## 4. Sum of Two Firms’ Profits

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### I. Research question, approach, and key modeling assumptions
- Research question: Is there a specific role for banks in economic growth when production externalities are present?
- Method: Extend a canonical growth model with production externalities (Romer (1986)–style) by replacing the Walrasian auctioneer with competing banks that set both deposit and loan terms.
- Modeling restrictions (banks’ other roles shut down):
  - no exogenous stochastic shocks, no informational problems, no illiquid projects, and no transaction costs.
- Conceptual contribution: Treat strategic intermediation as the equilibrium mechanism in a growth model with production externalities to identify whether bank-oriented systems can outperform market-oriented systems.

### II. Production technology, agents, and equilibrium benchmarks
- Production and firms:
  - Firm j output: y_jt = f(k_jt, K_jt) ≡ A K_jt^η k_jt^α with A ∈ R_+, α ∈ [0,1], η ∈ [0,1]. (Equation (2))
  - Population average capital for firm j: K_jt ≡ 1/(#Ψ_t −1) Σ_{l∈−j_t} k_lt. (Equation (1))
  - Firm profit: π^f_jt = A K_jt^η k_jt^α − R_jt k_jt. (Equation (3))
- Focus case: η = 1 − α (constant returns to accumulated capital) ⇒ planner treats each firm equally and production becomes linear in capital Ak, delivering perpetual growth.
- Consumers:
  - Lifetime utility U(c_i) = Σ_{t=1}^∞ β^t u(c_it) with β ∈ (0,1); u twice continuously differentiable, u′>0, u′′<0, Inada conditions hold.
  - Budget constraint: c_it + s_it ≤ m_it. (Equation (5))
  - Wealth law: m_{i,t+1} = r_it s_it + w_it, with w_it ≡ Σ_{j=1}^J ψ^f_ij π^f(k_jt, K_jt, R_jt). (Equations (7)–(8))
- Pareto-first-best (social planner):
  - Planner sets k_t = k_jt = K_jt so marginal return = A.
- Walrasian benchmark:
  - Equilibrium coupon/interest rate equals private marginal product αA and is lower than social marginal return A, so Walrasian allocation is not Pareto optimal.

### III. High-level game-theoretic results and existence issues
- Intuition on discontinuities and nonexistence without intermediation:
  - Replacing auctioneer by firms issuing bonds directly to consumers yields no Nash equilibrium:
    - A firm can marginally outbid αA, capture all funds, internalize externalities, and realize returns up to A; free-riding creates discontinuous payoff functions (Appendix V; Lemma 11, Proposition 5).
- Need for organized intermediation:
  - Banks competing in both deposit and loan markets are modeled; banks’ balance-sheet identity: k_h ≤ s_h + B_h. (Equation (11))
  - Bank profit: π^b_h = ∑_{j=1}^J R_hj k_hj − ρ_h B_h − ∑_{i=1}^I r_hi s_hi. (Equation (12))
- Two-stage within-period game:
  - Stage 1: banks compete for deposits (deposit contracts z^b_Dhi ≡ (r^b_hi, s^b_hi)).
  - Stage 2: banks compete for loans (loan contracts z^b_hj ≡ (R^b_hj, k^b_hj)).

### IV. Monopoly bank solution and benchmark consequences
- Monopoly bank (captures all savings) optimality (Lemma 1):
  - ∑_{j=1}^J k^{*}_{Mj} = ∑_{i=1}^I s_{Mi}. (Equation (28))
  - k^{*}_{Mj} = (∑_{i=1}^I s_{Mi})/J. (Equation (29))
  - R^{b*}_{Mj} = A. (Equation (30))
  - Firms’ optimal decision is to “accept.”
- Implication: a monopolist bank internalizes externalities and charges loan rate A with symmetric allocation of funds across firms.

### V. Weak link between sources and uses of funds; Assumptions 2–3
- Fundamental nonexistence problem for banks:
  - Two profitable but opposing strategies create discontinuities:
    1. Bid up deposit rates to capture funds (compete in deposit market).
    2. Collect small deposits and free-ride on others’ investments (shrink size and exploit externalities).
- Definition and role of active banks:
  - D(z^D) ≡ { h ∈ H : ∑_{i=1}^I s_{hi} > 0 }, #D = number of active banks.
  - If #D = 1 → monopoly in loan market; if #D ≥ 2 → competitive loan market.
- Assumption 2 (Weak link):
  - In competitive second stage, equilibrium distribution over bank loan strategies is the same for all active banks irrespective of each bank’s own deposit-market performance; loan strategies depend only on aggregate savings S, not on individual bank fund levels. (Equation (34))
- Assumption 3 (no firm discrimination):
  - Firms’ best responses do not discriminate among active banks when banks offer identical contracts. (Equation (35))
- Consequence: permits tractable characterization of deposit-stage competition and unique equilibrium candidate.

### VI. Unique equilibrium candidate with Assumptions 2–3 (Theorem 1)
- Theorem 1 — equilibrium implications if existence holds:
  - (i) equilibrium offers always specify deposit rate equal to A (r = A);
  - (ii) specifying no deposit amount ({N.S.}) and specifying depositor’s willingness to supply at rate A (s^b_hi = S(m,(A, N.S.))) are dominant strategies;
  - (iii) depositors face rate A and deposit willingness-to-supply S(m,(A, N.S.)); aggregate savings = S(m,(A, N.S.));
  - (iv) equilibrium loan offer by any bank is (A, S(m,(A, N.S.))/J), same as monopolist’s; and
  - (v) firms accept this offer.
- Result: candidate equilibrium is Pareto-optimal allocation with aggressive deposit competition and many active banks (subject to existence).

### VII. Strategic tâtonnement, interbank market, and existence (Theorem 2)
- Strategic tâtonnement institution:
  - Banks have free recontracting opportunities across T ≥ 2 sessions within the period (e.g., morning and afternoon).
  - Sessions τ ∈ {1,2,···, T} include tentative loan offers, tentative firm responses, interbank rate/quantity submissions, matching, confirmations, and possible re-actions.
  - Interbank market clearing requirement: for all ρ ∈ R_+, there exists τ^* ≤ T such that ˜B_{τ^*}(ρ) = 0. (Equation (42))
- Loan-contract canonical forms:
  - Walrasian contract: z_w ≡ (αA, N.S.). (Equation (52))
  - Pareto-optimal contract: z_p ≡ (A, S/J). (Equation (53))
  - Target contract: z_φ ≡ (φA, S/J) with φ ∈ (α, 1]. (Equation (59))
- Lemma 2: repeated z_w can be sustained as a Nash equilibrium in loan sessions.
- Lemma 3: target contracts z_φ can be enforced by threat of reverting to z_w after deviations; with T finite provides Nash equilibrium; T = ∞ gives subgame perfect equilibrium. The feasible loan-rate range satisfies αA ≤ φA ≤ A.
- Deposit-stage selection:
  - Deposit competition across banks selects φ = 1 (loan rate A) because deposit-market incentives push r up to A when strategic tâtonnement makes loan-stage responses contingent on aggregate S.
- Theorem 2 (Existence and characterization):
  - With strategic tâtonnement, an equilibrium exists for the strategically intermediated economy Γ.
  - Equilibrium outcome (uniquely determined outcome, possibly supported by multiple strategy profiles) is Pareto-optimal with many active banks and:
    - (i) no arbitrage: r_hi = R_hj = ρ_h = A for all i, h, j. (Equation (66))
    - (ii) symmetric investment per firm: k_j = S(m,(A, N.S.))/J for all j. (Equation (67))
    - (iii) each firm j borrows from only one bank: k_hj = k_j, k_lj = 0 for l ≠ h. (Equation (68))
    - (iv) interbank market clears: ˜B(ρ) = 0. (Equation (69))

### VIII. Robustness, generalizations, and limitations
- Failure when firms issue bonds directly to consumers:
  - If firms issue bonds directly, no Nash equilibrium exists because firms can slightly outbid A, attract funds, and free-ride, undermining bank-based enforcement (Appendix V; Lemma 11, Proposition 5).
- Alternative production specifications (Appendix IV):
  - Decreasing returns to accumulated capital (η < 1 − α): equilibrium loan rate equals the monopolist rate r_m = A K_m^{α+η−1}; equilibrium need not be socially optimal but can be Pareto-superior to Walrasian (Proposition 3).
  - Labor inputs (Cobb-Douglas with elastic labor): equilibrium capital share ˆα = (α + 1/ε_{L,w}) / (1 + 1/ε_{L,w}) and loan rate r_chi = ˆα A; outcome need not be first-best but can outperform Walrasian when labor supply elasticity small (Proposition 4).
- Industrial-organization implication:
  - Equilibrium loan contracts create bank-organized firm groups where banks coordinate client firms’ investments to internalize externalities.
- Policy implications:
  - For the canonical constant-returns-to-capital case, no government intervention necessary: strategic intermediation with institutional features (interbank market and free recontracting) can implement Pareto-optimal allocation.
  - Comparative-system implication: bank-oriented systems outperform market-oriented systems when internalizing production externalities is crucial (development settings); advantage fades as informational frictions decline during industrial maturation.

### IX. Relation to discontinuous-game literature and technical appendices
- Discontinuous-game issues:
  - Private direct finance game is seriously discontinuous; better-reply secure condition (Reny, 1999) fails in key examples (Appendix VI).
  - Mixed-strategy existence techniques (Dasgupta and Maskin (1986b)) are inapplicable because discontinuities occur in off-diagonal neighborhoods; payoff correspondences lack upper hemicontinuity in quantity dimensions (k^f_j).
- Appendices provide:
  - Appendix I: proofs of Lemma 1, Theorem 1, Lemmas 2–3 and supporting lemmas (detailed structure and steps).
  - Appendix II: existence of an optimal plan under Assumption 1 (parts (i) and (ii)).
  - Appendix III: explicit Pareto-optimal allocation and Walrasian equilibrium characterization (Proposition 1 and 2; CRRA corollaries: Walrasian growth (β αA)^{1/σ}, Pareto growth (βA)^{1/σ}).
  - Appendix IV: alternative production specifications (decreasing returns, labor inputs).
  - Appendix V: formal nonexistence results for private direct finance (no pure or mixed equilibrium).
  - Appendix VI: relation to discontinuous-game literature and why standard existence fixes do not apply.

*Source: _wp06264 - 4. Sum of Two Firms’ Profits (PDF chapter/section).*

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

### _wp06264 - References  ............................................................................................................

### Appendices
- I.  Proofs ...........................................................................................................................41
  - A.   Proof of Lemma 1 .................................................................................................41
  - B.   Proof of Theorem 1...............................................................................................42
  - C.   Proof of Lemma 2 .................................................................................................48
  - D.   Proof of Lemma 3 .................................................................................................50
- II. Existence of an Optimal Plan.......................................................................................52
- III. Pareto-Optimal Allocation and Walrasian Equilibrium...............................................53
  - A.   Allocation by a Benevolent Social Planner ..........................................................53
  - B.   Allocation by a Walrasian Corporate Bond Market .............................................56
- IV. Allocations under Other Production Functions............................................................60
  - A.   Decreasing Returns to Accumulated Capital ........................................................60
  - B.   Labor Inputs..........................................................................................................61
- V. Economy with Private Direct Finance .........................................................................64
- VI. Relation to Discontinuous Game Literature ................................................................71

### Figures
- 1. Iso-Profit Curve (IPC) and Indifference Curve (IDC) .........................................................69
- 2. Profit of Firm 1 ....................................................................................................................72
- 3. Profit of Firm 2 ....................................................................................................................73

*Source: _wp06264 - References  ............................................................................................................*

### 4. Sum of Two Firms’ Profits ...........................................................................................

### 4. Sum of Two Firms’ Profits

### I. Introduction — research question and approach
- Research question: Is there a specific role for banks in economic growth when production externalities are present?
- Method: Extend a canonical growth model with production externalities (Romer (1986)–style) by replacing the Walrasian auctioneer with competing banks that set both deposit and loan terms.
- Key modeling restrictions (banks’ other roles shut down): no exogenous stochastic shocks, no informational problems, no illiquid projects, and no transaction costs.
- Conceptual contribution: Treat strategic intermediation as the equilibrium mechanism in a growth model with production externalities to identify whether bank-oriented systems can outperform market-oriented systems.

### Main insights from the introduction
- Historical intuition: Bank-oriented systems historically (e.g., The House of Morgan) promoted industrial development and exercised direct control/monitoring over firms’ operational decisions.
- Production externalities: Investment by one firm increases productivity/returns for other firms via quasi-public goods and spillovers (examples: railroads, shipping lines, invention of tools, specialization of labor).
- Walrasian benchmark: The Walrasian equilibrium (competitive bond market) is not Pareto optimal because firms ignore externalities; the Walrasian interest rate equals the private marginal product of capital, which is lower than the social marginal product.
- Strategic banks can potentially internalize externalities by specifying both loan rates and loan amounts in contracts, forcing firms to invest more than private incentives would dictate.

### High-level game-theoretic result summaries (intuitive)
- If an auctioneer is simply replaced by banks with no additional institutional features:
  - At the social-optimal deposit/loan rate, banks face discontinuous profit functions: a bank that obtains full deposits can capture monopoly rents that internalize externalities.
  - Result: Banks bid up deposit rates to the social-optimal return but then prefer to limit deposit amounts (free-riding), producing no Nash equilibrium in this decentralized game.
- Remedy: introduce a "weak link" between sources and uses of funds (i.e., allow banks’ loan market behavior to be less tightly tied to their collected deposits).
  - With a weak link, the unique equilibrium candidate is identified: banks compete aggressively for deposits without worrying about loan-side fund positions, leading to a Pareto-optimal allocation that internalizes externalities.
- Institutional implementation needed:
  - Interbank market is necessary to break the tight link between collected deposits and loan allocations.
  - A price adjustment mechanism (free-recontracting opportunity in the loan market) is necessary so the interbank market can clear after deviations.
    - Example mechanism: two-session day where banks can adjust loan quotes in the afternoon after morning offers.
  - With interbank market plus free-recontracting, deviations that invest less than the social-optimal level cause the interbank market not to clear, and anticipation of afternoon recontracting eliminates incentives to deviate; equilibrium with social-optimal allocation is sustained.

### Contribution relative to literature
- Bridges strategic intermediation literature (Townsend (1983), Stahl (1988), Yanelle (1998)) and growth-with-externalities literature (Romer (1986)), where intermediaries previously delivered allocations same as or inferior to Walrasian outcomes.
- Contrasts with microeconomic banking roles emphasized in previous growth models (e.g., Greenwood and Jovanovic (1990), Bencivenga and Smith (1991), Greenwood and Smith (1997)) where banks mitigate informational frictions, provide liquidity, or economize transaction costs—here, banks’ role is to internalize production externalities through contract design.
- Notes similarity in spirit to Da Rin and Hellmann (2002) static coordination game, but extends to a general-equilibrium growth setting and identifies sustaining institutions for a persistent bank-oriented system.

### II. Model setting and characteristics — core assumptions and structure
- Agents and timing:
  - Consumers indexed i ∈ {1,···, I}; firms indexed j ∈ {1,···, J}.
  - Consumers maximize infinite discounted sum of period utilities U(c_i) = Σ_{t=1}^∞ β^t u(c_it) with β ∈ (0,1).
  - Firms are established and dismantled each period; they borrow capital at the beginning of period t, invest, and return outputs equivalent to capital and interest at end of period t.
- Production technology:
  - Firm j’s capital: k_jt ∈ R_+.
  - Set of active firms at date t: Ψ_t; set excluding firm j: −j_t; number of firms #Ψ_t.
  - Population average capital for firm j: K_jt ≡ 1/(#Ψ_t −1) Σ_{l∈−j_t} k_lt. (Equation (1))
  - Output of firm j at date t: y_jt = f(k_jt, K_jt) ≡ A K_jt^η k_jt^α, where A ∈ R_+, α ∈ [0,1], η ∈ [0,1]. (Equation (2))
  - Firm profit: π^f_jt = A K_jt^η k_jt^α − R_jt k_jt, where R_jt is average gross borrowing rate paid by firm j at date t. (Equation (3))
  - Focus on η = 1 − α (case of constant returns to accumulated capital). Under η = 1 − α, social planner treats each firm equally so production becomes linear in capital Ak; this case delivers perpetual growth.
- Preferences and consumer problem:
  - Period utility u is twice continuously differentiable, u′>0, u′′<0, satisfies Inada conditions: lim_{c→0} u′(c)=∞ and lim_{c→∞} u′(c)=0.
  - Budget constraint per period: c_it + s_it ≤ m_it. (Equation (5))
  - Initial wealth m_i1 equal across consumers; initial ownership of firms allocated equally and remains unchanged.
  - Ownership share of firm j by consumer i at t: ψ^f_ij ∈ [0,1] with Σ_{i=1}^I ψ^f_ij = 1. (Equation (6))
  - Total profit income of consumer i at t: w_it ≡ Σ_{j=1}^J ψ^f_ij π^f(k_jt, K_jt, R_jt). (Equation (7))
  - Wealth evolution: m_{i,t+1} = r_it s_it + w_it, where r_it is average gross return on savings at t. (Equation (8))
- Assumption 1 (existence conditions for consumer problems):
  - (i) There exists σ < ∞ and c < ∞ such that, for c_it > c, dlnu(c_it)/dlnc_it ≤ 1 − σ and β A^{1−σ} < 1.
  - (ii) β A > 1.
- Economywide constraints:
  - (i) Capital formed from savings each period: Σ_{j=1}^J k_jt ≤ Σ_{i=1}^I s_it. (Equation (9))
  - (ii) Consumption and savings in period t bounded by total output at t−1: Σ_{i=1}^I (c_it + s_it) ≤ Σ_{j=1}^J y_{j,t−1}. (Equation (10))
  - (iii) Fixed-point condition (Equation (1)) applies for all j = 1,2,···,Ψ_t.

### Pareto-optimal allocation and Walrasian equilibrium (summary)
- Social planner (symmetric first-best): internalizes externalities by setting each firm’s capital equal (k_t = k_jt = K_jt), so planner faces linear Ak production and accumulates capital at the social marginal return A.
- Walrasian economy (bond market cleared by auctioneer): firms finance investment via competitive bond issuance; auctioneer adjusts coupon rates to match supply and demand.
- Key contrast: Walrasian interest rate equals private marginal product of capital and is lower than the social marginal return, so Walrasian allocation is not Pareto optimal (replicates Romer (1986) result).

*Source: _wp06264 - 4. Sum of Two Firms’ Profits (PDF chapter/section)*

### Appendix III for a detailed analysis). Each firm determines its investment level based on its private

### Appendix III — Unique Equilibrium Candidate with Strategic Intermediation

### Walrasian versus Pareto-optimal growth
- Firms choose investment based on private marginal return and do not internalize spillovers.
- The equilibrium coupon rate is αA, which is lower than the social marginal return A.
- Capital accumulates at slower speeds in the Walrasian equilibrium than in the Pareto-optimal growth path.
- With a constant relative risk aversion (CRRA) utility function, u(c) = c^{1−σ}/(1−σ) with σ ∈ R_{++}, closed-form solutions are obtainable.
- Growth rates:
  - Walrasian equilibrium growth rate: (βαA)^{1/σ}.
  - Pareto-optimal allocation growth rate: (βA)^{1/σ}.
- These results are essentially the same as in Romer (1986).

### Nonexistence of a Nash equilibrium without intermediation
- If there is no organized bond market or no intermediation, firms finance inputs by issuing bonds privately and directly to consumers.
- In that regime, the first-order conditions coincide so only the Walrasian interest rate αA supports the fixed point condition k_j = K_j for all firms.
- αA cannot be an equilibrium interest rate because a firm can profitably deviate by offering a coupon rate slightly higher than αA, capture all funds, become a monopolist that internalizes externalities, and realize returns as high as A.
- Consequently, no Nash equilibrium exists when bonds are issued privately. (See proof in Appendix V.)
- Free-riding opportunities make firms’ payoff functions severely discontinuous, violating assumptions needed for the existence of a Nash equilibrium.

### Need for financial intermediation and model formulation
- Some form of financial intermediation is required to clear the market.
- The economy is formulated as a game among consumers, firms, and banks in the style of Arrow and Debreu (1954).
- Banks compete in both the deposit and loan markets.
- Competition concept: competition in both price and quantity; banks tailor deposit contracts for depositors and loan contracts for client firms.

### Equilibrium concept and dynamics
- Dynamics are defined recursively in the style of Prescott and Mehra (1980).
- The equilibrium concept is stationary Markov equilibrium, following Duffie and others (1994):
  - Consumers adopt the same strategies over time, conditional only on state variables.
  - Firms and banks emerge and are dissolved in each period as in Prescott and Mehra (1980), yielding stationary Markov strategies for these agents.
- For consumers, restricting strategies to stationary Markov strategies puts a restriction on the strategy space but does not lose much generality.
- As strategies are stationary, time subscripts are dropped when not creating confusion.

### Two-stage within-period game and approach to equilibrium
- The within-period economy is modeled, without loss of generality, as a two-stage game:
  - Stage 1: Banks compete for deposits.
  - Stage 2: Banks compete for loans.
- Traditional approach would search for subgame perfect equilibrium by finding a Nash equilibrium in the second-stage game; however, without additional institutional settings, the economy still suffers from a lack of equilibrium.
- Alternative approach taken:
  - Identify a condition, described as a weak link between sources and uses of funds, to support a unique deposit market outcome.
  - Then characterize the unique equilibrium candidate for the whole within-period game.
- Existence issues are deferred to the next section.

*Source: _wp06264 - Appendix III for a detailed analysis).*

### Appendix VI for further discussion related to the discontinuous game literature.

### Appendix VI for further discussion related to the discontinuous game literature.

### A. Banking sector — model setup and objectives
- There are H banks and the number of banks is assumed to be smaller than the number of firms.
- Bank h ∈ {1,···, H} offers:
  - deposit contract to consumer i: (r^b_hi ∈ R_+, s^b_hi ∈ R_+);
  - loan contract to firm j: (R^b_hj ∈ R_+, k^b_hj ∈ R_+);
  - interbank loan submission B^b_h with interbank rate ρ^b_h.
- Balance-sheet identity for bank h: k_h ≤ s_h + B_h. (Equation (11))
  - k_h ≡ ∑_{j=1}^J k_hj (assets);
  - s_h ≡ ∑_{i=1}^I s_hi (deposits).
- Bank h’s objective: maximize profit
  - π^b_h = ∑_{j=1}^J R_hj k_hj − ρ_h B_h − ∑_{i=1}^I r_hi s_hi. (Equation (12))
- Ownership of bank h is allocated equally to consumers; consumer i’s profit income:
  - w_i(z^c, z^b, z^f) ≡ ∑_{j=1}^J ψ^f_ij π^f_j(z^b, z^f) + ∑_{h=1}^H ψ^b_hi π^b_h(z^c, z^b, z^f). (Equation (13))

### B. Deposit market — timing, strategy spaces, and consumer problem
- First-stage (deposit market) bank strategy for consumer i: z^b_Dhi ≡ (r^b_hi, s^b_hi).
- Strategy set for banks in deposit market: Z^b_D ≡ (R_+ ∪ {N.S.})^2. (Equation (14))
  - "N.S." (Not Specified) is an allowed element.
- Consumer i pure strategy: z^c_i = ({r^c_hi}_{h=1}^H, {s^c_hi}_{h=1}^H) ∈ Z^c ≡ R_+^{2H}.
- Constrained consumer choice sets G^c_hi(z^b_Dhi):
  - = r^b_hi × R_+ if bank specifies r^b_hi only;
  - = R_+ × (s^b_hi ∪ {0}) if bank specifies s^b_hi only;
  - = r^b_hi × (s^b_hi ∪ {0}) if bank specifies both. (Equation (15))
- Feasible set for consumer i combines constrained choice and budget:
  - B_i = { z^c_i : z^c_i ∈ G^c_i(z^b_Di) and ∑_{j=1}^J s^c_hi ∈ [0, m_i] }. (Equation (18))
- Next-period wealth:
  - m^+_i = g(z^c, z^b, z^f) = ∑_{h=1}^H r_hi s_hi + w_i(z^c, z^b, z^f). (Equation (17))
- Mixed strategies and measurability:
  - Borel σ-algebra B(X) and Λ(X) denote probability measures; mixed strategies λ^D_h (banks), μ_i (households), λ^L_h (banks in stage 2), q_j (firms) are defined as probability measures over corresponding pure-strategy spaces and conditioned on wealth distribution m and expectations.
- Consumer value function (functional equation) — lifetime utility maximization given beliefs:
  - V(m_i | m_{−i}, λ^D, μ, λ^L, q) given by integral equation (19) (explicit integral form retained in source).
- Within-period (current-period) consumer problem written with given expectations on future stationary strategies (ˆλ^D, ˆμ, ˆλ^L, ˆq) as ˆV(·) (Equation (21)).
- Definition 1: deposit market game Γ_1(m | ˆλ^D, ˆμ, ˆλ^L, ˆq) ≡ (H+I, (Λ(Z^b_D), Λ(B_i)), (Eπ^b_h, V), (m, g)). (Equation (22))
- Individual savings function S_i(m, z^b_D, S_{−i}) and aggregate savings S(m, z^b_D) ≡ ∑_{i=1}^I S_i(m, z^b_D, S_{−i}).

### C. Second stage with a monopoly bank — characterization
- If a single bank M captures all savings (M ∈ {1,2,···, H} with s_{Mi} per consumer), its loan-market strategy z^b_M ≡ (R^b_Mj, k^b_Mj)_{j=1}^J with strategy set Z_M ≡ (R_+ ∪ {N.S.})^{2J}.
- Firm j constrained choices G^f_M(z^b_M) defined analogously to consumers; no borrowing (k_j = 0) is always in choice set. (Equation (23))
- Firm best response BR^f_M(z^b_M, k^f_{−j}) = arg max_{z^f_M ∈ G^f_M(z^b_M)} π^f(k^f_j, K^f_j, R^f_j). (Equation (24))
- Monopoly bank profit maximization (given deposit outcomes z^D):
  - max_{z^b_M ∈ Z_M} π^b_M(z^b_M | z^D) ≡ ∑_{j=1}^J R^{f*}_{Mj}(·) k^{f*}_{Mj}(·) − ∑_{i=1}^I r_{Mi} s_{Mi}, subject to ∑_{j=1}^J k^{f*}_{Mj} ≤ ∑_{i=1}^I s_{Mi}. (Equations (25)–(26))
- Definition 2: second stage with monopoly bank Γ_M(z^D) ≡ (1+J, (Λ(Z^b_M), Λ(G^f_M)), (Eπ^b_M, Eπ^f_j)). (Equation (27))
- Lemma 1 (Nash equilibrium of Γ_M):
  - Optimal monopoly decisions satisfy:
    - ∑_{j=1}^J k^{*}_{Mj} = ∑_{i=1}^I s_{Mi}. (Equation (28))
    - k^{*}_{Mj} = (∑_{i=1}^I s_{Mi})/J. (Equation (29))
    - R^{b*}_{Mj} = A. (Equation (30))
  - Firms’ optimal decision is to “accept.” (Interpretation: full utilization of funds, symmetric allocation across firms, loan rate equals social planner return A.)

### D. A weak link between sources and uses of funds — motivations and formalization
- Fundamental nonexistence problem (without further institutional settings):
  - Banks face two opposing profitable strategies:
    1. Compete aggressively in deposit market to become monopolist (bid up r).
    2. Collect small deposits and free-ride by investing in a firm, profiting from others’ investments.
  - This creates a discontinuity: until deposit rates reach monopoly loan rate, aggressive competition is profitable; at that rate banks prefer to shrink size.
- To eliminate the dilemma, one can (i) impose strict interest-rate control at Walrasian rate or (ii) allow banks to adjust funds via an interbank market to discourage the second strategy.
- Definition of active banks and #D:
  - D(z^D) ≡ { h ∈ H : ∑_{i=1}^I s_{hi} > 0 }, #D denotes the number of active banks.
  - If #D = 1 → monopoly in loan market; if #D ≥ 2 → competitive loan market.
- Definition 3: competitive second stage Γ_C(z^D) ≡ (#D + J, (Λ(Z^b_C), Λ(G^f_C)), (Eπ^b_h, Eπ^f_j)). (Equation (31))
- Definition 4: Γ_2(z^D) ≡ Γ_M(z^D) if D(z^D) singleton, else ≡ Γ_C(z^D). (Equation (32))
- Definition 5: strategically intermediated economy within a period Γ(m | ˆλ^D, ˆμ, ˆλ^L, ˆq) collects Γ_1, all histories for second stage, and Γ_2(z^D). (Equation (33))
- Definitions 6–7: within-period equilibrium of Γ and stationary Markov equilibrium over infinite period (consistency μ^* = ˆμ, λ^*_D = ˆλ^D, λ^*_L = ˆλ^L, q^* = ˆq).
- Assumption 2 (Weak link between sources and uses of funds):
  - In competitive second stage, there exists an equilibrium where the probability distribution over second-stage strategies is the same for all active banks, regardless of each bank’s own deposit-market performance.
  - Formally, for any active pair (l, h):
    - λ^*_h(z^b_h | z^D, λ^*_{−h}, q^*) = λ^{*\;l\h}_l(z^b_l | ̃z^D, λ^{*\;l\h}_{−l}, q^{*\;l\h}), (Equation (34))
    - where z^D and ̃z^D deliver same active banks and same aggregate savings S.
  - Interpretation: bank loan-market strategies depend only on aggregate savings S, not individual bank fund positions.
- Assumption 3 (no firm discrimination): firms’ best responses do not discriminate among active banks when banks offer same contracts. (Equation (35))
- Joint equilibrium distribution over (z^b, z^f) conditional on aggregate savings S:
  - Q^*(z^b, z^f | S) ≡ λ^*(z^b | z^D, q^*) q^*(z^f | z^b). (Equation (36))
  - Permutation invariance property: Q^{*\;l\h}_{lj}(·|S) = Q^*_{hj}(·|S). (Equation (37))

### E. Unique equilibrium candidate under Assumptions 2 and 3 — Theorem 1
- Intuition: banks bid deposit rate up to monopolist loan return A because competitive second-stage expected profits are at most monopoly profit; arbitrage prevents deposit-loan rate gaps.
- Theorem 1 (Unique Equilibrium Outcome):
  - If an equilibrium exists for an economy with competitive second stage satisfying Assumptions 2 and 3, then:
    - (i) equilibrium offers always specify deposit rate equal to A (r = A);
    - (ii) both specifying no deposit amount ({N.S.}) and specifying depositor’s willingness to supply at rate A (s^b_hi = S(m,(A, N.S.))) are dominant strategies;
    - (iii) depositors face deposit rate offer A and deposit their willingness to supply at A, so aggregate savings = S(m,(A, N.S.));
    - (iv) equilibrium loan offer by any bank is (A, S(m,(A, N.S.))/J), same as monopolist’s; and
    - (v) firms accept this offer.
  - Result: equilibrium candidate is Pareto-optimal allocation with aggressive competition by many active banks (subject to existence).

### F. Strategic tâtonnement — institution that supports Assumption 2 and existence
- To ensure existence and implement Assumption 2, the paper introduces strategic tâtonnement:
  - Banks have free recontracting opportunities across T ≥ 2 sessions within the period (e.g., morning and afternoon).
  - Sessions τ ∈ {1,2,···, T} have five phases: (1) banks offer tentative loan contracts; (2) firms submit tentative decisions; (3) banks submit tentative interbank rates and net borrowing amounts and tentative match is undertaken; (4) banks confirm to firms if bank profit satisfied; (5) firms respond to confirmations.
  - Agreed bank-firm pairs repeat/ honor agreements; interbank market clearing condition: for all ρ ∈ R_+, there exists τ^* ≤ T such that ˜B_{τ^*}(ρ) = 0. (Equation (42))
- Definition 9–10: formal structure of strategic tâtonnement sessions and equilibrium of competitive second stage (set of strategies {z^*_τ}_{τ=1}^T that are Nash and clear interbank market).
- Two canonical contracts used in constructive equilibrium arguments:
  - Walrasian contract: z_w ≡ (αA, N.S.). (Equation (52))
  - Pareto-optimal contract: z_p ≡ (A, S/J). (Equation (53))
  - Target contract: z_φ ≡ (φA, S/J) with φ ∈ (α, 1]. (Equation (59))

### G. Equilibria in the loan market with strategic tâtonnement — Lemmas 2–3
- Lemma 2 (Walrasian contract is sustainable):
  - If all banks repeatedly offer z_w, firms best-respond by accepting one bank’s (αA, S/J) and rejecting others (equations (54)–(55)); interbank and confirmation strategies clear markets and confirm (equations (56)–(58)). Repeated z_w is a Nash equilibrium for sessions and supports clearing.
- Lemma 3 (Target contract enforcement via punishment):
  - Strategies supporting a target contract z_φ across sessions constitute a Nash equilibrium for finite T and a subgame perfect equilibrium for T = ∞:
    - Banks offer z_φ unless deviations already occurred; otherwise offer z_w. (Equation (60))
    - Firms accept z_φ if all offers are z_φ, else accept z_w if at least one bank offers z_w. (Equations (61)–(62))
    - Interbank strategies adjust (ρ, B) = (φA, ∑_j k^f_τhj − s_h) when bank offered z_φ, and (αA, ·) otherwise. (Equation (63))
    - Banks and firms confirm only when interbank market clears. (Equations (64)–(65))
  - Immediate clearing of loan and interbank markets is achieved; the equilibrium loan rate range satisfies αA ≤ φA ≤ A.
- Consequence: for competitive second stage with strategic tâtonnement, many second-stage equilibria exist (φ ∈ [α,1]) but deposit-market competition selects φ = 1 (i.e., loan rate A) when integrated with deposit stage.

### H. Equilibrium for the whole game — Theorem 2 (existence and characterization)
- Lemma 3 with φ = 1 supports an equilibrium for the whole game:
  - strategic tâtonnement prevents free-riders; loan rates realized at A; banks can aggressively compete for deposits without fear.
  - Assumption 2 is satisfied (banks’ equilibrium lending strategies depend only on aggregate savings S).
- Theorem 2 (Existence of an Equilibrium):
  - With strategic tâtonnement, an equilibrium exists for the strategically intermediated economy Γ. The equilibrium outcome is uniquely determined and is a Pareto-optimal allocation with many active banks, with properties:
    - (i) no arbitrage: r_hi = R_hj = ρ_h = A for all i, h, j. (Equation (66))
    - (ii) symmetric investment per firm: k_j = S(m,(A, N.S.))/J for all j. (Equation (67))
    - (iii) each firm j borrows from only one bank: k_hj = k_j, k_lj = 0 for l ≠ h. (Equation (68))
    - (iv) interbank market clears: ˜B(ρ) = 0. (Equation (69))
  - Uniqueness of equilibrium outcome (prices and allocations) even though multiple strategy profiles may support it.

### I. Discussion — interpretation, robustness, and policy implications
- Role of interbank market and recontracting:
  - Strategic tâtonnement + interbank adjustment essential to deter free-riding deviations; without interbank price-adjustment process or with single interbank chance, Lemma 3–type enforcement fails.
- Failure when firms can issue bonds directly to consumers:
  - If firms also freely issue bonds to consumers, no Nash equilibrium exists because firms can attract funds with coupons slightly above A and free-ride; consumers have incentive to accept such offers, undermining the bank-based enforcement mechanism.
- Generalizations (Appendix IV summary in text):
  - For more general production functions (non-constant returns), the Nash equilibrium loan rate equals the monopoly loan rate; equilibrium savings and investment are determined by consumer choice at monopoly rate. The equilibrium need not be socially optimal but can be Pareto-superior to Walrasian when production externalities sizable.
  - With labor inputs and disutility of labor, monopolist cannot capture all revenue; competitive banks can mimic allocation that is Pareto superior to Walrasian but not necessarily fully Pareto optimal; closer to Pareto optimal when labor supply elasticity is small.
- Industrial-organization implication:
  - Equilibrium loan contracts lead to bank-organized firm groups (trust/cartel/conglomerate–like structures) where banks coordinate client firms’ investments and internalize externalities.
- Policy implications:
  - For the canonical constant-returns-to-capital case, government intervention unnecessary; strategic intermediation can yield Pareto-optimal allocation even with production externalities.
  - Comparative implications for bank-oriented vs market-oriented financial systems:
    - Bank-oriented systems perform better when internalizing production externalities is important (development context).
    - As industrial development matures and informational frictions decline, bank-oriented advantage weakens (consistent with Japanese/German transitions).
    - Different policy prescriptions follow depending on whether bank role is primarily to internalize externalities or to mitigate information/transaction costs.
- Contribution:
  - The paper formalizes banks’ active development role: banks competitively internalize production externalities, control firms’ investment via quantity-and-price loan contracts, and endogenously form firm groups; equilibrium requires institutional features (interbank adjustment and recontracting) to exist.

*Source: _wp06264 - Appendix VI for further discussion related to the discontinuous game literature.*

### REFERENCES

### _wp06264 - REFERENCES

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### Appendix I — Proofs (major lemmas and theorem conclusions)
- Lemma 1 (proof summary): A monopolist bank sets loan rate and amount to exhaust deposits; equal capital allocation across firms is revenue-maximizing, yielding symmetric lending k_bMj = k = (∑_{i=1}^I s_{Mi})/J and the highest chargeable interest rate A with firms earning zero profit at participation constraint equality.
- Theorem 1 (structure of proof): established via six lemmas:
  - Lemma 4: Under Assumptions 2 and 3, expected revenue in the competitive loan market is the same for all active banks; expected revenue expression (A11) defines R_e.
  - Lemma 5: A bank’s expected revenue is maximized when active banks mimic monopolist loan allocation; this yields expected return R_e ≤ A and when maximized R_e = A (equations (A12)–(A14)).
  - Lemma 6: Equilibrium deposit rates are nondiscriminatory across depositors: r_hi = r_h for all i; and the no-arbitrage condition r_h = r = R_e holds (equation (A15)).
  - Lemma 7: The deposit rate r equals A in any Nash equilibrium of the whole game (r = A), so combined with Lemma 6 implies R_e = r = A (equation (A19)).
  - Lemma 8: Equilibrium loan contracts are Pareto-optimal and mimic monopolist’s allocation: each firm receives the unique offer (A, S(m,(A, N.S.))/J) and repays at gross loan rate A; symmetrical loan amount S(m,(A, N.S.))/J is the sole equilibrium offer.
  - Lemma 9: In equilibrium a depositor faces at least one unspecified offer, or combinations of specified offers replicate willingness to supply capital at A. Dominant bank strategies include (A, N.S.) and (A, S(m,(A, N.S.))/I).
- Conclusion: Theorem 1 completed; equilibrium features R_e = r = A and symmetric, Pareto-optimal allocation via strategic intermediation.

### Appendix II — Existence of an optimal plan
- Assumption summary: Assumption 1 parts (i) and (ii) ensure lifetime utility boundedness and perpetual growth in Walrasian economy.
- Lemma 10: Under Assumption 1 and r_it ∈ [αA, A] for all t, an optimal plan c*_i exists that attains the maximum of U(c_i); proof sketch relies on compactness via Tychonoff, upper semi-continuity, and Weierstrass theorem; Claims A and B:
  - Claim A: With r_it ≤ A, lifetime utility U(c_i) < ∞, using asymptotic elasticity 1−σ and condition βA^{1−σ} < 1 (equations (A1)–(A3)).
  - Claim B: With r_it ≥ αA, lifetime utility bounded below (U(c_i) > −∞) and consumption grows perpetually (equations (A4)–(A8)).

### Appendix III — Pareto-optimal allocation and Walrasian equilibrium
- Proposition 1 (First-best symmetric Pareto-optimal allocation):
  - (i) return on investment is A;
  - (ii) household wealth evolution: m^p_{it+1} = A s^p(m^p_{it});
  - (iii) savings function satisfies Euler fixed point: u'(m^p_{it} − s^p(m^p_{it})) = βA u'(A s^p(m^p_{it}) − s^p(A s^p(m^p_{it})));
  - (iv) firm investment: k^p(m^p_{it}) = (∑_{i=1}^I s^p(m^p_{it}))/J.
  - Symmetric capital allocation is uniquely optimal (Lagrangian (A10) and first-order condition (A11)); bordered Hessian negative definite (equations (A13)–(A16)).
- Corollary 1 (CRRA utility): growth rate in first-best is (βA)^{1/σ} each period and associated savings rate is β^{1/σ} A^{(1−σ)/σ}.
- Walrasian corporate bond market (recursive formulation):
  - Auctioneer best response BR_a as in (A19).
  - Firms’ best response BR_fj given by (A21): BR_fj(K_j, r) = (αA/r)^{1/(1−α)} K_j.
  - Definition of recursive Walrasian economy Γ_W(m) and equilibrium as Nash fixed point (A29).
- Proposition 2 (Walrasian equilibrium uniqueness):
  - (i) r_w = αA (equation (A30));
  - (ii) wealth law: m^w_{i+1} = αA s^w_i + w_i(αA), with s^w_i = BR^c_i(r_w | m_i) (equations (A31)–(A32));
  - (iii) investment k^w_j = (∑_{i=1}^I s^w_i)/J (A33).
- Corollary 2 (CRRA utility): growth rate in Walrasian equilibrium is (β αA)^{1/σ}; savings rate β^{1/σ} (αA)^{(1−σ)/σ}; hence Walrasian growth < Pareto-optimal growth.

### Appendix IV — Alternative production specifications
- Decreasing returns to accumulated capital (production y_{hj} = A K_j^{η} k_j^{α}):
  - Proposition 3: For η < 1−α, equilibrium of strategically intermediated economy is characterized by interest r_m = A K_m^{α+η−1} (= r_chi = R_fhj = ρ_h) and K_m = S(m,(r_m, N.A.))/J; equilibrium interest, investment, and growth may exceed Pareto-optimal allocation when banks capture average product (equations (A1)–(A3)).
- Labor inputs (Cobb-Douglas F(k_j,l_j,K_j) = A K_j^{η} k_j^{α} l_j^{1−α}):
  - With elastic labor supply, equilibrium capital share ˆα = (α + 1/ε_{L,w}) / (1 + 1/ε_{L,w}) (equation (A9)).
  - Proposition 4: strategically intermediated equilibrium yields r_chi = ρ = R_fhj = ˆα A and k_j = S(m,(ˆα A, N.A.))/J; outcome not Pareto-optimal but Pareto-superior to Walrasian when wages lower and investment higher relative to Walrasian.
  - Limits: as ε_{L,w} → 0 (inelastic labor), capital share → 1; as ε_{L,w} → ∞, capital share → α (Walrasian).

### Appendix V — Economy with private direct finance (corporate bonds issued directly to consumers)
- Strategy objects: z^f_{ijt} = (r^f_{ijt}, s^f_{ijt}) ∈ (R_+ ∪ {N.S.})^2; consumer choice sets constrained accordingly (equations (A1)–(A4)).
- Recursive game Γ_D(m) defined; equilibrium defined as stationary Markov strategies (Definitions 14–15).
- Nonexistence results:
  - Lemma 11: No pure strategy equilibrium exists with private direct finance (illustrated with two firms / representative household; symmetric candidate r = αA, investments k = S(m,(αA, N.S.))/2 unstable to profitable deviations) (equations (A9)–(A16)).
  - Proposition 5: No mixed strategy equilibrium exists in private direct finance economy; proof shows any nondegenerate mixed strategy can be profitably improved by constructing weighted-average offers that yield strictly higher firm profits while being weakly preferred by consumers (arguments use iso-profit and indifference curve geometry; see (A18)–(A19) and accompanying discussion and figures).

### Appendix VI — Relation to discontinuous game literature
- Key observations:
  - The private direct finance game lacks general existence guarantees for Nash equilibria because payoff correspondences are seriously discontinuous; better-reply secure condition (Reny, 1999) fails in key examples (e.g., deviation around Pareto-optimal bond z^f* = (A, N.S.) can make a deviator monopolist but with negative profit).
  - Simon and Zame (1990) and Yanelle (1998) constructions (endogenous sharing rules or lotteries at tie prices) require exogenous tie-breaking rules and do not resolve off-diagonal discontinuities present here.
  - Discontinuities here are not confined to diagonal sets: they occur in off-diagonal neighborhoods (e.g., (k_1 = k, k_2 = S(m,(A, N.S.))/J)), making mixed-strategy existence techniques (Dasgupta and Maskin (1986b) style) inapplicable.
  - Upper hemicontinuity of payoff correspondences fails in quantity dimensions (k^f_j), though price dimensions (r^f_j) may be upper hemicontinuous; aggregate profit correspondences can be open at critical strategy vectors (figures discussed in text).

*Italic: Source: _wp06264 - REFERENCES (PDF chapter/section).*

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