## Section 4.3 ## Source details **Canonical URL:** [Section 4.3](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021254-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021254-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021254-print-pdf.pdf.json) --- ### Core contributions and positioning - Adds a new angle to heterogeneous-agent macroeconomics by treating capital income more carefully than existing studies on wealth inequality and elaborating the mechanism behind a structural change in capital income. - Embeds global financial markets into a Pareto-inequality framework originating with Champernowne (1953) to connect changes in financial variables (risk premium, Sharpe ratio, capital gains, portfolio frontier) to wealth distribution dynamics over different horizons. - Framework is applicable to structural changes beyond financial globalization, such as financial innovation and capital tax reform. ### Model setup and key definitions - Economy: continuum of households (measure normalized to unity), time continuous, initial wealth distribution drawn from density g0. - Closed-economy assets: - Risk-free asset yields r* dt. - Domestic risky asset yields (r* + σ1 s*1) dt + σ1 dz1t. - Sharpe ratio decomposition: risk premium = σ1 s*1 where σ1 is standard deviation and s*1 is Sharpe ratio. - Household problem (born at time 0) maximizes: - max_{θ1it, cit} E0 [ ∫_0^∞ e^{−(δ+m)t} log(cit − κ) dt ] - Budget constraint: dait = [ (r* + σ1 s*1 θ1it) ait − cit ] dt + σ1 θ1it ait dz1t - Parameters and interpretations: - m captures death probability (perpetual youth model). Fraction m of households die and are replaced by newborns whose wealth is re-drawn from g0. - κ > 0 in flow utility captures decreasing relative risk aversion (HARA class). - a ≡ κ / r* denotes the wealth cutoff; households retain wealth above a to avoid negative consumption. - Closed-economy optimal solutions (Merton 1971): - cit = (δ + m) ait + (r* − δ − m) a - θ1it = (s*1 / σ1) (1 − a / ait) - Property: θ1it increases with ait and ∂^2 θ1it / ∂ait ∂s*1 > 0 (wealthier households respond more elastically to changes in s*1). - Asset-price specification for tractability: - Risky share pays dividend x dt and stochastic cash flow σ1 dz1t. - Focus on case x = r* + σ1 s*1 and pt = 1 for all t; unanticipated changes in required expected return generate capital gains/losses. - Wealth-distribution evolution (zero trajectory dz1t = 0, ex ante volatility Var[σ1 dz1t] = σ1^2 dt): - Kolmogorov Forward Equation (KFE): - d/dt g_t(a) = − m g_t(a) + m g_0(a) − d/da [{ (r* + σ1 s*1 θ1(a)) a − c(a) } g_t(a)] - Stationary distribution g_∞ defined by d/dt g_∞(a) = 0; g_t → g_∞ as t → ∞. - Measures of concentration: - Ω_t = ( ∫_{G_t^{-1}(0.99)}^∞ a g_t(a) da ) / ( ∫_{−∞}^∞ a g_t(a) da ) (top one percent wealth share in g_t) - Ω_∞ ≡ 100 / (ξ − 1) where ξ is Pareto exponent from lim_{a→∞} g_∞(τ a) / g_∞(a) = τ^{−(1+ξ)} (approximation exact if g_∞ is Pareto) - Use d log Ω_T to capture immediate increase and d log Ω_∞ for long-run increase after integration at time T. ### Financial globalization experiment and assumptions - Financial globalization transforms portfolio frontier: - { r*, (s*1, σ1) } ⇒ { r, (s1, σ1), (s2, σ2) } with correlation between dz1t and dz2t equal to ρ ∈ [0,1). - Example assumptions (financial center as small open economy): - (a) r < r* - (b) s1 = s*1 - (c) σ2 > 0 - From time T onward, household budget constraints incorporate both domestic and foreign risky assets: - dait = ( ( r + [θ1it θ2it] [σ1 s1; σ2 s2] + m ) ait − cit ) dt + ait θ1it σ1 dz1t + ait θ2it σ2 dz2t - Two simplifying assumptions to avoid pathologies: - Optimal θ1it and θ2it are non-negative. - Wealth cutoff a is constant (prevents negative consumption for bottom households after an unanticipated drop in r). ### Main analytical results (Proposition 1 and Corollary 1) - Short-run (Revaluation Effect) and long-run (Decline-in-return and Rebalancing Effects) decomposition for top one percent wealth share: - d log Ω_T = − φ1 d log ( r + σ1 s1 ) (i) Revaluation Effect - d log Ω_∞ = φ2 d log ( r + σ1 s1 ) (ii) Decline-in-return Effect + φ3 d log s2 − φ4 d log ρ (iii) Rebalancing Effect - φ1, φ2, φ3, φ4 are all positive coefficients. - Corollary 1: - Financial globalization widens wealth inequality immediately via capital gains. - Wealth inequality is increased permanently only if s2 and 1/ρ are sufficiently large. ### Mechanisms and intuition - Revaluation Effect (short run): - Unexpected global integration lowering the market required return r + σ1 s1 while pledged cash flows remain unchanged causes immediate price appreciation of domestic risky asset p_T. - Because affluent households have larger exposure to the domestic risky asset pre-integration, Ω_T rises immediately; magnitude linked to the size of the drop in required expected return. - Decline-in-return Effect (long run): - Lower discount rates that generate capital gains imply lower future expected returns on domestic assets. - Through within-generation consumption smoothing and imperfect intergenerational wealth transmission (m > 0), the increase in concentration is transitory; stationary distribution Pareto tail exponent satisfies 1/ξ ≡ ( r + R′ Σ^{-1} R − δ ) / m, so dr = d(r + σ1 s1) < 0 exerts downward force on long-run concentration. - Rebalancing Effect (persistence condition): - Financial globalization expands the portfolio frontier and raises the slope of the capital allocation line, incentivizing portfolio reallocation toward risky domestic and foreign assets. - Wealthy households, due to decreasing relative risk aversion, reallocate more elastically, widening capital-return inequality across households. - Rebalancing effect larger when s2 is high and ρ is low (large 1/ρ); persistence requires sufficiently large s2 and small ρ. ### Summary implication (Section 4.3 conclusion) - Financial globalization produces an immediate increase in measured wealth concentration through asset revaluation. - Long-run effect depends on two opposing forces: - Decline-in-return Effect: lower expected returns reduce long-run wealth concentration. - Rebalancing Effect: asymmetric portfolio responses (wealthier households increasing risky exposure more) can raise long-run concentration if s2 and 1/ρ are large enough to offset the decline-in-return. - Permanent increases in top wealth shares require strong rebalancing incentives from newly accessible foreign assets. *Source: wpiea2021254-print-pdf - Section 4.3.* ### Section 5 — model extensions and numerical exercises ### Model extensions and qualitative implications (Section 5 overview) - Allow some households to have negative net worth; numerical simulations show indebted households begin to take on more debts as r drops upon financial globalization. - Consequence: Net worth of households in the bottom decile falls, which raises the top one percent’s wealth share more than the illustrative model suggests. - Extension: every household has unrestricted access to risky investment; entrepreneurial households may hold a large share of stock including private equity. - Planned extension: model allows for private equity in the next section; extension strengthens main results: global integration of capital markets increases entrepreneurial income and inflates its market-equivalent valuation. ### Back-of-the-envelope calculation: setup and calibration - Stationary wealth distribution formula invoked: - g∞(a) = C(a)(a − a)^(−1−m/(˜r+˜s/2−δ)) - Closed economy interest rates: - ˜r = r∗ and ˜s = s∗1 - Open economy Sharpe ratio expression: - ˜r = r and ˜s = √[σ1 s1, σ2 s2] Σ^(−1) [σ1 s1, σ2 s2]′ - Approximation for top one percent’s wealth share: - Ω∞ = 100^(−1−m/(˜r+˜s/2−δ)) - Calibration choices (benchmark year 1989): - r∗ = 0.027 - s∗1 = 0.029 - δ = 0.05 - m left as free parameter to fit top one percent’s wealth share in 1989 from SCF - Average portfolio weights for top 1 percent, 1–9 percent, and bottom 90 percent calibrated from SCF: 0.5, 0.35, 0.15 respectively. ### Back-of-the-envelope scenarios and numerical results (Table 1 summary) - Common parameters (Autarky / baseline): - Real risk-free interest (Autarky) = 0.027 - Sharpe ratio (Autarky) = 0.29 - m = 0.085 - δ = 0.05 - Portfolio weight in equity by wealth groups = 0.5, 0.35, 0.15 (top 1%, top 1–9%, bottom 90%) - Shock from financial globalization: three scenarios reported - Scenario (1) - Real risk-free interest (Open) = 0.01 - Sharpe ratio (Open) = 0.327 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +1.8%p - Open (stationary state) = +8.8%p - Data, 2016 = +8.7%p - Interpretation: Top 1% wealth share immediately increases by 1.8%p due to capital gains; share continues to increase because portfolio rebalancing effect outweighs decline in return on domestic assets; model-implied increase, 8.8%p, is comparable to actual increase. - Scenario (2) - Real risk-free interest (Open) = 0.01 - Sharpe ratio (Open) = 0.310 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +3.1%p - Open (stationary state) = −6.5%p - Data, 2016 = +8.7%p - Interpretation: Sharpe ratio does not increase as much; model exhibits inverse-U shaped transitional dynamics: top one percent share first rises due to revaluation, but eventually reverts; new stationary state ends up having lower wealth inequality than initial state. - Scenario (3) - Real risk-free interest (Open) = 0.0 - Sharpe ratio (Open) = 0.327 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +7.5%p - Open (stationary state) = −5.5%p - Data, 2016 = +8.7%p - Interpretation: Similar pattern as scenario (2); current rising trend in wealth concentration can reverse if expansion of foreign investment no longer increases the Sharpe ratio as much (scenario (1) ⇒ scenario (3)) or expected return on domestic assets falls too sharply (scenario (1) ⇒ scenario (2)). - Sensitivity remark: - The back-of-the-envelope calculation is highly sensitive to change in the Sharpe ratio. - Sensitivity is alleviated if utility function has higher risk aversion than log utility; example: u(c_it) = (c_it − κ)^(1−γ)/(1−γ) with γ = 3; in this case the Sharpe ratio should rise to 0.37 to generate the same magnitude as in Scenario (1). ### Banking, equilibrium expressions, and comparative statics (core mechanism linking finance and prices) - Aggregate saving relations (closed economy): - S_t ≡ A_t ≡ ∫ a_it di - S_1t ≡ ∫ θ_1it a_it di - Using θ_1it = s∗1/σ1 (1 − a/a_it), obtain S_1t = s∗1/σ1 (A_t − κ) - Bank production and funding: - Bank generates dπ_t = Φ(K_t) dt + ̄σ K_t dz_1t - Funding tranching yields constraints: K_t ≡ D_t + E_t, σ1 = ̄σ K_t / E_t, D_t ≤ λ K_t - Investment supply curves: - I_t = Φ′^(−1)(r∗_t + ̄σ s∗1t + τ − τλ) - I_1t = (1 − λ) I_t - Market clearing and equilibrium rates as functions of aggregate wealth A: - s∗1(A) = ̄σ A/(A − a) - r∗(A) = Φ′(A) − ̄σ^2 A/(A − a) − τ + τλ - Proposition 2 (comparative statics): - d log r∗ = φ5 d log λ − φ6 d log ̄σ - d log s∗1 = φ7 d log ̄σ - d log(r∗ + ̄σ s∗1) = φ8 d log λ - d log(V∗) = φ9 d log λ - φ5, φ6, φ7, φ8, φ9 > 0 - Core message: Peripheral economies (small λ) tend to have lower risk-free interest rate r∗, lower expected required return on risky asset, and higher Sharpe ratio s∗1. ### Financial globalization, security market liberalization, and FDI (Propositions 3 and 4) - Proposition 3 (security market liberalization effects): - After liberalization US becomes exporter of safe asset and net importer of global risky assets; US households face: - (a) r < r∗ - (b) r + σ1 s1 < r∗ + σ s∗1 - (c) V > V∗ - (d) s_mix ≥ s1 - Domestic Sharpe ratio rises (s1 > s∗1) iff ̄σ1 < ρ ̄σ2. - FDI extension: - FDI modeled via monitoring contract with FO entrepreneur misbehavior parameter πL ∈ [0,1). - Empirical note: FDI outflow of US multinational firms currently accounts for 43 percent of foreign equity holdings by American households in terms of estimated market value. - Proposition 4 (FDI + liberalization): - (a) s(iii)1 > s(ii)1 > s(i)1 - (b) (V + VFDI)(iii) > V(iii) > V(i) - Condition: if ̄σUS < ρ ̄σFO and πL is sufficiently small. - Intuition: FDI unlocks FO investment via US monitoring, expands risky investment supply, raises s1, and increases excess profits to US banks. ### Cross-country heterogeneous outcomes - Financial globalization increases wealth concentration most prominently within a financially-developed economy (short run) via changes in equilibrium interest rates. - Effects in peripheral economies depend on: - If FDI is shut off and foreign risky asset is a perfect substitute, FO experiences opposite changes. - If foreign and domestic risky assets provide diversification benefits, both regions can experience heterogeneous portfolio rebalancing. - If FDI dominates, wealth inequality can increase in both regions. - Integration of symmetric countries: diversification drives capital flows; decline-in-return effect tends to dominate unless asymmetries exist. ### Quantitative extension: taking model to data (Section 5.1) and numerical method (Section 5.2) - Three-step quantitative approach: - Extend baseline model by adding labor income and housing wealth. - Present target moments and estimates for key variables. - Present results (calibration and simulations). - Household lifetime utility: - Eτ[∫∞τ e−(δ+m)t u(cit) dt] - Autarky budget constraint includes labor income w∗t lit and housing return rh t hit; individual wealth = ait + hit. - Portfolio choice dimensionality reduction (Assumption 1): - In autarky: θ1it = s1t χ1 / σ1 (1 − χ2 ait − χ3 ait `i) - Constants χ1, χ2, χ3 calibrated from data; households below χ2 + χ3 `i have θit = 0. - Numerical algorithm (continuous-time Krusell-Smith analogue): 1. Guess law of motion for state variable (e.g., d log A1t = (ψ1 − 1) log A1t + ψ2). 2. Solve HJB under guess to compute individual saving decisions. 3. Compute evolution of wealth distribution via Kolmogorov Forward. 4. Verify and iterate until consistency. ### Calibration strategy (Section 5.3) - Benchmark year for financial autarky: 1989. - Target moments: match top 1%, top 5% and bottom 90% wealth shares in Survey of Consumer Finances (1989). - Parameter block to adjust portfolio frontier and fit moments: - {λ_US, ̄σ_US, λ_FO, ̄σ_FO, ρ; μ_a, Σ_aa, Σ_al} - Features used to generate a modest risk premium: - Many households do not own risky assets; indebted households reduce demand for risky assets. - χ_1 can be set differently from intertemporal elasticity of substitution; ̄σ_US and ̄σ_FO free to match Sharpe ratios. - Estimation of returns and Sharpe ratios: - Average realized returns constructed from national accounts (Gross National Income flows / Fed’s Financial Accounts market values). - Capital gains computed as increase in market value that exceeds net issuance. - Sharpe ratio measurement: sample period 1982-2017; mixed portfolio (U.S. households) shows higher risk-return trade-off than domestic portfolio. - Numerical implementation details and simulation algorithm provided for closed and open economy cases; transition to open economy implements capital gains via updated equity price p^{new}_T and then evolves wealth distribution forward. *Source: wpiea2021254-print-pdf - section 5 and subsequent appendices.* *Source: wpiea2021254-print-pdf* ### Section 4.3. ### Section 4.3 ### Core contributions and positioning - Adds a new angle to heterogeneous-agent macroeconomics by treating capital income more carefully than existing studies on wealth inequality and elaborating the mechanism behind a structural change in capital income. - Embeds global financial markets into a Pareto-inequality framework originating with Champernowne (1953) to connect changes in financial variables (risk premium, Sharpe ratio, capital gains, portfolio frontier) to wealth distribution dynamics over different horizons. - Framework is applicable to structural changes beyond financial globalization, such as financial innovation and capital tax reform. ### Model setup and key definitions - Economy: continuum of households (measure normalized to unity), time continuous, initial wealth distribution drawn from density g0. - Assets in closed economy: a risk-free asset yielding r* dt and a domestic risky asset yielding (r* + σ1 s*1) dt + σ1 dz1t. - Sharpe ratio decomposition: risk premium = σ1 s*1 where σ1 is standard deviation and s*1 is Sharpe ratio. - Household problem (born at time 0) maximizes - max_{θ1it, cit} E0 [ ∫_0^∞ e^{−(δ+m)t} log(cit − κ) dt ] - budget constraint: dait = [ (r* + σ1 s*1 θ1it) ait − cit ] dt + σ1 θ1it ait dz1t - Parameters and interpretations: - m captures death probability (perpetual youth model). Fraction m of households die and are replaced by newborns whose wealth is re-drawn from g0. - κ > 0 in flow utility captures decreasing relative risk aversion (HARA class). - a ≡ κ / r* denotes the wealth cutoff; households retain wealth above a to avoid negative consumption. - Closed-economy optimal solutions (Merton 1971): - cit = (δ + m) ait + (r* − δ − m) a - θ1it = (s*1 / σ1) (1 − a / ait) - Property: θ1it increases with ait and ∂^2 θ1it / ∂ait ∂s*1 > 0 (wealthier households respond more elastically to changes in s*1). - Asset-price specification for tractability: - Each risky share pays dividend x dt and stochastic cash flow σ1 dz1t. - Focus on case x = r* + σ1 s*1 and pt = 1 for all t, so price constant absent unanticipated changes; unanticipated changes in required expected return generate capital gains/losses. - Wealth-distribution evolution (zero trajectory dz1t = 0, but ex ante volatility Var[σ1 dz1t] = σ1^2 dt): - Kolmogorov Forward Equation (KFE): - d/dt g_t(a) = − m g_t(a) + m g_0(a) − d/da [{ (r* + σ1 s*1 θ1(a)) a − c(a) } g_t(a)] - Stationary distribution g_∞ defined by d/dt g_∞(a) = 0; g_t → g_∞ as t → ∞. - Measures of concentration: - Ω_t = ( ∫_{G_t^{-1}(0.99)}^∞ a g_t(a) da ) / ( ∫_{−∞}^∞ a g_t(a) da ) (top one percent wealth share in g_t) - Ω_∞ ≡ 100 / (ξ − 1) where ξ is Pareto exponent from lim_{a→∞} g_∞(τ a) / g_∞(a) = τ^{−(1+ξ)} (approximation exact if g_∞ is Pareto) - Use d log Ω_T to capture immediate increase and d log Ω_∞ for long-run increase after integration at time T. ### Financial globalization experiment and assumptions - Financial globalization transforms portfolio frontier: - { r*, (s*1, σ1) } ⇒ { r, (s1, σ1), (s2, σ2) } with correlation between dz1t and dz2t equal to ρ ∈ [0,1). - Example assumptions for exposition (financial center as small open economy): - (a) r < r* - (b) s1 = s*1 - (c) σ2 > 0 - From time T onward, household budget constraints incorporate both domestic and foreign risky assets: - dait = ( ( r + [θ1it θ2it] [σ1 s1; σ2 s2] + m ) ait − cit ) dt + ait θ1it σ1 dz1t + ait θ2it σ2 dz2t - Two simplifying assumptions to avoid pathologies: - Optimal θ1it and θ2it are non-negative. - Wealth cutoff a is constant (prevents negative consumption for bottom households after an unanticipated drop in r). ### Main analytical results (Proposition 1 and Corollary 1) - Proposition 1 (short-term and long-term effects on top one percent wealth share): - d log Ω_T = − φ1 d log ( r + σ1 s1 ) (i) Revaluation Effect - d log Ω_∞ = φ2 d log ( r + σ1 s1 ) (ii) Decline-in-return Effect + φ3 d log s2 − φ4 d log ρ (iii) Rebalancing Effect - φ1, φ2, φ3, φ4 are all positive coefficients. - Corollary 1: - Financial globalization widens wealth inequality immediately via capital gains. - Wealth inequality is increased permanently only if s2 and 1/ρ are sufficiently large. ### Mechanisms and intuition - Revaluation Effect (short run): - An unexpected global integration lowering the market required return r + σ1 s1 while pledged cash flows remain unchanged causes immediate price appreciation of domestic risky asset p_T. - Because affluent households have larger exposure to the domestic risky asset pre-integration, Ω_T rises immediately; magnitude linked to the size of the drop in required expected return (see equation for d log Ω_T). - Decline-in-return Effect (long run): - Lower discount rates that generate capital gains imply lower future expected returns on domestic assets. - Through within-generation consumption smoothing and imperfect intergenerational wealth transmission (m > 0), the increase in concentration is transitory; in the long run the stationary distribution has a thicker or thinner tail depending on returns. - Stationary distribution in open economy: - g(a) = C(a) (a − a)^{−1− (m / ( r + R′ Σ^{-1} R − δ )) } (expression (9) summarized) - Pareto exponent relation: 1/ξ ≡ ( r + R′ Σ^{-1} R − δ ) / m - Lower expected returns dr = d(r + σ1 s1) < 0 exert downward force on wealth concentration (term (ii) has opposite sign to term (i)). - Rebalancing Effect (persistent increase condition): - Financial globalization expands the portfolio frontier and raises the slope of the capital allocation line, incentivizing portfolio reallocation toward risky domestic and foreign assets. - Wealthy households, due to decreasing relative risk aversion, reallocate more elastically, widening capital-return inequality across households. - The rebalancing effect in (8) is larger when the newly-accessed foreign risky asset has higher reward-to-risk ratio s2 and lower correlation with domestic assets ρ; thus persistence requires sufficiently large s2 and small ρ (large 1/ρ). ### Summary implication - Financial globalization produces an immediate increase in measured wealth concentration through asset revaluation. - The long-run effect depends on two opposing forces: - Decline-in-return Effect: lower expected returns reduce long-run wealth concentration. - Rebalancing Effect: asymmetric portfolio responses (wealthier households increasing risky exposure more) can raise long-run concentration if s2 and 1/ρ are large enough to offset the decline-in-return. - Thus permanent increases in top wealth shares require strong rebalancing incentives from newly accessible foreign assets. *Source: wpiea2021254-print-pdf - Section 4.3.* ### section 5, I extend the model such that some households have negative net worth. Numerical ### wpiea2021254-print-pdf - section 5, I extend the model such that some households have negative net worth. Numerical ### Model extensions and qualitative implications - Extension: allow some households to have negative net worth; numerical simulations show indebted households begin to take on more debts as r drops upon financial globalization. - Consequence: Net worth of households in the bottom decile falls, which raises the top one percent’s wealth share more than the illustrative model suggests. - Extension: every household has unrestricted access to risky investment; entrepreneurial households may hold a large share of stock including private equity. - Planned extension: model is extended to allow for private equity in the next section; the extension strengthens the main results: global integration of capital markets increases entrepreneurial income and inflates its market-equivalent valuation. ### Back-of-the-envelope calculation: setup and calibration - Stationary wealth distribution formula (as invoked): - g∞(a) = C(a)(a − a)^(−1−m/(˜r+˜s/2−δ)) (formula presented in text) - Closed economy interest rates: - ˜r = r∗ and ˜s = s∗1 - Open economy Sharpe ratio expression: - ˜r = r and ˜s = √[σ1 s1, σ2 s2] Σ^(−1) [σ1 s1, σ2 s2]′ (indicates Sharpe ratio of open economy portfolio) - Approximation for top one percent’s wealth share: - Ω∞ = 100^(−1−m/(˜r+˜s/2−δ)) - Calibration choices (benchmark year 1989): - r∗ = 0.027 - s∗1 = 0.029 - δ = 0.05 - m left as free parameter to fit top one percent’s wealth share in 1989 from SCF - Average portfolio weights for top 1 percent, 1–9 percent, and bottom 90 percent calibrated from SCF (values used in table; see scenarios below) - Note: r∗ is calibrated from the 1-year treasury yield after inflation around 1989. s∗1 stems from the estimated Sharpe ratio for the domestic portfolio. δ is standard discount rate. ### Back-of-the-envelope scenarios and numerical results (Table 1 summary) - Common parameters (Autarky / baseline): - Real risk-free interest (Autarky) = 0.027 - Sharpe ratio (Autarky) = 0.29 - m = 0.085 - δ = 0.05 - Portfolio weight in equity by wealth groups = 0.5, 0.35, 0.15 (top 1%, top 1–9%, bottom 90%) - Shock from financial globalization: three scenarios reported - Scenario (1) - Real risk-free interest (Open) = 0.01 - Sharpe ratio (Open) = 0.327 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +1.8%p - Open (stationary state) = +8.8%p - Data, 2016 = +8.7%p - Interpretation: Top 1% wealth share immediately increases by 1.8%p due to capital gains; share continues to increase because portfolio rebalancing effect outweighs decline in return on domestic assets; model-implied increase, 8.8%p, is comparable to actual increase. - Scenario (2) - Real risk-free interest (Open) = 0.01 - Sharpe ratio (Open) = 0.310 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +3.1%p - Open (stationary state) = −6.5%p - Data, 2016 = +8.7%p - Interpretation: Sharpe ratio does not increase as much; model exhibits inverse-U shaped transitional dynamics: top one percent share first rises due to revaluation, but eventually reverts; new stationary state ends up having lower wealth inequality than initial state. - Scenario (3) - Real risk-free interest (Open) = 0.0 - Sharpe ratio (Open) = 0.327 - Results: Top 1% wealth share - Autarky (= Data, 1989) = 27.3% - Open (after capital gains) = +7.5%p - Open (stationary state) = −5.5%p - Data, 2016 = +8.7%p - Interpretation: Similar pattern as scenario (2); current rising trend in wealth concentration can reverse if expansion of foreign investment no longer increases the Sharpe ratio as much (scenario (1) ⇒ scenario (3)) or expected return on domestic assets falls too sharply (scenario (1) ⇒ scenario (2)). - Sensitivity remark: - The back-of-the-envelope calculation is highly sensitive to change in the Sharpe ratio. - Sensitivity is alleviated if utility function has higher risk aversion than log utility; example given: u(c_it) = (c_it − κ)^(1−γ)/(1−γ) with γ = 3; in this case the Sharpe ratio should rise to 0.37 to generate the same magnitude as in Scenario (1). ### Closed economy: banks, market clearing, and equilibrium expressions - Aggregate saving definitions: - S_t ≡ A_t ≡ ∫ a_it di - S_1t ≡ ∫ θ_1it a_it di - Using θ_1it = s∗1t σ1 (1 − a/a_it), closed economy saving curves yield: - S_t = A_t - S_1t = s∗1/σ1 (A_t − κ) (equation (10)) - Aggregate savings in domestic risky asset increases with the Sharpe ratio. - Banking sector production and funding (representative bank): - Bank generates dπ_t = Φ(K_t) dt + ̄σ K_t dz_1t - Funding decisions convert future cash flow into risk-free and risky tranches. - Bank maximizes contemporaneous profit (private equity income not distributed to households): - Objective incorporates Debt Income = r∗_t D_t dt, Public Equity Income = (r∗_t + σ1 s∗1t + τ) E_t dt, and Public Equity Volatility = σ1 E_t dz_1t (equation (11)) - Balance sheet constraints: - K_t ≡ D_t + E_t - σ1 = ̄σ K_t / E_t - D_t ≤ λ K_t (equation (12)) - Interpretation: - V∗_t dt represents excess profit; entrepreneur consumes V∗_t dt immediately. - Debt holders receive risk-free rate; equity holders get risk premium proportional to risk per unit of equity plus deadweight transaction cost τ. - λ indicates maximum leverage, country capacity to create safe assets by tranching; peripheral countries assumed to have lower λ than financial center. - Bank optimization solutions and investment supply curves: - I_t = K_t and I_1t = E_t with σ1 = ̄σ/(1 − λ) - Solving optimization yields (equation (13)): - I_t = Φ′^(−1)(r∗_t + ̄σ s∗1t + τ − τλ) - I_1t = (1 − λ) I_t - Both investment curves are downward sloping in market funding cost r_t + ̄σ s_1t and Sharpe ratio s∗1t. ### Market clearing and equilibrium rates as functions of aggregate wealth - Closed economy equilibrium defined by S_t = I_t and S_1t = I_1t for all t. - When aggregate wealth A_t = A fixed, equilibrium solutions (equations (14) and (15)): - s∗1(A) = ̄σ A/(A − a) - r∗(A) = Φ′(A) − ̄σ^2 A/(A − a) − τ + τλ - Interpretation: - Market clearing interest rates expressed as function of total wealth stock A; A_t acts as state variable; domestic risky asset dividend x_t ≡ r∗(A_t) + s∗1(A_t) σ1 depends on A_t. ### Comparative statics: financial friction and output volatility effects (Proposition 2) - Setup: two countries US and FO identical in size (A_1t = A_2t = A). Assume λ_US > λ_FO. - Proposition 2 (comparative statics; log-differentiation results): - d log r∗ = φ5 d log λ (Financial Friction) − φ6 d log ̄σ (Output Volatility) - d log s∗1 = φ7 d log ̄σ (Output Volatility) - Both φ5, φ6, φ7 are positive coefficients. - Further results: - d log(r∗ + ̄σ s∗1) = φ8 d log λ (Financial Friction) - d log(V∗) = φ9 d log λ (Financial Friction) - φ8 and φ9 are positive coefficients. - Core message: - Peripheral economies tend to have equilibrium with lower risk-free interest rate, lower expected required return on risky asset, and higher Sharpe ratio of domestic risky asset when λ is small. - Pledgeability of future cash flows and limited supply of safe contractual claims are central drivers. - Remark on dynamics: - As time passes, wealth stocks grow; stationary state wealth stock is lower in FO than in US given small λ; many results in Proposition 2 remain intact when comparing stationary state interest rates provided λ sufficiently small. ### Financial globalization: security market liberalization (definition and Proposition 3) - Security market liberalization: allows US households to invest in assets issued by foreign bank; introduces foreign risky asset characterized by (s_2t, σ2). - Open economy market clearing definitions: - S_k1t ≡ ∫ θ_k1it a_kit di, S_k2t ≡ ∫ θ_k2it a_kit di, S_k t ≡ ∫ a_kit di for k ∈ {US, FO} - Open economy equilibrium clears global financial markets: - ∑_{k∈{FO,US}} (S_k t − I_k t) = 0 - ∑_{k∈{FO,US}} S_k1t = I_US1t - ∑_{k∈{FO,US}} S_k2t = I_FO2t - Proposition 3 (effects after security market liberalization): - (i) After liberalization, US becomes exporter of safe asset and net importer of global risky assets. US households face: - (a) r < r∗ - (b) r + σ1 s1 < r∗ + σ s∗1 - (c) V > V∗ - (d) s_mix ≥ s1 where s_mix is the Sharpe ratio of optimal portfolio combining foreign and domestic risky assets. - (ii) The Sharpe ratio of the domestic risky asset rises (i.e., s1 > s∗1) if and only if ̄σ1 < ρ ̄σ2. - Intuition and mechanisms: - Security market liberalization offers new risky investment opportunities for US households and decreases required expected returns on US domestic assets. - If FO has limited supply of safe assets and higher overall cost of capital (due to τ), FO demands safe assets, leading US to become net debtor; excess demand from FO exerts downward pressure on r + σ1 s1 and r in US. - Excess profit V increases in US (V > V∗) as US bank faces lower average cost of capital. - Change in domestic Sharpe ratio depends on relative riskiness of outside world; if FO is significantly riskier (̄σ_US < ρ ̄σ_FO), US households bear more risk and domestic Sharpe ratio rises to clear market. *Italicized source: wpiea2021254-print-pdf - section 5, I extend the model such that some households have negative net worth. Numerical* ### 1. If this is not the case (e.g. ̄σ ### 1. If this is not the case (e.g. ̄σUS = ̄σFO andρ <1), the domestic Sharpe ratio falls after global integration ### Open Economy Equilibrium — mechanisms and short/long run channels - Diversification from the foreign risky asset reduces portfolio risk, so a lower value of s1 is sufficient to induce US households to clear the markets. - Notation consolidation used: t ≡ r(A1t + A2t), s1t ≡ s1(A1t + A2t), s2t ≡ s2(A1t + A2t) and Vt ≡ V(A1t + A2t). - Simplified statement: r < r* is shorthand for r(A1t + A2t) < r*(A1t) when A1t = A2t = A (US and FO identical in size). - Figure 4 highlights: - (a) Total Assets: Savings & Investment equations and market clearing across US and EM. - (b) Risky Assets (if ρ = 1 and ̄σ1 < ̄σ2): risky savings & investment allocations and Σi πiUS = Σi πiEM relations. - Core channels identified: - Short run: domestic asset price inflation can raise wealth concentration in the US via capital gains to wealthy households. - Long run: asymmetric portfolio rebalancing can further increase wealth concentration. - Calibration/numerical need: microfoundation provided here requires numerical simulations to quantify effects on wealth inequality; to be revisited in Section 5. ### FDI Liberalization — model extension and proposition - FDI adds additional risky investment opportunities and can further concentrate wealth in the US by expanding foreign risky asset supply. - Model embedding: simplified Holmstrom and Tirole (1997) / Antràs et al. (2009) contract model to capture US multinationals as de facto financial intermediaries in the FO when FO has poor contracting environment. - Three additional ingredients: - FO entrepreneur misbehavior lowers expected earnings from Φ(K2t) dt to πL Φ(K2t) dt where πL ∈ [0,1). - US bank can invest KFDI2t to create a joint venture, monitor the FO entrepreneur, and receive a designated profit share. - Remaining investment KLocal2t ≡ K2t − KFDI2t is funded through local FO banks; banks raise funds from investors as before. - Empirical note: FDI outflow of US multinational firms currently accounts for 43 percent of foreign equity holdings by American households in terms of estimated market value. (Bureau of Economic Analysis, 2017) - Comparative equilibrium stages considered: (i) autarky, (ii) security market liberalization, (iii) FDI and security market liberalization. Equilibrium interest rates denoted with superscripts (i), (ii), (iii). - Proposition 4 (preserved statement): - (a) s(iii)1 > s(ii)1 > s(i)1 - (b) (V + VFDI)(iii) > V(iii) > V(i) - Condition: if ̄σUS < ρ ̄σFO and πL is sufficiently small. Furthermore, the US becomes the exporter of safe assets, and the net importer of the foreign portfolio and direct investment assets. - Intuition: - Even after security market liberalization, low pledgeability in FO (πL < 1) limits FO investment. - FDI lets the US bank act as parent monitor, unlocking FO investment potential; FO investment is riskier than US investment. - A higher s1 is required when both FDI and security markets are liberalized to induce US households to bear more risk. - The US bank gains additional excess profit in the form of VFDI dt. ### Cross-country implications — heterogeneous outcomes across central and peripheral economies - Main implication: financial globalization increases wealth concentration most prominently within a financially-developed economy (short run) via changes in equilibrium interest rates. - Two channels reiterated: domestic asset-price inflation (short run) and portfolio reallocation (long run). - Explanation for US-specific rise in wealth concentration: US special role ("world’s banker") transforms domestic interest rates through its global finance architecture. - Effects in peripheral economies depend on circumstances: - If FDI is shut off and foreign risky asset is a perfect substitute for domestic risky asset, FO experiences the opposite of US changes. - If foreign and domestic risky assets provide diversification benefits to each other, both US and FO experience heterogeneous portfolio rebalancing between rich and poor. - If FDI is the dominant component of financial globalization, wealth inequality can increase in both regions because FDI expands risky investment opportunities in advanced and emerging markets. - Remark 2 — integration of symmetric countries: - When integrating identical countries, diversification is the sole driver of capital flows. - Proposition 3(ii) implies the US domestic risky asset risk premium falls, generating capital gains for rich households, followed by heterogeneous portfolio rebalancing that offsets part of the return decline. - Lower ρ makes the decline-in-return and rebalancing effects cancel more strongly, but the decline-in-return effect always dominates; a persistent increase in wealth inequality requires an extra driver such as asymmetry between US and peripheral economies. ### Taking the model to the data — quantitative extension (Section 5.1) - Objective: assess whether global financial flows between central and peripheral economies explain the observed increase in US wealth inequality and its persistence. - Three-step quantitative approach: - Extend baseline model by adding labor income and housing wealth. - Present target moments and estimates for key variables. - Present results (calibration and simulations). - Lifetime utility for households born at time τ: - Eτ[∫∞τ e−(δ+m)t u(cit) dt] (equation (16) referenced) - Autarky budget constraint (detailed terms preserved): - dait = [(r∗t + θ1it σs∗t + m) ait + w∗t lit ︸︷︷︸ (i) Labor Income − cit + rh t hit ︸︷︷︸ (ii) Housing Return) dt + σ1 θ1it ait dz1t - w∗t denotes wage, lit labor productivity, hit value of housing endowment, rh t return on housing assets. - Individual wealth defined as ait + hit. - Modeling simplifications: - Labor productivity lit ≡ `i + εit with `i permanent skill (lognormal at birth) and εit temporary AR(1) with jumps: dεit = −βε εit + qit dJit; jumps rate ζε; qit ∼ N(με, σ2ε); φ(ε) denotes normal p.d.f. - Housing: hit ≡ βh1 ait + βh2 ; rh t exogenously taken from data and unaffected by financial globalization; βh1 and βh2 calibrated to fit data. - Standard functional forms: - Utility: u(cit) = (cit − κ)1−γ − 1 / (1 − γ) - Technology: Φ(Kt) = Z Kαt L1−αt - Endowment log joint distribution: [loga i0 log`i] ∼ N([μa μ`], [Σaa Σa` Σ`a Σ``]) - Lt ≡ ∫[0,1] lit di total labor. - Distribution notation: gt(a,`,ε) denotes cross-sectional density over assets a, skill ` and temporary shock ε; g0(a0i,`,ε) newborn distribution; ∫ g0(a0i,`,ε) dε is bivariate lognormal. - Transitional dynamics approach: two differential equations — HJB and Kolmogorov forward equation — govern household saving decisions and wealth distribution evolution; focus on trajectories with dz1t = dz2t ≡ 0 in autarky and dz1t = dz2t = 0 in open economy. - Proposition 5 (differential system for wealth distribution evolution; preserved labels): - (1.HJB) (δ+m) Jt = max_{c,θ1} { u(c) + ∂Jt/∂a vt(a,`,ε) + 1/2 ∂^2 Jt/∂a^2 (σ1 θ1 a)^2 + ∂Jt/∂` (−β` sit) + ζ ∫∞−∞ (Jt(a,`,x) − Jt(a,`,ε)) φ(x) dx + 1 dt Et[dJt] } - (2.Kolmogorov) d/dt gt(a,`,ε) = −m gt(a,`,ε) + m g0(a,`,ε) − d/da [ vt(a,`,ε) gt(a,`,ε) ] − ζ gt(a,`,ε) + ζ φ(ε) ∫∞−∞ ∫ g t(a,`,x) dx d` - vt(a,`,ε) represents saving function; 1 dt Et[dJt] denotes lim_{s↘0} Et[Jt+s − Js] / s. - Shock experiment: US transitions from financial autarky to open economy from period T onwards; aim to quantify effect on wealth distribution; US and FO differ in banking technology λUS > λFO and output volatility ̄σUS < ̄σFO. ### Numerical method (Section 5.2) — solution strategy and approximation - Core numerical challenge: household saving decisions depend on the infinite-dimensional cross-sectional distribution gt(a,`,ε). - Dimensionality reduction: impose a functional form approximation for portfolio choice (bounded rationality / aggregation). - Assumption 1 — portfolio choice functional form: - In autarky: θ1it = s1t χ1 / σ1 (1 − χ2 ait − χ3 ait `i) (equation (17)) - Constants χ1, χ2, χ3 calibrated from data. In open economy, s1 σ1 replaced with Σ−1 [σ1 s1; σ2 s2]. - Households with wealth below χ2 + χ3 `i have θit = 0. - Rationale and accuracy: - In baseline model with no labor income, approximation is exact (χ1 equals risk-aversion parameter, χ2 = a, χ3 = 0). - With labor income, approximation simplifies interaction between labor income and portfolio choice for tractability; consistent with empirical facts: wealthier households invest more in equity; marginal increase in equity share is diminishing; share converges to an upper bound. - Advantages of Assumption 1: - Aggregation: total demand for risky assets can be expressed via a finite number of state variables (mean of wealth distribution acts as sufficient statistic in autarky). - Good empirical fit to portfolio shares. - Numerical algorithm — continuous-time Krusell-Smith analogue: - Step 1: Guess law of motion for state variable (in autarky d log A1t = (ψ1 − 1) log A1t + ψ2 works well); guess ψ1 and ψ2. - Step 2: Under guessed law, numerically solve HJB to compute individual saving decisions. - Step 3: Use saving decisions to compute evolution of wealth distribution. - Step 4: Verify consistency of guessed law with Step 3; iterate if inconsistent. - Further technical details and handling of household debt referred to Appendix C. *Source: IMF working paper content (excerpt provided).* ### 5.3 Calibration Strategy ### 5.3 Calibration Strategy ### Notable target moments and calibration approach - Benchmark year for financial autarky: 1989. - Target moments for matching the model-implied stationary wealth distribution to Survey of Consumer Finances (1989): top 1%, top 5% and bottom 90% wealth shares. - Parameter block used to adjust portfolio frontier and fit moments: - {λ_US, ̄σ_US, λ_FO, ̄σ_FO, ρ; μ_a, Σ_aa, Σ_al} - Calibration steps: - Choose {λ_US, ̄σ_US, λ_FO, ̄σ_FO, ρ} to adjust the equilibrium portfolio frontier in autarky and in open economy—{r*_t, s*_{1t}, r_t, s_{1t}, s_{2t} }—to fall within a reasonable range according to historical patterns. - Pick μ_a, Σ_aa, and Σ_al to match top 1%, top 5% and bottom 90% wealth shares in 1989. - Modeling assumption: only a finite number of moments of the wealth distribution matter in the law of motion for the state variables (as in Krusell and Smith (1998)). ### Features used to generate a modest risk premium - Three model features helping to partially reconcile the equity risk premium puzzle: - Many households do not own risky assets; households below a threshold (in equation (17)) take on debt, reducing demand for risky assets and increasing supply of safe assets (mechanism similar to Mankiw and Zeldes (1991)). - χ_1 in (17) can be set differently from the elasticity of intertemporal substitution implied by the utility function, adding one degree of freedom to inhibit investment in risky assets. - ̄σ_US and ̄σ_FO are left as free parameters to match Sharpe ratios. ### Estimation of interest rates, capital gains, expected returns, and Sharpe ratios - Construction of average realized returns: - Use national accounts: macroeconomic yield for each asset class = flow payments reported in Gross National Income divided by market value in the Fed’s Financial Accounts (following Saez and Zucman (2016)). - Example for equities: dividend yield in 2005 = total dividend paid to households during 2005 / total value of equity holdings at end-2004. - Capital gains computed as increase in market value that exceeds net issuance of equities during the year. - Average return on equities = average dividend yield + average rate of capital gain. - Smoothing and sample windows: - Figure 5 panel (a): average realized returns on equities, fixed income and 3-month treasury bills in the United States, smoothed over twenty years (geometric average of real returns over a twenty-year horizon centered on the x coordinate). - Sharpe ratio calculations restricted to 1982-2017 due to data constraints. - Estimation of expected returns and capital gains: - Employ simple estimation methods proposed by Fama and French (2002), Campbell (2008) and Campbell and Thompson (2008) that use fundamentals (dividends, earnings, profitability) to estimate ex ante expected stock returns (simplest form: dividend yield + expected dividend growth rate). - Panel (b) and (c) of Figure 5 indicate a substantial portion of realized returns on equities stems from cumulative capital gains over past decades; estimated capital gains account for a significant part of realized returns across estimation methods. - Sharpe ratio evidence: - Panel (d) of Figure 5 compares Sharpe ratios for two equity portfolios: - Mixed portfolio (actual portfolio owned by U.S. households; dividends and earnings originate from foreign and domestic firms, reported in Gross National Income). - Domestic portfolio (based on profits generated by domestic investment, reported in Gross Domestic Product). - Measurement: Sharpe ratio = estimated risk premium / sample standard deviation of realized returns (simple average = mean return over standard deviation, no consideration of capital gains). - Sample period for Sharpe ratios: 1982-2017. - Finding: U.S. households enjoy a higher risk-return trade-off in the mixed (global) portfolio than in the domestic portfolio—benefit stems from higher return on global investment and diversification effect. ### Numerical calibration context and fit (overview relevant to calibration strategy) - Smoothing window for return series: 20-year geometric average. - Data constraint: right-hand side of the time horizon is shorter in last ten data points for smoothed series. - Sharpe ratio sample window: 1982-2017. - The calibration strategy feeds into subsequent quantitative analysis where: - The model generates a Pareto tail in the wealth distribution consistent with 1989 top wealth shares. - The model can match wage inequality and capture its effect on top wealth shares. - Portfolio choices vary by wealth: affluent households invest more heavily in equity; middle-class invest more in safe assets and housing. *Source: 5.3 Calibration Strategy (excerpt) from wpiea2021254-print-pdf* ### References ### References ### Bibliographic entries - [1] Y. Achdou, J. Han, J.-M. Lasry, P.-L. Lions, and B. Moll, “Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach,” National Bureau of Economic Research, Working Paper 23732, August 2017. - [2] S. Ahn, G. Kaplan, B. Moll, T. Winberry, and C. Wolf, “When Inequality Ma‹ers for Macro and Macro Ma‹ers for Inequality,”NBER Macroeconomics Annual, vol. 32, no. 1, pp. 1–75, 2018. - [3] S. R. Aiyagari, “Uninsured Idiosyncratic Risk and Aggregate Saving,”Œe ‹arterly Journal of Economics, vol. 109, no. 3, pp. 659–684, 1994. - [4] P. Antr`as and R. J. Caballero, “On the Role of Financial Frictions and the Saving Rate during Trade Liberalizations,”Journal of the European Economic Association, vol. 8, no. 2-3, pp. 442–455, 2010. - [5] P. Antr`as, M. A. Desai, and C. F. Foley, “Multinational Firms, FDI Flows, and Imperfect Capital Markets,”Œe ‹arterly Journal of Economics, vol. 124, no. 3, p. 1171, 2009. - [6] L. Bach, L. E. Calvet, and P. Sodini, “Rich Pickings? Risk, Return, and Skill in Household Wealth,” Tech. Rep. FIN-2016-1126, 2018. - [7] J. Benhabib and A. Bisin, “Skewed Wealth Distributions: ‘eory and Empirics,” 2018. - [8] T. Bewley, “A Diculty with the Optimum antity of Money,”Econometrica, vol. 51, no. 5, pp. 1485–1504, 1983. - [9] O. J. Blanchard, “Debt, De€cits, and Finite Horizons,”Journal of Political Economy, vol. 93, no. 2, pp. 223–247, 1985. - [10] R. J. Caballero, E. Farhi, and P.-O. Gourinchas, “An Equilibrium Model of Global Imbalances and Low Interest Rates,”American Economic Review, vol. 98, no. 1, pp. 358–93, March 2008. - [11] ——, “Rents, Technical Change, and Risk Premia Accounting for Secular Trends in Interest Rates, Returns on Capital, Earning Yields, and Factor Shares,”American Economic Review, vol. 107, no. 5, pp. 614–20, May 2017. - [12] M. Cage‹i and M. De Nardi, “Entrepreneurship, Frictions, and Wealth,”Journal of Political Economy, vol. 114, no. 5, pp. 835–870, 2006. - [13] J. Y. Campbell, “Viewpoint: Estimating the equity premium,”Canadian Journal of Economics, vol. 41, no. 1, pp. 1–21, 2008. - [14] J. Y. Campbell and J. H. Cochrane, “By Force of Habit: A Consumption€Based Explanation of Aggregate Stock Market Behavior,”Journal of Political Economy, vol. 107, no. 2, pp. 205–251, 1999. - [15] J. Y. Campbell and S. B. ‘ompson, “Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?”Œe Review of Financial Studies, vol. 21, no. 4, pp. 1509–1531, 2008. - [16] D. G. Champernowne, “A Model of Income Distribution,”Œe Economic Journal, vol. 63, no. 250, pp. 318–351, 1953. - [17] T. Chesnokova, “Immiserizing deindustrialization: A dynamic trade model with credit con- straints,”Journal of International Economics, vol. 73, no. 2, pp. 407 –420, 2007. - [18] J. C. Cox, J. E. Ingersoll, and S. A. Ross, “An Intertemporal General Equilibrium Model of Asset Prices,”Econometrica, vol. 53, no. 2, pp. 363–384, 1985. - [19] S. E. Curcuru, C. P. ‘omas, and F. E. Warnock, “On Returns Di‚erentials,” National Bureau of Economic Research, Working Paper 18866, March 2013. - [20] W. W. Dou and A. Verdelhan, “‘e Volatility of International Capital Flows and Foreign Assets,” Tech. Rep., 2015. - [21] B. Eichengreen,Exorbitant Privilege: Œe Rise and Fall of the Dollar and the Future of the International Monetary System, 1st ed. Oxford University Press, 2011. - [22] A. Fagereng, L. Guiso, D. Malacrino, and L. Pistaferri, “Heterogeneity and Persistence in Returns to Wealth,”IMF Working Papers, 2018. - [23] E. F. Fama and K. R. French, “‘e Equity Premium,”Œe Journal of Finance, vol. 57, no. 2, pp. 637–659, 2002. - [24] J. Fern´andez-Villaverde, S. Hurtado, and G. Nuno, “Financial Frictions and the Wealth Dis- tribution,” 2018. - [25] D. Furceri and P. Loungani, “Capital Account Liberalization and Inequality,”IMF Working Papers, 2015. - [26] D. Furceri, P. Loungani, and J. D. Ostry, “‘e Aggregate and Distributional E‚ects of Finan- cial Globalization: Evidence from Macro and Sectoral Data,” 2017. - [27] X. Gabaix, “Power Laws in Economics and Finance,”Annual Review of Economics, vol. 1, no. 1, pp. 255–294, 2009. - [28] X. Gabaix, J.-M. Lasry, P.-L. Lions, and B. Moll, “‘e Dynamics of Inequality,”Econometrica, vol. 84, no. 6, pp. 2071–2111, 2016. - [29] M. Gomez, “Asset Prices and Wealth Inequality,” 2019. - [30] ——, “Displacement and the Rise in Top Wealth Inequality,” 2019. - [31] P.-O. Gourinchas and H. Rey,From World Banker to World Venture Capitalist: US External Adjustment and the Exorbitant Privilege. ‘e University of Chicago Press, 2007, pp. 11–15. - [32] ——, “Exorbitant Privilege and Exorbitant Duty,”IMES Discussion Paper Series, 2010. - [33] C. R. Harvey, “Predictable Risk and Returns in Emerging Markets,”Œe Review of Financial Studies, vol. 8, no. 3, pp. 773–816, 1995. - [34] E. Helpman,Globalization and Inquality. Harvard University Press, 2018. - [35] B. Holmstrom and J. Tirole, “Financial Intermediation, Loanable Funds, and ‘e Real Sector,” Œe ‹arterly Journal of Economics, vol. 112, no. 3, pp. 663–691, 1997. - [36] J. Hubmer, P. Krusell, and A. A. Smith Jr, “A Comprehensive antitative ‘eory of the US Wealth Distribution,” Working Paper, 2018. - [37] M. Hugge‹, “‘e Risk-free Rate in Heterogeneous-agent Incomplete-insurance Economies,” Journal of Economic Dynamics and Control, vol. 17, no. 5, pp. 953–969, 1993. - [38] T. L. Hungerford, “Changes in the Distribution of Income Among Tax Filers Between 1996 and 2006: ‘e Role of Labor Income, Capital Income, and Tax Policy,” CRS Report for Con- gres, 2011. - [39] F. Jaumo‹e, S. Lall, and C. Papageorgiou, “Rising Income Inequality: Technology, or Trade and Financial Globalization?”IMF Working Papers, no. 08/185, 2008. - [40] C. I. Jones, “Pareto and Pike‹y: ‘e Macroeconomics of Top Income and Wealth Inequality,” Journal of Economic Perspectives, vol. 29, no. 1, pp. 29–46, February 2015. - [41] O. Jord`a, K. Knoll, D. Kuvshinov, M. Schularick, and A. M. Taylor, “‘e Rate of Return on Everything, 1870-2015,” National Bureau of Economic Research, Working Paper 24112, De- cember 2017. - [42] M. Kacperczyk, J. Nosal, and L. Stevens, “Investor Sophistication and Capital Income In- equality,”Journal of Monetary Economics, 2018. - [43] G. Kaplan, B. Moll, and G. L. Violante, “Monetary Policy According to HANK,”American Economic Review, vol. 108, no. 3, pp. 697–743, March 2018. - [44] C. P. Kindleberger, “Balance-of-Payments De€cits and the International Market for Liquid- ity,”Essays in International Finance, vol. 106, no. 46, 1965. - [45] P. Krusell and A. A. Smith, “Income and Wealth Heterogeneity in the Macroeconomy,”Jour- nal of Political Economy, vol. 106, no. 5, pp. 867–896, 1998. - [46] P. R. Lane, “Financial Globalisation and the Crisis,”Open Economic Review, vol. 24, no. 3, pp. 555–580, 2013. - [47] P. R. Lane and G. M. Milesi-Ferre‹i, “‘e External Wealth of Nations Mark II: Revised and Extended Estimates of Foreign Assets and Liabilities, 1970-2004,”Journal of International Economics, vol. 73, no. 2, pp. 223–250, 2007. - [48] R. Litzenberger and M. Rubinstein, “‘e Strong Case for the Generalized Logarithmic Utility Model as the Premier Model of Financial Markets,”Œe Journal of Finance, vol. 31, no. 2, pp. 551–571, 1976. - [49] M. Maggiori, “Financial Intermediation, International Risk Sharing, and Reserve Curren- cies,”American Economic Review, vol. 107, no. 10, pp. 3038–71, October 2017. - [50] N. G. Mankiw, “Yes, r>g. So What?”American Economic Review: Papers & Proceedings, vol. 105, no. 5, pp. 43–47, 2015. - [51] E. G. Mendoza, V. adrini, and J.-V. Rios-Rull, “Financial Integration, Financial Develop- ment, and Global Imbalances,”Journal of Political Economy, vol. 117, no. 3, pp. 371–416, 2009. - [52] R. C. Merton, “Lifetime Portfolio Selection under Uncertainty: ‘e Continuous-Time Case,” Œe Review of Economics and Statistics, vol. 51, no. 3, pp. 247–257, 1969. - [53] ——, “Optimum Consumption and Portfolio Rules in a Continuous-time Model,”Journal of Economic Œeory, vol. 3, no. 4, pp. 373–413, 1971. - [54] ——,Continuous-Time Finance. Wiley-Blackwell, 1992. - [55] A. Mian, K. Rao, and A. Su€, “Household Balance Sheets, Consumption, and the Economic Slump,”Œe ‹arterly Journal of Economics, vol. 128, no. 4, pp. 1687–1726, 2013. - [56] M. Obstfeld, “Risk-Taking, Global Diversi€cation, and Growth,”Œe American Economic Re- view, vol. 84, no. 5, pp. 1310–1329, 1994. - [57] M. Ogaki and Q. Zhang, “Decreasing Relative Risk Aversion and Tests of Risk Sharing,” Econometrica, vol. 69, no. 2, pp. 515–526, 2001. - [58] T. Pike‹y,Capital in the Twenty-First Century. Harvard University Press, 2014. - [59] T. Pike‹y, E. Saez, and G. Zucman, “Distributional National Accounts: Methods and Esti- mates for the United States,”Œe ‹arterly Journal of Economics, vol. 133, no. 2, pp. 553–609, 2018. - [60] V. adrini, “Entrepreneurship, Saving, and Social Mobility,”Review of Economic Dynamics, vol. 3, no. 1, pp. 1–40, 2000. - [61] M. Rognlie, “A Note on Pike‹y and Diminishing Returns,” 2014. - [62] E. Saez and G. Zucman, “Wealth Inequality in the United States since 1913: Evidence from Capitalized Income Tax Data,”Œe ‹arterly Journal of Economics, vol. 131, no. 2, p. 519, 2016. - [63] W. F. Sharpe, “Capital Asset Prices: A ‘eory of Market Equilibrium under Conditions of Risk ,”Œe Journal of Finance, vol. 19, no. 3, pp. 425–442, 1964. - [64] J. C. Stein, “Monetary Policy as Financial Stability Regulation,”Œe ‹arterly Journal of Eco- nomics, vol. 127, no. 1, pp. 57–95, 2012. - [65] L. H. Summers, “U.S. Economic Prospects: Secular Stagnation, Hysteresis, and the Zero Lower Bound,”Business Economics, vol. 49, no. 2, 2014. - [66] M. E. Yaari, “Uncertain Lifetime, Life Insurance, and the ‘eory of the Consumer,”Œe Review of Economic Studies, vol. 32, no. 2, pp. 137–150, 04 1965. *wpiea2021254-print-pdf - References* ### Appendix A Detailed Proofs ### Appendix A Detailed Proofs ### A.1 Proof of Proposition 1 — Revaluation of Top Wealth Share (dlog Ω_T) and Stationary Pareto (dlog Ω_∞) - Result 1 (dlog Ω_T) - Ω_T expressed using pre-shock wealth c.d.f. and p.d.f. G∗(·) and g∗(·): - Ω_T = ∫_{a=G∗^{-1}(0.99)}^{∞} { a(1−θ_1(a)) + ( r∗ + σ_1 s∗_1 / (r+σ_1 s_1) ) a θ_1(a) } g∗(a) da divided by ∫_{a=a}^{∞} { a(1−θ_1(a)) + ( r∗ + σ_1 s∗_1 / (r+σ_1 s_1) ) a θ_1(a) } g∗(a) da - Taking logs and differentiating with respect to x ≡ r+σ_1 s_1 yields - dlog Ω_T = −φ_1 dlog(r+σ_1 s_1) - φ_1 defined as - φ_1 = − (r+σ_1 s_1) [ ∫_{a=G∗^{-1}(0.99)}^{∞} { a(1−θ_1(a)) } g∗(a) da / ∫_{a=G∗^{-1}(0.99)}^{∞} { a(1−θ_1(a))(r+σ_1 s_1) + a θ_1(a) (r∗+σ_1 s∗_1) } g∗(a) da + ∫_{a=a}^{∞} { a(1−θ_1(a)) } g∗(a) da / ∫_{a=a}^{∞} { a(1−θ_1(a))(r+σ_1 s_1) + a θ_1(a) (r∗+σ_1 s∗_1) } g∗(a) da ] - Equivalently φ_1 = − 1 / [ 1 + (r∗+σ_1 s∗_1)/(r+σ_1 s_1) ∫_{a=G∗^{-1}(0.99)}^{∞} a θ_1(a) g∗(a) da / ∫_{a=G∗^{-1}(0.99)}^{∞} a(1−θ_1(a)) g∗(a) da ] + 1 / [ 1 + (r∗+σ_1 s∗_1)/(r+σ_1 s_1) ∫_{a=a}^{∞} a θ_1(a) g∗(a) da / ∫_{a=a}^{∞} a(1−θ_1(a)) g∗(a) da ] > 0 - Positive sign of φ_1 follows from Lemma 1: top 1 percent households on average invest more in risky assets prior to the shock and thus obtain larger revaluation gains. - Lemma 1 - For all x ≥ a: - d/dx [ ∫_{a=x}^{∞} a θ_1(a) g∗(a) da / ∫_{a=x}^{∞} a (1−θ_1(a)) g∗(a) da ] > 0 - Proof sketch uses ratio manipulation and θ_1(a)/θ_1(x) > 1 for a ≥ x. - Result 2 (dlog Ω_∞) — Pareto exponent and openness effects - Closed economy stationary distribution g(a) solves Kolmogorov Forward equation; Pareto exponent in closed economy equals - (r∗ + (s∗_1)^2 − δ) / m - In open economy, portfolio weights characterized by Lemma 2: - [ θ_1it ; θ_2it ] = Σ^{−1} [ σ_1 s_1 ; σ_2 s_2 ] (1 − a/a_it) (Equation (18)) - Σ ≡ [ σ_1^2, ρ σ_1 σ_2 ; ρ σ_1 σ_2, σ_2^2 ] - Relative portfolio weight explicit: - Σ^{−1} [ σ_1 s_1 ; σ_2 s_2 ] = 1/(1−ρ^2) [ s_1/σ_1 − ρ s_2/σ_1 ; s_2/σ_2 − ρ s_1/σ_2 ] - Non-negativity requires condition: - ρ < min{ s_1/s_2, s_2/s_1 } (Equation (19)) - Open economy Pareto exponent 1/ξ given by - 1/ξ = ( r + R′ Σ^{−1} R − δ ) / m - = ( r + 1/(1−ρ^2) ( s_1^2 − 2ρ s_1 s_2 + s_2^2 ) − δ ) / m - Total differential of Ω_∞ = 100^{1/ξ − 1} yields - dlog Ω_∞ = φ_2 dlog(r+σ_1 s_1) + φ_3 dlogs_2 − φ_4 dlogρ - with - φ_2 = (r+σ_1 s_1) log 100 / m > 0 - φ_3 = 2 s_2 log 100 / [ m(1−ρ^2) ( s_2 − ρ s_1 ) ] > 0 - φ_4 = 2 ρ log 100 / [ m(1−ρ^2)^2 ( ρ^2 s_1 s_2 − ρ(s_1^2 + s_2^2) + s_1 s_2 ) ] > 0 - Positivity of φ_3 and φ_4 uses condition (19). ### A.2 More Details on Dynamics and Stationary Wealth Stock - Wealth stock evolution (Equation (20)): - dA_t = [ ( r∗(A_t) + s∗_1(A_t)^2 − δ − m )(A_t − a) + m(A_t − A_0) ] dt + s∗_1(A_t) A_t dz - r(·) and s(·) given by (14) and (15); A_0 is mean wealth of newborns. - Stationary state A_s defined by E_t[dA_t] = 0 and A_s > a. - Corollary 2 (stationary state implications when a = 0) - In the stationary state, a developing country exhibits: - a smaller wealth stock A_s - a lower risk-free rate r∗(A_s) - a higher Sharpe ratio s∗_1(A_s) - a lower cost of capital r∗(A_s) + ̄σ s∗_1(A_s) - a lower excess profit V∗(A_s) - Stated result: these hold when a = 0. ### A.3 Proof of Corollary 2 (Stationary State Comparative Statics) - Wealth dynamics restated with functional forms: - s_1(A) = ̄σ A/(A−a) (Equation (21)) - r(A) = Φ′(A) − ̄σ^2 A/(A−a) − τ + τ λ (Equation (22)) - Stationary condition: - r(A_s) + s(A_s)^2 = δ + m + m(A_s − A_0)/(A_s − a) - When a = 0, condition reduces to: - Φ′(A_s) − τ + τ λ = 2m + δ − m A_0 / A_s - Left-hand side decreasing in A_s; right-hand side increasing in A_s → unique A_s. - Implicit differentiation: - ( Φ′′(A_s) − m A_0 / (A_s)^2 ) ∂A_s/∂λ + τ = 0 → ∂A_s/∂λ > 0 given diminishing marginal returns. - Comparative statics: - FO (autarky) has lower A_s, higher s_1(A) = ̄σ, lower r(A_s) = − ̄σ^2 − τ + τ λ, and higher cost of capital r(A_s) + ̄σ s(A_s) + τ − τ λ = Φ′(A_s) → leading to lower V_1t. ### A.4 Proof of Proposition 3 — Financial Globalization Effects (Two Cases) - A.4.1 Case 1: ρ ∈ (0,1) - Portfolio choices: - [ θ_1it ; θ_2it ] = Σ^{−1} [ σ_1 s_1t ; σ_2 s_2t ] (1 − a/a_it) - Market clearing and first-order conditions (Equations (23)-(27)): - r_t = Φ′(K_1t) − ̄σ_1 s_1t − τ(1−λ_1) (23) - r_t = Φ′(K_2t) − ̄σ_2 s_2t − τ(1−λ_2) (24) - K_1t = (s_1t − ρ s_2t) ̄σ_1 (1−ρ^2)^{-1} (2A − a_1 − a_2) (25) - K_2t = (s_2t − ρ s_1t) ̄σ_2 (1−ρ^2)^{-1} (2A − a_1 − a_2) (26) - K_1t + K_2t = 2A (27), conditional on A_1t = A_2t = A - Solve for s_1t (Equation (28)): - s_1t = ̄σ_1 ( 2A − (1 − ρ ̄σ_2/̄σ_1) K_2t / (2A − a_1 − a_2) ) - With ρ ̄σ_2 > ̄σ_1 and a_1 < a_2, obtain s_1t > s∗_1t. - Show r_t < r∗_t when A_1t = A_2t = A (contradiction method using Φ′′(·) < 0 and market-clearing). - Show r_t + ̄σ_1 s_1t < r∗_t + ̄σ_1 s∗_1t and V_1t > V∗_1t: - r_t + ̄σ_1 s_1t = Φ′(K_1t) − τ(1−λ_1) < Φ′(A) − τ(1−λ_1) = r∗_t + ̄σ_1 s∗_1t - Envelope theorem: V_1t = max_{K_1t} { Φ(K_1t) − (r_t + ̄σ_1 s_1t) K_1t } and dV_1t/dx < 0 ⇒ V_1t > V∗_1t. - A.4.2 Case 2: ρ = 1 (perfect correlation between risky returns) - Common Sharpe ratio s_t ≡ s_1t = s_2t. - Market-clearing equations become (31)-(34): - r_t = Φ′(K_1t) − ̄σ_1 s_t − τ(1−λ_1) (31) - r_t = Φ′(K_2t) − ̄σ_2 s_t − τ(1−λ_2) (32) - (2A − a_1 − a_2) s_t = ̄σ_1 K_1t + ̄σ_2 K_2t (33) - 2A = K_1t + K_2t (34) - s_t = ω_1 ̄σ_1 + ω_2 ̄σ_2 / (1 − (a_1 + a_2)/(2A)) > s∗_t ⇒ s_t > s∗_t. - From K_1t > K_2t and first-order conditions, derive r_t + ̄σ_1 s_t < r∗_t + ̄σ_1 s∗_t and thus r_t < r∗_t. Envelope theorem implies V_1t > V∗_1t when A_1t = A_2t = A. ### A.5 More Details on Foreign Direct Investment (FDI) — US Firm Problem and Contract Constraints - US firm total payoff: V_1t + V_{FDI 1t} - V_{FDI 1t} dt ≡ max_{ { φ_Local, φ_FDI, c, K_Local_2t, K_FDI_2t } } [ φ_FDI Φ(K_2t) − R_1 K_FDI_2t − c Φ(K_2t) ] dt (Equation (35)) - Subject to: - K_2t = K_Local_2t + K_FDI_2t (36) - (1−π_L)(1−φ_FDI − φ_Local) Φ(K_2t) ≥ B(η) Φ(K_2t) (37) (FO entrepreneur incentive constraint) - (1−π_L) φ_FDI Φ(K_2t) ≥ η Φ(K_2t) (38) (US firm incentive/monitoring constraint) - (1−φ_Local − φ_FDI) Φ(K_2t) ≥ 0 (39) (FO entrepreneur participation) - φ_Local Φ(K_2t) ≥ R_2 K_Local_2t (40) (local intermediary break-even) - Funding costs defined: - R_1 ≡ r_t + ̄σ_2 s_2t + τ_{FDI} (1 − λ_1) - R_2 ≡ r_t + ̄σ_2 s_2t + τ (1 − λ_2) - Assumption: τ_{FDI} > τ so that R_1 > R_2; equilibrium chooses K_Local_2t and K_FDI_2t to equalize marginal benefit of FDI with opportunity cost. - Open economy equilibrium definition: stochastic process { r_t, (s_1t, σ_1), (s_2t, σ_2) }_{t≥0} clearing global financial markets: - ∑_{k∈{FO,US}} ( S_k_t − I_k_t ) = 0 - ∑_{k∈{FO,US}} S_{k1t} = I_{US 1t} - ∑_{k∈{FO,US}} S_{k2t} = I_{FO 2t} where I_{FO 2t} = K_Local_2t + K_FDI_2t ### A.6 Proof of Proposition 4 — Optimal FDI Contract and Comparative Equilibrium - Lagrangian for US entrepreneur's FDI problem (constraints binding except participation (39) which does not bind): - L = φ_FDI Φ(K_2t) − R_1 K_FDI_2t − η Φ(K_2t) + μ_1 [ K_Local_2t + K_FDI_2t − K_2t ] + μ_2 [ (1−π_L)(1−φ_FDI − φ_Local) − B(η) ] + μ_3 [ φ_FDI − η/(1−π_L) ] + μ_5 [ φ_Local Φ(K_2t) − R_2 K_Local_2t ] - First-order conditions yield system (41)-(42) and multipliers (43): - μ_1 = R_1 > 0 - μ_5 = R_1 / R_2 > 0 - μ_2 = R_1 / R_2 Φ(K_2t) / (1−π_L) > 0 - μ_3 = Φ(K_2t) ( R_1 / R_2 − 1 ) > 0 (uses R_1 > R_2) - Constraints (36), (37), (38), (40) bind in optimum. - Monitoring choice pinned down by: - B′(η) = − R_2 / R_1 ( R_1 / R_2 − π_L ) < 0 → determines η∗ - Investment condition (Equation (44)): - Φ′(K_2t) = R_1 η π_L / (1−π_L) + R_1 / R_2 ( 1 − B(η) + η / (1−π_L) ) - Solution yields K∗_2t and η∗ - Optimal contract variables: - φ_FDI∗ = η∗ / (1−π_L) (45) - φ_Local∗ = 1 − ( B(η∗) + η∗ ) / (1−π_L) (46) - K_Local∗_2t = φ_Local∗ Φ(K∗_2t) / R_2 (47) - K_FDI∗_2t = K∗_2t − K_Local∗_2t (48) - US entrepreneur payoff under FDI (Equation (49)): - V_{FDI 1t} = R_1 ( Φ(K∗_2t) / Φ′(K∗_2t) − K∗_2t ) - Case without FDI (e.g., investment barriers/security market liberalization): - φ_Local = 1, φ_FDI = 0, and investment satisfies: - Φ′(K∗∗_2t) = R_2 / π_L - As π_L → 0, K∗∗_2t → 0, whereas K∗_2t > 0 under FDI for B(η∗)+η∗ < 1. - Comparison of equilibrium prices: - Supply condition under limited integration: r_t = π_L Φ′(K_2t) − ̄σ_2 s_2t − τ(1−λ_2) (50) - Under full integration: r_t = (1−B(η∗) − η∗) Φ′(K_2t) − ̄σ_2 s_2t − τ(1−λ_2) (51) - Since K∗_2t > K∗∗_2t > 0 for small π_L, obtain ordering of s_1t: - s_1t^{(iii)} > s_1t^{(ii)} > s_1t^{(i)} - Conclude V_1t^{(iii)} > V_1t^{(i)} and V_{FDI 1t} > 0 under FDI. ### A.7 More Details on the Money Premium and Banking Interpretation - Alternative household budget constraint with explicit money premium τ: - da_it = [ ( r∗_t + σ_1 s∗_{1t} θ_1it ) a_it − c_it ] dt + τ(1−θ_1it) a_it (Money Premium) + σ_1 θ_1it a_it dz_{1t} - τ interpreted as money premium / convenience yield of money-like claims (monetary value of banking services). - Savings curves: - S_t ≡ A_t ≡ ∫ a_it di - S_{1t} ≡ ∫ θ_1it a_it di - Using θ_1it = s∗_{1t} σ_1^{−1} − τ σ_1^{−2} (1 − a/a_it), obtain - S_t = A_t - S_{1t} = s∗_{1t} σ_1^{−1} − τ σ_1^{−2} (A_t − a) - Define ̄s∗_{1t} ≡ s∗_{1t} (σ_1^{−1} − τ σ_1^{−2}); then S_{1t} = ̄s∗_{1t} σ_1 (A_t − a) consistent with baseline model. - Bank optimization (private equity income vs. public claims): - V∗_t dt ≡ max_{K_t, D_t, E_t} { dπ_t − r∗_t D_t dt − ( r∗_t + σ_1 s∗_{1t} ) E_t dt − σ_1 E_t dz_{1t} } - Equivalently, using ̄s∗_{1t} and τ: - V∗_t dt ≡ max_{K_t, D_t, E_t} { dπ_t − r∗_t D_t dt − ( r∗_t + σ_1 ̄s∗_{1t} + τ ) E_t dt − σ_1 E_t dz_{1t} } - This coincides with the bank’s optimization problem in Section 3, linking τ to the money premium interpretation. *Italic: Source — wpiea2021254-print-pdf - Appendix A Detailed Proofs* ### Appendix B HJB Equations ### Appendix B HJB Equations ### B.1 Section 2. Exogenous Prices — Closed Economy - Value function: J(a_it) = max_{c_it, θ_1it} E[∫_0^∞ e^{−(δ+m)t} log(c_it − κ) dt] subject to budget constraint (2). - HJB representation: - (δ+m)J(a_it) = max_{c_it, θ_1it} { log(c_it − κ) + J_a{[r^* + σ_1 s^*_1 θ_1it] a_it − c_it} + 1/2 J_{aa} σ_1^2 θ_{1it}^2 a_it^2 } - Transversality: lim_{t→∞} e^{−(δ+m)t} J(a_it) = 0 - First-order conditions: - c_it = (J_a)^{−1} + κ (equation (52)) - θ_{1it} = − s_1 J_a σ_1 a_it / J_{aa} (equation (53)) - Candidate solution: - Pick J = 1/(δ+m) log(a_it − κ/r^*) + const., where const. ≡ log(δ+m)/(δ+m) + (r^*+m)/(δ+m)^2 − 1/(δ+m) − s^{*2}_1 / [2(δ+m)^2]. - Implied derivatives: J_a = 1/[(δ+m)(a_it − κ/r^*)], J_{aa} = −1/[(δ+m)(a_it − κ/r^*)^2]. - Verify substitution into (54) yields equality. - Closed-economy policy functions: - c_it = (δ+m)(a_it − a) + κ = (δ+m)a_it + (r^* − m − δ) a, where a ≡ κ / r^*. - θ_{1it} = s^*_1 σ_1^{−1} (1 − a / a_it). ### B.1 Section 2. Exogenous Prices — Open Economy - Portfolio frontier: {r, (s_1, σ_1), (s_2, σ_2)} with correlation ρ between dz_{1t} and dz_{2t}. - Lemma 2 (portfolio choice): - θ_it = Σ^{−1} [σ_1 s_1; σ_2 s_2] (1 − a / a_it) - Σ ≡ [σ_1^2, ρ σ_1 σ_2; ρ σ_1 σ_2, σ_2^2] - HJB representation: - (δ+m)J(a_it) = max_{c_it, θ_it} { log(c_it − κ) + J_a{[σ_1 s_1 θ_1it + σ_2 s_2 θ_2it + r] a_it − c_it} + 1/2 J_{aa}[(σ_1 θ_1it)^2 + (σ_2 θ_2it)^2 + 2ρ σ_1 σ_2 θ_1it θ_2it] a_it^2 } - First-order conditions: - c_it = (J_a)^{−1} + κ - θ_it = Σ^{−1} [σ_1 s_1; σ_2 s_2] (− J_a a_it / J_{aa}) - Candidate solution: - J = 1/(δ+m) log(a_it − κ/r) + const., const. ≡ log(δ+m)/(δ+m) + (r+m)/(δ+m)^2 − 1/(δ+m) − (1/[2(δ+m)^2])(s_1^2 + s_2^2 − 2ρ s_1 s_2 / (1−ρ^2)). - Second-order condition: holds if Ω positive definite (i.e. 1 > ρ > 0). ### B.2 Section 3. Endogenous Prices — Closed Economy - Set-up: r^*_t ≡ r^*(A_t), s^*_{1t} ≡ s^*_1(A_t); κ = r^*_t a where a is constant. - Guess for individual policies: - c_it = (δ+m) a_it − (δ+m) ā - θ_{1it} = s^*_{1t} σ_1^{−1} (1 − a / a_it) - Aggregate wealth dynamics (integrating i): - dA_t = [ (r^*(A_t) + s^*_1(A_t)^2 − δ − m)(A_t − a) + m(A_t − A_0) ] dt + s^*_1(A_t) A_t dz_{1t} ≡ μ_A dt + σ_A dz_{1t} (equation (57)) - HJB for household with state variables (a_it, A_t): - (δ+m) J dt = max_{c_it, θ_{1it}} { log c_it + J_a{(r^*_t + σ_1 s^*_{1t} θ_{1it}) a_it − c_it − r^*_t a} + 1/2 J_{aa} σ_1^2 θ_{1it}^2 a_it^2 + J_A μ_A + 1/2 J_{AA} σ_A^2 + J_{Aa} σ_1 θ_{1it} a_it σ_A } dt (equation (58)) - Transversality: lim_{t→∞} e^{−δt} J(a_it, A_t) → 0 - First-order conditions: - c_it = (J_a)^{−1} - θ_{1it} = − s^*_{1t} J_a σ_1 a_it / J_{aa} + J_{Aa} σ_1 a_it σ_A / J_{aa} - Candidate value function: - J(a_it, A_t) ≡ 1/(δ+m) log(a_it − a) + C(A_t), where C(·) solves: - (δ+m) C(A_t) = log(δ+m) + [r^*(A_t) − δ − m]/(δ+m) − 1/2 [s_1(A_t)]^2/(δ+m) + C'(A_t) μ_A + 1/2 C''(A_t) σ_A^2 - Equilibrium policy functions and value function: - c_it = (δ+m)(a_it − a) - θ_{1it} = s^*_{1t} σ_1^{−1} (1 − a / a_it) - J(a_it, A_t) = 1/(δ+m) log(a_it − a) + C(A_t) - Note: log utility allows additive separability J(a_it, A_t) = function(a_it) + function(A_t). ### B.2 Section 3. Endogenous Prices — Open Economy - Guess household solutions: - c_it = (δ+m)(a_it − a) (equation (59)) - θ_{1it} = (s_{1t} − ρ s_{2t}) / [σ_1 (1 − ρ^2)] (1 − a / a_it) (equation (60)) - θ_{2it} = (s_{2t} − ρ s_{1t}) / [σ_2 (1 − ρ^2)] (1 − a / a_it) (equation (61)) - Market clearing conditions: - Φ'(K_{1t}) − \bar{σ}_1 s_{1t} − τ_1 (1 − λ_1) = r_t - Φ'(K_{2t}) − \bar{σ}_2 s_{2t} − τ_2 (1 − λ_2) = r_t - s_{1t} − ρ s_{2t} over σ_1 (1 − ρ^2) (A_{1t} + A_{2t} − 2a) = K_{1t} - s_{2t} − ρ s_{1t} over σ_2 (1 − ρ^2) (A_{1t} + A_{2t} − 2a) = K_{2t} - K_{1t} + K_{2t} = A_{1t} + A_{2t} - State variable: \bar{A}_{1t} ≡ A_{1t} + A_{2t}; denote market-clearing prices r(\bar{A}_t), s_1(\bar{A}_t), s_2(\bar{A}_t). - Aggregate dynamics: - d\bar{A}_t = μ_{\bar{A}} dt + σ_{\bar{A},1} dz_{1t} + σ_{\bar{A},2} dz_{2t} - μ_{\bar{A}} = ( \bar{A}_t − 2a )/(1 − ρ^2) [ s_1(\bar{A}_t)^2 + s_2(\bar{A}_t)^2 − 2ρ s_1(\bar{A}_t) s_2(\bar{A}_t) ] + r(\bar{A}_t) − δ − m + m(A_t − A_0) - σ_{\bar{A},1} = [s_1(\bar{A}_t) − ρ s_2(\bar{A}_t)]/(1 − ρ^2) \bar{A}_t - σ_{\bar{A},2} = [s_2(\bar{A}_t) − ρ s_1(\bar{A}_t)]/(1 − ρ^2) \bar{A}_t - HJB and candidate value function: - J(a_it, \bar{A}_t) ≡ 1/(δ+m) log(a_it − a) + C(\bar{A}_t), where C solves: - (δ+m) C(\bar{A}_t) = log(δ+m) + [r_t − δ − m]/(δ+m) − 1/2 [s_1^2 + s_2^2 − 2ρ s_1 s_2]/[(δ+m)(1 − ρ^2)] + C' μ_{\bar{A}} + 1/2 C'' σ_{\bar{A}}^2 - Substitution verifies HJB equality; transversality holds. --- ### Appendix C Quantitative Analysis ### C.1 Closed Economy — Overview and Simulation - (i) Households' Problem: - Utility: u(c_it) = c_it^{1−γ}/(1−γ). - Budget constraint: da_it = [(r^*_t + σ_1 s^*_{1t} θ_{1it} + m) a_it + w^*_{1t} l_it + r_h h_it − c_it − κ r^*_t] dt + σ_1 θ_{1it} a_it dz_{1t}. - Portfolio choice (taken as given approximate functional form): - θ_{1it} = max{ s^*_{1t} χ_1 / σ_1 (1 − χ_2 a_it − χ_3 a_it l_it), 0 } (equation (62)) - Note: χ_1 = γ, χ_2 = κ, χ_3 = 0 would arise if no labor income and housing assets. - (ii) Market Clearing Conditions (closed economy): - r^*_t = α Z A_{1t}^{α−1} L^{1−α} − \bar{σ}_1 s^*_{1t} − τ(1 − λ_1) - (1 − λ) A_{1t} = ∫_i a_it θ_{1it} di - w^*_{1t} = (1 − α) Z A_{1t}^{α} L^{−α} - Rewriting aggregate portfolio condition: - (1 − λ) A_{1t} = s_{1t} \bar{σ} χ_1 (A_{1t} − F^a_{1t} − F_i), where F^a_{1t} ≡ ∫_{a_it < χ_2 + χ_3/l_i} a_it di, F_i ≡ ∫_{a_it ≥ χ_2 + χ_3/`i} (χ_2 + χ_3/`i) di. - Practical expressions: - r^*_t ≡ r^*(A_{1t}, F^a_{1t}) = α Z A_{1t}^{α−1} L^{1−α} − \bar{σ}_1 s^*_1(A_{1t}) − τ(1 − λ_1) - s^*_{1t} ≡ s^*_1(A_{1t}, F^a_{1t}) = \bar{σ}_1 χ_1 A_{1t} / (A_{1t} − F^a_{1t} − F_i) - w^*_{1t} ≡ w^*_1(A_{1t}, F^a_{1t}) = (1 − α) Z A_{1t}^α L^{−α} - (iii) Simulation Algorithm (closed economy) — four steps: 1. Guess law of motion for state variables. Suggested form: - dA_{1t}/A_{1t} = ((ψ_2 − 1) log A_{1t} + ψ_1) dt + \bar{σ}_1 dz_{1t} (equation (63)) - Interpretation: discrete-time analogue log A_{1,t+1} = ψ_2 log A_{1,t} + ψ_1 + \bar{σ}_1 ε_t. - If indebted households exist, consider law for F^a_{1t}; in practice dF^a_{1t} = 0 around stationary state. 2. Given initial guess (ψ_1, ψ_2), solve differential equations for saving decisions and wealth distribution: - HJB (time-dependent) and Kolmogorov Forward Equation determine J_t ≡ J(a, `, ε, A_t) and g_t(a, `, ε). - HJB (equation (64)): - (δ+m) J_t dt = max_{c, θ_1} { u(c) + ∂J_t/∂a v_t(a,`,ε) + 1/2 ∂^2 J_t/∂a^2 (σ_1 θ_1 a)^2 + ∂J_t/∂ε (−β ε) + ζ ∫ (J_t(a,`,x) − J_t(a,`,ε)) φ(x) dx + 1 dt E_t[dJ_t] } dt - Kolmogorov Forward (equation (65)): - d/dt g_t(a,`,ε) = − m g_t + m g_0 − d/da[v_t g_t] − ζ g_t + ζ φ(ε) ∫ g_t dx d` - Individual savings v_t(a, l, ε) ≡ [ (r^*_t + σ_1 s^*_{1t} θ_1 + m) a + w^*_{1t} l + r_h h − c − κ(r^*_t + m) ]. - Solve numerically via finite-difference Upwind Scheme. 3. Check consistency of guessed (ψ_1, ψ_2) with OLS estimates \hat{ψ}_1, \hat{ψ}_2 obtained from simulated series A_{1,0},...,A_{1,T}. Iterate until convergence. 4. Once converged, compute model fit to observed data (1989 benchmark) and calibrate parameters so stationary wealth distribution matches data. - (cf) Upwind Scheme — numerical details: - Discretize a, `, A into grids a_i, `_j, A_k with I, J, K points. - Define θ_{1,i,j,k} = max{ s^*_1(A_k) χ_1 / σ_1 (1 − χ_2 a_i − χ_3 a_i ` _j), 0 }. - Compute incomes I_{i,j,k} and housing h_{i,j,k}. - Value function grid J_{i,j,k}. Initialization: J^0_{i,j,k} ≡ u(I_{i,j,k})/(δ+m). - Compute forward/backward slopes s^n,F_{i,j,k}, s^n,B_{i,j,k} via (66)-(67) and construct (J^n_{i,j,k})'. - Consumption c^n_{i,j,k} = (u')^{−1}((J^n_{i,j,k})'). - Update J via implicit time iteration: - J^{n+1}_{i,j,k} − J^n_{i,j,k} / Δ + (δ+m) J^{n+1}_{i,j,k} = u(c^n_{i,j,k}) + discrete spatial derivative terms + diffusion cross-terms + \bar{σ}_1^2/2 second difference in A + mixed partial term (σ_1 θ_1 a) \bar{σ}_1 cross-difference / (Δa ΔA). - Iterate until ||J^{n} − J^{n−1}|| small. - Kolmogorov Forward discretization: - g^{n+1}_{i,j} − g^{n}_{i,j} / Δt = − (s^{n,F}_{i,j,k} g^{n}_{i,j} − s^{n,F}_{i−1,j,k} g^{n}_{i−1,j})/Δa − (s^{n,B}_{i+1,j,k} g^{n}_{i+1,j} − s^{n,B}_{i,j,k} g^{n}_{i,j})/Δa, using (·)^+, (·)^−. ### C.2 Open Economy — Overview and Simulation - (i) Households' Problem (open economy): - Budget constraint: - da_it = [(r_t + σ_1 s_{1t} θ_{1it} + σ_2 s_{2t} θ_{2it} + m) a_it + w_{1t} l_it + r_h h_it − c_it − κ(r_t + m)] dt + σ_1 θ_{1it} a_it dz_{1t} + σ_2 θ_{2it} a_it dz_{2t}. - Portfolio choice approximations from time T onward: - θ_{1it} = max{ (s_{1t} − ρ s_{2t}) χ_1 / [σ_1 (1 − ρ^2)] (1 − χ_2 a_it − χ_3 a_it l_it), 0 } - θ_{2it} = max{ (s_{2t} − ρ s_{1t}) χ_1 / [σ_1 (1 − ρ^2)] (1 − χ_2 a_it − χ_3 a_it l_it), 0 } - (ii) Market Clearing Conditions (open economy): - α Z K_{1t}^{α−1} L^{1−α}_1 − \bar{σ}_1 s_{1t} − τ_1 (1 − λ_1) = r_t (equation (68)) - α Z K_{2t}^{α−1} L^{1−α}_2 − \bar{σ}_2 s_{2t} − τ_2 (1 − λ_2) = r_t (equation (69)) - s_{1t} − ρ s_{2t} over \bar{σ}_1 (1 − ρ^2) χ_1 (A_{1t} + A_{2t} − F^a_{1t} − F^a_{2t} − 2 F_i) = K_{1t} (equation (70)) - s_{2t} − ρ s_{1t} over \bar{σ}_2 (1 − ρ^2) χ_1 (A_{1t} + A_{2t} − F^a_{1t} − F^a_{2t} − 2 F_i) = K_{2t} (equation (71)) - K_{1t} + K_{2t} = A_{1t} + A_{2t} (equation (72)) - (1 − α) Z K_{1t}^{α} L^{−α}_1 = w_{1t} (equation (73)) - Define \bar{A}_t ≡ A_{1t} + A_{2t}, ζ(\bar{A}_t) ≡ A_{1t} + A_{2t} − 2 F^a_{1t} − 2 F_i. - Reduced system (merging equations): - [ (s_{1t} − ρ s_{2t}) \bar{σ}_1/(1 − ρ^2) χ_1 + (s_{2t} − ρ s_{1t}) \bar{σ}_2/(1 − ρ^2) χ_1 ] ζ(\bar{A}_t) = \bar{A}_t (equation (74)) - α Z [ (s_{1t} − ρ s_{2t}) \bar{σ}_1/(1 − ρ^2) χ_1 ζ(\bar{A}_t) ]^{α−1} L^{1−α}_1 − \bar{σ}_1 s_{1t} − τ_1 (1 − λ_1) = α Z [ (s_{2t} − ρ s_{1t}) \bar{σ}_2/(1 − ρ^2) χ_1 ζ(\bar{A}_t) ]^{α−1} L^{1−α}_2 − \bar{σ}_2 s_{2t} − τ_2 (1 − λ_2) (equation (75)) - Thus s_{1t} = s_1(\bar{A}_t), s_{2t} = s_2(\bar{A}_t). - r_t = r(\bar{A}_t) given by (76); w_{1t} = w_1(\bar{A}_t) given by (77). - (iii) Simulation Algorithm (open economy): 1. Guess law of motion for \bar{A}_t. Suggested form (when all households retain a_i ≥ χ_2 + χ_3 / `i): - d\bar{A}_t / \bar{A}_t = ((ψ_2 − 1) log \bar{A}_t + ψ_1) dt + σ_{\bar{A},1} dz_{1t} + σ_{\bar{A},2} dz_{2t} (equation (78)) - σ_{\bar{A},1} = [s_1(\bar{A}_t) − ρ s_2(\bar{A}_t)]/(1 − ρ^2) \bar{A}_t, σ_{\bar{A},2} = [s_2(\bar{A}_t) − ρ s_1(\bar{A}_t)]/(1 − ρ^2) \bar{A}_t 2. Given initial guess (ψ_1, ψ_2), solve HJB and Kolmogorov Forward (as in closed economy) with v_t(a,l,ε) updated to use r_t, s_{1t}, s_{2t}, w_{1t}. 3. Compute series \bar{A}_{1},...,\bar{A}_{T}, estimate \hat{ψ}_1, \hat{ψ}_2 by OLS, iterate until (ψ_1, ψ_2) consistent. - (iv) Transition to open economy (capital gains): - Price p_t for equity paying σ_1 dz_{1t} must satisfy no-arbitrage: - dp_t / p_t + (r^*_t + σ_1 s^*_{1t}) dt p_t = (r_t + σ_1 s_{1t}) dt - Discrete-time approximation for simulations: - E_t[ p_{t+1} − p_t + r^*_t + σ_1 s^*_{1t} p_t ] = r_t + σ_1 s_{1t} - Since p_t = 1 for t < T, compute p^{new}_T ≈ E_T[ ∑_{t=T}^∞ (r^*(A_{1t}) + σ s^*_{1}(A_{1t})) / (1 + r(A_{1t} + A_{2t}) + σ s_1(A_{1t} + A_{2t}))^t ] by Monte Carlo. - Immediately after shock, update wealth distribution g^{n+1}_{i,j,k} incorporating capital gains in income term: - Replace I_{i,j,k} by (r^*(A_k) + σ_1 s^*_{1}(A_k) θ_{1,i,j,k} + m + p^{new}_T) a_i + ` _j w^*_1(A_k) + r_h h_{i,j,k} − κ (r^*(A_k) + m) - Then evolve distribution forward using Kolmogorov Forward without capital gains for subsequent periods. *Source: Appendix B and Appendix C (HJB Equations and Quantitative Analysis) of the provided document.* ### Appendix D Additional Details ### Appendix D Additional Details ### D.1 Estimation of the Risk Premium - Definitions and return decomposition: - Let d_{t+1} denote the dividend for year t+1, P_t denote the price at the end of year t and R_{t+1} denote the return for year t+1. - Return decomposition (equation (79)): - R_{t+1} = d_{t+1}/P_t + (P_{t+1} − P_t)/P_t - Key identifying assumption: - Stationarity (mean reversion) of valuation ratios (e.g., dividend-price ratio d_t/P_t or earning-price ratio E_t/P_t). - Approaches considered to estimate expected capital gains / time-varying equity premium: - FF Dividends: Fama and French (2002) — use dividend growth rate (d_{t+1} − d_t)/d_t under stationary dividend-price ratio. - FF Earnings: use earnings growth rate (E_{t+1} − E_t)/E_t under stationary earning-price ratio. - CT RoE: Campbell and Thompson (2008) / Campbell (2008) — combines steady-state relation between dividend growth and accounting return on equity; express return (equation (80)): - R_{t+1} = (d_{t+1}/e_{t+1}) (e_{t+1}/P_t) + (1 − d_{t+1}/e_{t+1}) (e_{t+1}/B_t) - where B_t is the book value of equity. - Implementation detail: five-year smoothed dividend growth rates and earnings growth rates are used for FF Dividends and FF Earnings. Three-year smoothed return on equity, dividend yields, and payout ratios are used for CT RoE. - Purpose: - These methods judge whether the average realized return is high or low relative to the expected return implied by fundamentals. ### D.2 Kolmogorov Forward Equation (Section 2) - Individual wealth dynamics (autarky) — Ito diffusion (equation (81)): - da_t = [(r^* + s^{*2}_1 − δ − m)(a_{it} − a)] dt + s^*_1 a_t dz_{1t} - Integral form (equation (82)): - a_t(ω) = a_0 + ∫_0^t [(r^* + s^{*2}_1 − δ − m)(a_τ(ω) − a)] dτ + ∫_0^t s^*_1 a_τ(ω) dz_{1τ}(ω) - Construction of Ito integral via sequence of elementary functions ζ_n(t, ω). - Existence of deterministic trajectory ̄ω with z_{1t}(̄ω) = z_0 for all sampled times yields ordinary differential equation (equation (83)): - a_t(̄ω) = a_0 + ∫_0^t [(r^* + s^{*2}_1 − δ − m)(a_τ(̄ω) − a)] dτ - Along this trajectory: da_{it} = [(r^* + s^{*2}_1 − δ − m)(a_{it} − a)] dt - Cross-sectional wealth distribution definitions: - G_t(a) = ∫_{[0,1]} I{i∈[0,1]:a_{it} ≤ a} di - g_t(a) = ∂G_t(a)/∂a (density) - Small-interval update with death and newborn replacement rate m: - G_{t+dt}(a) = (1 − m dt) G_t(a − [(r + s^2 − δ − m)(a − a)] dt) + m dt G_0(a) - In the limit dt → 0: - dG_t(a)/dt = −m G_t(a) + m G_0(a) − [(r^* + s^{*2} − δ − m)(a − a)] g_t(a) - Differentiating wrt a yields evolution for density (displayed in source): - d/dt g_t(a) = −m g_t(a) + m g_0(a) − d/da [ (r^* + σ_1 s^*_1 θ_1(a)) a − c(a) ] g_t(a) - where θ_1(a) = s^*_1/σ_1 (1 − a/a_{it}) and c(a) = (δ + m) a + (r^* − δ − m) a - Note: analogous derivation applies for the open economy. ### D.3 Convergence of the Wealth Distribution - Stationary distribution in autarky solves ordinary differential equation: - 0 = −m g(a) + m g_0(a) − ∂[(r^* + s^{*2}_1 − δ − m)(a − ā )] g(a) / ∂a - Subject to normalization: ∫_a^∞ g(a) = 1 - Convergence claim: - ∫_κ^∞ | g_t(a) − g(a) | da ≤ e^{−m t} - Proof strategy: - Use smoothing via z(q) = √(ε^2 + q^2) and let ε → 0. - Lemma 3 (stated): For any twice continuously differentiable q(a,t), - ∂|q(a,t)|/∂t ≤ −m |q(a,t)| + m |g_0(a)| − (r^* + s^{*2}_1 − δ − m) ∂|(a − ā) q(a,t)| / ∂a - Substitute q(a,t) = g_t(a) − g(a), integrate over [κ, ∞), use Gronwall’s lemma to obtain: - ∫_κ^∞ | g_t(a) − g(a) | da ≤ e^{−m t} - Conclusion: - The wealth distribution g_t(a) converges to the stationary distribution g(a) as t → ∞. ### D.4 Discrete Time Model - Asymptotic notation used: - f_1(h) = O[f_2(h)] if lim_{h→0} f_1(h)/f_2(h) is bounded. - f_1(h) = o[f_2(h)] if lim_{h→0} f_1(h)/f_2(h) = 0. - f_1(h) ∼ f_2(h) if f_1(h) = O[f_2(h)] but f_1(h) ≠ o[f_2(h)]. - Market clearing times: 0, h, 2h, ..., n h with T ≡ n h and h the minimum spacing. - Production and shocks: - Representative firm: investing K_t units in period t produces Φ(K_t) h + ̄σ K_t ε_{t+h} in period t+h, where ε_{t+h} is unanticipated productivity change. - Assumptions on ε_{t+h}: - (A1) ε_{t+h} takes any one of n_ε distinct values ε(k) with probabilities p(k); ε(k) ∼ h^{1/2} and p(k) = O(1). - (A2) E_t[ε_{t+h}] = 0 and lim_{h→0} ∑_{k=1}^{n_ε} p(k) ε(k)^2 / h = 1. - (A3) {ε_{k h}}_{k=1}^n are i.i.d. across times k. - Firm’s optimization (per-period value V_t^h): - V_t^h ≡ max_{K_t, D_t, E_t} { Φ(K_t) − r^*(S_t) D_t − ( r^*(S_t) + σ_1 s^*_1(S_t) + τ ) E_t } h - Subject to: - K_t = D_t + E_t - σ_1 = ̄σ K_t / E_t - D_t ≤ λ K_t - Equilibrium r^*(S_t) and s^*_1(S_t) pinned down by market clearing conditions. - Household balance sheet and portfolio: - a_{it} = n_{it} p_t + n^D_{it} (equation (84)) - Portfolio weight on risky assets θ_{1it} = n_{it} p_t / a_{it} - Deposits pay r_t^h in period t+h with no uncertainty. - Each equity share pays x(S_t) h + σ_1 p_t ε_{t+h} in period t+h and pledges future payoffs. - After realization of ε_{t+h}, rebalancing and budget flow yields (equation merging (84) and (85)): - a_{i,t+h} − a_{it} = (( r^*(S_t) + θ_{1it} ( x(S_t)/p_t − r^*(S_t) ) ) a_t − c_{t+h} ) h + (p_{t+h} − p_t)/p_t a_{it} + σ_1 θ_{1it} a_{it} ε_{t+h} - Financial intermediary no-arbitrage condition: - (p_{t+h} − p_t)/p_t + x(S_t) h / p_t + σ ε_{t+h} = ( r(S_t) + σ s(S_t) ) h + σ ε_{t+h} - Under assumption x(S_t) = r^*(S_t) + σ_1 s^*_1(S_t), p_t = 1 for all t as long as there is no unanticipated change. - Continuous-time limit (h → 0): - Household wealth a_{it} follows diffusion: - da_{it} = ( ( r^*_t + σ s^*_{1t} θ_{it} ) a_{it} − c_{it} ) dt + σ_1 θ_{1it} a_{it} dz_{1t} - This follows from (A1), (A2), and (A3). *Appendix D Additional Details — source PDF: wpiea2021254-print-pdf* --- _Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021254-print-pdf.pdf_