## wpiea2023051-print-pdf - References

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---

### Main findings and contributions
- Novel channel: monetary policy affects financial stability through a downward nominal wage rigidity interacting with a financial friction; expansionary counter-cyclical monetary policy can prevent involuntary unemployment but lowers asset prices and amplifies inefficient capital reallocation.
- Three principal results:
  - Monetary policy impacts financial stability via the nominal-wage channel.
  - Monetary policy by itself can significantly mitigate the wedge between the constrained efficient and the decentralized allocation induced by a pecuniary externality.
  - Regardless of the availability of macroprudential tools, fully stabilizing demand is usually not optimal for monetary policy; optimal monetary policy often deviates from full employment when financial-friction costs are large.
- Policy implication: The design and intensity of optimal macroprudential interventions are critically shaped by the monetary policy regime; paradoxically, the need for macroprudential regulation is largest under a constant price level regime because expansionary monetary policy induces banks to self-insure more.

### Model setup and agents
- Environment:
  - Three-period model with t = 0, 1, 2.
  - Two aggregate states θ ∈ {g, b}.
  - Four agent types: Households, Banks (BS), Entrepreneurs in the Traditional Sector (TS), Central Bank (CB).
- Households:
  - Utility: U = c0 + δ E c2(θ).
  - Period-0 nominal endowment: X. Period-0 price level set P0 = 2.
  - First-order condition: E [Rd(θ) / P2(θ)] ≤ 1/δ.
  - Labor supply: ˜h = 1.
- Traditional Sector (TS):
  - Endowment ̄l = 1 and e units of consumption good in period 1.
  - Production: y(k(θ), h(θ), l) = A k(θ)^α h(θ)^β l^{1−α−β}.
  - Entrepreneurs obtain r_L with r_H > r_L > 1 if they manage projects themselves; banks return r_H if managed by banks.
  - Real period-1 price of a bank project: q(θ).
  - FOCs: w(θ) = A β h(θ)^{β−1} k(θ)^α l^{1−α−β}; r_L q(θ) ≤ A α h(θ)^β k(θ)^{α−1}.
- Nominal wage rigidity:
  - Constraint: W(θ) ≥ ̄W = A β e^α.
  - Labor demand under rigidity: h(θ) = [ k(θ)^α P2(θ) / e^α ]^{1/(1−β)}.
  - Full employment price level: P2(θ) = ( e / k(θ) )^α.
- Period-1 asset price:
  - q(θ) = r_L / [ A α h(θ)^β k(θ)^{α−1} ] — decreasing in k(θ) and h(θ).

### Banking sector, shocks, and technology
- Banks:
  - Initial equity: b units (worth B in currency). Balance-sheet: I + L = D + B.
  - Invest in bank projects I (return r_H if held) and liquid assets L (zero net return but liquid).
  - Liquidity shock in period 1: with probability p a liquidity shock requires additional funds c per unit of investment; with probability 1−p no extra funds required.
  - In bad state constraint: c I ≤ L + (1 − γ(θ)) I q(θ).
  - Analysis assumes c I > L (fire-sales occur) unless otherwise characterized (Lemma 1).
- Technology assumption (Assumption 2):
  - 1/δ + p c < r_H < c(1 − p) / (c − (1 + c) p).
- Scarce resources assumption (Assumption 1):
  - A α e^{α−1} h^β ≥ 1.

### Key mechanisms and equilibrium implications
- Fire-sales and real effects:
  - In bad states banks sell projects to TS; TS is less efficient (r_L < r_H) so fire-sales crowd out productive capital, reducing k and lowering employment due to nominal wage rigidity.
  - Net effect: with α + β < 1, a larger liquidity shortage (c I* − L*) strictly reduces q*(b).
- Monetary policy channel:
  - Expansionary counter-cyclical monetary policy raises P2(b), lowering real wages, increasing employment h(b), raising return on capital, and thereby lowering q(b); lower q(b) increases expected fire-sale losses for banks, inducing them to hold more liquidity and invest less in risky projects ex-ante.
  - Monetary policy can reduce the pecuniary externality gap between constrained-efficient and competitive allocations.
- Time consistency and ex-post effects:
  - Monetary policy is time consistent: investment decisions are made in period 0, so ex-post monetary actions affect outcomes but do not change financial stability determined by prior choices.
- Optimal policy interaction:
  - Optimal macroprudential interventions target banks’ liquid assets and investment projects; their necessity and magnitude depend on the monetary policy regime.
  - Even with full macroprudential tools, if financial-friction costs are large, optimal monetary policy may deviate from full employment; absent macroprudential tools the CB “leans against the wind” via expansionary counter-cyclical policy.

### Unregulated competitive equilibrium (key expressions)
- γ*(b) = 1 − c I* − L* / [ q*(b) I* ].
- h*(b) = [ ( e − c I* + L* )^α P2*(b) / e^α ]^{1/(1−β)}.
- q*(b) = r_L / [ α A ( e^α P2*(b) )^{β/(1−β)} [ e − (c I* − L*) ]^{(1−α−β)/(1−β)} ].
- Banks’ FOCs (period-0):
  - 1/δ + p c + Φ′(I + L) = r_H − p c [ r_H / q(b) − 1 ].
  - 1/δ + Φ′(I + L) ≥ 1 + p [ r_H / q(b) − 1 ].

### Lemmas, propositions, and planner's problem (high level)
- Lemma 1: Define ˆe such that r_H − (1−p) r_L pr_H (1+c) = y'_k(ˆe,1,1). Private banks take on fire-sale risk (cI* > L*) iff e ≥ ˆe. ˆe increases with employment, (r_H − r_L), p, and c.
- Lemma 2: There exists ̄e such that banks hold positive liquid assets iff e < ̄e. ̄e increases with P2(b) and is maximized under full employment.
- Proposition 1: A more expansionary counter-cyclical monetary policy (higher P2(b)) ⇒ banks hold more L, invest less in I, and k in TS falls.
- Planner internalizes effects of investment on q and labor demand. Planner FOCs include:
  - For I: 1/δ + p c + Φ'(I+L) = r_H − p c [ r_H Q(.)^{-1} ] + p (cI − L) Q(.)^{-2} ∂Q(.)/∂I (r_H − r_L).
  - For L: 1/δ + Φ'(I+L) ≥ 1 + p [ r_H Q(.)^{-1} ] + p (cI − L) Q(.)^{-2} ∂Q(.)/∂L (r_H − r_L) + p y'_h(.) ∂H(.)/∂k − p ζ.
  - For P2(b): y'_h(.) ∂H(.)/∂P2 ≥ − (cI − L) Q(.)^{-2} ∂Q(.)/∂P2 (r_H − r_L).
- Proposition 2: There exists ̃e such that optimal monetary policy does not fully stabilize employment iff e < ̃e; in that case P∗∗2 < Pfull2 and h∗∗ < 1.
- Lemma 5: Unregulated competitive equilibrium is constrained inefficient: planner holds weakly more L and invests less in I; q(b) and k(b) larger under constrained efficiency.
- Lemma 6: Constrained efficient allocation can be decentralized with a positive investment tax τ∗∗ and a positive subsidy on liquid assets r∗∗ (τ∗∗ < r∗∗) with explicit expressions given in source.
- Proposition 3: τ∗∗ and r∗∗ necessary to implement constrained efficiency fall as P2(b) increases; tax and subsidy are lowest under full employment.
- Proposition 4: With no macroprudential tools, CB strictly prefers to implement the full employment regime (CB “leans against the wind”).

### Optimal monetary policy with limited macroprudential instruments
- Case A — No macroprudential tools:
  - CB can only set P2(b); optimal response is to commit to full employment regime ex ante; “leaning against the wind” is third-best.
- Case B — One instrument:
  - If efficient investment can be implemented but liquidity cannot, planner reduces balance-sheet size (I+L) and values more expansionary P2(b).
  - If liquidity can be set but investment cannot, planner increases (I+L) and also implements more expansionary P2(b) compared to full regulation.

### Numerical illustration (parameter choices and quantitative observations)
- Time: model period set to 2 years (total length = 4 years).
- Cost function: Φ(I,L) = ξ (I+L)^2 with ξ = .01.
- Liquidity needs c = .1.
- Initial equity b = 1.
- Final period return of bank projects r_H = 1.6.
- Biennial probability of crisis p = 9%.
- Domestic real interest rate Rb = 1.2155.
- TS parameters: α = .35, β = .6, e = 1.
- A chosen so marginal return on capital = 1 under full employment.
- r_L varied in [.7, rH].
- Numerical findings:
  - Private agents and planner never fully insure against fire-sale risk: e > ˆeDE and e > ˆeCEA.
  - Both invest a positive amount in liquid assets: e < ̄eDE and e < ̄eCEA.
  - Full employment regime: more L, less I, less TS capital crowding out.
  - For very low rL full employment can display a lower share of bank projects remaining in BS.
  - As rL increases agents insure less, increasing the price-level necessary to restore full employment.
  - Welfare: full employment regime is optimal for rL > .83; wedge between competitive and constrained-efficient allocations minimized in full employment regime.
  - Optimal macroprudential instruments τ∗∗ and r∗∗ are larger in fixed price regime than in full employment regime.

### Extensions and robustness
- Appendix A — First-best allocation:
  - q(b)^{FB} = r_H, γ^{FB} = 1, k^{FB} = e, L^{FB} = 0, h^{FB} = 1, I^{FB} = (Φ′(I+L))^{−1} [ r_h − 1/δ − c ].
  - Monetary policy neutrality under first-best.
- Appendix B.1 — Endogenous TS endowment:
  - Main results with exogenous e carry over when e chosen privately; Proposition 5 expression for e^* given in source; implication e^* = e^{**}.
- Appendix B.2 — Stochastic TFP:
  - Introducing A ∈ [A_H, A_L] leaves the core policy trade-off arising from restructuring costs intact; socially optimal k determined by k^{**}(b) [ r_L + α(r_H − r_L) ] = e^α (r_H − r_L) independent of TFP.
  - Declines in TFP increase need for countercyclical monetary interventions and possibly tighter macroprudential regulation when correlated with liquidity shocks.

### Technical results and closed-form expressions (selected)
- Closed-form expressions (when I^* > 0 and L^* > 0) include:
  - q(b)^* = r_H p(1+c) / [ r_H − (1−p) ].
  - (I+L)^* = (Φ′)^{−1} [ r_H − pc( r_H q(b)^* − 1 ) − 1/δ − pc ].
  - k^*_{full}(b) = [ r_L q(b)^* / (α A) ]^{1/(α−1)}.
- Closed-form constrained-efficient solution when condition (22) holds with equality:
  - k^{**} = e^α (r_H − r_L) / [ r_L + α (r_H − r_L) ].
  - (I+L)^{**} = (Φ′)^{−1} [ r_H − 1/δ (1+c) + c(1−p)/(1+c) ].
- Proposition 3 provides exact expression for τ^{**} with sign ∂τ/∂P_2(b) < 0.

### Conceptual conclusions
- Integrated nominal rigidity and financial friction produce a policy trade-off: expansionary counter-cyclical monetary policy stabilizes employment but increases fire-sale discounts and financial-friction distortions.
- Monetary policy alone cannot fully restore constrained efficiency; macroprudential regulation and monetary regime interact critically.
- Anticipation of expansionary monetary policy encourages private precautionary liquidity holdings, reducing need for macroprudential interventions.
- In absence of macroprudential tools, CB strictly prefers full employment policy to better insure the economy against financial distress.

*Italic: Content unit: wpiea2023051-print-pdf - References (includes Appendices A–C and numerical illustration).*

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

### wpiea2023051-print-pdf - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Main findings and contributions
- Novel channel: monetary policy affects financial stability through a downward nominal wage rigidity interacting with a financial friction; expansionary counter-cyclical monetary policy can prevent involuntary unemployment but lowers asset prices and amplifies inefficient capital reallocation.
- Three principal results:
  - Monetary policy impacts financial stability via the nominal-wage channel.
  - Monetary policy by itself can significantly mitigate the wedge between the constrained efficient and the decentralized allocation induced by a pecuniary externality.
  - Regardless of the availability of macroprudential tools, fully stabilizing demand is usually not optimal for monetary policy; optimal monetary policy often deviates from full employment when financial-friction costs are large.
- Policy implication: The design and intensity of optimal macroprudential interventions are critically shaped by the monetary policy regime; paradoxically, the need for macroprudential regulation is largest under a constant price level regime because expansionary monetary policy induces banks to self-insure more.

### Model setup and agents
- Environment:
  - Three-period model with t = 0, 1, 2.
  - Two aggregate states θ ∈ {g, b} (good and bad).
  - Four agent types: Households, Banks (BS), Entrepreneurs in the Traditional Sector (TS), Central Bank (CB).
- Households:
  - Utility: U = c0 + δ E c2(θ). (equation (1))
  - Period-0 nominal endowment: X. Period-0 price level set P0 = 2 so nominal and real variables coincide in period 0. (text)
  - First-order condition: E [Rd(θ) / P2(θ)] ≤ 1/δ. (equation (4))
  - Labor supply: ˜h = 1 (no disutility of labor).
- Traditional Sector (TS):
  - Entrepreneurs endowed with ̄l = 1 units of land and e units of the consumption good in period 1.
  - Production: y(k(θ), h(θ), l) = A k(θ)^α h(θ)^β l^{1−α−β}. (text)
  - Entrepreneurs can instead buy bank projects returning r_H if managed by banks, but entrepreneurs obtain r_L with r_H > r_L > 1.
  - Real period-1 price of a bank project: q(θ). (text)
  - First-order conditions for labor and capital:
    - w(θ) = A β h(θ)^{β−1} k(θ)^α l^{1−α−β}. (equation (5))
    - r_L q(θ) ≤ A α h(θ)^β k(θ)^{α−1} (equality if entrepreneurs buy projects). (equation (6))
- Nominal wage rigidity:
  - Constraint: W(θ) ≥ ̄W = A β e^α. (equation (7))
  - Labor demand under rigidity: h(θ) = [ k(θ)^α P2(θ) / e^α ]^{1/(1−β)}. (equation (8))
  - Full employment price level that implements h = 1: P2(θ) = ( e / k(θ) )^α. (text)
- Period-1 asset price:
  - q(θ) = r_L / [ A α h(θ)^β k(θ)^{α−1} ]. (text) — decreasing in k(θ) and in employment h(θ).

### Banking sector, shocks, and technology
- Banks:
  - Initial equity: b units (worth B in currency). Can raise D by issuing deposits so initial funds I + L = D + B. (equation (9))
  - Invest in bank projects I (return r_H if held) and liquid assets L (zero net return, liquid anytime).
  - Liquidity shock in period 1: with probability p a liquidity shock occurs requiring additional funds c per unit of investment; with probability 1−p no extra funds required.
  - In bad state, banks finance c I via liquid assets and by selling fraction 1 − γ(θ) of projects to TS:
    - c I ≤ L + (1 − γ(θ)) I q(θ). (equation (10))
  - Assume c I > L in analysis and later characterize when this holds (lemma 1).
- Technology assumption (Assumption 2):
  - 1/δ + p c < r_H < c(1 − p) / (c − (1 + c) p). (text)
  - First inequality: net expected return on bank projects is positive.
  - Second inequality: ensures scrapping is never optimal.
- Scarce resources assumption (Assumption 1):
  - A α e^{α−1} h^β ≥ 1. (text) — ensures period-1 price of bank-projects is always below the fundamental value.

### Key mechanisms and equilibrium implications
- Fire-sales and real effects:
  - In bad states banks sell projects to TS; TS is less efficient (r_L < r_H) so fire-sales crowd out productive capital, reducing k and lowering employment due to nominal wage rigidity.
  - Two opposing effects on q*(b):
    - Lower k raises marginal productivity of capital and lowers q*(b).
    - Lower employment reduces real return on capital and raises q*(b).
  - Net effect: with α + β < 1, overall effect of larger liquidity shortage (c I* − L*) on q*(b) is strictly negative. (text)
- Monetary policy channel:
  - Expansionary counter-cyclical monetary policy raises P2(b), lowering real wages, increasing employment h(b), raising return on capital, and thereby lowering q(b). Lower q(b) increases expected fire-sale losses for banks, inducing them to hold more liquidity and invest less in risky projects ex-ante.
  - As a result, monetary policy can reduce the pecuniary externality gap between constrained-efficient and competitive allocations.
- Time consistency and ex-post effects:
  - Monetary policy is time consistent regardless of macroprudential availability; because investment decisions are made in period 0, ex-post monetary actions affect macroeconomic outcomes but do not change financial stability determined by prior investment choices. Thus ex-post monetary policy can focus on macroeconomic stability without generating a trade-off with financial stability in this model.
- Optimal policy interaction:
  - Optimal macroprudential interventions target banks’ holdings of liquid assets and investment projects, but their necessity and magnitude depend on the monetary policy regime: more expansionary monetary policy reduces the wedge and induces banks to self-insure, yet the need for macroprudential tools actually increases as monetary policy focuses more on financial-stability regulation and is largest under constant-price regimes.
  - Even with full macroprudential tools, if financial-friction costs are large (large return difference between BS and TS), optimal monetary policy may deviate from full employment; absent macroprudential tools the CB “leans against the wind” via expansionary counter-cyclical policy to mitigate ex-ante excessive risk taking.

### Unregulated competitive equilibrium (key equilibrium expressions)
- In equilibrium bad state, share of projects remaining in BS:
  - γ*(b) = 1 − c I* − L* / [ q*(b) I* ]. (equation (11))
- Equilibrium labor demand in bad state:
  - h*(b) = [ ( e − c I* + L* )^α P2*(b) / e^α ]^{1/(1−β)}. (text)
- Equilibrium period-1 asset price in bad state:
  - q*(b) = r_L / [ α A ( e^α P2*(b) )^{β/(1−β)} [ e − (c I* − L*) ]^{(1−α−β)/(1−β)} ]. (equation (12), presented in text form)
- Banks’ first-order conditions (period-0) with respect to I and L:
  - 1/δ + p c + Φ′(I + L) = r_H − p c [ r_H / q(b) − 1 ]. (equation (13))
  - 1/δ + Φ′(I + L) ≥ 1 + p [ r_H / q(b) − 1 ]. (equation (14))
  - Interpretation: marginal expected real costs of bank projects equal marginal return net of expected fire-sale losses; liquid assets’ net return rises as q(b) falls.

*Source: wpiea2023051-print-pdf - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . .*

### 1.  The optimality conditions of households (4), entrepreneurs (5)-(6) and banks (13) -

### 1. The optimality conditions of households (4), entrepreneurs (5)-(6) and banks (13) -

### Competitive equilibrium: key characteristics
- Fire-sales are harmful to an individual bank because the fire-sale price is strictly below the return of bank projects: q(b) < r_H.
- Fire-sale losses arise because of lower productivity r_L and limited interim-period resources.
- Banks can fully insure against fire-sale risk by accumulating liquid assets so that L* = cI*. In that case, the BS need not sell projects to the TS and the economy operates at full capacity.
- Private banks never optimally choose full insurance (L* = cI*) when the TS endowment is not too small (Lemma 1).
- Marginal derivatives notation: y'_k(.) and y'_h(.) denote marginal derivatives of the production function with respect to capital and labor, respectively.

### Lemma 1 (cut-off for fire-sale risk taking)
- Define ˆe such that
  r_H − (1−p) r_L pr_H (1+c) = y'_k(ˆe,1,1).
- Private banks take on fire-sale risk (cI* > L*) if and only if e ≥ ˆe.
- The cut-off value ˆe:
  - Is an increasing function of the level of employment and hence of the price level P_2(b).
  - Increases with the productivity loss of the TS (r_H − r_L).
  - Increases with the probability of a liquidity crisis p and with liquidity needs c.
- Assumption used thereafter: the TS endowment is sufficiently large such that e > ˆe and banks never fully insure against fire-sale risk.

### Banks' investment in liquid assets
- Private banks always invest in bank projects given positive net return (assumption 2).
- Banks invest in liquid assets only if the net return compensates for holding-costs: 1/δ + Φ'(I+L).
- Net return of liquid assets rises with higher probability of liquidity shock p and with lower asset price in the aggregate bad state q(b).
- Fire-sale price q(b) increases with entrepreneurs’ endowment e and productivity r_L; thus large e or r_L can lead banks to hold zero liquid assets.
- Monetary policy affects net return: a CB commitment to the full employment regime maximizes the real return on capital, implying a lower fire-sale price and increasing banks' incentive to hold liquid assets.

### Lemma 2 (threshold for positive liquid assets)
- There exists a real number ̄e such that banks hold a positive amount of liquid assets if and only if e < ̄e.
- The threshold ̄e:
  - Increases with the price level P_2(b).
  - Is maximized under the full employment regime.
- Consequence: expansionary monetary policy in response to liquidity shocks increases employment, lowers expected asset price, makes fire-sales more expensive, induces banks to increase liquid holdings and reduce investment in bank projects.

### Proposition 1 (monetary policy and private investment decisions)
- If the CB commits to a more expansionary counter-cyclical monetary policy (i.e., a higher price level), then:
  - Private banks hold more liquid assets L.
  - Private banks invest less in bank projects I.
  - Less capital k is invested in the TS (capital is crowded out from the TS).
- Note: the fire-sale price is not continuous at cI* = L* since r_L < r_H and y'(e,1) ≥ 1.

### Interpretation
- Monetary policy shapes private bank decisions: by committing to expansionary counter-cyclical policy the CB can "lean against the wind" and reduce ex-ante risk taking in the BS.
- Whether this monetary policy intervention improves welfare is addressed in section 6 (not included here).

---

### The constrained-efficient allocation: planner's problem
- Planner: benevolent constrained planner who sets price level P_2(b) in the initial period and instructs banks on initial choices {D, L, I}, while all remaining markets clear competitively.
- Planner anticipates equilibrium responses of households and entrepreneurs and maximizes social welfare U (the discounted sum of household consumption).
- Period-1 total consumption: c_0 = x − d.
- Period-2 consumption in the good state:
  c_2(g) = r_H I + y(e,1,1) + L − Φ(I+L).  (15)
- Period-2 consumption in the bad state:
  c_2(b) = γ(b) r_H I + (1−γ(b)) r_L I + L − cI − Φ(I+L) + y(k(b),h(b),1) + cI − L.  (16)
- Social welfare expression:
  U =
  1/δ (X − D) + r_H I + L − Φ(I+L) + y(e,1,1)
  + p [ y(k(b),h(b),1) − y(e,1,1) − cI − (1−γ(b)) (r_H − r_L) I + cI − L ].
- Planner is constrained by the same market constraints as private agents: wage rigidity, cash-in-advance constraint of banks in the bad state, private decisions by households and the TS. Unlike private banks, the planner internalizes how initial investment affects asset price and labor demand.

### Competitive pricing and labor demand functions (used by planner)
- Asset price:
  q(θ) = Q((e − cI + L), h((e − cI + L), P_2(b))) =
  - r_L y'_k((e − cI + L), h((e − cI + L), P_2(b)), 1) if cI > L,
  - r_H if cI = L.  (18)
- Labor demand:
  h(θ) = H((e − cI + L), P_2(θ)) =
  - h((e − cI + L) α P_2(b) e^α)^{1/(1−β)} if cI > L,
  - 1 if cI = L.  (19)

### Planner's Lagrangian and first-order conditions
- Using constraint (10) and balance-sheet condition (9) to replace γ(b) and D, the planner's problem summarized by Lagrangian:
  L_P max_{I,L,P_2(b)} =
  1/δ (X − I − L + B) + r_H I + L − Φ(I+L) + y(e,1,1)
  + p [ y(k,H(.),1) − y(e,1,1) − cI − (cI − L) Q(.) (r_H − r_L) + cI − L ]
  − ζ (L − cI),
  with ζ > 0 the multiplier for k ≤ e.
- First-order conditions (derivatives w.r.t. {I, L, P_2(b)}):
  - For I:
    1/δ + p c + Φ'(I+L) = r_H − p c [ r_H Q(.)^{-1} ] + p (cI − L) Q(.)^{-2} ∂Q(.)/∂I (r_H − r_L).  (20)
  - For L:
    1/δ + Φ'(I+L) ≥ 1 + p [ r_H Q(.)^{-1} ] + p (cI − L) Q(.)^{-2} ∂Q(.)/∂L (r_H − r_L) + p y'_h(.) ∂H(.)/∂k − p ζ.  (21)
  - For P_2(b):
    y'_h(.) ∂H(.)/∂P_2 ≥ − (cI − L) Q(.)^{-2} ∂Q(.)/∂P_2 (r_H − r_L).  (22)

### Planner vs private bank: internalized effects
- The planner internalizes:
  1) More investment (less liquidity) crowds out productive capital in the TS, lowering the asset price and increasing the share of projects held by the less productive TS.
  2) Lower capital investment in the TS reduces employment and output in the TS, further increasing the costs of bank projects and the relative gain of liquid assets.
- Condition (22) captures the trade-off when setting P_2(b): expansionary monetary policy increases employment and TS output (LHS), but also raises the real return on capital in the TS and lowers the fire-sale price, increasing the share of projects in the less productive TS and crowding out real capital investment (RHS).

### Equilibrium capital and policy implications
- If condition (22) holds with equality, equilibrium capital stock:
  k**(b) = e^α (r_H − r_L) / [ r_L + α (r_H − r_L) ].
- Closed-form efficient allocation exists with optimal employment level below one in that case.
- If LHS > RHS in (22), the planner prefers the full employment regime.
- Full-employment regime is optimal if r_H = r_L (no cost to higher price level).
- If financial friction is sufficiently costly, optimal monetary policy deviates from fully stabilizing employment in favor of financial stabilization even with full macroprudential tools.
- The RHS of (22) can never exceed the LHS because the planner can choose arbitrarily low P_2(b) to raise the fire-sale price and minimize redistributed projects.
- Because y(.) satisfies the Inada condition, there exists a sufficiently low P_2(b) such that the marginal cost of lowering P_2(b) dominates marginal gains.
- Condition (22) simplifies to:
  k ≥ (e − k)^α (r_H − r_L) / r_L,
  which is independent of the price level P_2(b), q(b), and employment h(b).
- Implication: once banks’ balance sheet (and entrepreneur capital k = e − cI + L) is fixed in period 1, monetary policy has no effect on the trade-off in (22); monetary policy is time-consistent in this framework.

### Definition of constrained efficient allocation
- The constrained efficient allocation is the collection of prices {W** , q**(b), R** , P_2(b)** } and allocations { c_0** , c_2(g)** , c_2(b)** , I** , L** , D** , ̃D** , γ(b)** , k(b)** , h(b)** , ̃h(b)** } that satisfy:
  1. The optimality condition of households (4),
  2. The equilibrium asset price (18) and labor demand function (19),
  3. The planner’s optimality conditions (20)-(22),
  4. Bank’s period one budget constraint (10),
  5. Nominal wage rigidity (7),
  6. The resource constraints of households (c_0 = x − d), (15), (16) and the BS (9).

*Source: wpiea2023051-print-pdf - 1.  The optimality conditions of households (4), entrepreneurs (5)-(6) and banks (13) -*

### 7.  Market clearing:

### 7.  Market clearing:

### Characteristics of the Constrained Efficient Allocation
- Planner’s allocations satisfy  ̃h∗∗ = h∗∗, ̃D∗∗ = D∗∗, and k∗∗ = e−cI∗∗ + L∗∗ given {A, α, β, ̄l, ̄h, e, b, x, rH, rL, δ, p, Φ(.)}.
- The social planner can fully insure against fire-sale risk by setting cI∗∗ = L∗∗.
- Fire-sales reduce period-2 consumption via three channels:
  - Purchases of bank assets crowd out productive capital investment in the TS (period 1 resource constraint).
  - A lower capital stock in the TS decreases labor demand and further reduces TS output.
  - A positive share of bank projects γ is held by the less efficient TS; efficiency loss given by rH − rL; lower fire-sale price increases the share sold to the TS and further depresses prices.
- Lemma 3:
  - Define ˆeCEA such that ( (rH − p c−1) rL (1+c) rH / p ) = y′k(ˆeCEA,1,1). (expression as in source)
  - The planner takes on fire-sale risk iff e ≥ ˆeCEA.
  - ˆeCEA is an increasing function of the level of employment and of the price level P2(b).
  - ˆeCEA > ˆe.
- Planner invests in liquid assets if their gain compensates for costs 1/δ + Φ′(I+L).
- Lemma 4:
  - There exists ̄eCEA such that the planner instructs banks to hold positive liquid assets iff e < ̄eCEA, where ̄eCEA > ̄e.
  - The planner certainly instructs banks to hold liquidity in states where unregulated banks hold liquidity.
- Proposition 2:
  - There exists ̃e such that optimal monetary policy does not fully stabilize employment iff e < ̃e.
  - In that case condition (22) holds as strict equality, P∗∗2 < Pfull2 and h∗∗ < 1.
- Intuition: When resources in TS are small, planner values liquidity more; when TS endowment is below cut-off, constrained efficient allocation can exhibit involuntary unemployment because monetary policy equalizes costs of fire-sales and unemployment.

### Social Inefficiencies and Optimal Policy
- Lemma 5:
  - The unregulated competitive equilibrium is constrained inefficient.
  - Compared to private agents, a constrained social planner holds (weakly) more liquid assets (L∗∗ ≥ L∗) and invests less in bank projects (I∗∗ < I∗).
  - The fire-sale price and TS capital stock in the bad state are larger in the constrained efficient allocation: k(b)∗∗ > k(b)∗ and q(b)∗∗ > q(b)∗.
- Sources of inefficiency:
  - Atomistic private banks ignore incremental effect of individual investment on period-1 asset price; when rH > rL this reduces period-2 consumption by shifting projects to less efficient TS.
  - Nominal wage rigidity adds a channel: fire-sales crowd out capital in TS and reduce employment; if CB does not commit to full employment this introduces further wedge even when rH = rL.
- Monetary policy:
  - Under rH > rL, unregulated equilibrium is constrained inefficient independent of monetary policy; monetary policy cannot fully restore constrained efficiency but can mute fire-sales’ employment effects by committing to full employment.

### Implementing the Constrained Efficient Allocation (Macroprudential Instruments)
- Lemma 6:
  - Constrained efficient allocation can be decentralized using:
    - A positive tax on investment τ∗∗ > 0,
    - A positive subsidy on liquid assets r∗∗ > 0,
    - Set to satisfy:
      - τ∗∗ = δ p c [ −(c I∗∗ − L∗∗) / rL ] [ y′′k(.) + y′′k,h(.) ∂H(.)/∂L ] (rH − rL) + y′h(.) ∂H(.)/∂L,
      - r∗∗ = τ∗∗ / (c δ),
      - with τ∗∗ < r∗∗.
  - Two independent instruments and detailed fundamentals are required to restore constrained efficiency (the “full regulation regime”).
- One instrument suffices only if CB can directly target the liquidity shortage cI − L:
  - Set tax on liquidity shortage τLS = p [ y′h(.) − ( y′′k(.) + y′k,h(.) ∂H(. )/∂k ) (rH − rL) / rL ] to restore constrained efficiency.

### Social Inefficiencies and Monetary Policy (Interaction)
- Proposition 3:
  - The macroprudential tax τ∗∗ and subsidy r∗∗ necessary to implement constrained efficiency fall as the price level P2(b) increases.
  - Tax and subsidy are lowest when CB commits to the full employment regime.
- Intuition:
  - A more expansionary counter-cyclical monetary policy reduces exposure to fire-sale risk, increases available TS capital after a liquidity shock, lowers marginal return on capital y′K(.), and reduces marginal effects of projects and liquid assets on fire-sale price.
  - Both the price-level dependence of y′k(.) and y′h(.) ∂H(. )/∂k decrease with P2(b), shrinking the wedge between constrained efficient and decentralized allocations as monetary policy is more expansionary in crises.

### Optimal Monetary Policy with Limited Macroprudential Instruments
- Context: Macroprudential regulation requires detailed information, active tool management, and legal ability. Uninformed/ad-hoc macroprudential rules can cause welfare costs.
- Assumption in analysis: e < ̄e so liquidity holdings are strictly positive in decentralized (L∗ > 0) and constrained-efficient allocation (L∗∗ > 0).

A. No Access to Macroprudential Policy Tools
- CB can only set period-2 price level P2(b) and anticipates effects on bank choices {I, L, D} via expected P2(b) and asset price q(b).
- Implementability constraints (13) and (14) bind for planner because social cost/gain of investment/liquidity differ from private costs/gains when rH > rL.
- Without macroprudential tools, CB can “lean against the wind” by committing ex ante to a more expansionary monetary policy in bad state to influence initial bank choices and minimize fire-sales.
- Proposition 4:
  - In absence of macroprudential tools, CB strictly prefers to implement the full employment regime.
- Note: “Leaning against the wind” is third-best and optimal only when no macroprudential tools are available; monetary policy trade-off (condition (22)) remains.

B. Limited Macroprudential Policy (one instrument only)
- Case (1): CB can implement efficient investment level but cannot regulate liquidity holdings (liquidity chosen by banks).
  - Planner’s FOCs include multipliers ζL > 0 on implementability constraint for liquidity.
  - Result: Bank balance sheets (I + L) are strictly smaller compared to full regulation regime because planner reduces investment to offset low private liquidity holdings.
  - Planner values expansionary counter-cyclical monetary policy more compared to full regulation regime.
- Case (2): CB sets liquidity holdings but cannot regulate investment.
  - Planner’s FOCs include multiplier ζI < 0 on implementability constraint for investment.
  - Result: Balance sheet size (I + L) is larger compared to full regulation regime; planner accumulates more liquid assets to mitigate unregulated investment choices.
  - Planner again implements more expansionary monetary policy compared to full regulation regime when evaluated at same allocation.

### Numerical Illustration (parameter choices and key quantitative observations)
- Model period set to 2 years (total length = 4 years).
- Cost function Φ(I,L) = ξ (I+L)^2 with ξ = .01.
- Liquidity needs c = .1.
- Initial equity b = 1.
- Final period return of bank projects = 1.6 (≈ annual return 12%).
- (Biennial) probability of crisis p = 9%.
- Domestic real interest rate Rb = 1.2155 (annual interest rate 5%).
- TS parameters: α = .35, β = .6, e = 1.
- A chosen so marginal return on capital = 1 under full employment.
- rL varied in interval [.7, rH] so efficiency loss in TS varies between 0.9 and 0 percentage points.
- Numerical findings:
  - For chosen functions/parameters both private agents and planner never fully insure against fire-sale risk: e > ˆeDE and e > ˆeCEA.
  - Both invest a positive amount in liquid assets: e < ̄eDE and e < ̄eCEA.
  - Full employment regime: both private banks and planner accumulate more liquid assets and invest less in bank projects; fire-sales crowd out less TS capital.
  - In constrained efficient allocation, fire-sale price can be smaller under full employment regime due to larger employment level; for very low rL full employment can display a lower share of bank projects remaining in BS.
  - As rL increases agents insure less against fire-sale risk, increasing severity of potential liquidity shock and increasing price-level necessary to restore full employment.
  - Welfare:
    - Full employment regime is optimal monetary policy regime for rL > .83 (condition (22) strict inequality; CB prefers stabilizing employment).
    - Fixed price regimes become more costly as efficiency loss in TS becomes smaller.
    - Wedge between competitive and constrained-efficient allocations is minimized in full employment regime.
  - Optimal macroprudential instruments τ∗∗ and r∗∗ are larger in fixed price regime than in full employment regime.

### Conclusion (key conceptual takeaways)
- Integrated model with nominal rigidity and financial friction yields policy trade-off: expansionary counter-cyclical monetary policy reduces real wages and stabilizes employment in crises but increases fire-sale discounts and exacerbates financial-friction distortions.
- Monetary policy alone is generally insufficient to restore constrained efficiency.
- Optimal macroprudential regulation critically depends on monetary policy regime: more employment-targeted monetary policy reduces need for macroprudential interventions because anticipation of expansionary policy increases private precautionary insurance against crises.
- In absence of macroprudential tools, monetary authorities strictly prefer to focus on macroeconomic stability (full employment) to better insure economy against financial distress.

*Italic: Content unit: wpiea2023051-print-pdf - 7.  Market clearing:*

### REFERENCES

### wpiea2023051-print-pdf - REFERENCES

### Major bibliographic sources
- Lists literature on monetary policy, financial stability, macroprudential policy, sudden stops, international reserves, exchange rate policy, credit cycles, and related theoretical and empirical contributions. Key cited authors include Acharya; Adrian and Liang; Aizenman; Amador et al.; Barro and Gordon; Benigno et al.; Bernanke; Bianchi; Bocola and Lorenzoni; Borio; Boz et al.; Caballero and Krishnamurthy; Calvo; Cavallino; Cerutti et al.; Chang and Velasco; Collard et al.; Coulibaly; Curdia; Dell’Ariccia et al.; Dávila and Korinek; Eichengreen and Sachs; Fanelli and Straub; Farhi and Werning; Fornaro; Gabaix and Maggiori; Gabriel Jimenez et al.; Gersbach et al.; Gertler et al.; Itskhoki and Mukhin; Jeanne; Jeanne and Sandri; Kara and Ozsoy; Kiley and Sim; Kiyotaki and Moore; Korinek; Kydland and Prescott; Lane and Xu; Lawrence Christiano et al.; Loisel; Lorenzoni; Lutz and Pichler; Mendoza; Obstfeld, Shambaugh, and Taylor; Olivei and Tenreyro; Otrok et al.; Ottonello; Richter, Schularick, and Shim; Schmitt-Grohé and Uribe; Stein; Svensson; Van der Ghote; Woodford.
- The references span journal articles, NBER and IMF working papers, BIS papers, and edited volumes relevant to integrated monetary and macroprudential frameworks.

### Appendix A — First-best allocation (summary)
- Environment: markets complete, no fire-sales, planner has access to short-term funding; no redistribution to less efficient Traditional Sector (TS); no capital crowding out in TS; employment and production at full capacity; no need to invest in liquid assets.
- Closed-form first-best allocation (as presented):
  - q(b)^{FB} = r_H
  - γ^{FB} = 1
  - k^{FB} = e
  - L^{FB} = 0
  - h^{FB} = 1
  - I^{FB} = (Φ′(I+L))^{−1} [ r_h − 1/δ − c ]
- Monetary policy neutrality: because employment is always at potential, monetary policy is neutral in this setup.

### Appendix B — Model extensions (summary)

B.1 Endogenous Traditional Sector endowment
- Relaxed assumption: TS endowment e chosen endogenously by borrowing in period 0.
- Main robustness result: all main results with exogenous e carry over when e chosen privately; in particular, when L^* > 0 private agents choose e^* = e^{**} (socially efficient) and assumption 1 (y′(e^*,1,1) > 1) holds.
- Proposition 5 (stated result):
  - Private optimum with endogenous e:
    e^* = ( (1/δ(1+c) − r_H + (1−p)(1+c) / ( (1−p)(1+c) α_A ) ) )^{1/(α−1)}
    (expression presented in source exactly)
  - Implication: e^* = e^{**}; results on dependence of private choices on monetary policy, social inefficiencies, and dependence of optimal macroprudential policy on monetary regime continue to hold.
- Additional conditional statement: for parameter range CEÂe < e < ̄e both agents hold positive liquid assets and full-insurance not optimal; previous results apply.

B.2 Stochastic Total Factor Productivity (TFP)
- Extension: two-state TFP A ∈ [A_H, A_L] with A_H > A_L; uncorrelated with liquidity shock θ; four possible states at t=1: { [θ=g, A=A_H], [θ=g, A=A_L], [θ=b, A=A_H], [θ=b, A=A_L] }.
- Key implications:
  - When θ = g and e = k (no capital crowding out), if A = A_H optimal monetary policy is non-action; if A = A_L employment falls below potential and optimal monetary policy raises the price level to restore full employment — no trade-off in this case.
  - The policy trade-off arises in states with restructuring costs (fire-sale risk); that trade-off is independent of TFP. Condition (22) rearranged:
    k^{**}(b) [ r_L + α(r_H − r_L) ] = e^α (r_H − r_L)
    - This determines socially optimal k independent of TFP, employment, and price level.
  - Low TFP reduces marginal productivity of labor and capital in TS, raising the period-1 asset price; banks face lower fire-sale losses, invest more in bank projects, less in liquid assets, and more capital is crowded out from TS.
  - If declines in TFP are expected to coincide with liquidity shocks, the wedge between constrained efficient allocation and competitive equilibrium increases; CB commitment to more expansionary policy reduces that wedge.
  - Policy implication: declines in TFP increase need for countercyclical monetary interventions; if TFP declines coincide with liquidity shocks, tighter macroprudential regulation is needed to restore second-best. Monetary-policy trade-off and dependence of social inefficiencies on monetary regime remain unaffected.

### Appendix C — Proofs and key technical results (selected summaries)

- Lemma 1 (intuition):
  - There exists a cutoff ˆe for TS endowment above which banks will not fully insure (cI = L not optimal); ˆe increases with employment because higher employment raises marginal productivity of capital and private marginal costs of fire-sales.

- Lemma 2 (equilibrium with I^* > 0 and L^* > 0 — closed form):
  - q(b)^* = r_H p(1+c) / [ r_H − (1−p) ]
  - (I+L)^* = (Φ′)^{−1} [ r_H − pc( r_H q(b)^* − 1 ) − 1/δ − pc ]
  - k^*_{full}(b) = [ r_L q(b)^* / (α A) ]^{1/(α−1)}
  - Other expressions for k^*_P2(b), I^*_{full}, I^*_{P2}, L^*_{full}, L^*_{P2} are provided in closed form in the source.
  - Equilibrium exists with L^* > 0 if e < (I+L)^* c + k^*(b) = ̄e.
  - Capital in TS largest in full-employment regime; ̄e_{full} > ̄e_{P2}.

- Proposition 1 (existence and uniqueness of decentralized investment I^*):
  - Given P_2(b), there is a unique I^* solving the decentralized FOC for investment due to monotonicity properties; as P_2(b) increases, equilibrium I^* decreases and, when L^* = 0, capital in TS in bad state increases.

- Lemma 3 (planner preferences over insurance):
  - Planner prefers full insurance against fire-sale risk at a higher TS endowment level than decentralized agents; cutoff denoted ˆe_{CEA} where condition y′_k(e,1,1) ≥ [r_H − pc − 1] r_L / [ (1+c) r_H p ] holds.

- Lemma 4 (thresholds for L = 0 allocations):
  - Defines a threshold ˆk such that planner’s conditions imply constraints on k and hence on e; provides expression (31) bounding k^{**}(b) and conditions under which planner strictly prefers L^{**} > 0, yielding threshold ̄e_{CEA}.

- Proposition 2 (constrained efficient allocation closed form when condition (22) equality holds):
  - Closed-form CEA solution:
    - k^{**} = e^α (r_H − r_L) / [ r_L + α (r_H − r_L) ]
    - (I+L)^{**} = (Φ′)^{−1} [ r_H − 1/δ (1+c) + c(1−p)/(1+c) ]
    - I^{**}, L, P^{**}_2, h expressions given in source (see Appendix).
  - Equilibrium requires h^{**} ≤ 1, i.e., constraint on P^{**}_2 relating to r_L and α as in source; holds if e ≤ ̃e with ̃e defined by equality condition shown in source.

- Lemma 5 (comparison decentralized vs constrained efficient):
  - When L^* > 0 and L^{**} > 0:
    - (I+L)^* = (I+L)^{**} = (Φ′(I+L))^{−1} [ r_H − 1/δ(1+c) + c(1−p)/(1+c) ]
    - q(b)^* < q(b)^{**} ⇒ k^{**}(b) > k^*(b); with fixed balance sheet size implies L^{**} > L^*, I^{**} < I^*.
  - Various cases (L^* = 0, L^{**} > 0; L^* = L^{**} = 0) produce monotonic comparisons between k, I, q consistent with planner internalizing externalities.

- Lemma 6 (equivalence via tax r and τ):
  - Bank and planner FOCs coincide if τ = τ^{**} and r = r^{**}; provides expressions linking Φ′(I+L), 1/δ, τ and r to equilibrium pricing terms.

- Proposition 3 (optimal tax τ^{**} and its dependence on price level P_2(b)):
  - τ^{**} = δ q c y′_k(.) [ (e − k^{**}) r_L / k^{**} ( (1−α−β)/(1−β) (r_H − r_L) + β/(1−β) ) ]
    - (expression provided exactly as in source)
  - ∂τ/∂P_2(b) < 0: equilibrium tax rate decreases as P_2(b) increases. Proof uses decomposition of ∂k/∂P_2(b) and sign properties under α+β < 1.

- Proposition 4 (central bank prefers full employment regime):
  - Decentralized FOCs determine q(b)^* and (I+L)^* independent of P_2(b):
    - q(b)^* = r_H p(1+c) / [ r_H − (1−p) ]
    - (I+L)^* = (Φ′)^{−1} [ r_H − pc( r_H q(b)^* − 1 ) − 1/δ − c ]
  - CB objective U as function of P_2(b) only; derivative condition simplifies to p y′_h(.) ∂h/∂P_2(b) ≥ 0. Since lhs > 0, CB strictly prefers the full employment regime; higher P_2(b) increases k(b)^* in bad state, reduces I^*, raises L^*.

- Proposition 5 (endogenous e result — proof sketch)
  - TS first-order condition for e:
    - 1/δ − 1 = (1−p)( y′(e,1,1) − 1 ) + p ( r_H q(θ) − 1 )
    - Rearranged: 1/δ − 1 − p( r_H q(θ) − 1 ) / (1−p) + 1 = y′(e,1,1)
  - Using q(b) for L^* > 0 yields the expression in Proposition 5; planner’s FOC delivers same condition leading to e^* = e^{**}.

_Italic: Content unit "wpiea2023051-print-pdf - REFERENCES" (pages: REFERENCES, Appendix A–C) from the supplied PDF._

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023051-print-pdf.pdf_