## wpiea2025204-source-pdf

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---

### Equilibrium definition
- Competitive equilibrium defined given:
  - sequences of foreign prices {i
∗
t
, P
∗
t
}, monetary policy rule it = r
∗
+ φ
π
π
t
, initial wealth distribution D
0
(a,e), and initial capital K
0.
  - Equilibrium objects: household policy paths {n
c
i
H,t
(a,e), c
i
F,t
(a,e), c
i
t
(a,e), a
i
t+1
(a,e)}_{i∈F I,FE}, distributions D
t
(a,e), prices {E
t
, Q
t
, P
t
, P
H,t
, P
F,t
, W
t
, p
t
, i
t
, r
t
, r
k
t}, and aggregates {C
t
, C
H,t
, C
F,t
, C A
t
, TB
t
, K
t
, I
t
, Y
t
, A
t
, nfa
t} such that agents and firms optimize and markets clear.
- Market clearing conditions:
  - Goods market: Y
t
= C
H,t
+ C
∗
H,t
+ G
t
+ I
t
+ Ξ
t
. (31)
  - Labor market: N
t
= ∫ n
i,t
e
i,t
di = L
t
. (32)
  - Asset market: A
t
= ∫_{i∈Rich} a
i,t
di = B
t
+ p
t
+ n f a
t
= B
t
+ q
t
K
t+1
+ n f a
t
. (33)
- Definition: C
H,t
≡ ∑_{e} π
e
R c
H,t
(a,e) D
t
(a,e) denotes aggregate consumption of home goods; C
t
, C
F,t
, A
t
are defined analogously.
- Focus: equilibria where Q
∞
= Q
ss
 (long-run exchange rate returns to steady state).

### Calibration (quarterly frequency and key parameter values)
- Time period: a quarter.
- Discount factor:
  - β = 0.955 (chosen to be consistent with an annual interest rate of 6%).
- Preference and labor parameters:
  - σ = 2
  - φ = 2
  - ψ = 7.04 (Labor disutility coefficient; to normalize quarterly output to be equal to 1).
- Idiosyncratic productivity process:
  - log e
t
= ρ
l
 log e
t−1
+ ε
t
, with σ
l
= std(ε
t
).
  - ρ
l
= 0.97 (Own calc., see appendix C).
  - σ
l
= 0.14 (Own calc., see appendix C).
- Goods substitution and home weight:
  - η = η
∗
= 1.01
  - χ = 0.484
- Subsistence imported consumption:
  - ̄c = 0.058 (Own calc., appendix B).
- Financial access and borrowing:
  - λ = 33%
  - a = 0
- Firm parameters:
  - α = 0.33
  - ε = 11
  - κ = 0.1
  - ζ = 17
- Monetary and fiscal calibration:
  - φ
π
 = 1.5
  - G/Y = 15%
  - B/Y = 20%
  - K/Y = 47%
  - n f a/Y = −35%
- Additional calibration notes:
  - MPC target: Mean annual MPC = 0.632 (used to set K/Y = 47%).
  - Real rate: obtained from average nominal interest rate on domestic bonds minus inflation in Ghana between 2009 and 2018 (IMF IFS Statistics).
  - Data sources for specific calibrations: ECG-ISSER Ghana Socioeconomic Panel Survey (GSPS), FAO Food Balances, World Bank Consumption database, Findex Database.
- Selected parameter table entries (preserved exactly):
  - β Discount factor 0.955 — Interest rate r = 6%
  - σ Inverse EIS 2 — Standard value
  - φ Inverse Frisch 2 — Standard value
  - ψ Labor disutility coefficient 7.04 — Y = 1
  - ρ
l
 Persistence, labour productivity 0.97 — Own calc. (see appendix C)
  - σ
l
 St. deviation, labour productivity 0.14 — Own calc. (see appendix C)
  - η Elasticity of substitution across goods 1.01 — Feenstra et al. (2018)
  - χ Weight of home good in consumption 0.484 — Melina and Portillo (2018)
  - ̄c Subsistence foreign consumption 0.058 — Own calc. (see appendix B)
  - λ Financially excluded 33% — Findex Database (see appendix D)
  - a Borrowing constraint 0 — Standard value
  - α Capital share 0.33 — Standard value
  - ε Steady-state markup 11 — De Ferra et al. (2020)
  - κ Slope of Phillips curve 0.1 — Standard value
  - ζ Capital adjustment cost coefficient 17 — De Ferra et al. (2020)
  - φ
π
 Taylor rule coefficient 1.5 — Standard value
  - G/Y Government spending (% GDP) 15% — Melina and Portillo (2018)
  - B/Y Domestic debt (% GDP) 20% — Melina et al. (2016)
  - K/Y Domestic capital (% GDP) 47% — Mean annual MPC = 0.632
  - n f a/Y Net Foreign Asset Position (% GDP) −35% — WEO and IFS databases (last year available)

### Positive implications — shock to import prices (quantitative experiments)
- Shock specification:
  - P
F,t
 = E
t
P
∗
t
(1 + ε
t
) > E
t
P
∗
t
. (34)
  - ε
t
 interpreted as a transportation cost, follows AR(1) with persistence 0.8.
  - Calibrated so that, if exchange rate were kept constant, P
F,t
 would increase by 1% on impact.
- Simulation notes:
  - Shocks up to 10 times baseline simulated; qualitative results similar with higher variation.
- Aggregate equilibrium responses:
  - Higher import prices → increase in domestic inflation (consistent with equation (5)).
  - Central bank raises nominal interest rates via Taylor rule; with φ
π
 = 1.5 this raises the real interest rate.
  - Real exchange rate Q appreciates (consistent with UIP (24)), contracting external demand and exports.
  - Government raises taxes to keep debt constant (equation 27).
  - Initial GDP drop due to lower exports and domestic demand; subsequent recovery as expenditure switching favors domestic goods.
  - Real wage falls: inflation reduces real wages mechanically; drop in output reduces labor demand; nominal wage stickiness produces hump-shaped real wage response.
- Consumption dynamics:
  - Subsistence assumption (̄c > 0) mutes demand drop for foreign goods.
  - FE households:
    - Large initial drop in domestic goods consumption due to lower wages and negative income effect.
    - Subsequent recovery mainly from imported price reduction over time and expenditure switching.
  - FI households:
    - Smaller initial consumption drop due to access to saving instruments.
  - Subsistence needs require a higher policy rate increase to stabilize inflation compared to ̄c = 0.
- Distributional impacts:
  - Consumption inequality increases and slowly reverts.
  - Lower productivity households disproportionately affected via larger real wage drops and lower asset holdings (for FI).
  - FE households more affected than FI households.
  - Cash-on-hand (COH) distribution: immediate increase in mass below-mean steady-state COH; gradual return to steady state.
- Baseline summary:
  - Imported goods price increase → higher inflation, output contraction, reduced consumption (especially for lower-productivity and financially excluded households), increased consumption inequality, more households with below-average steady-state COH.

### Normative implications — welfare analysis
- Welfare metric:
  - Λ solves: E
0
[∑_{t=0}^∞ β
t
 U(c
t
, n
t
)] = E
0
[∑_{t=0}^∞ β
t
 U((1 + Λ) c
ss
t
, n
ss
t
)]. (35)
  - Λ is the percentage increase in steady-state consumption needed to match lifetime utility after the shock; Λ < 0 implies welfare loss.
  - Computation approximates ∞ by T = 50 periods given β.
- Welfare effects of import price shock:
  - All households experience welfare losses (Λ < 0).
  - Losses larger for households with lower initial COH.
    - FE agents suffer most: reliant solely on labor income; low-productivity agents have largest labor income decline.
    - FI agents: bottom of COH distribution suffer most; top quantile suffers slightly more than middle/high quantiles because asset returns decline (investment firm equity falls).
  - Counterfactual (no negative asset revaluation): welfare losses strictly higher the lower the initial COH position.

### Monetary policy rules — welfare implications (section 3.4)
- Less aggressive monetary policy (φ
π
 from 1.5 to 1.15):
  - Macros (relative to baseline):
    - Inflation increases by more.
    - GDP drops by less.
    - Real wage experiences a larger decline due to sticky nominal wages and higher inflation.
  - Distribution/welfare:
    - Slightly lower welfare drop for bottom COH households: welfare decline is 0.005 percentage points smaller than baseline.
    - High COH households face more pronounced negative impacts due to larger real wage drop and smaller hours decline.
  - Counterfactual isolations:
    - Fixing real wages to baseline: all households would suffer less under less aggressive policy.
    - Fixing labor demand to baseline: low COH households suffer more while high COH households suffer less.
- Interest rate smoothing (it = ρ i_{t−1} + (1−ρ)[r* + φπ πt], ρ = 0.5):
  - Dynamics:
    - Slower initial response but same aggressiveness; higher initial inflation, faster subsequent decline.
    - Households decrease consumption less on impact but more in subsequent periods.
    - GDP drops less on impact and rebounds less.
    - Real wage drops more on impact but recovers faster.
  - Welfare:
    - Decline in welfare smaller for all agents relative to baseline due to improved real wage behavior and equity dynamics.
- Targeting imported goods price inflation (it = r* + φπ πF,t):
  - Dynamics:
    - Stronger policy rate hikes to contain imported price component.
    - Foreign goods inflation increases significantly less; domestic inflation declines more.
    - Higher real rate triggers deeper recession and larger decrease in demand for home-produced goods.
    - Aggregate inflation mostly unchanged on impact; increases in subsequent periods.
    - Output drops more; smaller immediate rise in inflation leads to lower decline in real wage despite larger labor demand drop.
  - Welfare:
    - Generally larger welfare losses compared to baseline.
    - Heterogeneous effects: FE households benefit from real wage dynamics but low COH FE households harmed by labor demand drop; high COH FI households face larger equity value declines offsetting higher real rates.
- Targeting domestic price inflation (it = r* + φπ πH,t):
  - Motivation: stabilize “core” prices; see literature on subsistence and sticky prices.
  - Dynamics:
    - Smoother interest rate path as central bank “sees through” imported shock.
    - Lower initial increase in interest rates → smaller intertemporal substitution of domestic goods → smaller GDP drop.
  - Welfare:
    - Smaller welfare loss for all households compared to baseline.
    - Mechanism: better consumption smoothing; higher consumption of foreign good; higher real wages for FE agents; higher interest rates benefit FI asset returns.
    - Among monetary options, this rule yields the smallest welfare losses, though poorer households still suffer most.

### Fiscal transfers (role of transfers)
- Policy experiment: maintain zero deficit but provide positive transfers to bottom-income households financed by higher taxes on richer households.
- Macroe effects:
  - Macroeconomic variables almost unchanged relative to baseline.
  - Transfers offset a large part of the shock for bottom-income households.
- Distributional/welfare effects:
  - Appears Pareto improving in model: other groups also exhibit reduced consumption-equivalent losses due to general equilibrium effects (higher consumption among poor, reduced savings among rich → smaller decline in output and real wages).
- LIC applicability caveats:
  - Targeted transfers feasibility limited by large informal sectors, narrow tax bases, weak administrative capacity in LICs.
  - Emphasizes importance of improving fiscal targeting capacity in LICs.

### Summary of key findings and policy implications (section conclusion)
- Model features: open economy HANK with (i) fraction of poor households without financial market access and (ii) poorer households spending a larger share on imported food items via subsistence consumption ̄c.
- Three main findings:
  - Imported goods price shock causes a consumption-led recession, higher inflation, and drop in real wages. Consumption inequality rises because poor households cannot insure and face subsistence exposures.
  - Monetary policy can shape macro outcomes but does not meaningfully alter distributional consequences across alternative rules (less aggressive stance, import/domestic targets, interest-rate smoothing). Targeting domestic/core prices yields the smallest welfare losses among monetary options, but poorer households still suffer most.
  - Fiscal transfers to poorer households are more effective at addressing distributional concerns than monetary policy in the model and can be implemented without major welfare decreases for other groups.
- Broader notes:
  - Framework calibrated to an average LIC; other idiosyncratic features (informality, incomplete markets) not modeled.
  - Future research: full fiscal policy treatment including fiscal dominance possibility.

### Model solution method, current account identity, and investment firm FOCs
- Transitional dynamics algorithm (quantitative implementation):
  1. Guess paths for Y_t, π_t, K_t.
  2. Use monetary policy (29) and π_t to back out r_t.
  3. UIP (24) with r_t and exogenous i^*_t gives Q_t.
  4. Q_t → P_{H,t}/P_t, P^*_{H,t}, P_{F,t}/P_t via (18)-(20).
  5. P_{H,t}/P_t and π_t → π^H_t.
  6. r_t and P_{H,t}/P_t → T_t via tax rule (27) (G_t fixed).
  7. Y_t, π^H_t, K_t → r^k_t, w_t, L_t via (11), (12), and firm production function.
  8. π_t and w_t → π^W_t.
  9. K_t → I_t, investment adjustment cost, and q_t via law of motion and (22).
  10. r_t, w_t, P ratios, T_t → household problems → A_t, C_t, I^M_t (tax incidence assumption required).
  11. P^*_{H,t} → C^*_{H,t} via (14).
  12. r_t and TB_t → n^f a_t via (30).
  - Check NKPC (10), assets market (33), investment firm optimality (23); update guesses via Newton’s method (Auclert et al. (2021)) until convergence.
- Current account identity (final expression):
  - −TB_t + nfa_t = (1+r_{t-1}) nfa_{t-1} ⇒ CA_t = nfa_t − nfa_{t-1} = TB_t + r_{t-1} nfa_{t-1}
- Investment firm problem and FOCs:
  - Π^{Inv}_t = [ r^k_t K_t − I_t − Ξ_t ] (P_H,t / P_t)
  - Ξ_t(I_t,K_t) = ζ/2 [ ((1−δ)K_t + I_t) / K_{t-1} ]^2 K_t
  - FOC w.r.t. I_t:
    - q_t = [ 1 + ζ (K_{t+1}/K_{t} − 1) ] (P_H,t / P_t)
  - FOC w.r.t. K_{t+1} (exact expression preserved):
    - q_t = (P_H,t+1 / P_{t+1}) (1+r_{t+1}) [ r^k_{t+1} − ζ/2 (K_{t+2}/K_{t+1} − 1)^2 − ζ (K_{t+2}/K_{t+1} − 1)(−I_{t+1}/K_{t+1}) ] + q_{t+1} (1−δ) (1+r_{t+1})
  - Compact equivalent:
    - q_t = (P_H,t+1 / P_{t+1}) (1+r_t) [ r^k_{t+1} − dΞ_{t+1}(K_{t+1},I_{t+1})/dK_{t+1} ] + q_{t+1} (1−δ) (1+r_t)

### Calibration appendices and sensitivity highlights
- Calibration of subsistence c̄ (Appendix B):
  - FAOSTAT Food Balances (2015–2019) used to construct staple food baskets; imports of staple foods ≈ 10% of consumption basket in LIDCs.
  - Model baseline uses c̄ = 0.058; alternatives: c̄ = 0.068 (high) and c̄ = 0.
  - Key stylized facts:
    - LIDCs’ imports of staple foods ≈ 10% of consumption basket; AEs ≈ 2.5%.
    - Heterogeneity: at least 10% of LIDCs import a quarter of staple foods.
    - Regression results: LIDCs dependence ≈ three times EMEs and ≈ 10% of consumption basket; adding rule of law makes group differences non-significant.
- Calibration of labor productivity process (Appendix C):
  - AR(1): log e_t = ρ_l log e_{t−1} + ε_t.
  - GMM results (Table A4):
    - ρ estimate = 0.922*** (0.0377)
    - σ^2_ε = 0.0503 (0.0321)
    - σ^2_ξ = 0.135* (0.0693)
    - χ^2_obs = 18.38; p−value = 0.0004
  - Conversion: annual → quarterly AR(1) using variance/covariance expressions for yearly averages.
- Financial exclusion (Appendix D):
  - Global Findex Database (2024 vintage) stylized facts: median account share in LIDCs ≈ 58%.
  - Model uses λ = 33% financially excluded (conservative relative to Findex medians).
- Sensitivity and impulse-response findings (Appendices G–N):
  - Subsistence c̄ variations:
    - Lower c̄ → FE households switch more to domestic goods; central bank needs smaller rate increase; consumption inequality rises less.
    - Higher c̄ → larger drops in domestic consumption and domestic price inflation on impact; consumption Gini rises dramatically.
    - Threshold identified around 0.03 for investment stability behavior.
  - Alternative shocks and MP specifications produce different IRFs; exogenous UIP premium shock yields milder consumption effects and smaller welfare losses (~half relative to main import price shock for most COH groups).
  - Figures and IRFs referenced across appendices illustrate comparative dynamics (Figures 25–34).

_Italic: Source — wpiea2025204-source-pdf - Appendix A.2 shows the derivation of the identity._

### Appendix A.2 shows the derivation of the identity.

### Appendix A.2 shows the derivation of the identity.

### Equilibrium definition
- Given sequences of foreign prices {i
∗
t
, P
∗
t
}, a monetary policy rule given by i
t
= r
∗
+ φ
π
π
t
, an initial wealth distribution D
0
(a,e), and an initial capital level K
0
, a competitive equilibrium is a path of policies for households
  - {n
c
i
H,t
(a,e), c
i
F,t
(a,e), c
i
t
(a,e), a
i
t+1
(a,e)}_{i∈F I,FE}, distributions D
t
(a,e), prices {E
t
, Q
t
, P
t
, P
H,t
, P
F,t
, W
t
, p
t
, i
t
, r
t
, r
k
t}, and aggregate quantities {C
t
, C
H,t
, C
F,t
, C A
t
, TB
t
, K
t
, I
t
, Y
t
, A
t
, nfa
t} such that all agents and firms optimize and markets clear.
- Goods market clearing:
  - Y
t
= C
H,t
+ C
∗
H,t
+ G
t
+ I
t
+ Ξ
t
. (31)
- Labor market clearing:
  - N
t
= ∫ n
i,t
e
i,t
di = L
t
. (32)
- Asset market clearing:
  - A
t
= ∫_{i∈Rich} a
i,t
di = B
t
+ p
t
+ n f a
t
= B
t
+ q
t
K
t+1
+ n f a
t
. (33)
- Definitions:
  - C
H,t
≡ ∑_{e} π
e
R c
H,t
(a,e) D
t
(a,e) denotes aggregate consumption of home goods; C
t
, C
F,t
, A
t
 are defined similarly.
- Focus: equilibria in which the long-run exchange rate returns to its steady state level, Q
∞
= Q
ss
.

### Calibration
- Time period: a quarter.
- Discount factor:
  - β = 0.955 (chosen to be consistent with an annual interest rate of 6%).
- Preference and labor parameters:
  - σ = 2 (Inverse EIS).
  - φ = 2 (Inverse Frisch).
  - ψ = 7.04 (Labor disutility coefficient; to normalize quarterly output to be equal to 1).
- Idiosyncratic productivity process:
  - log e
t
= ρ
l
 log e
t−1
+ ε
t
, with σ
l
= std(ε
t
).
  - ρ
l
= 0.97 (Own calc., see appendix C).
  - σ
l
= 0.14 (Own calc., see appendix C).
- Goods substitution and home weight:
  - η = η
∗
= 1.01 (Feenstra et al. (2018)).
  - χ = 0.484 (Weight of home good in consumption; Melina and Portillo (2018)).
- Subsistence imported consumption:
  - ̄c = 0.058 (Own calc., appendix B).
  - Rationale: consumption of imported staple goods accounts, on average, for 10% of the overall households’ consumption basket in LICs (appendix B).
- Financial access and borrowing:
  - λ = 33% (Fraction of financially excluded; Findex Database, appendix D).
  - a = 0 (Borrowing constraint; financially included households unable to lend among themselves).
- Firm parameters:
  - α = 0.33 (Capital share).
  - ε = 11 (Steady-state markup; implies a 10% steady-state markup).
  - κ = 0.1 (Slope of Phillips curve).
  - ζ = 17 (Capital adjustment cost coefficient).
- Monetary and fiscal calibration:
  - φ
π
 = 1.5 (Taylor rule coefficient; benchmark, experiments with different values).
  - G/Y = 15% (Government spending to output; Melina and Portillo (2018)).
  - B/Y = 20% (Domestic debt to GDP; Melina et al. (2016)).
  - K/Y = 47% (Domestic capital to GDP; chosen to match Mean annual MPC = 0.632).
  - n f a/Y = −35% (Net Foreign Asset Position to GDP; WEO and IFS databases).
- Additional calibration notes:
  - MPC target: Mean annual MPC = 0.632 (used to set K/Y = 47%).
  - Real rate: obtained from average nominal interest rate on domestic bonds minus inflation in Ghana between 2009 and 2018 (IMF IFS Statistics).
  - Subsample and data sources: ECG-ISSER Ghana Socioeconomic Panel Survey (GSPS) used to estimate labour productivity process; FAO Food Balances and World Bank’s Consumption database used for ̄c estimation.

- Table 1: Parameter Values (selected entries reproduced exactly)
  - β Discount factor 0.955 — Interest rate r = 6%
  - σ Inverse EIS 2 — Standard value
  - φ Inverse Frisch 2 — Standard value
  - ψ Labor disutility coefficient 7.04 — Y = 1
  - ρ
l
 Persistence, labour productivity 0.97 — Own calc. (see appendix C)
  - σ
l
 St. deviation, labour productivity 0.14 — Own calc. (see appendix C)
  - η Elasticity of substitution across goods 1.01 — Feenstra et al. (2018)
  - χ Weight of home good in consumption 0.484 — Melina and Portillo (2018)
  - ̄c Subsistence foreign consumption 0.058 — Own calc. (see appendix B)
  - λ Financially excluded 33% — Findex Database (see appendix D)
  - a Borrowing constraint 0 — Standard value
  - α Capital share 0.33 — Standard value
  - ε Steady-state markup 11 — De Ferra et al. (2020)
  - κ Slope of Phillips curve 0.1 — Standard value
  - ζ Capital adjustment cost coefficient 17 — De Ferra et al. (2020)
  - φ
π
 Taylor rule coefficient 1.5 — Standard value
  - G/Y Government spending (% GDP) 15% — Melina and Portillo (2018)
  - B/Y Domestic debt (% GDP) 20% — Melina et al. (2016)
  - K/Y Domestic capital (% GDP) 47% — Mean annual MPC = 0.632
  - n f a/Y Net Foreign Asset Position (% GDP) −35% — WEO and IFS databases (last year available)

### Positive implications — shock to import prices (quantitative experiments)
- Shock specification:
  - P
F,t
 = E
t
P
∗
t
(1 + ε
t
) > E
t
P
∗
t
. (34)
  - ε
t
 is interpreted as a transportation cost and follows an AR(1) process with persistence 0.8.
  - Calibrated so that, if the exchange rate were kept constant, P
F,t
 would increase by 1% relative to its steady-state value on impact.
- Simulation notes:
  - We simulate shocks of up to 10 times larger than the baseline; results are qualitatively similar but with higher variation.
- Aggregate equilibrium responses:
  - Higher import prices → increase in domestic inflation (consistent with equation (5)).
  - Central bank raises nominal interest rates following the Taylor rule (equation (28)); with φ
π
 = 1.5 this raises the real interest rate.
  - Real exchange rate Q appreciates (consistent with UIP in equation (24)), triggering contraction in external demand and decline in exports.
  - Government raises taxes to cover higher interest payments to keep debt constant (equation 27).
  - Initial drop in GDP due to decline in exports and domestic demand; GDP subsequently recovers as expenditure switching favors domestic goods.
  - Real wage falls: inflation mechanically reduces real wages; drop in aggregate output reduces labor demand; nominal wages are sticky so the decline in nominal wages is gradual producing a hump-shaped real wage response.
- Consumption dynamics:
  - Subsistence assumption (̄c > 0) mutes the drop in demand for foreign goods; subsistence needs imply households still devote substantial resources to imported goods consumption.
  - For FE (financially excluded) households:
    - Initial drop in consumption of domestic goods is large due to lower wages (their only income source) and negative income effect of higher import prices.
    - Subsequent recovery mainly from imported price reduction over time and expenditure switching.
  - For FI (financially included) households:
    - Initial drop in consumption is smaller; access to saving instruments allows smoothing.
  - Subsistence needs require a higher policy rate increase to stabilize inflation compared to the case with no subsistence needs (̄c = 0).
- Distributional impacts:
  - Consumption inequality increases following the shock and slowly reverts toward steady state.
  - Lower productivity households are disproportionately affected:
    - Channel 1: larger drop in real wages compresses labor income more for low-productivity households.
    - Channel 2 (FI households): low-productivity households have accumulated less assets and thus have less ability to smooth the shock.
  - FE households are more affected than FI households given lack of assets.
  - Cash-on-hand (COH) distribution shifts:
    - Immediately after the shock, mass of agents with below-mean steady-state COH increases and mass with above-mean steady-state COH decreases.
    - Over time distributions gradually return to steady state.
- Summary baseline result:
  - Exogenous increase in imported goods prices leads to a rise in inflation, contraction in output, reduced consumption (especially for lower-productivity and financially excluded households), increased consumption inequality, and more households with below-average steady-state COH.

### Normative implications — welfare analysis
- Welfare metric:
  - Λ solves: E
0
[∑_{t=0}^∞ β
t
 U(c
t
, n
t
)] = E
0
[∑_{t=0}^∞ β
t
 U((1 + Λ) c
ss
t
, n
ss
t
)]. (35)
  - Λ captures the percentage increase in steady-state consumption necessary for households to achieve the same lifetime utility as after the shock; negative Λ implies welfare loss.
  - Computational approach: approximate ∞ by T = 50 periods given calibrated β.
- Welfare effects of the import price shock:
  - All households experience welfare losses (Λ < 0).
  - Losses are larger for households with lower initial COH.
    - FE agents suffer most: dependent solely on labor income, and low-productivity agents have largest labor income decline.
    - FI agents also suffer: bottom of COH distribution suffer most; top quantile suffers slightly more than middle/high quantiles because asset returns decline (investment firm equity value falls with the shock), reducing investment income for asset-rich households.
  - Counterfactual: if shock did not affect equity of the investment firm (no negative asset revaluation), the welfare losses are strictly higher the lower the initial COH position of the household (Figure 7 comparison).

- Policy experiments (briefly reported):
  - Varying Taylor rule aggressiveness (φ
π
) affects inflation, output, and real wage responses (Figure 8 contrasts baseline and less aggressive monetary policy).

_Italic: Source — wpiea2025204-source-pdf - Appendix A.2 shows the derivation of the identity._

### 3.4    Welfare Implications of Monetary Policy Rules

### 3.4    Welfare Implications of Monetary Policy Rules

### Different monetary policy aggressiveness
- Policy experiment: change the coefficient φπ in the Taylor equation (28) from 1.5 to 1.15.
- Key macro effects (relative to baseline):
  - Inflation increases by more when the central bank responds less aggressively.
  - GDP drops by less because the smaller rise in policy rates provides less incentive for households to save and domestic demand decreases less.
  - Real wage experiences a larger decline, mostly due to stickiness of nominal wages and the larger increase in inflation.
- Distributional/welfare effects:
  - A less aggressive monetary policy stance leads to a slightly lower welfare drop for households at the bottom of the COH distribution.
  - In consumption equivalents, the welfare decline is only 0.005 percentage points smaller than in the baseline.
  - The negative impact of the shock is more pronounced for households with higher COH positions due to two offsetting effects:
    - Real wages drop more because of higher inflation.
    - The smaller decline in output is associated with a lower drop in hours worked.
  - Counterfactual isolations:
    - If real wages were fixed to the baseline, all households would have suffered less under the less aggressive policy.
    - If labor demand were fixed to the baseline, low COH households would suffer more while high COH households would suffer less. Rationale: low COH households gain less from higher income relative to the negative effect from more hours worked and need resources to sustain consumption after the shock.

### Monetary policy with interest rate smoothing
- Policy rule used (equation 36): it = ρ i_{t−1} + (1−ρ)[r* + φπ πt], with ρ set to 0.5 in the calibration.
- Key dynamics:
  - Central bank is slower to respond but as aggressive as in the baseline.
  - Inflation increases more than baseline on impact, but declines faster subsequently.
    - Higher initial inflation because of slower response.
    - Faster subsequent decline due to shifted timing of aggregate demand movements.
  - Households decrease consumption less on impact, but consumption decline is more pronounced in subsequent periods.
  - GDP drops by less on impact and has a more muted rebound.
  - Real wage drops more on impact (when inflation is higher) but recovers faster as inflationary pressures ease.
- Welfare effects:
  - Decline in welfare of all agents is smaller under the interest rate smoothing rule.
  - Mechanism: improved behavior of the real wage benefits all households; for FI agents, the lower drop in output implies a lower decline in equity value, benefiting high COH households.

### Monetary policy targeting imported goods price inflation
- Policy rule used (equation 37): it = r* + φπ πF,t, where πF,t = PF,t / PF,t−1 − 1 (foreign price inflation).
- Key dynamics:
  - Central bank hikes policy rate by more to contain the most affected part of the price index.
  - Foreign goods price inflation increases significantly less compared to baseline; domestic price inflation declines more.
  - Higher increase in the real rate triggers a deeper recession and a larger decrease in demand for home-produced goods.
  - Net effect on aggregate inflation: mostly unchanged on impact (domestic inflation falls while import inflation rises), increasing only in subsequent periods.
  - Output drops more since demand must be depressed further to contain import price rise.
  - Despite larger drop in labor demand (lower wages), the smaller immediate rise in inflation leads to a lower decline in the real wage.
- Welfare effects:
  - Targeting foreign goods inflation generally leads to larger welfare losses compared to the baseline.
  - Heterogeneous effects:
    - FE households benefit from new real wage dynamics, but the drop in labor demand is particularly harmful to low COH FE households.
    - For high COH FI households, two opposing effects keep welfare losses similar to baseline: larger decline in equity values versus relatively higher real interest rates leaving investment income roughly unchanged.

### Monetary policy targeting domestic price inflation
- Policy rule used (equation 38): it = r* + φπ πH,t, with πH,t defined as domestic price inflation.
- Motivation: corresponds to stabilizing “core” prices (domestic goods) and relates to literature on monetary policy and subsistence (e.g., stabilization of sticky prices non-food inflation).
- Key dynamics:
  - Path of interest rates is smoother than baseline as the central bank “sees through” imported goods price shock and reacts to potential contamination of core inflation.
  - Lower initial increase in interest rates leads to smaller intertemporal substitution of domestic goods (especially by FI households), producing a smaller drop in GDP.
- Welfare effects:
  - Rule achieves a smaller welfare loss for all households compared to baseline.
  - Mechanism: consumers are able to smooth consumption more; consumption of the foreign good is higher.
  - Associated outcomes: higher real wages for FE agents and higher interest rates that benefit returns on assets of FI agents.
  - While targeting core/domestic inflation yields the smallest welfare losses among monetary options considered, poorer households still suffer the most.

### The role of transfers (fiscal policy)
- Experiment: government collects same amount of taxes to sustain a zero deficit (as baseline) but provides positive transfers to households at the bottom of the income distribution, financed by higher taxes on richer households.
- Macroe effects:
  - Macroeconomic variables are almost the same as in baseline.
  - Transfers offset a large part of the shock for those at the bottom of the income distribution.
- Distributional/welfare effects:
  - Because of general equilibrium effects, this policy appears Pareto improving: other groups also present a reduced loss of consumption equivalents.
  - Mechanisms: transfers increase consumption among lower-income households and reduce savings among higher-income households; in general equilibrium these effects result in a smaller decline in output and real wages, benefiting households across the income distribution.
- Caveats for LIC applicability:
  - Feasibility of finely targeted transfers in LICs is limited by structural constraints: large informal sectors, narrow tax bases, weak administrative capacity that restricts the state’s ability to reach the poorest households.
  - The transfer experiments underscore the importance of improving targeting capacity of fiscal systems in LICs.

### Summary of key findings and policy implications (from section conclusion)
- Model features: open economy HANK with (i) a fraction of poor households with no access to financial markets and (ii) poorer households spending a larger share on imported food items modeled as a subsistence level of consumption for tradable goods.
- Three main findings:
  - The shock to imported goods prices causes a consumption-led recession, an increase in inflation, and a drop in real wages. Consumption inequality increases because poor households cannot insure against the shock and are more exposed due to subsistence consumption needs.
  - Monetary policy can shape macroeconomic effects to some extent, but none of the alternative monetary policy reactions analyzed meaningfully alter the distributional consequences. Alternatives considered include: less aggressive stance, targeting import or domestic prices, and interest-rate smoothing. Targeting domestic/core prices is associated with the smallest welfare losses and increases in inequality among the monetary alternatives, but poorer households still suffer most.
  - Fiscal transfers to poorer households seem more effective at addressing distributional concerns than monetary policy and can be implemented without major decreases in welfare of other income groups (in the model).
- Broader notes:
  - The framework is calibrated to an average LIC and highlights characteristics important for LICs, though other idiosyncratic features (informality, incomplete markets) are not modeled.
  - Future research: full treatment of fiscal policy including possibility of fiscal dominance.

*Source: IMF working paper chapter "3.4    Welfare Implications of Monetary Policy Rules" (wpiea2025204-source-pdf).*

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### Notable datasets and technical reports cited
- The global findex database 2021.
- Technical reports and IMF working papers cited across entries (specific titles preserved above).

### A Model and Solution Method

### Quantitative Implementation: Transitional Dynamics
- After we send a shock into the model, the algorithm proceeds in the following manner to find a solution:
  1. Guess a path for Y_t, π_t, K_t
  2. Given a path for π_t, we can use monetary policy (29) to back out a path for the real rate r_t
  3. Given a path for r_t and one for i^*_t, which is exogenous in our framework, the UIP condition (24) gives us a path for the real exchange rate, Q_t
  4. Given a path for Q_t, we can back out a path for P_{H,t}/P_t, P^*_{H,t}, P_{F,t}/P_t using equations (18), (19), (20)
  5. Given a path for P_{H,t}/P_t and one for π_t, we can back out the path of domestic goods inflation, π^H_t
  6. Given a path for r_t and P_{H,t}/P_t, the tax rule (27) gives us a path for T_t (importantly, we keep G_t fixed to its steady state level)
  7. Given a path for Y_t, π^H_t, K_t, we can back out a path for the marginal product of capital r^k_t, the real wage w_t, and labor demand L_t using equations (11), (12) and the firm’s Cobb-Douglas production function
  8. Given a path for π_t and w_t, we can solve for the path of π^W_t using the definition of the real wage
  9. Given a path for K_t we can back out a path for investment I_t using the law of motion of capital, a path for the investment firm’s adjustment cost using its definition, and a path for q_t using equation (22)
  10. Given a path for r_t, w_t, P_{H,t}/P_t, P_{F,t}/P_t, T_t, we can solve the household problems and obtain paths of A_t, C_t and I^M_t
    - To solve the household problems we need to make an assumption about how taxes are levied on differ-
  11. Given a path for P^*_{H,t}, equation (14) gives us a path for C^*_{H,t}
  12. Given a path for r_t and TB_t, equation (30) allows us to back out a path for n^f a_t

*Source: wpiea2025204-source-pdf - References*

### 13.  Finally, given a path forC

### 13.  Finally, given a path forC

### Checking equilibrium conditions and solution method
- Given a path for C_t, L_t, π^W_t, w_t, we check whether the NKPC (10) holds and whether the assets market in equation (33) and the investment firm’s optimality condition (23) hold; by Walras’ law the goods market will clear as well.
- If guesses for Y_t, π_t and K_t are incorrect, we update them using a Newton’s method as specified in Auclert et al. (2021), which lets the code converge in a few steps.

### Current account identity (A.2) — derivation and final expression
- Aggregate budget constraints of financially included and excluded households, use assets market clearing, government budget constraint (26), labor market clearing, production function, and firm zero-profit condition.
- Key intermediate identity (after substitutions and simplifications):
  - α [ (P_H,t / P_t) C^P_H,t + (P_F,t / P_t) C^P_F,t ] + (1−α) [ (P_H,t / P_t) C^{R}_H,t + (P_F,t / P_t) C^{R}_F,t ] + B_t + nfa_t + q_t K_t
    = (1+r_{t-1})(B_{t-1} + nfa_{t-1} + q_{t-1} K_{t-1}) + w_t ∫_n i,t e_{i,t} di − (P_H,t / P_t) Tax_t
- Using goods market clearing and the firm’s first-order conditions, re-arranging and substituting the law of motion for capital yields:
  - −TB_t + nfa_t + q_t K_t = (1+r_{t-1})(nfa_{t-1} + q_{t-1} K_{t-1}) − (P_H,t / P_t) r^k_t K_{t-1} + (P_H,t / P_t) [ K_t − (1−δ)K_{t-1} + Ξ_t ]
- Using investment firm FOC rearranged as:
  - q_{t-1} (1−r_{t-1}) K_{t-1} = (P_H,t / P_t) [ r^K_t K_t / K_{t-1} − Ξ_t − K_t + (1−δ)K_{t-1} ] + q_t K_t
- Final current account identity obtained:
  - −TB_t + nfa_t = (1+r_{t-1}) nfa_{t-1} ⇒ CA_t = nfa_t − nfa_{t-1} = TB_t + r_{t-1} nfa_{t-1}

### Investment firm problem and first-order conditions (A.3)
- Per-period profits and adjustment cost:
  - Π^{Inv}_t = [ r^k_t K_t − I_t − Ξ_t ] (P_H,t / P_t)
  - Ξ_t(I_t,K_t) = ζ/2 [ ((1−δ)K_t + I_t) / K_{t-1} ]^2 K_t
- Firm problem with R_t := ∏_{i=0}^t 1/(1+r_i):
  - max_{ {I_t, K_{t+1}}_{t=0}^∞ } ∑_{t=0}^∞ [ r^k_t K_t − I_t − Ξ_t(K_t,I_t) ] (P_H,t / P_t) R_t
    subject to K_{t+1} = (1−δ)K_t + I_t
- FOC w.r.t. I_t:
  - q_t = [ 1 + ζ (K_{t+1}/K_{t} − 1) ] (P_H,t / P_t) = [ 1 + dΞ_t(I_t,K_t)/dI_t ] (P_H,t / P_t)
- FOC w.r.t. K_{t+1} (exact expression preserved):
  - q_t = (P_H,t+1 / P_{t+1}) (1+r_{t+1}) [ r^k_{t+1} − ζ/2 (K_{t+2}/K_{t+1} − 1)^2 − ζ (K_{t+2}/K_{t+1} − 1)(−I_{t+1}/K_{t+1}) ] + q_{t+1} (1−δ) (1+r_{t+1})
- Equivalent compact form:
  - q_t = (P_H,t+1 / P_{t+1}) (1+r_t) [ r^k_{t+1} − dΞ_{t+1}(K_{t+1},I_{t+1})/dK_{t+1} ] + q_{t+1} (1−δ) (1+r_t)

### Calibration of subsistence parameter c̄ (Appendix B) — methodology and stylized facts
- c̄ is calibrated based on relative imports of staple foods by LIDCs.
- Main empirical steps:
  - Use FAOSTAT Food Balances (2015–2019) to rank food items by contribution to caloric intake and construct country-specific baskets of staple foods needed to reach just above undernourishment thresholds (FAO country-specific minimum calories).
  - Compute countries’ imports of those staple items and express as a share of the consumption basket (from IMF IFS).
- Key empirical findings:
  - LIDCs’ imports of staple foods ≈ 10% of their consumption basket (similar to EMEs), and three times higher than for AEs.
  - Advanced economies (AEs): average imports of staple foods = 2.5% of consumption basket (most AEs import < 10%).
  - EMEs and LIDCs: averages and medians closer to 10% of the consumption basket.
  - LIDCs heterogeneity: at least 10% of countries import a quarter of their staple foods.
- Subsample analysis using World Bank consumption database by income strata:
  - Lowest level of income in LIDCs imports ≈ 13% of consumption basket in staple foods vs 5.5% for the same stratum in AEs.
  - Higher income level in LIDCs: average imports ≈ 4.8% vs almost 2% in AEs.
- Regression analysis (coefficients summarized in figure 22):
  - Baseline: LIDCs dependence on imports ≈ three times higher than EMEs and around 10% of consumption basket.
  - Adding controls (population in log; geography: small islands indicator, % desert, distance to ice-free coast, ruggedness) does not materially change results; all coefficients significant at 5% with expected signs. Population has negative coefficient.
  - Adding an Africa dummy: African countries import on average more 3 percentage points of food for subsistence in their consumption basket.
  - Adding “rule of law” (World Bank WGI): differences between groups (LIDCs and EMEs) become non-significant — rule of law captures much of the cross-group variation.
- Robustness:
  - Replacing group indicators by GDP per capita yields identical pattern: higher GDP → lower dependence on food imports, except when controlling for rule of law which makes GDP per capita non-significant.
- Calibration implication:
  - The model uses c̄ = 0.058 in baseline; alternative values explored: c̄ = 0.068 (high) and c̄ = 0 (no subsistence).

### Calibration of labor productivity process (ρ_l and σ_l) using Ghana GSPS (Appendix C)
- Labor productivity process assumed AR(1): log e_t = ρ_l log e_{t−1} + ε_t with σ_l = std(ε_t).
- Data and procedure:
  - ECG-ISSER Ghana Socioeconomic Panel Survey (GSPS), waves 2009/2010, 2013/2014, 2017/2018; construct hourly wages from paid amount and paid other; lower bound for wages = 10% of average wage; exclude agents with < 1000 hours.
  - Regress log hourly wages (wave 1) on age, age^2, male dummy, education dummies, occupation dummies to extract permanent component ψ̂_i and compute transformed series x̃_{i,t} = x_{i,t} − ψ̂_i.
  - Use 3 waves to form 6 unconditional moment conditions per Floden and Lindé (2001) and estimate ρ, σ^2_ε, σ^2_ξ by GMM.
- Key descriptive statistics:
  - Relative wages W in waves: Wave 1 W = 0.87, Wave 2 W = 2.18, Wave 3 W = 3.65; Std(w_i): 1.06, 1.75, 1.95; Min(w_i) = 0.10 in all waves.
  - After removing permanent component x̃: Std(w_i) = 1.20 (Wave 1), 4.76 (Wave 2), 3.84 (Wave 3).
  - Variance of estimated permanent differences σ^2_ψ = 0.1843.
- OLS for initial wage (selected coefficients from Table A2):
  - Age: 0.0723*** (0.0139)
  - Age^2/100: −0.0675*** (0.0159)
  - D_male: 0.197*** (0.0595)
  - Constant: −3.115*** (0.811)
  - Observations: 847; Adjusted R-squared: 0.317; F-test: 19.69; Prob>F: 0.000
- GMM estimates for AR(1) wage process (Table A4):
  - ρ estimate = 0.922*** (0.0377)
  - σ^2_ε = 0.0503 (0.0321)
  - σ^2_ξ = 0.135* (0.0693)
  - χ^2_obs = 18.38; p−value = 0.0004
- Interpretation and conversion:
  - Persistence of wages is high and similar to estimates for advanced economies.
  - Variance of temporary shocks higher than for US/Sweden, consistent with less developed markets.
  - Annual estimates converted to quarterly AR(1) using expressions for variance and covariance of yearly average of quarterly AR(1) (Wöhrmüller et al. (2022)).

### Financial exclusion (λ) — evidence and calibration (Appendix D)
- Data source: Global Findex Database (World Bank), 2024 vintage.
- Stylized facts:
  - AEs: financial inclusion near universal; rates concentrated close to 100%.
  - LIDCs: median share of respondents with an account ≈ 58%; 25th percentile ≈ 43%; 75th percentile ≈ 63%.
  - Mobile accounts: median in LIDCs = 32% (can reach 87% in some countries); mobile accounts have driven recent inclusion increases.
  - Model calibration uses λ = 33% financially excluded; this underestimates financial exclusion relative to median survey statistics.
- Implication: λ is a key parameter given prevalence of limited financial market participation in LIDCs; model calibration errs on the conservative side relative to Findex medians.

### Sensitivity and impulse-response findings (selected highlights from appendices G–N)
- Subsistence parameter c̄ (Appendix G):
  - Baseline c̄ = 0.058; alternative high c̄ = 0.068; alternative zero c̄ = 0.
  - Main effects of changing c̄:
    - Lower c̄ (less subsistence): financially excluded households switch between foreign and domestic goods more; fall in domestic good consumption by FE households is less pronounced; weaker external trade contribution; central bank needs to raise interest rates less for the same foreign price shock.
    - Higher c̄: larger drops in domestic consumption and domestic price inflation on impact; lower foreign demand/exports drop (i.e., exports relatively stronger), boosting GDP after first period; initial boost to investment followed by decline.
    - Threshold: with lower subsistence, investment becomes more stable and shrinks for most of the period; threshold around 0.03 identified in the text.
  - Inequality dynamics:
    - Higher c̄: inflation on real wages compresses labor income distribution (Gini falls for labor income) but investment boosts dividends to rich households (Gini for cash-on-hand increases); consumption Gini rises dramatically (poor households consume more imported goods and are hit harder).
    - c̄ = 0: rise in consumption inequality is much lower.
- Alternative monetary policy exercises and shocks (Appendices F, H–N):
  - IRFs to an exogenous increase in import prices reported (Figures 26, 27, 29–31).
  - Alternative MP specifications (interest-rate smoothing, import inflation targeting, domestic inflation targeting) and counterfactuals (foreign price inflation target) produce different welfare and distributional outcomes; figures and discussions present comparative dynamics.
  - Alternative shock to UIP premium (Appendix N):
    - Shock modeled via (1+r_t) = (1+i^*_t)*(1+ε_t)*Q_{t+1}/Q_t > (1+i^*_t) Q_{t+1}/Q_t, with ε_t AR(1) persistence 0.8 calibrated to match real interest rate response in main text.
    - Central bank raises nominal rates and partially offsets risk-premium increase; real exchange rate depreciates and trade surplus expands; GDP reacts positively despite consumption dynamics.
    - Consumption effects milder than foreign price shock; FE households benefit from higher working hours; inequality increases but by roughly half the variation compared to main text shock; consumption-equivalent welfare losses are about half the size for most COH groups relative to main text.
- Figures and IRFs referenced in the text illustrate these dynamics (Figures 25–34).

*External Shocks and Monetary Policy Trade-offs in Low-Income Countries — Working Paper No. WP/2025/204*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025204-source-pdf.pdf_