## _wp0899 - 6. Transition to Steady State when (and accompanying sections)

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---

### I. Research objective and core mechanisms
- Research objective: propose a simple neoclassical real growth model (no monetary/exchange-rate channel) that captures key real features of the Chinese economy and assesses implications for transition to steady state.
- Two crucial features incorporated:
  - Imperfect capital mobility with two complementary capital types:
    - Domestic capital Z (must be financed by domestic saving).
    - Foreign/tradable capital K (borrowable on international markets at a fixed rate).
    - Production uses both types as complements: raising domestic capital increases the marginal product of foreign capital and attracts FDI.
  - Surplus rural labor and migration into the urban production sector; speed of migration depends on labor mobility parameter χ.
- Key insight: an increase in domestic capital raises marginal products, attracting rural labor and foreign capital until marginal products return to initial levels — explaining coexistence of high saving, large FDI inflows, high investment, low wages, and competitive goods prices.
- Transition implication: if substantial surplus labor remains, the transition to steady state could last an additional decade.

### II. Key calibrated stylized facts motivating the model (quoted exactly)
- per capita growth has averaged 8.5 percent over the last decade;
- investment has averaged over 36 percent of GDP since 1996;
- the national saving rate (with both households and entreprises contributing) has been extraordinarily high by any standard of comparison — in 2006, it was 54 percent (Table1);
- notwithstanding the high national saving rate, there has been a significant inflow of foreign direct investment (Figure1);
- wages in manufacturing have been very low by any international standard (Table2) and rural incomes have been even lower than urban incomes — by a considerable margin (Table3).

### III. Stylized empirical facts (section II)
- Fact 1: Saving, Investment and FDI are high
  - China’s saving rate is substantially higher than average; around 40 percent for the past twelve years (Figure2).
  - Households savings are high, at about 20% of GDP.
  - Investment has averaged around 40% over the past decade.
  - China accounts for 60 percent of FDI going to emerging markets (Figure1).
  - Firms’ retained earnings, government capital transfers, bank loans, and FDI finance non-financial enterprise investment.
- Fact 2: Manufacturing wages and urban-rural gap
  - Average hourly compensation of China’s manufacturing workers was 57 cents in 2002 (Banister (2005)).
  - The hourly manufacturing wage was $0.95 in cities while it was only $0.41 in rural area.
  - This is 3% of the equivalent American wage (Table2).
  - Urban-rural income gap has been rising steadily since the late 1980s (Table3).
- Fact 3: Rural surplus labor and migration frictions
  - Estimates of rural surplus labor: 100 to 200 million (Banister (2005), Brooks and Tao (2003)).
  - OECD (2002) estimates rural hidden unemployment range from 150-275 million.
  - Ministry of Agriculture estimates rural underemployment at about 150 million.
  - Alternative unemployment-based estimates yield a surplus between 15 and 85 million (in 2004 labor force ≈ 768 million), i.e. between 2% and 11% of the labor force willing to migrate.
  - Urban unemployment estimates vary: unemployment according to NBS quarterly labor force survey = 2% of total labor force in 2002; Giles, Park and Zhang (2004) estimate unemployment among urban permanent residents = 11.1 percent in September 2002.
  - Hukou registration system restricts mobility and implies migration costs that can lower effective labor mobility and lengthen the transition.

### IV. Model structure and exact core equations
- Production function (exact form preserved):
  - Y_t = F(K_t, Z_t, L_t) = a_t K_t^α Z_t^η (E_t L_t)^{1−α−η}
  - y_t = a_t k_t^α z_t^η l_t^{1−α−η} (per efficiency units)
- Consumption CES and price index (exact forms preserved):
  - C_t = [γ^{1/ρ} C_{ht}^{(ρ−1)/ρ} + (1−γ)^{1/ρ} C_{ft}^{(ρ−1)/ρ}]^{ρ/(ρ−1)}
  - P_t = [γ p_{ht}^{1−ρ} + (1−γ) p_{ft}^{1−ρ}]^{1/(1−ρ)}
  - Individual demands: C_{ht}(i) = (p_{ht}(i)/p_{ht})^{−θ} C_{ht}, C_{ft}(i) = (p_{ft}(i)/p_{ft})^{−θ} C_{ft}
- Intertemporal budget constraint (exact form preserved):
  - (1 + g)(1 + n)(k_{t+1} + z_{t+1} − d_{t+1}) = (1 + R_{kt} − δ) k_t + (1 + R_{zt} − δ) z_t − (1 + r) d_t + w_t − c_t P_t
- Asset constraint and collateral: d_t ≤ k_t and under small-open-economy r = R_k − δ; equilibrium implies d_t = k_t.
- Euler and first-order conditions (exact preserved), migration and labor law of motion:
  - b_t = Σ_{s=t}^∞ β^{∗ s} w_s = B_t / E_t with (1 + g) b_{t+1} = (1/β^{∗}) b_t − w_t
  - b̄ = Σ_{s=t}^∞ β^{∗ s} w̄ = w̄/(1−β^{∗})
  - log l_{t+1} − log l_t = χ( log b_t − log b̄ )
- Firm factor pricing and markup (exact preserved):
  - p_{ht}(i) = θ/(θ−1) mc_t
  - w_t = mc_t (1−α−η) y_t / l_t
  - R_{zt} = mc_t η y_t / z_t
  - R_{kt} = mc_t α y_t / k_t
  - π_t = (P_{ht} − mc_t) y_t
- Exogenous shocks (exact preserved):
  - a_t = a_t^{ρ_a} exp ε^a_t
  - r_t = r_t^{ρ_r} exp ε^r_t
  - y^*_t = y^*_{t−1}^{ρ_{y∗}} exp ε^{y∗}_t

### V. Calibration, shock processes, and parameters
- Preferences: U(c) = c^(1−σ) − 1 / (1−σ) with σ = 1.
- Growth and steady-state parameters:
  - g = 5.7%
  - β = 0.9972 chosen so steady state interest rate = 6%
  - θ = 6 so steady state markup = 20%
  - γ chosen so sum of imports and exports as a percentage of GDP in steady state = 54.4% (Chinese ratio in 2004, Penn World Tables)
  - ρ = 3 (elasticity of substitution between domestic and foreign consumption)
- Migration parameter χ (calibration notes and ranges):
  - Barro and Sala-i-Martin (1991) imply χ = 1/25.
  - Greenwood et al. imply χ ≈ 1/5.
  - Gallin (2004) estimates imply 1.5 ≤ χ ≤ 2 (Rappaport (2004) discussion).
  - Model exercises consider range between high (χ = 4) and low (χ = 0) labor mobility.
  - Transition exercises assume size of urban labor is 10% below eventual level once all rural labor has moved to cities.
- Shock processes and variances:
  - ρ_a = ρ_i* = ρ_y* = 0.9
  - Standard deviation of all shocks normalized to 1%

### VI. Impulse responses and qualitative mechanisms
- Complementarity between foreign and domestic capital (k and z) explains comovements between FDI flows and saving.
- Migration flows allow competitiveness to be maintained via labor adjustments (wage differential) rather than monetary policy.
- Productivity shock (Figure 3) — qualitative dynamics:
  - Temporary increase in productivity → positive and persistent effect on output.
  - Higher productivity → reduces marginal costs → depreciation of relative price of domestic good.
  - Foreign capital inflows rise to offset increased marginal product of k; foreign inflows induce rise in domestic capital via complementarity.
  - Migration amplifies responses: present value of urban–rural wage differential rises → labor flows into urban production → downward pressure on wage → faster wage adjustment → larger output movements → more investment.
- Foreign interest rate shock (Figure 4) — qualitative dynamics:
  - Fall in world interest rate → increased inflow of foreign capital → lower marginal costs → real depreciation → rise in foreign demand for domestic good.
  - Foreign capital inflows raise return to domestic capital → domestic investment rises; output, consumption, and savings increase; migration amplifies effects.
- Foreign demand shock (Figure 5) — qualitative dynamics:
  - Rise in foreign demand → firms increase prices → higher profits → redistributed into domestic investment and savings.
  - Output, consumption, and both domestic and foreign capital increase; migration amplifies impulse responses.

### VII. Simulation design and selected quantitative results
- Simulations: model simulated 5000 times for combinations of shocks (all shocks and individual shocks); all shocks normalized standard deviation = 1%.
- General findings:
  - Foreign–domestic capital complementarity and labor mobility lower variability of wages relative to output and lower wage level relative to output.
  - Relative variability of real exchange rate p_h is lower when labor mobility is higher (χ = 4), driven largely by foreign demand shocks.
  - If foreign demand shocks dominate and labor mobility is high, model predicts more stable real exchange rate.
  - Labor mobility may be more important for competitiveness than monetary policy; nominal exchange rate policy not necessary to ensure real competitiveness per the model.
- Selected numeric simulation excerpts (Table 7 excerpts; values preserved exactly):
  - All shocks:
    - χ = 0.000: ̄w = 0.982, ̄y = 0.721, σ_w = 0.972, σ_y = 0.328
    - χ = 4.000: ̄w = 0.792, ̄y = 0.347, σ_w = 0.178, σ_y = 0.058
  - Technology shock:
    - χ = 0.000: ̄w = 0.980, ̄y = 0.658, σ_w = 0.974, σ_y = 0.312
    - χ = 4.000: ̄w = 0.796, ̄y = 0.323, σ_w = 0.179, σ_y = 0.058
  - Foreign demand shock:
    - χ = 0.000: ̄w = 1.003, ̄y = 1.478, σ_w = 0.998, σ_y = 0.534
    - χ = 4.000: ̄w = 0.830, ̄y = 0.596, σ_w = 0.186, σ_y = 0.063
  - Foreign interest rate shock:
    - χ = 0.000: ̄w = 0.999, ̄y = 0.676, σ_w = 0.999, σ_y = 0.325
    - χ = 4.000: ̄w = 0.832, ̄y = 0.344, σ_w = 0.186, σ_y = 0.061

### VIII. Transition to steady state: dynamics, convergence speeds, and half lives
- Transition narrative when initial labor and domestic capital are 90 percent of steady state (l_0/l* = z_0/z* = 0.9):
  - Low initial z and l imply high wage and high rental price for z → high marginal cost → high domestic price.
  - High initial wage attracts rural labor → downward wage pressure during transition.
  - As domestic capital rises, rental price of domestic capital falls → foreign capital flows in (complementarity) → marginal costs fall → downward pressure on terms of trade.
  - Profits feed saving and investment throughout transition.
- Speed of convergence and half life (Tables 5 and 6; values preserved exactly):
  - For α = 1/3:
    - χ = 0.00: convergence = 0.04, half life = 17.78
    - χ = 2.00: convergence = 0.05, half life = 12.48
    - χ = 4.00: convergence = 0.06, half life = 10.55
  - For α = 1/2:
    - χ = 0.00: convergence = 0.06, half life = 11.16
    - χ = 2.00: convergence = 0.07, half life = 9.02
    - χ = 4.00: convergence = 0.08, half life = 8.09
- Implication: depending on labor mobility, the transition could potentially last over a decade. Openness (α) and labor mobility (χ) raise speed of convergence and lower transition half life.

### IX. Policy-relevant conclusions and implications
- Model explains qualitatively:
  1. Saving, Investment and FDI are high.
  2. Wages are low.
  3. There is a large reserve of surplus labor driving internal migration.
- Two central model features:
  - Complementarity between domestic saving and foreign investment: increasing domestic saving raises marginal product of foreign capital and attracts capital flows.
  - Wage movements largely determined by mobility of low-earning rural workers.
- Policy implications emphasized:
  - Real factors (e.g., lowering migration barriers across regions) may be critical to sustaining competitiveness and growth; nominal exchange rate manipulation is not necessary per the model to ensure competitiveness.
  - Nominal exchange rate policy is not required by the model and may be fraught with dangers (e.g., inflationary pressures).
  - The model does not claim that monetary policy had no role historically, only that if money is neutral in the long run, persistent performance cannot be explained solely by an undervalued currency.
- Limitations noted:
  - Model calibrated and simulated; none of the parameters are estimated, so quantitative assertions are limited.
  - Model is stylized and cannot explain accumulation of reserves or observed level of the exchange rate.

### X. Appendices — steady state, linear system, reserve accumulation (key expressions preserved)
- Appendix A — Steady State highlights (selected exact relations and procedural note):
  - Choose p_h first to compute all variables except y*/y, which is computed at the end.
  - Core steady-state equations include:
    - c*_h + c_h + x = y  (.41)
    - mc = (θ−1)/θ p_h  (.45)
    - y_z = 1/(η mc) (1/β* − (1−δ))  (.46)
    - y_k = (δ + i*)/(α mc)  (.47)
    - l = 1  (.52)
    - b̄ = w̄ / (1/β* − (1 + g))  (.58)
- Appendix B — Linear system (selected exact dynamic and static linearized equations):
  - ˆb_{t+1} = 1/β* ˆb_t − w̄/b̄ ˆmc_t − w̄/b̄ ˆy_t + w̄/b̄ ˆl_t  (.59)
  - ˆl_{t+1} − ˆl_t = χ ˆb_t  (.60)
  - ˆy_t − α ˆk_t = η ˆz_t + (1−α−η) ˆl_t + ρ_a ˆa_{t−1} + ˆμ_{at}  (.68)
  - Matrix representation objectives:
    - U_X X_t = U_Y Y_{t−1} + U_≤ ≤_t + U_η η_t  (.79)
    - G_O Y_t + G_0 X E_t X_{t+1} = G_1 Y_{t−1} + G_1 X X_t + G_≤ ≤_t + G_η η_t  (.80)
- Appendix C — Reserve accumulation (core modeling observation and exact modifications):
  - Introduce exogenous allocation τ_t to a foreign instrument q_t with depreciation φ:
    - q_{t+1} = (1−φ) q_t + τ_t
    - Let τ_t = s̄ y_t so only (1−s̄) is left for consumption and investment.
  - Modified current account with reserve accumulation:
    - Under original model: ca_t = −[(1 + n)(1 + g) k_{t+1} − (1−δ) k_t] < 0
    - With reserve accumulation: ca_t = −[(1 + n)(1 + g) k_{t+1} − (1−δ) k_t] + s̄ y_t
    - Example reduction when private saving fraction s used: ca_t = s̄ y_t − ((1 + n)(1 + g) k_{t+1} − (1−δ) k_t)

*Source: _wp0899 - 6.    Transition to Steady State when (IMF working paper chapter as provided).*

### References..............................................................................................................

### _wp0899 - References..............................................................................................................

### Appendix and Sections
- References.............................................................................................................................28
- Appendix...............................................................................................................................31

### Tables
- 1. National Saving Rate, 2006...........................................................................................19
- 2.     Relative Hourly Wage in Manufacturing, Selected Economies, 2002 ..........................19
- 3.    Income of Urban and Rural Households and the Urban-Rural Gap (RMB) .................20
- 4.     Summary Indicators of Saving and Investment ............................................................20
- 5.     Convergence and Transition Half Life 
3
1
=
α..............................................................27
- 6.    Convergence and Transition Half Life 
2
1
=
α
..............................................................27
- 7.    Simulation    Results.........................................................................................................27

### Figures
- 1.     Net Capital Flows into China ........................................................................................21
- 2.    Saving and Investment ..................................................................................................22
- 3.    Productivity    Shock ........................................................................................................23
- 4.    Foreign Interest Rate Schock ........................................................................................24
- 5.    Foreign Output Shock ...................................................................................................25

*Source: _wp0899 - References..............................................................................................................*

### 6.    Transition to Steady State when

### _wp0899 - 6.    Transition to Steady State when

### I. Introduction — research objective and core mechanisms
- Research objective: propose a simple neoclassical real growth model (no monetary/exchange-rate channel) that captures key real features of the Chinese economy and assesses implications for transition to steady state.
- Two crucial features incorporated:
  - Imperfect capital mobility with two complementary capital types:
    - Domestic capital Z (must be financed by domestic saving).
    - Foreign/tradable capital K (borrowable on international markets at a fixed rate).
    - Production uses both types as complements: raising domestic capital increases the marginal product of foreign capital and attracts FDI.
  - Surplus rural labor and migration into the urban production sector; speed of migration depends on labor mobility parameter χ.
- Key insight: an increase in domestic capital raises marginal products, attracting rural labor and foreign capital until marginal products return to initial levels — explaining coexistence of high saving, large FDI inflows, high investment, low wages, and competitive goods prices.
- Transition implication: if substantial surplus labor remains, the transition to steady state could last an additional decade.

### Key calibrated stylized facts motivating the model (quoted exactly)
- per capita growth has averaged 8.5 percent over the last decade;
- investment has averaged over 36 percent of GDP since 1996;
- the national saving rate (with both households and entreprises contributing) has been extraordinarily high by any standard of comparison — in 2006, it was 54 percent (Table1);
- notwithstanding the high national saving rate, there has been a significant inflow of foreign direct investment (Figure1);
- wages in manufacturing have been very low by any international standard (Table2) and rural incomes have been even lower than urban incomes — by a considerable margin (Table3).

### Interpretive observation on monetary/exchange-rate policy
- Paper is largely silent on reserve accumulation unrelated to FDI flows.
- Claim: an appreciation of the currency now would not change the underlying real phenomena produced by imperfect capital mobility and excess labor supply; but policies that restrain an equilibrating real appreciation may trigger a serious inflationary spiral.

### Compact summary of competing explanations in literature (preserved)
- Explanations for high saving: life-cycle/retirement effects, one-child policy substitution of financial saving for children, aging offset by higher share of high savers, precautionary saving due to lack of social insurance/health/government spending, credit constraints, distorted financial incentives encouraging low consumption/high investment, uncertainty from transition to market economy, lack of international portfolio diversification.
- Explanations for FDI: tax/legal policy, open economic zones and infrastructure, preferential bank credit to SOEs pushing private firms to joint ventures with foreigners.

---

### II. Stylized facts (section II) — enumerated empirical features
- Fact 1: Saving, Investment and FDI are high
  - China’s saving rate is substantially higher than average; around 40 percent for the past twelve years (Figure2).
  - Households savings are high, at about 20% of GDP.
  - Investment has averaged around 40% over the past decade.
  - China accounts for 60 percent of FDI going to emerging markets (Figure1).
  - Firms’ retained earnings, government capital transfers, bank loans, and FDI finance non-financial enterprise investment.
- Fact 2: Manufacturing wages are low relative to international levels but high relative to rural levels
  - Average hourly compensation of China’s manufacturing workers was 57 cents in 2002 (Banister (2005)).
  - The hourly manufacturing wage was $0.95 in cities while it was only $0.41 in rural area.
  - This is 3% of the equivalent American wage (Table2).
  - Urban-rural income gap has been rising steadily since the late 1980s (Table3).
- Fact 3: China’s agricultural sector has a large reserve of surplus labor which drives internal migration
  - Estimates of rural surplus labor: 100 to 200 million (Banister (2005), Brooks and Tao (2003)).
  - OECD (2002) estimates rural hidden unemployment range from 150-275 million.
  - Ministry of Agriculture estimates rural underemployment at about 150 million.
  - Alternative unemployment-based estimates yield a surplus between 15 and 85 million (in 2004 labor force ≈ 768 million), i.e. between 2% and 11% of the labor force willing to migrate.
  - Urban unemployment estimates vary: unemployment according to NBS quarterly labor force survey = 2% of total labor force in 2002; Giles, Park and Zhang (2004) estimate unemployment among urban permanent residents = 11.1 percent in September 2002.
  - Legal constraints on migration: Hukou registration system restricts mobility and implies migration costs that can lower effective labor mobility and lengthen the transition.

---

### III. The model — structure and key equations
- Economy: small open urban economy with identical infinitely-lived households; monopolistically competitive domestic firms; three inputs to production: labor L, domestic capital Z, foreign capital K.
- Production function (exact form preserved):
  - Y_t = F(K_t, Z_t, L_t) = a_t K_t^α Z_t^η (E_t L_t)^{1−α−η}
    - where 0< α <1, 0< η <1, E_t is labor efficiency growing at deterministic rate g, and a_t is temporary technology shock.
  - Aggregate output in efficiency units: y_t = a_t k_t^α z_t^η l_t^{1−α−η} with lowercase denoting units per efficiency.
- Interpretations of Z and K include human capital vs physical capital, non-tradable vs tradable sector capital, or informal vs formal sector capital.
- Capital mobility assumption: K is borrowable internationally; Z cannot be borrowed internationally (credit constraint). Both capital types and labor are necessary and complementary.
- Household decisions:
  - Intratemporal consumption: CES aggregate between domestic and foreign goods:
    - C_t = [γ^{1/ρ} C_{ht}^{(ρ−1)/ρ} + (1−γ)^{1/ρ} C_{ft}^{(ρ−1)/ρ}]^{ρ/(ρ−1)} with 0<γ<1 and ρ elasticity of substitution.
    - Price index: P_t = [γ p_{ht}^{1−ρ} + (1−γ) p_{ft}^{1−ρ}]^{1/(1−ρ)}.
    - Individual demands: C_{ht}(i) = (p_{ht}(i)/p_{ht})^{−θ} C_{ht}, C_{ft}(i) = (p_{ft}(i)/p_{ft})^{−θ} C_{ft}.
  - Intertemporal budget constraint (exact form preserved):
    - (1 + g)(1 + n)(k_{t+1} + z_{t+1} − d_{t+1}) = (1 + R_{kt} − δ) k_t + (1 + R_{zt} − δ) z_t − (1 + r) d_t + w_t − c_t P_t
    - notation: R_{zt}, R_{kt} rental prices; r constant world interest rate on debt; δ depreciation; n population growth.
  - Asset constraint and collateral: debt cannot exceed foreign capital (d_t ≤ k_t). Under small-open-economy assumption r = R_k − δ; equilibrium implies d_t = k_t.
  - Euler and first-order conditions:
    - U_{c_t} = β (1 + n)(1 + g) E_t[P_t/P_{t+1} U_{c_{t+1}} (1−δ + R_{z,t+1}/P_{t+1})]
    - R_{kt} − δ = r_t
    - P_t c_t + x_t = w_t + R_{zt} z_t + π_t
    - x_t = (1 + n)(1 + g) z_{t+1} − (1−δ) z_t
- Migration decision and law of motion for labor (exact forms preserved):
  - Present value of urban wage income: b_t = Σ_{s=t}^∞ β^{∗ s} w_s = B_t / E_t with law (1 + g) b_{t+1} = (1/β^{∗}) b_t − w_t.
  - Rural present value: b̄ = Σ_{s=t}^∞ β^{∗ s} w̄ = w̄/(1−β^{∗}).
  - Labor entry/exit rate proportional to log difference:
    - log l_{t+1} − log l_t = χ( log b_t − log b̄ ) where χ measures labor mobility.
- Firms:
  - Monopolistic competition with constant markup:
    - p_{ht}(i) = θ/(θ−1) mc_t.
  - Profit-maximizing factor payments equal marginal products:
    - w_t = mc_t (1−α−η) y_t / l_t
    - R_{zt} = mc_t η y_t / z_t
    - R_{kt} = mc_t α y_t / k_t
  - Profits: π_t = (P_{ht} − mc_t) y_t
- Exogenous shocks (exactly preserved):
  - a_t = a_t^{ρ_a} exp ε^a_t
  - r_t = r_t^{ρ_r} exp ε^r_t
  - y^*_t = y^*_{t−1}^{ρ_{y∗}} exp ε^{y∗}_t
- Equilibrium defined as a sequence of quantities {y_t, x_t, mc_t, c_{ht}, c^*_{ht}, c_{ft}, k_t, b_t, c_t, z_{t+1}, l_{t+1}, a_t, y^*_t} and prices {P_t, p_{ht}, r_t, R_{zt}, R_{kt}, w_t} satisfying household, firm, migration, and market conditions.

---

### IV. Model implications, experiments and transition dynamics (qualitative summary)
- Model can explain coexistence of:
  - High national saving and high investment and yet continued FDI inflows.
  - Low manufacturing wages (internationally) alongside higher urban wages relative to rural wages.
  - Competitively priced domestic goods on world markets due to domestic monopoly pricing with low marginal costs.
- Responses to shocks (illustrative experiments summarized qualitatively):
  - Productivity, foreign demand, and interest rate shocks affect wages, capital inflows, and migration; higher labor mobility reduces variance of wages and the real exchange rate, especially for foreign-demand shocks.
- Transition to steady state:
  - Characterization of transition from initial conditions with substantial surplus labor (close to current Chinese observations).
  - If substantial surplus labor persists, transition could last an additional decade (statement preserved from calibration/simulation results).
  - Labor mobility parameter χ and migration costs (Hukou system) play a crucial role in lengthening the transition.
- Limitation: model is stylized and not intended as a full quantitative calibration of the Chinese economy; it provides plausible channels and qualitative predictions.

---

*Italicized attribution: Source: _wp0899 - 6.    Transition to Steady State when (IMF working paper chapter as provided).*

### 1. Domestic and foreign household decisions are optimal, i.e. equations (2), (  3), (  4), (  5), (  6), (  7),

### _wp0899 - 1. Domestic and foreign household decisions are optimal, i.e. equations (2), (  3), (  4), (  5), (  6), (  7),

### Equilibrium conditions and model equations
- Domestic and foreign household optimality: equations (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), and (18) hold.
- Firm optimality: equations (19), (20), (21), and (22) hold.
- Market clearing: equations (12) and (14) hold and
  - P_t c_t + x_t = ( (θ/(θ−1)) − α ) mc_t y_t  (equation (26))
  - c*_ht + c_ht + x_t = y_t  (equation (27))
- Numeraire: p_f is the numeraire. Equilibrium conditions include:
  - P_t = ( γ p_ht^(1−%ht) + (1−γ) )^(1/(1−%) )  (equation (24))
  - p_ht = θ/(θ−1) mc_t  (equation (25))
  - c_t = ( γ^(1/ρ) c_ht^(ρ−1)/ρ + (1−γ)^(1/ρ) c_ft^(ρ−1)/ρ )^(ρ/(ρ−1))  (equation (28))
  - c_ht = γ (p_ht / P_t)^−ρ c_t  (equation (29))
  - c_ft = (1−γ) (1 / P_t)^−ρ c_t  (equation (30))
  - c*_ht = p_ht^−ρ y*_t  (equation (31))
  - y_t = a_t k_t^α z_t^η  (equation (32))
  - mc_t α y_t / k_t − δ = i*_t  (equation (33))
  - x_t = (1 + g) z_t+1 − (1−δ) z_t  (equation (34))
  - U_c_t = β* E_t [ P_t / P_t+1 U_c_t+1 (1−δ + mc_t+1 η y_t+1 / z_t+1) ]  (equation (35))
  - log l_t+1 − log l_t = χ ( log b_t − log ̄b )  (equation (36))
  - (1 + g) b_t+1 = 1/β* b_t − ( mc_t (1−α−η) y_t l_t )  (equation (37))
  - log a_t = ρ_a log a_t−1 + μ_a_t  (equation (38))
  - log i*_t = ρ_i* log i*_t−1 + μ_i*_t  (equation (39))
  - log y*_t = ρ_y* log y*_t−1 + μ_y*_t  (equation (40))
- β* = β (1+g)(1+n)

### Calibration
- Solution method: log-linear approximation around a balanced-trade steady state.
- Preferences: U(c) = c^(1−σ) − 1 / (1−σ) with σ = 1.
- Growth and steady-state parameters:
  - Growth rate of technological progress set at g = 5.7%.
  - β = 0.9972 chosen so steady state interest rate = 6%.
  - θ = 6 so steady state markup = 20%.
  - γ chosen so sum of imports and exports as a percentage of GDP in steady state = 54.4% (Chinese ratio in 2004, Penn World Tables).
  - Elasticity of substitution between domestic and foreign consumption ρ = 3.
- Migration parameter χ:
  - Barro and Sala-i-Martin (1991) imply χ = 1/25.
  - Greenwood et al. imply χ ≈ 1/5.
  - Gallin (2004) estimates imply 1.5 ≤ χ ≤ 2 (Rappaport (2004) discussion).
  - Model exercises consider range between high (χ = 4) and low (χ = 0) labor mobility.
  - Transition exercises assume size of urban labor is 10% below eventual level once all rural labor has moved to cities.
- Shock processes and variances:
  - Persistence: ρ_a = ρ_i* = ρ_y* = 0.9.
  - Standard deviation of all shocks normalized to 1%.

### Impulse Response Functions — qualitative dynamics and mechanisms
- Key mechanisms highlighted:
  - Complementarity between foreign and domestic capital (k and z) explains comovements between FDI flows and saving in China.
  - Migration flows allow the economy to remain competitive (low real wages or competitive terms of trade) without relying on monetary policy.
- Productivity shock (Figure 3):
  - Temporary increase in productivity → positive and persistent effect on output.
  - Higher productivity → reduces marginal costs → depreciation of relative price of domestic good (terms of trade).
  - With world interest rate unchanged, inflows of foreign capital rise to offset increased marginal product of k.
  - Foreign capital inflows induce rise in domestic capital investment due to complementarity.
  - Higher profits and savings reinforce investments.
  - Migration amplifies responses: present value of urban–rural wage differential rises → labor flows into urban production → downward pressure on wage → faster wage adjustment → larger output movements → more investment.
- Foreign interest rate shock (Figure 4):
  - Fall in world interest rate → increased inflow of foreign capital to close marginal product gap.
  - Lower foreign capital cost → reduces marginal costs → real depreciation → rise in foreign demand for domestic good.
  - Foreign capital inflows raise return to domestic capital → domestic investment rises via higher profits.
  - Output, consumption, and savings increase; migration amplifies these effects.
- Foreign demand shock (Figure 5):
  - Rise in foreign demand → upward pressure on terms of trade → firms increase prices → higher profits → redistributed into domestic investment and savings.
  - Output, consumption, and both domestic and foreign capital increase.
  - Migration amplifies impulse responses.

### Simulation results and statistical implications
- Simulations: model simulated 5000 times for combinations of shocks (all shocks and individual shocks); all shocks normalized standard deviation = 1%.
- General findings:
  - Foreign–domestic capital complementarity and labor mobility lower variability of wages relative to output and lower wage level relative to output.
  - When subjected to all shocks, relative variability of real exchange rate p_h is lower when labor mobility is higher (χ = 4); driven largely by foreign demand shocks.
  - If foreign demand shocks dominate and labor mobility is high, model predicts more stable real exchange rate.
  - Labor mobility may be more important for competitiveness than monetary policy; nominal exchange rate policy not necessary to ensure real competitiveness per the model.
- Simulation tables and selected numeric results (Table 7 excerpts):
  - All shocks:
    - χ = 0.000: ̄w = 0.982, ̄y = 0.721, σ_w = 0.972, σ_y = 0.328
    - χ = 4.000: ̄w = 0.792, ̄y = 0.347, σ_w = 0.178, σ_y = 0.058
  - Technology shock:
    - χ = 0.000: ̄w = 0.980, ̄y = 0.658, σ_w = 0.974, σ_y = 0.312
    - χ = 4.000: ̄w = 0.796, ̄y = 0.323, σ_w = 0.179, σ_y = 0.058
  - Foreign demand shock:
    - χ = 0.000: ̄w = 1.003, ̄y = 1.478, σ_w = 0.998, σ_y = 0.534
    - χ = 4.000: ̄w = 0.830, ̄y = 0.596, σ_w = 0.186, σ_y = 0.063
  - Foreign interest rate shock:
    - χ = 0.000: ̄w = 0.999, ̄y = 0.676, σ_w = 0.999, σ_y = 0.325
    - χ = 4.000: ̄w = 0.832, ̄y = 0.344, σ_w = 0.186, σ_y = 0.061

### Transition to steady state
- Key assumptions driving transition dynamics:
  - Existence of mobile surplus rural labor.
  - Complementarity of saving and investment (domestic and foreign capital complements).
- Dynamics when initial labor and domestic capital are 90 percent of steady state (l_0/l* = z_0/z* = 0.9; Figure 6):
  - Low initial z and l imply high wage and high rental price for z → high marginal cost → high domestic price.
  - High initial wage attracts rural labor → downward wage pressure during transition.
  - As domestic capital rises, rental price of domestic capital falls → foreign capital flows in (complementarity) → marginal costs fall → downward pressure on terms of trade.
  - Profits feed saving and investment throughout transition.
- Speed of convergence and half life (Tables 5 and 6):
  - For α = 1/3:
    - χ = 0.00: convergence = 0.04, half life = 17.78
    - χ = 2.00: convergence = 0.05, half life = 12.48
    - χ = 4.00: convergence = 0.06, half life = 10.55
  - For α = 1/2:
    - χ = 0.00: convergence = 0.06, half life = 11.16
    - χ = 2.00: convergence = 0.07, half life = 9.02
    - χ = 4.00: convergence = 0.08, half life = 8.09
- Implication: depending on labor mobility, the transition could potentially last over a decade. Openness (α) and labor mobility (χ) raise speed of convergence and lower transition half life.

### Policy-relevant conclusions and implications
- The model offers qualitative explanations for three stylized facts of the Chinese economy:
  1. Saving, Investment and FDI are high.
  2. Wages are low.
  3. There is a large reserve of surplus labor driving internal migration.
- Two central model features account for these facts:
  - Complementarity between domestic saving and foreign investment: increasing domestic saving raises marginal product of foreign capital and attracts capital flows.
  - Wage movements largely determined by mobility of low-earning rural workers.
- Policy implications emphasized by the model:
  - Real factors (e.g., lowering migration barriers across regions) may be critical to sustaining competitiveness and growth; nominal exchange rate manipulation is not necessary per the model to ensure competitiveness.
  - Nominal exchange rate policy is not required by the model and may be fraught with dangers (e.g., inflationary pressures).
  - The model does not claim that monetary policy had no role historically, only that if money is neutral in the long run, persistent performance cannot be explained solely by an undervalued currency.
- Limitations:
  - Model calibrated and simulated; none of the parameters are estimated, so quantitative assertions are limited.
  - Model is stylized and cannot explain accumulation of reserves or observed level of the exchange rate.

*Source: _wp0899 - 1. Domestic and foreign household decisions are optimal, i.e. equations (2), (  3), (  4), (  5), (  6), (  7), (PDF chapter/section).*

### REFERENCES

### _wp0899 - REFERENCES

### References list
- Bibliographic citations included (selected authors and works as listed): Au and Henderson (2002) Working Paper 8707; Appleton, Knight, Song, Xia (2002) China Economic Review13, pp. 252-275; Aziz (2006) IMF WP06-291; Banister (2005) Beijing Javelin Investment Consulting Company and Monthly Labor Review; Banister and Taylor (1989) Asia-Pacific Population Journal, 4, 3–20; Barnett and Brooks (2006) IMF WP06-265; Barro and Sala-i-Martin (1991) Brookings Papers on Economic Activity, 1991, 107–182; Barro, Mankiw, Sala-i-Martin (1995) American Economic Review, 85, 103–115; Blanchard and Giavazzi (2005) MIT WP05-32; Brooks and Tao (2003) IMF WP03/210; Bernanke and Gürkaynak (2001) NBER Macroeconomics Annual, 16, 11–57; Braun (1993) Ph.D. thesis, Harvard University; Chamon and Prasad (2005) IMF, mimeo; Dooley, Folkerts-Landau, Garber (2004a) NBER Working Paper 10626; Dooley, Folkerts-Landau, Garber (2004b) NBER Working Paper 10332; Easterly (1993) Journal of Monetary Economics, 32, 187–212; Eichengreen (2004) CEPR Working Paper 4375; Gallin (2004) Journal of Labor Economics, 22, 1–21; Goldstein and Lardy (2005) IIE Working Paper WP 05-2; Knight and Xue (2004) Journal of Chinese Economic and Business Studies4, pp. 91-107; Kuijs (2006) World Bank Policy Research Working Paper3958; Lane (2001) Oxford Economic Papers, 53, 221–240; McKinnon (2007) MIT Press; Modigliani and Shi Cao (2004) Journal of Economic Literature42(1), pp. 145-170; Mundell (2004) speech at IMF/People’s Bank of China High-Level Seminar; Prasad and Rajan (2006) IMF Policy Discussion Paper 06/3; Prasad and Wei (2005) NBER WP11306; Rappaport (2004) RWP 99-12, Federal Reserve Bank of Kansas City; Tseng and Zebregs (2002) IMF Policy Discussion Paper02/3; Verdier (2008) Journal of International Economics74, pp. 120-142.

### Appendix A — Steady State
- Key statement: To determine the steady state one must choose the ratio y*/y or the relative price p_h. Example choice: p_h = 1 implies balanced trade in the steady state.
- Core steady-state equations (preserving original numbering and notation):
  - c*_h + c_h + x = y  (.41)
  - p_h^−ρ y* + γ (p_h / P)^−ρ c + x = y  (.42)
  - p_h^−ρ y* = y − γ (p_h / P)^−ρ c − x  (.43)
  - y*/y = p_h^ρ − γ (1 / P)^−ρ c y − p_h^ρ x / y  (.44)
  - mc = (θ−1)/θ p_h  (.45)
  - y_z = 1/(η mc) (1/β* − (1−δ))  (.46)
  - y_k = (δ + i*)/(α mc)  (.47)
  - x/y = ((1 + n)(1 + g) − (1−δ)) z/y  (.48)
  - c/y = mc/P (θ/(θ−1) − α) − x/Py  (.49)
  - c_h/y = γ (p_h / P)^−ρ c y  (.50)
  - c_f/y = (1−γ) (1 / P_t)^−ρ c y  (.51)
  - l = 1  (.52)
- Notes on computing levels (preserving algebraic expressions and numbering):
  - y = k^α z^η l^(1−α−η) = 1  (level relationships and equations .53, .54)
  - w = w̄  (.55)
  - w̄ = mc(1−α−η)y  (.56)
  - b̄ = w̄ / (1/β* − (1 + g))  (.58)
- Procedural note: Choose p_h first to compute all variables except y*/y, which is computed at the end.

### Appendix B — Linear system
- Additional dynamic equations:
  - ˆb_{t+1} = 1/β* ˆb_t − w̄/b̄ ˆmc_t − w̄/b̄ ˆy_t + w̄/b̄ ˆl_t  (.59)
  - ˆl_{t+1} − ˆl_t = χ ˆb_t  (.60)
- Static equations (selected, preserving numbering):
  - ˆP_t − γ p^{1−ρ}_h ˆp_ht = 0  (.61)
  - ˆp_ht − ˆmc_t = 0  (.62)
  - Pc/(Pc+x) ˆP_t + Pc/(Pc+x) ˆc_t + x/(Pc+x) ˆx_t − ˆmc_t − ˆy_t = 0  (.63)
  - c*_h/y ˆc*_ht + c_h/y ˆc_ht + x/y ˆx_t − ˆy_t = 0  (.64)
  - ˆc_ht + ρ ˆp_ht − ρ ˆP_t − ˆc_t = 0  (.65)
  - ˆc_ft − ρ ˆP_t − ˆc_t = 0  (.66)
  - ˆc*_ht + ρ ˆp_ht = ρ_{y*} ˆy*_{t−1} + ˆμ_{y*t}  (.67)
  - ˆy_t − α ˆk_t = η ˆz_t + (1−α−η) ˆl_t + ρ_a ˆa_{t−1} + ˆμ_{at}  (.68)
  - (δ + i*)/i* ˆmc_t + (δ + i*)/i* ˆy_t − (δ + i*)/i* ˆk_t = ˆi*_t  (.69)
  - ˆb_t = E_{t−1} ˆb_t + η^b_t  (.70)
  - ˆc_t = E_{t−1} ˆc_t + η^c_t  (.71)
- Dynamic system (selected):
  - (1 + n)(1 + g)/z x ˆz_{t+1} = ˆx_t + (1−δ)/z x ˆz_t  (.72)
  - ˆb_{t+1} = 1/β* ˆb_t − w̄/b̄ ˆmc_t − w̄/b̄ ˆy_t + w̄/b̄ ˆl_t  (.73)
  - ˆl_{t+1} = χ ˆb_t + ˆl_t  (.74)
  - E_t ˆP_{t+1} + σ E_t ˆc_{t+1} − [1−β*(1−δ)]E_t ˆmc_{t+1} − [1−β*(1−δ)]E_t ˆy_{t+1} + [1−β*(1−δ)]E_t ˆz_{t+1} = σ ˆc_t + ˆP_t  (.75)
  - ˆa_t = ρ_a ˆa_{t−1} + ˆμ_{at}  (.76)
  - ˆi*_t = ρ_{i*} ˆi*_{t−1} + ˆμ_{i*}  (.77)
  - ˆy*_t = ρ_{y*} ˆy*_{t−1} + ˆμ_{y*}  (.78)
- Matrix representation objectives:
  - U_X X_t = U_Y Y_{t−1} + U_≤ ≤_t + U_η η_t  (.79)
  - G_O Y_t + G_0 X E_t X_{t+1} = G_1 Y_{t−1} + G_1 X X_t + G_≤ ≤_t + G_η η_t  (.80)
- State vectors (as defined):
  - X_t = (ˆy_t, ˆx_t, ˆmc_t, ˆc_ht, ˆc*_ht, ˆc_ft, ˆk_t, ˆp_ht, ˆP_t, ˆb_t, ˆc_t)
  - Y_t = (ˆz_{t+1}, ˆl_{t+1}, ˆa_t, ˆy*_t, ˆi*_t, E_t ˆb_{t+1}, E_t ˆc_{t+1})
  - ≤_t = (ˆμ_A t, ˆμ_{i* t}, ˆμ_{Y* t})
  - η_t = (η^b_t, η^c_t)

### Appendix C — Reserve accumulation
- Core modeling observation:
  - The model predicts China as a net recipient of capital flows (FDI, portfolio, debt) — implying net capital inflows — but does not by itself model the current account or the motivations for reserve accumulation.
  - The model can be made technically consistent with an observed current account surplus by introducing exogenous allocation τ_t to a foreign instrument q_t with depreciation of that instrument at rate φ per period.
- Key equations and definitions (preserving exact expressions):
  - q_{t+1} = (1−φ) q_t + τ_t
  - Let τ_t = s̄ y_t so only (1−s̄) is left for consumption and investment.
- Modified household problem (objective and constraints) — preserving constraint structure:
  - Max_{c_t, z_{t+1}, d_{t+1}, k_{t+1}} Σ_{t=0}^∞ β^t U(c_t)
  - Budget and sectoral constraints (as listed):
    - P_t c_t + x_t + (1 + g)(1 + n)k_{t+1} − (1−δ)k_t − (1 + g)(1 + n)d_{t+1} + d_t + τ_t = (1−s̄) (w_t + R_{z t} z_t + R_{k t} k_t) − r_t d_t + π_t
    - x_t = (1 + n)(1 + g) z_{t+1} − (1−δ) z_t
    - d_t ≤ k_t
    - τ_t = s̄ (w_t + R_{z t} z_t + R_{k t} k_t)
    - q_{t+1} = (1−φ) q_t + τ_t
- Implications for the current account:
  - Under the original model: ca_t = −[(1 + n)(1 + g) k_{t+1} − (1−δ) k_t] < 0
  - Under the modified model with reserve accumulation: ca_t = −[(1 + n)(1 + g) k_{t+1} − (1−δ) k_t] + s̄ y_t, which can be positive.
  - Example with fixed fraction private saving s:
    - ca_t = s(1−s̄) y_t + s̄ y_t − x_t − ((1 + n)(1 + g) k_{t+1} − (1−δ) k_t)
    - Since s(1−s̄) y_t = x_t = (1 + n)(1 + g) z_{t+1} − (1−δ) z_t (private saving used for domestic capital accumulation), the current account reduces to:
      - ca_t = s̄ y_t − ((1 + n)(1 + g) k_{t+1} − (1−δ) k_t)

*Content unit: _wp0899 - REFERENCES*

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