## 5. Pooling Equilibrium with Low Credit Volume

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---

### Model setup and key mechanisms
- Banks characterized by loan-technology parameter α (0≥α≥1). Choice variable: amount of credit C.
- Economy states (independent): good (probability p1), slowdown (probability p2), recession (probability 1 – p1 – p2).
  - Good state: credit yields u.
  - Slowdown: credit yields α v(C) u, with 1≥α v≥0 and v’<0 (A1).
  - Recession: credit yields g(α) w(C) u, with α≥g(α), g’(α)>0, 1≥α v(C) ≥ g(α) w(C) ≥0, and w’<0 (A2).
- Financing rate i determined by financers who have only an estimate α̃ of the bank’s true α; financers require returns equal to safe alternative s.
- Bankruptcy only in severe recession ensured by: for all α, α v(C) u > i > g(α) w(C) u (A3).
- Bank value function V(α, α̃, C) equals expected profits net of financing costs (equation (2)); first-order condition for maximization with respect to C given by equation (3). Regularity conditions (A4) and (4) ensure interior optimum and second-order conditions.

### Signaling logic and comparative statics
- V and V3 (marginal value of credit) are increasing in true α and believed α̃: better loan technology raises bank value and marginal benefit of credit.
- Credit volume C is a credible signal of loan-technology because loan technology is proprietary and unobservable; higher C is costlier (higher default probability and loss given default) particularly for poor-technology banks.
- Two-type case setup:
  - Types αH and αL, proportion θ have αH, with 1>αH>αL>0.
  - Full information benchmark: optimal credit C_F(α) solves FOC (5).
  - Under partial information, equilibria can be pooling or separating.

### Pooling equilibrium (Proposition 1) — characteristics and effects
- Definition: both types disburse same credit C_P so financers infer only average α = θαH + (1–θ)αL.
- Objective functions in pooling use α as financers’ belief (equations (6) and (7)).
- Proposition 1 (under A1–A4 and (4)):
  - Banks of type αH disburse less credit than in full information: C_P(αH) < C_F(αH).
  - Banks of type αL disburse more credit than in full information: C_P(αL) > C_F(αL).
- Consequences:
  - High-technology banks reduce volumes to avoid bearing financing costs associated with pooled beliefs; low-technology banks expand.
  - Market share shifts toward low-technology banks.
  - Total expected loan losses typically increase unless total credit falls substantially; whether total credit is higher or lower than under full information is model-dependent (depends on v(.), w(.), g(.)).

### Separating equilibrium (Proposition 2) — characteristics and effects
- Separating equilibrium: high-type banks choose C_S(αH) sufficiently large so low-type banks do not imitate.
- Incentive-compatibility constraints:
  - (10) low-type prefers not to imitate best outcome when financiers infer low type.
  - (11) high-type must prefer separating outcome to pooling outcome.
- Proposition 2 (under A1–A4, (4), and IC constraints):
  - Banks of type αH disburse more credit than in full information.
  - Banks of type αL disburse same amount as under full information.
  - Total credit is higher than under full information: C_S(αH)+C_S(αL) > C_F(αH)+C_F(αL).
- Intuition: high-quality banks extend additional credit to credibly signal superiority; marginal net benefit of lending larger for high-technology banks.

### Existence and multiplicity of equilibria (Proposition 3 and comparative statics)
- Proposition 3: parameterizations of v(.), w(.), g(.) exist for which either pooling or separating equilibria obtain.
- Boundary between pooling and separating determined by whether (11) holds; small shifts in parameters (p1, p2, s, or loan-technology functions) can move the system abruptly between equilibria.
- Example parameterization in source:
  - αH = 0.9, αL = 0.4, θ = 0.5, u = 1.5, s = 1.1, p1 = 0.7, p2 = 0.2 (recession probability 1 – p1 – p2 = 0.1). Under these values, a separating equilibrium obtains with a threshold C = 5.0 where αL is indifferent to imitating αH.
- Aggregate responses to shocks depend on equilibrium type:
  - Under pooling: total credit reaction determined by how high-technology banks are affected and by α via indirect channels.
  - Under separating: reactions constrained by incentive-compatibility and can produce non-monotonic or “perverse” comparative statics (e.g., an increase in expected return on lending may reduce credit volumes under some elasticities).

### Continuum of types — full separation result
- With α ∈ [α, α()], if regularity and single-crossing conditions hold (V3/V2 strictly monotonic in α) and an initial-value condition (worst-type bank chooses same C as under full information), Mailath (1987) implies existence of a strictly monotonic separating strategy τ(α).
- In a fully separating equilibrium:
  - All banks disburse more credit than under full symmetric information, except the bank with the very worst technology, which disburses the same amount as under full information.
  - The expected proportion of loans that fail is higher than under symmetric information.
- Single-crossing is met because expression (16) is monotonically increasing in α under assumptions v+v’C>0 and g’>0.

### Extensions
- Investment in loan technology:
  - Signaling raises incentives to invest in loan technology because investment lowers signaling costs and can move banks from pooling to separating.
  - Under asymmetric information, high-technology banks may undertake more expensive investment (higher I) than under full information. Approximate investment conditions presented in equations (17)–(19).
  - Low-technology banks may invest to avoid being left behind, potentially restoring pooling after mutual investment.
- Pervasive moral hazard and low-credit outcomes:
  - Introduce capital K and capital adequacy constraint 0 < C < ρ K (A.5). Value function then includes capital terms as in equation (21).
  - If second-order conditions fail (inequalities in (4) reversed), V can be minimized in interior range and banks may choose corner solutions (C = 0 or C = ρ K).
  - Signaling can produce equilibria with very low aggregate credit:
    - Separating equilibrium where high-technology bank chooses C = 0 to signal quality and low-technology bank chooses high C.
    - Pooling equilibrium where both types choose C = 0 (both prefer no risky lending under pooling), leading to sound banks but under-provision of finance.
  - Small parameter shifts can cause abrupt transitions across equilibria and large swings in aggregate intermediation.

### Policy implications and recommendations
- Key policy conclusions:
  - Over-extension of credit can be a rational signaling response to informational asymmetries about proprietary loan technology; both pooling and separating equilibria can generate excessive lending relative to full information, and expected loan losses are generally higher under asymmetric information.
- Supervisory and regulatory interventions to mitigate signaling-driven excess credit supply:
  - Detailed supervisory inspection of banks’ loan technology and practices to assess and improve quality.
  - Use of supervisory powers (moral suasion, Pillar 2 provisions) to impose additional capital and provisioning requirements where loan technology is weak.
  - Require disclosure about loan technology quality (without forcing proprietary methods to be revealed) and verify accuracy; even imperfect disclosure reduces pooling incentives and the incentive to signal via over-extension.
- Supervisors’ role is crucial because banks are unlikely to voluntarily reveal proprietary loan-technology details that constitute their informational rents.

### Appendix I: Expected loan losses in a pooling equilibrium — key expressions and conclusions
- Expected loan losses L for a bank of type α_i providing C composed of slowdown and recession components (expressions as provided).
- Relation via value function: (,) (    )(, ,). iii LCusCVCααα=− −
- Difference between combined expected losses of both types in pooling equilibrium and expected losses in full information equilibrium derived via substitutions and a second-order Taylor approximation around C_i^F.
- Final approximate expression: 22 11 3333 22 (  )(2)() (,,)() (,,). PFFPFFPFF HLHHHHLLLL usCCCCCVCCCVCαααα−−−−−−−
- Conclusion: the last two terms are certainly negative by second-order conditions; only if the first expression is sufficiently negative can total expected losses be smaller in pooling than in full information, which requires that total credit volume be smaller.

### Appendix II: Regularity conditions on the objective function with a continuum of bank types
- Mailath (1987, page 1352) regularity conditions (as listed):
  1. Smoothness: v, w, and g twice continuously differentiable implies linear equation (1) is twice continuously differentiable.
  2. Belief monotonicity: expression (22) is always positive because g’ and w are always positive.
  3. Type monotonicity: expression (23) is always positive provided v+v’C is positive.
  4. “Strict” quasiconcavity: V_33(α,α, φ(α))<0 given assumption in equation (4), so unique solution to (5).
  5. Boundedness: since V_33 is strictly negative, this condition is fulfilled.

*Italic source: IMF Working Paper — 5. Pooling Equilibrium with Low Credit Volume (chapter/section content).*

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

### _wp08188 - References................................................................................................................................25

### Tables
- 1. Expected Payoffs in Different States .....................................................................................8
- 2. Investment Decision Starting From and Ending at Pooling Equilibria................................19
- 3. Investment Decision Starting From and Ending at Separating Equilibria ...........................20

### Figures
- 1. Change in the ratio of credit to GDP, 2003–2007 .................................................................4
- 2. The Value Function for Different Types: Separating Equilibrium ......................................15
- 3. Credit Volumes and Bank Characteristics for a Continuum of Types.................................18
- 4. Separating Equilibrium with Low Credit Volume...............................................................22

*Source: _wp08188 - References..............................................................................................................*

### 5. Pooling Equilibrium with Low Credit Volume ...................................................................23

### 5. Pooling Equilibrium with Low Credit Volume

### Model setup and key mechanisms
- Banks characterized by loan-technology parameter α (0≥α≥1). Choice variable: amount of credit C.
- Economy has three states realized independently: good (probability p1), slowdown (probability p2), and recession (probability 1 – p1 – p2).
  - Good state: credit yields u.
  - Slowdown: credit yields α v(C) u, with 1≥α v≥0 and v’<0 (A1).
  - Recession: credit yields g(α) w(C) u, with α≥g(α), g’(α)>0, 1≥α v(C) ≥ g(α) w(C) ≥0, and w’<0 (A2).
- Financing rate i determined by financers who have only an estimate α̃ of the bank’s true α; financers require returns equal to safe alternative s.
- Assumption ensuring bankruptcy only in severe recession: for all α, α v(C) u > i > g(α) w(C) u (A3).
- Bank value function V(α, α̃, C) equals expected profits net of financing costs as in equation (2) of the source; first-order condition for maximization with respect to C given by equation (3). Regularity conditions (A4) and (4) ensure interior optimum and second-order conditions.

### Signaling logic and comparative statics
- V and V3 (marginal value of credit) are increasing in both true α and believed α̃: better loan technology increases both bank value and marginal benefit of extending credit.
- Credit volume C acts as a credible signal of loan-technology because loan technology is proprietary and unobservable; higher credit volume is costly (in terms of higher default probability and loss given default) especially for banks with poor loan technology.
- Two-type case: banks are αH (high) and αL (low), proportion θ have αH, 1>αH>αL>0.
  - Full information benchmark: optimal credit C_F(α) solves FOC (5).
  - Under partial information, equilibria can be pooling or separating.

### Pooling equilibrium (Proposition 1) — characteristics and effects
- Definition: both types disburse same credit C_P so financers infer only average α = θαH + (1–θ)αL.
- Objective functions under pooling for high and low types use α as financers’ belief (equations (6) and (7)).
- Proposition 1: under (A1)-(A4) and (4), in a pooling equilibrium:
  - Banks of type αH disburse less credit than in full information: C_P(αH) < C_F(αH).
  - Banks of type αL disburse more credit than in full information: C_P(αL) > C_F(αL).
- Consequences:
  - High-technology banks shrink volumes to avoid transferring financing costs to themselves; low-technology banks expand.
  - Market share shifts in favor of low-technology banks.
  - Total expected loan losses typically increase unless total credit falls substantially; whether total credit is higher or lower than under full information is model-dependent (depends on v(.), w(.), g(.)).

### Separating equilibrium (Proposition 2) — characteristics and effects
- Separating equilibrium: high-type banks choose C_S(αH) sufficiently large so low-type banks do not imitate.
- Two incentive-compatibility constraints:
  - (10) low-type prefers not to imitate best outcome when financiers infer low type.
  - (11) high-type must prefer separating outcome to pooling outcome.
- Proposition 2: under (A1)-(A4), (4), and the IC constraints:
  - Banks of type αH disburse more credit than in full information.
  - Banks of type αL disburse same amount as under full information.
  - Total credit is higher than under full information: C_S(αH)+C_S(αL) > C_F(αH)+C_F(αL).
- Intuition: high-quality banks extend additional credit to credibly signal superiority; marginal net benefit of lending is higher for high-technology banks.

### Existence and multiplicity of equilibria (Proposition 3 and comparative statics)
- Proposition 3: there exist parameterizations of v(.), w(.), g(.) for which either pooling or separating equilibria obtain.
- Boundary between pooling and separating determined by whether (11) holds; small parameter shifts (e.g., in p1, p2, s, or loan-technology functions) can move the system abruptly from pooling to separating or vice versa.
- Example parameterization in source:
  - αH = 0.9, αL = 0.4, θ = 0.5, u = 1.5, s = 1.1, p1 = 0.7, p2 = 0.2 (recession probability 1 – p1 – p2 = 0.1). Under these values, a separating equilibrium obtains with a threshold C = 5.0 where αL is indifferent to imitating αH.
- Aggregate responses to shocks depend on equilibrium type:
  - Under pooling: total credit reaction determined by how high-technology banks are affected and by α via indirect channels.
  - Under separating: reactions reflect incentive-compatibility constraints, and can produce non-monotonic or “perverse” comparative statics (e.g., an increase in expected return on lending may reduce credit volumes under some elasticities).

### Continuum of types — full separation result
- With α ∈ [α, α()], if regularity and single-crossing conditions hold (V3/V2 strictly monotonic in α) and an initial-value condition (worst-type bank chooses same C as under full information), Mailath (1987) implies existence of a strictly monotonic separating strategy τ(α).
- In a fully separating equilibrium:
  - All banks disburse more credit than under full symmetric information, except the bank with the very worst technology, which disburses the same amount as under full information.
  - The expected proportion of loans that fail is higher than under symmetric information.
- The single-crossing condition is met in the model because expression (16) is monotonically increasing in α under model assumptions (v+v’C>0, g’>0).

### Extensions
- Investment in loan technology:
  - Signaling raises incentives to invest in loan technology because investment lowers signaling costs and can move banks from pooling to separating.
  - Under asymmetric information, high-technology banks may undertake more expensive investment (higher I) than under full information. Approximate investment conditions presented in equations (17)–(19).
  - Low-technology banks may invest to avoid being left behind, potentially restoring pooling after mutual investment.
- Pervasive moral hazard and low-credit outcomes:
  - Introduce capital K and capital adequacy constraint 0 < C < ρ K (A.5). Value function then includes capital terms as in equation (21).
  - If second-order conditions fail (inequalities in (4) reversed), V can be minimized in interior range and banks may choose corner solutions (C = 0 or C = ρ K).
  - Signaling can produce equilibria with very low aggregate credit:
    - Separating equilibrium where high-technology bank chooses C = 0 to signal quality and low-technology bank chooses high C.
    - Pooling equilibrium where both types choose C = 0 (both prefer no risky lending under pooling), leading to sound banks but under-provision of finance.
  - Small parameter shifts can cause abrupt transitions across equilibria and large swings in aggregate intermediation.

### Policy implications and recommendations
- Over-extension of credit can be a rational signaling response to informational asymmetries about proprietary loan technology; both pooling and separating equilibria can generate excessive lending relative to full information, and expected loan losses are generally higher under asymmetric information.
- Supervisory and regulatory interventions can mitigate signaling-driven excess credit supply:
  - Detailed supervisory inspection of banks’ loan technology and practices to assess and improve quality.
  - Use of supervisory powers (moral suasion, Pillar 2 provisions) to impose additional capital and provisioning requirements where loan technology is weak.
  - Require disclosure about loan technology quality (without forcing proprietary methods to be revealed) and verify accuracy; even imperfect disclosure reduces pooling incentives and the incentive to signal via over-extension.
- Supervisors’ role is crucial because banks are unlikely to voluntarily reveal proprietary loan-technology details that constitute their informational rents.

*Italic source: IMF Working Paper — 5. Pooling Equilibrium with Low Credit Volume (chapter/section content).*

### REFERENCES

### _wp08188 - REFERENCES

### References
- Backé, P., B. Égert, and T. Zumer (2006), “Credit Growth in Central and Eastern Europe: New (Over)Shooting Stars?,” Focus on European Economic Integration 1/06 (Vienna: Oesterreichische Nationalbank).
- Barajas, Adolfo, Giovanni Dell’Ariccia, and Andrei Levchenko (2007), “Credit Booms: The Good, The Bad, and The Ugly,” mimeo.
- Bolt, Wilko, and Alexander F. Tieman (2004), “Banking Competition, Risk and Regulation,” Scandinavian Journal of Economics 106: 783-804.
- Cottarelli, Carlo, Giovanni Dell’Ariccia, and Ivana Vladkova-Hollar (2005), “Early Birds, Late Risers, and Sleeping Beauties: Bank Credit Growth in Central and Eastern Europe and in the Balkans,” Journal of Banking and Finance 29: 83-104.
- Dell’Ariccia, Giovanni, and Robert Marquez (2005), “Lending Booms and Lending Standards,” CEPR Discussion Paper No. 5095.
- Dell’Ariccia, Giovanni, Deniz Igan, and Luc Leaven (2008), “Credit Booms and Lending Standards: Evidence from the Subprime Mortgage Market,” IMF Working Paper WP/08/106.
- International Monetary Fund (2006), World Economic Outlook: Globalization and Inflation, Washington D.C.
- Jimenez, Gabriel, and Jesus Saurina (2005), “Credit Cycles, Credit Risk, and Prudential Regulation,” Bank of Spain Working Paper No. 0531.
- Kraft, Evan, and Ljubinko Jankov (2005) “Does speed kill? Lending booms and their consequences in Croatia,” Journal of Banking and Finance 29: 105–129.
- Mailath, George J. (1987), “Incentive Compatibility in Signaling Games with a Continuum of Types,” Econometrica 55, No. 6 (Nov.), pp. 1349–1365.
- Milgrom, Paul, and John Roberts (1982), “Limit Pricing and Entry Under Incomplete Information: An Equilibrium Analysis,” Econometrica, Vol. 50, pp. 443–459.
- Enoch, Charles, and Inco Otker-Robe, Editors (2007), Rapid credit growth in Central and Eastern Europe: endless boom or early warning?, Palgrave Macmillan, Hampshire and New York.
- Riley, John G. (1979), Information Equilibria,” Econometrica, Vol. 47, pp. 331–359.
- Rothschild, Michael, and Joseph E. Stiglitz (1976), “Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information,” Quarterly Journal of Economics, Vol. 90, pp. 629–649.
- Schaffer, S. (1998), “The Winner’s Curse in Banking,” Journal of Financial Intermediation 7: 359-392.
- Spence, Michael A. (1973), “Job Market Signaling,” Quarterly Journal of Economics, Vol. 87, pp. 355–374.

*References list as provided in the source PDF.*

### Appendix I: Expected loan losses in a pooling equilibrium — key expressions and conclusions
- Definition: Consider the expected loan losses L of a bank of type α_i providing C in credit. The total is composed of expected losses in the slowdown state and the expected losses in the recession, including accumulated interest:
  - 212iii
    (    ,   )(1(   ))(1)(1(    )   (   )).LCpvCuCpp   g wCuCααα=−+−− −
- Relation via value function:
  - (,) (    )(, ,).
    iii
    LCusCVCααα=− −
- Difference between combined expected losses of both types in pooling equilibrium and expected losses in full information equilibrium:
  - (,)(,)(,())(,())
    HLHHLL
    LCLCLCLCαααααα+−    +   =
    PP FF
    2()(,,)(,,)
    PPP
    HHLL
    usC  VC   VCαααα−−−−
    **
    (,,).
    FF
    LLL
    V Cαα

    (,,)()(,,)()
    PFF
    LLHH H HL
    VCusCV   CusCαααα−−  +−−  +
- Second order Taylor approximation of V(α_i, α_i, C_P) around C_i^F:
  - 2
    1
    33
    2
    )   (, ,  )()(, ,  )  ()
    PFPFFP
    3
    (, ,(, ,  ),
    FF
    i
    VCVCCCV CCCV  Cαααααααα≈+−   +− 
    iiii...
  - Because V_3(α_i, α_i, C_P)=0, hence:
    - 2
      1
      33
      2
      (, ,  )   (, ,  )  ()   (, ,  ).
      PFPFF
      iiiiiiii
      VCVCCCV  Cαααααα≈+−
- Substitution yields approximate difference:
  - (,)(,)(,())(,())
    PP FF
    HLHHLL
    LCLCLCLCαααααα+−    +   ≈
- Final substituted expression for the difference between expected losses:
  - 22
    11
    3333
    22
    (  )(2)() (,,)() (,,).
    PFFPFFPFF
    HLHHHHLLLL
    usCCCCCVCCCVCαααα−−−−−−−
- Conclusion:
  - The two last terms are certainly negative by the second order conditions for a maximum, and they enter with negative signs. Hence, only if the first expression is sufficiently negative can total expected losses be smaller in the pooling equilibrium than in the full information equilibrium, which requires that total credit volume be smaller.

### Appendix II: Regularity conditions on the objective function with a continuum of bank types
- Mailath (1987, page 1352) regularity conditions listed:
  1. Smoothness:
     - Assuming that v, w, and g are all twice continuously differentiable, the linear equation (1) is also twice continuously differentiable.
  2. Belief monotonicity:
     - 212
       , , ) (1) ()(VC ppgααα
       ′
       =− −
       %

       %
                                       (22)
     - This is always positive (and thus never zero) because g’ and w are always positive.
  3. Type monotonicity:
     - 132
       (, , )()VCpvvCuαα
       ′
       =+
       %
                                                   (23)
     - This is always positive (and thus never zero) provided that v+v’C is positive, as has been assumed.
  4. “Strict” quasiconcavity:
     - The condition V_3(α,α,C)=0 is provided in equation (5).
     - Furthermore,
       33212
       (   ,   ,   )(2)(1)   (   )(2)VCpvvCuppgwwCuαααα
       ′′′′   ′′
       =++−−   +                  (24)
     - This expression is strictly negative given the assumption in equation (4). Hence, V_33(α,α, φ(α))<0, so there is a unique solution to (5).
  5. Boundedness:
     - Since V_33 is strictly negative, this condition is always fulfilled.

*Appendices and mathematical expressions as provided in the source PDF.*

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