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### 2.1 Labor market — structure and matching
- Jobs created through sectoral matching functions as in Pappa et al.(2015) and Anand and Khera(2016).
- Employed fractions: n_{i,j,t} for household type i = L,B in sector j = F,I.
- Total employees during period t: survivors after exogenous separation plus new hires m_{i,j,t}.
- Formal sector law of motion:
  - n_{i,F,t} = (1−σ_F) n_{i,F,t−1} + m_{i,F,t}, where σ_F is the exogenous separation rate.
- Informal sector law of motion:
  - n_{i,I,t} = (1−ξ−σ_I) n_{i,I,t−1} + m_{i,I,t}, where σ_I is exogenous job destruction and ξ is audit job-loss probability.
- Aggregate notation:
  - n_{j,t} ≡ Γ_L n_{L,j,t} + Γ_B n_{B,j,t}; m_{j,t} ≡ Γ_L m_{L,j,t} + Γ_B m_{B,j,t}.
- Beginning-of-period unemployment pool for type i:
  - u_{i,t} = 1 − n_{i,F,t−1} − n_{i,I,t−1} + σ_F n_{i,F,t−1} + (ξ+σ_I) n_{i,L,t−1}
  - Equivalent: u_{i,t} = 1 − (1−σ_F) n_{i,F,t−1} − (1−ξ−σ_I) n_{i,L,t−1}.
- Symmetry assumption: n_{L,j,t} = n_{B,j,t} = n_{j,t} ⇒ u_{L,t} = u_{B,t} = u_{t}, m_{L,j,t} = m_{B,j,t} = m_{j,t}.
- Jobseeker hiring probability:
  - x_{j,t} ≡ m_{j,t} / u_{t}.
- Hiring (matching) cost for entrepreneurs in sector j:
  - g_{j,t} = A_{j,t} κ_j (x_{j,t})^{η}, where κ_j is matching efficiency and η is elasticity of matching to vacancies.

### 2.2 Households and entrepreneurs — preferences, constraints, production
- Patient households (L):
  - Utility: max E_0 ∑_{t=0}^∞ β_L^t [ log(c_{L,t} − ν_L c_{L,t−1}) + ι log(h_{L,t}) − μ (n_{L,t})^{1+φ} / (1+φ) ], with n_{L,t} ≡ n_{L,F,t} + n_{L,I,t}.
  - Budget constraint (real terms) includes c_{L,t}, q_{h,t} Δh_{L,t}, d_{L,t}, s_t b^*_t, ψ^*/2 (s_t (b^*_t − b^*))^2 on LHS; labor income, ε w_{F,t} u_{L,F,t}, asset returns, Π_{L,t}, T_{L,t} on RHS.
  - Definitions: Δh_{L,t} = h_{L,t} − (1−δ_h) h_{L,t−1}; s_t = e_t P^*_t / P_t; foreign asset adjustment cost ψ^*/2 (b^*_t − b^*)^2; unemployed transfer ε w_{F,t} u_{L,t}.
- Impatient households (B):
  - Utility: max E_0 ∑_{t=0}^∞ β_B^t [ log(c_{B,t} − ν_B c_{B,t−1}) + ι log(h_{B,t}) − μ (n_{B,t})^{1+φ} / (1+φ) ], with n_{B,t} ≡ n_{B,F,t} + n_{B,I,t}.
  - Budget constraint includes c_{B,t}, q_{h,t} Δh_{B,t}, R_{B,t−1} b_{B,t−1} / (Γ_B π_t) on LHS; labor income, ε w_{F,t} u_{B,F,t}, b_{B,t} / Γ_B, T_{B,t} on RHS.
  - Borrowing constraint (LTV and carryover ζ):
    - b_{B,t} / Γ_B ≤ (1−ζ) b_{B,t−1} / (Γ_B π_t) + π_t q_{h,t} Δh_{B,t}.
  - Inflation process: π_t follows AR(1): (π_t / π̄) = (π_{t−1} / π̄)^{ρ_π} exp(ε_{π,t}), 0<ρ_π<1, ε_{π,t} ∼ N(0,σ_π).
- Entrepreneurs (E):
  - Utility: max E_0 ∑_{t=0}^∞ β_E^t log(c_{E,t} − ν_E c_{E,t−1}).
  - Choice variables: k_{E,t}, n_{E,t}, b_{E,t}.
  - Production:
    - y_{E,F,t} = z_{F,t} (k_{E,t−1})^{α} (Z_t n_{E,F,t})^{1−α}.
    - y_{E,I,t} = z_{I,t} (Z_t n_{E,I,t})^{1−α}.
  - Aggregate productivity process:
    - Z_t = Z_{t−1} exp(z_t), z_t AR(1): (z_t / z̄) = (z_{t−1}/z̄)^{ρ_z} exp(ε_{z,t}), 0<ρ_z<1, ε_{z,t} ∼ N(0,σ_z).
  - Idiosyncratic sectoral tech shocks:
    - (z_{i,t} / z̄_i) = (z_{i,t−1} / z̄_i)^{ρ_{z_i}} exp(ε_{z_i,t}).
  - Borrowing: b_{E,t} at rate R_{E,t}; inspection probability ξ and penalty fraction τ_p on total revenues if caught.
  - Formal and informal goods are perfect substitutes sold at same price p_{E,t}.
  - Intertemporal budget constraint includes costs (wages, loan repayments, capital investment, hiring costs g_{F,t} m_{E,F,t} + g_{I,t} m_{E,I,t}) and revenues (1−ξ τ_p) p_{E,t} (y_{E,F,t} + y_{E,I,t}), b_{E,t}/Γ_E, T_{E,t}.
  - Borrowing constraint collateralized by physical capital:
    - R_{E,t} b_{E,t} / Γ_E ≤ ω_t E_t q_{k,t+1} π_{t+1} (1−δ_k) k_{E,t}.
  - ω_t follows AR(1): (ω_t / ω̄) = (ω_{t−1}/ω̄)^{ρ_ω} exp(ε_{ω,t}), 0<ρ_ω<1, ε_{ω,t} ∼ N(0,σ_ω).

### 2.3 Wage Nash bargaining — value functions and wage rule
- Worker marginal values (patient households) for formal sector:
  - V_{L,F}^{n,t} and V_{L,F}^{u,t} specified with dependence on (1−τ_t) w_{F,t}, μ_t (n_{L,t})^{φ} λ_{L,t}, β_L, λ_{L,t+1}/λ_{L,t}, separation σ_F, hiring probabilities x_{F,t+1}.
- Informal-sector analogues:
  - V_{L,I}^{n,t} and V_{L,I}^{u,t} with wage w_{I,t}, separation ξ+σ_I, hiring probabilities x_{L,I,t+1}.
- Equivalent expressions for impatient households B with λ_{B,1,t} and β_B.
- Surpluses:
  - Worker surplus = V_{j}^{n,t} − V_{j}^{u,t}, where V_{j}^{n,t} = Γ_L V_{L,j}^{n,t} + Γ_B V_{B,j}^{n,t}.
  - Firm’s surplus (simplifying assumption) = real cost per hire κ_j (x_{j,t})^{η}.
- Nash bargaining wage rule for sector j:
  - (1−Υ_j) A_{j,t} κ_j (x_{j,t})^{η} = Υ_j (V_{j}^{n,t} − V_{j}^{u,t}), where Υ_F and Υ_I are workers’ bargaining powers.

### 2.4 Producers — retailers, importers, final goods, and exports
- Domestic retailers:
  - Continuum of retailers purchase intermediate goods Γ_E y_{E,t}(i) = y_{D,t}(i) + y_{X,t}(i), brand and sell to aggregators.
  - Aggregator CES for domestic goods:
    - y_{D,t} = [ ∫_0^1 y_{D,t}(i)^{(θ_D−1)/θ_D} di ]^{θ_D/(θ_D−1)}; θ_D is elasticity of substitution.
    - Price index P_{D,t} = [ ∫_0^1 P_{D,t}(i)^{1−θ_D} di ]^{1/(1−θ_D)}.
    - Demand y_{D,t}(i) = [ P_{D,t}(i) / P_{D,t} ]^{−θ_D} y_{D,t}.
  - Export price: P_{X,t}(i) = e_t P^*_t (small open economy assumption).
  - Retailers face quadratic price-adjustment cost ψ_D/2 [ (P_{D,t}(i)/(π_D P_{D,t−1}(i)) − 1 ) ]^2.
- Importers:
  - Imported intermediates aggregator:
    - y_{M,t} = [ ∫_0^1 y_{M,t}(i)^{(θ_M−1)/θ_M} di ]^{θ_M/(θ_M−1)}; P_{M,t} defined similarly.
    - Demand y_{M,t}(i) = [ P_{M,t}(i) / P_{M,t} ]^{−θ_M} y_{M,t}.
  - Imported goods purchased at world price e_t P^*_t and resold with quadratic adjustment cost ψ_M/2 [ (P_{M,t}(i)/(π_M P_{M,t−1}(i)) − 1 ) ]^2.
  - Importer profit Π_{M,t}(i) = (p_{M,t}(i) − s_t) y_{M,t}(i) − ψ_M/2 [ (π_{M,t}(i)/π_M − 1)^2 p_{M,t}(i) y_{M,t}(i) ].
- Final-good producers and exports:
  - Final-good CES aggregator:
    - y_t = [ (1−a)^{1/γ} y_{D,t}^{(γ−1)/γ} + a^{1/γ} y_{M,t}^{(γ−1)/γ} ]^{γ/(γ−1)}, where a is steady-state import share and γ is inverse of elasticity between domestic and imported goods.
  - Zero-profit condition:
    - 1 = (1−a) (p_{D,t})^{1−γ} + a (p_{M,t})^{1−γ}.
  - Foreign demand for exports:
    - y_{X,t} = s_t^{−ρ_1/ρ_2} y^*_t, where y^*_t is foreign total revenue, ρ_1 elasticity of substitution and ρ_2 share of exports in rest-of-world demand.

### 2.5 Banks — structure, branches, and frictions
- Banks owned by patient households; monopolistic competition in deposit and loan markets.
- Real balance-sheet identity:
  - b_t = d_t + k_{b,t}, where b_t loans, d_t deposits, k_{b,t} bank capital.
- Wholesale branch:
  - Issues wholesale loans b_t; finances with wholesale deposits d_t and bank capital k_{b,t}.
  - Quadratic cost when capital-to-risk-weighted-assets ratio deviates from regulated ratio ψ_{k1,t}:
    - ψ_{k2}/2 [ k_{b,t}/b_{rw,t} − ψ_{k1,t} ]^2 k_{b,t} term in objective.
  - Regulated capital ratio stochastic process:
    - (ψ_{k1,t} / ψ_{k1}) = (ψ_{k1,t−1} / ψ_{k1})^{ρ_ψ} exp(ε_{ψ,t}), 0<ρ_ψ<1, ε_{ψ,t} ∼ N(0,σ_ψ).
  - Bank capital accumulation:
    - k_{b,n,t}(j) = (1−δ_b) k_{b,n,t−1}(j) + ς Π_{b,n,t−1}(j), where ς retained profits share, δ_b depreciation.
  - Constraint: b_t(j) = b_{B,t}(j) + b_{E,t}(j) = d_t(j) + k_{b,t}(j); b_{rw,t}(j) = θ_B b_{B,t}(j) + θ_E b_{E,t}(j) (risk weights).
- Loan and deposit demand (retail branches):
  - CES bundling with elasticities θ_B and θ_E yields demand functions:
    - b_{B,t}(j) = [ (R_{B,t}(j)^{−1} / R_{B,t}^{−1}) ]^{−θ_B} b_{B,t}.
    - Representative demand result: b_{B,t}(j) = [ (R_{B,t}(j)^{−1} / R_{B,t}^{−1}) − 1 ]^{−θ_B} b_{B,t} (Equation (30)); b_{E,t}(j) analogous (Equation (31)).
  - No-arbitrage: R_{D,t} = R_t.
- Retail banking optimization:
  - Loan branch sets lending rates R_{B,t}(j), R_{E,t}(j) as markups over wholesale rates ̄R_{B,t}, ̄R_{E,t}, with quadratic adjustment costs:
    - φ_i/2 [ (R_{i,t}(j)^{−1} / R_{i,t−1}(j)^{−1} − 1) ]^2 (R_{i,t}(j)^{−1}) b_{i,t}(j), for i = B,E.
  - Retail loan bank maximizes discounted profits subject to loan demand Equations (30) and (31).

### 2.6 Capital and housing producers — accumulation and adjustment costs
- Capital accumulation:
  - k_t = (1−δ) k_{t−1} + [ 1 − ψ_k/2 (i_{k,t} / i_{k,t−1} − 1)^2 ] i_{k,t}.
  - Capital-good producer maximizes expected discounted value with price q_{k,t} and adjustment cost ψ_k.
- Housing accumulation:
  - h_t = (1−δ_h) h_{t−1} + [ 1 − ψ_h/2 (i_{h,t} / i_{h,t−1} − 1)^2 ] i_{h,t}.
  - Housing producer maximization analogous with q_{h,t} and adjustment cost ψ_h.

### 2.7 Monetary and tax policies — rule, fiscal processes, and estimation
- Monetary policy rule (augmented Taylor):
  - R_t / R = (R_{t-1}/R)^{ρ_R} (π_t/π)^{ρ_π} (y_t/y_{t-1})^{ρ_y} (s_t/s_{t-1})^{ρ_s} exp(ε_{R,t}).
  - ρ_R ∈ (0,1); ρ_π, ρ_y, ρ_s > 0; ε_{R,t} ∼ N(0, σ_R).
- Government budget and tax processes:
  - Government expenditures: distributed social benefits and lump-sum transfers.
  - Revenues: fines following audits, payroll and labor income taxes.
  - Tax rates τ_t and τ^s_t follow AR(1) exogenous processes:
    - τ_t/τ = (τ_{t-1}/τ)^{ρ_τ} exp(ε_{τ,t}), τ^s_t/τ^s = (τ^s_{t-1}/τ^s)^{ρ_{τ^s}} exp(ε_{τ^s,t}).
    - ρ_τ and ρ^s_τ ∈ (0,1); ε_{τ,t} ∼ N(0, σ_τ); ε_{τ^s,t} ∼ N(0, σ_{τ^s}).
- Estimation strategy:
  - Bayesian maximum likelihood with Kalman filter; posterior via priors and likelihood.
  - Calibration: small open economy (Tunisia), quarterly data 2000–15.
  - Observables: log difference of real GDP; log difference of tax revenues; log difference of real deposits; log difference of real borrowing; log difference of real effective exchange rate; CPI inflation rate; housing price inflation rate.
- Key calibrated values:
  - Share of impatient households and entrepreneurs: 0.6 (to fit loans to output ratio of 70 percent).
  - Discount factors: patient households 0.99; impatient households and entrepreneurs 0.975.
  - Housing stock depreciation rate: 0.01.
  - Housing weight in utility, ι: 0.05 (to match steady state housing stock to output ratio of 1.5).
  - Average unemployment rate matched: 10 percent (through labor weight μ).
  - Inverse Frisch elasticity: 1.
  - Elasticity of output w.r.t. labor in both sectors, 1−α_F and 1−α_I: 0.65.
  - Formal sector technology steady state z_F: 1.
  - Elasticities of substitution among intermediary goods, θ_D and θ_M: 6 (implying intermediary firms’ steady-state mark-up of 20 percent).
  - Depreciation rate of physical capital: 0.025 (implying 10 years to completely depreciate).
  - Exogenous separation rate formal sector: 0.05.
  - Exogenous separation probability informal sector: 0.1.
  - Households’ bargaining power both sectors: 0.5.
  - Income tax and social security contribution steady states τ and τ^s: 0.35 and 0.15.
  - Elasticity of substitution between intermediary loans to impatient households and entrepreneurs, θ_B and θ_E: 3 and 4.
  - Share of non-distributed profits: 15 percent.
  - Bank risk weights on impatient households’ and entrepreneurs’ borrowing: 0.5 and 0.25.
  - Depreciation rate of bank capital: 0.035.
  - Prior average of informal technology steady state z_I: 3 with standard deviation 1.5 (implying implicit informal sector size around 30 percent when other priors at averages).

### Estimation results — posterior averages and structural inferences
- Estimation details:
  - Metropolis-Hastings algorithm with 500,000 draws; first 20 percent discarded.
- Notable posterior estimates:
  - Probability of being audited for tax evasion, ξ: 0.052.
  - Penalty posterior average τ_p: 0.280 (28 percent of period production).
  - Implied average size of the informal sector: around 45 percent.
  - Price rigidity: domestic prices more sluggish than imported prices → mild low pass-through of imported prices to domestic inflation.
  - Bank interest rate stickiness: entrepreneurial loan rates adjust more rapidly than mortgage loan rates.
  - Capital-to-asset ratio adjustment cost estimated below prior mean.
  - Housing vs productive capital adjustment costs:
    - ψ_h posterior average: 5.976.
    - ψ_k posterior average: 4.866.
- Extended Taylor rule posterior averages:
  - ρ_y: 0.268.
  - ρ_π: 1.771.
  - ρ_s: 0.743.
  - Interpretation: nominal interest rate reacts more strongly to exchange rate fluctuations than to output.
- Selected posterior parameter estimates (posterior medians / averages):
  - ν_L = 0.317 [0.201,0.454]; ν_B = 0.283 [0.217,0.348]; ν_E = 0.842 [0.787,0.894].
  - ξ = 0.052 [0.042,0.062].
  - τ_p = 0.280 [0.248,0.318].
  - η = 0.757 [0.695,0.811].
  - κ_F = 0.648 [0.582,0.718]; κ_I = 0.412 [0.369,0.460].
  - z_I = 3.623 [3.213,4.055].
  - ρ_R = 0.426 [0.326,0.519]; ρ_π (monetary) = 1.771 [1.661,1.879]; ρ_y (monetary) = 0.268 [0.178,0.366]; ρ_s (monetary) = 0.743 [0.678,0.809].
  - Autocorrelations: ρ_{z_F} = 0.869 [0.801,0.941]; ρ_{z_I} = 0.656 [0.514,0.795]; ρ_π = 0.912 [0.865,0.959]; ρ_ω = 0.955 [0.929,0.981].
  - Shock standard deviations: σ_{z_F} = 0.015 [0.012,0.018]; σ_τ = 0.077 [0.064,0.089]; σ_R = 0.009 [0.007,0.011]; σ_π = 0.332 [0.234,0.422].

### Impulse-response functions — financial shocks (selected)
- Households’ LTV shock (temporary 1 percent increase in impatient households’ LTV):
  - Household borrowing: hump-shaped increase with a maximum of about 6 percent reached after six quarters.
  - Interest rate spreads increase due to rising loan demand.
  - Aggregate consumption: drops on impact, then increases subsequently.
  - Housing prices increase on impact.
  - Inflation: moderate increase on impact as rising borrowing costs for entrepreneurs pass through to goods.
  - Real exchange rate: initial appreciation, then mild drop in interest rates due to augmented Taylor rule reaction.
  - Once deposits return to steady state, aggregate consumption becomes mildly positive.
- Entrepreneurs’ LTV shock (temporary 1 percent increase):
  - Lending to entrepreneurs rises; entrepreneurs increase demand for consumption goods, labor, and physical capital.
  - Impatient households face higher banking interest rate spread → lower demand for bank loans.
  - Credit demand shifts from households to businesses; deposits by savers increase.
  - Aggregate consumption falls persistently below steady state.
  - Shock behaves like a negative aggregate demand shock: sectoral and aggregate production decline, unemployment rises, final good prices drop.
  - Monetary authority loosens policy; real exchange rate depreciates.

### Variance decomposition — drivers of fluctuations (selected averages)
- Output (ΔY_t):
  - Formal sector technology shocks explain the largest fraction of output variance.
  - Foreign shocks and financial sector shocks account for 15.6 and 10.6 percent of real GDP growth fluctuations, respectively.
- Loans and financial variables:
  - Financial market shocks explain approximately half of the long-run fluctuations in loans of patient households and entrepreneurs.
  - LTV shocks' role in domestic inflation and exchange rate fluctuations: 5.7 percent for inflation and 0.5 percent for exchange rate (averages).
  - For real activity, financial market disturbances account for between 10 and 20 percent of sectoral production dynamics.
- Exchange rate volatility:
  - Real exchange rate volatility mainly explained by the foreign interest rate shock: 85.1 percent on average.
  - The foreign interest rate shock generates 9.1 percent of the variance of output in the long run.
  - Domestic monetary policy shock contributes very little to exchange rate fluctuations.
- Inflation drivers:
  - Foreign interest accounts for 22.4 percent of inflation fluctuations (average).
  - Foreign inflation accounts for 59.8 percent of inflation fluctuations (average).

### Monetary and macroprudential optimal policies — definitions and welfare metric
- Objective: evaluate optimality of alternative simple monetary and macroprudential rules using aggregate households’ welfare; compare to historical average welfare from benchmark model (posterior averages).
- Monetary rule: augmented Taylor rule as in Equation (34).
- Macroprudential rules considered (notation preserved):
  - LTV-B (loan-to-value on households’ borrowing) — Equation (36).
  - LTV-E (loan-to-value on entrepreneurs’ borrowing) — Equation (37).
  - BKR (bank capital requirement rule) — Equation (38).
- Policy search:
  - Feedback coefficients numerically searched to maximize present value of lifetime utility:
    - W_t = Γ_L E_t ∑_{s=1}^∞ β_L U(c^L_{t+s}, h^L_{t+s}, n^L_{t+s}) + Γ_B E_t ∑_{s=1}^∞ β_B U(c^B_{t+s}, h^B_{t+s}, n^B_{t+s}) + Γ_E E_t ∑_{s=1}^∞ β_E U(c^E_{t+s}).
  - Welfare reported per household type via consumption equivalent measure; methodology accounts for aggregate uncertainty via second-order approximation.

### Policy experiments and welfare findings — independent and cooperative settings
- Independent monetary policy (optimized ATR) — main findings:
  - Welfare-maximizing monetary rule: augmented Taylor rule that stabilizes inflation, reacts to the exchange rate, and disregards output fluctuations.
  - Optimal ATR yields welfare gain equivalent to average permanent increase in consumption of 2.06 percent.
  - Main beneficiaries: borrowing households and entrepreneurs.
  - Optimized exchange rate response coefficient: 1.97.
  - Targeting output gaps is welfare detrimental in this model.
- Macroprudential rules (independent optimization):
  - Optimized macroprudential LTV rules display pronounced countercyclical reactions to output and/or borrowing.
  - Macroprudential rules yield relatively small welfare impacts compared to optimized ATR.
  - Highest average consumption gain under optimized macroprudential rules: 0.33 percent (LTV-E rule).
  - Including the informal sector reduces macroprudential welfare gains drastically:
    - Welfare gain drops by 95 percent for LTV-B,
    - by 85 percent for LTV-E,
    - and by 70 percent for BKR rules.
- Cooperative setting (joint optimization of monetary and macroprudential):
  - Counterfactual macroprudential rules have more pronounced impact under cooperation.
  - Reported average consumption improvements (with and without informality in parentheses):
    - Optimized LTV-B: 0.02 (0.05) percent.
    - Optimized LTV-E: 0.19 (0.8) percent.
  - Under cooperative optimization:
    - Adopting LTV-B implies aggregate welfare gain of 2.08 percent (cooperative ATR+LTV-B).
    - Adopting BKR implies aggregate welfare gain of 2.25 percent (cooperative ATR+BKR).
  - Under cooperation, performances of LTV-B, LTV-E, and BKR on aggregate social welfare become relatively similar.
- Role of the informal sector:
  - Increasing steady-state informal sector size reduces welfare impact of macroprudential policies.
  - Welfare gains decline sharply when share of informal sector reaches about 20 percent; higher shares can lead to welfare losses under historical monetary rule.
  - Mechanisms: informal production relies less on capital, reduces entrepreneurs’ borrowing, lowers wages and housing accumulation, reduces access to financial services for impatient households; banks face reduced credits, lower spreads, lower profitability, and balance-sheet effects that slow credit expansion.
- Role of nominal and real rigidities:
  - Financial frictions combined with other market imperfections reduce macroprudential efficiency.
  - Abstracting from price stickiness may more than double welfare gains from counterfactual macroprudential policies.
  - Stronger inflation targeting substantially enhances macroprudential outcomes.
  - Labor market frictions can be dampened by rules such as LTV-B or BKR.

### Policy implications and recommendations (from analysis)
- Monetary policy should prioritize inflation stabilization and react to exchange rate movements in settings with costly foreign asset adjustment; targeting output gaps is not advisable in this model.
- Macroprudential tools (countercyclical LTV, bank capital ratio rules) can yield welfare benefits, but standalone effects are modest compared with an optimized ATR.
- Joint optimization of monetary and macroprudential policies substantially enhances welfare gains attributable to macroprudential instruments.
- Effectiveness of macroprudential policies critically depends on:
  - aggressiveness of monetary policy towards inflation,
  - size of the informal sector (negative effect of informality),
  - presence of nominal rigidities (price stickiness) and labor market frictions.
- In economies with sizable informality, strengthening monetary commitment to inflation stabilization improves macroprudential policy effectiveness; attention to informality is crucial when designing macroprudential regimes.
- Suggested extensions: optimizing the role of government (public spending, transfers, tax rates) and richer specifications of external financing for banks or private borrowers.

*Source: IMF Working Paper chapter sections 2.1–2.7 and 4.1–4.4 (wpiea2019255-print-pdf).*

### 2.1  Labor market

### 2.1  Labor market

### Labor market structure and matching
- Jobs are created through sectoral matching functions as in Pappa et al.(2015) and Anand and Khera(2016).
- Fraction of household members of type i = L,B employed in sector j = F,I is n_{i,j,t}.
- Total employees during period t: survivors after exogenous separation plus new hires m_{i,j,t}.
- Formal sector law of motion:
  - n_{i,F,t} = (1−σ_F) n_{i,F,t−1} + m_{i,F,t}, where σ_F is the exogenous separation rate.
- Informal sector law of motion:
  - n_{i,I,t} = (1−ξ−σ_I) n_{i,I,t−1} + m_{i,I,t}, where σ_I is exogenous job destruction and ξ is audit job-loss probability.
- Aggregate notation:
  - n_{j,t} ≡ Γ_L n_{L,j,t} + Γ_B n_{B,j,t}; m_{j,t} ≡ Γ_L m_{L,j,t} + Γ_B m_{B,j,t}.
- Beginning-of-period unemployment pool for type i:
  - u_{i,t} = 1 − n_{i,F,t−1} − n_{i,I,t−1} + σ_F n_{i,F,t−1} + (ξ+σ_I) n_{i,L,t−1}
  - Equivalent form: u_{i,t} = 1 − (1−σ_F) n_{i,F,t−1} − (1−ξ−σ_I) n_{i,L,t−1}.
- Symmetry assumption between patient and impatient households on sectoral working shares:
  - n_{L,j,t} = n_{B,j,t} = n_{j,t} ⇒ u_{L,t} = u_{B,t} = u_{t}, m_{L,j,t} = m_{B,j,t} = m_{j,t}.
- Jobseeker hiring probability in sector j:
  - x_{j,t} ≡ m_{j,t} / u_{t}.
- Hiring cost (matching cost) for entrepreneurs in sector j:
  - g_{j,t} = A_{j,t} κ_j (x_{j,t})^{η}, where κ_j is matching efficiency and η is elasticity of matching to vacancies.

---

### 2.2  Households

### Patient households (L)
- Representative utility:
  - max E_0 ∑_{t=0}^∞ β_L^t [ log(c_{L,t} − ν_L c_{L,t−1}) + ι log(h_{L,t}) − μ (n_{L,t})^{1+φ} / (1+φ) ],
  - where n_{L,t} ≡ n_{L,F,t} + n_{L,I,t}; φ is inverse Frisch elasticity; μ and ι weight housing and leisure.
- Intertemporal budget constraint (real terms):
  - c_{L,t} + q_{h,t} Δh_{L,t} + d_{L,t} + s_t b^*_t + ψ^*/2 (s_t (b^*_t − b^*))^2
    ≤ (1−τ_t) w_{F,t} n_{L,F,t} + w_{I,t} n_{L,I,t} + ε w_{F,t} u_{L,F,t}
    + R_{D,t−1} d_{L,t−1} / π_t + s_t R^*_{t−1} b^*_{t−1} / π^*_t + Π_{L,t} + T_{L,t}.
- Definitions:
  - Δh_{L,t} = h_{L,t} − (1−δ_h) h_{L,t−1}, δ_h is housing depreciation.
  - s_t = e_t P^*_t / P_t (real exchange rate).
  - q_{h,t} is real price of housing.
  - Foreign asset adjustment cost: ψ^*/2 (b^*_t − b^*)^2.
  - Unemployed transfer: ε w_{F,t} u_{L,t}.
  - Dividends Π_{L,t} and lump-sum transfers T_{L,t} included.

### Impatient households (B)
- Representative utility:
  - max E_0 ∑_{t=0}^∞ β_B^t [ log(c_{B,t} − ν_B c_{B,t−1}) + ι log(h_{B,t}) − μ (n_{B,t})^{1+φ} / (1+φ) ],
  - where n_{B,t} ≡ n_{B,F,t} + n_{B,I,t}.
- Budget constraint:
  - c_{B,t} + q_{h,t} Δh_{B,t} + R_{B,t−1} b_{B,t−1} / (Γ_B π_t) ≤ (1−τ_t) w_{F,t} n_{B,F,t} + w_{I,t} n_{B,I,t} + ε w_{F,t} u_{B,F,t} + b_{B,t} / Γ_B + T_{B,t}.
- Borrowing constraint (LTV and carryover ζ):
  - b_{B,t} / Γ_B ≤ (1−ζ) b_{B,t−1} / (Γ_B π_t) + π_t q_{h,t} Δh_{B,t},
  - π_t follows AR(1): (π_t / π̄) = (π_{t−1} / π̄)^{ρ_π} exp(ε_{π,t}), 0<ρ_π<1, ε_{π,t} ∼ N(0,σ_π).

### Entrepreneurs (E)
- Utility:
  - max E_0 ∑_{t=0}^∞ β_E^t log(c_{E,t} − ν_E c_{E,t−1}).
- Choice variables: k_{E,t}, n_{E,t}, b_{E,t}.
- Production:
  - y_{E,F,t} = z_{F,t} (k_{E,t−1})^{α} (Z_t n_{E,F,t})^{1−α}.
  - y_{E,I,t} = z_{I,t} (Z_t n_{E,I,t})^{1−α}.
- Total factor productivity:
  - Z_t = Z_{t−1} exp(z_t), z_t follows AR(1): (z_t / z̄) = (z_{t−1}/z̄)^{ρ_z} exp(ε_{z,t}), 0<ρ_z<1, ε_{z,t} ∼ N(0,σ_z).
- Idiosyncratic sectoral tech shocks:
  - (z_{i,t} / z̄_i) = (z_{i,t−1} / z̄_i)^{ρ_{z_i}} exp(ε_{z_i,t}).
- Entrepreneurs borrow b_{E,t} at rate R_{E,t}; face inspection probability ξ and penalty fraction τ_p on total revenues if caught.
- Formal and informal goods are perfect substitutes and sold at same price p_{E,t}.
- Intertemporal budget constraint:
  - c_{E,t} + (1+τ_{s,t}) w_{F,t} n_{E,F,t} + w_{I,t} n_{E,I,t} + R_{E,t−1} b_{E,t−1} / (Γ_E π_t) + q_{k,t} [k_{E,t} − (1−δ_k) k_{E,t−1}] + g_{F,t} m_{E,F,t} + g_{I,t} m_{E,I,t}
    ≤ (1−ξ τ_p) p_{E,t} (y_{E,F,t} + y_{E,I,t}) + b_{E,t}/Γ_E + T_{E,t}.
- Borrowing constraint (collateralized by physical capital):
  - R_{E,t} b_{E,t} / Γ_E ≤ ω_t E_t q_{k,t+1} π_{t+1} (1−δ_k) k_{E,t},
  - ω_t follows AR(1): (ω_t / ω̄) = (ω_{t−1}/ω̄)^{ρ_ω} exp(ε_{ω,t}), 0<ρ_ω<1, ε_{ω,t} ∼ N(0,σ_ω).

---

### 2.3  Wage Nash bargaining

### Value functions and bargaining
- Worker marginal values for patient households in formal sector:
  - V_{L,F}^{n,t} = (1−τ_t) w_{F,t} − μ_t (n_{L,t})^{φ} λ_{L,t} + β_L (λ_{L,t+1}/λ_{L,t}) [ (1−σ_F) (1−x_{F,t+1}) V_{L,F}^{n,t+1} + σ_F (1−x_{F,t+1}) V_{L,F}^{u,t+1} ].
  - V_{L,F}^{u,t} = ε w_{F,t} + β_L (λ_{L,t+1}/λ_{L,t}) [ x_{F,t+1} V_{L,F}^{n,t+1} + (1−x_{F,t+1}) V_{L,F}^{u,t+1} ].
- Informal sector analogues:
  - V_{L,I}^{n,t} = w_{I,t} − μ_t (n_{L,t})^{φ} λ_{L,t} + β_L (λ_{L,t+1}/λ_{L,t}) [ (1−(ξ+σ_I)) (1−x_{L,I,t+1}) V_{L,I}^{n,t+1} + (ξ+σ_I) (1−x_{L,I,t+1}) V_{L,I}^{u,t+1} ].
  - V_{L,I}^{u,t} = β_L (λ_{L,t+1}/λ_{L,t}) [ x_{L,I,t+1} V_{L,I}^{n,t+1} + (1−x_{L,I,t+1}) V_{L,I}^{u,t+1} ].
- Equivalent value-function equations for impatient households B with λ_{B,1,t} replacing λ_{L,t} and β_B replacing β_L; analogous formal and informal sector expressions given (Equations (16)–(19)).
- Total worker and firm surplus:
  - Worker surplus = V_{j}^{n,t} − V_{j}^{u,t}, where V_{j}^{n,t} = Γ_L V_{L,j}^{n,t} + Γ_B V_{B,j}^{n,t}, V_{j}^{u,t} similar.
  - Firm’s surplus (simplifying assumption) = real cost per hire κ_j (x_{j,t})^{η}.
- Nash bargaining wage rule for sector j:
  - (1−Υ_j) A_{j,t} κ_j (x_{j,t})^{η} = Υ_j (V_{j}^{n,t} − V_{j}^{u,t}), where Υ_F and Υ_I are workers’ bargaining powers.

---

### 2.4  Producers

### Domestic retailers and aggregators
- Continuum of domestic retailers (measure one) purchase intermediate goods from entrepreneurs Γ_E y_{E,t}(i) = y_{D,t}(i) + y_{X,t}(i), brand them and sell to aggregators.
- Aggregator domestic goods CES:
  - y_{D,t} = [ ∫_0^1 y_{D,t}(i)^{(θ_D−1)/θ_D} di ]^{θ_D/(θ_D−1)}, θ_D is elasticity of substitution.
  - Price index P_{D,t} = [ ∫_0^1 P_{D,t}(i)^{1−θ_D} di ]^{1/(1−θ_D)}.
  - Demand: y_{D,t}(i) = [ P_{D,t}(i) / P_{D,t} ]^{−θ_D} y_{D,t}.
- Exports pricing:
  - P_{X,t}(i) = e_t P^*_t (small open economy assumption).
- Retailers face quadratic price-adjustment cost ψ_D/2 [ (P_{D,t}(i)/(π_D P_{D,t−1}(i)) − 1 ) ]^2.
- Retailer profit expression includes domestic and export components and adjustment cost (expression given).

### Importers
- Aggregate imported intermediates:
  - y_{M,t} = [ ∫_0^1 y_{M,t}(i)^{(θ_M−1)/θ_M} di ]^{θ_M/(θ_M−1)}; P_{M,t} defined similarly.
  - Demand: y_{M,t}(i) = [ P_{M,t}(i) / P_{M,t} ]^{−θ_M} y_{M,t}.
- Imported goods purchased at world price e_t P^*_t and resold with quadratic adjustment cost ψ_M/2 [ (P_{M,t}(i)/(π_M P_{M,t−1}(i)) − 1 ) ]^2.
- Importer profit Π_{M,t}(i) = (p_{M,t}(i) − s_t) y_{M,t}(i) − ψ_M/2 [ (π_{M,t}(i)/π_M − 1)^2 p_{M,t}(i) y_{M,t}(i) ].

### Final-good producers and exports
- Final-good CES aggregator:
  - y_t = [ (1−a)^{1/γ} y_{D,t}^{(γ−1)/γ} + a^{1/γ} y_{M,t}^{(γ−1)/γ} ]^{γ/(γ−1)}, where a is steady-state import share and γ is inverse of elasticity between domestic and imported goods.
- Zero-profit condition:
  - 1 = (1−a) (p_{D,t})^{1−γ} + a (p_{M,t})^{1−γ}.
- Foreign demand for exports:
  - y_{X,t} = s_t^{−ρ_1/ρ_2} y^*_t, where y^*_t is foreign total revenue, ρ_1 elasticity of substitution and ρ_2 share of exports in rest-of-world demand.

---

### 2.5  Banks

### Aggregate and balance-sheet identity
- Banks (owned by patient households) are monopolistically competitive in deposit and loan markets.
- Real balance-sheet identity:
  - b_t = d_t + k_{b,t}, where b_t are loans, d_t deposits, k_{b,t} bank capital (equity).

### Wholesale branch
- Wholesale bank issues wholesale loans b_t and finances them with wholesale deposits d_t and bank capital k_{b,t}.
- Quadratic cost when capital-to-risk-weighted-assets ratio k_{b,t}/b_{rw,t} deviates from regulated ratio ψ_{k1,t}:
  - ψ_{k2}/2 [ k_{b,t}/b_{rw,t} − ψ_{k1,t} ]^2 k_{b,t} term in objective.
- Regulated capital ratio stochastic process:
  - (ψ_{k1,t} / ψ_{k1}) = (ψ_{k1,t−1} / ψ_{k1})^{ρ_ψ} exp(ε_{ψ,t}), 0<ρ_ψ<1, ε_{ψ,t} ∼ N(0,σ_ψ).
- Bank capital accumulation:
  - k_{b,n,t}(j) = (1−δ_b) k_{b,n,t−1}(j) + ς Π_{b,n,t−1}(j), where ς is retained profits share, δ_b capital depreciation.
- Wholesale bank chooses b_{B,t}, b_{E,t}, d_t to maximize discounted profits subject to:
  - b_t(j) = b_{B,t}(j) + b_{E,t}(j) = d_t(j) + k_{b,t}(j).
  - b_{rw,t}(j) = θ_B b_{B,t}(j) + θ_E b_{E,t}(j) (risk weights).

### Loan and deposit demand (retail branches)
- Loans bundled in CES baskets with elasticities θ_B and θ_E; demand functions:
  - b_{B,t}(j) = [ (R_{B,t}(j)^{−1} / R_{B,t}^{−1}) ]^{−θ_B} b_{B,t}.
  - Representative demand result used yields:
    - b_{B,t}(j) = [ (R_{B,t}(j)^{−1} / R_{B,t}^{−1}) − 1 ]^{−θ_B} b_{B,t} (as presented in Equation (30)).
    - b_{E,t}(j) analogous (Equation (31)).
- No-arbitrage: deposit rate paid by wholesale to deposit branch equals policy rate: R_{D,t} = R_t.

### Retail banking optimization
- Loan branch sets lending rates R_{B,t}(j), R_{E,t}(j) as markups over wholesale rates ̄R_{B,t}, ̄R_{E,t}, facing quadratic adjustment costs:
  - φ_i/2 [ (R_{i,t}(j)^{−1} / R_{i,t−1}(j)^{−1} − 1) ]^2 (R_{i,t}(j)^{−1}) b_{i,t}(j), for i = B,E.
- Retail loan bank maximizes discounted profits (objective specified) subject to loan demand Equations (30) and (31).

---

### 2.6  Capital and housing producers

### Capital producers
- New capital accumulation with adjustment cost:
  - k_t = (1−δ) k_{t−1} + [ 1 − ψ_k/2 (i_{k,t} / i_{k,t−1} − 1)^2 ] i_{k,t}.
- Capital-good producer maximization (owned by patient households):
  - max_{i_{k,t}} E_t ∑_{s=0}^∞ β_L^{t+s} (λ_{L,t+s} / λ_{L,t}) [ q_{k,t+s} ((1−δ) k_{t+s−1} + [1 − ψ_k/2 (i_{k,t+s}/i_{k,t+s−1} − 1)^2] i_{k,t+s}) − i_{k,t+s} ].

### Housing producers
- Housing accumulation:
  - h_t = (1−δ_h) h_{t−1} + [ 1 − ψ_h/2 (i_{h,t} / i_{h,t−1} − 1)^2 ] i_{h,t}.
- Housing producer maximization:
  - max_{i_{h,t}} E_t ∑_{s=0}^∞ β_L^{t+s} (λ_{L,t+s}/λ_{L,t}) [ q_{h,t+s} ((1−δ_h) h_{t+s−1} + [1 − ψ_h/2 (i_{h,t+s}/i_{h,t+s−1} − 1)^2] i_{h,t+s}) − i_{h,t+s} ].

*Source: IMF Working Paper chapter section 2.1–2.6 (wpiea2019255-print-pdf).*

### 2.7  Monetary and tax policies

### 2.7 Monetary and tax policies

### Monetary policy rule
- The central bank targets the short term policy rate according to an augmented Taylor type policy rule that allows reaction to changes in the real exchange rate:
  R_t / R = (R_{t-1}/R)^{ρ_R} (π_t/π)^{ρ_π} (y_t/y_{t-1})^{ρ_y} (s_t/s_{t-1})^{ρ_s} exp(ε_{R,t}).
- Parameters and shocks:
  - ρ_R ∈ (0,1) is the degree of interest rate smoothing.
  - ρ_π, ρ_y, ρ_s > 0 are reaction coefficients to inflation deviation, output growth, and real exchange rate change, respectively.
  - ε_{R,t} ∼ N(0, σ_R) is an uncorrelated exogenous shock.

### Government budget and tax processes
- Government expenditures: distributed social benefits and lump-sum transfers.
- Government revenues: collected fines following audits, payroll and labor income taxes.
- Balanced budget rule (equation notation preserved in source).
- Tax rates τ_t and τ^s_t are not policy instruments but follow AR(1) exogenous processes:
  - τ_t/τ = (τ_{t-1}/τ)^{ρ_τ} exp(ε_{τ,t}), τ^s_t/τ^s = (τ^s_{t-1}/τ^s)^{ρ_{τ^s}} exp(ε_{τ^s,t}).
  - ρ_τ and ρ^s_τ ∈ (0,1); ε_{τ,t} ∼ N(0, σ_τ); ε_{τ^s,t} ∼ N(0, σ_{τ^s}).

### Estimation strategy and calibration
- Estimation method: Bayesian maximum likelihood with Kalman filter; posterior via priors and likelihood.
- Calibration: small open economy (Tunisia), quarterly data 2000–15.
- Observables: log difference of real GDP; log difference of tax revenues; log difference of real deposits; log difference of real borrowing; log difference of real effective exchange rate; CPI inflation rate; housing price inflation rate.
- Key calibrated values:
  - Share of impatient households and entrepreneurs: 0.6 (to fit loans to output ratio of 70 percent).
  - Discount factors: patient households 0.99; impatient households and entrepreneurs 0.975.
  - Housing stock depreciation rate: 0.01.
  - Housing weight in utility, ι: 0.05 (to match steady state housing stock to output ratio of 1.5).
  - Average unemployment rate matched: 10 percent (through labor weight μ).
  - Inverse Frisch elasticity: 1.
  - Elasticity of output w.r.t. labor in both sectors, 1−α_F and 1−α_I: 0.65.
  - Formal sector technology steady state z_F: 1.
  - Elasticities of substitution among intermediary goods, θ_D and θ_M: 6 (implying intermediary firms’ steady-state mark-up of 20 percent).
  - Depreciation rate of physical capital: 0.025 (implying 10 years to completely depreciate).
  - Exogenous separation rate formal sector: 0.05.
  - Exogenous separation probability informal sector: 0.1.
  - Households’ bargaining power both sectors: 0.5.
  - Income tax and social security contribution steady states τ and τ^s: 0.35 and 0.15.
  - Elasticity of substitution between intermediary loans to impatient households and entrepreneurs, θ_B and θ_E: 3 and 4.
  - Share of non-distributed profits chosen to match capital to total asset ratio: 15 percent.
  - Bank risk weights on impatient households’ and entrepreneurs’ borrowing: 0.5 and 0.25.
  - Depreciation rate of bank capital: 0.035.
  - Prior average of informal technology steady state z_I: 3 with standard deviation 1.5 (implying implicit informal sector size around 30 percent when other priors at averages).

### Estimation results — key posterior averages and interpretations
- Posterior estimation details:
  - Metropolis-Hastings algorithm with 500,000 draws; first 20 percent discarded.
- Notable posterior estimates and findings:
  - Probability of being audited for tax evasion, ξ: 0.052.
  - Penalty posterior average: 28 percent of period production.
  - Implied average size of the informal sector: around 45 percent (consistent with Schneider et al. (2010) shadow economy ≈ 40 percent).
  - Price rigidity: domestic prices more sluggish than imported prices → mild low pass-through of imported prices to domestic inflation.
  - Bank interest rate stickiness: entrepreneurial loan rates adjust more rapidly than mortgage loan rates.
  - Capital-to-asset ratio adjustment cost estimated below prior mean (implying low deposit-loan spread needed to offset profit losses).
  - Housing investment adjustment cost larger than productive capital:
    - Posterior average for housing adjustment cost ψ_h: 5.976.
    - Posterior average for productive capital adjustment cost ψ_k: 4.866.
  - Extended Taylor rule posterior averages:
    - Reaction coefficient to output growth, ρ_y: 0.268.
    - Reaction coefficient to inflation deviations, ρ_π: 1.771.
    - Reaction coefficient to exchange rate changes, ρ_s: 0.743.
    - Interpretation: nominal interest rate reacts more strongly to exchange rate fluctuations than to output, consistent with managed exchange policy.
- Selected posterior parameter estimates from tables (posterior medians / averages where reported):
  - ν_L = 0.317 [0.201,0.454]; ν_B = 0.283 [0.217,0.348]; ν_E = 0.842 [0.787,0.894].
  - ξ = 0.052 [0.042,0.062].
  - τ_p = 0.280 [0.248,0.318].
  - η = 0.757 [0.695,0.811].
  - κ_F = 0.648 [0.582,0.718]; κ_I = 0.412 [0.369,0.460].
  - z_I = 3.623 [3.213,4.055].
  - ρ_R = 0.426 [0.326,0.519]; ρ_π (monetary) = 1.771 [1.661,1.879]; ρ_y (monetary) = 0.268 [0.178,0.366]; ρ_s (monetary) = 0.743 [0.678,0.809].
  - Autocorrelation examples: ρ_{z_F} = 0.869 [0.801,0.941]; ρ_{z_I} = 0.656 [0.514,0.795]; ρ_π = 0.912 [0.865,0.959]; ρ_ω = 0.955 [0.929,0.981].
  - Shock standard deviation examples: σ_{z_F} = 0.015 [0.012,0.018]; σ_τ = 0.077 [0.064,0.089]; σ_R = 0.009 [0.007,0.011]; σ_π = 0.332 [0.234,0.422].

### Impulse-response functions — financial shocks
- Households’ LTV shock (temporary 1 percent increase in impatient households’ LTV):
  - Household borrowing: hump-shaped increase with a maximum of about 6 percent reached after six quarters.
  - Interest rate spreads increase due to rising loan demand.
  - Aggregate consumption: drops on impact, then increases subsequently (positive revenue effect).
  - Housing prices increase on impact.
  - Rising borrowing cost for entrepreneurs passes through to intermediate and final goods → moderate increase in inflation on impact.
  - Initial interest rate rise ⇒ real exchange rate appreciation; then augmented Taylor rule reaction leads to a mild drop in interest rates.
  - Once deposits return to steady state, aggregate consumption becomes mildly positive.
- Entrepreneurs’ LTV shock (temporary 1 percent increase):
  - Lending to entrepreneurs rises; entrepreneurs increase demand for consumption goods, labor, and physical capital.
  - Impatient households face higher banking interest rate spread → lower demand for bank loans.
  - Credit demand shifts from households to businesses; deposits by savers increase.
  - Aggregate consumption falls persistently below steady state.
  - Shock behaves like a negative aggregate demand shock: sectoral and aggregate production decline, unemployment rises, final good prices drop.
  - Monetary authority loosens policy; real exchange rate depreciates.

### Variance decomposition — drivers of fluctuations
- Output (ΔY_t):
  - Formal sector technology shocks explain the largest fraction of output variance.
  - Foreign shocks and financial sector shocks account for 15.6 and 10.6 percent of real GDP growth fluctuations, respectively (reported averages).
- Loans and financial variables:
  - Financial market shocks explain approximately half of the long-run fluctuations in loans of patient households and entrepreneurs.
  - Impatient households’ and entrepreneurs’ LTV shocks significantly affect sectoral productions and aggregate consumption.
  - LTV shocks’ role in domestic inflation and exchange rate fluctuations is relatively limited: 5.7 percent for inflation and 0.5 percent for exchange rate (averages reported).
  - For real activity, financial market disturbances account for between 10 and 20 percent of sectoral production dynamics.
- Exchange rate as shock absorber:
  - Real exchange rate volatility mainly explained by the foreign interest rate shock: 85.1 percent in average.
  - The same foreign interest rate shock generates 9.1 percent of the variance of output in the long run.
  - Domestic monetary policy shock provides a very small contribution to exchange rate fluctuations.
- Inflation drivers:
  - Majority of inflation forecast error variance attributed to external shocks:
    - Foreign interest accounts for 22.4 percent of inflation fluctuations (average).
    - Foreign inflation accounts for 59.8 percent of inflation fluctuations (average).

### Monetary and macroprudential optimal policies — definitions and welfare metric
- Objective: evaluate optimality of alternative simple monetary and macroprudential rules using aggregate households’ welfare as metric; compare counterfactual rules to historical average welfare from benchmark model (posterior averages).
- Monetary rule: same specification as Equation (34) (augmented Taylor rule).
- Macroprudential policy rules considered (notation preserved):
  - Loan-to-value applied on households’ borrowing (LTV-B):
    - π^t_π = (π_{t-1}/π)^{ρ_π} (y_t/y_{t-1})^{ρ_y^π} [b_B_t / b_B_{t-1}]^{ρ_{b_B}^π} exp(ε_{π,t}). (Equation (36))
  - Loan-to-value applied on entrepreneurs’ borrowing (LTV-E):
    - ω_t/ω = (ω_{t-1}/ω)^{ρ_ω} (y_t/y_{t-1})^{ρ_y^ω} [b_E_t / b_E_{t-1}]^{ρ_{b_E}^ω} exp(ε_{ω,t}). (Equation (37))
  - Bank capital requirement (BKR):
    - ψ_{k,t}/ψ_k = [ψ_{k,t-1}/ψ_k]^{ρ_ψ} (y_t/y_{t-1})^{ρ_y^ψ} [b_B_t / b_B_{t-1}]^{ρ_{b_B}^ψ} [d_E_t / d_E_{t-1}]^{ρ_{b_E}^ψ} exp(ε_{ψ,t}). (Equation (38))
- Policy search:
  - Feedback coefficients in monetary and macrofinancial rules are numerically searched to maximize present value of lifetime utility:
    - W_t = Γ_L E_t ∑_{s=1}^∞ β_L U(c^L_{t+s}, h^L_{t+s}, n^L_{t+s}) + Γ_B E_t ∑_{s=1}^∞ β_B U(c^B_{t+s}, h^B_{t+s}, n^B_{t+s}) + Γ_E E_t ∑_{s=1}^∞ β_E U(c^E_{t+s}).
  - Welfare is reported per type of household via a consumption equivalent measure: percentage increase in the historical average consumption that makes each agent's welfare under the estimated policy equal to welfare under the optimized rule.
  - Methodology accounts for aggregate uncertainty via a second-order approximation.

*Source: 2.7 Monetary and tax policies (chapter content as provided).*

### 4.1  Independent policy setting

### 4.1 Independent policy setting

### Monetary policy: augmented Taylor rule (ATR) — main findings
- Welfare-maximizing monetary rule is an augmented Taylor rule that stabilizes inflation, reacts to the exchange rate, and disregards output fluctuations.
- Considering all shocks of the benchmark model, the optimal ATR yields a welfare gain equivalent to an average permanent increase in consumption of 2.06 percent.
- The main beneficiaries of the ATR welfare gain are borrowing households and entrepreneurs.
- Optimized exchange rate response in the benchmark model: coefficient equal to 1.97.
- Optimal ATR departs from pure inflation targeting when monetary and financial shocks or external shocks are considered; the rule implies relatively significant departures from inflation targeting in its parameters.
- Targeting output gaps is welfare detrimental in this model because output gaps partly signal inefficiency; targeting unemployment fluctuations would be expected to be welfare improving.

### Monetary policy — model variants and drivers
- Results are presented for two model versions: (a) benchmark model (with informality) and (b) one-sector model that abstracts from informality.
- Abstracting from the informal sector increases the welfare gain from optimal monetary policy regardless of the source of fluctuations, because the one-sector model exhibits higher fluctuations (stronger financial accelerator) and tax/financial shocks are more influential.
- When volatilities are higher (one-sector model), the welfare gain from optimizing the monetary rule relative to the historical rule is larger.

### Macroprudential rules (independent optimization) — summary effects
- Optimized macroprudential LTV rules display pronounced countercyclical reaction functions to output and/or borrowing dynamics.
- Macroprudential rules yield relatively small welfare impacts compared to the optimized ATR.
- Highest average consumption gain under optimized macroprudential rules: 0.33 percent, achieved by the LTV-E rule that stabilizes output and entrepreneurs’ borrowing.
- Welfare gains arise via: reduced fluctuations of production factors and prices, smaller interest rate premium due to lower lending rate adjustment costs, higher expected credit leading to higher investment, capital, housing, and consumption; housing accumulation contributes to welfare for borrowing households and entrepreneurs; patient households gain but to a smaller extent.

### Distributional and informality effects on independent macroprudential gains
- Including the informal sector in the benchmark model drastically reduces welfare gains from macroprudential rules:
  - Welfare gain drops by 95 percent for LTV-B,
  - by 85 percent for LTV-E,
  - and by 70 percent for BKR rules.
- Rationale: informal production relies less on capital and has less efficient labor matching; agents in informal production have reduced access to financial services; distortionary penalties on controlled agents amplify sensitivity to informality.
- Banks face reduced credits to entrepreneurs and households, lower long-term interest rate spreads, declining profitability, and a balance sheet effect that slows credit market expansion (less accumulation of capital and lower deposits).

### 4.2 Cooperative setting (monetary and macroprudential set jointly)

### Cooperative optimization — enhanced macroprudential effectiveness
- Under cooperation, one institution sets both monetary policy and the macroprudential tool to maximize social welfare.
- Counterfactual macroprudential rules have a more pronounced impact in the cooperative case than in the independent case.
- Reported improvements in average consumption under cooperative optimization (model with and without informal sector):
  - Optimized LTV-B: 0.02 (0.05) percent,
  - Optimized LTV-E: 0.19 (0.8) percent.
- The impact on welfare attributed to macroprudential rules is about twice as large under cooperative setting once ATR parameters are set to optimized values.
- Intuition: optimized interest-rate rule (stabilized inflation, mitigated frictions) raises average output, which increases housing and physical capital accumulation and makes macroprudential policy more operative; financial stability makes monetary policy’s price stabilization task easier by containing excessive credit accumulation, asset price excesses, and the procyclical financial accelerator.
- Example aggregate welfare outcomes under cooperative setting:
  - Adopting LTV-B implies an aggregate welfare gain of 2.08 percent (cooperative ATR+LTV-B),
  - Adopting BKR implies an aggregate welfare gain of 2.25 percent (cooperative ATR+BKR).
- Under cooperation, performances of LTV-B, LTV-E, and BKR on aggregate social welfare become relatively similar.

### 4.3 Role of the informal sector

### Sensitivity of macroprudential effectiveness to informality
- Increasing the steady-state size of the informal sector reduces the welfare impact of macroprudential policies.
- Mechanisms:
  - Entrepreneurs’ capital accumulation is negatively correlated with informality (informal production technology relies less on physical capital), reducing entrepreneurs’ access to borrowing and diminishing LTV-E effectiveness.
  - Lower wages in presence of informality for both savers and borrowers yield lower average housing accumulation; impatient households have less access to financial services and less consumption-smoothing ability, reducing LTV-B effectiveness.
  - From banks’ perspective: lower credit levels to entrepreneurs and households reduce long-term interest rate spreads and profitability, accelerating slowdown in credit markets via balance sheet effects.
- Welfare gains decline sharply when the share of the informal sector reaches about 20 percent; for higher (but still reasonable) informality shares, welfare losses can occur in several cases (e.g., LTV-B with respect to γ^π_b_B and LTV-E with respect to γ^ω_y) under the historical monetary rule.
- Note: these results are conditional on the historical (less aggressive) monetary rule; stronger inflation targeting can compensate negative effects of informality on macroprudential efficiency.

### 4.4 Role of nominal and real rigidities (sensitivity)
- Combining financial frictions with other market imperfections generally reduces the efficiency of macroprudential policies; quantitative reductions in welfare gains can be substantive (notably for LTV-B).
- Abstracting from price stickiness may more than double the welfare gain from counterfactual macroprudential policies.
- This is consistent with cooperative-setting results: stronger inflation targeting substantially enhances macroprudential outcomes.
- Labor market frictions can be dampened by policy rules such as LTV-B or BKR.

### Policy implications and recommendations (from analysis)
- Monetary policy should prioritize inflation stabilization and can improve welfare further by reacting to exchange rate movements in settings with costly foreign asset adjustment; targeting output gaps is not advisable in this model.
- Macroprudential tools (countercyclical LTV, bank capital ratio rules) can yield welfare benefits, but their standalone effect is modest compared with an optimized ATR.
- Jointly optimizing monetary and macroprudential policies (cooperative setting) substantially enhances the welfare gains attributable to macroprudential instruments.
- The effectiveness of macroprudential policies critically depends on:
  - the aggressiveness of monetary policy towards inflation,
  - the size of the informal sector (negative effect of informality),
  - the presence of nominal rigidities (price stickiness) and labor market frictions.
- In economies with sizable informality, strengthening monetary commitment to inflation stabilization improves the effectiveness of macroprudential policies; attention to informality is crucial when designing macroprudential regimes.

### Conclusion (summary)
- Macroprudential policies can play an important role in advanced and developing economies by stabilizing financial indicators and improving household welfare through reduced financial friction effects.
- In a New Keynesian small open DSGE with housing, financial frictions, and informality, countercyclical LTV rules are less effective when informality is large and monetary policy is passive.
- Cooperative implementation of monetary and macroprudential policies, particularly with an aggressive interest-rate response to inflation, drastically enhances the welfare-improving role of LTV and BKR rules.
- Extensions worth exploring include optimizing the role of government (public spending, transfers, tax rates) and richer specifications of external financing for banks or private borrowers.

*Source: wpiea2019255-print-pdf — 4.1 Independent policy setting (chapter section).*

### References

### References

### Monetary policy, macroprudential policy, and banking
- Angelini, P., S. Neri, and F. Panetta (2012).  Monetary and macroprudential policies.  European Central Bank Working Paper 1449.
- Angeloni, I. and E. Faia (2013).  Capital regulation and monetary policy with fragile banks. Journal of Monetary Economics 60, 311–324.
- De Paoli, B. and M. Paustian (2017).  Coordinating monetary and macroprudential policies. Journal of Money, Credit, and Banking 49, 319–349.
- Laureys,  L. and R. Meeks (2018).   Monetary and macroprudential policies under rules and discretion. Economics Letters 170, 104–108.
- Lewis, V. and S. Villa (2016).  The interdependence of monetary and macroprudential policy under the zero lower bound. National Bank of Belgium Working Paper 310.
- Ozkan, F. G. and D. F. Unsal (2014). On the use of monetary and macroprudential policies for small open economies. International Monetary Fund Working Paper 14/112.
- Quint, D. and P. Rabanal (2014).  Monetary and macroprudential policy in an estimated DSGE model of the euro area. International Journal of Central Banking 37, 169–236.
- Silvo, A. (2018).  The interaction of monetary and macroprudential. Journal of Money, Credit, and Banking forthcoming, 233–285.
- Unsal, D. F. (2013). Capital flows and financial stability: Monetary policy and macroprudential responses. International Journal of Central Banking 9, 233–285.
- Angelini, P., S. Neri, and F. Panetta (2012).  Monetary and macroprudential policies.  European Central Bank Working Paper 1449.

### Credit, housing, and boom–bust cycles
- Gerali, A., S. Neri, L. Sessa, and F. M. Signoretti (2010).  Credit, and banking in a DSGE model of the euro area. Journal of Money, Credit and Banking 42, 107–141.
- Iacoviello, M. and S. Neri (2010). Housing market spillovers: Evidence from an estimated DSGE model. American Economic Journal: Macroeconomics 2, 125–164.
- Lambertini, L., C. Mendicino, and M. T. Punzi (2013).  Leaning against boom–bust cycles in credit and housing prices. Journal of Economic dynamics and Control 37, 1500–1522.

### DSGE models and optimal policy rules
- Clarida, R., J. Galí, and M. Gertler (2002). A simple framework for international monetary policy analysis. Journal of Monetary Economics 49, 879–904.
- Corsetti, G., L. Dedola, and S. Leduc (2010). Optimal monetary policy in open economies. In B. Friedman and M. Woodford (Eds.), Handbook of Monetary Economics Vol.3B, Chapter 16, pp. 862–933. Elsevier.
- Smets, F. and R. Wouters (2003). An estimated dynamic and stochastic general equilibrium model for the euro area. Journal of the European Economic Association 5, 1123–1175.
- Schmitt-Grohe, S. and M. Uribe (2007).  Optimal simple and implementable monetary and fiscal rules. Journal of Monetary Economics 54, 1702–1725.
- Mumtaz, H. and F. Zanetti (2017).  The effect of labor and financial frictions on aggregate fluctuations. Macroeconomic Dynamics, Forethcoming.
- De Paoli, B. and M. Paustian (2017).  Coordinating monetary and macroprudential policies. Journal of Money, Credit, and Banking 49, 319–349.

### Informality, labor markets, and development
- Batini, N., P. Levine, E. Lotti, and B. Yang (2011).  Informality, frictions and monetary policy.  School of Economics, University of Surrey, School of Economics Discussion Papers 0711.
- Blanchard, O. and J. Galí (2010).  Labor markets and monetary policy:  A new keynesian model with unemployment. American Economic Journal: Macroeconomics 2, 1–30.
- Anand, R. and P. Khera (2016). Macroeconomic impact of product and labor market reforms on informality and unemployment in india. International Monetary Fund Working Paper 16/47.
- Shapiro, Alan, F. and A. Gonzalez (2015). Macroprudential policy and labor market dynamics in emerging economies. International Monetary Fund Working Paper 15/18.
- Jouini, N. and N. Rebei (2014). The welfare implications of services liberalization in a developing country. Journal of Development Economics 106, 1–14.
- Pappa, E., R. Sajedi, and E. Vella (2015).  Fiscal consolidation with tax evasion and corruption. Journal of International Economics 96, 56–75.

### Shadow economy estimates and tax evasion
- Schneider, F., A. Buehn, and M. C. Punzi (2010).  New estimates for the shadow economies all over the world. International Economic Journal 24, 443–461.
- Medina, L. and F. Schneider (2018).  Shadow economies around the world:  What did we learn over the last 20 years?  International Monetary Fund Working Paper 18/17.
- Pappa, E., R. Sajedi, and E. Vella (2015).  Fiscal consolidation with tax evasion and corruption. Journal of International Economics 96, 56–75.

*References (as listed in wpiea2019255-print-pdf - References).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019255-print-pdf.pdf_