## _wp06178

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### I. Introduction — puzzles, approach, and key mechanisms
- Background:
  - Cross-border capital flows rose to nearly $6 trillion in 2004, but less than 10 percent of them go to developing countries.
  - Two opposing theoretical puzzles:
    - Lucas (1990) paradox: “too small” capital flows in one-sector models where marginal returns to capital are implied to be much higher in poor countries.
    - Opposite paradox in two-sector neoclassical (Heckscher-Ohlin-Samuelson) models: factor price equalization (FPE) implies factor returns are pinned by goods prices, so observed capital flows would be excessive.
- Core modeling innovation:
  - Embed Holmström–Tirole (1998) style financial contracts and heterogeneous entrepreneurs into a two-good, two-factor, two-country HOS framework.
  - Key micro-foundations:
    - Return to financial investment can differ from marginal product of physical capital because financial investors must share returns with entrepreneurs.
    - Financial development governs the slice of returns allocated to financial investors: more developed financial systems allocate a larger slice to financial investors.
    - Entrepreneur heterogeneity in management ability generates endogenous sector-level decreasing returns to scale (marginal entrepreneurs are less able), while maintaining constant returns at the firm level.
- Principal implications:
  - Goods-price equalization does not imply factor-price equalization: interest rates and wages can differ across countries even under free trade in goods.
  - Model reconciles one-sector intuition (higher marginal product of capital in poor, capital-scarce countries) with small net capital flows and the possibility of two-way gross flows (financial capital outflows and inward FDI).
  - Institutions matter differently:
    - Financial development: inefficient financial systems can be bypassed (savings flow out as financial capital while FDI flows in).
    - Property rights / expropriation risk: high expropriation risk deters FDI; poor property rights can create financial capital outflows without compensating FDI inflows.
  - Robust result: wage rate is always higher in the country with better financial or property rights institutions, irrespective of initial endowments.

### II. Two paradoxes revisited and why FPE can be broken
- One-sector (Lucas) result summary:
  - Diminishing marginal product of capital implies much higher r in capital-scarce countries (Lucas calculation: India vs United States scale stated as an illustrative paradox).
- Two-sector HOS setup and factor price equalization:
  - Zero-profit conditions yield p1 = c1(w; r) and p2 = c2(w; r); for given product prices, factor prices (w; r) are determined and independent of endowments (FPE).
  - Lemma: with m factors and countries linked via production of common sets of m products, factor prices equalize across linked countries under free trade.
- Limits of prior escape routes:
  - Labor “differentiation”, adding human capital, or sovereign risk do not escape the chain-rule implications of FPE in multi-product linkages unless additional mechanisms alter how factor prices depend on variables beyond goods prices.
- Key insight motivating the model:
  - Marginal product of physical capital need not equal return to financial investment when financial contracts and entrepreneur incentives determine revenue shares. This distinction provides a plausible channel to break FPE without imposing firm-level DRS.

### III. Model structure and contract-level results
- Timing and technology:
  - Two-period production; sector i first-period production y1_i = G_i(L1_i; K1_i) with fixed labor-capital ratio a_i = L1_i/K1_i.
  - At date 2 a liquidity shock requires additional financing equal to ζ_i K1_i; ζ_i has distribution F_i(ζ) with density f_i(ζ).
- Moral hazard and continuation:
  - Entrepreneur exerts effort e ∈ {e_H, e_L}; θ ≡ θ(e_H) (normalized) is success probability when working; θ(e_L) = 0.
  - Project continues only if continuation payment covers ζ_i K1_i; first-best cutoff ζ1_i = θ R_i.
- Financial contracts and participation:
  - Contracts specify initial total investment K1_ni and continuation policy μ_ni(ζ_i); entrepreneur revenue per unit is R_E_ni(ζ_i).
  - Incentive compatibility with effort implies R_E_ni = c_ni(e_H)/θ and ζ_max_ni = θ R_i − c_ni(e_H).
  - Participation constraint yields initial investment formula:
    - K1_ni(:) = (1 + r)/((1 + r) − ∫_0^{bζ_ni} (ζ_max_ni − ζ_i) f_i(ζ_i) dζ_i).
  - Net return to internal capital when continuation cutoff is bζ_ni:
    - U_ni(bζ_ni) = [θ R_i − h(bζ_ni)]/[h(bζ_ni) − ζ_max_ni], where h_i(bζ_ni) = (1 + r) + ∫_0^{bζ_ni} ζ_i f_i(ζ_i) dζ_i / F_i(bζ_ni) is the expected unit cost of total investment.
- Financial development parameter ϑ (notation in source):
  - Only shocks up to bζ ≤ ϑ ζ_opt can be met by the financial system. Set bζ_ni = ϑ ζ_opt.
  - For uniform f(ζ) on [0; ζ], ζ_opt = [2 (1 + r) / ζ]^{1/2}.
  - h(r; ϑ) is decreasing in ϑ (∂h/∂ϑ < 0) and increasing in r (∂h/∂r > 0).
- Entrepreneur heterogeneity and sector allocation:
  - Sector 1 entrepreneurs heterogeneous in cost c_n1 = c1 n (ranked by n). U_n1 decreases in n; marginal entrepreneur N1 defined by U_{N1}^1 = U2.
  - Free entry yields capital revenue-sharing conditions (rewritten):
    - θ R1 = h(r; ϑ) + [f c1_{N1}]/(1 + f)
    - θ R2 = h(r; ϑ) + [f c2]/(1 + f)
  - Lemma 2: as r increases fewer capitalists choose entrepreneurship; more productive entrepreneurs enter the heterogeneous sector and manage more capital.

### IV. Aggregation, equilibrium conditions, and comparative statics
- Aggregation and market clearing:
  - Full employment and capital market clearing produce a system linking (w; r; p; N1; N2).
  - Relative supply condition yields product relative supply y1/y2 = D(p) (equation (33) form).
- Determination of factor prices (Proposition 1 — “Stolper-Samuelson plus”):
  - Curves z_i in (w; r) space are convex and downward sloping; a1 < a2 (sector 2 more labor intensive).
  - Comparative statics (exact statements preserved):
    - Decrease in price of a good decreases return to factor used intensively in that good and increases return to the other factor.
    - Increase in number of entrepreneurs in the heterogeneous sector decreases return to factor used intensively there and increases return to the other factor.
    - Improvement in financial development increases interest rate but has no effect on wage.
    - If the highest entrepreneur effort cost in the heterogeneous sector > homogeneous sector, lower expropriation risk increases interest rate but reduces wage.
  - Corollary 1 (factor price responses):
    - Increase in capital-labor ratio reduces interest rate but raises wage.
    - Improvement in financial system raises interest rate but leaves wage unchanged.
    - Reduction in expropriation risk raises interest rate but reduces wage.
- Rybczynski-type results (Proposition 2):
  - Under modified nonreversal (sector1 remains capital-intensive):
    - Increase in capital endowment increases N1 and decreases relative price of good1.
    - Improvement in financial development raises outputs in both sectors proportionally, leaving N_i and relative price unchanged.
    - If highest effort cost in heterogeneous sector > homogeneous sector, lower expropriation risk raises output but reduces number of entrepreneurs proportionally and does not change relative product prices.
- Important qualitative point:
  - Two-sector heterogeneity and financial-contract frictions allow a capital-abundant country to have a lower return to financial capital (r) than a capital-scarce country — consistent with one-sector intuition but with much smaller implied differentials.

### V. Free trade in goods and international capital flows
- Setup: two countries identical in tastes, technologies, shock distributions and manager behavior, but differ in L, K, ϑ, θ; labor immobile.
- Goods-only free trade (no capital flows):
  - Financial development and expropriation risk do not affect relative product prices; standard Heckscher-Ohlin result holds (Proposition 3): each country exports the good intensive in its abundant factor.
- Financial capital mobility only (no FDI):
  - Interest rate difference br = A_L b_L − A_K b_K + A_ϑ b_ϑ + A_θ b_θ with A_L, A_K, A_ϑ, A_θ > 0.
  - Proposition 4 (three polar cases):
    - If identical ϑ and θ but different endowments: financial capital flows from capital-abundant into labor-abundant country.
    - If identical K/L and θ but different ϑ: financial capital flows from less-developed financial system to more-developed one.
    - If identical K/L and ϑ but different θ: financial capital flows from higher expropriation risk to lower expropriation risk country.
- FDI allowed, entrepreneurs mobile (no international financial capital flows):
  - Entrepreneurs engage in FDI when net returns abroad exceed domestic returns; with identical θ and no financial capital flows, entrepreneurs from capital-abundant country undertake outbound FDI to exploit lower labor cost abroad (Proposition 5).
- Both financial capital and FDI mobile — “complete bypass” equilibrium:
  - With identical expropriation risk, perfectly mobile entrepreneurs and identical populations, unique equilibrium can be “complete capital bypass circulation” (Proposition 6):
    - All financial capital owned by investors in the country with the less-developed financial system leaves as financial capital outflow.
    - Physical capital/projects reenter as FDI.
    - The less-developed financial system serves no capital in equilibrium; the capital-abundant country can suffer net capital outflow (trade surplus).
  - Magnitude considerations: interest rate differentials depend on market imperfections f, effort cost parameters (c1), and heterogeneity c_n1(:). Small frictions can explain small interest differentials despite large K/L differences — addressing the Lucas paradox quantitatively without invoking extreme wedges.
- Expropriation risk and relocation costs:
  - When expropriation risk differs and entrepreneurs incur relocation cost d, corner or interior solutions arise depending on whether U_domestic ≤ U_foreign − d; expropriation risk can block FDI even when financial capital leaves.

### VI. Institutions, patterns of gross and net capital flows, and welfare notes
- Principal theoretical result (Proposition 7):
  - With free mobility of capital and diversified equilibria, the wage rate is higher in the country with a more efficient financial system (higher ϑ) or better property rights protection (higher θ), irrespective of initial endowments.
- Contrasting mechanisms:
  - Less efficient financial system:
    - Lowers domestic return to financial investment → financial capital outflow.
    - Financial outflow lowers domestic wage → creates incentive for inward FDI (financial outflow can be partly offset by FDI inflows).
  - Worse property rights protection:
    - Lowers financial returns → financial outflow.
    - Also lowers entrepreneurs’ expected profits → deters inward FDI.
    - Thus poor property rights can produce financial outflow without compensating FDI.
- Empirical consistency noted:
  - Evidence cited: poorer financial institutions associated with higher FDI share in inflows (Albuquerque 2003; Wei 2005); poor property rights or corruption deter FDI (Wei 2000 and 2005).
- Welfare remark:
  - Because financial investors capture part of the marginal product of capital via contracts, removing barriers to capital flows does not necessarily raise welfare unambiguously — entrepreneurs may lose sufficiently large rents that overall welfare could fall. Formal welfare analysis is deferred.

### VII. Conclusions and extensions
- Achievements:
  - Offers a micro-founded solution to the “too small” and “too large” capital flow puzzles by distinguishing financial returns from marginal products and introducing heterogeneous entrepreneurs.
  - Provides a framework for analyzing how financial development and property rights differentially shape patterns of gross and net capital flows, including the possibility of complete bypass of weak financial systems.
- Key takeaways:
  - Entrepreneur heterogeneity and revenue-sharing in financial contracts allow goods-price equalization while factor prices differ across countries.
  - Better financial systems or property rights raise equilibrium wages and influence directions and composition of capital flows in distinct ways.
- Suggested extensions:
  - Dynamic extension of the static model.
  - Empirical implementation linking gross and net capital flow patterns to institutional measures.

*Italic source: _wp06178 - Section VII concludes. An appendix provides the formal proofs for the propositions in the (PDF chapter/section).*

### References.................................................................................................28

### I.  INTRODUCTION

### Background and motivating puzzles
- Cross-border capital flows worldwide have risen substantially, reaching nearly $6 trillion in 2004, but less than 10 percent of them go to developing countries.
- Two opposing puzzles in standard trade and growth models:
  - Lucas (1990) paradox of too small flows: in a one-sector model, large implied marginal return gaps between rich and poor countries but limited capital flows.
  - Paradox of too large flows in two-sector neoclassical trade models: factor price equalization (FPE) implies returns to factors are equalized between countries with free trade in goods, so any observed capital flows would be excessive.

### Existing explanations and their limits
- Existing proposed solutions include:
  - Viewing a worker in a rich country as equivalent to multiple workers in a poor country.
  - Adding human capital as a new factor of production.
  - Allowing for sovereign risk.
  - Adding costs of goods trade.
- The paper argues that these explanations cannot escape the "tyranny of the FPE," and that few proposed reasons for FPE failure imply capital flow patterns that resolve the Lucas paradox.

### Model overview and key mechanisms
- New micro-founded theory: introduce a financial contract model following Holmstrom and Tirole (1998) and heterogeneous firms (entrepreneurs) into the Heckscher-Ohlin-Samuelson framework.
- Core features:
  - Return to financial investment generally differs from return to physical investment.
  - Financial investors (savers) obtain only a slice of the return to physical capital because they must share returns with entrepreneurs; more developed financial systems allocate a greater slice to financial investors.
  - Entrepreneurs are heterogeneous in their ability to manage capital; as a sector expands, marginal entrepreneur ability declines, endogenously generating decreasing returns to scale at the sector level while retaining constant returns to scale at the firm level.
- Implications:
  - Even with free trade in goods (equal product prices), factor returns can differ across countries: the interest rate is lower and the wage rate is higher in the capital-abundant country.
  - The model restores one-sector results (different factor returns across countries) while still predicting a small net capital flow between rich and poor countries.
  - The model can generate two-way gross capital flows (financial capital outflows and inward FDI) producing a small net inflow or even net outflow for a poorer country.

### Role of institutions: financial development vs. property rights
- Financial sector efficiency and property rights protection are modeled as distinct institutions with contrasting effects:
  - Financial development: more developed systems give a larger slice of returns to financial investors; an inefficient financial system can be bypassed — savings may flow out as financial capital while FDI flows in, possibly causing a complete exodus of savings from the less-developed financial system.
  - Property rights (expropriation risk): high expropriation risk can deter FDI; poor property rights protection may create financial capital outflows without compensating FDI inflows.
- Equilibrium result: the wage rate is always higher in the country with better financial or property rights institutions, irrespective of initial endowment.

### How this model breaks FPE and differs from other literature
- To break FPE, factor prices must depend on variables beyond product prices. Prior approaches:
  - Assume decreasing returns to scale (DRS) at the firm level — hard to justify in the long run.
- This paper instead:
  - Retains constant returns to scale at the firm level but endogenously generates sector-level decreasing returns via heterogeneous entrepreneurs.
  - Produces factor price differences across countries despite goods-price equalization.
- Distinctions from related literature:
  - Many previous papers use a one-sector model; their predictions do not generally survive a two-sector, two-factor extension.
  - This model:
    - Endogenously generates two-way gross capital flows with a small net flow.
    - Is the first in the literature to study contrasting effects of financial development and expropriation risk on capital flow within a two-sector, two-factor framework.
  - Related works cited include: Gertler and Rogoff (1990); Gordon and Bovenberg (1996); Shleifer and Wolfenzon (2002); Matsuyama (2004, 2005); Aoki, Benigno, and Kiyotaki (2006); Stulz (2005); Caballero, Farhi, and Gourinchas (2005); Obstfeld and Rogoff (1997); Ventura (1997); Melitz (2003); Bernard, Redding, and Schott (2005).

### Key model predictions and equilibrium properties
- With identical expropriation risk across countries, perfect entrepreneur mobility, and uneven financial sector efficiency:
  - The unique equilibrium in the world capital market may completely bypass the less-developed financial system.
  - The less-developed financial system’s country may experience complete outflow of savings as financial capital to the country with a better financial system, accompanied by FDI inflows from that country.
- If expropriation risk differs across countries:
  - Financial capital can still leave a country with an inefficient financial system, but FDI may be deterred by high expropriation risk despite low labor costs.
- Risk-neutral agents: the baseline model assumes entrepreneurs and financial investors are risk-neutral; adding risk-sharing motives would enrich patterns of capital flow but is not necessary for the core mechanisms.
- In all equilibria analyzed, the wage rate is always higher in the country with better financial or property rights institutions, regardless of endowment.

### Paper roadmap
- Section II reviews the two paradoxes of capital flow in neoclassical theory.
- Section III sets up the model.
- Sections IV and V study aggregation, equilibrium conditions, and key comparative statics.
- Section VI analyzes different forms of international capital flow under free trade in goods.

* _wp06178 - References.................................................................................................28*

### Section VII concludes. An appendix provides the formal proofs for the propositions in the

### _wp06178 - Section VII concludes. An appendix provides the formal proofs for the propositions in the

### II. PARADOXES OF INTERNATIONAL CAPITAL FLOWS
- One-sector (Lucas) setup:
  - Production: y = f(L; K). Firm profit maximization gives:
    - r = p @f(L; K)/@K = @K(1) (as in text).
  - Law of Diminishing Marginal Product implies r higher in country with lower per capita capital.
  - Lucas (1990) calculation: return to capital in India should be 58 times as high as in the United States — the "Lucas paradox" (too small capital flows).
- Two-sector (Heckscher-Ohlin-Samuelson) setup and the "opposite paradox":
  - Zero profits imply p1 = c1(w; r) and p2 = c2(w; r) (equation (2)).
  - Marginal returns in each sector:
    - r = p_i @f_i(1; K_i/L_i) = @K for i = 1,2 (equation (3)).
  - For given product prices, factor prices (w; r) and capital-labor ratios a_iK = a_iL are determined and independent from factor endowments (factor price insensitivity / factor price equalization (FPE)).
  - Lemma 1 (chain rule of factor price equalization):
    - If number of factors = m and any two countries can be linked by a sequence of country pairs where each pair produces a common set of m products, then factor prices are equalized among all these countries in a free-trade world, even without international factor movement.
- Reassessment of Lucas's three explanations:
  - Ineffective labor differentiation: In two-sector model with production y_i = f_i(E L_i; K_i), zero-profit conditions give p1 = c1(w/E; r), p2 = c2(w/E; r) and a unique solution (w/E; r). Increasing E raises w proportionally; r unaffected.
  - Missing factors (human capital): With chain rule, returns to capital, labor, and human capital equalize across countries if at least three common products link countries; abundance of human capital changes output composition but not return to capital.
  - Sovereign risk: If FPE already equalizes return to capital under free trade, sovereign risk has no further role in affecting return to capital in that two-factor, two-sector model.
- Limits to FPE and TFP explanations:
  - Trade costs break FPE but declining trade costs over decades should lead to factor return convergence — contradicted by data if it implied declining international capital flows.
  - Cross-country TFP differences (B_i) can change factor returns, but direction of capital flows may be ambiguous: equations (6) show p1 = B1 c1(w'; r') and p2 = B2 c2(w'; r'), with B_i < 1 possibly affecting w' and r' differently across sectors.
  - No general equilibrium in HOS model when technology differs and capital freely flows unless knife-edge or specialization cases (contradiction argument in text).
- Key insight: Marginal product of physical capital need not equal return to financial investment; model to follow makes this distinction.

### III. THE MODEL
- Overview:
  - Financial contracts between investors and entrepreneurs a la Holmstrom and Tirole (1998) embedded in two-good, two-factor, two-country HOS framework.
- A. Basic setup:
  - Two-period production; sector i first-period production: y1_i = G_i(L1_i; K1_i), with fixed labor-capital ratio a_i = L1_i/K1_i.
  - Timing: Date 1 initial investment K1_i injected; at beginning of date 2 liquidity shock occurs: additional financing needed = ζ_i K1_i, where ζ_i > 0 distributed F_i(ζ) with density f_i(ζ).
  - Project continues if ζ_i K1_i is paid; if not, terminated and yields no output.
  - Moral hazard: entrepreneur exerts effort e ∈ {e_H, e_L}. Success probability when working = θ = θ(e_H) (normalized), when shirking = 0. Entrepreneur's utility for managing one unit:
    - V_ni(e) = θ_i(e) R_E_ni − c_ni(e) (equation (7)); normalize θ(e_L) = 0 and θ(e_H) = θ.
  - Firm zero-profit gives total return to one unit of initial capital if succeeds:
    - R_i determined by p_i y1_i − w L1_i = [p_i G_i(a_i;1) − w a_i] K1_i = R_i K1_i (equation (8)).
  - Continuation decision: continue iff θ R_i − ζ_i ≥ 0. First-best cutoff ζ1_i = θ R_i (equation (9)).
  - Assume projects have positive NPV if entrepreneur works, negative if she shirks; focus on contracts implementing high effort.
- B. Financial contracts:
  - K capitalists each born with 1 unit capital and index n (observable cost of effort). Choose entrepreneur or financial investor at date1.
  - Entrepreneur invests 1 unit internal capital, raises K_X1_ni external capital; total initial investment K1_ni = 1 + K_X1_ni. Contract C_ni = {K1_ni; μ_ni(ζ_i); R_E_ni(ζ_i)} where μ_ni is continuation policy (1 continue, 0 stop), R_E_ni entrepreneur's revenue per unit investment.
  - Entrepreneur optimization:
    - Max U_ni = [1/(1 + r)] K1_ni ∫ θ R_E_ni(ζ_i) μ_ni(ζ_i) f_i(ζ_i) dζ_i − 1 (equation (10))
    - s.t. [1/(1 + r)] K1_ni ∫ f θ [R_i − R_E_ni(ζ_i)] μ_ni(ζ_i) − ζ_i g f_i(ζ_i) dζ_i ≥ K_X1_ni (participation) (equation (11))
    - and θ R_E_ni − c_ni(e_H) ≥ 0 (incentive compatibility) (equation (12))
  - Optimal continuation μ_ni is cutoff rule at ζ = bζ_ni. Incentive constraint binding yields:
    - R_E_ni = c_ni(e_H)/θ and ζ_max_ni = θ R_i − c_ni(e_H) (equation (13)).
  - Participation binding gives initial investment:
    - K1_ni(:) = (1 + r)/((1 + r) − ∫_0^{bζ_ni} (ζ_max_ni − ζ_i) f_i(ζ_i) dζ_i) (equation (14)).
  - Net return to internal capital:
    - U_ni(bζ_ni) = [θ R_i − h(bζ_ni)]/[h(bζ_ni) − ζ_max_ni] (equation (15)) where
    - h_i(bζ_ni) = (1 + r) + ∫_0^{bζ_ni} ζ_i f_i(ζ_i) dζ_i / F_i(bζ_ni) (equation (16)), the expected unit cost of total investment.
  - First-order condition minimizing h gives:
    - ∫_0^{ζ_opt_ni} F_i(ζ_i) dζ_i = 1 + r (equation (17)), implying ζ_opt_ni independent of n ⇒ ζ_opt_i = ζ_opt.
  - Assume f(ζ) uniform in [0; ζ], then solution:
    - ζ_opt = [2 (1 + r) / ζ]^{1/2} (equation (18)).
  - Financial development parameter ϑ (denoted as  in source):
    - Only shocks bζ_ni ≤ ϑ ζ_opt_i can be met by system. Interpretations: each firm financed up to bζ = ϑ ζ_opt or fraction ϑ firms financed up to ζ_opt.
    - Set bζ_ni = ϑ ζ_opt. Then h_i(bζ_ni) = h(r; ϑ) = [ϑ/(1 + ϑ)] (1 + r) + [2 ϑ/(p 2 ϑ)] [(1 + r) ζ]^{1/2} (equation (19)) as in text.
    - ∂h/∂r > 0 and ∂h/∂ϑ < 0.
- C. Allocation of capital and entrepreneurs' market:
  - Sector 1: entrepreneurs heterogeneous in cost of work; rank by n with c_n1 = c1 n (assumption).
  - ζ_max_n1 = θ R1 − c1 n decreases in n, so U_n1 decreases in n.
  - Sector 2: entrepreneurs identical cost c2; uniform profit U2.
  - Number of firms in Sector 1, N1, solves U_{N1}^1 = U2 (equation (22)).
  - Entry cost f units of numeraire to become entrepreneur; U2 = f and marginal entrepreneur in Sector 1 also gets f. Free entry conditions (equation (23)):
    - U_{N1}^1 = [θ R1 − h(r; ϑ)]/[h(r; ϑ) − (θ R1 − c1_{N1})] = f
    - U2 = [θ R2 − h(r; ϑ)]/[h(r; ϑ) − (θ R2 − c2)] = f
  - Lemma 2:
    - As r increases, fewer capitalists choose to be entrepreneurs at date1.
    - More productive entrepreneurs enter the heterogeneous sector; less productive enter the homogeneous sector.
    - In the heterogeneous sector, more productive entrepreneurs manage more capital.

### IV. AGGREGATION AND EQUILIBRIUM CONDITIONS
- Free entry conditions rewritten (capital revenue-sharing conditions) (equation (24)):
  - θ R1 = h(r; ϑ) + [f c1_{N1}]/(1 + f)
  - θ R2 = h(r; ϑ) + [f c2]/(1 + f)
  - Left-hand sides are expected marginal products of physical capital in two sectors.
- Full employment conditions (labor and capital) (equations (25)–(29)):
  - Labor-capital ratio identical within sector: a_ni = a_i (equation (21)).
  - Total capital usages given by (20); following algebra yields:
    - a1 ∫_1^{N1} K_{n1}(:) dn + a2 K2(:) N2 = L (equation (28) restated)
    - a1 K ∫_1^{N1} K_{n1}(:) dn + a2 K K2(:) N2 = K (equation (29) restated)
  - Definitions (equation (30)):
    - a1L = a1 h(r; ϑ)/c1, a1K = h(r; ϑ)/c1
    - a2L = a2 (1 + f) h(r; ϑ)/c2, a2K = (1 + f) h(r; ϑ)/c2
- Product market clearing and relative supply condition (equation (33)):
  - y1/y2 = [G1(a1;1) c2 / (G2(a2;1) (1 + f) c1)] ln[N1/(1 + f − f N1)] / N2 = D(p), where p = p1/p2 and good 2 is numeraire (p2 = 1).

### V. COMPARATIVE STATICS
- Free entry conditions rewritten (equations (34)–(35)):
  - θ a1 w + (1 + ϑ)/ (2 ϑ) [2 (1 + r) ζ]^{1/2} = θ p G1(a1;1) − f c1_{N1}/(1 + f) (equation (34))
  - θ a2 w + (1 + ϑ)/ (2 ϑ) [2 (1 + r) ζ]^{1/2} = θ G2(a2;1) − f c2/(1 + f) (equation (35))
- Endogenous variables (w; r; p; N1; N2) determined by (28), (29), (33), (34), (35).
- A. Determination of factor prices:
  - Curves z_i in (w; r) space convex and downward sloping; slopes given by (36):
    - dr/dw = − [θ 2^{3/2} (1 + r)^{1/2} ϑ ζ^{1/2} − (1 + ϑ)^2]/(1 + ϑ)^2 × a_i (as in text formulation; source equation (36)).
  - Assume a1 < a2 (Sector 2 more labor intensive).
  - Comparative movements:
    - Increase in N1 (or decrease in relative price of good1) shifts z1 inward: wage up, interest rate down.
    - Increase in ϑ shifts both z1 and z2 out: equilibrium moves vertically up from A to B: wage unchanged, interest rate increases.
    - Increase in θ shifts both z1 and z2 out differently: under condition highest cost in heterogeneous sector > homogeneous sector, interest rate increases and wage declines.
  - Proposition 1 (Stolper-Samuelson plus):
    - Decrease in price of a good decreases return to factor used intensively in that good and increases return to the other factor.
    - Increase in number of entrepreneurs in heterogeneous sector decreases return to factor used intensively there and increases return to other factor.
    - Improvement in financial development increases interest rate but has no effect on wage.
    - If highest entrepreneur effort cost in heterogeneous sector > homogeneous sector, lower expropriation risk increases interest rate but reduces wage.
  - Factor Price Equalization does not hold; ϑ and θ differences make factor prices differ across countries.
- B. Changes in endowment and institutions (Rybczynski plus):
  - Totally differentiating full employment conditions yields linear system (equation (37)) connecting percentage changes in N1, N2 to changes in L, K, a_ij.
  - Direct and feedback effects described: increase in K shifts KK outward increasing N1, decreasing N2; relative price p decreases; by Proposition 1, r decreases and w increases; feedback reduces factor usage per unit and shifts curves further.
  - Under modified nonreversal of factor intensity, overall effect of increase in K/L is to increase N1; effect on N2 ambiguous; relative price p declines and relative output y1/y2 increases.
  - Change in ϑ: raises r, leaves w unchanged; h(r; ϑ) must stay constant so a_ij constant; N1, N2, and p unaffected; but outputs y1 and y2 increase proportionally.
  - Change in θ: raises r and factor usages per unit; both LL and KK shift back; N1 and N2 decline; if highest effort cost in heterogeneous sector > homogeneous sector, N1 and N2 decrease proportionally and p unchanged; lower expropriation risk reduces number of firms but increases firm size and output.
  - Proposition 2 (Rybczynski plus):
    - If modified nonreversal condition holds so sector1 always capital-intensive:
      - Increase in capital endowment increases N1 and decreases relative price of good1.
      - Improvement in financial development raises outputs in both sectors proportionally, leaving N_i and relative price unchanged.
      - If highest effort cost in heterogeneous sector > homogeneous sector, lower expropriation risk raises output but reduces number of entrepreneurs in both sectors proportionally and has no effect on relative product price.
  - Corollary 1:
    - Increase in capital-labor ratio reduces interest rate but raises wage.
    - Improvement in financial system raises interest but leaves wage unchanged.
    - Reduction in expropriation risk raises interest but reduces wage.
  - Note: Contrary to one-sector intuition, a rich country may have a lower return to financial capital; but cross-country difference in returns is smaller in two-sector model.

### VI. FREE TRADE AND CAPITAL FLOWS
- Setup: Two countries identical in tastes, technologies, liquidity shocks and manager behavior, but differ in endowments, financial development ϑ, and expropriation risk θ. Labor immobile.
- A. Free trade in goods only (no capital flows):
  - Percentage difference in autarky prices bp = (p' − p)/p satisfies (equation (39)):
    - A_p bp = b_L − b_K where A_p = −|ε| ζ_D / ζ_N > 0 in text notation; b_L, b_K, b_ϑ, b_θ denote percentage differences in labor, capital, financial development, and expropriation risk.
  - Financial development and expropriation risk have no effect on relative product prices; Heckscher-Ohlin pattern holds.
  - Proposition 3: With capital flows prohibited, model yields standard Heckscher-Ohlin result: each country produces and exports good that uses its relatively abundant factor intensively.
- B. Financial capital flows (only financial capital mobile; no FDI):
  - Direction determined by br = (r' − r)/r. Corollary 1: relatively labor-abundant country, more financially developed country, or country with lower expropriation risk has higher interest rate absent international capital flows.
  - With free trade in goods and only financial capital flow, br expressed (equation (40)):
    - br = A_L b_L − A_K b_K + A_ϑ b_ϑ + A_θ b_θ where A_L, A_K, A_ϑ, A_θ > 0.
  - Proposition 4 (three polar cases):
    - If countries same in ϑ and θ but different in endowment: financial capital flows from capital-abundant country into labor-abundant country.
    - If same K/L and θ but different ϑ: financial capital flows from less developed financial system into the more developed one.
    - If same K/L and ϑ but different θ: financial capital flows from country with higher expropriation risk into one with lower expropriation risk.
- C. Foreign Direct Investment (FDI) allowed, entrepreneurs mobile:
  - Entrepreneur net return expression (equation (41)):
    - U_ni(w; r; ϑ; θ) = θ [p_Ti G_i(a_i;1) − w a_i] − ( (1 + ϑ)/(2 ϑ) ) [ (1 + r) ζ ]^{1/2} (1 + ϑ)/(2 ϑ) [ (1 + r) ζ ]^{1/2} − θ [p_Ti G_i(a_i;1) − w a_i] + c_ni (source equation (41) as presented).
  - For b_θ = 0 and no international financial capital flows, entrepreneurs from capital-abundant country will engage in outbound FDI to take advantage of lower labor cost abroad.
  - Proposition 5: With free trade in goods, identical expropriation risk but prohibition of international financial capital flow, FDI goes from capital-abundant country to labor-abundant one.
- D. Complete bypass of inefficient financial system (both financial capital flow and FDI allowed):
  - With identical θ and perfectly mobile entrepreneurs, unique equilibrium is "complete capital bypass circulation":
    - All capital owned by financial investors in country with less developed financial system leaves in financial capital outflow.
    - Physical capital and projects reenter as FDI.
    - Less developed financial system serves no capital in equilibrium.
  - Graphical intuition (parallelogram representation): equalization of r and w implies N1 and N1' must be same; factor usages lie on middle line AA'; result is capital bypass circle FH–EF in Figure 5 as described.
  - Proposition 6:
    - With identical expropriation risk and perfectly mobile entrepreneurs across countries with identical populations, in unique equilibrium the less developed financial system is completely bypassed: all financial capital leaves, but FDI comes in; capital-abundant country incurs net capital outflow (trade surplus).
  - Magnitude of interest rate differential:
    - With b_L = 0, substituting (64) into (55) yields br = − ξ_1N ϑ ξ_2w (ε_2L + ϕ_2) / |ξ| |β| × b_K (formula as in text). Key point: br depends not only on b_K but also on market imperfections f, effort cost c1, and heterogeneity c_n1(:). Thus, even with large K/L differences, small frictions (e.g., small f or small c1) can make interest differentials small — unlike one-sector Lucas result where extremely large implicit friction needed.
  - Interpretation: For Lucas example, while a one-sector model might require an enormous friction (e.g., 5800% tax equivalent) to stop capital flows, in this two-sector heterogeneous-entrepreneur model only a small friction (e.g., 5% tax equivalent) might suffice.
- E. Role of expropriation risk when institutions differ and entrepreneurs incur relocation cost:
  - Entrepreneurs pay fixed cost d to locate abroad. With financial capital flows equalizing r, an entrepreneur produces abroad if U_ni ≤ U_d_ni = U_ni(w'; r; θ'; ϑ) − d. Corner solutions arise depending on these comparisons (text stops at this point).

*Italic source: _wp06178 - Section VII concludes. An appendix provides the formal proofs for the propositions in the (PDF chapter/section).*

### 2. Suppressing the notations ofrandfor convenience, all Örms in Sector 2 produce at

### 2. Suppressing the notations ofrandfor convenience, all Örms in Sector 2 produce at

### Model setup and equilibrium characterization
- Entrepreneurs are heterogeneous; marginal entrepreneur in sector 1 is N
d
1 defined by
  - U
N
d
1
1
(w; ) = U
N
d
1
1
(w

; 

)d (equation (43)), implying more efficient firms choose FDI while less efficient produce at home (intervals [1; N
d
1
] and (N
d
1
; N
1
]).
- Capital usage for an FDI firm n:
  - k
d
(n) = h(r; ) / [ h(r; )

R

1
+ c
1
n ] (derived similar to expression (20)).
- Aggregate capital usage by all FDI firms:
  - 
d
=
Z
N
d
1
1
k
d
(n)dn =
h(r; )
c
1
ln
h(r; )

R

1
+ c
1
N
d
1
h(r; )

R

1
+ c
1
(equation (44)).
- Expected output of all FDI firms:
  - y
d
1
= F
1
(
opt
1
)

G
1
(a
1
;1)
Z
N
1
1
K
1
n1
(w

; 

)dn =
G
1
(a
1
;1)

(1 + r)
c
1
ln
h(r; )

R

1
+ c
1
N
d
1
h(r; )

R

1
+ c
1
(equation (45)).

- The equilibrium is characterized by 10 non-linear equations: (34), (35), (46), (47), (48), (49), (50), (51), (52), and (43) with ten endogenous variables:
  - p
T
; w; r(= r

); N
1
; N
2
; w

; N

1
; N

2
; N
d
1
; and 
f
.

### Full employment, capital flows, and market clearing
- Domestic full employment conditions (labor and capital):
  - a
1
Z
N
1
N
d
1
K
n1
(:)dn + a
2
K
2
(:)N
2
= L (equation (46))
  - 
d
+
Z
N
1
N
d
1
K
n1
(:)dn + K
2
(:)N
2
= K + 
f
(equation (47)), where 
f
is the amount of financial capital flow; 
f
> 0 represents financial capital inflow while 
f
< 0 represents outflow.
- Foreign full employment conditions:
  - a
1


d
+
Z
N

1
1
K

n1
(:)dn

+ a
2
K

2
(:)N

2
= L

(equation (48))
  - Z
N

1
1
K

n1
(:)dn + K

2
(:)N

2
= K

 
f
(equation (49)).
- World product market clearing:
  - y
1
+ y
d
1
+ y

1
y
2
+ y

2
= D(p
T
) (equation (52)).

### Comparative statics: financial efficiency, property rights, and factor prices
- From (35) it is verified that @w/@ > 0 and @w/@ > 0: a better financial system (higher ) reduces the investment cost h(r; ) and raises the wage rate; better property rights protection (higher ) increases expected revenue and raises the wage.
- Relative wage across countries:
  - w > w

if  > 

or  > 

. The relative wage in equilibrium is determined by institutional parameters  and , independent of initial endowments. A country with low initial capital-to-labor ratio but higher  or  can attract more capital and command a higher wage in equilibrium.
- Dynamics when capital flows are allowed (illustrated by Figure 3 description):
  - If home has lower expropriation risk ( > 

), financial capital and possibly direct investment leave the foreign country to the home country; N

1
declines and N
1
increases; interest rates are equalized across countries but w > w

.

### Principal theoretical result (Proposition 7) and patterns of capital flows
- Proposition 7:
  - Suppose the two countries are diversified in the equilibrium. With free mobility of capital, the wage rate is higher in the country with a more efficient financial system or better property rights protection, irrespective of the initial endowment.
- Distinct effects of financial development versus property rights on capital flows:
  - Less efficient financial system:
    - Depresses domestic return on financial investment → leads to an outflow of financial capital (financial outflow).
    - Financial outflow lowers the wage rate, which encourages inward FDI (so FDI may rise).
  - Worse property rights protection:
    - Depresses domestic financial returns → leads to financial outflow.
    - Also depresses firm profits → discourages inward FDI.
    - Therefore poor property rights protection may produce financial outflow without compensating inward FDI.
- Empirical consistency cited in text:
  - Albuquerque (2003) and Wei (2005) found that poor financial institutions are associated with a higher share of FDI in inward capital inflow.
  - Wei (2000 and 2005) found that poor property rights protection or severe bureaucratic corruption deters inward FDI.
- Welfare implication note:
  - Unlike many models where removing barriers to capital flows improves welfare due to improved efficiency (return to investment equals marginal product of physical capital), in this model financial investors can gain at the expense of entrepreneurs; sufficiently large entrepreneur losses due to financial capital outflow can reduce welfare. A formal welfare analysis is deferred to a companion paper (Ju and Wei, 2006).

### Conclusions and research directions
- Two objectives achieved:
  - Provide a solution to two opposing puzzles about international capital flows.
  - Offer a framework to analyze roles of financial and property rights institutions in determining patterns of gross and net capital flows.
- Key model mechanisms and insights:
  - Entrepreneur heterogeneity helps reconcile one-sector intuition in a two-sector setting: the interest rate can be lower in a capital-abundant country.
  - Revenue-sharing between financial investors and entrepreneurs, together with marginal product of capital, determine the interest rate.
  - Interest rate is higher in the country with a better financial system or lower expropriation risk.
  - Financial capital flows and FDI can move in the same or opposite directions, producing rich patterns of gross capital flows; in a frictionless world with identical expropriation risks, the less developed financial system can be bypassed entirely.
  - Better financial system or property rights leads to higher equilibrium wages, but their impacts on cross-border capital flows differ: lower financial development → lower interest rate → financial outflow → lower wages → more inward FDI; higher expropriation risk → lower profits → less FDI (and financial outflow).
- Suggested extensions:
  - Dynamic extension of the static model.
  - Empirical implementation linking patterns of gross and net capital flows to institutional variables.

### Appendix highlights (selected analytical results)
- Total differentiation of equilibrium conditions (equations (34) and (35)) yields linear relations among percentage changes in wages (bw), interest rate (br), institutional parameters (b, b), price (bp), and entrepreneur counts (bN
1
) (equations (53)–(55)).
- Expressions for factor allocation responses to shocks, including:
  - ba
iL
= ba
iK
=
r
2(1 + r)
br

1
2
1 +
2

b
 (equation (56)).
- Relations determining responses of entrepreneur counts bN
1
and bN
2
to changes in factor endowments, financial development, and expropriation risk given determinant sign conditions and modified non-reversal of factor intensity assumptions (equations (57)–(65)).
- Impact of a change in  on sectoral output by
  - by
1
=
b
 +
r
1 + r
br + 
N
b
N
1
(equation (66)), with subsequent substitutions yielding by
1
> b
 > 0 and by
2
> b
 > 0 under the specified parameter relations.

*Source: _wp06178 - 2. Suppressing the notations ofrandfor convenience, all Örms in Sector 2 produce at (IMF Working Paper content provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06178.pdf_