).
- Financial wealth, human capital, and return definitions introduced (equations (3.7)–(3.12)), including the share of foreign equity in portfolio γ (3.12).
- Firms: flexible-price firms’ optimal prices given by (3.16); preset-price firms’ optimal prices given by (3.17)–(3.18). Aggregate profit expressions unchanged in form (3.22)–(3.23).
- Stochastic processes and persistence:
  - Money processes: 11 m ttt mmv ++ =+ and symmetric assumptions (3.24).
  - World and relative technology processes: 11WWW tWtt aavθ ++ =+, 11RRR tRtt aavθ ++ =+ (3.25) with θW, θR in [0,1) indicating persistence in world and relative technology levels and v shocks i.i.d.
  - Variance notation: e.g., 2var()W W vσ=, 2var()R R vσ=, cov(,) 0 WR vv =, and symmetric initial conditions 0 0 R a=, 0 0 R m=.

### Dynamic model — qualitative implications (from text)
- Allowing for price flexibility (lower τ) and persistent productivity shocks reduces home bias compared with the static sticky-price case:
  - When elasticity of substitution between Home and Foreign goods is more than unity (1ω>), the optimal Home portfolio is less home biased than in the static model because households consider future adjusted prices.
- The dynamic model enables study of how differential price stickiness and persistence of technology shocks affect portfolio allocation over time; results depend on τ, θW, θR, and β (textual statement: “The degree of home bias depends on the persistence of price stickiness, the persistence of productivity shocks, and the weight that households assign to future consumption.”).

### Key analytical identities and equilibrium conditions emphasized in the source
- Portfolio constraint: hf γγ += (equation (2.7)).
- Household asset first-order conditions highlighting exchange-rate hedging: equations (2.12), (2.13), (3.4)–(3.6), and compact summary (3.15).
- Complete-home-bias result in static symmetric sticky-price model: 0γ= (derived around equations (2.26)–(2.31)).
- World consumption / marginal utilities equality in linearized model: *tt    t
cs  cρρ=+ (equation (2.33)).
- Exchange rate and money relation (linearized): *ttt smm=− (equation (2.35)).

*Source: _wp05165 - introduction of nontradable risk generally has not been helpful in explaining home bias.*

### Appendix presents the solution to the model. There, the equilibrium is defined and solutions

### _wp05165 - Appendix presents the solution to the model. There, the equilibrium is defined and solutions

### Salient features of the solution
- The linearly approximated dynamic model replicates the allocation achieved when a full set of nominal/state-contingent claims are traded. Key replicated condition:
  - **()tt    t  t  t cc  s p pρ−=+−**. (3.26)
- Purchasing power parity holds in expectation because prices are sticky for at most one period:
  - ***11 ()   () tttt EcEc ++ =.** (3.27)
- Stationarity property: even if contemporaneous Home and Foreign consumption differ, forward-looking expected consumption is equalized due to equilibrium asset and human-capital wealth equalization.
  - Relative total wealth equality result (with ζ defined in Appendix):
    - **(1)0 RR ttt vhsζζ+−   −= .** (3.28)
  - Interpretation: tV is the value of equities the Home household carries into period t+1 and tH is the expected value at time t of returns to work from t+1 onward; equation (3.28) implies Home and Foreign total wealth at end of period t are equal.

### Mechanism driving home bias and portfolio allocation
- Home and Foreign households can hold different equity portfolios, and conditional expected returns on equities depend on shock realizations (0R t v≠ and 0R t h≠ in general).
- Example: a positive relative technology shock (0R t a> ) with zero monetary shocks can leave consumption unchanged but alters current labor income and profits. Under parameter configurations delivering home bias, a relative decline in Home current income is exactly offset by increases in the value of Home human wealth and equities carried into t — preserving relative total wealth.
- As a consequence of stationarity, certain coefficients are constant over time:
  - **11 (1) 2 t δδτ ρ ≡=  −,** (3.29)
  - ** *1 2(1) tt γγ γ ζζ Α ≡== Β+  −  Α ,** (3.30)
    - with definitions for Α and Β given in the Appendix text.
- Comparative statics for γ (share of equity portfolio held in foreign assets):
  - γ increases in Α and decreases in Β.
  - Demand is decreasing in ζ when Β>Α; to have home bias (1 2 γ<) generally requires Β>Α, which implies condition (3.31) (preserved verbatim):
    - **1(1)1 0 1(1) 1    1 R R βθ ωτω ωτψ  ωψ βθ −−    − − > +−   + − .** (3.31)
  - The condition (3.31) does not depend on ρ or ζ, while ζ determines the level of home bias.

### Role of parameters and special cases
- Interpretation of parameters:
  - Rβθ: weight the future receives in portfolio decision / persistence of shocks.
  - τ: fraction of firms setting price in advance (price stickiness).
  - ω: elasticity of substitution between Home and Foreign aggregates.
  - ζ: labor’s share in national income.
  - ψ: elasticity of labor supply.
- Extremes and limits:
  - If all prices are sticky (1τ=) and the future does not matter (0R βθ=), there is complete home bias (0γ=) — the static model result.
  - If all goods prices flexible (0τ=), optimal portfolio approaches: **11   1 212 γ ζ => −** (matches Baxter and Jermann (1997) result).
  - Increasing τ (more price stickiness) tends to make home bias more likely; γ is decreasing in τ when 1ω>.
  - As Rβθ increases (future matters more), Α increases and share of Foreign equities increases; in the limit 1 R βθ→ the portfolio approaches the flexible-price value **11 21 γ ζ = −**.
  - When 1ω= and price stickiness absent, 100 percent home bias arises: **0γ=**; if 1ω= and prices flexible, γ is indeterminate (Cole and Obstfeld (1991); Obstfeld and Rogoff (2002) analog).

### Calibration and quantitative results (properties of the model)
- Calibration choices and rationale:
  - Set 1τ= and calibrate period length by half-life of price adjustment; choose half-life of 1 year → one period = two years.
  - Parameter values (baseline):
    - 23 ζ = .
    - 8 (0.855)0.286 R θ = ≈ .
    - 8 (0.99)0.923 β = ≈ .
    - 1.5 ω = .
    - 1 ψ = .
- Baseline numerical result:
  - With baseline parameters, foreign equity share γ ≈ 0.052. (Model symmetric so unbiased portfolio would be 0.5 γ = .)
- Discussion on correlations:
  - Negative conditional correlation between labor hours and productivity (conditioning on productivity shock) is the key driver of home bias in the model; unconditional correlation can be positive due to hedging via forward contracts.
  - Empirical literature: Gali (1999) finds negative conditional correlation between labor hours and productivity for technology shocks; literature contested by Christiano, Eichenbaum and Vigfusson (2003), Francis and Ramey (2003, 2004), Gali, Lopez-Salido and Valles (2003).
- Numerical illustration table (correlation between return on domestic equities and human capital under alternative technology shock assumptions):
  - Table entries (rows labelled cor(a t , a t *) and columns Standard Deviation of Home Productivity Shock relative to Home Monetary Shock = 0.01, 0.5, 1, 2, 4, 8, 100):
    - For cor(a t , a t *) = 0.00: 0.990 0.959 0.875 0.614 0.117 -0.314 -0.569
    - For cor(a t , a t *) = 0.25: 0.990 0.960 0.878 0.625 0.137 -0.296 -0.560
    - For cor(a t , a t *) = 0.50: 0.990 0.961 0.882 0.637 0.159 -0.287 -0.579
    - For cor(a t , a t *) = 0.75: 0.990 0.962 0.886 0.651 0.186 -0.290 -0.656
    - For cor(a t , a t *) = 0.99: 0.990 0.963 0.889 0.666 0.219 -0.317 -0.960
  - Interpretation: with the parameter set (including ρ = 5, W θ = 0.75, and time period = two years), these parameter values do not necessarily imply a negative unconditional correlation between returns to human capital and Home equities; only when productivity shock standard deviation is large relative to monetary shock standard deviation does unconditional correlation become negative.
- Sensitivity to speed of price adjustment:
  - Foreign equity share under alternative half-life values (and comparison for 1τ= and 0.8τ=):
    - Half-life (quarters) / Foreign equity share (1τ=) / Foreign equity share (0.8τ=; implied aggregate half-life in parentheses)
      - 0.5  (0.375) : 0.533 / 0.734
      - 1    (0.75)  : 0.303 / 0.482
      - 2    (1.5)  : 0.143 / 0.284
      - 4    (3)    : 0.052 / 0.162
      - 6    (4.5)  : 0.023 / 0.121
      - 8    (6)    : 0.011 / 0.104
  - Interpretation: with faster price adjustment (shorter half-life) foreign equity share rises; a half-life of 2 quarters gives foreign share 14.3% (1τ=), half-life 1 quarter gives 30.3%, and half-life 0.5 quarter yields anti-home bias consistent with Baxter and Jermann (1997).

### Main conclusions and implications
- The model provides a general-equilibrium theory where home bias emerges naturally in a symmetric dynamic economy with partial price stickiness and nontradable claims to labor income.
- Despite replicating complete-market allocations in the linear approximation, the model does not resolve other international-finance puzzles (e.g., high exchange-rate volatility, consumption-real exchange-rate anomaly described in Chari et al. (2002)).
- Potential extensions and alternative mechanisms:
  - Embedding credit constraints (short-sale constraints) as in Julliard (2004) could produce stronger home bias and relax the tight link between real exchange rate and relative consumption.
  - The model is compatible with other explanations for home bias (information costs, institutional frictions); the forces identified here do not exclude those channels.
- Model strengths: analytically tractable, yields closed-form expressions (e.g., equation (3.30) for foreign equity share) and offers a tractable framework to study valuation effects, portfolio adjustment, stock prices and exchange rate interactions.

### Formal definition and linearized equilibrium (Appendix content)
- Formal Definition A: an equilibrium is a sequence of variables (list provided in Appendix) solving the full system of (50) equations (equations (2.5), (2.8), (2.9), (3.3), (3.4), (3.7)-(3.22), their foreign counterparts, plus 3 asset-market clearing conditions), given stochastic sequences { A t t , M t t , A t * , M t * } and specified initial conditions.
- Approximated system: log-linearization around unconditional means (real variables stationary under stationary productivity processes; nominal variables have unit roots).
- Key log-linearized first-order conditions and definitions appear as equations (A.1)–(A.6), (A.9)–(A.17).
- Approximated equilibrium (Definition B): sequences solving equations (A.1)-(A.6), (A.14)-(A.17) given shocks and initial conditions; solutions for relative variables R t x and world variables W t x are presented.
- Conjectured equilibrium allocation given in closed-form log-linear solutions (equations (A.18)–(A.37)), including:
  - Stationary constants: **11 (1) 2 t δδτ ρ ≡=  −** (A.32)
  - Foreign equity share expression: ** *1 2(1) tt γγ γ ζζ Α ≡== Β+  −  Α** (A.33)
  - Replication of full contingent-claims allocation: ** **()tt    t  t  t cc  s p pρ−=+−.** (A.38)
- Verification steps: proofs provided that first-order conditions, goods and labor market clearing, returns on assets (human capital and equity), asset allocation first-order conditions (A.3)–(A.5), human wealth (A.15), and relative and world budget constraints are satisfied by the conjectured allocation (detailed substitutions and algebra summarized in Appendix D).

*Source: _wp05165 - Appendix presents the solution to the model. There, the equilibrium is defined and solutions*

### References

### References

### International portfolio diversification and home bias
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- Andersen, Torben, Tim Bollerslev, Francis X. Diebold, and Clara Vega, 2005, “Real Time Price Discovery in Stock, Bond, and Foreign Exchange Markets,” NBER Working Paper No. 11312 (Cambridge, Massachusetts: National Bureau of Economic Research).  
- Baxter, Marianne, and Mario Crucini, 1995, “Business Cycles and the Asset Structure of Foreign Trade,” International Economic Review, Vol. 36, pp. 821-854.  
- Baxter, Marianne, and Urban J. Jermann, 1997, “The International Diversification Puzzle Is Worse Than You Think,” American Economic Review, Vol. 87 (March), pp. 170-80.  
- Baxter, Marianne, and Robert G. King, 1998, “Nontraded Goods, Nontraded Factors, and International Non-Diversification,” Journal of International Economics, Vol. 44, pp. 211-29.  
- Bottazzi, Laura, Paolo Pesenti, and Eric van Wincoop, 1996, “Wages, Profits and the International Portfolio Puzzle,” European Economic Review, Vol. 40 (February), pp. 219-54.  
- Butler, Kirt C., and Domingo C. Joaquin, 2002, “Are the Gains from International Portfolio Diversification Exaggerated? The Influence of Downside Risk in Bear Markets,” Journal of International Money and Finance, Vol. 21, pp. 981-1011.  
- Cole, Harold L., and Maurice Obstfeld, 1991, “Commodity Trade and International Risksharing: How Much Do Financial Markets Matter?,” Journal of Monetary Economics, Vol. 28, pp. 3-24.  
- French, Kenneth R., and James M. Poterba, 1991, “Investor Diversification and International Equity Markets,” American Economic Review, Vol. 81 (May), pp. 222-26.  
- Heathcote, Jonathan, and Fabrizio Perri, 2004, “The International Diversification Puzzle is Not as Bad as You Think,” New York University, manuscript.  
- Jermann, Urban J., 2002, “International Portfolio Diversification and Endogenous Labor Supply Choice,” European Economic Review, Vol. 46, pp. 507-22.  
- Julliard, Christian, 2002, “The International Diversification Puzzle is Not Worse than Your Think,” Princeton University, manuscript.  
- Julliard, Christian, 2004, “Human Capital and International Portfolio Choice,” Princeton University, manuscript.  
- Kang, Jun Koo, and Rene M. Stulz, 1997, “Why Is There a Home Bias? An Analysis of Foreign Portfolio Equity Ownership in Japan,” Journal of Financial Economics, Vol. 46, pp. 3-28.  
- Lewis, Karen K., 1999, “Trying to Explain Home Bias in Equities and Consumption,” Journal of Economic Literature, Vol. 37 (June), pp. 571-608.  
- Lewis, Karen K., 2000, “Why Do Stocks and Consumption Imply Such Different Gains from International Risk Sharing?,” Journal of International Economics, Vol. 52 (October), pp. 1-35.  
- Matsumoto, Akito, 2004, “Essays in International Finance,” Ph.D. dissertation, University of Wisconsin (May).  
- Palacios-Huerta, Ignacio, 2001, “The Human Capital of Stockholders and the International Diversification Puzzle,” Journal of International Economics, Vol. 54, pp. 309-31.  
- Pastor, Lubos, 2000, “Portfolio Selection and Asset Pricing Models,” Journal of Finance, Vol. 55, pp. 179-223.  
- Pesenti, Paolo and Eric van Wincoop, 2002, “Can Nontradables Generate Substantial Home Bias?,” Journal of Money, Credit, and Banking, Vol. 34, pp. 25-50.  
- Serrat, Angel, 2001, “A Dynamic Equilibrium Model of International Portfolio Holdings,” Econometrica, Vol. 69, pp. 1467-1489.  
- Stockman, Alan C., and Harris Dellas, 1989, “International Portfolio Nondiversification and Exchange Rate Variability,” Journal of International Economics, Vol. 26, pp. 271-89.  
- Tesar, Linda L., 1993, “International Risk-Sharing and Nontraded Goods,” Journal of International Economics, Vol. 35, pp. 69-89.  
- Tesar, Linda L., 1995, “Evaluating the Gains from International Risksharing,” Carnegie-Rochester Conference Series on Public Policy, Vol. 42, pp. 95-143.  
- Tesar, Linda L., and Ingrid M. Werner, 1995, “Home Bias and High Turnover,” Journal of International Money and Finance, Vol. 14, No. 4, pp. 467-92.  
- van Nieuwerburgh, Stijn and Laura Veldkamp, 2005, “Information Immobility and the Home Bias Puzzle,” working paper, New York University.  
- van Wincoop, Eric, 1994, “Welfare Gains from International Risksharing,” Journal of Monetary Economics, Vol. 34, pp. 175-200.  
- van Wincoop, Eric, 1999, “How Big Are Potential Welfare Gains from International Risksharing,” Journal of International Economics, Vol. 47, pp. 109-135.  
- Warnock, Francis E., 2002, “Home Bias and High Turnover Reconsidered,” Journal of International Money and Finance, Vol. 21, pp. 795-805.  
- Rowland, Patrick F., and Linda L. Tesar, 2004, “Multinationals and the Gains from International Diversification,” Review of Economic Dynamics, Vol. 7 (October), pp789-826.

### Exchange rates, current account, and international adjustment
- Backus, David K., Patrick J. Kehoe, and Finn E. Kydland, 1992, “International Real Business Cycles,” Journal of Political Economy, Vol. 100 (August), pp. 745-75.  
- Backus, David K., Patrick J. Kehoe, and Finn E. Kydland, 1994, “Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?,” American Economic Review, Vol. 84 (March), pp. 84-103.  
- Bergin, Paul R, 2004, “How Well Can the New Open Economy Macroeconomics Explain the Exchange Rate and the Current Account?” NBER Working Paper No. 10356 (Cambridge, Massachusetts: National Bureau of Economic Research).  
- Chari, V. V., Patrick J. Kehoe, and Ellen R. McGrattan, 2002, “Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?,” Review of Economic Studies, Vol. 69 (July), pp. 533-63.  
- Gourinchas, Pierre-Olivier, and Helene Rey, 2005, “International Financial Adjustment,” NBER Working Paper No. 11155 (Cambridge, Massachusetts: National Bureau of Economic Research).  
- Pavlova, Anna, and Roberto Rigobon, 2003, “Asset Prices and Exchange Rates,” NBER Working Paper No. 9834 (Cambridge, Massachusetts: National Bureau of Economic Research).  
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- Tille, Cedric, 2004, “Financial Integration and the Wealth Effect of Exchange Rate Fluctuations,” Federal Reserve Bank of New York, manuscript.

### Technology shocks, business cycles, and macro dynamics
- Christiano, Lawrence J., Martin Eichenbaum, and Robert Vigfusson, 2003, “What Happens after a Technology Shock?,” International Finance Discussion Papers 768 (June).  
- Dotsey, Michael, 1999, “Structure from Shocks,” Federal Reserve Bank of Richmond Working Paper.  
- Eldor, Rafael, David Pines, and Abba Schwarz, 1988, “Home Asset Preference and Productivity Shocks,” Journal of International Economics, Vol. 25, pp. 165-176.  
- Francis, Neville, and Valerie A. Ramey, 2003, “Is the Technology-Driven Real Business Cycle Hypothesis Dead? Shocks and Aggregate Fluctuations Revisited,” manuscript UCSD.  
- Francis, Neville, and Valerie A. Ramey, 2004, “A New Measure of Hours Per Capita with Implications for the Technology-Hours Debate,” manuscript UCSD.  
- Gali, Jordi, 1999, “Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?,” American Economic Review, Vol. 89 (March), pp. 249-71.  
- Gali, Jordi, and Pau Rabanal, 2005 “Technology Shocks and Aggregate Fluctuations: How Well Does the RBC Model Fit Postwar U.S. Data?,” in NBER Macroeconomics Annual, 2004, ed. by Mark Gertler and Kenneth Rogoff (Cambridge, Massachusetts: MIT Press).  
- Gali, Jordi, J. David Lopez-Salido, and Javier Valles, 2003, “Technology Shocks and Monetary Policy: Assessing the Fed's Performance,” Journal of Monetary Economics, Vol. 50, pp. 723-43.  
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### Asset prices, monetary policy, and financial market dynamics
- Bernanke, Ben S., and Kenneth N. Kuttner, 2004, “What Explains the Stock Market’s Reaction to Federal Reserve Policy?” Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series: 2004-16.  
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- Pavlova, Anna, and Roberto Rigobon, 2003, “Asset Prices and Exchange Rates,” NBER Working Paper No. 9834 (Cambridge, Massachusetts: National Bureau of Economic Research).  
- Warnock, Francis E., 2002, “Home Bias and High Turnover Reconsidered,” Journal of International Money and Finance, Vol. 21, pp. 795-805.

*References list as presented in the source PDF _wp05165 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05165.pdf_