## wpiea2023056-print-pdf

## Source details

**Canonical URL:** [wpiea2023056-print-pdf](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023056-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023056-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023056-print-pdf.pdf.json)

---

### Background and motivation
- Public goods (infrastructures and utilities) can be provided by government or private firms; public provision keeps strategic management, financing, and most investment and operation risks in the public sector, while private provision (PPP) bundles phases and transfers a larger share of investment and operational risks to the private sector (Engel, Fischer and Galetovic 2014).
- Choice between public and private provision driven by differentials in capital outlays, operating margins, risk management, and cost of capital.
- No academic consensus on whether government financing of a project that generates a stream of net social benefits should imply a lower discount rate than private financing (Lind 1982).

### Survey of VfM discount-rate practices
- Jurisdiction — Discount rate
  - Australia — PPP vs Public-Sector Comparator (PSC) systematic risk-adjusted rates
  - Netherlands — PPP's weighted average cost of capital (WACC)
  - Canada, BC — PPP's weighted average cost of capital (WACC)
  - Canada, OT — Government borrowing rate
  - France — Government borrowing rate
  - Germany — Government borrowing rate
  - Ireland — Government borrowing rate
  - South Africa — Government borrowing rate
  - United Kingdom — Social discount rate
- Note: Almost all jurisdictions apply a common discount rate to project flows of net social benefits regardless of the source of financing. Only Australia applies different systematic risk-adjusted rates with PSC lower than the PPP rate.

### Literature and normative arguments on public versus private discount rates
- Arguments for a lower public discount rate:
  - Public sector can better absorb and spread risks among a greater number of individuals (Arrow and Lind, 1970; Fisher, 1973).
  - Public projects complementary to private investment should apply lower discount rates to partially undo distortions that discourage private investment (Ogura and Yohe, 1977).
  - Private firms cannot internalize all externalities and may lead to inefficient piecemeal decision-making (Flemming and Mayer, 1997).
- Arguments for equal or higher public discount rates:
  - Public sector's lower borrowing cost may reflect lack of default and taxing power rather than superior risk management; some argue the social discount rate should equal public and private discount rates (Bailey and Jensen, 1972; Brealey, Cooper and Habib, 1997; Diamond, 1967).
  - Recent research emphasizes accounting for market risk premium in public-sector discount rates to avoid downward bias (Hirshleifer 1964; 1966; Lucas 2012).
- In incomplete markets, differential risk premia arise; Grout (2003) argued PPP payments to investors are more volatile and may justify higher PPP discount rates. Lind (1990) and Spackman (2004) suggested government borrowing rates as candidates for public discounting.

### Terminal value, predictability, and state-contingency — core contribution
- Financial valuation decomposition:
  - Predictable but risky flows within a forecast period (t ∈ (0,n]); and
  - Unpredictable flows beyond the forecast period (t ∈ (n, ∞)), whose discounted sum at time tn is the terminal value, E(TV_i).
- Terminal value is state-contingent due to Knightian uncertainty beyond the forecast period; unpredictable contingencies make true terminal value contingent on realized states.
- Contract incompleteness beyond the forecast period prevents credible ex-ante, binding arrangements to fully allocate state-contingent contingencies between government and private firms.
- Strategic interaction under contract incompleteness implies a systematically larger expected terminal value for publicly financed utilities than for privately financed ones, unless government reactions to contingencies are very inefficient because of economic or political failures.
- This state-contingent terminal-value channel provides a new argument in favor of using a lower discount rate to assess infrastructure projects that are publicly financed versus PPPs.

### Financial model and key equations
- Three-stage investment cycle:
  1. At t = 0, I_i > 0 invested with certainty by government (i = g) or private firm (i = p).
  2. For t ∈ (0,n], predictable but risky net social benefits A_it are realized.
  3. For t ∈ (n,∞), Knightian uncertainty yields unpredictable flows; expected discounted terminal value at tn is E(TV_i).
- Decomposition:
  - A_it = CS_it + PS_it  (equation (1))
  - PS_it = τ_it A_it  (equation (2)), τ_it ∈ (0,1)
  - CS_it = (1 − τ_it) A_it  (equation (3))
- Time-invariant case net present value:
  - NPV_i = −I_i + ∫_0^n A_i e^{−r_i t} dt + E(TV_i) e^{−r_i n}  (equation (4))
- Condition for private financing socially optimal (equation (5)):
  - −I_p + A_p (1 − e^{−r_p n})/r_p + E(TV_p) e^{−r_p n} > −I_g + A_g (1 − e^{−r_g n})/r_g + E(TV_g) e^{−r_g n}
- If E(TV_p) = E(TV_g) and r_p = r_g = r (equation (6)):
  - I_g − I_p + (A_p − A_g) (1 − e^{−r n})/r > 0

### Quantitative importance of terminal value (examples and illustration)
- Terminal value accounts for around 40-70 percent of the present value of the asset, depending on predictable flows, forecast-period duration, and discount rate.
- Stylized examples:
  - Series of $100 annuities in perpetuity discounted at r = 10 percent: first ten years (n) account for 61 percent of present value; terminal value accounts for 39 percent.
  - For n = 5 and r = 7 percent, terminal value represents 71 percent of present value.
- Market illustration (Credit Suisse):
  - Vinci valued at €77 a share: €26.07 (33.8 percent) from discounted dividends in next 15 years (2016-2030) and €50.96 (66.2 percent) from discounted terminal value after 2030.

### Terminal value and contract incompleteness — model of unpredictable contingencies
- Knightian uncertainty model: Bernoulli distribution with states B and G:
  - B: probability π ∈ [0,1], state “bad,” requiring revamping effort E to mitigate long-run loss L so A_it = A_TV − [L − h(E)] for t ∈ (n,∞), with h(0)=0, h′>0, h′′<0, lim_{E→∞} h(E)=L>0.
  - G: probability 1−π, state “good,” no revamping needed, A_it = A_TV for t ∈ (n,∞).
- Analysis abstracts from long-term efficiency gaps so A_TV and τ are same for public and private.

### Public financing: enforcement and efficiency cases
- Government can enforce desired level of E after state realization when implementation is public.
- Efficient government (case A.1):
  - Government maximizes W(E) = A_TV − (L − h(E))/r − E in bad state; first-order condition: h′(E_g)/r = 1.
  - Expected terminal value under government revamping: E(TV_g) = π W(E_g) + (1−π) W = A_TV / r − π (L − h(E_g)) / r + π E_g.
- Inefficient public revamping (marginal public cost δ ≥ 0):
  - Ŵ(E) = A_TV − (L − h(E))/r − E(1+δ); first-order condition h′(Ê_g)/r = 1 + δ.
  - E(TV̂_g) = π Ŵ(Ê_g) + (1−π) W = A_TV / r − π [ (L − h(Ê_g)) / r + Ê_g(1+δ) ].
  - Proposition 1: Expected terminal value of public projects is decreasing in δ; ∂/∂δ E(TV̂_g) = −π Ê_g < 0.

### Private financing: objectives, unverifiable and verifiable E
- Private discounted producer surplus: PS(E)+S = τ [ A_TV − (L − h(E)) / r ] − E + S.
- Unverifiable E (B.1):
  - Government cannot subsidize (S = 0). Private FOC: τ h′(E_p)/r = 1.
  - E(TV_p) = π W(E_p) + (1−π) W = A_TV / r − π (L − h(E_p)) / r + π E_p.
  - Proposition 2: If E is ex-post unverifiable, E(TV_g) > E(TV_p).
- Verifiable E (B.2):
  - Ex post negotiation over E and subsidy S with participation constraints:
    - Government: W(E) − λ S ≥ W(E_p).
    - Private firm: PS(E) + S ≥ PS(E_p).
  - Nash bargaining FOC yields h′(E_pb)/r = 2 / (1 + τ).
  - E(TV_p_b) = π (W(E_pb) − λ S_b) + (1−π) W = A_TV / r − π [ (L − h(E_pb)) / r + E_pb + λ S_b ].
  - Proposition 3: Minimum subsidy S required is strictly positive whenever E ≠ E_p.
  - Proposition 4: When E is ex-post verifiable, E(TV_g) > E(TV_p_b).
  - Proposition 5: E(TV_p_b) ≥ E(TV_p) (verifiability weakly increases private expected terminal value).

### Public versus private financing when public revamping is inefficient (B.3)
- Comparison of Ê_g and E_p: Ê_g > E_p iff δ < (1 − τ) / τ.
- Proposition 6: With ex-post unverifiable revamping, public expected terminal value is larger (smaller) than private iff δ < δ̄ (δ > δ̄), where δ̄ is characterized analytically:
  - Condition (23): h(Ê_g) / r − Ê_g (1+δ) > h(E_p) / r − E_p.
  - Second-order Taylor approximation yields threshold δ̄ < (1 − τ) / τ (detailed expression in text).
- Interpretation:
  - If government revamping inefficiency δ is sufficiently large (δ > δ̄), public financing yields a lower expected terminal value than private financing, potentially implying a larger discount rate for public financing.
  - Ex post verifiability and bargaining parameters (subsidy size and government bargaining power γ) affect private E_pb and the δ threshold for public advantage.

### Financial assessment and policy implications
- Revised feasibility inequality (equation (24)):
  - I_g − I_p + (A_p − A_g) (1 − e^{−r n}) / r > [ E(TV_g) − E(TV_p) ] e^{−r n} > 0.
- Key policy implications:
  - If public revamping is not too inefficient and E(TV_g) > E(TV_p), the discount rate for publicly financed projects should be lower than for privately financed projects: r_g < r_p.
  - Investment efficiency gains (I_g − I_p) and operating efficiency gains (A_p − A_g) from private provision must be sufficiently large to offset the lower expected terminal value under private financing for PPPs/BOTs to be preferred.
  - Policymakers should consider the role of state-contingent terminal values, contract incompleteness, and whether discount rates or explicit modeling of state-contingent terminal values should be used in VfM analysis.
  - Relevance for climate change and long-horizon projects: treatment of tail risks and ex-post endogeneity of future economic flows supports arguments for decreasing or time-varying discount rates in far-distant uncertain outcomes.

### Quantitative and illustrative takeaways
- Terminal value share examples:
  - Terminal value accounts for around 40-70 percent of present value depending on parameters.
  - Example: n = 10, r = 10 percent → terminal = 39 percent; n = 5, r = 7 percent → terminal = 71 percent.
  - Vinci share example: €26.07 (33.8 percent) from 2016-2030 dividends; €50.96 (66.2 percent) from terminal value after 2030.

*Source — IMF Working Papers: "Public versus Private Cost of Capital with State-Contingent Terminal Value" (wpiea2023056-print-pdf).*

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

### References (IMF Working Paper: Public versus Private Cost of Capital with State-Contingent Terminal Value)

### Background and motivation
- Public goods (infrastructures and utilities) can be provided by government or private firms; public provision keeps strategic management, financing, and most investment and operation risks in the public sector, while private provision (PPP) bundles phases and transfers a larger share of investment and operational risks to the private sector (Engel, Fischer and Galetovic 2014).
- The choice between public and private provision is driven by differentials in capital outlays, operating margins, risk management, and cost of capital.
- There is no academic consensus on whether government financing of a project that generates a stream of net social benefits should imply a lower discount rate than private financing; the profession remains divided on theory, computation, and the rate itself (Lind 1982).

### Survey of VfM discount-rate practices (Table 1)
- Jurisdiction — Discount rate
  - Australia — PPP vs Public-Sector Comparator (PSC) systematic risk-adjusted rates
  - Netherlands — PPP's weighted average cost of capital (WACC)
  - Canada, BC — PPP's weighted average cost of capital (WACC)
  - Canada, OT — Government borrowing rate
  - France — Government borrowing rate
  - Germany — Government borrowing rate
  - Ireland — Government borrowing rate
  - South Africa — Government borrowing rate
  - United Kingdom — Social discount rate
- Note: Almost all jurisdictions apply a common discount rate to project flows of net social benefits regardless of the source of financing. Only Australia applies different systematic risk-adjusted rates with PSC lower than the PPP rate.

### Literature and normative arguments on public versus private discount rates
- Arguments for a lower public discount rate:
  - Public sector can better absorb and spread risks among a greater number of individuals (Arrow and Lind, 1970; Fisher, 1973).
  - Public projects complementary to private investment should apply lower discount rates to partially undo distortions that discourage private investment (Ogura and Yohe, 1977).
  - Private firms cannot internalize all externalities and may lead to inefficient piecemeal decision-making (Flemming and Mayer, 1997).
- Arguments for equal or higher public discount rates:
  - Public sector's lower borrowing cost may reflect lack of default and taxing power rather than superior risk management; some argue the social discount rate should equal public and private discount rates (Bailey and Jensen, 1972; Brealey, Cooper and Habib, 1997; Diamond, 1967; others).
  - Recent research emphasizes accounting for market risk premium in public-sector discount rates to avoid downward bias that distorts incentives (Hirshleifer 1964; 1966; Lucas 2012).
- In incomplete markets, differential risk premia arise; Grout (2003) argued PPP payments to investors are more volatile and may justify higher PPP discount rates. Lind (1990) and Spackman (2004) suggested government borrowing rates as candidates for public discounting.

### Terminal value, predictability, and state-contingency — core contribution
- Financial valuation decomposes future net social benefits into:
  - Predictable but risky flows within a forecast period (t ∈ (0,n]); and
  - Unpredictable flows beyond the forecast period (t ∈ (n, ∞)), whose discounted sum at time tn is the terminal value, E(TV_i).
- Terminal value is state-contingent due to Knightian uncertainty beyond the forecast period; unpredictable contingencies (e.g., alternative climate scenarios) make true terminal value contingent on realized states.
- Contract incompleteness beyond the forecast period prevents credible ex-ante, binding arrangements to fully allocate state-contingent contingencies between government and private firms.
- Strategic interaction under contract incompleteness implies a systematically larger expected terminal value for publicly financed utilities than for privately financed ones, unless government reactions to contingencies are very inefficient because of economic or political failures (e.g., lack of innovation management, tax-and-subsidy distortions, corruption, lobbying).
- This state-contingent terminal-value channel provides a new argument in favor of using a lower discount rate to assess infrastructure projects that are publicly financed versus PPPs.

### Financial model and key equations
- Cycle of investment (three-stage process):
  1. At t = 0, I_i > 0 is invested with certainty by the government (i = g) or private firm (i = p).
  2. For t ∈ (0,n], predictable but risky net social benefits A_it are realized.
  3. For t ∈ (n,∞), Knightian uncertainty yields unpredictable flows; expected discounted terminal value at tn is E(TV_i).
- Decomposition of net social benefits at time t:
  - A_it = CS_it + PS_it  (equation (1))
  - PS_it = τ_it A_it  (equation (2)), where τ_it ∈ (0,1) is the share of net social benefit captured by producer surplus.
  - CS_it = (1 − τ_it) A_it  (equation (3))
- For time-invariant A_i and τ_i, net present value:
  - NPV_i = −I_i + ∫_0^n A_i e^{−r_i t} dt + E(TV_i) e^{−r_i n}  (equation (4))
- Condition for private financing to be socially optimal (equation (5)):
  - −I_p + A_p (1 − e^{−r_p n})/r_p + E(TV_p) e^{−r_p n} > −I_g + A_g (1 − e^{−r_g n})/r_g + E(TV_g) e^{−r_g n}
- If E(TV_p) = E(TV_g) and r_p = r_g = r, simplifies to (equation (6)):
  - I_g − I_p + (A_p − A_g) (1 − e^{−r n})/r > 0
  - Interpretation: I_g − I_p reflects private-sector higher investment efficiency; A_p − A_g captures private-sector higher operating efficiency.

### Quantitative importance of terminal value (stylized numerical examples and empirical illustration)
- Terminal value accounts for around 40-70 percent of the present value of the asset, depending on predictable flows, forecast-period duration, and discount rate.
- Stylized example:
  - Series of $100 annuities in perpetuity, discounted at r = 10 percent annually.
  - First ten years (n) account for 61 percent of present value; terminal value (cash flows from year 11 onwards) accounts for 39 percent.
  - For n = 5 and r = 7 percent, the terminal value represents 71 percent of the present value.
- Market illustration (Credit Suisse example):
  - Vinci valued at €77 a share, of which €26.07 (33.8 percent) came from discounted dividends in the next 15 years (2016-2030) and €50.96 (66.2 percent) from the discounted terminal value realized after 2030.

### Implications for VfM appraisal and policy
- When predictable forecast-period flows are independent of financing source, VfM comparisons often use a common discount rate; however, differing expected terminal values and contract incompleteness beyond the forecast period challenge that practice.
- If public financing yields systematically larger expected terminal values due to state-contingent adaptive responses, using the same discount rate for PPPs and public-sector comparators can bias VfM conclusions against public financing.
- Policymakers should consider:
  - The role of state-contingent terminal values in valuation and VfM analysis.
  - Whether and how discount rates should reflect the incompleteness of contracts and differential expected terminal values under public versus private financing.
  - The potential need to adjust discount rates or incorporate explicit modeling of state-contingent terminal values when comparing public and private provision options.

*Italic: Source — IMF Working Papers: "Public versus Private Cost of Capital with State-Contingent Terminal Value" (References and selected sections).*

### conclusions than the plain model.

### wpiea2023056-print-pdf - conclusions than the plain model.

### Terminal value and contract incompleteness
- When the project's lifespan extends beyond the forecast period, the terminal value becomes of key importance for project feasibility; the terminal value tends to account for a large share of the present value.
- Beyond the forecast period, flows of net social benefits are risky and unpredictable; hedging against such risks relies upon state-contingent investments that can be designed and implemented only after unpredictable events are realized (i.e., after the state of the world is observable).
- Such risks linked to unpredictable contingencies cannot be ex-ante insured nor ex-ante shared between the government and the private firm through an enforceable private financing contract; contracts beyond the forecast period are therefore incomplete.
- The model represents unpredictable risk beyond the forecast period with a Bernoulli distribution:
  - B: with probability 휋∈[0,1], the state is “bad,” requiring a state-contingent investment (revamping effort) to hedge an adverse event that squeezes net social benefits, i.e., 퐴푖푡 = 퐴푇푉 − [퐿 − ℎ(퐸)], for all 푡∈(푛,∞), where ℎ(0)=0, ℎ′>0, ℎ′′<0, and 푙푖푚퐸→∞ ℎ(퐸)=퐿>0;
  - G: with probability 1−휋, the state is “good,” so no state-contingent investment is needed, i.e., 퐴푖푡 = 퐴푇푉, for all 푡∈(푛,∞).
- For simplicity, the analysis abstracts from any long-term efficiency gap in private versus public operation; thus 퐴푇푉 is the same for public and private financing, and consistently 휏푖푡 = 휏 for any 푖∈{푝,푔}.
- 퐿 − ℎ(퐸) is the average loss in long-run net social benefits in state B and can be mitigated by a revamping effort 퐸≥0 implemented by the private firm or the government depending on financing structure.
- The cost of the revamping effort is assumed to fully accrue just after the forecast period (i.e., at time 푛); the state of the world (G or B) is ex post observable but not ex ante contractible, so the level of 퐸 depends on the financing framework.

### Public financing: enforcement and efficiency cases
- Under public financing the government can enforce the desired level of 퐸 after the state of the world is observed; this applies when implementation and operation are by a governmental agency or publicly owned firm.
- Two cases analyzed:
  - Efficient government (A.1):
    - If the government faces the same costs as a private firm in implementing revamping, then in state B the government maximizes discounted net social benefit including revamping cost:
      - 푊(퐸)=∫[퐴푇푉∞0 − (퐿 − ℎ(퐸))]푒−푟푡 푑푡 − 퐸 = 퐴푇푉 − (퐿 − ℎ(퐸)) / 푟 − 퐸; (7)
    - In state G no revamping is needed (퐸=0) and the government obtains the discounted net social benefit accordingly.
  - The analysis later considers the case in which the government is less efficient than private firms in implementing the revamping effort (Section 3.2.1) — details beyond the provided excerpt.

### Managerial equivalence when project life is finite
- If the lifespan of the project is close to the forecast period (e.g., a finite-life infrastructure project with no terminal value), the problem reduces to a managerial exercise as in equation (6): assign the project to the party that can achieve overall better investment and operating efficiency.
- If no investment and operation efficiency margins exist (i.e., 퐼푔 = 퐼푝 and 퐴푝 = 퐴푔), the value-for-money (VfM) from private (compared to public) financing is equal to zero.
- Example cited: the case of the Atlas Arteria road (Atlas Arteria's road to growth), referenced in the text.

*Source: wpiea2023056-print-pdf - conclusions than the plain model.*

### 31.0 in four grants flood mitigation for water and wastewater facilities, ca. 8.5 in two grants in wind mitigation for e

### wpiea2023056-print-pdf - 31.0 in four grants flood mitigation for water and wastewater facilities, ca. 8.5 in two grants in wind mitigation for e

### Theory and model setup (state-contingent terminal value)
- Terminal value expression (equation (8)): 푊 = ∫_0^∞ 퐴_푇푉 e^{−푟푡} dt − 퐸 = 퐴_푇푉 / 푟.
- Government optimization in bad state 퐵 yields first-order condition (equation (9)): ℎ′(퐸_푔) / 푟 = 1.
- Expected terminal value under government revamping (equation (10)):
  - 퐸(푇푉_푔) = 휋 푊(퐸_푔) + (1−휋) 푊 = 퐴_푇푉 / 푟 − 휋 (퐿 − ℎ(퐸_푔)) / 푟 + 휋 퐸_푔.

### Inefficient public revamping (additional marginal public cost 훿)
- Government objective with additional marginal cost (equation (11)): 
  - 푊̂(퐸) = 퐴_푇푉 − (퐿 − ℎ(퐸)) / 푟 − 퐸(1+훿), where 훿 ≥ 0.
- First-order condition with inefficient public revamping (equation (12)): ℎ′(퐸̂_푔) / 푟 = 1 + 훿.
- Expected terminal value with inefficient public revamping (equation (13)):
  - 퐸(푇푉̂_푔) = 휋 푊̂(퐸̂_푔) + (1−휋) 푊 = 퐴_푇푉 / 푟 − 휋 [ (퐿 − ℎ(퐸̂_푔)) / 푟 + 퐸̂_푔(1+훿) ].
- Proposition 1: Under public financing, the expected terminal value of public projects is decreasing in the marginal cost of public revamping.
  - Proof note: By the Envelope Theorem, ∂/∂훿 푊̂(퐸̂_푔) = −퐸̂_푔, hence ∂/∂훿 퐸(푇푉̂_푔) = −휋 퐸̂_푔 < 0.

### Private financing: objectives and producer surplus
- Private firm discounted producer surplus (equation (14)):
  - 푃푆(퐸)+푆 = ∫_0^∞ 휏[퐴_푇푉 − (퐿 − ℎ(퐸))] e^{−푟푡} dt − 퐸 + 푆 = 휏 [퐴_푇푉 − (퐿 − ℎ(퐸)) / 푟 ] − 퐸 + 푆.
- In good state (no revamping, 퐸 = 푆 = 0) producer surplus (equation (15)):
  - 푃푆 + 푆 = ∫_0^∞ 휏 퐴_푇푉 e^{−푟푡} dt − 퐸 + 푆 = 휏 퐴_푇푉 / 푟.
- Two polar cases for observability of 퐸 are considered:
  1. 퐸 is observed by both parties but ex post unverifiable.
  2. 퐸 is ex-post verifiable.

### Unverifiable state-contingent investment (B.1)
- With unverifiable 퐸, government cannot subsidize (푆 = 0). Private firm chooses 퐸 to maximize (14).
- Private first-order condition (equation (16)): 휏 ℎ′(퐸_푝) / 푟 = 1.
- Expected terminal value under private unverifiable effort (equation (17)):
  - 퐸(푇푉_푝) = 휋 푊(퐸_푝) + (1−휋) 푊 = 퐴_푇푉 / 푟 − 휋 (퐿 − ℎ(퐸_푝)) / 푟 + 휋 퐸_푝.
- Proposition 2: When state-contingent revamping efforts beyond the forecast period are ex-post unverifiable, the expected terminal value of public projects under public financing is always larger than under private financing.
  - Proof note: Given 휏 < 1 and comparing (9) and (16) implies 퐸_푔 > 퐸_푝, hence 퐸(푇푉_푔) > 퐸(푇푉_푝).

### Verifiable state-contingent investment (B.2)
- Parties can negotiate ex post over 퐸 and subsidy 푆, subject to participation constraints:
  - Government participation (equation (18)): 푊(퐸) − 휆 푆 ≥ 푊(퐸_푝).
  - Private firm participation (equation (19)): 푃푆(퐸) + 푆 ≥ 푃푆(퐸_푝).
- Proposition 3: Minimum subsidy 푆 required is strictly positive whenever 퐸 ≠ 퐸_푝.
- Nash bargaining setup (equation (20)) maximizes:
  - (푊(퐸_푝푏) − 휆 푆 − 푊(퐸_푝))^{훾} (푃푆(퐸_푝푏) + 푆 − 푃푆(퐸_푝))^{1−훾}.
- First-order condition from bargaining yields (equation (21)):
  - ℎ′(퐸_푝푏) / 푟 = 2 / (1 + 휏).
- Expected terminal value under negotiated private outcome (equation (22)):
  - 퐸(푇푉_푝푏) = 휋 (푊(퐸_푝푏) − 휆 푆_푏) + (1−휋) 푊 = 퐴_푇푉 / 푟 − 휋 [ (퐿 − ℎ(퐸_푝푏)) / 푟 + 퐸_푝푏 + 휆 푆_푏 ].
- Proposition 4: When state-contingent revamping efforts are ex-post verifiable, the expected terminal value under public financing is always larger than under private financing.
  - Proof note: Given 휏 < 1 and comparing (9) and (21) implies 퐸_푔 > 퐸_푝푏, hence 퐸(푇푉_푔) > 퐸(푇푉_푝푏).
- Proposition 5: The expected terminal value of privately financed public projects is larger (or at least equal) under ex-post verifiability than under ex-post unverifiability.
  - Proof note: Government participation constraint implies 푊(퐸_푝푏) − 휆 푆_푏 ≥ 푊(퐸_푝), hence 퐸(푇푉_푝푏) ≥ 퐸(푇푉_푝).

### Public versus private financing when public revamping is inefficient (B.3)
- Comparison between government with marginal revamping cost 훿 and private unverifiable outcome:
  - From (12) and (16): 퐸̂_푔 > 퐸_푝 if and only if 훿 < (1 − 휏) / 휏.
- Proposition 6: When state-contingent revamping efforts beyond the forecast period are ex-post unverifiable, the expected terminal value of public projects under public financing is larger (smaller) than under private financing if and only if 훿 < 훿 < (1 − 휏) / 휏 (훿 > 훿).
  - Analytical condition (equation (23)) for public advantage: ℎ(퐸̂_푔) / 푟 − 퐸̂_푔 (1+훿) > ℎ(퐸_푝) / 푟 − 퐸_푝.
  - Second-order Taylor approximation yields threshold 훿 ≡ (1 − 휏) / 휏 − (1 − 휏) / 휏 (퐸_푝 / 퐸̂_푔) + h′′(퐸_푝) / (2푟) (퐸̂_푔 − 퐸_푝)^2 / 퐸̂_푔 < (1 − 휏) / 휏.
- Interpretation:
  - If government revamping is sufficiently inefficient (훿 > 훿), public financing yields a lower expected terminal value than private financing, potentially implying a larger discount rate for public financing.
  - When private financing includes ex post verifiability and bargaining, the subsidy size and government bargaining power 훾 affect thresholds: smaller 훾 leads to larger subsidies, raising 퐸_푝푏 and lowering the 훿 threshold for public advantage.

### Financial assessment and policy implications (C and Conclusion)
- Revised feasibility inequality (equation (24)):
  - 퐼_푔 − 퐼_푝 + (퐴_푝 − 퐴_푔) (1 − e^{−푟푛}) / 푟 > [ 퐸(푇푉_푔) − 퐸(푇푉_푝) ] e^{−푟푛} > 0.
- Key policy implication:
  - If public revamping is not too inefficient and 퐸(푇푉_푔) > 퐸(푇푉_푝), the discount rate for publicly financed projects should be lower than for privately financed projects, i.e., 푟_푔 < 푟_푝.
  - The efficiency gains in investment (퐼_푔 − 퐼_푝) and operations (퐴_푝 − 퐴_푔) from private provision must be sufficiently large to offset the lower expected terminal value under private financing if one is to prefer private provision (PPPs/BOTs).
- Relevance to climate change and long-horizon projects:
  - The analysis emphasizes treatment of tail risks and ex-post endogeneity of future economic flows due to revamping efforts.
  - The authors note compatibility with arguments for decreasing or time-varying discount rates in cost-benefit analysis for far-distant uncertain outcomes.

*International Monetary Fund — IMF Working Papers: Public versus Private Cost of Capital with State-Contingent Terminal Value*

### References

### References

### Journal articles and book chapters
- Andrews, R. and T. Entwistle, 2010, “Does Cross-Sectoral Partnership Deliver? An Empirical Exploration of Public Service Effectiveness, Efficiency, and Equity,” Journal of Public Administration Research & Theory, Vol. 20(3), pp. 679–701.  
- Arrow, K., M. Cropper, C. Gollier, B. Groom, G. Heal, R. Newell, W. Nordhaus, R. Pindyck, W. Pizer, P. Portney, T. Sterner, R. S. J. Tol, and M. Weitzman, 2013, “Determining benefits and costs for future generations,” Science, Vol. 341(6144), pp. 349–350.  
- Arrow, K. J. and R. C. Lind, 1970, “Uncertainty and the evaluation of public investment decisions,” American Economic Review, Vol. 60(3), pp. 364–378.  
- Bailey, M. J. and M. C. Jensen, 1972, “Risk and the discount rate for public investment,” In M. C. Jensen, Ed.), Studies in the Theory of Capital Markets. Praeger Publishers.  
- Baumol, W. J., 1968, “On the social rate of discount,” American Economic Review, Vol. 58(4), pp. 788–802.  
- Binmore, K., A. Rubinstein, and A. Wolinsky, 1986, “The Nash bargaining solution in economic modelling,” RAND Journal of Economics, Vol. 17(2), pp. 176–188.  
- Bozeman, B., 1993, “A theory of government ‘red tape’,” Journal of Public Administration Research & Theory, Vol. 3(3), pp. 273–303.  
- Brealey, R. A., I. A. Cooper, and M. A. Habib, 1997, “Investment appraisal in the public sector,” Oxford Review of Economic Policy, Vol. 13(4), pp. 12–28.  
- Bruggink, T. H., 1982, “Public versus regulated private enterprise in the municipal water industry: a comparison of operating costs,” Quarterly Review of Economics & Business, Vol. 22, pp. 111–125.  
- Chopra, A., 2014, Innovative State: How New Technologies Can Transform Government. New York, NY: Atlantic Monthly Press.  
- Coulson, A., 2008, “Value for money in PFI proposals: A commentary on the UK Treasury Guidelines for Public Sector Comparators,” Public Administration, Vol. 86(2), pp. 483–498.  
- Demsetz, H., 1968, “Why regulate utilities?” Journal of Law & Economics, Vol. 11(1), pp. 55–65.  
- Diamond, P. A., 1967, “The role of a stock market in a general equilibrium model with technological uncertainty,” American Economic Review, Vol. 57(4), pp. 759–776.  
- Drèze, J. H., 1974, “Discount rates and public investment: A post-scriptum,” Economica, Vol. 41(161), pp. 52–61.  
- Engel, E., R. Fischer, and A. Galetovic, 2013, “The Basic Public Finance of Public-Private Partnerships,” Journal of the European Economic Association, Vol. 11, pp. 83–111.  
- Fisher, A. C., 1973, “Environmental externalities and the Arrow-Lind public investment theorem,” American Economic Review, Vol. 63(4), pp. 722–725.  
- Flemming, J. and C. Mayer, 1997, “The assessment: public-sector investment,” Oxford Review of Economic Policy, Vol. 13(4), pp. 1–11.  
- Gordon, M. J., 1959, “Dividends, earnings and stock prices,” Review of Economics & Statistics, Vol. 41(2), pp. 99–105.  
- Gordon, M. J. and E. Shapiro, 1956, “Capital equipment analysis: The required rate of profit,” Management Science, Vol. 3(1), pp. 102–110.  
- Green, A., T. Koller, and R. Palter, 2015, “A hidden roadblock in public-infrastructure projects,” Corporate Finance Practice, McKinsey & Company.  
- Grout, P. A., 2003, “Public and private sector discount rates in public-private partnerships,” Economic Journal, Vol. 113(486), pp. C62–C68.  
- Grout, P. A., 2005, “Value-for-Money measurement in public–private partnerships,” EIB Papers, Vol. 10(2), pp. 33–56.  
- Hart, O., A. Shleifer, and R. W. Vishny, 1997, “The proper scope of government: Theory and an application to prisons,” Quarterly Journal of Economics, Vol. 112(4), pp. 1127–1161.  
- Hirshleifer, J., 1964, “Efficient allocation of capital in an uncertain world,” American Economic Review, Vol. 54(3), pp. 77–85.  
- Hirshleifer, J., 1966, “Investment decision under uncertainty: Applications of the state-preference approach,” Quarterly Journal of Economics, Vol. 80(2), pp. 252–277.  
- Hodge, G. A. and C. Greve, 2007, “Public-private partnerships: An international performance review,” Public Administration Review, Vol. 67(3), pp. 545–558.  
- Iossa, E. and D. Martimort, 2012, “Risk allocation and the costs and benefits of public–private partnerships,” RAND Journal of Economics, Vol. 43(3), pp. 442–474.  
- Jeong, S.-Y., S.-G. Moon, and S. Bae, 2017, “A comparative study of the effects of institutional arrangements in municipal solid waste services in Korea,” Journal of Comparative Policy Analysis, Vol. 19(1), pp. 17–39.  
- Karp, L., 2005, “Global warming and hyperbolic discounting,” Journal of Public Economics, Vol. 89(2-3), pp. 261–282.  
- Kay, J., 1993, Efficiency and Private Capital in the Provision of Infrastructure. Paris: OECD.  
- Klein, M., 1997, “The risk premium for evaluating public projects,” Oxford Review of Economic Policy, Vol. 13(4), pp. 29–42.  
- Kurland, N. B. and T. D. Egan, 1999, “Public v. private perceptions of formalization, outcomes, and justice,” Journal of Public Administration Research & Theory, Vol. 9(3), pp. 437–458.  
- Lan, Z. and H. G. Rainey, 1992, “Goals, rules, and effectiveness in public, private, and hybrid organizations: More evidence on frequent assertions about differences” Journal of Public Administration Research & Theory, Vol. 2(1), pp. 5–28.  
- Le Lannier, A. and S. Porcher, 2014, “Efficiency in the public and private French water utilities: prospects for benchmarking,” Applied Economics, Vol. 46(5), pp. 556–572.  
- Lind, R. C., 1982, Discounting for Time and Risk in Energy Policy, Resources for the Future. Baltimore: Johns Hopkins University Press.  
- Lind, R. C., 1990, “Reassessing the government’s discount rate policy in light of new theory and data in a world economy with a high degree of capital mobility,” Journal of Environmental Economics & Management, Vol. 18(2), pp. S8–S28.  
- Lucas, D., 2012, “Valuation of government policies and projects,” Annual Review of Financial Economics, Vol. 4(1), pp. 39–58.  
- Lucas, D. and M. Phaup, 2008, “Reforming credit reform,” Public Budgeting & Finance, Vol. 28(4), pp. 90–110.  
- Meier, K. J. and L. J. O’Toole, 2011, “Comparing public and private management: Theoretical expectations,” Journal of Public Administration Research & Theory, Vol. 21(suppl 3), pp. i283–i299.  
- Ogura, S. and G. Yohe, 1977, “The complementarity of public and private capital and the optimal rate of return to government investment,” Quarterly Journal of Economics, Vol. 91(4), pp. 651–662.  
- Peltzman, S., 1976, “Toward a more general theory of regulation,” Journal of Law & Economics, Vol. 19(2), pp. 211–240.  
- Rainey, H. G. and B. Bozeman, 2000, “Comparing public and private organizations: Empirical research and the power of the a priori,” Journal of Public Administration Research & Theory, Vol. 10(2), pp. 447–470.  
- Samuelson, P. A., 1964, “Principles of efficiency: Discussion,” American Economic Review, Vol. 54(3), pp. 93–96.  
- Sharpe, W. F., 1964, “Capital asset prices: A theory of market equilibrium under conditions of risk,” Journal of Finance, Vol. 19(3), pp. 425–442.  
- Solow, R. M., 1965, Capital Theory and the Rate of Return. Chicago: Rand McNally.  
- Spackman, M., 2004, “Time discounting and of the cost of capital in government,” Fiscal Studies, Vol. 25(4), pp. 467–518.  
- Van Damme, E., 1986, “The Nash bargaining solution is optimal,” Journal of Economic Theory, Vol. 38(1), pp. 78–100.  
- Vickrey, W., 1964, “Principles of efficiency: Discussion,” American Economic Review, Vol. 54(3), pp. 88–92.  
- Weitzman, M. L., 1998, “Why the far-distant future should be discounted at its lowest possible rate,” Journal of Environmental Economics & Management, Vol. 36(3), pp. 201–208.  

### Reports, technical papers, and institutional publications
- Blanc-Brude, F. and M. Hasan, 2015, “The valuation of privately-held infrastructure equity investments,” Technical report, EDHEC-Risk Institute.  
- Boardman, A. E. and M. Hellowell, 2017, “Comparison and analysis of specialist PPP units’ methodologies for conducting value for money analysis,” Journal of Comparative Policy Analysis, Vol. 19(3), pp. 191–206.  
- CBO, 2003, “Evaluating and accounting for federal investment in corporate stocks and other private securities,” CBO Study, The Congress of the United States, Congressional Budget Office, Washington, DC.  
- Green, A., T. Koller, and R. Palter, 2015, “A hidden roadblock in public-infrastructure projects,” Corporate Finance Practice, McKinsey & Company.  
- Multi-Hazard Mitigation Council, 2019, Natural Hazard Mitigation Saves: 2019 Report. Washington, DC: National Institute of Building Sciences.  

*Public versus Private Cost of Capital with State-Contingent Terminal Value — Working Paper No. WP/2023/056*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023056-print-pdf.pdf_