## wp18146

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### Main findings and policy implications
- Low-income countries: average tax to GDP ratio is 15% compared to 30% in advanced economies.
- Quantitative exercise calibrated to Ethiopia evaluates welfare costs of raising additional tax revenues equal to 2% GDP via VAT, CIT, and PIT (tax reforms are unexpected and permanent).
- Steady-state comparative results on output change:
  - VAT leads to a 1.5% loss in output.
  - CIT leads to a 4.9% decline in output.
  - PIT leads to a 6.2% decline in output.
- Consumption equivalence of total welfare costs (whole economy):
  - VAT: 3.89%.
  - PIT: 3.31%.
  - CIT: 2.24%.
- Decomposition of VAT welfare cost:
  - Distributional component: 1.32% (the main driver of VAT’s high consumption-equivalence cost).
  - Aggregate component of VAT: 2.61% (higher than CIT’s aggregate component of 2.52%).
- Cross-region incidence effects:
  - VAT substantially worsens the urban-rural gap; welfare cost borne by rural households is 8 times that borne by urban households (benchmark model where government expenditure is on manufacturing goods produced in the urban area).
  - VAT implicitly transfers income from rural to urban households because VAT revenues partially rebate to urban households via government demand for manufacturing goods.
- Policy implications:
  - VAT is least efficient on aggregate output loss but has the highest consumption-equivalent welfare cost due to distributional effects; best accompanied by transfer programs to offset cross-region regressivity.
  - CIT and PIT generate larger aggregate consumption losses but are more uniformly distributed across regions; pro-growth policies pair better with CIT and PIT.
  - Reducing idiosyncratic labor-market risks or improving insurance can materially increase formal labor supply and capital accumulation, yielding larger welfare gains in low-income countries than suggested by advanced-economy studies.

### Role of idiosyncratic risks
- Eliminating idiosyncratic risks results in:
  - Output and capital stock doubling.
  - Labor supply to the formal sector almost tripling.
- Welfare-cost comparison with and without idiosyncratic risks:
  - Aggregate components of tax reform welfare costs are similar across risk and no-risk scenarios.
  - Distributional components differ substantially because in the no-risk case households are wealthier and savings constitute a higher fraction of disposable income; price changes in consumption goods affect indirect utility differently given utility concavity.
- Modeling treatment:
  - Idiosyncratic shocks make households hesitate to work in higher-productivity formal sectors; removing shocks increases formal sector participation and capital accumulation.

### Model structure and key elements
- Economy type and calibration:
  - Small open economy calibrated to Ethiopia as representative of low-income countries.
  - Two regions: rural and urban (population shares μr and μu).
  - Four sectors: food (domestic agriculture), exported cash crops, manufacturing, and services.
  - Unified capital market; interest rate r is endogenously determined domestically.
- Households:
  - Continuum of infinitely lived households in each region; one unit of time allocated between formal labor markets and informal self-employment.
  - Persistent idiosyncratic productivity shocks ε (εu for urban formal, εr for rural informal food); shocks are Markovian.
  - Households choose consumption across food, manufacturing goods, services and have access to one risk-free bond for partial insurance.
  - Labor markets segmented across regions; migration precluded.
- Production and sectoral assignment:
  - Rural: informal household production of food; share μf of large farmers hire formal labor to produce food and cash crops.
  - Urban: households provide services informally; a representative neoclassical firm hires formal labor and capital to produce manufacturing goods.
  - Cash crops modernized and employ capital for exports; food and services for domestic consumption.
  - Manufacturing goods are the numeraire; relative prices pa (food) and ps (services).
- Government:
  - Utilitarian objective; balanced budget each period.
  - Three Ramsey linear taxes: VAT (τa) on food and manufacturing consumption, PIT (τw) on formal wage income, CIT (τr) on manufacturing revenues and cash-crop profits.
  - Government expenditure G is on manufacturing goods (not directly valued by households in benchmark); government can implement lump-sum transfers in alternatives.
  - Government budget constraint:
    - G + μf τr δkf,t = τa(pa Ca,t + Cm,t) + μf τr(πf,t + π∗,t) + τr ym,t + τw(μu wm Hu,t + μr wf Hr,t)
    - where μf τr δkf,t is tax deduction to agricultural machinery investment.
- Preferences and aggregation:
  - Utilitarian expected discounted utility U = E[∑_{t=0}^∞ β^t u(ct)] with per-period utility:
    - u(ct) = log ca,t + γ log cm,t + ψ log cs,t
  - Homothetic (log-linear) preferences imply fixed expenditure shares: 1/(1+γ+ψ) on food, γ/(1+γ+ψ) on manufacturing, ψ/(1+γ+ψ) on services.
  - Aggregate consumption expenditure Ct relates to per-period utility as u(Ct) = Ut + (1+γ+ψ) log Ct, where Ut depends on taxes and prices.
- Technologies (selected functional forms):
  - Informal rural food: ya,t = za εr,t (1 − hr,t)^{1−αa}.
  - Informal urban services: ys,t = zs (1 − hu,t)^{1−αs}.
  - Urban formal labor income per unit: wm εu,t.
  - Manufacturing: ym,t = zm (km,t)^{αm} (hm,t)^{1−αm}.
  - Large farmers’ food: ya,f,t = za (ha,t)^{1−αa}.
  - Cash crops: y∗t = z∗ (kf,t)^{α∗1} (h∗t)^{α∗2} with α∗1 + α∗2 < 1; no idiosyncratic risk in rural formal market.
  - Capital depreciates at rate δ; manufacturing and cash crops use capital.

### Quantitative experiments and key numerical calibration
- Exogenously calibrated parameters (selected):
  - Discount Rate β 0.96
  - Depreciation Rate δ 0.06
  - Labor Share in Food Production α_a 0.49
  - Labor Share in Services Production 1−α_s 0.37
  - Capital Share in Manufacturing Production α_m 0.37
  - Land Share in Cash Crop Production α^*_1 0.19
  - Labor share in Cash Crop Production α^*_2 0.32
  - Persistence of Urban Income Shocks ρ_u 0.92
  - Persistence of Rural Income Shocks ρ_r 0.92
  - Urban Population Share μ_u 0.28
  - Rural Population Share μ_r 0.69
  - Large Farmer Share μ_f 0.03
- Endogenously calibrated fit (selected entries from Table 2):
  - Manufacturing Share in Consumption γ 0.82 → Data 0.33, Model 0.35
  - Services Share in Consumption ψ 0.50 → Data 0.21, Model 0.22
  - Rural Consumption Gini σ^2_r 0.23 → Data 0.26, Model 0.26
  - Urban Consumption Gini σ^2_u 0.63 → Data 0.40, Model 0.40
  - Tax to GDP Ratio τ_a 0.06 → Data 0.08, Model 0.08
  - CIT in Total Tax Revenues τ_r 0.12 → Data 0.30, Model 0.30
  - PIT in Total Tax Revenues τ_w 0.06 → Data 0.17, Model 0.19
  - Food Share in Output z_a 0.39 → Data 0.42, Model 0.34
  - Manufacturing Share in Output z_m 10.19 → Data 0.33, Model 0.38
  - Export Share in Output z^* 14.17 → Data 0.08, Model 0.10
  - Note: model slightly overshoots manufacturing and cash crops shares while undershooting food share.
- Experiment design (long-run and transition):
  - Government raises additional tax revenue of 2% GDP in benchmark steady state via VAT, CIT, and PIT.
  - Additional tax revenue used as government expenditure not directly valued by households.
  - Compute consumption equivalence welfare changes; decompose welfare into aggregate and distributional components.
- Benchmark effective tax rates and new steady-state rates:
  - Benchmark effective rates: VAT 6.45%, CIT 11.55%, PIT 5.55%.
  - New steady-state rates: VAT 10.5%, CIT 24.0%, PIT 15.2%.
  - Percentage increases w.r.t benchmark: VAT 62%, CIT 108%, PIT 174%.
- Aggregate outcomes (percentage changes from benchmark steady state):
  - VAT: aggregate output −1.49%, aggregate consumption −4.66%, aggregate investment +1.16%.
  - CIT: aggregate output −4.85%, aggregate consumption −7.55%, aggregate capital −7.15%.
  - PIT: aggregate output −6.19%, aggregate consumption −9.50%, aggregate capital −5.51%.

### Welfare results (consumption-equivalence decomposition, regional detail)
- VAT (whole economy = 3.89%):
  - Urban Total −0.68%
    - Urban Aggregate −0.29%
    - Urban Distributional −0.46%
  - Rural Total −5.17%
    - Rural Aggregate −5.26%
    - Rural Distributional −0.10%
  - Whole −3.89% (Aggregate −2.61%, Distributional −1.32%)
- CIT (whole economy = 2.24%):
  - Urban Total −2.80%
    - Urban Aggregate −2.76%
    - Urban Distributional −0.04%
  - Rural Total −2.02%
    - Rural Aggregate −2.25%
    - Rural Distributional 0.24%
  - Whole −2.24% (Aggregate −2.52%, Distributional 0.28%)
- PIT (whole economy = 3.31%):
  - Urban Total −3.77%
    - Urban Aggregate −4.65%
    - Urban Distributional 0.92%
  - Rural Total −3.13%
    - Rural Aggregate −3.14%
    - Rural Distributional 0.02%
  - Whole −3.31% (Aggregate −3.95%, Distributional 0.66%)
- Key interpretations:
  - VAT has the lowest efficiency cost on macro aggregates but largest overall welfare cost because of large distributional cost driven by rural households (rural welfare loss much larger: 5.17% vs urban 0.68%).
  - CIT and PIT produce small distributional gains because incidence falls more on richer capital owners and urban formal wage earners.
  - VAT increases overall consumption Gini by 2.3%.
  - VAT’s distortions to relative prices generate sizable aggregate-component welfare losses despite smaller aggregate consumption decline.

### Transitional dynamics (short-run vs long-run)
- Transition setup: taxes unexpectedly and permanently increased to new steady-state levels; simulate forward until convergence.
- Short-run vs long-run welfare costs (examples):
  - VAT:
    - Urban Steady State Total −0.68%; Transition Total −0.95%
    - Rural Steady State Total −5.17%; Transition Total −5.22%
  - CIT:
    - Urban Steady State Total −2.80%; Transition Total −2.17%
    - Rural Steady State Total −2.02%; Transition Total −1.85%
  - PIT:
    - Urban Steady State Total −3.77%; Transition Total −3.33%
    - Rural Steady State Total −3.13%; Transition Total −3.16%
- Findings:
  - Differences between long-run and short-run results are small because capital stock and household savings are low → fast convergence.
  - CIT and PIT transitions take longer than VAT because capital stock decreases more under CIT and PIT.
  - Short-run total welfare costs smaller for CIT (1.94% vs 2.24%) and PIT (3.21% vs 3.31%), but larger for VAT (4.01% vs 3.89%).

### Role of idiosyncratic risks (quantitative comparison with no-risk case)
- Experiment: shut down idiosyncratic shocks by setting ε_u = ε_r = 1; include transitional dynamics.
- Main aggregate change:
  - Capital stock and savings increase by 47% (absence of precautionary saving in this model specification).
- Welfare costs (Whole economy, Table 5):
  - VAT:
    - Benchmark −4.01%; No Risk −0.59%
    - Aggregate: Benchmark −2.58%; No Risk −2.55%
    - Distributional: Benchmark −1.46%; No Risk 2.01%
  - CIT:
    - Benchmark −1.94%; No Risk −0.33%
    - Aggregate: Benchmark −1.91%; No Risk −2.21%
    - Distributional: Benchmark −0.02%; No Risk 1.92%
  - PIT:
    - Benchmark −3.21%; No Risk 1.91%
    - Aggregate: Benchmark −3.37%; No Risk −2.74%
    - Distributional: Benchmark 0.17%; No Risk 4.77%
- Interpretation:
  - Idiosyncratic risks generate large distributional costs of fiscal consolidations in the benchmark.
  - Without risks, distributional components often become welfare gains due to higher average wealth and different price/income effects.

### Lump-sum transfers and VAT (cases and quantitative mitigation)
- Experiments (VAT additional revenue = 2% GDP):
  - Case 1: Half of additional tax revenue used for lump-sum transfers to rural households.
  - Case 2: Half used for lump-sum transfers to the whole population.
  - Case 3: All additional tax revenue used for lump-sum transfers to the whole population (or equivalently households derive utility directly from additional government expenditure).
- Long-run and transition welfare outcomes for Case 1 (Table 6):
  - Transition Total: Urban −1.43%, Rural −1.29%, Whole −1.33% (Aggregate −1.51%, Distributional 0.19%)
  - Steady State Total: Urban −1.29%, Rural −1.30%, Whole −1.29% (Aggregate −1.72%, Distributional 0.32%)
- Key finding: Lump-sum transfers to rural households mitigate about 66.8% of the welfare costs from the VAT reform.
- Comparison across cases:
  - Case 1 (Half to rural) yields higher increase in output and consumption and larger improvement in consumption inequality versus Case 2 (Half to whole population).
  - Case 2 → larger decline in aggregate investment (urban households less risky → precautionary saving declines).
  - Case 3 amplifies effects roughly doubling Case 2’s impacts.

### Computational algorithm, transition computation, and welfare decomposition
- Steady-state solution algorithm (households and farmer):
  - Discretize state spaces with geometric grids B = {b1, ···, bn}; approximate shock distributions using Tauchen (1986).
  - Initial guess for price vector P0 = {pa, ps, wm, wf, r}.
  - Solve recursive household problems using FSQP and Monotone Piecewise Cubic Hermite Interpolation (PCHI); obtain policy functions c_j(b, ε), b'_j(b, ε), h_j(b, ε).
  - Solve large farmer using steady-state Euler (A.6) and budget constraint (A.7).
  - Construct invariant joint density functions φi,j via Monte Carlo simulation; aggregate and iterate prices until market clearing using Powell’s Hybrid Method.
  - Parallelization: MPI over shock states εj when solving Bellman equations.
- Transition path computation:
  - Taxes change unexpectedly at t = 2 from τo to τn; simulate T periods until converge to new steady state.
  - Solve perfect-foresight household problems (A.8) and (A.9) backward with terminal conditions V^u_T = V^u_n and V^r_T = V^r_n.
  - Farmer and firms solved backward using steady-state terminal conditions; simulate distributions forward; iterate on price paths using hybrd (Powell’s Hybrid Method) until goods and factor markets clear each period.
  - Practical notes: example scale 5 prices and 30 periods → 150 unknowns; Jacobian 150 × 150; diagonally-dominating requirement often violated; initial guesses crucial.
  - Convergence criterion for distributions (A.10):
    - max { ‖Γu T−1(bu, εu) − Γu n(bu, εu)‖, ‖Γr T−1(br, εr) − Γr n(br, εr)‖ } < εd.
- Welfare decomposition (consumption-equivalence, homothetic preferences):
  - Indirect utilities V0 and V1 lead to consumption-equivalence λ solving:
    - 1 + λ = exp { (W1 − W0) / (1 − β) / (1 + γ + ψ) }, where Wj = ∫ Vj(x0) dΓ0(x0).
  - Decompose λ into aggregate ˆλ and distributional ˜λ:
    - 1 + ˆλ = exp { (ˆW1 − W0) / (1 − β) / (1 + γ + ψ) } with ˆW1 based on aggregate-scaled consumptions.
    - (1 + ˆλ)(1 + ˜λ) = 1 + λ.
  - Decompositions computed separately for urban area, rural area, and whole economy using respective Γ0(x).
- Model without idiosyncratic risks:
  - Set εrt = εut = 1; homothetic utility implies aggregation and representative-agent equivalence at aggregate level.
  - Wealth distribution indeterminate; scale benchmark distribution to match aggregate saving when computing distributional effects.

### Policy implications (selected)
- The regressiveness of consumption taxes (VAT) in this setting is driven importantly by between-region redistribution as well as within-region effects — mismatch between tax incidence and government expenditure causes implicit redistribution.
- CIT and PIT, though more distortive in aggregate, may have lower welfare costs because their incidence is distributed more evenly across regions.
- VAT distorts households’ optimal consumption bundle; aggregate consumption decline generates large welfare losses.
- Low capital stock in low-income countries implies policy reforms realize long-run effects relatively quickly.
- Regional lump-sum transfers most effectively remedy negative impacts from VAT; pro-growth policies most effective for PIT and CIT.

*Source: IMF Working Paper (wp18146)*

### REFERENCES .............................................................................................................

### REFERENCES

### Main findings and policy implications
- Low-income countries: average tax to GDP ratio is 15% compared to 30% in advanced economies.
- Quantitative exercise calibrated to Ethiopia used to evaluate welfare costs of raising additional tax revenues equal to 2% GDP via VAT, CIT, and PIT (tax reforms are unexpected and permanent).
- Steady-state comparative results on output change:
  - VAT leads to a 1.5% loss in output.
  - CIT leads to a 4.9% decline in output.
  - PIT leads to a 6.2% decline in output.
- Consumption equivalence of total welfare costs:
  - VAT: 3.89%.
  - PIT: 3.31%.
  - CIT: 2.24%.
- Decomposition of VAT welfare cost:
  - Distributional component: 1.32% (the main driver of VAT’s high consumption-equivalence cost).
  - Aggregate component of VAT: 2.61% (higher than CIT’s aggregate component of 2.52%).
- Cross-region incidence effects:
  - VAT substantially worsens the urban-rural gap; welfare cost borne by rural households is 8 times that borne by urban households (in the benchmark model where government expenditure is on manufacturing goods produced in the urban area).
  - VAT implicitly transfers income from rural to urban households because VAT revenues partially rebate to urban households via government demand for manufacturing goods.
- Policy implications:
  - VAT is least efficient on aggregate output loss but has the highest consumption-equivalent welfare cost due to distributional effects; best accompanied by transfer programs to offset cross-region regressivity.
  - CIT and PIT generate larger aggregate consumption losses but are more uniformly distributed across regions; pro-growth policies pair better with CIT and PIT.
  - Reducing idiosyncratic labor-market risks or improving insurance can materially increase formal labor supply and capital accumulation, yielding larger welfare gains in low-income countries than suggested by advanced-economy studies.
- Transitional dynamics:
  - Transitional dynamics are less important in low-income countries due to low household savings (low capital stock) leading to fast convergence; steady-state comparisons are reasonably informative.
  - Short-run transition matters more for urban households who hold more savings relative to rural households.
  - Inclusion of transition dynamics increases VAT’s welfare cost but reduces welfare costs for CIT and PIT because CIT and PIT reduce steady-state capital stock, allowing higher consumption during transition.

### Role of idiosyncratic risks
- Eliminating idiosyncratic risks results in:
  - Output and capital stock doubling.
  - Labor supply to the formal sector almost tripling.
- Welfare-cost comparison with and without idiosyncratic risks:
  - Aggregate components of tax reform welfare costs are similar across risk and no-risk scenarios.
  - Distributional components differ substantially because in the no-risk case households are wealthier and savings constitute a higher fraction of disposable income; price changes in consumption goods affect indirect utility differently given utility concavity.
- Modeling treatment:
  - Idiosyncratic shocks make households hesitate to work in higher-productivity formal sectors; removing shocks increases formal sector participation and capital accumulation.

### Model structure and key elements
- Economy type:
  - Small open economy calibrated to Ethiopia as representative of low-income countries.
  - Two regions: rural and urban (population shares μr and μu).
  - Four sectors: food (domestic agriculture), exported cash crops, manufacturing, and services.
  - Unified capital market; interest rate r is endogenously determined domestically.
- Households:
  - Continuum of infinitely lived households in each region; each endowed with one unit of time divided between formal labor markets and informal self-employment.
  - Persistent idiosyncratic productivity shocks ε (εu for urban formal, εr for rural informal food production); shocks are Markovian.
  - Households choose consumption across food, manufacturing goods, services and have access to one risk-free bond for partial insurance.
  - Labor markets are segmented across regions and migration is precluded.
- Production and sectoral assignment:
  - Rural area: informal household production of food; a share μf of large farmers hire formal labor to produce food and cash crops.
  - Urban area: households provide services informally; a representative neoclassical firm hires formal labor and capital to produce manufacturing goods.
  - Cash crops production is modernized and employs capital; cash crops serve international markets; food and services are for domestic consumption.
  - Manufacturing goods are the numeraire; relative prices pa (food) and ps (services).
- Government:
  - Utilitarian objective; balanced budget each period.
  - Has access to three Ramsey linear taxes:
    - Value added tax (VAT, τa) on food and manufacturing goods consumption.
    - Personal income tax (PIT, τw) on formal wage income.
    - Corporate income tax (CIT, τr) on revenues of manufacturing firms and profits from cash crops production.
  - Government expenditure G is on manufacturing goods (not directly valued by households in benchmark), and government can implement lump-sum transfers in alternative scenarios.
  - Government budget constraint:
    - G + μf τr δkf,t = τa(pa Ca,t + Cm,t) + μf τr(πf,t + π∗,t) + τr ym,t + τw(μu wm Hu,t + μr wf Hr,t)
    - where μf τr δkf,t is tax deduction to agricultural machinery investment.
- Preferences:
  - Utilitarian expected discounted utility U = E[∑_{t=0}^∞ β^t u(ct)] with per-period utility:
    - u(ct) = log ca,t + γ log cm,t + ψ log cs,t
  - Homothetic (log-linear) preferences imply fixed expenditure shares on goods: 1/(1+γ+ψ) on food, γ/(1+γ+ψ) on manufacturing, ψ/(1+γ+ψ) on services.
  - Aggregate consumption expenditure Ct relates to per-period utility as u(Ct) = Ut + (1+γ+ψ) log Ct, where Ut is a time-varying constant depending on taxes and prices.
- Technologies:
  - Informal rural food production: ya,t = za εr,t (1 − hr,t)^{1−αa} with decreasing returns to scale.
  - Informal urban services: ys,t = zs (1 − hu,t)^{1−αs}.
  - Urban formal labor income per unit: wm εu,t.
  - Manufacturing firm production: ym,t = zm (km,t)^{αm} (hm,t)^{1−αm}.
  - Large farmers’ food production: ya,f,t = za (ha,t)^{1−αa}.
  - Cash crops (modernized): y∗t = z∗ (kf,t)^{α∗1} (h∗t)^{α∗2} with α∗1 + α∗2 < 1; no idiosyncratic risk in the rural formal market in the model.
  - Capital depreciates at rate δ; both manufacturing and cash crops use capital.

### Optimization and equilibrium objects
- Household recursive problems:
  - Urban households: V^u(b^u, ε^u) = max_{C^u, b'^u, h^u} { u(C^u) + β E[V^u(b'^u, ε'^u)|ε^u] } subject to
    - C^u + b'^u = (1 − τw) ε^u wm h^u + ps zs (1 − h^u)^{1−αs} + (1 + r) b^u.
  - Rural households: V^r(b^r, ε^r) = max_{C^r, b'^r, h^r} { u(C^r) + β E[V^r(b'^r, ε'^r)|ε^r] } subject to
    - C^r + b'^r = (1 − τw) wf h^r + pa za ε^r (1 − h^r)^{1−αa} + (1 + r) b^r.
  - Policy functions are c_j(b, ε), b'_j(b, ε), h_j(b, ε) for j ∈ {u, r}.
- Firms and farmers:
  - Large farmer maximizes discounted utility with budget constraint involving after-tax profits and tax credits for capital investment; farmer faces πf,t and π∗,t profit expressions.
  - Manufacturing firm maximizes (1 − τr) zm (km,t)^{αm} (hm,t)^{1−αm} − wm hm,t − (r + δ) km,t.
- Aggregates and government accounting:
  - Aggregate consumption of good x: Cx,t = μu ∫ c_{x,u,t} dΓu + μr ∫ c_{x,r,t} dΓr + μf c_{x,f,t}, for x ∈ {a,m,s}.
  - Total efficient formal labor units: Hu,t = ∫ εu,t hu,t dΓu, Hr,t = ∫ hr,t dΓr.
  - Total manufacturing revenue ym,t = zm (km,t)^{αm} (hm,t)^{1−αm}.
  - Government budget constraint (repeated for clarity):
    - G + μf τr δkf,t = τa(pa Ca,t + Cm,t) + μf τr(πf,t + π∗,t) + τr ym,t + τw(μu wm Hu,t + μr wf Hr,t).

*Source: IMF Working Paper (wp18146) — content units: Figures, Tables, Sections I–II, and model equations as provided in the source.*

### Appendix A.1 provides details of the solution to the large farmer’s problem.

### Appendix A.1 provides details of the solution to the large farmer’s problem.

### Recursive Competitive Equilibrium (stationary equilibrium)
- Definition 1. (Recursive Competitive Equilibrium) A recursive competitive equilibrium consists of equilibrium prices p = {p_a, p_s, w_m, w_f, r}, value functions V_j(b_j, ε_j), consumer decision rules {c_x,j(b_j, ε_j), b_j′(b_j, ε_j), h_j(b_j, ε_j)}, cumulative distribution functions Γ_j(b_j, ε_j), farmer’s decision rules {c_x,f, k_f, h_r, h^*}, where j ∈ {u, r} and x ∈ {a, m, x}, and firm’s decisions {k_m, h_m}, for any given policies {τ_a, τ_r, τ_w}, such that:
  - (i) Given p, V_j(b_j, ε_j) and {c_x,j(b_j, ε_j), b_j′(b_j, ε_j), h_j(b_j, ε_j)} solve the households’ optimization problems (4) and (5).
  - (ii) Given p, {c_x,f, k_f, h_r, h^*} solve the large farmer’s optimization problem (6).
  - (iii) Given p, {k_m, h_m} solve the firms’ optimization problem (7).
  - (iv) (Aggregate Consistency) Γ_j(b_j, ε_j)s are stationary distributions corresponding to the joint transition matrices Π_j constructed from b_j′(b_j, ε_j) and the transition matrices of ε_j, j = u, r:
    - Π_j = Pr[b_{t+1} = b′, ε_{t+1} = ε′ | b_t = b, ε_t = ε] = Pr[b = b^{-1}_{t+1}(b′, ε_t = ε)] Pr[ε_{t+1} = ε′ | ε_t = ε],
    - where b^{-1}_{t+1}(·) is the inverse function of saving b′(b, ε).
  - (v) Government budget (8) is balanced.
  - (vi) Prices p clear all markets:
    - Urban Labor Market:
      - μ_u ∫ ε_u h_u dΓ_u(b_u, ε_u) = h_m.
    - Rural Labor Market:
      - μ_r ∫ h_r dΓ_r(b_r, ε_r) = μ_f (h_a + h^*).
    - Capital Market:
      - μ_u ∫ b_u′ dΓ_u(b_u, ε_u) + μ_r ∫ b_r′ dΓ_r(b_r, ε_r) = k_m.
    - Food:
      - C_a = μ_r ∫ z_a ε_r (1−h_r)^{1−α_a} dΓ_r(b_r, ε_r) + μ_f z_a (h_a)^{1−α_a}.
    - Services:
      - C_s = μ_u ∫ z_s (1−h_u)^{1−α_s} dΓ_u(b_u, ε_u).
    - Manufacturing Goods:
      - C_m + δ(k_m + μ_f k_f) + G = z_m (k_m)^{α_m} (h_m)^{1−α_m} + μ_f R^*,
      - where R^* = z^* (k_f)^{α^*_1} (h^*)^{α^*_2} is the revenue from the export sector.
- Note: p_m = 1 (manufacturing goods are the numeraire). Five independent market-clearing conditions solved endogenously; R^* substitutes the balanced current account condition.

### Simplified static economy (illustrative example)
- Assumptions:
  - No capital.
  - Households homogeneous within each region.
  - μ_r = 2/3, μ_u = 1/3, μ_f = 0.
  - α_s = 1/2.
  - Agricultural and manufacturing production functions are linear.
- Key analytical results:
  - Result 1. The urban-rural income gap is increasing in τ_a.
    - Intuition: Tax burden imposed on all sectors while tax revenue purchases goods produced only in urban area → redistribution from rural to urban; requires different sectors hosted in different regions and difference between goods taxed and purchased by government.
  - Result 2. If government uses VAT revenue to purchase the same good, then VAT has zero efficiency cost.

### Exogenously calibrated parameters (Table 1 — values)
- Discount Rate β 0.96
- Depreciation Rate δ 0.06
- Labor Share in Food Production α_a 0.49
- Labor Share in Services Production 1−α_s 0.37
- Capital Share in Manufacturing Production α_m 0.37
- Land Share in Cash Crop Production α^*_1 0.19
- Labor share in Cash Crop Production α^*_2 0.32
- Persistence of Urban Income Shocks ρ_u 0.92
- Persistence of Rural Income Shocks ρ_r 0.92
- Urban Population Share μ_u 0.28
- Rural Population Share μ_r 0.69
- Large Farmer Share μ_f 0.03

### Endogenously calibrated parameters and model fit (Table 2 — parameters and data/model)
- Manufacturing Share in Consumption γ 0.82 → Data 0.33, Model 0.35
- Services Share in Consumption ψ 0.50 → Data 0.21, Model 0.22
- Rural Consumption Gini σ^2_r 0.23 → Data 0.26, Model 0.26
- Urban Consumption Gini σ^2_u 0.63 → Data 0.40, Model 0.40
- Tax to GDP Ratio τ_a 0.06 → Data 0.08, Model 0.08
- CIT in Total Tax Revenues τ_r 0.12 → Data 0.30, Model 0.30
- PIT in Total Tax Revenues τ_w 0.06 → Data 0.17, Model 0.19
- Food Share in Output z_a 0.39 → Data 0.42, Model 0.34
- Manufacturing Share in Output z_m 10.19 → Data 0.33, Model 0.38
- Export Share in Output z^* 14.17 → Data 0.08, Model 0.10
- Note: model slightly overshoots manufacturing and cash crops shares while undershooting food share.

### Quantitative experiments (long-run and transition)
- Experiment design:
  - Government raises additional tax revenue of 2% GDP in benchmark steady state via VAT, CIT, and PIT.
  - Additional tax revenue used as government expenditure not directly valued by households.
  - Compute consumption equivalence welfare changes; decompose welfare into aggregate and distributional components.
- Benchmark effective tax rates and new steady-state rates:
  - Benchmark effective rates: VAT 6.45%, CIT 11.55%, PIT 5.55%.
  - New steady-state rates: VAT 10.5%, CIT 24.0%, PIT 15.2%.
  - Percentage increases w.r.t benchmark: VAT 62%, CIT 108%, PIT 174%.
- Aggregate outcomes (percentage changes from benchmark steady state):
  - VAT: aggregate output −1.49%, aggregate consumption −4.66%, aggregate investment +1.16%.
  - CIT: aggregate output −4.85%, aggregate consumption −7.55%, aggregate capital −7.15%.
  - PIT: aggregate output −6.19%, aggregate consumption −9.50%, aggregate capital −5.51%.
- Overall welfare costs (consumption equivalence, Table 3 — regional and whole economy)
  - VAT:
    - Urban Total −0.68%
    - Urban Aggregate −0.29%
    - Urban Distributional −0.46%
    - Rural Total −5.17%
    - Rural Aggregate −5.26%
    - Rural Distributional −0.10%
    - Whole −3.89% (Aggregate −2.61%, Distributional −1.32%)
  - CIT:
    - Urban Total −2.80%
    - Urban Aggregate −2.76%
    - Urban Distributional −0.04%
    - Rural Total −2.02%
    - Rural Aggregate −2.25%
    - Rural Distributional 0.24%
    - Whole −2.24% (Aggregate −2.52%, Distributional 0.28%)
  - PIT:
    - Urban Total −3.77%
    - Urban Aggregate −4.65%
    - Urban Distributional 0.92%
    - Rural Total −3.13%
    - Rural Aggregate −3.14%
    - Rural Distributional 0.02%
    - Whole −3.31% (Aggregate −3.95%, Distributional 0.66%)
- Key interpretation:
  - VAT has the lowest efficiency cost on macro aggregates but largest overall welfare cost because of large distributional cost driven by rural households (rural welfare loss much larger: 5.17% vs urban 0.68%).
  - CIT and PIT produce distributional gains (small) because incidence falls more on richer capital owners and urban formal wage earners.
  - VAT increases overall consumption Gini by 2.3%.
  - VAT’s distortions to relative prices generate sizable aggregate-component welfare losses despite smaller aggregate consumption decline.

### Transitional dynamics (short-run vs long-run — Table 4)
- Transition setup: taxes unexpectedly and permanently increased to new steady-state levels; simulate forward until convergence.
- Short-run vs long-run welfare costs (examples from Table 4):
  - VAT:
    - Urban Steady State Total −0.68%; Transition Total −0.95%
    - Rural Steady State Total −5.17%; Transition Total −5.22%
  - CIT:
    - Urban Steady State Total −2.80%; Transition Total −2.17%
    - Rural Steady State Total −2.02%; Transition Total −1.85%
  - PIT:
    - Urban Steady State Total −3.77%; Transition Total −3.33%
    - Rural Steady State Total −3.13%; Transition Total −3.16%
- Findings:
  - Differences between long-run and short-run results are small because capital stock and household savings are low → fast convergence.
  - CIT and PIT transitions take longer than VAT because capital stock decreases more under CIT and PIT.
  - Short-run total welfare costs smaller for CIT (1.94% vs 2.24%) and PIT (3.21% vs 3.31%), but larger for VAT (4.01% vs 3.89%).

### Role of idiosyncratic risks (Table 5)
- Experiment: shut down idiosyncratic shocks by setting ε_u = ε_r = 1; include transitional dynamics.
- Main differences when risks are removed:
  - Capital stock and savings increase by 47% (absence of precautionary saving in this model specification).
  - Wealth distribution indeterminate in no-risk case; scale wealth distribution to match aggregate saving.
- Welfare costs (Whole economy, Table 5):
  - VAT:
    - Benchmark −4.01%; No Risk −0.59%
    - Aggregate: Benchmark −2.58%; No Risk −2.55%
    - Distributional: Benchmark −1.46%; No Risk 2.01%
  - CIT:
    - Benchmark −1.94%; No Risk −0.33%
    - Aggregate: Benchmark −1.91%; No Risk −2.21%
    - Distributional: Benchmark −0.02%; No Risk 1.92%
  - PIT:
    - Benchmark −3.21%; No Risk 1.91%
    - Aggregate: Benchmark −3.37%; No Risk −2.74%
    - Distributional: Benchmark 0.17%; No Risk 4.77%
- Interpretation:
  - Idiosyncratic risks generate large distributional costs of fiscal consolidations in the benchmark.
  - Without risks, distributional components often become welfare gains due to higher average wealth and different price/income effects.

### Lump-sum transfers and VAT (Cases and Table 6)
- Experiments (VAT additional revenue = 2% GDP):
  - Case 1: Half of additional tax revenue used for lump-sum transfers to rural households.
  - Case 2: Half used for lump-sum transfers to the whole population.
  - Case 3: All additional tax revenue used for lump-sum transfers to the whole population (or equivalently households derive utility directly from additional government expenditure).
- Long-run and transition welfare outcomes for Case 1 (Table 6):
  - Transition Total: Urban −1.43%, Rural −1.29%, Whole −1.33% (Aggregate −1.51%, Distributional 0.19%)
  - Steady State Total: Urban −1.29%, Rural −1.30%, Whole −1.29% (Aggregate −1.72%, Distributional 0.32%)
- Key finding: Lump-sum transfers to rural households mitigate about 66.8% of the welfare costs from the VAT reform.
- Comparison of Cases 1–3 (Figure 4 summary):
  - Case 1 (Half to rural) yields higher increase in output and consumption and larger improvement in consumption inequality versus Case 2 (Half to whole population).
  - Case 2 → larger decline in aggregate investment (urban households less risky → precautionary saving declines).
  - Case 3 amplifies effects roughly doubling Case 2’s impacts.

### A.1 The large farmer’s problem — sequential formulation and steady-state solution
- Sequential problem (transition path) for the large farmer:
  - max_{ {C^f_t, k^f_{t+1}, h^a_t, h^*_t} } Σ_{t=0}^∞ β^t u(C^f_t) (A.1)
  - s.t.
    - C^f_t + k^f_{t+1} = (1−τ^r_t)(π^f_t + π^*_t) + (1−δ) k^f_t + τ^r_t δ k^f_t,
    - π^f_t = p^a_t z_a (h^a_t)^{1−α_a} − w^f_t h^a_t,
    - π^*_t = z^*(k^f_t)^{α^*_1} (h^*_t)^{α^*_2} − w^f_t h^*_t.
- Intratemporal first-order conditions:
  - Labor in domestic food production h^a_t:
    - (1−α_a) p^a_t z_a (h^a_t)^{−α_a} = w^f_t. (A.2)
  - Labor in cash crops h^*_t:
    - α^*_2 z^* (h^*_t)^{α^*_2 − 1} (k^f_t)^{α^*_1} = w^f_t. (A.3)
- Euler equation (after substitutions):
  - 1/(p^s_t c^s_t) = 1/(p^s_{t+1} c^s_{t+1}) β [ (1−δ) + δ τ^r_{t+1} + (1−τ^r_{t+1}) ∂π^*_{t+1} / ∂k^f_{t+1} ]. (A.4)
  - Marginal product of capital:
    - ∂π^*_{t+1} / ∂k^f_{t+1} = α^*_1 z^* (h^*_{t+1})^{α^*_2} (k^f_{t+1})^{α^*_1 − 1}.
- Budget constraint:
  - (1 + γ + ψ)/ψ p^s_t c^s_t + k^f_{t+1} = (1−τ^r_t)(π^a_t + π^*_t) + (τ^r_t δ + 1 − δ) k^f_t. (A.5)
- Steady-state simplifications:
  - Euler equation becomes:
    - 1/β = (1−τ^r) ∂π^*/∂k^f + 1 − δ + τ^r δ. (A.6)
  - Budget constraint becomes:
    - (1 + γ + ψ)/ψ p^s c^s = (1−τ^r)(π^a + π^*) + (τ^r δ − δ) k^f. (A.7)
  - Substitute (A.3) into (A.6) to solve for k^f; use (A.7) to find c^s. This completes characterization of the large farmer’s problem in steady state.

### Policy implications (selected)
- The regressiveness of consumption taxes (VAT) in this setting is driven importantly by between-region redistribution as well as within-region effects — mismatch between tax incidence and government expenditure causes implicit redistribution.
- CIT and PIT, though more distortive in aggregate, may have lower welfare costs because their incidence is distributed more evenly across regions.
- VAT distorts households’ optimal consumption bundle; aggregate consumption decline generates large welfare losses.
- Low capital stock in low-income countries implies policy reforms realize long-run effects relatively quickly.
- Regional lump-sum transfers most effectively remedy negative impacts from VAT; pro-growth policies most effective for PIT and CIT.

*Source: Appendix A.1 and related sections of wp18146 — "Appendix A.1 provides details of the solution to the large farmer’s problem."*

### 1.  Discretize the state spaces. We useifor the index of the saving grids, andjfor the index of

### 1.  Discretize the state spaces. We useifor the index of the saving grids, andjfor the index of

### Discretization and solution of household problems (steady state algorithm)
- Step 1: Discretize state spaces
  - Construct n geometric grids B = {b1, ···, bn} ⊆ B.
  - Approximate shock distributions of productivity in rural informal and urban formal sectors using Tauchen (1986)’s method. Denote the m-states Markov chain by εx = {ε1,x, ···, εm,x} ⊆ Ex, x ∈ {r, u}.
  - Rationale: Geometric grids put more grids when b is small because policy functions are likely to be nonlinear in that region.
- Step 2: Initial guess
  - Make an initial guess for the price vector P0 = {pa, ps, wm, wf, r}.
- Step 3: Solve recursive household problems given P0 (using urban as example)
  - Use Feasible Sequential Quadratic Programming (FSQP) [Zhou, Tits and Lawrence (1997)] to solve the Bellman equation by approximating f(x) with a quadratic form and solving constrained quadratic problems via first order conditions.
  - Approximate expected continuation value E[V(b′, ε′ | ε)] by Monotone Piecewise Cubic Hermite Interpolation (PCHI) [Fritsch and Carlson (1980), Fritsch and Butland (1984)].
  - Sub-steps for each node (bi, εj):
    - (i) Calculate optimal labor supply decision (intra-temporal; independent of intertemporal consumption-saving decision).
    - (ii) Initial guess for value function V(bi, εj): household uses all resources for consumption.
    - (iii) Construct PCHI from the Lagrangian table (polynomial values and derivatives at each node); compute derivative of constraint w.r.t. choice variables c and b′.
    - (iv) Use FSQP to solve optimization at (bi, εj).
    - (v) Loop over all nodes in B and εu to obtain V(b, ε), c(b, ε) and b′(b, ε).
    - (vi) Compare initial guess and solved value and policy functions; iterate until convergence.
- Step 4: Large farmer’s problem
  - Calculate solution using Equations (A.6) and (A.7).
- Step 5: Construct joint density functions of stationary asset-shock distributions φi,j for urban and rural households
  - Cumulative distributions are Γu(bi, εj) and Γr(bi, εj).
  - Use Monte Carlo simulation on density functions to compute invariant joint density functions.
  - Sub-steps:
    - (i) For dimension i of φi,j, construct denser nodes on B: linear grids B1 = {b1, ···, bn1}, where n1 > n.
    - (ii) Initial distribution: marginal density on i set to invariant distribution of idiosyncratic shocks; marginal density on j set to uniform.
    - (iii) For agents at (bi, εj), use saving policy b′(bi, εj) and shock transition matrix to determine share at (b′(bi, εj), ε′j). If b′ not on B1, locate containing interval in B1 and distribute population share to endpoints based on relative distances. ε′j always on εx.
    - (iv) Repeat (iii) until successive density iterations converge to invariant distribution.
    - (v) From φi,j construct stationary distributions over consumption, income, and savings via the same Monte Carlo method.
- Step 6: Aggregate and market clearing
  - Aggregate individual decisions using invariant distributions φi,j to compute aggregate demand and supply; if not equal, return to Step 2 and iterate until market clearing P is found.
  - Use Powell’s Hybrid Method to solve for market-clearing P.
- Parallelization: Model is parallelized using MPI over shock states εj when solving Bellman equations (Step 3).

### Perfect-foresight equilibria along the transition path (definitions and household problems)
- Transition framework and timing
  - Tax rates: status quo τo = {τao, τwo, τro}; after reform τn = {τan, τwn, τrn}.
  - Transition assumed to take T periods. Economy at old steady state at t = 1 and at new steady state from t = T.
  - Agents expect τo until t = 1 included; at t = 2 taxes change to τn as a surprise and remain forever; from t = 2 agents correctly anticipate τn and actual market-clearing price paths {pt}∞t=1 are realized.
- Time-varying household dynamic programming problems (perfect foresight)
  - Urban households (A.8):
    - Vut(but, εut) = max {Cut, but+1, hut} { u(Cut) + β Et[Vut+1(but+1, εut+1 | εut)] }
    - Constraint: Cut + but+1 = (1 − τwt) εut wmt hut + pst zs (1 − hut)1−αs + (1 + rt) but
  - Rural households (A.9):
    - Vrt(brt, εrt) = max {Crt, brt+1, hrt} { u(Crt) + β Et[Vrt+1(brt+1, εrt+1 | εrt)] }
    - Constraint: Crt + brt+1 = (1 − τwt) wft hrt + pat za εrt (1 − hrt)1−αa + (1 + rt) brt
  - V u T(·) and V r T(·) are value functions in the new steady state.
- Large farmer and firms solved backward using steady-state terminal conditions (e.g., kfT is new steady-state level).

### Formal definition of Perfect Foresight Competitive Equilibria during Transition (Definition 2)
- Equilibrium objects for t = 1..T:
  - Prices {pt}Tt=1 = {pat, pst, wmt, wft, rt}Tt=1.
  - Value functions {V j t (bj, εj)}Tt=1 and consumer decision rules {c x, j t (bj, εj), bj t+1 (bj, εj), hj t (bj, εj)}Tt=1 for j ∈ {u, r}, x ∈ {a, m, x}.
  - Cumulative distribution functions {Γ j t (bj, εj)}Tt=1.
  - Farmer decision rules {c x, f t , k f t+1 , h r t , h ∗ t }Tt=1.
  - Firm decisions {km t , hm t }Tt=1.
- Equilibrium conditions:
  - (i) Economy at old steady state at t = 1 and at new steady state from t = T.
  - (ii) Given pt, value functions and decision rules solve household problems (A.8) and (A.9).
  - (iii) Given pt, farmer rules solve large farmer’s problem (A.1).
  - (iv) Given pt, firm rules solve firms’ problem (7) at each t.
  - (v) Aggregate consistency: joint transition matrices Π j t constructed from bj t+1 (bj, εj) and shock transition matrices:
    - Π j t = Pr[b t+1 = b′, ε t+1 = ε′ | b t = b, ε t = ε] = Pr[b = b−1 t+1(b′, ε t = ε)] Pr[ε t+1 = ε′ | ε t = ε], where b−1 t+1(·) is inverse of saving function.
  - (vi) Government budget balance for each t:
    - Gt + μf τ r t δkf t = τa t (pa t Ca t + Cm t) + μf τ r t (π f t + π ∗ t) + τr t ym t + τw t (μu w m t hu t + μr w f t hr t).
  - (vii) Prices clear markets each period:
    - Urban labor: μu ∫ εu t hu t dΓu t (bu, εu) = hm t.
    - Rural labor: μr ∫ hr t dΓr t (br, εr) = μf (ha t + h ∗ t).
    - Capital: μu ∫ bu t+1 dΓu t + μr ∫ br t+1 dΓr t = km t+1.
    - Food: Ca t = μr ∫ za εr t (1 − hr t)1−αa dΓr t + μf za (ha t)1−αa.
    - Services: Cs t = μu ∫ zs (1 − hu t)αs dΓu t.
    - Manufacturing: Cm t + (km t+1 + μf kf t+1) + (1 − δ)(km t + μf kf t) + Gt = zm (km t)αm (hm t)1−αm + μf R ∗ t, where R ∗ t = z ∗ (kf t)α∗1 (h ∗ t)α∗2.

### Computing the transition path (numerical algorithm and practical considerations)
- Problem dimension and solver choice
  - Solve N paths of length T; price paths live in P ⊆ ∏N i=1 R T i.
  - Construct operator T : P → P to update price paths and operator E : P → ∏N i=1 R T i to calculate excess demands.
  - Fixed-point tatonnement not used because no straightforward update for pa t and ps t; use standard nonlinear solver.
  - Example scale: 5 prices and 30 periods → nonlinear system with 30 × 5 = 150 unknowns.
  - Use hybrd from MINPACK (Powell’s Hybrid Method) to solve the nonlinear system.
  - Numerical caveats: Jacobian of size 150 × 150; diagonally-dominating requirement often violated because markets are tightly linked; algorithm sensitive to initial guesses.
- Numerical approximation to transition path
  - Convergence to new steady state is asymptotic; compute approximation by choosing T long enough so distributions at T − 1 are close to new steady-state distributions Γ j n(·) within tolerance εd under a chosen norm:
    - (A.10) max { ‖Γu T−1(bu, εu) − Γu n(bu, εu)‖, ‖Γr T−1(br, εr) − Γr n(br, εr)‖ } < εd.
- Pseudo-code summary to compute transition path
  - 1. Choose transition length T.
  - 2. Make initial guesses of price paths {Pt}Tt=1 = {ps t, pa t, wm t, wf t, r t}, with P1 = Po and P T = Pn.
  - 3. Solve steady state equilibrium at t = 1 and t = T using Appendix A.2 algorithm; obtain Γ j 1 (·) = Γ j o (·) and Γ j T (·) = Γ j n (·), j = u, r.
  - 4. Given Pt and terminal V u T = V u n and V r T = V r n, solve household problems (A.8) and (A.9) backward to get {V u t , V r t }T−1 t=2 using FSQP and PCHI.
  - 5. Given Pt and k f T = k f n, solve large farmer’s problem backward using (A.4) and (A.5) to get {cf t , kf t }T−1 t=2, using hybrd for (A.4) and (A.5).
  - 6. Starting from Γ j 1 (·), simulate forward using policy functions until t = T − 1; compute aggregate demand and supply for the five markets in Definition 2 for each t. If demands ≠ supplies, return to Step 2 and update {Pt}T−1 t=2 using Powell’s Hybrid Method.
  - 7. For price paths that clear markets, compare Γu T−1 and Γr T−1 with Γu n and Γr n respectively; if (A.10) not satisfied, increase T and repeat until tolerance εd achieved.
- Additional numerical notes
  - Jacobian size grows as 150 × 150 for T = 30 (and expands exponentially with T); diagonally-dominating issues persist and initial guesses crucial.
  - When prices stabilize, algorithm effectively reaches Step 5 of steady-state solver.

### Welfare decomposition: consumption equivalence and aggregate vs distributional components
- Consumption equivalence for homothetic preferences (B.1)
  - Indirect utilities V0(x0) and V1(x0) defined as infinite discounted sums of expected log consumptions of three goods (log ca + γ log cm + ψ log cs) under π0 and π1 transition probabilities.
  - Consumption equivalence λ for household state x0 solves V1(x0) = ∑s=0∞ βs ∑xs|x0 π0(xs | x0) [log(1 + λ) ca,0 s + γ log(1 + λ) cm,0 s + ψ log(1 + λ) cs,0 s].
  - Using log properties and geometric summation:
    - (B.11) V1(x0) = (1 + γ + ψ) log(1 + λ) / (1 − β) + V0(x0).
  - λ is independent of x0. Integrating over Γ0(x):
    - (B.12) 1 + λ = exp { (W1 − W0) / (1 − β) / (1 + γ + ψ) }, where Wj = ∫ Vj(x0) dΓ0(x0), j = 0, 1.
  - For non-homothetic preferences, Monte Carlo simulation is required to find λ numerically.
- Decomposing λ into aggregate and distributional components (B.2)
  - Construct indirect utility for aggregate component by scaling individual consumptions by corresponding aggregate consumption levels C a,j t , C m,j t , C s,j t , j = 0, 1.
  - Define ˆV1(x0) (aggregate-only indirect utility) leading to aggregate component ˆλ:
    - 1 + ˆλ = exp { (ˆW1 − W0) / (1 − β) / (1 + γ + ψ) }, where ˆW1 = ∫ ˆV1(x0) dΓ0(x0).
  - Distributional component ˜λ defined residually:
    - (B.13) (1 + ˆλ)(1 + ˜λ) = 1 + λ.
  - For steady-state comparisons:
    - 1 + λss = exp { (W1 − W0) / (1 − β) / (1 + γ + ψ) }.
    - 1 + ˆλss = exp { 1 / (1 + γ + ψ) ( log Ca1 / Ca0 + γ log Cm1 / Cm0 + ψ log Cs1 / Cs0 ) }.
    - (1 + ˜λss)(1 + ˆλss) = 1 + λss.
  - Decompositions are computed for urban area, rural area, and whole economy separately using different Γ0(x).
- Note: Alternative scaling by overall aggregate consumption yields similar results due to log-linear specification; VAT case sensitive to relative price changes.

### Model without idiosyncratic risks (aggregation and computation)
- Specification: Set idiosyncratic shocks {εrt, εut} to one.
- Homothetic utility (1) implies aggregation: model is indistinguishable at aggregate level from representative agent models.
- Implication: Cannot separately solve rural and urban consumption and savings; can only pin down aggregate levels:
  - CH t = μu Cu t + μr Cr t
  - bH t+1 = μu bu t+1 + μr br t+1
- Computational approach:
  - First solve representative agent version to characterize equilibrium price paths.
  - Then solve dynamic programming problem given equilibrium prices and perform welfare decompositions.

*Italic: wp18146 - appendix content (computational algorithm, transition path, and welfare decomposition).*

### 1.  Choose the transition lengthT.

### 1.  Choose the transition lengthT.

### Computational algorithm for transitional equilibrium
- Initialization
  - Choose transition length T.
  - Set initial guesses for {p_s_t, p_a_t}_{t=1}^T, with t = 1 and t = T corresponding to the two steady states respectively.
  - Set initial guesses for {r_t}_{t=1}^T conditional on the guesses of {p_s_t, p_a_t}_{t=1}^T.
- Firm and labor market back-out
  - Use the first order conditions of the manufacturing firm to back out urban wages and capital-labor ratio {w_t, κ_m_t}_{t=1}^T.
  - Use the labor supply decision of the urban household and the urban labor market clearing condition to calculate capital demand {k_m_t}_{t=1}^T.
- Household Euler equations and consumption
  - Given interest rates, use the Euler equations of the households to solve average household consumption {C_H_t}_{t=2}^{T−1} by backward induction from C_H_T (households are not distinguished rural/urban at this stage).
- Forward simulation and interest rate updating
  - With average consumption, find average income {I_H_t}_{t=2}^{T−1} using the fact that the economy is in steady state at t = 1 (so b_H_2 is given).
  - Simulate forward the sequence of capital supply {b_H_{t+1}}_{t=2}^{T−1} using household budget constraints.
  - Compare {b_H_{t+1}}_{t=1}^{T−2} with {k_m_t}_{t=2}^{T−1} to pin down interest rates during the transition {r_t}_{t=2}^{T−1}.
  - Iteration detail:
    - (a) Urban income: I_u_t = p_s_t z_s (1 − h_u_t)^{1−α_s} + (1 − τ_{w_t}) w_m_t h_u_t.
    - (b) Rural allocations solved by backward induction: for each t, use Euler equation of the large farmer to solve c_f_t; with c_f_t, the large farmer budget constraint defines an equation with 4 unknowns {k_f_t, h_a_t, h^*_t, w_f_t}; the two labor demand functions plus rural labor market clearing and rural household labor supply close the system. Solve {k_f_t, h_a_t, h^*_t, w_f_t} jointly and extend from t = T−1 to t = 2.
    - (c) Rural income: I_r_t = p_a_t z_a (1 − h_r_t)^{1−α_a} + (1 − τ_{w_t}) w_f_t h_r_t; average household income I_H_t = μ_u I_u_t + μ_r I_r_t.
    - (d) Update guesses for {r_t}_{t=2}^{T−1} until {b_H_{t+1}} equals {k_m_t}.
- Price iteration and goods market clearing
  - Use average consumption {c_H_t}_{t=2}^{T−1} to solve average consumption for food and services.
  - Update guesses for {p_s_t, p_a_t}_{t=2}^{T−1} until the food and services goods markets are cleared.
- Convergence and transition length adjustment
  - With all equilibrium price paths, compare simulated b_H_T with that in the new steady state.
  - If the difference is large, return to Step 1 and increase transition length T.

### Value functions and welfare computation
- Bellman equations (steady state illustration; extension to transition is straightforward)
  - Rural:
    - V_r(b) = max_{C_r, b_r′, h_r} { u(C_r) + β V_r(b_r′) } s.t. C_r + b_r′ = (1 − τ_w) w_f h_r + p_a z_a (1 − h_r)^{1−α_a} + (1 + r) b_r.
  - Urban:
    - V_u(b) = max_{C_u, b_u′, h_u} { u(C_u) + β V_u(b_u′) } s.t. C_u + b_u′ = (1 − τ_w) w_m h_u + p_s z_s (1 − h_u)^{1−α_s} + (1 + r) b_u.
- Welfare decomposition
  - Use value functions to calculate the welfare decomposition following procedures in Appendix B.2 with a caveat about aggregation and wealth distribution (see next subsection).

### Wealth distribution adjustment in the no-idiosyncratic-risk case
- Aggregation with idiosyncratic risks shut down makes the wealth distribution indeterminate.
- Assumption: rural and urban wealth distributions have the “same shape” as in the benchmark case; levels adjusted because aggregate capital differs across cases.
- Example: in the status quo steady state, capital in the no-risk case is approximately 40% higher than the benchmark case.
  - To be consistent, shift the benchmark distribution (for instance by a ratio of 1.4) so aggregated capital equals that in the representative agents model.

### Analytical results (Section D — Proofs)
- Result 1: The urban-rural income gap is increasing in τ_a.
  - Total income of rural households is p_a; urban income I_u is given by (D.14):
    - I_u = w_m (1 − h_s) + p_s y_s = z_m [1 − (p_s)^2 / 4(z_m)^2] + (p_s)^2 / 2 z_m = z_m + (p_s)^2 / 4 z_m.
  - Household demand functions (as derived): rural c_a_r = 1 / [3(1 + τ_a)], c_m_r = p_a / [3(1 + τ_m)], c_s_r = p_a / [3 p_s]; urban c_a_u = I_u / [3(1 + τ_a) p_a], c_m_u = I_u / [3(1 + τ_m)], c_s_u = I_u / [3 p_s].
  - Food market clearing yields (after algebra) (D.15):
    - I_u = 2[3(1 + τ_a) − 1] p_a,
    - from which the urban-rural income gap I_u / p_a is increasing in τ_a.
- Result 2: If government uses VAT revenue to purchase the same good, VAT has zero efficiency cost (illustrated with τ_m).
  - Rural output is always 2/3 unit of agricultural good and is not distorted by taxes.
  - For urban households, labor supply equates formal and informal marginal returns:
    - (1 − τ_w) w_m = ∂ p_s y_s / ∂ h_s = 1/2 p_s (h_s)^{−1/2}, yielding (D.16):
      - h_s = (p_s)^2 / [4 ((1 − τ_w) z_m)^2].
  - It suffices to show p_s does not change with τ_m.
  - Combining (D.14) and (D.15) gives (D.17):
    - p_a / p_s = 1 / M [ z_m p_s + p_s^4 / z_m ], with M = 2[3(1 + τ_a) − 1].
  - Services market clearing (D.18) and substitution yield (D.19):
    - p_s = [ 4(M + 2) / (5M − 2) ]^{1/2} z_m,
    - which does not depend on τ_m, implying urban production unchanged by τ_m when VAT revenue is used to buy the same good.

### Quantitative and welfare results (selected)
- Wealth aggregation example:
  - Capital in the no-risk case ≈ 40% higher than benchmark; adjust benchmark distribution by ratio 1.4.
- Table E.1 — THE WELFARE COSTS WHEN RURAL HOUSEHOLDS ARE VALUED MORE (excerpted entries; government doubles weights on each rural household)
  - VAT: Total −4.01% (Equal) and −4.51% (Rural)
  - CIT: Total −1.94% (Equal) and −1.90% (Rural)
  - PIT: Total −3.21% (Equal) and −3.19% (Rural)
  - Aggregate and Distributional rows reported similarly:
    - Aggregate VAT: −2.58% (Equal) and −3.34% (Rural)
    - Aggregate CIT: −1.91% (Equal) and −1.85% (Rural)
    - Aggregate PIT: −3.37% (Equal) and −3.20% (Rural)
    - Distributional VAT: −1.46% (Equal) and −1.21% (Rural)
    - Distributional CIT: −0.02% (Equal) and −0.05% (Rural)
    - Distributional PIT: 0.17% (Equal) and 0.01% (Rural)
- Government objective
  - Benchmark experiments assume a utilitarian government.
  - Doubling weights assigned to each rural household yields small changes because rural population share is large: 69%.

### Toolkit for quantitative exercises (Section F)
- A graphical-interface toolkit accompanies the paper to bridge findings and policy prescriptions.
- Toolkit features:
  - Calibrate the model to different target economies.
  - Conduct analyses similar to the paper without deep programming knowledge.
  - Export many intermediate results in Microsoft Excel .xlsx format for additional analysis.
- Note: the latest version referenced in the source is version 2.0 and is distributed via GitHub (link provided in the source document).

*Source: wp18146 - 1.  Choose the transition lengthT.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18146.pdf_