## 9. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model: Time

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### Introduction and research questions
- Context: debate over loosening capital requirements during recessions and tightening during expansions; critiques of the 1988 Basel Accord and discussions leading to Basel II. Key citations mentioned: Kashyap and Stein (2003 and 2004); Goodhart, Hoffmann, and Segoviano (2004); Pennacchi (2004); Borio (2003); Danielsson and others (2001).
- Core questions:
  - How should bank regulators set capital requirements in different phases of the business cycle?
  - Should policy responses differ depending on whether shocks are loan supply (loan write-offs) or productivity (loan demand) shocks?
  - Should regulators respond differently when productivity (loan demand) shocks are anticipated rather than unanticipated?
- High-level preview of main results:
  - Increase capital requirements in response to negative loan supply shocks.
  - Decrease capital requirements in response to negative productivity (loan demand) shocks.
  - Preemptively lower capital requirements when a negative productivity shock is anticipated to smooth consumption and deposits.

### Unrestricted model: structure and mechanism
- Economy and agents:
  - Closed economy populated by households, firms, banks, deposit insurers, and government.
  - Households: supply labor inelastically (h_t^n = 1 for all t), own banks, consume a single storable good, hold portfolio of bank equity (h_t^k) and deposits (h_t^d) with b_t^h = k_t^h + d_t^h.
  - Firms: produce y_t = A_t f(l_t, n_t) with idiosyncratic productivity shocks A_it ∈ {A, 0} (high with probability p, low with probability 1 − p).
  - Banks: lend to firms (each bank i lends to single firm i), hold capital and deposits, buy deposit insurance.
  - Deposit insurers: offer full deposit insurance, charge fair premiums, incur operational costs C_t(d_t, l_t) homogeneous of degree one and convex.
  - Government: imposes full deposit insurance and capital requirements to internalize cross-bank externalities via deposit insurance channel.
- Households — preferences and first-order conditions:
  - Lifetime utility: ∫_0^∞ e^{−βt} u(c_t, d_t) dt, with u(.) homogeneous of degree one, strictly increasing and concave; subjective discount rate β > 0.
  - First-order conditions (as given):
    - ( ,1)_t c u = λ_t  (equation (5))
    - ( ,1)_t d u = (r_t − r_t^d) λ_t  (equation (6))
    - λ̇_t = (r_t − β) λ_t  (equation (7))
  - Consumption-deposit ratio c_t/d_t is uniquely determined by the equity-deposit spread (r_t − r_t^d); assumption ∂u/∂d > 0 implies complementarity between consumption and deposits.
  - Deposit demand: d_t = δ( r_t − r_t^d, c_t ).
- Firms:
  - Production: y_it = A_it f(l_it, n_it).
  - Two productivity states: high A and low 0; expected productivity E[A_it] = A_t.
  - In high state firm labor choice: ( , )_t n f = w_t A_t (equation (9)); low state yields zero labor and returns.
- Banks:
  - Balance sheet: k_it + d_it = l_it (equation (11)).
  - Deposit insurance premium per unit of loan τ_it(l_it, k_it, p) in high-revenue state; τ decreasing in k and increasing in l.
  - Expected bank profit per bank and first-order conditions for loans and capital defined by equations (12)–(16); optimal d, k, l are functions of (r_t, r_t^d, w_t, A_t, p).
- Deposit insurers:
  - Zero-expected-profit condition: p l_t τ_t − (1−p) d_t r_t^d − (1−p) C_t(d_t, l_t) = 0 (equation (17)); C homogeneous degree one.
  - Operational cost function: C_t(d_t, l_t) = l_t c(d_t/l_t, 1).
- Government and externalities:
  - Cross-bank externalities through insurer costs cause private banks to hold less capital than socially optimal.
  - Government can impose capital requirements x_it to internalize external effects; x_it derived from social marginal costs (equations (18) and (19)).

### Equilibrium and dynamic solution (unrestricted case)
- Market clearing and aggregate definitions:
  - Aggregate household assets = aggregate bank loans; h_b_t = l_t; λ_t is the shadow value of loans.
  - Aggregate bank profit Ω_t given by equation (24).
  - Aggregate flow constraint: ḃ_t = A_t f(l_t, n_t) − c_t − p C(d_t, l_t) (equation (25)).
- Key relationships:
  - Loan-deposit and equity-deposit spreads are decreasing in capital-asset ratio x_t (equations (26) and (27)).
  - c_t and d_t characterized via function z(λ_t) (equation (28)); capital requirement x_t expressed as χ(p, λ_t) (equation (30)).
  - Dynamics summarized by two differential equations in (λ_t, l_t, A_t): equations (33) and (34).
  - Steady state conditions for constant A: equations (35) and (36).
  - System exhibits saddle-path stability: λ_t is a jump variable; l_t is predetermined.

### Restricted case: exogenous constant capital requirement
- Government sets x_t = x ∈ (0,1) for all t.
- Dynamic system simplifies to equations (37) and (38) in (λ_t, l_t).
- Steady state: equations (39) and (40).
- Linearization yields saddle-path stability.
- Comparison benchmark: assume initial steady state's capital requirement satisfies x = χ(p, λ*).
- Restricted economy’s saddle path is steeper than unrestricted economy’s saddle path (Appendix III).

### Cyclical implications, dynamic scenarios, and policy prescriptions
- General policy message:
  - Policy response should depend on shock type:
    - Raise capital requirements in response to adverse loan supply shocks (loan write-offs).
    - Lower capital requirements in response to adverse productivity (loan demand) shocks.
  - If a negative productivity shock is anticipated, regulators should preemptively lower capital requirements to reduce the required subsequent cut and improve welfare.
- Scenario A — Unanticipated and permanent reductions in productivity (A drops from A^1 to A^2 < A^1 at t = 0)
  - Unrestricted economy dynamics:
    - λ_t jumps down on impact and travels along saddle path U_SP to new steady state.
    - Capital requirement x_t jumps down on impact and increases over time back to long-run level.
    - Bank loans l_t decrease monotonically to new lower steady state.
    - Output y_t jumps down on impact (due to discrete fall in productivity) then decreases monotonically to lower long-run level.
    - Deposits d_t, consumption c_t, and consumption-deposit ratio c_t/d_t jump up on impact and decrease smoothly toward lower steady state levels.
  - Mechanism summary:
    - Lower A → lower marginal productivity of loans and wage → loan, deposit, and equity returns fall → households dissave and increase deposits (complement to consumption) → deposit supply expands, deposit rate falls, equity-deposit spread widens.
    - Banks cannot change loans on impact → increased household deposits imply equal reduction in bank equity and fall in capital-asset ratio.
    - During transition interest rates rise and output and consumption fall as loans decline.
  - Comparison with restricted economy:
    - Shadow value λ_t falls more in restricted economy on impact.
    - Restricted economy: larger increases in consumption-deposit ratio and equity-deposit spread; loans and output decline more slowly in low-productivity period.
  - Welfare implication and recommendation:
    - Lowering capital requirement in response to negative and permanent productivity shocks amplifies output decline but enhances welfare by releasing liquidity that complements higher household consumption. Net welfare gains offset insurer operational costs; hence, lowering capital requirements is recommended for productivity shocks.
- Scenario B — Unanticipated reductions in loan supply (discrete fall in l at t = 0)
  - Unrestricted economy dynamics:
    - λ_t jumps up on impact, then declines along saddle path U_SP to steady state.
    - Capital requirement x_t jumps up on impact and declines over time.
    - Bank loans l_t increase monotonically after impact as economy returns to steady state.
    - Output y_t jumps down on impact (due to discrete fall in loans) then rises monotonically toward long-run level.
    - Deposits d_t, consumption c_t, consumption-deposit ratio c_t/d_t, and equity-deposit spread r_t − r_t^d jump down on impact and then increase smoothly during transition.
  - Mechanism summary:
    - Decline in loans and output raises marginal productivity of loans → lending, deposit, and equity rates increase → households reduce consumption and deposits → deposit supply falls, deposit rate increases, equity-deposit spread narrows.
    - Banks finance loans with more equity relative to deposits → capital-asset ratio rises.
    - During transition loans are rebuilt, output rises, interest rates and capital requirement decline; consumption and deposits increase.
  - Comparison with restricted economy:
    - Impact decline in λ_t larger in restricted case.
    - Restricted case: deposits decline less (liabilities cannot be restructured); equity-deposit spread declines more.
    - Restricted economy’s lower capital requirements lead to higher deposit insurance premiums and higher aggregate insurer operational costs during transition, shrinking intertemporal consumption possibilities.
  - Welfare implication and recommendation:
    - Raising capital requirements in response to adverse loan supply shocks is optimal: it increases household incentives to cut consumption and save, accelerates restoration of loan stock, and reduces insurer operational costs.
- Scenario C — Anticipated and permanent reductions in productivity (household learns at t = 0 that at t = T productivity will fall from A^1 to A^2 < A^1)
  - Unrestricted economy dynamics:
    - At t = 0, λ_t jumps down while l_t remains constant.
    - During [0, T] the economy follows a path intersecting the saddle path for steady state 2 exactly at t = T; l_t and λ_t decline meanwhile.
    - After realization at t = T, economy moves along saddle path to steady state 2 with loans declining and λ_t increasing over time.
  - Policy recommendation:
    - Regulators should lower capital requirements preemptively in response to an anticipated negative productivity shock to avoid a larger required reduction at shock realization and to enhance welfare by permitting smoother intertemporal consumption and deposit adjustments.

### Appendix IV — Anticipated/unanticipated permanent and temporary reductions (selected dynamics)
- Anticipated and unanticipated permanent reduction (Figure 7 highlights):
  - Capital requirement x_t:
    - jumps down on impact,
    - decreases over time to reach minimum when shock realized (at t = T),
    - thereafter increases and returns to steady state.
  - Bank loans l_t decrease monotonically toward steady state 2.
  - Output y_t decreases smoothly until t = T, jumps down discretely at t = T, then decreases smoothly to lower long-run level.
  - Deposits d_t and consumption c_t jump up on impact and decrease smoothly toward lower steady states.
  - Intuition: pre-shock dissaving (increase in consumption and deposits) driven by fall in future productivity; banks cannot adjust loans on impact, so increased deposits reduce equity and capital-asset ratio.
  - Comparison: anticipation allows preemptive regulator and household responses, enabling greater intertemporal smoothing; recommended policy is preemptive lowering of capital requirement.
- Unanticipated and temporary reduction in productivity (setup and Figure 8 and 9 findings):
  - Setup: initial steady state 1; at t = 0 productivity A_t falls temporarily from A^1 to A^2 < A^1 and returns to A^1 at t = T. Fall is unanticipated at t = 0; duration T known.
  - Phase-diagram outcomes:
    - On impact the economy moves to a point 0+ and evolves to converge to saddle path 1_SP exactly at t = T.
    - Short-lasting temporary shock (S): trajectory intersects 1_SP at t = S = T without crossing line l̇_t^{,2} = 0.
    - Long-lasting temporary shock (L): trajectory intersects 1_SP at t = L = T after having crossed l̇_t^{,2} = 0 at some t' < L.
  - Time-paths (short-lasting case):
    - Capital requirement x_t:
      - jumps down on impact,
      - increases during low-productivity interval,
      - reaches maximum at t = T,
      - thereafter converges from above to steady state.
    - Output y_t:
      - jumps down on impact,
      - decreases until t = T as loans decline,
      - jumps up at t = T when productivity returns,
      - then rises smoothly following loans.
    - Deposits d_t and consumption c_t:
      - jump up on impact,
      - decrease during low-productivity period to lowest levels at t = T,
      - thereafter recover to initial steady state levels.
  - Intuition:
    - Temporary shock is combination of unanticipated permanent decline and anticipated future increase; initial shadow value of loans falls, then as t approaches T the discounted future reversal dominates and households save to rebuild loans.
    - Reduction of capital requirement on impact is larger when decline is permanent rather than temporary; shorter low-productivity interval → smaller initial reduction in capital requirement.
  - Policy implication for temporary shocks:
    - Lower capital requirements at beginning of temporary low-productivity period and increase above steady state before period end. Time-varying policy provides high deposit liquidity when dissaving is optimal and low deposit liquidity when saving is optimal.

### Main conclusions and policy takeaways (Section V)
- Answers to the three policy questions:
  1. How set capital requirements across business cycle?
     - For negative loan supply shocks: tighten capital requirements and lower deposit insurance premiums.
     - For negative productivity shocks: loosen capital requirements and increase deposit insurance premiums.
     - For anticipated productivity shocks: preemptively lower capital requirements and increase deposit insurance premiums.
  2. Should policy differ by shock type?
     - Yes: loan supply shocks call for tighter capital; productivity shocks call for looser capital.
  3. Should responses differ when shocks are anticipated?
     - Yes: anticipation justifies preemptive easing of capital requirements to smooth consumption and deposits.
- Welfare emphasis:
  - Amplification of output fluctuations by capital standards is not necessarily welfare-reducing; lowering capital requirements in response to negative productivity shocks can deepen recessions yet improve welfare via consumption smoothing.
  - Caveat: model omits channels (e.g., employment fluctuations) that in practice may make output fluctuations costly; intertemporal leisure smoothing could favor dampening output and employment volatility.
- Limitations and future research:
  - A stochastic model could extend insights.
  - Current framework assumes bankruptcy costs are borne within banking system via fairly priced deposit insurance; real-world crises often involve general taxation.
  - Future work should explore jointly optimal fiscal and capital requirement policies in stochastic general equilibrium settings.
- Policy caution:
  - Authors caution against reflexively relaxing capital requirements during recessions; careful evaluation is needed before implementation.

*Source: _wp05168 - 9. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model: Time*

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

### _wp05168 - References

### Figures
- 1. Phase Diagrams: Unrestricted (U) and Restricted (R) Models ............................................29
- 2. Unanticipated and Permanent Reduction in Productivity in Unrestricted (U) and  
    Restricted (R) Models ..........................................................................................................29
- 3. Unanticipated and Permanent Reduction in Productivity in Unrestricted (U) and  
     Restricted (R) Models: Time Paths of Selected Variables..................................................30
- 4. Loan Write-Offs in Unrestricted (U) and Restricted (R) Models ........................................31
- 5. Loan Write-Offs in Unrestricted (U) and Restricted (R) Models: Time  
    Paths of Selected Variables..................................................................................................31
- 6. Anticipated and Permanent Reduction in Productivity in Unrestricted Model ...................32
- 7. Anticipated (AN) and Unanticipated (UN) Permanent Reduction in Productivity in  
    Unrestricted Model: Time Paths of Selected Variables .......................................................32
- 8. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model ...............33

*Source: _wp05168 - References.*

### 9. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model: Time

### 9. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model: Time

### Introduction and research questions
- Context: debate over loosening capital requirements during recessions and tightening during expansions; critiques of the 1988 Basel Accord and discussions leading to Basel II. Cited works include Kashyap and Stein (2003 and 2004); Goodhart, Hoffmann, and Segoviano (2004); Pennacchi (2004); Borio (2003); Danielsson and others (2001).
- Core questions addressed:
  - How should bank regulators set capital requirements in different phases of the business cycle?
  - Should policy responses differ depending on whether shocks are loan supply (loan write-offs) or productivity (loan demand) shocks?
  - Should regulators respond differently when productivity (loan demand) shocks are anticipated rather than unanticipated?

- Main high-level results previewed:
  - Increase capital requirements in response to negative loan supply shocks.
  - Decrease capital requirements in response to negative productivity (loan demand) shocks.
  - Preemptively lower capital requirements when a negative productivity shock is anticipated to smooth consumption and deposits.

### Unrestricted model: structure and mechanism
- Economy: closed, populated by households, firms, banks, deposit insurers, and government.
- Key features:
  - Households supply labor inelastically (htn = 1 for all t), own banks, consume a single storable good, and hold a portfolio of bank equity (htk) and deposits (htd) with bth = kth + dth.
  - Firms produce yt = A_t f(l_t, n_t) with idiosyncratic productivity shocks itA ∈ {A, 0} (high with probability p, low with probability 1 − p).
  - Banks lend to firms, hold capital and deposits, and buy deposit insurance; each bank i lends to a single firm i (specialization).
  - Deposit insurers offer full deposit insurance, charge fair premiums, and incur operational costs C_t(d_t, l_t) homogeneous of degree one and convex.
  - Government imposes full deposit insurance and capital requirements; the capital requirement is used to internalize cross-bank externalities via the deposit insurance channel.

- Households:
  - Lifetime utility: ∫_0^∞ e^{−βt} u(c_t, d_t) dt, with u(.) homogeneous of degree one, strictly increasing and concave. Subjective discount rate is β > 0.
  - First-order conditions:
    - ( ,1)_t c u = λ_t  (equation (5))
    - ( ,1)_t d u = (r_t − r_t^d) λ_t  (equation (6))
    - λ̇_t = (r_t − β) λ_t  (equation (7))
  - Consumption-deposit ratio c_t/d_t is uniquely determined by the equity-deposit spread (r_t − r_t^d); assuming ∂u/∂d > 0 indicates complementarity between consumption and deposits.
  - Deposit demand can be written d_t = δ( r_t − r_t^d, c_t ).

- Firms:
  - Production: y_it = A_it f(l_it, n_it).
  - Two productivity states: high A and low 0; expected productivity E[A_it] = A_t.
  - In high state, firm chooses labor according to ( , )_t n f = w_t A_t (equation (9)); in low state labor and returns are zero.

- Banks:
  - Balance sheet: k_it + d_it = l_it (equation (11)).
  - Bank pays deposit insurance premium per unit of loan τ_it(l_it, k_it, p) in high-revenue state; τ is decreasing in k and increasing in l.
  - Expected bank profit per bank: equation (12).
  - First-order conditions for loans and capital given by (13) and (14); optimal d, k, l are functions of (r_t, r_t^d, w_t, A_t, p) (equation (15)).
  - Equity return in high state given by a formula reflecting zero expected profits and state contingent returns (equation (16)).

- Deposit insurers:
  - Zero-expected-profit condition: (1) p l_t τ_t − (1)(1−p) d_t r_t^d − (1)(1−p) C_t(d_t, l_t) = 0 (equation (17)), with C homogeneous degree one.
  - Operational cost function written C_t(d_t, l_t) = l_t c(d_t/l_t, 1) (notation and properties described).

- Government and externalities:
  - Cross-bank externalities through insurer costs lead private banks to hold less capital than socially optimal.
  - Government can impose capital requirements to internalize external effects: requirement x_it is minimum capital-asset ratio derived from social marginal costs (equations (18) and (19)).

### Equilibrium and dynamic solution (unrestricted case)
- Aggregate market clearing: aggregate household assets = aggregate bank loans; hb_t = l_t; equilibrium implies λ_t is shadow value of loans.
- Key equilibrium relationships:
  - Aggregate bank profit Ω_t given by equation (24).
  - Aggregate flow constraint (25): ḃ_t = A_t f(l_t, n_t) − c_t − (1) p C(d_t, l_t) with notation ξ(.,.) described.
  - Loan-deposit and equity-deposit spreads are decreasing in the capital-asset ratio x_t (equations (26) and (27)).
  - c_t and d_t characterized via function z(λ_t) (equation (28)), and capital requirement x_t expressed as χ(p, λ_t) (equation (30)).
  - Two differential equations characterize dynamics in (λ_t, l_t, A_t): equations (33) and (34).
  - Steady state conditions for constant A: equations (35) and (36).
  - System exhibits saddle-path stability; λ_t is a jump variable, l_t is predetermined.

### Restricted case: exogenous constant capital requirement
- Government sets x_t = x ∈ (0,1) for all t.
- Dynamic system simplifies to equations (37) and (38) in (λ_t, l_t).
- Steady state satisfies (39) and (40).
- Linearization around steady state yields saddle-path stability.
- For comparison, assume initial steady state's capital requirement satisfies x = χ(p, λ*).
- Restricted economy’s saddle path is steeper than unrestricted economy’s saddle path (Appendix III).

### Cyclical implications and policy prescriptions
- General message:
  - Policy response should depend on shock type:
    - Raise capital requirements in response to adverse loan supply shocks (loan write-offs).
    - Lower capital requirements in response to adverse productivity (loan demand) shocks.
  - If a negative productivity shock is anticipated, regulators should preemptively lower capital requirements to reduce the required subsequent cut and improve welfare.

- A. Unanticipated and permanent reductions in productivity (A drops from A^1 to A^2 < A^1 at t = 0)
  - Unrestricted economy dynamic responses:
    - λ_t jumps down on impact, travels along saddle path U_SP to new steady state.
    - Capital requirement x_t jumps down on impact and increases over time back to long-run level.
    - Bank loans l_t decrease monotonically to new lower steady state.
    - Output y_t jumps down on impact (due to discrete fall in productivity) then decreases monotonically to lower long-run level.
    - Deposits d_t, consumption c_t, and consumption-deposit ratio c_t/d_t jump up on impact and decrease smoothly toward lower steady state levels.
  - Mechanism:
    - Lower A reduces marginal productivity of loans and wage rate → loan, deposit, and equity returns fall → households dissave (increase consumption) and increase deposits (complementary with consumption), expanding deposit supply and lowering deposit rate, widening equity-deposit spread.
    - Banks cannot change loans on impact, so increased household deposits imply an equal reduction in bank equity and a fall in capital-asset ratio.
    - During transition, interest rates rise and output and consumption fall as loans decline.
  - Comparison with restricted economy:
    - Shadow value λ_t falls more in restricted economy on impact.
    - Because banks cannot change liability composition, restricted economy exhibits larger increases in consumption-deposit ratio and equity-deposit spread.
    - Loans and output decline more slowly in restricted economy during low-productivity times.
  - Welfare implication and policy recommendation:
    - Lowering the capital requirement in response to negative and permanent productivity shocks amplifies output decline but enhances welfare by releasing liquidity that complements higher household consumption. Net welfare gains offset insurer operational costs; thus lowering capital requirements is recommended for productivity shocks.

- B. Unanticipated reductions in loan supply (discrete fall in l at t = 0: initial l_0 to l_0 + l_00 with l_0 + l_00 > l_0)
  - Unrestricted economy dynamic responses:
    - λ_t jumps up on impact, then declines along saddle path U_SP to steady state.
    - Capital requirement x_t jumps up on impact and declines over time.
    - Bank loans l_t increase monotonically after impact as economy returns to steady state.
    - Output y_t jumps down on impact (due to discrete fall in loans) then rises monotonically toward long-run level.
    - Deposits d_t, consumption c_t, consumption-deposit ratio c_t/d_t, and equity-deposit spread r_t − r_t^d jump down on impact and then increase smoothly during transition.
  - Mechanism:
    - Decline in loans and output raises marginal productivity of loans and lending, deposit, and equity rates → households reduce consumption and deposits → deposit supply falls → deposit rate increases and equity-deposit spread narrows.
    - Banks respond by financing loans with more equity relative to deposits, raising capital-asset ratio.
    - During transition loans are rebuilt, output rises, interest rates and capital requirement decline; consumption and deposits increase.
  - Comparison with restricted economy:
    - Impact decline in λ_t larger in restricted case.
    - Because liabilities cannot be restructured, deposits decline less in restricted case; equity-deposit spread declines more in restricted case.
    - Restricted economy has lower capital requirements, leading to higher deposit insurance premiums and higher aggregate insurer operational costs during transition, shrinking intertemporal consumption possibilities.
  - Welfare implication and policy recommendation:
    - Raising capital requirements in response to adverse loan supply shocks is optimal: it increases households’ incentives to cut consumption and save (via easier deposit reduction), accelerates restoration of loan stock, and reduces insurer operational costs. Thus, increase capital requirements for loan supply shocks.

- C. Anticipated and permanent reductions in productivity (household learns at t = 0 that at t = T productivity will fall from A^1 to A^2 < A^1)
  - Unrestricted economy dynamic responses:
    - On impact (t = 0), λ_t jumps down while l_t remains constant.
    - During [0, T], economy follows a path that intersects the saddle path associated with steady state 2 exactly at t = T; l_t and λ_t decline meanwhile.
    - After realization at t = T, economy moves along saddle path to steady state 2 with loans declining and λ_t increasing over time.
  - Policy recommendation:
    - Regulators should lower capital requirements preemptively in response to an anticipated negative productivity shock to avoid a larger required reduction at shock realization and to enhance welfare by permitting smoother intertemporal consumption and deposit adjustments.

### Summary of policy rules and intuition
- For negative loan supply shocks (loan write-offs): increase capital requirements. Rationale: higher capital requirements widen incentives for households to save (easier to reduce deposits), narrow equity-deposit spread, reduce deposit insurance costs, and speed recovery of loans and output.
- For negative productivity (loan demand) shocks: decrease capital requirements. Rationale: lower requirements release deposit liquidity that supports household dissaving and consumption smoothing; despite amplifying output decline, welfare improves through enhanced liquidity and consumption smoothing.
- For anticipated negative productivity shocks: preemptively lower capital requirements to improve welfare via smoother intertemporal adjustment and to reduce the magnitude of required post-shock cuts.

*Source: _wp05168 - 9. Unanticipated and Temporary Reduction in Productivity in Unrestricted Model: Time*

### Appendix IV:

### _wp05168 - Appendix IV:

### Anticipated and Unanticipated Permanent Reduction in Productivity (Unrestricted Economy)
- Dynamic responses (Figure 7):
  - The capital requirement t x:
    - jumps down on impact,
    - decreases over time to reach its minimum level when the productivity shock has been realized (at t T =),
    - thereafter increases and returns to its steady state level in the long run.
  - The stock of bank loans t l decreases monotonically as the economy converges to steady state 2.
  - Output t y:
    - decreases smoothly until t T =,
    - jumps down discretely at t T = due to the discrete fall in productivity,
    - then decreases smoothly toward its (lower) long-run level.
  - Deposits t d and consumption t c jump up on impact and decrease smoothly during their transitions to lower steady state levels.
- Intuition and mechanism:
  - On impact, the decline in future productivity reduces the shadow value of loans t λ, inducing households to dissave and to increase consumption and deposits via an intertemporal substitution effect.
  - The wage rate is unchanged at 0 t =, so there is no income effect on impact.
  - As deposit supply expands, a lower deposit rate and a wider equity-deposit spread restore equilibrium.
  - Because banks cannot change loans on impact, the discrete increase in household deposits implies a reduction in bank equity and in the capital-asset ratio.
  - At t T = (when the productivity shock hits):
    - the wage rate, lending, deposit, and equity rates fall discretely,
    - the capital requirement, the shadow value of loans, consumption, and deposits do not jump (they were perfectly foreseen).
  - After t T =, loans, output, consumption, and deposits continue decreasing toward lower long-run values.
- Comparison: anticipated vs. unanticipated shocks (same size at t T =)
  - Anticipation allows preemptive responses by bank regulators and households, enabling greater intertemporal smoothing of consumption and deposit holdings.
  - If regulators anticipate a negative productivity shock, they should preemptively lower the capital requirement to avoid a larger reduction that would be required in the unanticipated case.
  - Smoother time paths of consumption and deposits accompany smoother paths of the capital requirement, loans, and output.
- Welfare implication:
  - Lowering capital requirements in anticipation of future negative productivity shocks brings welfare benefits by smoothing household consumption and deposits and by inducing forward-looking dissaving.

### Unanticipated and Temporary Reduction in Productivity (Unrestricted Economy)
- Scenario setup:
  - Initially at steady state 1 (Figure 8).
  - At time 0 t =, the productivity parameter t A decreases temporarily from 1 A to 22 1 () A A A <, returning to the initial level 1 A at time t T =.
  - The temporary fall is unanticipated as of date 0 t =; the duration t T is known with certainty.
  - Steady states 1 and 2 correspond to productivity levels 1 A and 2 A.
- Phase-diagram behavior (Figure 8):
  - On impact, the economy moves to a point 0 + and evolves to converge to saddle path 1 SP exactly at time t T =.
  - Short-lasting temporary shock (S):
    - Trajectory intersects 1 SP at time S t T = without crossing the line ,2 0 t l • =.
  - Long-lasting temporary shock (L):
    - Trajectory intersects 1 SP at time L t T = after having crossed the line ,2 0 t l • = at some time ' L T <.
- Time-paths for the short-lasting case (Figure 9):
  - Capital requirement t x:
    - jumps down on impact,
    - increases during the low-productivity times,
    - reaches its maximum level at t T =,
    - thereafter converges from above to its steady state value.
  - Output t y:
    - jumps down on impact due to the discrete fall in productivity,
    - decreases until t T = as loans decline,
    - jumps up at t T = owing to the discrete increase in productivity,
    - finally rises smoothly following loans.
  - Deposits t d and consumption t c:
    - jump up on impact,
    - decrease during the low-productivity period to their lowest levels at t T =,
    - thereafter recover smoothly to initial steady state levels.
- Intuition:
  - The temporary shock is a combination of an unanticipated permanent decline and an anticipated permanent increase in productivity (discounted future reversal).
  - On impact, the shadow value of loans falls as the current negative effect offsets the positive discounted future effect.
  - As t approaches T, the positive discounted effect of the productivity reversal becomes dominant before t T =; after t T = the economy behaves qualitatively like a positive unanticipated permanent productivity shock.
  - The reduction on impact of the capital requirement is larger when the unanticipated productivity decline is permanent rather than temporary.
  - The shorter the low-productivity interval, the smaller the reduction of the capital requirement at 0 t + =.
  - During the transition, as the shadow value of loans rises, households save to build a larger stock of loans anticipating the productivity reversal.
- Policy implication for temporary shocks:
  - Capital requirements should be lowered at the beginning of the temporary low-productivity period and increased above steady state before the end of the period.
  - Such time-varying policy provides high deposit liquidity when dissaving is optimal (beginning) and low deposit liquidity when saving is optimal (end).

### Main Conclusions (Section V)
- Three policy questions addressed:
  1. How should regulators set bank capital requirements across the business cycle?
  2. Should policy differ depending on whether shocks are loan supply or productivity driven?
  3. Should responses differ when shocks are anticipated versus unanticipated?
- Answers and prescriptions:
  - For negative loan supply shocks:
    - Capital requirements should be tightened and deposit insurance premiums lowered.
    - This policy outperforms a flat capital requirement policy by incentivizing households to reduce consumption and save, enabling a more rapid recovery of bank credit and aggregate output.
  - For negative productivity shocks:
    - Capital requirements should be loosened and deposit insurance premiums increased.
    - Although this amplifies the output decline, it enhances welfare by releasing liquidity that complements higher household consumption and optimally speeds dissaving during low-productivity times.
  - For anticipated productivity shocks:
    - Regulators should lower bank capital requirements and increase deposit insurance premiums preemptively.
    - This enhances welfare by allowing greater intertemporal smoothing of household consumption and deposit holdings.
- Emphasis on intertemporal welfare maximization:
  - Output fluctuation amplification by capital standards is not necessarily welfare-reducing; lowering capital requirements in response to negative productivity shocks can deepen recessions yet improve welfare.
  - Caveat: the model omits channels (e.g., employment fluctuations) that in practice may make output fluctuations costly; smoothing leisure intertemporally could favor dampening output and employment volatility.
- Limitations and future research directions:
  - A stochastic model could provide additional insights.
  - The current framework assumes bankruptcy costs are fully borne within the banking system via a reserve-targeting, fairly priced deposit insurance system; in practice many crises are financed with general taxation.
  - Future work should explore jointly optimal fiscal and capital requirement policies in stochastic general equilibrium environments.
- Policy caution:
  - The paper flags caution regarding the view that capital requirements should be relaxed during recessions; policymakers should be careful before implementing such policies.
  - The authors express less concern about preliminary quantitative evaluations of Basel II that point to more stringent effective capital requirements during downturns.

### Appendix I–IV: Stability, Comparative Statics, and Model Comparisons (selected results)
- Unrestricted model (Appendix I):
  - Linearizing the dynamic system around steady state **(,) l λ for t A A = yields Jacobian matrix * J.
  - Elements of * J include expressions with partial derivatives such as:
    - **
      1 , (   )(,)  [(,),][1  (,)]  [(),1]'()0
      t ld x t Afl pp pp  u  zz λ λ β ξ χ λ χ λ λ λ λ • ∂ = − ⋅ + − − ⋅ ⋅   >
      ∂ %
    - **
      ,() (,)0 t ll x t A flp l λ λ λ • ∂ = − ⋅ ⋅ >
      ∂ %
    - **
      ,(   )'( )   [1   ( ,)]( ,){( )    [( ,),]} 0 t x x t l zlplpzpp λ λ λ χ λ χ λ λ ξ χ λ λ • ∂ = − ⋅ ⋅ − + ⋅ ⋅ − >
      ∂
    - **
      ,(, ) ()[(,)]0 t l x t l flp Aflp ll λ • ∂ = ⋅ − <
      ∂ % %
  - Determinant *() Det J is strictly negative; eigenvalues 1 θ, 2 θ solve 2** ()()0 tr J Det J θ − + =, implying eigenvalues of opposite signs and saddle-path stability.
  - Comparative statics: steady state functions ** 0(,,) t G l A λ λ • ==, ** 0(,,) t l H l A λ • ==. Using Cramer's rule and noting determinant |M| < 0, results include:
    - ***
      (, ) {(,)[(,)]   (,)   (,)}  0 |||| Al  l A llll GH GH Aflp fl p  fl pfl p f l p AM  Ml λλ > < − ⋅ + ⋅ ∂ ⋅ == ⋅ ⋅ − − ⋅ ∂ %%%%%
    - *
      0 || AA GH  GH l AM λλ − ⋅ + ⋅ ∂ = > ∂ .
  - For Cobb-Douglas production 1 1 ( , )( ) = fl p l p γ γ − ⋅ , 0 A λ ∂ = ∂ .
- Restricted model (Appendix II):
  - Linearization around its steady state (,) l λ yields Jacobian J with elements:
    - 1 , ()[(,)   (,)] (1   )    [(),1] '() 0 t ld l t A fl p x p x u  zz λ λ β ξ λ λ λ • ∂ = − ⋅    +    − − ⋅ ⋅  >
      ∂ %
    - ()(,)0 t ll l t Af l p l λ λ λ • ∂ = − ⋅ ⋅ >
      ∂ %
    - ,()'()(1)0 t l t l zl x λ λ λ • ∂ = − ⋅ ⋅ − >
      ∂
    - ,(, ) ()[(,)]0 t l l t l fl p A fl p ll λ • ∂ = ⋅ − <
      ∂ % %
  - Determinant of J is strictly negative; solution exhibits saddle-path stability.
  - Comparative statics: 0' (,,, ) t G l A x λ λ • ==, 0' (,,, ) t l H l A x λ • ==; determinant ' || M < 0 and Cramer's rule yields analogous comparative static signs.
  - For Cobb-Douglas production, 0 A λ ∂ = ∂ .
- Comparison of saddle-path slopes (Appendix III):
  - The trace of the Jacobians in restricted and unrestricted models are equal: *()   () ; tr J tr J =
  - Determinant in constrained model is greater than in unconstrained model, both negative: *()()0; Det J Det J << 
  - First rows of both Jacobians are identical.
  - With saddle-path stability, negative characteristic roots 11 , UR θ θ satisfy 11 0 UR θ θ << .
  - Eigenvector components for the negative root imply 11 11 0 RU e e << and saddle-path equations:
    - 11 () U t t e l l λ λ − = ⋅ − ,
    - 11 () R t t e l l λ λ − = ⋅ − .
  - Therefore, the restricted model's saddle path is steeper than the unrestricted model's saddle path in the ( t λ vertical, t l horizontal) plane.
- Loan dynamics comparison (Appendix IV):
  - Show that 00 0 UR ll •• ++ << , where 0 + is time immediately after shock.
  - Equilibrium path dynamic equations for t l:
    - 1 202 () U t U t l l l e θ ⋅ = + − ⋅ ,
    - 1 202 () R t R t l l l e θ ⋅ = + − ⋅ .
  - Time derivatives:
    - 00 21 () U U l l l θ • + = − ⋅ ,
    - 00 21 () R R l l l θ • + = − ⋅ .
  - Because 11 0 UR θ θ << and 02 0 ll − >, it follows that 00 21 02 10 () () U U R R l l l l l l θ θ •• ++ = − ⋅ < − ⋅ = .
- Figures referenced (no downloads or file lists included): Figures 1–9 illustrate phase diagrams and time paths across scenarios, including:
  - Figure 2, 3, 4, 5, 6, 7, 8, 9: various combinations of anticipated/unanticipated and permanent/temporary productivity reductions in unrestricted and restricted models, showing time paths for productivity t A, shadow value of loans t λ, capital requirement t x, bank loans t l, output t y, deposits t d, consumption t c, consumption-deposit ratio, and equity–deposit spread.

*Source: _wp05168 - Appendix IV*

### REFERENCES

### REFERENCES

### Foundational studies on capital, deposit insurance, and bank regulation
- Berger, A., R. Herring, and G. Szego, 1995, “The Role of Capital in Financial Institutions,” Journal of Banking and Finance, Vol. 19 (June), pp. 393–430.
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### Banking crises, credit crunches, and transmission of financial shocks
- Bernanke, B., 1983, “Nonmonetary Effects of the Financial Crisis in Propagation of the Great Depression,” American Economic Review, Vol. 73, pp. 257–76.
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### Banking in macroeconomic models and empirical evidence on monetary transmission
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- Edwards, S., and C. Végh, 1997, “Banks and Macroeconomic Disturbances under Predetermined Exchange Rates,” Journal of Monetary Economics, Vol. 40, pp. 239–78.
- van den Heuvel, S., 2002, “Banking Conditions and the Effects of Monetary Policy: Evidence from the U.S.,” (unpublished; Philadelphia: University of Pennsylvania).

*Source: _wp05168 - REFERENCES*

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