## _wp1678 — Selected Excerpts (Introduction; 2.3 National Budget Constraint; 6.1 Distortionary Taxes Adjust; 6.2 Distributional Concerns)

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### Introduction — context, mechanisms, and research objective
- P3s (public-private partnerships) in less-developed countries (LDCs) may be more expensive than traditional procurement because of:
  - Higher cost of private finance due to non-diversifiable risk (citations: Vickrey, 1964; Arrow, 1966; Arrow and Lind, 1970).
  - Compensation for country risk where government is perceived as an “unreliable business partner” (Estache et al., 2015).
  - Administrative and transaction costs: complex long-term contracts, bid costs that limit ex ante competition, auction design difficulties, limited information on firm costs, and frequent costly renegotiations.
  - Monitoring costs required to verify private partner compliance.
- Empirical magnitudes cited (preserve numeric precision):
  - Non-diversifiable risk pushes interest rate on senior debt in P3s "2−3 percentage points" above government debt.
  - Transaction costs as percentage of capital costs: average "10%" for European governments (Dudkin and Valila, 2005); "2−10%" for private firms in Great Britain (House of Commons, 2002-03).
  - Monitoring costs in U.S. P3s magnify total costs "3−25%" (Torres and Pina, 2001).
  - Example renegotiation frequency: 35 Portuguese P3s produced "254" contract renegotiations between "1995 and 2012".
  - Typical P3 contracts last "20−30 years".
- Countervailing benefits emphasized:
  - Private partners may bring superior technical expertise, implementation capacity, and fewer agency problems → shorter construction periods and better, more productive infrastructure.
- Key question: Do speed and efficiency gains of P3s compensate for higher costs? Does P3 yield better "value for money" than own investment (OI)?
- Paper objective: compare P3 vs OI in a dynamic general equilibrium macro model featuring private capital accumulation and a labor market specification that nests full employment and involuntary unemployment (efficiency wages). Unifying finding: general equilibrium effects can shift the comparison in favor of P3 under several conditions.
- Quantitative statement from introduction: asymmetric macro externalities raise the social return in the P3 "2−9 percentage points" relative to OI depending on externalities and speed advantages.
- Illustrative conclusion: a P3 that pays a direct return of "2%" may be preferable to OI that pays a direct return of "10%" when general equilibrium effects are large.

### Model setup overview (as presented)
- Three sectors/components:
  - Government: builds infrastructure either by own investment (OI) or by partnering with a foreign firm (P3).
  - Domestic private sector: representative agent, efficiency wages, capital accumulation.
  - National budget constraint: capital flows finance current account deficit each period.
- Production technology (Cobb-Douglas with infrastructure ζ, capital κ, labor Λ):
  - For P3: θ = [α(ζ − ζ0) + ζ0] ψ κ^a (eΛ)^(1−a) with α > 1 for P3.
  - For OI: θ = ζ ψ κ^a (eΛ)^(1−a).
  - Infrastructure enters as a shift factor; effective labor depends on effort e.
- P3 contract: annuity where foreign partner receives (ρp + δ)(ζ − ζ0) in perpetuity.
- Government revenue expressions (lump-sum taxes T adjust in main variant):
  - P3: T = ρp(ζ − ζ0) + δζ − R[α(ζ − ζ0) + ζ0] − ηc.
  - OI: T = ρ(ζ − ζ0) + (δ − R)ζ − ηc, borrowing at world interest rate ρ.
- Speed of construction dynamics:
  - ż = iζ − δζ; in P3 iζ overshoots its steady-state and ζ increases faster than under OI.
  - P3 investment dynamics: diζ/dτ = σ(ȷiζ − iζ) with σ > 0 and iζ,o+ > ȷiζ.
  - For OI, iζ = ȷiζ = δȷζ.

### Private sector optimization (outline)
- Representative-agent utility:
  - U = ∫∞0 [ c^(1−1/τ) / (1−1/τ) − [e^(−β0 − β1 ln ω − β2 u) / 2] ] e^(−ρτ) dτ
  - Capital accumulation: ḱ = θ − T − (1 + η)c − R[α(ζ − ζ0) + ζ0] − δκ.
- Labor supply fixed at unity; unemployment u = 1 − Λ.
- Effort specification: e = β0 + β1 ln ω + β2 u; in equilibrium β1 = 1 and e = β1 = 1 (effort constant in general equilibrium).
- Euler equation on optimal path: ċ = τ c (θ_κ − ρ − δ).

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### 2.3 The National Budget Constraint — closure, efficiency wages, and long-run comparative statics

- National budget constraint (closure): substituting T into (6) yields (preserving equation numbers and forms):
  - For P3: ̇κ = θ − ρπ(ζ − ζo) − χ − μ(ζ + κ)  (equation (13))
  - For OI: ̇κ = θ − ρ(ζ − ζo) − χ − μ(ζ + κ)  (equation (13′))

- Efficiency wages: motivation and stylized facts
  - Efficiency wage framework is empirically relevant in non-agricultural sectors in LDCs; it can jointly explain:
    - firm-size wage premiums starting at very small establishment size (5+ employees);
    - persistent inter-industry wage differentials;
    - high correlation of industry wage premiums across occupations;
    - large wage premia for formal vs. informal employment and informal non-agricultural vs. agricultural employment;
    - large cyclical flows into and out of unemployment in both formal and informal sectors.
  - Empirical parameter ranges:
    - β2 (wage curve when dependent variable is log unemployment) clusters between 0.05 and 0.15 → evaluated at unemployment rate of 10% corresponds to β2 = 0.5−1.5 in equation (12).
    - South Africa example: national unemployment rate 25%; Kingdon and Knight (1999) estimate β2 = 0 for homeland regions and −.66 elsewhere (cluster means evaluated at an unemployment rate of 25%).
    - Illustrative extreme: enforce full employment by increasing β2 to 100 for irrigation in smallholder areas.
  - Author note: choice of labor-market specification depends on country/project location.

- Long-run P3 steady state and linear comparative statics (exact algebra retained)
  - Steady-state relations (preserve equation numbers and symbols):
    - θ/κ = ρ + μ  (equation (14))
    - χ = θ − ρπ(ζ − ζo) − μ(ζ + κ)  (equation (15))
  - Linear change relations:
    - δχ = ρ δκ + (Rπ − ρπ) δζ + ω δL  (equation (16))
    - δω = β2 ω δL  (equation (17))
    - δL = [(Rπ + μ)(1 − α) / (ω f1)] δζ + [(ρ + μ)(1 − α) / (ω f1)] δκ  (equation (18))
    - δκ = [(Rπ + μ)/(ρ + μ)] [α/(1 − α)] δζ + (κ/L) δL  (equation (19))
    - f1 ≡ α + β2 Λ = α + β2(1 − u); Rπ = θ/ζ − μ (return on infrastructure net of depreciation).
  - Long-run multipliers (effects of δζ):
    - δκ = [(Rπ + μ)/(ρ + μ)] [α/(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] δζ  (equation (20))
    - δL = [(Rπ + μ)/(ω f1)(1 − α)] [1 + (α/(1 − α)) 1+β2(1−u) / (β2(1−u))] δζ  (equation (21))
  - P3 vs OI differences:
    - δκ|P3 − δκ|OI = (Rπ − R)/(ρ + μ) [α/(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] δζ  (equation (22))
    - δL|P3 − δL|OI = (Rπ − R)/(ω f1)(1 − α) [1 + (α/(1 − α)) 1+β2(1−u) / (β2(1−u))] δζ  (equation (23))
  - Qualitative implications:
    - Direct positive effects on κ and L depend on Rπ relative to R; consensus view Rπ > R strengthens P3's case.
    - Feedback between κ and L magnifies direct effects. For empirical β2 in 0.5−1.5 range feedback multiplier ≈ 1.5−3 for κ and 2−4 for L.

- Consumption response and return-gap decomposition
  - For P3:
    - (δχ/δζ)|P3 = Rπ − ρπ + (Rπ + μ) Φ  (equation (24))
    - Φ ≡ [ρ + (ρ + μ)(1 − α) / (α + β2(1 − u))] [α/(ρ + μ)(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] + (1 − α) / [α + β2(1 − u)] > 0
  - For OI replace Rπ and ρπ with R and ρ (equation (25)).
  - Total Return Gap and Direct Return Gap:
    - (δχ/δζ)|P3 − (δχ/δζ)|OI = [Rπ − ρπ − (R − ρ)] + (Rπ − R) Φ  (equation (26))
    - Direct Return Gap (DRG) ≡ Rπ − ρπ − (R − ρ)

- Parameterization notes (as given in source, preserve exact tokens):
  - α = :0.30 − :0.50, μ = :0.05, ρ = :0.06, ρπ = :0.15, ρ = :0.06, ρ∗? (context), ρπ = :0.15 noted as pricey.
  - ρ (external borrowing rate) = :0.06, ρπ (P3 cost) = :0.15, R = :0.16, Rπ = :0.16 − :0.25, β2 = :0.5 − 1.5, ∞ (cases considered).
  - Private time preference rate ρ chosen at :0.10.
- Numerical insight highlighted:
  - With Rπ topping out at :0.25 and ρπ = :0.15, DRG ranges from zero to −9% (negative values mean OI has better direct return).
  - Feedback multipliers amplify effects substantially: point D (P3) is far north/east of points B and C (OI) in the paper's diagrams.

- Transition dynamics and eigenstructure (summary)
  - Linearized dynamics (deviations) and three-dimensional system in [χ − χ̄, κ − κ̄, ζ − ζ̄].
  - ζ dynamics when infrastructure investment constant at ζ̄: ̇ζ = −μ(ζ − ζ̄) (equation (32)).
  - Stationary equilibrium is a saddle point with two negative eigenvalues:
    - λ1 = −μ
    - λ2 = φ4 − sqrt(φ4^2 + 4 τ (1 − α)(ρ + μ)(χ/κ) φ2^2)
  - Definitions of φ2, φ3, φ4, φ5 retained from text (φ2 ≡ 1 − α/φ1 > 0; φ3 ≡ 1 + (1 − α)/φ1 > 0; φ4 ≡ ρ + (ρ + μ)(1 − α)/φ1 > 0; φ5 ≡ Rπ − ρπ + (Rπ + μ)(1 − α)/φ1 > 0).

- Welfare criterion and policy parameter ρ∗
  - Social welfare:
    - SW = ∫∞_0 [c^(1 − 1/τ) / (1 − 1/τ)] e^(−ρ∗ τ) dτ, with ρ∗ ≤ ρ  (equation (34)).
  - Lower ρ∗ raises present-value welfare gains from infrastructure and allows crowding-in effects to matter more (HM Treasury example: ρ∗ = 2 − 3.5% recommended in practice).

- Welfare comparisons: full employment vs unemployment (key analytical results)
  - Full employment welfare gain for P3 (units of consumption):
    - ΣΩ − ΣΩo |P3 = [ (Rπ − ρπ) / (ρ + μ) ] [μ / ρ∗] (1 + Hπ)  (equation (35))
    - Hπ expression retained (complex term involving Γ, Λ, λ2; Γ > 0, Λ > 0).
    - Special case ρ∗ = ρ: Hπ = 0 and welfare gain reduces to capitalized value:
      - ΣΩ − ΣΩo |P3,ρ∗=ρ = [ (Rπ − ρπ) / (ρ + μ) ] [μ / ρ] (ζ̄ − ζo)  (equation (36))
  - Condition for P3 to deliver higher social welfare than OI (full GE):
    - ΣΩ|P3 > ΣΩ|OI iff (preserve equation):
      - (Rπ − R) α(ρ − ρ∗) / [(ρ∗ − λ2)(1 − α)(ρ + μ)] Γ > R − ρ − (Rπ − ρπ)  (equation (37))
    - Numerical effect: when ρ∗ = :0.02 − :0.06 it lowers breakeven Rπ values from :0.25 to :0.228 − :0.239.
    - Conclusion: P3 may increase welfare more than OI despite lower direct return when crowding-in matters.

  - Unemployment (efficiency wages) case:
    - Multiplier on (Rπ − R) much larger when infrastructure reduces unemployment; the case favoring P3 is stronger.
    - Numerical highlights:
      - P3 produces an additional welfare gain of 27−34% in runs where direct returns are equal (Rπ = :0.25).
      - Breakeven Rπ decreases to :0.179 − :0.196 for some parameterizations.
      - In five of nine runs where P3 direct return = :0.04, P3 raises real income more than OI in some runs (indicative).
    - Practical takeaway: policy makers can be confident P3 is right when DRG ≤ 3 percentage points; with highly rigid wages (β2 = :1 − :1.5), P3 may dominate OI even when direct return is 7−8 points lower.

### Importance of construction speed (tempo)
- When P3 builds faster than OI, iζ becomes a state variable and the system expands to four dimensions.
- Define ν ≡ [iζ,o+ − iζ,o−] / (ζ̄ − iζ,o) and σ governing decay; ζ path:
  - ζ − ζo = [1 − e^(−μ τ) + (ν − 1) μ (e^(−σ τ) − e^(−μ τ))/(μ − σ)] (ζ̄ − ζo)
- Benchmarks and numerical calibration:
  - Government-only case: ν = 1; ζ traverses 50% of the gap in τ1 = −ln(:0.50)/μ years.
  - P3 benchmark: choose σ = :0.20 and ν so ζ reaches 50% benchmark in 25% less time.
- Key insights:
  - Speed matters a lot where infrastructure deficits are large (LICs, MICs, EMEs).
  - Faster P3 construction increases labor demand directly (θLζ > 0) and via faster private capital growth.
  - Numerical outcomes: when direct returns equal, welfare-advantage of P3 rises dramatically (examples: 0−34% → 35−147% in some runs); breakeven Rπ decreases to :0.186 − :0.212 (full employment) and :0.119 − :0.180 (efficiency wages).
  - For β2 = :1.5 − 4, ρ∗ ≥ :0.04, and DRG = 7−8 percentage points (Rπ = :0.17 − :0.18), P3 beats OI 62−88% of the time.

### The fiscal challenge — budget effects and fiscal stress coefficient (CFS)
- Lump-sum taxes adjust to pay private partner in P3 and foreign bondholders in OI. Although ρπ ≫ ρ suggests P3 creates greater fiscal strain, endogenous revenue effects (larger tax and fee bases due to higher κ and L) can offset that.
- Net national income example for P3: Y = θ − ρπ(ζ − ζo) − μ(κ + ζ).
- Define v ≡ T/Y and ̄Ω ≡ Ω/(R + μ) (ratio of user fee to monetary value of infrastructure services produced by OI).
- Government budget constraint linearized solutions:
  - (v − vo)|P3 = [ρπ + μ − (Rπ + μ) Ω − v (Rπ − ρπ)] (ζ − ζo)/Y − η (χ − χo)/Y − v Φ (κ − κo)/Y  (equation (39))
  - (v − vo)|OI = [ρ + μ − (R + μ) Ω − v (R − ρ)] (ζ − ζo)/Y − η (χ − χo)/Y − v Φ (κ − κo)/Y  (equation (39′))
  - When speed same for P3 & OI, difference simplifies to:
    - (v − vo)|P3 − (v − vo)|OI = (ρπ − ρ)(1 + v) (ζ − ζo)/Y − (Rπ − R)(Ω + v) (ζ − ζo)/Y − η [ (χ − χo)/Y |_P3 − (χ − χo)/Y |_OI ] − v Φ [ (κ − κo)/Y |_P3 − (κ − κo)/Y |_OI ]  (equation (40))
- Coefficient of fiscal stress (CFS) defined:
  - CFS ≡ (v − vo) / (ζ̄ − iζ,o) / Y ; interpretation: CFS(τ1) = :0.75 means government must collect an additional :0.75% of NNI in lump-sum taxes at time τ1 to finance a permanent increase in infrastructure equal to 1% of NNI.
- Numerical and qualitative results:
  - Diverse outcomes: sometimes P3 raises fiscal stress, sometimes reduces it; depends on (Rπ − R), tax structure, user fees, and labor market.
  - Examples from text:
    - When Rπ = :0.20, ρπ = :0.10, ̄Ω = :0.25 and DRG = 0: P3 tax line positive and large for decades (Figures cited).
    - If Rπ = :0.25 and unemployment exists, sign can change favoring P3 by year 13.
    - If user fees cover O+M costs plus 30−50% of interest/return payments (̄Ω = :0.40), P3 can pay for itself immediately or within a few years and yield larger fiscal dividend than OI.
  - Policy implication: fiscal impact of P3 should not be pre-judged; welfare-maximizing P3s can ease fiscal constraints and improve debt sustainability in many scenarios.

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### 6.1 Distortionary Taxes Adjust — setup, linearized impact, and magnitude
- Setup: χ is a CES aggregate of two consumer goods; only χ2 is taxed at rate η. Exact price index:
  - Π = [μ (Π*1)1−σ + (1−μ) Π1−σ2]1/(1−σ)  (equation (41))
  - Π2 = Π*2 (1 + η)  (equation (42))
- Units chosen so Π*1 = 1 and Π*2 = 1/(1 + ηo): initial equilibrium Π1 = Π2 = Π = 1.
- Budget constraints and demand:
  - η Π*2 χ2 = ρπ (ζ − ζo) + β ζ − Λ[α(ζ − ζo) + ζo] − T  (equation (43))
  - k̇ = q − T − Π χ − β k − Λ[α(ζ − ζo) + ζo]  (equation (44))
  - k̇ = q − ρπ (ζ − ζo) − Π χ − β (ζ + k) + η Π*2 χ2  (equation (45))
  - Demand for good 2: χ2 = (1 − μ) (Π2 / Π)−σ χ  (equation (46))
- η adjusts endogenously in (43) to satisfy government budget constraint.
- Linearized relations (deviations):
  - k̇ = φ4 δk + φ5 δζ − δχ + η Π*2 δχ2  (equation (47))
  - δχ2 / χ2 = δχ / χ − σ (1 − θ) δ η / (1 + η)  (equation (48))
  - δ η / (1 + η) = [ρπ + β − ̄Λ (Rπ + β)] / g0 δζ − [η / (χ g0)] δχ  (equation (49))
  - g0 ≡ 1 + η − σ η (1 − θ); θ ≡ Π2 χ2 / Π χ.
- Steady-state consumption relation (compact form preserved):
  - (δχ̄ / δζ̄)Π3 = (1 − η θ / g0)−1 { [ ρ + (ρ + β)(1 − α) / φ1 ] α2 + Rπ − ρπ
    + (Rπ + β)(1 − α) / φ1 − [η σ (1 − θ) / g0] [ρπ + β − ̄Λ (Rπ + β)] }  (equation (50))
  - α2 ≡ (Rπ + β) / (ρ + β) [ (α / (1 − α)) ] (1 + β2 (1 − υ) / (β2 (1 − υ)) ).
- Interpretation: the η-related term reflecting efficiency loss from distortionary taxation is extremely small because σ is small at high aggregation levels. The negative direct effect in the numerator is counterbalanced by a multiplier slightly larger than unity; net differences relative to lump-sum tax case are “pocket change.”
- Numerical example (preserve numbers exactly from text):
  - Parameter values: Rπ = :23, R = :16, ρπ = :15, ρ = :06, β = :05, σ = θ = :5, β2 = 1, α = :40, ̄Λ = :25, η = :15.
  - Computed η-related term = −:00261 versus .123 for the component mirroring lump-sum tax solution.
  - Conclusion: "little triangles" (differences between distortionary and lump-sum tax adjustments) are truly little; transition-path and welfare comparisons in the paper show numbers in Table 7 always very close and often within rounding error.

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### 6.2 Incorporating Distributional Concerns — social weights and quantitative impacts
- Social welfare augmented by real wage income term:
  - ΣΩ − ΣΩ₀ χ^(−1/τ) = ∫₀^∞ [χ − χ₀ + f(ωΛ − ω₀Λ₀)] e^(−ρ* τ) dτ  (Equation (52))
  - Interpretation: when real wage income decreases by one dollar, social welfare is unchanged if aggregate consumption increases by f dollars.
- Normative calibration for f (distributional weight):
  - Utilitarian calculations imply f = .47 − 2.05 when τ = .35 − 1 under usual parameter values.
  - Text treats f = .5 as conservative; f = 1 and f = 1.5 are reasonable exercises.
- Quantitative effects on breakeven direct return R_P:
  - Because P3 increases labor demand more than OI, P3 benefits more from distributional externality when real wage income enters welfare function.
  - Reported impacts for ρ* = .06:
    - Efficiency-wage runs: breakeven R_P drops from .202 to between .169 and .185 (for f values considered).
    - Full-employment regions: breakeven R_P drops from .239 to between .189 and .214 (for f values considered).
  - Distributional concerns increase social return for both P3 and OI, but the gain is much larger for P3, especially in full-employment case.
- Macro-welfare magnitudes and conclusion:
  - Macro externalities raise social return in P3 by 5−8 percentage points relative to OI.
  - Recommendation: partial-equilibrium comparisons are incomplete; general-equilibrium macro analysis is required in second-best settings.
  - Authors plan integration of P3 analysis into DIG (Debt, Investment, and Growth) framework to permit policy makers to specify contract terms (annuity or DBOT), rank welfare gains for P3 vs OI, and track key macro variables (sectoral wages, employment, output, private investment, real exchange rate, external debt, current account deficit).

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*Source: _wp1678 — Excerpts from IMF working paper (1. Introduction; 2.3 The National Budget Constraint; 6.1 Distortionary Taxes Adjust; 6.2 Incorporating Distributional Concerns)*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Context and motivation
- Public-private partnerships (P3s) to build and operate infrastructure assets are increasingly common in less-developed countries (LDCs) and are highly controversial.
- Microeconomic theory and case-study evidence warn that P3s may be much more expensive than traditional procurement because:
  - Higher cost of private finance: private sector cannot spread non-diversifiable risk as widely as the public sector (Vickrey, 1964; Arrow, 1966; Arrow and Lind, 1970).
  - Compensation for country risk in some LDCs where government is perceived as an “unreliable business partner” (Estache et al., 2015).
  - Administrative and transaction costs at procurement and during operations: complex long-term contracts, bid costs that limit ex ante competition, difficulties designing auctions that prevent collusion, limited information on firm costs, and frequent, costly renegotiations due to relationship-specific investments and contract complexity.
  - Monitoring costs required to verify compliance by private partners.

### Empirical magnitudes cited
- Non-diversifiable risk pushes interest rate on senior debt in P3s "2−3 percentage points" above government debt.
- Transaction costs measured as a percentage of capital costs:
  - Average "10%" for European governments (Dudkin and Valila, 2005).
  - "2−10%" for private firms in Great Britain (House of Commons, 2002-03).
- Monitoring costs in U.S. P3s magnify total costs "3−25%" (Torres and Pina, 2001).
- Renegotiation frequency example: 35 Portuguese P3s resulted in "254" contract renegotiations between "1995 and 2012".
- Typical P3 contracts last "20−30 years" (difficulty enumerating all contingencies over such horizons).

### Countervailing benefits of P3s emphasized
- Private partners can bring superior technical expertise, greater implementation capacity, and fewer agency problems.
- Advantages translate into:
  - Shorter construction periods (important where acute infrastructure bottlenecks exist).
  - Better, more productive infrastructure.
- Key question: Do gains in speed and efficiency compensate for higher cost — i.e., do P3s offer better "value for money" than own investment (OI) by the public sector?

### Gap in the literature and objective of the paper
- Macroeconomic literature has been almost mute; prior macro work mainly recommends integrating future payments to private partners into fiscal accounts and debt sustainability analysis (IMF, 2007).
- This paper gives macroeconomics a greater voice by comparing P3 vs. OI in a dynamic general equilibrium macro model that features:
  - Private capital accumulation.
  - A flexible labor market specification that nests full employment with flexible wages and involuntary unemployment with efficiency wages.
- Unifying finding: general equilibrium effects shift the comparison in favor of P3 under several conditions.

### Mechanisms and comparative-static insights
- Direct return comparison (partial equilibrium): direct return = return on infrastructure minus return paid to private partner (or interest on external debt). Partial equilibrium may pick a winner.
- General equilibrium considerations complicate welfare ranking because P3:
  - Builds better/higher-quality infrastructure → crowds in private investment more and increases labor demand more.
  - These effects help P3 when:
    - The private time preference rate exceeds the social time preference rate (private investment is suboptimal at the initial equilibrium).
    - Real wage rigidity prevents full employment.
    - Wage income carries a larger weight in the social welfare function than income of the representative agent.
- Quantitative statement from the introduction: the asymmetric impact on macro externalities raises the social return in the P3 "2−9 percentage points" relative to the social return to OI, depending on whether externalities operate singly or in combination and on whether P3 enjoys an advantage in speed of construction.
- Conclusion drawn: the ranking of direct returns is not a reliable proxy for the welfare ranking — e.g., a P3 that pays a direct return of "2%" may be preferable to OI that pays a direct return of "10%".

### Fiscal implications
- P3s are considerably more expensive than OI in fiscal terms.
- However, stronger positive effects on infrastructure services, private capital accumulation, and employment generate more revenue from taxes and user fees.
- This can reduce the need for supporting fiscal adjustment and can make debt sustainability easier to achieve for P3 despite higher upfront cost.
- In some cases, P3 requires less fiscal support than OI at every point in time.

### Structure of the paper (as presented)
- Sections 2 and 3: present the model and solve for the transition path and steady-state equilibrium.
- Sections 4–6: compare the effects of P3 and OI on welfare and fiscal sustainability.
- Section 7: concluding remarks and priorities for future research.

### Model overview (summary of setup introduced in the Introduction)
- Three components:
  - Government that builds infrastructure either on its own (OI) or by partnering with a foreign firm (P3).
  - Domestic private sector run by a representative agent who pays efficiency wages and accumulates capital.
  - National budget constraint requiring capital flows to finance the current account deficit each period.
- Production technology: Cobb-Douglas with infrastructure ζ, capital κ, and labor Λ:
  - For P3: θ = [α(ζ − ζ0) + ζ0] ψ κ^a (eΛ)^(1−a) with α > 1 for P3.
  - For OI: θ = ζ ψ κ^a (eΛ)^(1−a).
  - Infrastructure enters as a shift factor; effective labor depends on effort e.
- P3 contract structure: annuity; foreign partner receives (ρp + δ)(ζ − ζ0) in perpetuity, where δ covers depreciation and ρp is the return earned on the investment.
- Government revenue for P3 (lump-sum taxes T adjust in main variant):
  - T = ρp(ζ − ζ0) + δζ − R[α(ζ − ζ0) + ζ0] − ηc.
- Government revenue for OI:
  - T = ρ(ζ − ζ0) + (δ − R)ζ − ηc, where borrowing occurs at world interest rate ρ.
- Speed of construction:
  - Law of motion for infrastructure: ż = iζ − δζ.
  - In P3, iζ overshoots its steady-state level and ζ increases faster than under OI.
  - Dynamics for P3 investment: diζ/dτ = σ(ȷiζ − iζ) with σ > 0 and iζ,o+ > ȷiζ.
  - For OI, iζ = ȷiζ = δȷζ.

### Private sector optimization (outlined)
- Representative agent maximizes
  - U = ∫∞0 [ c^(1−1/τ) / (1−1/τ) − [e^(−β0 − β1 ln ω − β2 u) / 2] ] e^(−ρτ) dτ,
    subject to capital accumulation constraint: ḱ = θ − T − (1 + η)c − R[α(ζ − ζ0) + ζ0] − δκ.
- Labor supply fixed at unity; unemployment u = 1 − Λ.
- On optimal path, Euler equation: ċ = τ c (θ_κ − ρ − δ).
- Effort specification: e = β0 + β1 ln ω + β2 u; combined with the Solow condition and profit maximization yields effort constant in general equilibrium, with β1 = 1 and e = β1 = 1.

*Source: IMF working paper — 1. Introduction section*

### 2.3  The National Budget Constraint

### _wp1678 - 2.3  The National Budget Constraint

### National budget constraint (closure)
- The model is closed by the national budget constraint. Substituting for T in equation (6) produces:
  - For P3: ̇κ = θ − ρπ(ζ − ζo) − χ − μ(ζ + κ) (equation (13))
  - For OI: ̇κ = θ − ρ(ζ − ζo) − χ − μ(ζ + κ) (equation (13′))

### The case for efficiency wages (motivation and stylized facts)
- Efficiency wages are argued to be empirically relevant across non-agricultural sectors in LDCs.
- Stylized facts that efficiency wage models can jointly explain:
  - firm-size wage premiums starting at very small establishment size (5+ employees) and larger than in developed countries;
  - persistent, stable inter-industry wage differentials;
  - high correlation of industry wage premiums across occupations;
  - large wage premia for formal vs. informal sector employment and for informal non-agricultural vs. agricultural employment;
  - large cyclical flows into and out of unemployment in both formal and informal sectors.
- Empirical parameter ranges and country notes:
  - β2 empirical range for wage curve (when dependent variable is log unemployment): cluster between 0.05 and 0.15 -> evaluated at unemployment rate of 10% corresponds to β2 = 0.5−1.5 in equation (12).
  - South Africa example: national unemployment rate 25%; Kingdon and Knight (1999) wage curves estimate β2 = 0 for homeland regions and −.66 elsewhere (cluster means evaluated at an unemployment rate of 25%).
  - For irrigation in smallholder areas, enforce full employment by increasing β2 to 100,000 (illustrative).
- Author note: efficiency wages are not the only relevant framework; model choice depends on country and project location.

### Long-run solution (P3 steady state and comparative statics)
- At the new steady state under P3:
  - θ/κ = ρ + μ (equation (14))
  - χ = θ − ρπ(ζ − ζo) − μ(ζ + κ) (equation (15))
- Linear change relations (exact algebra delivered):
  - δχ = ρ δκ + (Rπ − ρπ) δζ + ω δL (equation (16))
  - δω = β2 ω δL (equation (17))
  - δL = [(Rπ + μ)(1 − α) / (ω f1)] δζ + [(ρ + μ)(1 − α) / (ω f1)] δκ (equation (18))
  - δκ = [(Rπ + μ)/(ρ + μ)] [α/(1 − α)] δζ + (κ/L) δL (equation (19))
  - where f1 ≡ α + β2 Λ = α + β2(1 − u), and Rπ = θ/ζ − μ is the return on infrastructure (net of depreciation).
- Long-run multipliers (effects of δζ):
  - δκ = [(Rπ + μ)/(ρ + μ)] [α/(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] δζ (equation (20))
  - δL = [(Rπ + μ)/(ω f1)(1 − α)] [1 + (α/(1 − α)) 1+β2(1−u) / (β2(1−u))] δζ (equation (21))
- Comparison P3 vs OI:
  - δκ|P3 − δκ|OI = (Rπ − R)/(ρ + μ) [α/(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] δζ (equation (22))
  - δL|P3 − δL|OI = (Rπ − R)/(ω f1)(1 − α) [1 + (α/(1 − α)) 1+β2(1−u) / (β2(1−u))] δζ (equation (23))
- Key qualitative points:
  - Direct positive effects on capital accumulation and employment depend on the return on infrastructure; consensus view Rπ > R strengthens the case for P3.
  - Feedback effects between κ and L magnify direct effects. Empirical β2 cluster between 0.5 and 1.5; for β2 in this range feedback multiplier is on the order of 1.5−3 for κ and 2−4 for L.

### Consumption response and return-gap decomposition
- Substituting solutions into (16) yields for P3:
  - (δχ/δζ)|P3 = Rπ − ρπ + (Rπ + μ) Φ (equation (24))
  - Φ ≡ [ρ + (ρ + μ)(1 − α) / (α + β2(1 − u))] [α/(ρ + μ)(1 − α)] [1 + β2(1 − u)] / [β2(1 − u)] + (1 − α) / [α + β2(1 − u)] > 0
- For OI replace Rπ and ρπ with R and ρ (equation (25)).
- Difference (Total Return Gap):
  - (δχ/δζ)|P3 − (δχ/δζ)|OI = [Rπ − ρπ − (R − ρ)] + (Rπ − R) Φ (equation (26))
  - Direct Return Gap (DRG) ≡ Rπ − ρπ − (R − ρ)
- Parameterization used to quantify effects:
  - α = :0.30 − :0.50, μ = :0.05, ρ = :0.06, ρπ = :0.15, ρ = :0.06, ρ∗? (context), ρπ = :0.15 noted as pricey.
  - ρ (external borrowing rate) = :0.06, ρπ (P3 cost) = :0.15, R = :0.16, Rπ = :0.16 − :0.25, β2 = :0.5 − 1.5, ∞ (cases considered).
  - Private time preference rate ρ chosen at :0.10.
- Numerical insights:
  - With Rπ topping out at :0.25 and ρπ = :0.15, DRG ranges from zero to −9% (i.e., negative values meaning OI has better direct return).
  - For β2 in empirical range, feedback multipliers yield substantial amplification: point D (P3) is far north/east of points B and C (OI).

### Transition path (dynamics and eigenstructure)
- Linearized dynamics (variables deviations from steady state):
  - ̇χ = τχ (θκκ δκ + θκζ δζ + θκL δL) (equation (28))
  - ̇κ = ρ δκ + (Rπ − ρπ) δζ + ω δL − δχ (equation (29))
- After substitution and simplification:
  - ̇χ = τ(χ/κ)(ρ + μ)(α − 1) φ2 (κ − κ̄) + τ(χ/κ) α (Rπ + μ) φ3 (ζ − ζ̄) (equation (30))
  - ̇κ = φ4 (κ − κ̄) + φ5 (ζ − ζ̄) − (χ − χ̄) (equation (31))
  - ̇ζ = −μ(ζ − ζ̄) (equation (32)) when infrastructure investment constant at ζ̄ and ̇ζ = −μ(ζ − ζ̄) for both P3 and OI.
  - Definitions: φ2 ≡ 1 − α/φ1 > 0, φ3 ≡ 1 + (1 − α)/φ1 > 0, φ4 ≡ ρ + (ρ + μ)(1 − α)/φ1 > 0, φ5 ≡ Rπ − ρπ + (Rπ + μ)(1 − α)/φ1 > 0.
- System matrix and eigenvalues:
  - System (33) is three-dimensional in deviations [χ − χ̄, κ − κ̄, ζ − ζ̄].
  - Stationary equilibrium is a saddle point with two state variables κ and ζ, and two negative eigenvalues:
    - λ1 = −μ
    - λ2 = φ4 − sqrt(φ4^2 + 4 τ (1 − α)(ρ + μ)(χ/κ) φ2^2)

### Welfare criterion (social welfare function)
- Social welfare measured as:
  - SW = ∫∞_0 [c^(1 − 1/τ) / (1 − 1/τ)] e^(−ρ∗ τ) dτ, with ρ∗ ≤ ρ (equation (34))
- Policy discussion:
  - Common practice: social time preference rate ρ∗ is set below private rate ρ (e.g., HM Treasury recommends ρ∗ = 2 − 3.5%).
  - Lower ρ∗ raises present-value welfare gains from infrastructure and allows crowding-in of private investment to matter.

### Welfare comparisons: full employment vs unemployment
- Full employment (closed-form results summarized):
  - Welfare gain (units of consumption) for P3:
    - ΣΩ − ΣΩo |P3 = [ (Rπ − ρπ) / (ρ + μ) ] [μ / ρ∗] (1 + Hπ) (equation (35))
    - Hπ ≡ (ρ − ρ∗)/ρ∗ + μ + (Rπ + μ) α (ρ − ρ∗)/[(ρ∗ − λ2)(1 − α)(ρ + μ)] Γ, with Λ, Γ, and J definitions (Γ > 0, Λ > 0).
  - For OI replace Rπ and ρπ with R and ρ (equation (35′)).
  - Special case ρ∗ = ρ: Hπ = H = 0 and welfare gain reduces to capitalized value:
    - ΣΩ − ΣΩo |P3,ρ∗=ρ = [ (Rπ − ρπ) / (ρ + μ) ] [μ / ρ] (ζ̄ − ζo) (equation (36))
    - Analogous for OI (equation (36′)).
- Condition for P3 to deliver higher social welfare than OI (full general equilibrium):
  - ΣΩ|P3 > ΣΩ|OI iff
    - (Rπ − R) α(ρ − ρ∗) / [(ρ∗ − λ2)(1 − α)(ρ + μ)] Γ > R − ρ − (Rπ − ρπ) (equation (37))
  - Left-hand side (LHS) is small but significant; when ρ∗ = :0.02 − :0.06 it lowers breakeven Rπ values from :0.25 to :0.228 − :0.239.
  - Conclusion: P3 may increase welfare more than OI despite lower direct return when general equilibrium crowding-in effects matter.

- Unemployment case (efficiency wages):
  - The multiplier on (Rπ − R) is much larger when infrastructure reduces unemployment; the case for P3 is much stronger.
  - Numerical highlights:
    - P3 produces an additional welfare gain of 27−34% in runs where direct returns are equal (Rπ = :0.25).
    - Breakeven Rπ decreases to :0.179 − :0.196 for some parameterizations.
    - In five of nine runs where P3 direct return = :0.04, P3 raises real income more than OI (not full welfare results but indicative).
  - Conclusion: policy makers can be confident P3 is right when DRG ≤ 3 percentage points; with highly rigid wages (β2 = :1 − :1.5), P3 may dominate OI even when direct return is 7−8 points lower.

### Importance of speed (construction tempo)
- When P3 builds faster than OI, iζ enters as a state variable and the system expands to four dimensions (equation (38)).
- Define ν ≡ [iζ,o+ − iζ,o−] / (ζ̄ − iζ,o), and σ governing decay; ν and σ determine iζ path and hence ζ path:
  - ζ − ζo = [1 − e^(−μ τ) + (ν − 1) μ (e^(−σ τ) − e^(−μ τ))/(μ − σ)] (ζ̄ − ζo)
- Benchmarks:
  - Government-only case: ν = 1, ζ traverses 50% of the gap in τ1 = −ln(:0.50)/μ years.
  - Comparison run for P3: choose σ = :0.20 and ν so ζ reaches 50% benchmark in 25% less time.
- Key insights:
  - Speed matter matters a lot, especially where infrastructure deficits are large (LICs, MICs, EMEs).
  - Faster P3 construction increases labor demand directly (θLζ > 0) and via faster private capital growth.
  - Numerical outcomes (Tables 5–6 vs Tables 3–4): when direct returns equal, welfare-advantage of P3 soars (0−34% → 35−147% in some runs); breakeven Rπ decreases to :0.186 − :0.212 (full employment) and :0.119 − :0.180 (efficiency wages).
  - For β2 = :1.5 − 4, ρ∗ ≥ :0.04, and DRG = 7−8 percentage points (Rπ = :0.17 − :0.18), P3 beats OI 62−88% of the time.

### The fiscal challenge (government budget and fiscal stress)
- Lump-sum taxes adjust to pay private partner in P3 and foreign bondholders in OI. Since ρπ ≫ ρ, intuition suggests P3 creates greater fiscal strain, but endogenous revenue effects can offset this.
- Net national income Y (example for P3): Y = θ − ρπ(ζ − ζo) − μ(κ + ζ).
- Define v ≡ T/Y and ̄Ω ≡ Ω/(R + μ) (ratio of user fee to monetary value of infrastructure services produced by OI).
- Government budget constraint solutions yield:
  - (v − vo)|P3 = [ρπ + μ − (Rπ + μ) Ω − v (Rπ − ρπ)] (ζ − ζo)/Y − η (χ − χo)/Y − v Φ (κ − κo)/Y (equation (39))
  - (v − vo)|OI = [ρ + μ − (R + μ) Ω − v (R − ρ)] (ζ − ζo)/Y − η (χ − χo)/Y − v Φ (κ − κo)/Y (equation (39′))
  - Ω ≡ ̄Ω + v(1 − α)/φ1 > 0, Φ ≡ ρ + (ρ + μ)(1 − α)/φ1 > 0
- When speed is same for P3 and OI, difference simplifies (equation (40)):
  - (v − vo)|P3 − (v − vo)|OI = (ρπ − ρ)(1 + v) (ζ − ζo)/Y − (Rπ − R)(Ω + v) (ζ − ζo)/Y − η [ (χ − χo)/Y |_P3 − (χ − χo)/Y |_OI ] − v Φ [ (κ − κo)/Y |_P3 − (κ − κo)/Y |_OI ]
- Define CFS (coefficient of fiscal stress):
  - CFS ≡ (v − vo) / (ζ̄ − iζ,o) / Y ; interpretation: CFS(τ1) = :0.75 means government must collect an additional :0.75% of NNI in lump-sum taxes at time τ1 to finance a permanent increase in infrastructure equal to 1% of NNI.
- Numerical and qualitative results:
  - Diversity of outcomes: sometimes P3 creates more fiscal stress, sometimes less; depends on (Rπ − R), tax structure, user fees, labor market.
  - Examples:
    - When Rπ = :0.20, ρπ = :0.10, ̄Ω = :0.25 and DRG = 0: P3 line positive and large for decades (Figures 5a, 5e). If Rπ = :0.25 and unemployment exists, sign can change favoring P3 by year 13 (Figures 5b, 5f).
    - If user fees cover O+M costs plus 30−50% of interest/return payments (̄Ω = :0.40), P3 can pay for itself immediately or within a few years (Figures 5d, 5g) and yield larger fiscal dividend than OI.
  - Policy implication: Do not pre-judge fiscal impact of P3. Welfare-maximizing P3s can ease fiscal constraints and improve debt sustainability in many scenarios.

### Extensions (model assumptions relaxed)
- Benchmark assumptions relaxed:
  - Lump-sum taxes adjust in government budget constraint (first relaxation).
  - Social welfare depends solely on representative agent consumption; relaxing this is potentially important (second relaxation).

*Italic: Source: _wp1678 - 2.3  The National Budget Constraint (excerpt). *

### 6.1  Distortionary Taxes Adjust

### 6.1  Distortionary Taxes Adjust

### Model setup and price index
- Treat χ as a CES aggregate of two consumer goods and assume only χ2 is subject to the tax η.
- Exact consumer price index:
  - Π = [μ (Π*1)1−σ + (1−μ) Π1−σ2]1/(1−σ)  (equation (41))
  - Π2 = Π*2 (1 + η)  (equation (42))
- Π* i is the world market price of good i; μ is a CES distribution parameter; and σ is the elasticity of substitution between χ1 and χ2.
- Units chosen so that Π*1 = 1 and Π*2 = 1/(1 + ηo). Thus Π1 = Π2 = Π = 1 at the initial equilibrium.

### Budget constraints and demand
- Public, private, and national budget constraints:
  - η Π*2 χ2 = ρπ (ζ − ζo) + β ζ − Λ[α(ζ − ζo) + ζo] − T  (equation (43))
  - k̇ = q − T − Π χ − β k − Λ[α(ζ − ζo) + ζo]  (equation (44))
  - k̇ = q − ρπ (ζ − ζo) − Π χ − β (ζ + k) + η Π*2 χ2  (equation (45))
- Demand for good 2:
  - χ2 = (1 − μ) (Π2 / Π)−σ χ  (equation (46))
- η adjusts endogenously in (43) to satisfy the government budget constraint.

### Linearization and key relations
- Efficiency gains/losses from the tax distortion appear in the term η Π*2 χ2 in the national budget constraint.
- Linearizing (45) and solving (43) and (46) for χ2 and η yields:
  - k̇ = φ4 δk + φ5 δζ − δχ + η Π*2 δχ2  (equation (47))
  - δχ2 / χ2 = δχ / χ − σ (1 − θ) δ η / (1 + η)  (equation (48))
  - δ η / (1 + η) = [ρπ + β − ̄Λ (Rπ + β)] / g0 δζ − [η / (χ g0)] δχ  (equation (49))
- Definitions and notation:
  - g0 ≡ 1 + η − σ η (1 − θ)
  - Differentials refer to deviations from the steady state.
  - θ ≡ Π2 χ2 / Π χ, the consumption share of good 2.

### Steady-state consumption solution and interpretation
- From (47)-(49) and the solution for δk in (20), the steady-state consumption relation is:
  - (δχ̄ / δζ̄)Π3 = (1 − η θ / g0)−1 { [ ρ + (ρ + β)(1 − α) / φ1 ] α2 + Rπ − ρπ
    + (Rπ + β)(1 − α) / φ1 − [η σ (1 − θ) / g0] [ρπ + β − ̄Λ (Rπ + β)] }  (equation (50))
- Where
  - α2 ≡ (Rπ + β) / (ρ + β) [ (α / (1 − α)) ] (1 + β2 (1 − υ) / (β2 (1 − υ)) ).
- The term ρπ + β − ̄Λ (Rπ + β) captures the direct effect on the budget (see (49)); it is negative for plausible values of ρπ, Rπ, and ̄Λ.
- Because σ is small at high levels of aggregation (a universal finding in estimates of demand systems with 5-10 goods), the coefficient on the direct effect, η σ (1 − θ) / g0, is extremely small.
- The small negative term in the numerator is counterbalanced by a multiplier slightly larger than unity (reflecting the gain from increases in χ that translate into increases in χ2).
- Consumption may increase more or less than in the lump-sum tax adjustment case, but the difference is "pocket change." The impact on the P3–OI ranking, which depends on the size difference in two little triangles, is even smaller.
- The solution for OI substitutes R and ρ for Rπ and ρπ in (50). Hence
  - (δχ̄ / δζ̄)Π3 − (δχ̄ / δζ̄)OĪ > 0 if and only if
    Rπ − ρπ − (R − ρ) + (Rπ − R) Φ + [η σ (1 − θ) / g0] [̄Λ (Rπ − R) − (ρπ − ρ)] > 0  (equation (51))
  - The bracketed term involving η functions essentially as a tiebreaker.

### Numerical example and magnitude of effects
- Using the parameter values:
  - Rπ = :23, R = :16, ρπ = :15, ρ = :06, β = :05, σ = θ = :5, β2 = 1, α = :40, ̄Λ = :25, η = :15
- The η-related term equals −:00261 versus .123 for the component that mirrors the lump-sum tax solution.
- Conclusion: the "little triangles" are truly little; differences between lump-sum and distortionary tax adjustments are very small.

### Transition path and welfare comparisons
- The transition path was derived (see Appendix C) and exact welfare comparisons for lump-sum vs. distortionary tax adjustment were computed.
- The numbers in Table 7 are always very close; in many cases the difference amounts to rounding error.

*Source: 6.1  Distortionary Taxes Adjust — _wp1678 - 6.1  Distortionary Taxes Adjust*

### 6.2  Incorporating Distributional Concerns

### _wp1678 - 6.2  Incorporating Distributional Concerns

### Social welfare specification with real wage income
- The paper incorporates real wage income into the social welfare function by augmenting the representative-agent consumption welfare flow with a term for real wage income:
  ΣΩ − ΣΩ₀ χ^(−1/τ) = ∫₀^∞ [χ − χ₀ + f(ωΛ − ω₀Λ₀)] e^(−ρ* τ) dτ. (Equation (52) in source)
- Interpretation: when real wage income decreases by one dollar, social welfare is unchanged if aggregate consumption increases by f dollars.

### Normative calibration for f (distributional weight)
- Benthamite utilitarian calculations provide an indicative range for f in the model:
  - In the model, a utilitarian welfare function implies f = .47 − 2.05 when τ = .35 − 1 and other parameters take their usual values.
- The text treats f = .5 as a (too) conservative value and f = 1 and f = 1.5 as reasonable values for exercises.

### Quantitative effects on breakeven direct return (R_P) and welfare arithmetic
- Key qualitative point: because P3 increases labor demand more than OI, P3 benefits more from the distributional externality when real wage income enters the welfare function.
- Reported numerical impacts (for ρ* = .06 and f values considered):
  - In runs with efficiency wages:
    - The breakeven value of R_P drops from .202 to between .169 and .185.
  - For projects in regions with full employment:
    - The breakeven value of R_P drops from .239 to between .189 and .214.
- Distributional concerns increase the social return for both P3 and OI, but the gain for P3 is much larger, especially in the full-employment case.

### Macro-welfare magnitudes and broader conclusion
- The paper emphasizes that partial-equilibrium comparisons are incomplete; general-equilibrium macro analysis is required in second-best settings.
- Welfare gains from ameliorating major macroeconomic externalities raise the social return in P3 by 5−8 percentage points relative to the return on OI.
- The authors plan to integrate P3 analysis into the DIG (Debt, Investment, and Growth) modeling framework to allow policy makers to specify contract terms (annuity or DBOT), rank welfare gains for P3 vs. OI, and track paths of key macro variables (sectoral wages, employment, output, private investment, real exchange rate, external debt, current account deficit).

### Representative numerical highlights drawn from tables and text
- Utilitarian-implied range for f: .47 − 2.05 (for τ = .35 − 1 under usual parameter values).
- Example breakeven R_P shifts for ρ* = .06:
  - Efficiency-wage runs: .202 → .169 − .185 (for f = .5 or f = 1,1.5 as considered).
  - Full-employment runs: .239 → .189 − .214 (for f = .5 or f = 1,1.5 as considered).
- Aggregate welfare premium for P3 vs. OI from macro externality reduction: 5−8 percentage points.

*Italic: Source — _wp1678 - 6.2  Incorporating Distributional Concerns (IMF PDF content unit)*

### References

### References

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*Source: _wp1678 - References*

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