## wpiea2019284-print-pdf

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---

### Model setup and lifecycle
- Overlapping generations of entrepreneurs living three periods: young, middle-aged, old.
- Young supply endowment e>0 (non-storable); consumption occurs only when old.
- Middle-aged invest in next-period projects of two types j ∈ {1,2} with constant returns R_j and pledgeable fraction θ_j so pledgeable return is θ_j R_j.
- Key parametric benchmark (Definition 1): R_1 > R_2 > 1 and 1 > θ_1 R_1 > θ_2 R_2 (productive type 1 dominates in return and liquidity).

### Middle-aged entrepreneur problem and reduced form
- Problem I objective:
  - maximize c_{o,t+1} ≡ max_{i_t,x_{1t},x_{2t}≥0} R_1 x_{1t} + R_2 x_{2t} − (1 + r_t) i_t
  - s.t. x_{1t} + x_{2t} ≤ (1 + r_{t−1}) e + i_t
  - and (1 + r_t) i_t ≤ θ_1 R_1 x_{1t} + θ_2 R_2 x_{2t}.
- Under 1 + r_t < R_1 for all t, borrowing constraint binds and Problem II reduces to:
  - max_{i_t} Λ(θ,R; r_t) i_t + Φ(θ,R; r_{t−1}) e
  - with i_t bounded by expressions involving θ_j R_j and (1 + r_{t−1})/(1 + r_t).
- Λ(θ,R; r_t) and Φ(θ,R; r_{t−1}) defined exactly:
  - Λ(θ,R; r_t) ≡ [ (θ_2 − θ_1) R_1 R_2 / (θ_2 R_2 − θ_1 R_1) ] − [ ( (1 − θ_1) R_1 − (1 − θ_2) R_2 ) / (θ_2 R_2 − θ_1 R_1) ] (1 + r_t)
  - Φ(θ,R; r_{t−1}) ≡ [ (θ_2 − θ_1) R_1 R_2 / (θ_2 R_2 − θ_1 R_1) ] (1 + r_{t−1})
- Indifference interest rate:
  - 1 + r_Λ(θ,R) ≡ (θ_2 − θ_1) R_1 R_2 / ( (1 − θ_1) R_1 − (1 − θ_2) R_2 ).

### Optimal demand for funds (i_t) and specialization
- i_t rule (exact):
  - If r_t < r_Λ(θ,R): i_t = [ θ_2 R_2 (1 + r_{t−1})/(1 + r_t) − θ_2 R_2 ] e
  - If r_t = r_Λ(θ,R): i_t ∈ [ [ θ_1 R_1 (1 + r_{t−1})/(1 + r_t) − θ_1 R_1 ] e , [ θ_2 R_2 (1 + r_{t−1})/(1 + r_t) − θ_2 R_2 ] e ]
  - If r_t > r_Λ(θ,R): i_t = [ θ_1 R_1 (1 + r_{t−1})/(1 + r_t) − θ_1 R_1 ] e
- Aggregate specialization at date t (Equation (6)):
  - If r_t < r_Λ: x_{1t} = 0, x_{2t} = (2 + r_{t−1}) e
  - If r_t = r_Λ: explicit x_{1t}, x_{2t} formulas (see source)
  - If r_t > r_Λ: x_{1t} = (2 + r_{t−1}) e, x_{2t} = 0

### Competitive equilibrium, interest-rate law of motion and steady-state regions
- Market clearing: i_t = e ∀ t ≥ 0.
- Interest-rate dynamics (exact):
  - r_t = θ_2 R_2 (2 + r_{t−1})^{−1} if θ_2 R_2 (2 + r_{t−1})^{−1} < r_Λ(θ,R)
  - r_t = θ_1 R_1 (2 + r_{t−1})^{−1} if θ_1 R_1 (2 + r_{t−1})^{−1} > r_Λ(θ,R)
  - r_t = r_Λ(θ,R) otherwise
- Parameter regions in F (Assumption 1 plus θ_1 R_1/(1 − θ_1 R_1) < R_1):
  - Liquid Region F_ℓ: [ θ_1 R_1 / (1 − θ_1 R_1) ] < [ θ_2 R_2 / (1 − θ_2 R_2) ] ≤ 1 + r_Λ(θ,R)
  - Mixed Region F_m: [ θ_1 R_1 / (1 − θ_1 R_1) ] < 1 + r_Λ(θ,R) < [ θ_2 R_2 / (1 − θ_2 R_2) ]
  - Illiquid Region F_i: 1 + r_Λ(θ,R) ≤ [ θ_1 R_1 / (1 − θ_1 R_1) ] < [ θ_2 R_2 / (1 − θ_2 R_2) ]
- Lemma 2 steady states (unique and stable):
  - If (θ,R) ∈ F_ℓ: r_{ss}^ℓ = ( θ_2 R_2 / (1 − θ_2 R_2) )^{−1}
  - If (θ,R) ∈ F_m: r_{ss}^m = r_Λ(θ,R)
  - If (θ,R) ∈ F_i: r_{ss}^i = ( θ_1 R_1 / (1 − θ_1 R_1) )^{−1}
- Proposition 1: unique competitive equilibrium for any (θ,R) ∈ F and initial 1 + r_{−1} < R_1 converging to corresponding steady state.

### Novel inefficiency: "inefficiently liquid" equilibria (mechanism and diagnostics)
- Main phenomenon: equilibria with overinvestment in the liquid (low-return/high-liquidity) type despite being neither first-best nor second-best.
- Pecuniary externality channel:
  - Additional investment in liquid type bids up the interest rate and raises other entrepreneurs’ debt payments.
  - When θ_1 and θ_2 are low, middle-aged initial wealth is low, so interest-rate response from extra liquid investment is small and fails to deter others, producing excessive investment in liquid type.
- Characteristic signatures:
  - Inefficiently liquid steady states can have non-positive interest rate (r_{ss} ≤ 0) while still featuring interest rates that are “too-high” relative to constrained efficient allocation.
  - Credit and investment booms concentrated in liquid but low-productivity assets (examples: large firms, real estate).

### Efficiency analysis and constrained Pareto results
- Definition 4: constrained Pareto efficient allocation respects feasibility and the liquidity constraints of I for all t ≥ 0.
- Proposition 3: Any competitive equilibrium in the benchmark economy is constrained Pareto efficient when R_1 > R_2 > 1 and 1 > θ_1 R_1 > θ_2 R_2 (Benchmark Economy).
- Proposition 4 (characterization of constrained inefficiency):
  - For (θ,R) ∈ F_ℓ ∪ F_m, if r_Λ(θ,R) ≤ 0 then the competitive equilibrium is constrained inefficient.
  - The equilibrium interest rate at steady state is strictly negative in the interior of the inefficient region {(θ,R) ∈ F_ℓ ∪ F_m | r_Λ(θ,R) ≤ 0}, and zero on part of its boundary in F_m.
- Reallocation thought experiment (steady state change in old consumption):
  - ∆Vss = R_1 ε − (ε + δ) R_2 + δ; if ∆Vss ≥ 0 the steady-state allocation is constrained inefficient.
- Lemma 4: inefficiently liquid equilibria:
  - Nonempty sets with strictly positive measure; exist arbitrarily close to origin (low θ).
  - At θ_1 = 0, maximum θ_2 producing inefficiency increases in R_1.
  - Countries with low financial development but high R_1 are more prone to constrained inefficiency (policy-relevant).

### Policy instruments to implement Pareto improvements
- Regulated portfolio shares (mandated α_ℓt fraction invested in liquid type):
  - Middle-aged choose it subject to mandated α_ℓt and liquidity constraint (IV).
  - Proposition 6: Any Pareto improving reallocation (for small δ) can be implemented by regulating investment portfolios (choosing lower liquid investment-to-total ratio); regulated interest rate is lower than unregulated.
  - Regulation analogous to a maximum liquid asset ratio in a competitive banking sector.
- Debt tax instrument:
  - Lemma 5: a debt tax τ>0 levied on middle-aged borrowing (with lump-sum reimbursement) can implement the constrained efficient allocation by penalizing excess borrowing and internalizing the pecuniary externality.
  - Modified indifference rate r_Λ(θ,R; τ) has ∂ r_Λ/∂τ < 0 and r_Λ → 0 as τ → ∞.
- Implementation features:
  - For inefficiently liquid equilibria, choose τ_t = 0 for t < T and τ_t = τ for t ≥ T large enough to stop liquid-type investment and reach constrained Pareto efficiency.

### Government bonds: effects on allocation and welfare
- Model extension: one-period risk-free government bonds b^y_t, b^m_t, perfectly liquid; normalize bond supply σ_t = b_t / e.
- Assumption 2: σ_t < min(1 − θ_2 R_2, 1 − θ_1 R_1 / (1 − θ_1)) for all t and σ = lim σ_t exists with same bound.
- Regions F(Σ) defined analogously with factors (1 − σ).
- Lemma 6: unique competitive equilibrium converging to unique steady state given (θ,R) and Σ; steady-state rates:
  - 1 + r^{ss}_ℓ(Σ) = [ (1 − σ) θ_1 / R_1 / ((1 − σ) − θ_1 / R_1) ] if (θ,R) ∈ F^ℓ(Σ)
  - 1 + r^{ss}_m(Σ) = 1 + r_Λ(θ,R) if (θ,R) ∈ F^m(Σ)
  - 1 + r^{ss}_i(Σ) = [ (1 − σ) θ_2 / R_2 / ((1 − σ) − θ_2 / R_2) ] if (θ,R) ∈ F^i(Σ)
- Lemma 7: marginal effects of σ on total private investment i^{ss}_x:
  - ∂ i^{ss}_x / ∂ σ |_{(θ,R)∈F^ℓ} = [ θ_2 / R_2 / ((1−σ) − θ_2 / R_2) ]^2 − 1  e
  - ∂ i^{ss}_x / ∂ σ |_{(θ,R)∈F^m} = − e
  - In F_m: ∂ x^{ss}_1 / ∂ σ = [ 1 + r_Λ(θ,R) − θ_2 / R_2 ] / [ θ_2 / R_2 − θ_1 / R_1 ] e > 0; ∂ x^{ss}_2 / ∂ σ = − [ 1 + r_Λ(θ,R) − θ_1 / R_1 ] / [ θ_2 / R_2 − θ_1 / R_1 ] e < − e
- Welfare derivatives at σ = 0 (Lemma 8, exact):
  - ∂ V^{ss}_ℓ(Σ) / ∂ σ |_{σ=0} = [ R_2 − 1 ] / [ 1 − θ_2 R_2 ] r^{ss}_ℓ e
  - ∂ V^{ss}_i(Σ) / ∂ σ |_{σ=0} = [ R_1 − 1 ] / [ 1 − θ_1 R_1 ] r^{ss}_i e
  - ∂ V^{ss}_m(Σ) / ∂ σ |_{σ=0} = − r^{ss}_m e
- Proposition 8 (welfare implications):
  - In inefficient part of F_ℓ, small long-run public liquidity σ (government bonds) harms long-term welfare and cannot Pareto improve the competitive equilibrium.
  - In inefficient part of F_m (and certain boundary cases where r = 0), a small sequence of government bonds Σ can Pareto improve the competitive equilibrium by crowding out liquid private investment more than proportionally, allowing substitution into the productive type without raising the interest rate.

### Comparative statics on financial development, output and welfare (Proposition 7)
- For mixed region steady-state utility Vss^m(θ,R):
  - Vss^m is increasing in θ2 iff 1 + r_Λ(θ,R) > 2 θ1 R1.
  - Consequently Vss^m is always increasing in θ2 in the efficient part of F_m.
  - Within the inefficient part of F_m, Vss^m can be decreasing in θ2 if θ2 is low enough for given θ1.
  - There exists threshold θ^*_1 such that for θ1 < θ^*_1, Yss^m is decreasing in θ2 if θ2 is low enough.
  - If Yss^m decreases in θ2 then Vss^m decreases in θ2 as well.
- Mechanism: increasing θ2 raises portfolio liquidity and investment size but induces substitution toward the liquid (low-return) type; in inefficient region the substitution effect dominates lowering output and welfare.
- Policy implication: certain financial-development policies that increase liquidity of already relatively liquid/low-productivity sectors (mortgage guarantees, securitization, stronger creditor rights favoring tangible collateral) can reduce long-run output and welfare if the economy lies in the inefficient region; conversely, improving liquidity for intangibles and information (accounting, credit records) can raise output and welfare when overinvestment in tangibles exists.
- Measurement note: financial development need not monotonically correlate with growth; non-monotonicities can appear at low levels of liquidity.

### Open-economy intuition (capital outflows)
- Two-country thought experiment: home with higher R_1 can reallocate credit toward productive projects, inducing capital outflows and potentially lowering the world interest rate when home is sufficiently large. Reallocation patterns parallel empirical observations of credit flow toward less constrained large firms versus constrained productive firms.

### Key policy recommendations (concise)
- Avoid blanket policies that primarily increase liquidity of low-return, already-liquid assets (e.g., large-scale bond-market development, securitization, seizure-facilitating reforms) in economies at risk of inefficiently liquid equilibria.
- Consider targeted regulation of portfolio composition (e.g., maximum liquid asset ratio) or a debt tax to internalize pecuniary externality and restore constrained efficiency.
- Use government bonds cautiously: can be Pareto-improving in mixed-region settings but harmful in inefficient liquid-region equilibria unless supply is sufficiently large and appropriately timed.
- Support young firms and SMEs through targeted policies that improve pledgeability/liquidity of productive projects (information sharing, accounting standards) rather than expanding liquidity of low-productivity tangibles.

*Source: wpiea2019284-print-pdf — https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019284-print-pdf.pdf*

### Introduction ------------------------------------------------------------- 01

### Introduction

### Model setup and lifecycle
- Overlapping generations of entrepreneurs who live for three periods: young, middle-aged and old.
- There is a unit measure of young, middle-aged, and old cohorts in each period.
- Entrepreneurs receive a fixed endowment e>0 of non-storable and homogenous consumption goods when young and no endowment thereafter; consumption occurs only when old.
- Middle-aged entrepreneurs invest in a portfolio of projects in the next period; investments pay off next period.
- Two types of constant return to scale investment technologies (projects) exist: type j ∈ {1,2}.
- A project of type j has return R_j and pledgeable fraction θ_j, so θ_j R_j is the pledgeable (liquid) return.

### Key parametric assumptions and benchmarks
- Assumption 1: R_1 > R_2 > 1 and θ_1 R_1 < θ_2 R_2 < 1.
  - Interpretation: type 1 is the more productive (productive type) while type 2 is the more liquid (liquid type).
- Definition 1 (Benchmark Economy): R_1 > R_2 > 1 and 1 > θ_1 R_1 > θ_2 R_2.
  - In the Benchmark Economy, type 1 dominates type 2 in both liquidity and return, and entrepreneurs never invest in type 2 in equilibrium.

### Novel inefficiency: "inefficiently liquid" equilibria
- Setting: high-return/low-liquidity (productive) vs low-return/high-liquidity (liquid) projects.
- Main finding: equilibria may exhibit overinvestment in the liquid type (low-return/high-liquidity) despite being neither first-best nor second-best.
- Pecuniary externality mechanism:
  - Middle-aged entrepreneurs choose portfolio given prevailing interest rate.
  - Additional investment in the liquid type bids up the interest rate and raises other entrepreneurs’ debt payments.
  - When liquidities of both types are low, initial middle-aged wealth is low; hence the increase in the interest rate from additional liquid investment is small and insufficient to discourage others from investing in the liquid type.
  - Result: too-high interest rate and investment, and too-low consumption relative to the constrained optimum (second best).
- Characteristics of ineffectively liquid steady states:
  - Non-positive interest rate (i.e., lower or equal to the growth rate) can coexist with interest rates that are “too-high” relative to the constrained efficient allocation.
  - Equilibria can feature credit and investment booms concentrated in liquid but low-productivity assets (e.g., large firms, real estate, state-owned firms, tangible assets).

### Policy implications and Pareto improvements
- Financial development can be non-monotonic: increasing liquidity of low-return projects can reduce output and welfare when the economy is in an inefficiently liquid equilibrium.
- Public interventions that may be harmful when they mainly increase liquidity of low-productivity assets:
  - Developing private bond markets, securitization, loan guarantees, and policies facilitating seizure of collateral (e.g., property registries, stronger creditor rights) — these can favor liquid but low-productivity investments and lower long-term output and welfare.
- Interventions that can improve welfare in inefficiently liquid equilibria:
  - Regulating the fraction of resources invested in liquid assets (e.g., a maximum liquid asset ratio).
  - A simple debt tax can restore the constrained efficient allocation.
  - Government bonds (assumed fully liquid due to taxation ability) can achieve Pareto improvement only when there is strictly positive long-run investment in both types: government bonds crowd out the liquid type and crowd in the productive type without raising the equilibrium interest rate, allowing entrepreneurs to substitute bonds for liquid assets and raise consumption.
- Stronger case for public support of young firms and SME financing arises because the decentralized allocation can fail to be even second-best; targeted regulation or debt taxation can implement the constrained efficient allocation within a competitive banking sector.

### Comparative statics and broader mechanisms
- An increase in liquidity or return of the productive type leads to a lower steady-state interest rate.
  - Two opposing effects: (i) higher liquidity for any given portfolio raises portfolio liquidity and demand for investment, increasing interest rates; (ii) greater attractiveness of the productive type induces substitution away from the liquid type, reducing interest rates — the second effect dominates.
- Open-economy implication (qualitative): if the home country’s productive type return rises, credit reallocates toward productive projects at home and can generate capital outflows and a lower world interest rate, with parallels to observed patterns in emerging markets.

### Relation to literature
- Model structure closely related to Farhi and Tirole (2012); novelty arises from heterogeneity in investment liquidity.
- Mechanism: pecuniary externality through interest rate parallels Kehoe and Levine (1993) and Lorenzoni (2008) but operates without aggregate uncertainty.
- Distinct from standard overlapping-generations inefficiency (Samuelson 1958, Diamond 1965):
  - Zero or negative interest rates can be part of inefficient equilibria here, whereas in standard models zero interest rate typically indicates efficiency.
- Empirical connections: explains patterns such as credit reallocation toward less constrained large firms when borrowing rates decline (Gopinath et al. 2017), construction-led booms (Dell’Ariccia et al. 2019), and credit misallocation in nontradables (Reis 2013).

### Paper structure (as described)
- Section 2: model description and characterization of competitive equilibria and steady states.
- Section 3: properties of equilibria and interpretations/applications.
- Section 4: efficiency analysis and Pareto improvements (including regulation and debt tax).
- Section 5: introduction of government bonds and welfare implications.
- Section 6: conclusion.

*Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019284-print-pdf.pdf*

### 2.2  The Problem of Middle-Aged Entrepreneurs

### 2.2  The Problem of Middle-Aged Entrepreneurs

### Model setup and optimization problem
- Competitive credit market each period: young and middle-aged entrepreneurs can lend and borrow.
- Young born in period t>0 inelastically supply all their endowments in the capital market.
- Middle-aged entrepreneur at time t has transferred wealth (1 + r_{t−1}) e from period t−1 and can borrow i_t from the young at interest rate r_t, but borrowing is limited by the liquidity of her investment portfolio.
- Investment choices: x_{1t} and x_{2t} (types 1 and 2), new funds borrowed i_t.
- Middle-aged entrepreneur solves (Problem I):
  - Objective: maximize c_{o t+1} ≡ max_{i_t,x_{1t},x_{2t}≥0} R_1 x_{1t} + R_2 x_{2t} − (1 + r_t) i_t
  - Subject to:
    - x_{1t} + x_{2t} ≤ (1 + r_{t−1}) e + i_t
    - (1 + r_t) i_t ≤ θ_1 R_1 x_{1t} + θ_2 R_2 x_{2t}
  - Interpretation:
    - First constraint: resource constraint; (1 + r_{t−1}) e is wealth transferred from period t−1; i_t is external funds.
    - Second constraint: limited pledgeable return (liquidity constraint); maximum promiseable per type j is θ_j R_j x_{j t}.
    - c_{o t+1} is consumption of the old entrepreneur in period t+1.
- Cross-pledging assumption not essential; alternative formulations (precommitment to type-specific investment, or invest in only one type) yield similar aggregate results.
- Resource constraint is always binding. If interest rate not too high, borrowing constraint binds as well.

### Reduced form (Lemma 1) and key functions
- Under 1 + r_t < R_1 for all t, borrowing constraint binds in every period and the middle-aged problem reduces to (Problem II):
  - max_{i_t} Λ(θ,R; r_t) i_t + Φ(θ,R; r_{t−1}) e
  - s.t. θ_1 R_1 (1 + r_{t−1})/(1 + r_t) − θ_1 R_1) e ≤ i_t ≤ (θ_2 R_2 (1 + r_{t−1})/(1 + r_t) − θ_2 R_2) e
- Definitions:
  - Λ(θ,R; r_t) ≡ [ (θ_2 − θ_1) R_1 R_2 / (θ_2 R_2 − θ_1 R_1) ] − [ ( (1 − θ_1) R_1 − (1 − θ_2) R_2 ) / (θ_2 R_2 − θ_1 R_1) ] (1 + r_t)
  - Φ(θ,R; r_{t−1}) ≡ [ (θ_2 − θ_1) R_1 R_2 / (θ_2 R_2 − θ_1 R_1) ] (1 + r_{t−1})
  - Vector bold (θ,R) denotes (θ_1, θ_2, R_1, R_2).
- Indifference interest rate between two types:
  - 1 + r_Λ(θ,R) ≡ (θ_2 − θ_1) R_1 R_2 / ( (1 − θ_1) R_1 − (1 − θ_2) R_2 ).  (Equation (1))

### Optimal demand for funds (characterization)
- Entrepreneur’s optimal i_t (Equation (2)):
  - If r_t < r_Λ(θ,R):
    - i_t = [ θ_2 R_2 (1 + r_{t−1})/(1 + r_t) − θ_2 R_2 ] e
  - If r_t = r_Λ(θ,R):
    - i_t ∈ [ [ θ_1 R_1 (1 + r_{t−1})/(1 + r_t) − θ_1 R_1 ] e , [ θ_2 R_2 (1 + r_{t−1})/(1 + r_t) − θ_2 R_2 ] e ]
  - If r_t > r_Λ(θ,R):
    - i_t = [ θ_1 R_1 (1 + r_{t−1})/(1 + r_t) − θ_1 R_1 ] e
- Figure 1 (described): supply (inelastic e) vs. middle-aged demand curve with two arms (invest only in type 1 or type 2) and a flat segment at r_t = r_Λ(θ,R) where entrepreneurs mix. Higher w_{t−1} = (1 + r_{t−1}) e shifts the arms right but does not affect the flat segment.

### Competitive equilibrium and market clearing
- Fixed supply of funds e each period. Market clearing:
  - i_t = e, ∀ t ≥ 0.  (Equation (3))
- Combining the demand characterization with market clearing gives equilibrium path of interest rates (Equation (4)):
  - r_t = θ_2 R_2 (2 + r_{t−1})^{−1} if θ_2 R_2 (2 + r_{t−1})^{−1} < r_Λ(θ,R)
  - r_t = θ_1 R_1 (2 + r_{t−1})^{−1} if θ_1 R_1 (2 + r_{t−1})^{−1} > r_Λ(θ,R)
  - r_t = r_Λ(θ,R) otherwise
- Dynamic upper and lower bounds on r_t (Equation (5)):
  - θ_1 R_1 (2 + r_{t−1})^{−1} ≤ r_t ≤ θ_2 R_2 (2 + r_{t−1})^{−1}
- Definition 2: Competitive equilibrium is sequence { i_t, x_{1t}, x_{2t}, r_t }_{t=0}^∞ and initial r_{−1} satisfying conditions 1 to 5 and 1 + r_t < R_1 for all t > 0.
- Aggregate investment portfolio at date t (Equation (6)):
  - If r_t < r_Λ(θ,R):
    - x_{1t} = 0, x_{2t} = (2 + r_{t−1}) e
  - If r_t = r_Λ(θ,R):
    - x_{1t} = [ θ_2 R_2 (2 + r_{t−1}) − (1 + r_Λ(θ,R)) θ_2 R_2 ] / (θ_2 R_2 − θ_1 R_1) e
    - x_{2t} = [ (1 + r_Λ(θ,R)) − θ_1 R_1 (2 + r_{t−1}) ] / (θ_2 R_2 − θ_1 R_1) e
  - If r_t > r_Λ(θ,R):
    - x_{1t} = (2 + r_{t−1}) e, x_{2t} = 0
- Entrepreneurs specialize in productive (liquid) type when interest rate is relatively high (low).

### Parameter regions and steady states (Definition 3 and Lemma 2)
- Define F as set of (θ,R) that satisfies Assumption 1 and θ_1 R_1 / (1 − θ_1 R_1) < R_1.
- Three regions in F:
  - Liquid Region F_ℓ: (θ,R) ∈ F such that [ θ_1 R_1 / (1 − θ_1 R_1) ] < [ θ_2 R_2 / (1 − θ_2 R_2) ] ≤ 1 + r_Λ(θ,R).
  - Mixed Region F_m: (θ,R) ∈ F such that [ θ_1 R_1 / (1 − θ_1 R_1) ] < 1 + r_Λ(θ,R) < [ θ_2 R_2 / (1 − θ_2 R_2) ].
  - Illiquid Region F_i: (θ,R) ∈ F such that 1 + r_Λ(θ,R) ≤ [ θ_1 R_1 / (1 − θ_1 R_1) ] < [ θ_2 R_2 / (1 − θ_2 R_2) ].
- Lemma 2: Each of F_ℓ, F_m, F_i has a unique and stable steady state equilibrium:
  - If (θ,R) ∈ F_ℓ:
    - r_{ss}^ℓ = ( θ_2 R_2 / (1 − θ_2 R_2) )^{−1}
  - If (θ,R) ∈ F_m:
    - r_{ss}^m = r_Λ(θ,R)
  - If (θ,R) ∈ F_i:
    - r_{ss}^i = ( θ_1 R_1 / (1 − θ_1 R_1) )^{−1}
- At steady state: entrepreneurs specialize in liquid type in F_ℓ and productive type in F_i; in F_m they invest in both types with amounts given by Equation (6).
- Proposition 1: Given any (θ,R) ∈ F and initial 1 + r_{−1} < R_1, there exists a unique competitive equilibrium that converges to the corresponding steady state from Lemma 2.

### Comparative statics and properties of equilibria (Proposition 2, Lemma 3)
- Proposition 2 (summary of general properties for given R):
  - When θ is small enough (close to origin), all three steady-state equilibria can exist.
  - For any θ in liquid region, θ ≤ ( 1/(1 + R_1), 1/(1 + R_1) ).
  - For any θ_1, the θ_2 values placing (θ_1, θ_2) in liquid region lie strictly above the θ_2 values placing (θ_1, θ_2) in illiquid region.
  - Boundary of liquid region is non-monotonic, cutting θ_1 = 0 line twice: at origin and at θ = (0, 1/(1 + R_1)).
  - Inner boundary of illiquid region is strictly increasing and convex in θ_1, reaching maximum θ_2 = 1/R_2.
  - Top right corner θ = (1/(1 + R_1), 1/R_2) belongs to illiquid region.
- Lemma 3 (mixed region comparative statics and non-monotonicity):
  - In F_m, steady state interest rate is:
    - strictly decreasing in θ_1 and R_1
    - strictly increasing in θ_2 and R_2
  - Steady-state fraction invested in liquid type, x_{ss 2}/(x_{ss 1} + x_{ss 2}), is non-monotonic in θ_2 and has an interior maximum for relatively low θ_1.
  - That fraction is always weakly decreasing in θ_1 and strictly decreasing in θ_1 and R_1 in F_m.
- Intuition:
  - Increasing θ_2 raises average portfolio liquidity and demand for funds at a given r, tending to raise steady-state r.
  - Increasing θ_1 or R_1 makes productive type more attractive at given r, inducing substitution towards productive type and lowering r.
  - In F_m the substitution effect dominates, lowering the interest rate when θ_1 or R_1 rise.
  - Non-monotonicity arises because higher θ_2 both encourages investment in liquid type and increases r via higher demand; for high θ_2 the latter dominates, reducing the liquid share.

### Capital outflow scenario (open-economy intuition)
- Two-country thought experiment:
  - Home and foreign initially same (θ,R) but different endowments e and e^* with e < e^*, net flows zero initially.
  - Interpret liquid type at home as large mature firms with easy external finance (e.g., state-owned firms); productive type as more productive but finance-constrained private firms.
  - In autarky, an increase in R_1 at home lowers interest rate and raises the fraction invested in productive type.
  - In two-country version, higher R_1 at home induces funds to flow out of the economy (capital outflow) as credit reallocation toward productive type occurs; if home is large enough (e/(e + e^*) not small), outflow lowers equilibrium world interest rate.
- Connection to literature: narrative similar to Zheng et al.(2011) reconciling high growth and high return to capital with growing foreign surplus; reallocation from less productive state-owned firms to productive but financially constrained private firms can make transition resemble AK-like behavior.

### Transition to welfare and efficiency analysis
- Section 4 introduces analysis of efficiency of competitive equilibria and policies that can Pareto improve inefficient allocations.
- Also examines effects of financial development on long-term welfare and implications for measurement and policy.

*Source: wpiea2019284-print-pdf - 2.2  The Problem of Middle-Aged Entrepreneurs (IMF working paper excerpt).*

### 4.1  Eciency of Competitive Equilibria

### 4.1  Eciency of Competitive Equilibria

### Definition and benchmark result
- Definition 4. An allocation in the overlapping generations economy is called constrained Pareto efficient if a social planner cannot reallocate the resources to make at least one entrepreneur strictly better o while keeping all others at least as well o and if the reallocation respects the liquidity constraint in I. More formally, an allocation {c∗t , x∗1t , x∗2t }∞t=0 is constrained Pareto efficient if it is feasible, i.e., it satisfies the series of constraints for all t≥0:
  - ct + x1t + x2t ≤ R1 x1,t−1 + R2 x2,t−1 + e,
  - x1t + x2t ≤ θ1 R1 x1,t−1 + θ2 R2 x2,t−1 + e
  (and there does not exist any feasible allocation {ct , x1t , x2t }∞t=0 such that ct ≥ c∗t for all t≥0 with at least one strict inequality, given initial xj,−1 = x∗j,−1 for j ∈ {1,2}.)
- Proposition 3. Any competitive equilibrium in the benchmark economy is constrained Pareto efficient.

### Characterization of constrained inefficiency
- Consider steady-state equilibria where investment in the liquid type is strictly positive. Suppose the planner reduces aggregate debt payments of all middle-aged entrepreneurs in every generation to the young by an amount of δ>0 by substituting the productive for the liquid type. Let the increase in the productive type be ε>0; then the resource constraint implies investment in the liquid type must be reduced by ε+δ.
- Change in consumption level of the old at t≥1 from this reallocation:
  - ∆Vss = R1 ε − (ε + δ) R2 + δ.
- If ∆Vss ≥ 0, the steady-state allocation is constrained inefficient.
- Proposition 4. Consider any competitive equilibrium with liquidities and returns given by (θ,R) ∈ Fℓ ∪ Fm. If rΛ(θ,R) ≤ 0, the competitive equilibrium is constrained inefficient. Moreover:
  - the equilibrium interest rate at the steady state is strictly negative when (θ,R) lies in the interior of the inefficient region, i.e., {(θ,R) ∈ Fℓ ∪ Fm | rΛ(θ,R) ≤ 0},
  - and zero on part of its boundary that lies in Fm.

### Mechanism: pecuniary externality and role of liquidities
- Key insight: entrepreneurs’ portfolio choices generate a pecuniary externality via middle-aged entrepreneurs’ leverage and portfolio composition.
- Intuition:
  - An additional unit of investment in the liquid type by an entrepreneur bids up the interest rate and raises debt payments for other middle-aged entrepreneurs.
  - When liquidities θ1 and θ2 are low, initial wealth of middle-aged is low and the borrowing constraint binds; the interest rate response is small and cannot deter others from investing in the liquid type.
  - When liquidities are high enough, investment in the liquid type would bid interest rates so high that entrepreneurs switch to productive type; thus low θ for both types is the environment prone to inefficiently liquid equilibria.
- Implication: ineciently liquid steady states have a non positive interest rate.

### Properties and comparative statics (Lemma 4)
- For any R satisfying Assumption 1:
  - The sets of inefficiently liquid competitive equilibria in F are nonempty with strictly positive measure.
  - There are inefficiently liquid equilibria in any arbitrarily small neighborhood of the origin.
  - At θ1 = 0, the maximum value of θ2 that results in constrained inefficient equilibria is increasing in R1.
  - The set of inefficiently liquid equilibria in Fℓ is a proper subset of Fℓ iff (R1 − R2) / (R2 − 1) ≤ 1.
  - Let Si denote the unique intersection of rΛ(θ,R) = 0 with the boundary of Fi. Then all (θ,R) which correspond to inefficiently liquid equilibria have liquidities, i.e., θ, less than Si.
- Illustration notes:
  - Figure 4: for R1 = 4, R2 = 3, shows an inefficiently liquid region (light gray) and the line rΛ = 0.
  - Figure 5: for R = (4,2), the inefficiently liquid region expands; in contrast with R = (4,3), all competitive equilibria in liquid region Fℓ are constrained Pareto inefficient.
- Policy-relevant comparative point:
  - Countries with low financial development but high growth opportunities (large R1) may be more prone to this constrained inefficiency; they can suffer shortages of stores of value due to low liquidity of returns to investment (example: overinvestment in real estate as a liquid but relatively unproductive store of value).
- Additional notes:
  - Investment in liquid assets such as real estate can be bubbly equilibria in some models; absent uncertainty such bubbly equilibria can be Pareto efficient because they transfer resources across periods. This paper shows investment in liquid assets may be inefficient even without uncertainty.

### Allocation across firm types and collateral channels
- The model implies allocation of credit between SMEs / young firms and old / large firms may be worse than second-best:
  - Young firms and SMEs face limited net worth and higher collateral requirements, which exacerbates misallocation.
  - Higher collateral requirements arise from inadequate records, accounts, and higher credit risk.
- Investment in tangible assets (land, machinery) eases pledgeability and raises borrowing capacity; hence economies with high shares of tangibles may overinvest in tangibles when liquidity of tangibles rises.

### Characterization of constrained Pareto frontier
- Proposition 5. Let (θ,R) ∈ F.
  - If rΛ(θ,R) > 0, any allocation {ct , x1t , x2t }∞t=0 that satisfies the feasibility constraints with equality for all t≥0 is constrained Pareto efficient. Consequently, any competitive equilibrium corresponding to (θ,R) is constrained Pareto efficient.
  - If rΛ(θ,R) ≤ 0, any allocation that satisfies feasibility with equality for all t≥0 and has x2t = 0 for t≥T for some T≥0 is constrained Pareto efficient. Hence any competitive equilibrium in Fi is constrained Pareto efficient.

---

### 4.2  Regulated Economy — policy instruments to implement Pareto improvements

### Regulatory instrument: mandate portfolio shares
- Consider a planner who can dictate fraction αℓt of total funds invested in the liquid type. Then middle-aged entrepreneurs choose only new funds raised it and solve:
  - max it ≥ 0 ((1 − αℓt) R1 + αℓt R2) (it + (1 + rt−1) e) − (1 + rt) it  (IV)
  - s.t. (1 + rt) it ≤ ( θ1 (1 − αℓt) R1 + θ2 αℓt R2 ) ( it + (1 + rt−1) e ).
- Proposition 6. Any Pareto improving reallocation of the type analyzed in Section 4.1, when δ is small enough in absolute value, can be implemented by regulating the investment portfolios of the entrepreneurs. In an inefficiently liquid equilibrium:
  - a planner chooses a lower liquid investment-to-total investment ratio,
  - the regulated interest rate is lower than in the unregulated equilibrium.
  - Moreover, given any inefficiently liquid equilibria where rΛ(θ,R) < 0 or one where rΛ(θ,R) = 0 in the mixed region, this regulation can implement a Pareto improvement reallocation that results in a constrained Pareto efficient allocation.
- Interpretation: this regulation is akin to a maximum liquid asset ratio in a perfectly competitive banking sector (banks required to keep fraction of assets invested in liquid type ≤ planner’s choice).

### Alternative instrument: debt tax
- Lemma 5. Given an inefficiently liquid competitive equilibrium, a social planner can make a Pareto improvement that reaches the Pareto frontier by levying a debt tax (and reimbursing via a lump sum transfer) where the middle-aged entrepreneur has to pay (1 + τ)(1 + rt) it at t+1 for all t≥T for some T≥0 and a constant τ>0.
- Rationale: the constrained inefficient equilibrium features excess borrowing by middle-aged entrepreneurs; a debt tax penalizes excess borrowing and internalizes the pecuniary externality.

### Additional regulatory observations
- Overinvestment in liquid assets is accompanied by too much aggregate investment relative to the constrained optimum: aggregate investment at any period is s(2 + rt−1) e, and any Pareto improvement would reduce aggregate investment at all periods in a constrained inefficient equilibrium.
- Empirical pointers: some emerging market economies (example given: China) are candidate cases of such inefficient investment booms.

---

### 4.3  Output and Welfare in the Long Run

### Key proposition on financial development effects
- Proposition 7. Let Vss m (θ,R) and Yss m (θ,R) denote the steady-state utility and aggregate output in the mixed region.
  - Vss m is increasing in θ2 iff 1 + rΛ(θ,R) > 2 θ1 R1.
  - Consequently, Vss m is always increasing in θ2 in the efficient part of the mixed region.
  - Within the inefficient part of the mixed region, if θ2 is low enough for any given θ1, Vss m is decreasing in θ2.
  - In the inefficient part of the mixed region, there exists a threshold θ∗1 such that given θ1 < θ∗1, Yss m is decreasing in θ2 if θ2 is low enough.
  - Finally, for any economy (θ1, θ2), if Yss m is decreasing in θ2, then Vss m is decreasing in θ2 as well.

### Mechanism and policy implications
- An increase in θ2 in the inefficient region has two opposing effects:
  - increases liquidity of any given portfolio and raises investment size;
  - makes the liquid type relatively more attractive, inducing substitution from productive to liquid type, which lowers output and welfare since liquid investment is inecient.
- Policy implications:
  - Certain financial market policies that raise liquidity of already relatively liquid sectors (e.g., mortgage loan guarantees, asset securitization, corporate bond market development) can reduce long-term output and welfare in economies that are in the inefficient region.
  - Policies that raise liquidity of tangible-asset investments (e.g., better creditor rights, public property registries) may worsen long-term welfare when the economy overinvests in tangibles.
  - Conversely, policies that increase liquidity of intangibles (human capital, labor-intensive production) — e.g., improving accounting standards, creating credit records, establishing information sharing platforms — can raise long-term output and welfare when there is overinvestment in tangibles.

### Measurement and benchmarking of financial development
- Financial development may not monotonically correlate with higher output and welfare; effects depend on:
  - composition of financial development,
  - initial level of development (low levels of liquidity make economies more likely to be in the inefficient region).
- For low values of liquidities, financial development can lead to lower aggregate output at the steady state — potentially explaining non-monotonic empirical relationships between financial development and growth at low levels of development.

*Source: Section 4.1–4.3 of the supplied IMF working paper content (wpiea2019284-print-pdf).*

### 5.1  Competitive Equilibrium with Government Bonds

### 5.1  Competitive Equilibrium with Government Bonds

### Model setup and market clearing
- Young and middle-aged entrepreneurs at any time t≥0 can purchase a one-period, risk free government bond sold at par, denoted by b^y_t and b^m_t.
- A unit of bond purchased at time t is a promise by the government to deliver one unit of consumption good plus the interest in period t+1.
- Middle-aged maximization problem (as long as 1 + r_t < R_1) with government bonds (comparable to II) is:
  - max_{i_t, b^m_t ≥ 0} Λ(θ,R;r_t)(i_t − b^m_t) + Φ(θ,R;r_{t−1}) e^{−τ^o_{t+1}}  (II_b)
  - s.t. θ_1/R_1 (1 + r_{t−1})/(1 + r_t) − θ_1/R_1 ≤ (i_t − b^m_t) ≤ θ_2/R_2 (1 + r_{t−1})/(1 + r_t) − θ_2/R_2.
- Investment in government bonds is perfectly liquid so bond purchases reduce total debt payments to (1 + r_t)(i_t − b^m_t).
- τ^o_{t+1} denotes the lump sum tax levied on old entrepreneurs before consumption.
- Government budget constraint (balances every period):
  - (1 + r_t) b_t = b_{t+1} + τ^o_{t+1}. (8)
- Market clearing conditions for all t≥0:
  - i_t + b^y_t = e. (9)
  - b^m_t + b^y_t = b_t. (10)

### Supply restriction and definitions of regions
- Normalize supply of bonds: σ_t = b_t / e for all t≥0.
- Assumption 2: σ_t < min(1 − θ_2 R_2, 1 − θ_1 R_1 / (1 − θ_1)) for all t≥0 and σ = lim_{t→∞} σ_t < min(1 − θ_2 R_2, 1 − θ_1 R_1 / (1 − θ_1)) exists.
- Definition 5: F(Σ) is set of (θ,R) satisfying Assumption 1 and Assumption 2 given sequence Σ = {σ_t}_{t=0}^∞. Three regions of F(Σ):
  - The Liquid Region: F^ℓ(Σ) = { (θ,R) ∈ F(Σ) | [ (1−σ) θ_1 / R_1 / ((1−σ) − θ_1 / R_1) ] < [ (1−σ) θ_2 / R_2 / ((1−σ) − θ_2 / R_2) ] ≤ 1 + r_Λ(θ,R) }.
  - The Mixed Region: F^m(Σ) = { (θ,R) ∈ F(Σ) | [ (1−σ) θ_1 / R_1 / ((1−σ) − θ_1 / R_1) ] < 1 + r_Λ(θ,R) < [ (1−σ) θ_2 / R_2 / ((1−σ) − θ_2 / R_2) ] }.
  - The Illiquid Region: F^i(Σ) = { (θ,R) ∈ F(Σ) | 1 + r_Λ(θ,R) ≤ [ (1−σ) θ_1 / R_1 / ((1−σ) − θ_1 / R_1) ] < [ (1−σ) θ_2 / R_2 / ((1−σ) − θ_2 / R_2) ] }.

### Existence, uniqueness, and steady-state interest rates (Lemma 6)
- For any (θ,R) and Σ satisfying Assumptions 1 and 2 and any given initial condition 1 + r_{−1} < R_1, there is a unique competitive equilibrium that converges to a unique and stable steady state corresponding to the region in Definition 5 containing (θ,R).
- Steady-state interest rates for the three regions:
  - 1 + r^{ss}_ℓ(Σ) = [ (1 − σ) θ_1 / R_1 / ((1 − σ) − θ_1 / R_1) ] if (θ,R) ∈ F^ℓ(Σ),
  - 1 + r^{ss}_m(Σ) = 1 + r_Λ(θ,R) if (θ,R) ∈ F^m(Σ),
  - 1 + r^{ss}_i(Σ) = [ (1 − σ) θ_2 / R_2 / ((1 − σ) − θ_2 / R_2) ] if (θ,R) ∈ F^i(Σ).
- Specialization at steady state:
  - Entrepreneurs specialize in the liquid type in F^ℓ(Σ).
  - Entrepreneurs specialize in the productive type in F^i(Σ).
  - Entrepreneurs invest strictly positive amounts in both types in F^m(Σ).

### Effects of government bonds on steady-state allocation (Lemma 7)
- Let i^{ss}_x ≡ x^{ss}_1 + x^{ss}_2 denote total resources invested in the two types by entrepreneurs at steady state.
- Marginal effects of σ:
  - ∂ i^{ss}_x / ∂ σ |_{(θ,R)∈F^ℓ} = [ θ_2 / R_2 / ((1−σ) − θ_2 / R_2) ]^2 − 1  e.
  - ∂ i^{ss}_x / ∂ σ |_{(θ,R)∈F^m} = − e.
- In the mixed region, component effects:
  - ∂ x^{ss}_1 / ∂ σ |_{(θ,R)∈F^m} = [ 1 + r_Λ(θ,R) − θ_2 / R_2 ] / [ θ_2 / R_2 − θ_1 / R_1 ] e > 0.
  - ∂ x^{ss}_2 / ∂ σ |_{(θ,R)∈F^m} = − [ 1 + r_Λ(θ,R) − θ_1 / R_1 ] / [ θ_2 / R_2 − θ_1 / R_1 ] e < − e.
- Comparative implications:
  - An increase in long-term public liquidity σ in the liquid region crowds out private investment when public liquidity is scarce and crowds in private investment when public liquidity is abundant.
  - The marginal effect of public liquidity on private investment is strictly increasing in σ: ∂^2 i^{ss}_x / ∂ σ^2 > 0 in the liquid region.
  - In the mixed region, government bonds always crowd out private investment one for one: ∂^2 i^{ss}_x / ∂ σ^2 = 0 and ∂ i^{ss}_x / ∂ σ = − e.
  - Public liquidity crowds out the liquid type and crowds in the productive type in the mixed region; the crowding out of the liquid type is more than proportional so that demand for funds and the interest rate remain unchanged.
  - Government bonds can crowd out private investment while having no effects on the interest rate.

### 5.2  Welfare Effects of Government Bond

### Steady-state welfare derivatives (Lemma 8)
- Let V^{ss}_z(Σ) denote steady-state utility level for region z ∈ {ℓ, m, i} given (θ,R) ∈ F_z(Σ) when long-run supply of government bonds is σ. Then at σ = 0:
  - ∂ V^{ss}_ℓ(Σ) / ∂ σ |_{σ=0} = [ R_2 − 1 ] / [ 1 − θ_2 R_2 ] r^{ss}_ℓ e.
  - ∂ V^{ss}_i(Σ) / ∂ σ |_{σ=0} = [ R_1 − 1 ] / [ 1 − θ_1 R_1 ] r^{ss}_i e.
  - ∂ V^{ss}_m(Σ) / ∂ σ |_{σ=0} = − r^{ss}_m e.
- r^{ss}_z denotes the steady-state interest rate for region z ∈ {ℓ, m, i} when there is no government bond in the economy.

### Interpretations and Pareto implications
- Introduction of government bonds in an economy lying in the inefficient part of the liquid region F^ℓ is harmful to long-term welfare because government bonds crowd out the more productive (relative to government bond) investment in the liquid type.
- Government bonds cannot Pareto improve constrained inefficient equilibria in the liquid region.
- In contrast, supply of government bonds enhances steady-state utility in the inefficient part of F^m.
- Proposition 8:
  - For any (θ,R) in the inefficient part of F^ℓ, there exists ε > 0 such that for any Σ satisfying Assumption 2 with long-term supply of bonds no more than ε, Σ cannot Pareto improve the competitive equilibrium corresponding to (θ,R).
  - For any inefficiently liquid equilibria in F^m and for constrained inefficient equilibria corresponding to the unique point (θ^*, R^*) ∈ F^ℓ where r(θ^*, R^*) = 0 (steady-state interest rate is zero), there exists a small enough sequence of government bonds Σ which Pareto improves the competitive equilibrium allocation.
- Mechanism in the mixed region:
  - Public liquidity crowds out liquid investment more than proportionally, keeping demand for funds and the interest rate unchanged.
  - Substituting one unit of investment in government bonds for one unit in the liquid type allows entrepreneurs to pledge more, relaxing the borrowing constraint.
  - Extra liquidity is used to invest in the productive type while the interest rate does not increase, raising consumption and producing a Pareto improvement.

### Policy-relevant insights
- Government bond issuance can have region-dependent welfare and allocation effects:
  - Harmful in inefficient liquid-region equilibria.
  - Potentially welfare-enhancing and Pareto-improving in mixed-region equilibria (and certain boundary cases where r = 0).
- Government bonds can crowd out private investment without raising interest rates in some regimes, implying conventional crowding-out predictions can fail in this framework.

*Source: Section 5.1–5.2, "Competitive Equilibrium with Government Bonds" (excerpt) from the provided PDF content.*

### 6. The only condition that remains is that 1+r

### 6. The only condition that remains is that 1+r_t < R_1 for all t ≥ 0

### Stability and convergence of interest-rate dynamics
- Under Assumption 1: 1 + r_Λ(θ,R) < min(R_1, R_2).
- Remaining condition: 1 + r_t < R_1 for all t ≥ 0 is satisfied in liquid, mixed, and illiquid regions (illiquid by Definition 3: θ_1 R_1/(1−θ_1 R_1) < R_1).
- Local stability of steady states in F_ℓ and F_i follows from θ_1 R_1 < θ_2 R_2 < 1 (Assumption 1).
- Convergence argument (for z ∈ {ℓ,i}):
  - If 1 + r_ss^z ≠ 1 + r_Λ(θ,R), suppose 1 + r_ss^z − ε < 1 + r_t < 1 + r_ss^z. For small enough ε > 0 the whole interval is either strictly below or above 1 + r_Λ(θ,R). Then 1 + r_t < 1 + r_{t+1} = θ_z R_z (2 + r_t) < 1 + r_ss^z, and by Assumption 1 rates converge to the steady state.
  - If 1 + r_ss^z = 1 + r_Λ(θ,R) (z ∈ {ℓ,i}) or the mixed region F_m with steady state 1 + r_Λ(θ,R), suppose wlog 1 + r_{t−1} < 1 + r_Λ(θ,R). If 1 + r_Λ(θ,R) ∈ [θ_1 R_1 (2 + r_{t−1}), θ_2 R_2 (2 + r_{t−1})], then 1 + r_t = 1 + r_Λ(θ,R). Otherwise, if θ_2 R_2 (2 + r_{t−1}) < 1 + r_Λ(θ,R), then 1 + r_t−1 < 1 + r_t = θ_2 R_2 (2 + r_{t−1}) < 1 + r_Λ(θ,R) and hence 1 + r_{t+k} converges to 1 + r_Λ(θ,R). Similar when 1 + r_Λ(θ,R) < θ_1 R_1 (2 + r_{t−1}).

### Proof of Lemma 3 (comparative statics of 1 + r_Λ and investment shares)
- Partial derivatives of r_Λ(θ,R):
  - ∂r_Λ(θ,R)/∂θ_1 = (1−θ_2) R_1 R_2 (R_2 − R_1) / (( (1−θ_1) R_1 − (1−θ_2) R_2 )^2) < 0. (Equation (11))
  - ∂r_Λ(θ,R)/∂θ_2 = (1−θ_1) R_1 R_2 (R_1 − R_2) / (( (1−θ_1) R_1 − (1−θ_2) R_2 )^2) > 0. (Equation (12))
- Define s_j(θ,R) = x_ss^j / (x_ss^1 + x_ss^2) for j ∈ {1,2}.
- For s_1(θ,R):
  - s_1(θ,R) = θ_2 R_2 − (1−θ_2 R_2)(1 + r_Λ(θ,R)) / ( (θ_2 R_2 − θ_1 R_1)(2 + r_Λ(θ,R)) ).
  - Rewritten: s_1(θ,R) = 1/(2 + r_Λ(θ,R)) − (1−θ_2 R_2)/(θ_2 R_2 − θ_1 R_1).
  - Numerator strictly increasing in θ_1 by Proposition 3; denominator strictly decreasing in θ_1 ⇒ s_1(θ,R) strictly increasing in θ_1 in F_m and hence monotone in θ_1 in all three regions.
- For s_2(θ,R):
  - ∂s_2(θ,R)/∂θ_2 expression leads to quadratic polynomial in θ_2: a(θ_1) θ_2^2 + b(θ_1) θ_2 + c(θ_1) (multiplied by positive scalar).
  - Coefficients:
    - a(θ_1) = −R_1 R_2^2 (1 + R_1) (1 − (1 + R_1) θ_1) ≤ 0 (since θ_1 ≤ 1/(1 + R_1) by Proposition 2).
    - b(θ_1) = R_1 R_2^2 (1 + R_1) + R_1 R_2 [ (R_1 − R_2)(1 + 2R_1) − R_1 (R_1 R_2 − 1) ] θ_1 − R_2^1 R_2 (1 + R_1)(2 + R_2) θ_1^2.
    - c(θ_1) = R_1 R_2 (R_1 − R_2) − R_1 R_2 (R_1 (2 + R_1) − R_2) θ_1 + R_2^1 (1 + R_2)(R_1 R_2 − R_1 + R_2) θ_1^2 + R_1^3 (1 + R_1) θ_1^3.
  - Show c(θ_1) > 0 in F by considering ec(θ_1) = R_1^{−1} [ c(θ_1) − R_1^3 (1 + R_1) θ_1^3 ] and analyzing its smallest root θ_1^* (explicit formula given). Conclude θ_1^* > 1/(1 + R_1) so c(θ_1) > 0 inside F.
  - Since a(θ_1) ≤ 0 and c(θ_1) > 0, at least one root is non-positive ⇒ at most one root inside F ⇒ s_2(θ,R) has at most one interior maximum.
- On boundary of F_m and F_i: s_2(θ,R) = 0 and ∂s_2/∂θ_2 > 0 for any θ_1.
- Existence of unique maximum for s_2(θ,R) in neighborhood of tangency value e_{θ_1} completes proof.

### Proof of Proposition 1 (existence, uniqueness, and convergence of equilibrium path)
- Step 1: For all t ≥ 0, 1 + r_t < R_1.
  - Suppose 1 + r_{t−1} < R_1. Consider window where 1 + r_t ∈ [θ_1 R_1 (2 + r_{t−1}), θ_2 R_2 (2 + r_{t−1})].
  - If θ_2 R_2 (2 + r_{t−1}) ≤ 1 + r_Λ(θ,R), then 1 + r_t = θ_2 R_2 (2 + r_{t−1}) ≤ 1 + r_Λ(θ,R) < R_1 (Assumption 1).
  - If θ_1 R_1 (2 + r_{t−1}) < 1 + r_Λ(θ,R) < θ_2 R_2 (2 + r_{t−1}), then 1 + r_t = 1 + r_Λ(θ,R) < R_1.
  - If 1 + r_Λ(θ,R) < θ_1 R_1 (2 + r_{t−1}), then 1 + r_t = θ_1 R_1 (2 + r_{t−1}) ≤ max(1 + r_{t−1}, θ_1 R_1/(1−θ_1 R_1)) < R_1. The first inequality holds since θ_1 R_1 (2 + r_{t−1}) is between 1 + r_{t−1} and θ_1 R_1/(1−θ_1 R_1).
  - By induction, 1 + r_{−1} < R_1 implies 1 + r_t < R_1 for all t ≥ 0.
- Step 2: Existence and uniqueness of path given (θ,R) ∈ F and initial 1 + r_{−1}.
  - Given 1 + r_{t−1} unique determination of 1 + r_t via cases: if θ_2 R_2 (2 + r_{t−1}) ≤ 1 + r_Λ or 1 + r_Λ ≤ θ_1 R_1 (2 + r_{t−1}) use formulas 1 + r_t = θ_2 R_2 (2 + r_{t−1}) or θ_1 R_1 (2 + r_{t−1}) respectively; otherwise 1 + r_t = 1 + r_Λ(θ,R). By induction uniqueness follows.
- Step 3: Convergence to unique steady state in Lemma 2.
  - Case (θ,R) ∈ F_ℓ:
    - If 1 + r_{t−1} ≤ 1 + r_Λ(θ,R), then 1 + r_t = θ_2 R_2 (2 + r_{t−1}) ≤ max(1 + r_{t−1}, θ_2 R_2/(1−θ_2 R_2)) ≤ 1 + r_Λ(θ,R) ⇒ if 1 + r_{−1} ≤ 1 + r_Λ path defined by 1 + r_t = θ_2 R_2 (2 + r_{t−1}) converges to 1 + r_ss^ℓ = θ_2 R_2/(1−θ_2 R_2).
    - If 1 + r_{−1} > 1 + r_Λ, define sequence 1 + \bar r_t = θ_2 R_2 (2 + \bar r_{t−1}) with 1 + \bar r_{−1} = 1 + r_{−1}. Show 1 + r_t ≤ 1 + \bar r_t for all t and since θ_2 R_2 < 1 (Assumption 1) there exists finite t_0 with 1 + r_{t_0} ≤ 1 + r_Λ and convergence follows.
  - Case (θ,R) ∈ F_m:
    - Definition 3 implies θ_1 R_1/(1−θ_1 R_1) < 1 + r_Λ(θ,R) < θ_2 R_2/(1−θ_2 R_2).
    - If 1 + r_{−1} > 1 + r_Λ, define 1 + r_t sequence with 1 + r_t = θ_1 R_1 (2 + r_{t−1}). There exists finite unique t_0 ≥ 0 s.t. 1 + r_{t_0} ≤ 1 + r_Λ < 1 + r_{t_0−1}. Then 1 + r_{t_0} = max(θ_1 R_1 (2 + r_{t_0−1}), 1 + r_Λ) = 1 + r_Λ ⇒ convergence in finite periods. Similar if 1 + r_{−1} < 1 + r_Λ. If 1 + r_{−1} = 1 + r_Λ economy is already in steady state.
  - Illiquid region proof is analogous to liquid region.

### Proof of Proposition 2 (geometry of regions and boundaries as functions of θ_1)
- Illiquid boundary characterized by 1 + r_Λ(θ,R) = θ_1 R_1/(1−θ_1 R_1). Solving for θ_2:
  - θ_i^2(θ_1) = [ θ_1 R_1 (1 − θ_1 (1 + R_2)) ] / [ R_2 (1 − θ_1 (1 + R_1)) ].
  - θ_i^2(θ_1) is strictly increasing and convex in θ_1; θ_i^2(0) = 0 ⇒ illiquid equilibria exist arbitrarily close to origin.
- Liquid region characterized by 1 + r_Λ(θ,R) = θ_2 R_2/(1−θ_2 R_2). Solving yields two curves:
  - θ_ℓ^2(θ_1) = ( (θ_1 R_1 (1 + R_2) + R_2) + sqrt( (θ_1 R_1 (1 + R_2) + R_2)^2 − 4 θ_1 R_1 R_2 (1 + R_1) ) ) / (2 R_2 (1 + R_1)),
  - θ_ℓ^2(θ_1) = ( (θ_1 R_1 (1 + R_2) + R_2) − sqrt( (θ_1 R_1 (1 + R_2) + R_2)^2 − 4 θ_1 R_1 R_2 (1 + R_1) ) ) / (2 R_2 (1 + R_1)).
  - θ_ℓ^2(θ_1) ≤ θ_ℓ^2(θ_1) and θ_ℓ^2(0) = 0 ⇒ lower boundary passes through origin.
- Let Δ(θ_1) ≡ (θ_1 R_1 (1 + R_2) + R_2)^2 − 4 θ_1 R_1 R_2 (1 + R_1). The two liquid curves touch when Δ(θ_1) = 0 with two roots given explicitly:
  - θ_1 = [ R_2 ( (2(1 + R_1) − (1 + R_2)) ± sqrt(4(1 + R_1)(R_1 − R_2)) ) ] / ( R_1 (1 + R_2)^2 ).
- Smaller root < 1/(1 + R_1); bigger root > 1/(1 + R_1). Thus for high θ_1 there is no liquid steady state.
- Monotonicity:
  - θ_ℓ^2(θ_1) is strictly decreasing; θ_ℓ^2(θ_1) is strictly increasing.
  - Derivative formulas provided; sign arguments use Assumption 1.
- Liquid region lies above illiquid region:
  - ∂r_Λ/∂θ_2 > 0, and a monotonicity argument shows r_Λ(θ_1,θ_2,R) cannot simultaneously satisfy r_Λ ≥ θ_2 R_2/(1−θ_2 R_2) and r_Λ(θ_1,θ_2′,R) ≤ θ_1 R_1/(1−θ_1 R_1) for θ_2 ≤ θ_2′ if (θ_1,θ_2,R) ∈ F.
- Existence of (1/R_2, 1/(1 + R_1)) ∈ F_i:
  - ∂r_Λ/∂θ_1 < 0.
  - θ_i^2(θ_1) strictly increasing, passes through origin, and diverges to infinity as θ_1 → 1/(1 + R_1) ⇒ θ_i^2 crosses θ_2 = 1/R_2 at some ̄θ_1 < 1/(1 + R_1). For θ_1 ∈ (̄θ_1, 1/(1 + R_1)) one has θ_i^2(θ_1) < θ_1 R_1/(1−θ_1 R_1) ⇒ θ_1 ∈ F_i.

### Proof of Proposition 3 (degenerate case R_1 > R_2 > 1 and θ_1 R_1 > θ_2 R_2)
- When R_1 > R_2 > 1 and 1 > θ_1 R_1 > θ_2 R_2, type 1 dominates in liquidity and return ⇒ entrepreneurs invest only in type 1.
- Economy reduces to Farhi and Tirole (2012) with single investment (θ_1, R_1) and no bubbles or outside liquidity.
- Farhi and Tirole(2012) Proposition 5 then implies all competitive equilibria are Pareto efficient (hence constrained Pareto efficient).

### Proof of Proposition 4 (comparative statics of steady-state utilities and regulation)
- Definitions:
  - V_ss^ℓ, V_ss^i: steady-state utility levels in liquid and illiquid regions.
  - For any (θ,R) ∈ F:
    - V_ss^ℓ(θ,R) − V_ss^i(θ,R) and r_Λ(θ,R) have the same sign.
    - V_ss^ℓ(θ,R) − V_ss^m(θ,R) > 0 iff r_Λ(θ,R) > 0 and (θ,R) ∈ F_ℓ.
    - V_ss^m(θ,R) − V_ss^i(θ,R) < 0 iff r_Λ(θ,R) < 0 and (θ,R) ∈ F_i.
- Steady-state utility for regulated portfolio with α_ℓ such that [γ_α /(1−γ_α)] < R_α:
  - V_ss^α = (R_α − γ_α /(1−γ_α)) e = ( (1−α_ℓ)(1−θ_1)R_1 + α_ℓ (1−θ_2)R_2 ) / ( (1−α_ℓ)(1−θ_1 R_1) + α_ℓ (1−θ_2 R_2) ) e.
- Unregulated steady states:
  - V_ss^ℓ = [ (1−θ_2) R_2 / (1 − θ_2 R_2) ] e.
  - V_ss^i = [ (1−θ_1) R_1 / (1 − θ_1 R_1) ] e.
  - V_ss^m = [ (θ_2 − θ_1)^2 R_2^1 R_2^2 / ( (θ_2 R_2 − θ_1 R_1) ( (1−θ_1) R_1 − (1−θ_2) R_2 ) ) ] e.
- For (θ,R) ∈ F_m define eα_ℓ = x_ss^2(θ,R) / (x_ss^1(θ,R) + x_ss^2(θ,R)). Then regulated economy with α_ℓ = eα_ℓ yields:
  - 1 + r_Λ(θ,R) = 1 + r_ss^{eα_ℓ} = [ (1−eα_ℓ) θ_1 R_1 + eα_ℓ θ_2 R_2 ] / ( (1−eα_ℓ) (1 − θ_1 R_1) + eα_ℓ (1 − θ_2 R_2) ).
  - V_ss^m(θ,R) = V_ss^{eα_ℓ} = [ (1−eα_ℓ) (1−θ_1) R_1 + eα_ℓ (1−θ_2) R_2 ] / ( (1−eα_ℓ) (1 − θ_1 R_1) + eα_ℓ (1 − θ_2 R_2) ) e.
- For regulated economy with α_ℓ small enough (neighborhood of zero where [γ_α /(1−γ_α)] < R_α), steady state interest rate is 1 + r_ss^α = γ_α /(1−γ_α) and V_ss^α = (R_α − γ_α /(1−γ_α)) e. V_ss^α lies between V_ss^ℓ and V_ss^i (numerator and denominator are weighted averages).
- Expression for eα_ℓ from equation 6:
  - eα_ℓ = ( (1 + r_Λ(θ,R)) − θ_1 R_1 (2 + r_Λ(θ,R)) ) / ( (θ_2 R_2 − θ_1 R_1) (2 + r_Λ(θ,R)) ).
  - 1 + r_ss^{eα_ℓ} = γ_{eα_ℓ} /(1−γ_{eα_ℓ}) = 1 + r_Λ(θ,R) (algebra shown in text).
  - Note 1 + r_Λ(θ,R) < min(R_1, R_2) ≤ R_{eα_ℓ} by Assumption 1 ⇒ utilities at steady state are the same.
- Comparative statics sign equivalence (V_ss^ℓ − V_ss^i and 1 + r_Λ > 1):
  - V_ss^ℓ(θ,R) > V_ss^i(θ,R) ⇔ (1−θ_2) R_2 /(1 − θ_2 R_2) > (1−θ_1) R_1 /(1 − θ_1 R_1)
    ⇔ 1 + r_Λ(θ,R) > 1.
- Definitions introducing welfare gain and maximum reduction:
  - Ω_ℓ(θ,R) ≡ [ (θ_2 − θ_1) R_1 R_2 − ( (1−θ_1) R_1 − (1−θ_2) R_2 ) ] / (θ_2 R_2 − θ_1 R_1) e.
  - Γ_ℓ(θ,R) ≡ [ θ_2 R_2 ((1−θ_1) R_1 − (1−θ_2) R_2) − (1−θ_2 R_2) (θ_2 − θ_1) R_1 R_2 ] / ( (1 − θ_2 R_2) ((1 − θ_1) R_1 − (1 − θ_2) R_2) ).
  - Denominators are strictly positive. Numerator of Γ_ℓ positive iff 1 + r_Λ(θ,R) > 1. Numerator of Ω_ℓ positive iff 1 + r_Λ(θ,R) < θ_2 R_2 /(1 − θ_2 R_2) = 1 + r_ss^ℓ(θ,R) (equivalently (θ,R) ∈ F_ℓ).
  - Interpretation: Ω_ℓ(θ,R) is welfare gains per unit reduction in x_1 from investing freed resources in x_2; Γ_ℓ(θ,R) is maximum possible reduction in x_1.

*Italic: Source: wpiea2019284-print-pdf (section 6, proofs and propositions).*

### Section 4.1). Given

### Section 4.1). Given

### Algebraic simplifications and steady-state utility comparisons
- V_ss_m(θ,R) is given by:
  - V_ss_m =
    (θ2−θ1)^2 R1 R2^2 (θ2 R2 − θ1 R1) ((1−θ1) R1 − (1−θ2) R2)^{-1} e.
- Common denominator and numerator for V_ss_m(θ,R)+Ω_ℓ(θ,R) Γ_ℓ(θ,R):
  - DEN = (1−θ2 R2)((1−θ1) R1 − (1−θ2) R2)(θ2 R2 − θ1 R1)
  - NUM simplifies to ((1−θ1) R1 − (1−θ2) R2)(1−θ2) R2(θ2 R2 − θ1 R1)
- Therefore:
  - V_ss_m(θ,R)+Ω_ℓ(θ,R) Γ_ℓ(θ,R) = θ2 R2 / (1−θ2 R2) = V_ss_ℓ(θ,R)
  - Implication: V_ss_ℓ(θ,R) − V_ss_m(θ,R) = Ω_ℓ(θ,R) Γ_ℓ(θ,R).
- Sign condition:
  - The sign of V_ss_ℓ(θ,R) − V_ss_m(θ,R) is positive if and only if r_Λ(θ,R) > 0 and (θ,R) < F_ℓ.

- Definitions for the interior case:
  - Ω_i(θ,R) ≡ ((1−θ1) R1 − (1−θ2) R2) − (θ2−θ1) R1 R2 (θ2 R2 − θ1 R1)^{-1} e,
  - Γ_i(θ,R) ≡ (1−θ1 R1)(θ2−θ1) R1 R2 − θ1 R1 ((1−θ1) R1 − (1−θ2) R2) ) / (1−θ1 R1)((1−θ1) R1 − (1−θ2) R2).
  - Note: Ω_i(θ,R) = −Ω_ℓ(θ,R).
  - Result: V_ss_i(θ,R) − V_ss_m(θ,R) = Ω_i(θ,R) Γ_i(θ,R).
  - Hence V_ss_i(θ,R) − V_ss_m(θ,R) is positive iff r_Λ(θ,R) < 0 and (θ,R) < F_i.

### Constrained Pareto inefficiency of competitive equilibrium
- By Proposition 2, the competitive equilibrium converges to a unique steady state corresponding to (θ,R) ∈ F_ℓ ∪ F_m.
- There exist T ≥ 0 and ε > 0 such that x_2t ≥ ε for t ≥ T.
- Reallocation construction (for t ≥ T+1):
  - reduce x_2t by δ + ε, increase x_1t by ε;
  - adjust x_2T and x_1T by 1/(θ2 R2 − θ1 R1) δ;
  - δ>0, ε>0 such that ε+δ<ε and
    - δ = (θ2 R2 − θ1 R1) ε + θ2 R2 δ,
    - ε = (1−θ2 R2)/(θ2 R2 − θ1 R1) δ.
- Effects:
  - Reduces debt payments of each generation from T onward by δ.
  - Leaves all middle-aged entrepreneurs at or after T strictly better off when r_Λ(θ,R) < 0.
  - If r_Λ(θ,R) = 0, utility of middle-aged after T unchanged but middle-aged at T improved.
- Conclusion: Competitive equilibrium is constrained Pareto inefficient whenever r_Λ(θ,R) ≤ 0.
- Interest-rate characterization:
  - If (θ,R) ∈ F_m and r_Λ(θ,R) < 0 ⇒ steady-state interest rate strictly negative.
  - If (θ,R) ∈ F_ℓ and r_Λ(θ,R) < 0 ⇒ θ2 R2 / (1−θ2 R2) < θ1 R1 / (1−θ1 R1) ≤ 1 + r_Λ(θ,R) < 1 ⇒ steady-state interest rate strictly negative (by Lemma 2).

### Proof summary of Lemma 4 (geometry of r_Λ(θ,R)=0)
- r_Λ(θ,R) = 0 line: θ_Λ2(θ1) = [R1(R2−1)/(R2(R1−1))] θ1 + [(R1 − R2)/(R2(R1−1))].
- Intersection with θ1 = 0: θ_Λ2(0) = (R1 − R2)/(R2(R1 − 1)) > 0.
- Implication:
  - A strictly positive neighborhood of the origin θ = 0 corresponds to inefficiently liquid equilibria.
  - The r_Λ(θ,R)=0 line passes through F_ℓ iff (R1 − R2)/(R2 − 1) ≤ 1.

### Proof summary of Proposition 5 (existence of weights and constrained efficiency)
- Feasible allocations {c_t, x_1t, x_2t} ∈ ℓ_∞ by inequalities:
  - x_1t + x_2t ≤ θ1 R1 x_1,t−1 + θ2 R2 x_2,t−1 + e ≤ θ1 R1 (x_1,t−1 + x_2,t−1) + e,
  - hence bounds: x_1t + x_2t ≤ i_−1 + e/(1−θ1 R1), c_t ≤ R1(i_−1 + e/(1−θ1 R1)) + e.
- If there exist strictly positive weights {λ_t} ∈ ℓ_1 such that allocation solves the stated weighted maximization, then allocation is constrained Pareto efficient (via complementary slackness and Ghate and Smith (2005) Theorem 2.6).
- Sufficient conditions (SC) from Lagrange multipliers {η_t, γ_t, δ_1t, δ_2t, δ_ct}:
  - λ_t − η_t + δ_ct = 0,
  - (R1 η_t+1 − η_t) + (θ1 R1 γ_t+1 − γ_t) + δ_1t = 0,
  - (R2 η_t+1 − η_t) + (θ2 R2 γ_t+1 − γ_t) + δ_2t = 0,
  - non-negativity and complementary slackness conditions for multipliers and variables,
  - {η_t, γ_t, δ_1t, δ_2t, δ_ct} ∈ ℓ_1.
- Construction for r_Λ(θ,R) > 0:
  - Set {δ_1t, δ_2t, δ_ct} = 0, obtain closed-form η_t = λ_t, γ_t+1 = (R1 − R2)/(θ2 R2 − θ1 R1) λ_t+1, and second-difference recursion for λ_t with coefficient (1 + r_Λ(θ,R))^{-1} < 1 ensuring {λ_t} ∈ ℓ_1.
- Construction for r_Λ(θ,R) < 0 with eventual x_2t = 0 for t ≥ T:
  - Introduce δ_2t sequence with small α, α′ to ensure λ_t sequence decays (m < 1) and all multipliers in ℓ_1, satisfying SC.
- Construction for r_Λ(θ,R) = 0 with eventual x_2t = 0:
  - Explicit δ_2T+j sequence defined so {λ_t} and multipliers lie in ℓ_1 and satisfy SC.
- Conclusion: In each case when construction feasible, allocation is constrained Pareto efficient.

### Proof summary of Proposition 6 (regulation implementing Pareto improvements)
- For an inefficiently liquid equilibrium with r_Λ(θ,R) ≤ 0, same reallocation as in Proposition 4 reduces debt payments from T onward by δ and yields a Pareto improvement.
- Define perturbations {δ_t, κ_t, ν_t} with:
  - δ_t = θ2 R2 ν_t − θ1 R1 κ_t,
  - δ_t−1 = ν_t − κ_t.
- Regulator sets regulated liquid share eα_ℓt = (x_2t − ν_t)/(x_1t + x_2t − δ_t−1).
- Define r*_t recursively:
  - 1 + r*_t = (eα_ℓt θ2 R2 + (1−eα_ℓt) θ1 R1)(2 + r*_{t−1}).
- r*_t is an upper bound for e r_t; show (1 + r*_t) e = θ1 R1 x_1t + θ2 R2 x_2t − δ_t holds by induction.
- Since 1 + r_t ≤ 1 + r_Λ(θ,R) < R2 ≤ R_αℓ,t and eα_ℓt ≥ α_ℓt, obtain 1 + r*_t < 1 + r_t and R_eα,t ≥ R_αℓ,t, hence borrowing constraints binding and er_t = r*_t; the regulated allocation coincides with the Pareto superior allocation.
- Second part (reaching Pareto frontier of Proposition 5):
  - For arbitrarily small ε > 0 choose T so equilibrium paths close to steady state for t ≥ T.
  - Set eα_ℓt = α_ℓt for t ≤ T−1 and eα_ℓt = 0 for t ≥ T (shut down liquid-type investment from T onward).
  - Regulated equilibrium converges to new steady state with 1 + e r_ss = θ1 R1 / (1−θ1 R1) which is below original steady-state values; middle-aged for t ≥ T strictly better off (if r_Λ(θ,R) < 0, V_ss_i(θ,R) > V_ss_z(θ,R) for z ∈ {m,ℓ}); if r_Λ(θ,R) = 0 and (θ,R) ∈ F_ℓ, at least middle-aged at T strictly better off.
  - Thus regulation via controlling portfolio composition can implement Pareto improvements up to the constrained Pareto frontier.

### Taxes on debt and Lemma 5 (debt tax implementation)
- Social planner levies taxes {τ_t} on old at t+1; middle-aged problem becomes:
  - max_{i_t, x_1t, x_2t ≥ 0} R1 x_1t + R2 x_2t − (1 + τ_t)(1 + r_t) i_t + T_t s.t. x_1t + x_2t ≤ (1 + r_{t−1}) e + i_t, (1 + r_t) i_t ≤ θ1 R1 x_1t + θ2 R2 x_2t.
  - Tax rebate T_t = τ_t (1 + r_t) i_t ensures feasibility with original constraints.
- After simplification, the linear maximization (II) over i_t with bounds:
  - i_t ∈ [θ1 R1 (1 + r_{t−1})/(1 + r_t) − θ1 R1, θ2 R2 (1 + r_{t−1})/(1 + r_t) − θ2 R2] e,
  - Objective coefficient Λ(θ,R; r_t, τ_t) and constant Φ(θ,R; r_{t−1}) defined as:
    - Λ(θ,R; r_t, τ_t) ≡ ( (θ2−θ1) R1 R2 /(θ2 R2 − θ1 R1) ) − ( (1−(1 + τ_t) θ1) R1 − (1−(1 + τ_t) θ2) R2 )/(θ2 R2 − θ1 R1) (1 + r_t),
    - Φ(θ,R; r_{t−1}) ≡ ( (θ2−θ1) R1 R2 /(θ2 R2 − θ1 R1) ) (1 + r_{t−1}).
- Optimal i_t rule (13):
  - i_t = [θ2 R2 (1 + r_{t−1})/(1 + r_t) − θ2 R2] e if r_t < r_Λ(θ,R; τ_t);
  - i_t ∈ [θ1 R1 (1 + r_{t−1})/(1 + r_t) − θ1 R1, θ2 R2 (1 + r_{t−1})/(1 + r_t) − θ2 R2] e if r_t = r_Λ(θ,R; τ_t);
  - i_t = [θ1 R1 (1 + r_{t−1})/(1 + r_t) − θ1 R1] e if r_t > r_Λ(θ,R; τ_t).
- Interest rate rule:
  - r_t = θ2 R2 (2 + r_{t−1})^{-1} if θ2 R2 (2 + r_{t−1}) < 1 + r_Λ(θ,R; τ_t),
  - r_t = θ1 R1 (2 + r_{t−1})^{-1} if θ1 R1 (2 + r_{t−1}) > 1 + r_Λ(θ,R; τ_t),
  - r_t = r_Λ(θ,R; τ_t) otherwise.
- Properties:
  - ∂ r_Λ(θ,R; τ)/∂τ < 0 and r_Λ(θ,R; τ) → 0 as τ → ∞.
- Tax policy implication:
  - For an inefficiently liquid equilibrium, choose τ_t = 0 for t < T and τ_t = τ for t ≥ T with τ large enough so that 1 + r_Λ(θ,R; τ) < θ1 R1 (2 + r_{t−1}) for t ≥ T, shutting down liquid-type investment and reaching constrained Pareto efficient allocation (by Proposition 5).

### Comparative statics and Proposition 7 (derivatives and steady-state aggregates)
- V_ss_m(θ,R) normalized with e = 1. Derivative w.r.t. θ2:
  - ∂V_ss_m(θ,R)/∂θ2 = Ω × [2(θ2 − θ1)(θ2 R2 − θ1 R1)((1−θ1) R1 − (1−θ2) R2) − (θ2 − θ1)^2 [R2((1−θ1) R1 − (1−θ2) R2) + R2(θ2 R2 − θ1 R1)]] where
  - Ω = [R1 R2 (θ2 R2 − θ1 R1)((1−θ1) R1 − (1−θ2) R2)]^{-2}.
- Sign condition reduction yields:
  - ∂V_ss_m(θ,R)/∂θ2 > 0 ⇐⇒ 1 + r_Λ(θ,R) > 2 θ1 R1.
- Boundary characterization between F_i and F_m:
  - Boundary defined by 1 + r_Λ(θ,R) = θ1 R1 /(1−θ1 R1).
  - On this boundary equilibrium is efficient iff θ1 R1 /(1−θ1 R1) > 1 ⇐⇒ θ1 R1 > 1/2 ⇐⇒ 1 + r_Λ(θ,R) > 2 θ1 R1.
- For small θ1 and θ2 near the boundary, 1 + r_Λ(θ,R) < 2 θ1 R1 can hold (hence ∂V_ss_m/∂θ2 < 0).
- Steady-state allocations in F_m:
  - x_ss1 = R2 [ θ2 (θ2 − θ1) R1 R2 − (1−θ2)(θ2 R2 − θ1 R1) ] / ((θ2 R2 − θ1 R1)((1−θ1) R1 − (1−θ2) R2))
  - x_ss2 = R1 [ (1−θ1)(θ2 R2 − θ1 R1) − θ1(θ2 − θ1) R1 R2 ] / ((θ2 R2 − θ1 R1)((1−θ1) R1 − (1−θ2) R2))
- Y_ss_m(θ,R) = R1 x_ss1 + R2 x_ss2 =
  - R1 R2 (θ2 − θ1) [ (θ2 R2 − θ1 R1) − (θ2 − θ1) R1 R2 ] / ((θ2 R2 − θ1 R1)((1−θ1) R1 − (1−θ2) R2))
  - Equivalent form: Y_ss_m(θ,R) = (1 + r_Λ(θ,R))(1 + ∆) where ∆ = (θ2 − θ1) R1 R2 /(θ2 R2 − θ1 R1).
  - Also ∆ = (1−θ1) R1 (1 + r_Λ(θ,R))/((1 + r_Λ(θ,R)) − θ1 R1).
- Derivative ∂Y_ss_m/∂r_Λ given explicitly:
  - ∂Y_ss_m/∂r_Λ = (1 +(1−θ1) R1) [ ((1 + r_Λ) − θ1 R1)^{-2} − (θ1 R1)^2 (1−θ1) R1 /(1 + (1−θ1) R1) ] ((1 + r_Λ) − θ1 R1)^{-2} (expression as in text).
- Comparative static signs:
  - ∂Y_ss_m/∂θ2 < 0 ⇐⇒ ∂Y_ss_m/∂r_Λ < 0 ⇐⇒ 1 + r_Λ(θ,R) < [1 + sqrt((1−θ1) R1 /(1 + (1−θ1) R1))] θ1 R1.
  - For small enough θ1 at the boundary of F_i and F_m the above holds; hence ∂Y_ss_m/∂θ2 < 0 in neighborhoods with small θ1 and θ2 near the boundary.
  - Also final inequality implies 1 + r_Λ(θ,R) < 2 θ1 R1.

### Additional proofs and lemmas (overview)
- Lemma 6: existence and uniqueness claims sketched for competitive equilibrium under assumptions on parameters and initial value; borrowing constraint always binding and interest-rate law of motion parallels earlier derivations (full details in text).

*Italic line: Source: wpiea2019284-print-pdf - Section 4.1).*

### 4. The rest of the proof is very similar to

### 4. The rest of the proof is very similar to Proposition 1 and Lemma 2

### Borrowing constraint, interest-rate bounds, and binding condition
- First note that 1 + r_t ≤ R_1 for any t. Suppose 1 + r_t > R_1. Then the middle-aged do not invest in any of the two investment types since their returns are strictly less than the interest rate. The resource constraint gives i_t − b^m_t = −(1 + r_{t−1}) e < 0. Using (9), in equilibrium one must have i_t − b^m_t = e − b_t, which is strictly positive given Assumption 2. Contradiction.
- Show 1 + r_t < R_1 by induction:
  - True for t = −1 by assumption.
  - Suppose 1 + r_t = R_1 while 1 + r_{t−1} < R_1. Then entrepreneurs at t do not invest in the liquid type. Resource constraint gives x_{1t} = (1 + r_{t−1}) e + (i_t − b^m_t). Substituting into the borrowing constraint yields:
    (1 + r_t − θ_1 R_1)(i_t − b^m_t) ≤ θ_1 R_1 (1 + r_{t−1}) e.
  - In equilibrium, by (9) and 1 + r_t = R_1 and 1 + r_{t−1} < R_1, one has:
    (1 − θ_1)(e − b_t) ≤ θ_1 (1 + r_{t−1}) e < θ_1 R_1 e ⇒ σ_t > 1 − θ_1 R_1 / (1 − θ_1),
    which contradicts Assumption 2. Hence 1 + r_t < R_1.
- Because 1 + r_t < R_1, the borrowing constraint must be binding at any t; otherwise entrepreneurs can raise consumption by increasing x_{1t}.
- Using borrowing and resource constraints to express x_{1t} and x_{2t} in terms of (1 + r_{t−1}) e and i_t − b^m_t, and noting x_{1t} ≥ 0 and x_{2t} ≥ 0, one gets bounds:
  θ_1 R_1 (1 + r_{t−1}) / (1 + r_t − θ_1 R_1) e ≤ (i_t − b^m_t) ≤ θ_2 R_2 (1 + r_{t−1}) / (1 + r_t − θ_2 R_2) e.
- Equation (9) implies i_t − b^m_t = e − b_t, hence:
  θ_1 R_1 (1 − σ_t) (1 + r_{t−1}) + θ_1 R_1 ≤ 1 + r_t ≤ θ_2 R_2 (1 − σ_t) (1 + r_{t−1}) + θ_2 R_2.

### Law of motion for the interest rate and stability
- The middle-aged problem in II_b is similar to II except entrepreneurs maximize with respect to i_t − b^m_t rather than i_t. The net gain of increasing net investment equals Λ(θ,R; r_t), so threshold interest rate at which entrepreneurs switch equals r_Λ(θ,R).
- Law of motion for r_t:
  r_t =
    - θ_2 R_2 (1 − σ_t) (1 + r_{t−1}) + θ_2 R_2 − 1   if θ_2 R_2 (1 − σ_t) (1 + r_{t−1}) + θ_2 R_2 − 1 < r_Λ(θ,R),
    - θ_1 R_1 (1 − σ_t) (1 + r_{t−1}) + θ_1 R_1 − 1   if θ_1 R_1 (1 − σ_t) (1 + r_{t−1}) + θ_1 R_1 − 1 > r_Λ(θ,R),
    - r_Λ(θ,R) otherwise.
- Assumption 2 implies existence of ε > 0 such that σ_t < 1 − θ_2 R_2 − ε for all t. Thus 1 − σ_t > θ_2 R_2 + ε for all t, guaranteeing stability of the difference equations and uniqueness of the path of interest rates.
- The remainder of the proof parallels Proposition 1 and Lemma 2 and is omitted.

### Lemma 7 (summary)
- The first two equations follow from Lemma 6 and market clearing condition (9).
- The last two equations derive from II_b at the steady state.

### Lemma 8: steady-state objective and derivatives
- Using II_b and conditions (8) and (9) for z ∈ {ℓ, m, i}:
  V^{ss}_z(σ) = [ (1 − σ) Λ(θ,R; r^{ss}_z(σ)) + Φ(θ,R; r^{ss}_z(σ)) − σ r^{ss}_z(σ) ] e.
- More explicit form (using definitions of Λ and Φ):
  V^{ss}_z(σ) =
    ( (1 − σ) [ (θ_2 − θ_1) R_1 R_2 − ((1 − θ_1) R_1 − (1 − θ_2) R_2) (1 + r^{ss}_z(σ)) ] / (θ_2 R_2 − θ_1 R_1) ) e
    + ( [ (1 + r^{ss}_z(σ)) (θ_2 − θ_1) R_1 R_2 − r^{ss}_z(σ) (σ) (θ_2 R_2 − θ_1 R_1) ] / (θ_2 R_2 − θ_1 R_1) ) e.
- Using expressions for r^{ss}_z(σ) in Lemma 6, take derivatives at σ = 0:
  - For z = ℓ:
    d(1 + r^{ss}_ℓ(σ))/dσ = [1/(1 − σ)]^2 (1 + r^{ss}_ℓ(σ)) = [ θ_2 R_2 (1 − σ) − θ_2 R_2 ]^2.
    dV^{ss}_ℓ(σ)/dσ |_{σ=0} simplifies to:
      dV^{ss}_ℓ(σ)/dσ |_{σ=0} = [ (R_2 − 1) / (1 − θ_2 R_2) ] r^{ss}_ℓ e,
    since r^{ss}_ℓ = θ_2 R_2 / (1 − θ_2 R_2).
  - For z = i: proof is very similar to z = ℓ.
  - For z = m: r^{ss}_m(σ) = r^{ss}_m for any small enough σ, so d(1 + r^{ss}_m(σ))/dσ = 0 and Λ(θ,R; r^{ss}_m(σ)) = 0. Hence:
    dV^{ss}_m(σ)/dσ |_{σ=0} = − r^{ss}_m e.

### Proposition 8: welfare implications and construction of bonds that improve Pareto
- First part:
  - By Lemma 8, for small enough σ > 0 the change in steady-state welfare is strictly negative in inefficient equilibria of the liquid region because the steady-state interest rate is strictly negative. Therefore for small enough σ > 0, a sequence of bonds Σ with a long-term supply of σ cannot make a Pareto improvement.
- Second part (constructive):
  - Let T ≥ 0 be such that r_t = r_Λ(θ,R) for t ≥ T. Consider Σ = {σ_t}^∞_{t=0} where σ_t = 0 for t ≤ T − 1 and σ_t = ε for t ≥ T and ε > 0 is small enough.
  - Using II_b and conditions (8) and (9):
    V_t(Σ) = [ (1 − σ_t) Λ(θ,R; r_t(Σ)) + Φ(θ,R; r_{t−1}(Σ)) − ((1 + r_t(Σ)) σ_t − σ_{t+1}) ] e.
  - For small enough ε, r_t(Σ) = r_t for all t, where r_t is the interest rate in the competitive equilibrium without government bonds. Thus consumption of the middle-aged at t ≤ T − 1 does not change.
  - For t ≥ T + 1, consumption increases exactly by −r_t ε ≥ 0 since the competitive equilibrium is inefficient.
  - The change in consumption of the middle-aged at T is ε > 0. Therefore Σ makes a Pareto improvement.

*Source: wpiea2019284-print-pdf — Section 4 (proof continuation)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019284-print-pdf.pdf_