Macroeconomic Dimensions of Public-Private Partnerships

Edward F Buffie; Michele Andreolli; Grace B Li; Luis-Felipe Zanna

doi:10.5089/9781484307700.001.A001

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Macroeconomic Dimensions of Public-Private Partnerships

Author: Mr. Edward F Buffie, Michele Andreolli, Ms. Grace B Li, and Luis-Felipe Zanna

Contributor Notes

Author’s E-Mail Address: ebuffie@indiana.edu, mandreolli@imf.org, bli2@imf.org, fzanna@imf.org

Publication Date:: 24 Mar 2016

eISBN:: 9781484307700

Language:: English

Keywords:: WP; return gap; market price; full-employment case; return in the investment program; infrastructure assets

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The voluminous literature comparing public-private partnerships (P3s) and own-investment (OI) by the public sector is dominated by contributions from microeconomic theory. This paper gives macroeconomics a voice in the debate by investigating the repercussions of P3 vs. OI in a dynamic general equilibrium model featuring private capital accumulation and involuntary unemployment with efficiency wages. Typically P3s cost more but produce higher-quality infrastructure and boast a better on-time completion record than OI; consequently, they are comparatively more effective in reducing underinvestment in private capital, underinvestment in infrastructure, unemployment and poverty. 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 the externalities operate singly or in combination and on whether P3 enjoys an advantage in speed of construction.

Abstract

1 Introduction

Public-private partnerships (P3s) to build and operate infrastructure assets are increasingly common in less-developed countries (LDCs). They are also highly controversial. Microeconomic theory and the evidence accumulated in case studies warn that P3s may be much more expensive than traditional procurement. The list of extra expenses is long and usually starts with the higher cost of private finance. P3s assign construction risk to the private partner to the tight complementarity between asset construction and quality of services.¹ But in a world with incomplete markets, the private sector cannot spread risk as widely as the public sector (Vickrey, 1964; Arrow, 1966; Arrow and Lind, 1970). The difference in the capacity to bear risk raises the return paid to the private partner above the interest rate on government debt.² In some LDCs, the return also includes compensation for country risk–for the perception that the government is an “unreliable business partner” (Estache et al., 2015).

There are numerous other costs to doing business with the private sector. At the procurement stage, the administrative costs of writing and tendering bids for complicated long-term contracts are non-trivial. High bid costs for private firms, in turn, limit ex ante competition (i.e., competition for the market). This, combined with the difficulty of designing auctions that prevent collusive behavior (Klemperer, 1999), limited information on firm costs, the urgent need to resolve infrastructure bottlenecks, and the illusory appeal of P3s as an off-budget operation, make it all too easy for inexperienced governments to overpay the private partner (IMF, 2007). Moreover, the operations phase is sure to bring additional costs. The complexity of the contracts, the impossibility of enumerating all contingencies in partnerships that last 20 − 30 years, cumbersome legal systems that impede quick, efficient resolution of disputes, and the strong incentive for opportunistic behavior that arise after each party has made relationship-specific investments lead inevitably to frequent, costly renegotiations of the original contract. Last but not least, even if the government bargains exceptionally well and minimizes bid, tendering, and renegotiation costs, it cannot avoid the extra expense of monitoring compliance of the private partner; trust without verification is inadvisable in P3s and nuclear disarmament agreements.³

To date no study has compared the total cost of P3s and own investment by the public sector. We know from various fragmentary sources, however, that the extra costs associated with P3s add up to something substantial.

Spackman (2002) and HM Treasury (2000) estimate that non-diversifiable risk pushes the interest rate on senior debt in P3s 2 − 3 percentage points above the interest rate on government debt; junior debt and equity finance are presumably even more expensive.
Measured as a percentage of capital costs, transaction costs average 10% for European governments (Dudkin and Valila, 2005) and 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).
In Portugal, government officials acknowledge that a dearth of competitive bids and overly generous payments to the private partner have been persistent problems (Monteiro, 2005; Sarmento and Renneboog, 2014).
While renegotiation costs have eluded quantification, the frequency of renegotiations is dismaying: the 35 Portuguese P3s studied by Sarmento and Renneboog (2014) gave birth to 254 contract renegotiations between 1995 and 2012; Chile’s widely praised P3 has also suffered from constant renegotiation costs (IMF, 2004).

Certainly more hard data is needed to justify firm conclusions, especially in the case of LDCs. But the evidence that exists paints a disturbing picture. The verdict in Dudkin and Valila (2005) that “high transaction costs are perhaps the worst, and least studied, drawbacks of PPPs” is entirely fair.

Costs are only one-half of the equation. The other half contains everything the private partner brings to the table: superior technical expertise, greater implementation capacity, and fewer agency problems. These advantages translate into shorter construction periods–important in countries plagued by acute bottlenecks in transport, power, telecommunications, irrigation, etc.–and better, more productive infrastructure. The critical issue is whether the gains in speed and efficiency compensate for the higher cost. In the language favored by government bureaucrats, do P3s offer enough “value for money”? More precisely, do P3s offer better value for money than own investment by the public sector?

Compared to the microeconomic literature with its rich, expansive insights into how asymmetric information, moral hazard, uncertainty, risk allocation, imperfect monitoring, and incomplete contracts affect the pros and cons of P3s, the macroeconomic literature is almost mute. Its total contribution to the debate consists of the trivial recommendation that future payments to the private partner be explicitly incorporated into the government’s fiscal accounts in order to highlight the point that P3s defer costs but create a greater overall (i.e., present value) fiscal burden than traditional procurement. Echoing the latter concern, the IMF (2007) has called for full integration of P3s into debt sustainability analysis.

This paper gives macroeconomics a greater voice in the conversation about how P3s compare with own investment (OI) by the public sector. We investigate the macroeconomic repercussions of P3 vs. OI in a dynamic general equilibrium macromodel featuring private capital accumulation and a flexible specification of the labor market that nests the cases of full employment with flexible wages and involuntary unemployment with efficiency wages.⁴ The unifying theme in our welfare results is that general equilibrium effects shift the comparison in favor of P3. The direct return in the investment program equals the return on infrastructure minus either the return paid to the private partner (a foreign firm) or the interest rate paid on external debt. In partial equilibrium, the direct return picks the winner. In general equilibrium, the welfare calculation is more complicated. Because P3 builds better, higher quality infrastructure than OI, it crowds in private investment more and increases labor demand more. This helps P3 in the welfare comparison when (i) the private time preference rate exceeds the social time preference rate (implying that private investment is suboptimal at the initial equilibrium); (ii) real wage rigidity prevents full employment; or (iii) wage income carries a larger weight in the social welfare function than income of the representative agent. We show specifically that 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 the externalities operate singly or in combination and on whether P3 enjoys an advantage in speed of construction.⁵ The ranking of direct returns is not therefore a reliable proxy for the welfare ranking: a P3 that pays a direct return of 2% may be preferable to OI that pays a direct return of 10%.

Macroeconomic effects also strongly condition the impact on the fiscal budget and the prospects for debt sustainability. P3s are considerably more expensive than OI. But their stronger positive effects on the supply of infrastructure services, private capital accumulation, and employment also generate more revenue from taxes and user fees. This can reduce the need for supporting fiscal adjustment to the point where debt sustainability is easier to achieve for P3 despite its higher cost. In some cases, P3 requires less fiscal support than OI at every point in time.

The rest of the paper is organized into six sections. Sections 2 and 3 present the model and solve for the transition path and the steady-state equilibrium. Following this, Sections 4-6 compare the effects of P3 and OI on welfare and fiscal sustainability. Section 7 concludes with a discussion of priorities for future research.

2 The Model

The model has three components: a government that builds new infrastructure on its own or by partnering with a foreign firm; the domestic private sector, run by a representative agent who pays efficiency wages and accumulates capital; and the national budget constraint, which requires that capital flows finance the current account deficit each period.

2.1 The Choice: P3 or OI?

Cobb-Douglas production functions convert infrastructure z, capital k, and labor L into output q:⁶

\begin{matrix} q = {[a (z - z_{o}) + z_{o}]}^{ψ} k^{α} {(eL)}^{1 - α}, & (1) \end{matrix}

with a > 1 for P3 and

\begin{matrix} q = z^{ψ} k^{α} {(eL)}^{1 - α}, & (1') \end{matrix}

for OI. Infrastructure enters the production function as a shift factor, while effective labor input depends on the amount of effort e that workers expend. The service flow from the infrastructure project is a(z − z_o), with a > 1 measuring the gain in efficiency when z is built by the private partner instead of the government. z_o is the initial stock level of infrastructure

For simplicity, we assume the P3 contract is structured like an annuity. The foreign partner receives (r_p + δ)(z − z_o) in perpetuity, where δ(z − z_o) covers depreciation costs and r_p is the return earned on the investment.⁷ The government levies a tax h on consumption and a user fee χ on infrastructure services. In the main variant of the model, lump-sum taxes T adjust to satisfy the government budget constraint:

\begin{matrix} T = r_{p} (z - z_{o}) + δz - \underset{\overset{⏟}{service flow}}{χ [a (z - z_{o}) + z_{o}] - hc,} & \begin{matrix} (2) \end{matrix} \end{matrix}

for P3.

The alternative to P3 is OI. In this case, the investment is financed by borrowing in the world capital market at the interest rate r. Analogous to the annuity-type P3 contract, the government rolls over the debt forever. The counterpart of (2) is⁸

\begin{matrix} T = r (z - z_{o}) + (δ - χ) z - hc, & (2') \end{matrix}

for OI.

Finally, P3 and OI differ in the speed of construction. The private partner not only builds a better road, it builds the road faster—a big selling point in countries suffering from large infrastructure deficits (Monteiro, 2005; Sarmento, 2010). To capture this, we specify different paths for investment i_z. The law of motion for the stock of infrastructure is

\begin{matrix} ż = i_{z} - δz, & (3) \end{matrix}

where ż represents the time derivative of z. In P3, i_z overshoots its new steady-state level ${\bar{ι}}_{z}$ and z increases faster than under OI.⁹ To capture this:¹⁰

\begin{matrix} {di}_{z} / dt = s ({\bar{ι}}_{z} - i_{z}), & (4) \end{matrix}

with s > 0 and i_z,o+ > ${\bar{ι}}_{z}$ where i_z,o is the post-jump value of i_z. For OI, i_z is constant at its new steady-state level ${\bar{ι}}_{z}$ :

\begin{matrix} i_{z} = {\bar{ι}}_{z} = δ \bar{z} . & (4') \end{matrix}

2.2 The Private Sector

The representative agent chooses consumption and effort to maximize

\begin{matrix} \begin{matrix} U = \int_{o}^{\infty} {\frac{c^{1 - 1 / τ}}{1 - 1 / τ} - {[e - b_{0} - b_{1} \ln ω - b_{2 u}]}^{2}} & e^{- pt} dt \end{matrix} & (5) \end{matrix}

subject to

\begin{matrix} \dot{k} = q - T - (1 + h) c - χ [a (z - z_{o}) + z_{o}] - δk, & (6) \end{matrix}

with a ≥ 1 and where b₀, b₁, and b₂ are constants; labor supply is fixed at unity; u = 1 − L is the unemployment rate; and w, ρ, and τ denote the real wage, the pure time preference rate, and the intertemporal elasticity of substitution. Following standard practice in the efficiency wage literature (e.g., Collard and de la Croix, 2000; Danthine and Kurmann, 2004, 2010), we assume that gratitude for having a job and the net utility loss from effort depends on how well the job pays and on whether jobs are easy or hard to come by (as determined by u).

On an optimal path,

\begin{matrix} \dot{c} = τc (qk - ρ - δ) & (7) \end{matrix}

and

\begin{matrix} e = b_{0} + b_{1} \ln w + b_{2 u} . & (8) \end{matrix}

Equation (7) is the standard Euler equation for consumption, while equation (8) says that employees work harder when they are paid more or feel threatened by a high unemployment rate.

Firms recognize the connection between labor productivity (q_L) and the real wage (w). Hence they optimize over L and w. The profit-maximizing choices satisfy, as usual,

\begin{matrix} qL = w & (9) \end{matrix}

and

\begin{matrix} \frac{\partial e / \partial w}{e / w} = 1. & (10) \end{matrix}

Equation (8) and the Solow condition in (10) imply, conveniently, that effort is constant in general equilibrium:¹¹

\begin{matrix} e = b_{1} . & (11) \end{matrix}

We set b₁ = 1 (without loss of generality), so the wage curve defined by (8) and (11) reads

\begin{matrix} \ln w= 1 - b_{0} - b_{2} \underset{\overset{⏟}{u}}{(1 - L)} . & (12) \end{matrix}

Unlike in Phillips Curve models, there is no natural rate of unemployment, just a curve relating the equilibrium wage to the unemployment rate.

2.3 The National Budget Constraint

The model is closed by the national budget constraint. Substituting for T in equation (6) produces¹²

\begin{matrix} \dot{k} = q - r_{p} (z - z_{o}) - c - δ (z + k) & (13) \end{matrix}

for P3 and

\begin{matrix} \dot{k} = q - r (z - z_{o}) - c - δ (z + k) & (13') \end{matrix}

for OI.

2.4 The Case for Efficiency Wages

Efficiency wages are rarely seen in development macromodels. This needs to change. Over the past twenty years, empirical studies have amassed abundant, compelling evidence that efficiency wages operate throughout the non-agricultural sector in LDCs. Estimates of the impact of unemployment on real wages confirm the existence of wage curves in the formal and informal sectors in Argentina, Turkey, Colombia, Uruguay, Chile, S. Africa, Cote d’Ivoire, Mexico, and a host of other developing countries (Blanchflower and Oswald, 2005). There is also powerful, if indirect evidence supportive of efficiency wages in the stylized facts documented in microeconomic studies of LDC labor markets. Across the development spectrum, wage and employment data exhibit the same patterns: (i) firm-size wage premiums that start at very small establishment size (5+ employees) and are much larger than in developed countries; (ii) persistent, remarkably stable inter-industry wage differentials; (iii) high correlation of industry wage premiums across occupations; (iv) large wage premia for formal vs. informal sector employment and for informal non-agricultural employment vs. agricultural employment; and (v) large cyclical flows into and out of unemployment in both the formal sector and the informal sector. At present, only efficiency wage models can explain all of these stylized facts. We do not have the space here to survey the literature in greater depth or to discuss myrial estimation issues. References and capsule summaries of the results for 47 studies are available, however, at http://mypage.iu.edu/~ebuffie/.

None of this denies the potential relevance of models that postulate full employment or alternative theories of wage rigidity. There is tremendous diversity in the structure of labor markets across and within LDCs. The right model depends therefore on the country and the type of infrastructure project under investigation. In South Africa, where the national unemployment rate (really) is 25%, trade unions are strong, and minimum wage laws are enforced in all regions, all industries, and at firms of all size, equation (12) with b₂ very small is probably a good fit.¹³ In the most common case, high unemployment rates are confined to urban areas (see Table 1). The appropriate model then depends on the location of the project. If the project increases the supply of power to a major urban center, b₂ in equation (12) should take a value between 0.5 and 1.5 (the range in empirical estimates);¹⁴ if it expands the irrigation network in an area populated by smallholder farms, full employment should be enforced by increasing b₂ to 100, 000.¹⁵ The wage curve is simple but flexible; it can represent any degree of real wage rigidity.

Table 1:

Unemployment rates in LDCs

Sources: International Labor Office, World Economic Outlook, World Bank, CIA World Factbook, and National Statistical Offices (Labor Force Surveys, Censuses, Poverty Surveys, AIDS Surveys, and Household Surveys). Unless otherwise noted, figures for the national unemployment rate are based on two sources that agree to within one percentage point.

^{^a}Unemployment rate for New Providence Island (where the capital city is located).

^{^b}Unemployment rate for the capital city.

^{^c}Unemployment rates in the capital city and other urban centers, respectively.

^{^d}Disagreement between two or more sources.

^{^e}Only one source (Labor Force Survey for Cote d’Ivoire and Namibia; Population Census for Guyana).

^{^f}Data cross-checked for a different year.

^{^g}Unemployment rate changed more than 10% across two years in the Labor Force Survey and the Census.

Table 1:

Unemployment rates in LDCs

Country	Year	National	Rural	Urban
Algeria	2013	9.8	8.1	10.6
Bahamas	2013	15.8	−	15.8^a
Bangladesh	2010	4.5	4.0	6.5
Belize	2013	11.7^d	11.8	11.5
Botswana	2010	17.9	−	9.8^b
Brazil	2009	8.4^d	3.4	9.3
Cameroon	2010	3.8	1.4	8.1
Colombia	2014	9.1	5.7	10.0
Costa Rica	2014Q2	8.6^d	8.8	8.5
Cyprus	2014	16.1	17.1	15.7
Cote d’Ivoire	2015	6.9^e	3.0	13.4, 7.7^c
Dominican Rep.	2013	15.0^d	14.2	15.4
Egypt	2013	13.2	10.7	16.5
Gabon	2010	20.4	19.0	21.0
Gambia	2012	29.8	31.1	28.4
Guyana	2002	−	−	11.8^b
India	2014	4.9^d	5.5	4.7
Indonesia	2012	6.1^f	4.7	7.7
Iran	2013	10.4	7.0	11.8
Jamaica	2013	15.2	−	−
Jordon	2012	12.2	14.2	11.8
Kyrgyzstan	2012	8.4^f	7.9	9.5
Malawi	2013	6.6	6.0	11.5
Malaysia	2014	2.9	3.2	3.1
Mauritania	2012	10.1	−	−
Mauritius	2014	7.8	−	−
Mexico	2014	4.8	2.8	5.3
Morocco	2014	9.7	4.2	15.0
Mozambique	2012	22.6^d	−	−
Namibia	2014	28.1^e	30.2	26.2
Nigeria	2015Q3	9.9^d,f	9.0	12.1
Philippines	2014	6.8	−	9.3^b
Senegal	2015	13.4^g	10.1	17.2
South Africa	2012	24.7	28.0	24.3
Tajikistan	2009	11.5	9.6	16.8
Thailand	2015	.9	1.0	.9
Tunisia	2012Q4	16.7^f	−	20.4^b
Turkey	2013	9.7	6.1	11.5
Uruguay	2014	6.6	4.7	6.7, 7.0^c
Zambia	2012	7.8	3.3	14.2

^{^a}Unemployment rate for New Providence Island (where the capital city is located).

^{^b}Unemployment rate for the capital city.

^{^c}Unemployment rates in the capital city and other urban centers, respectively.

^{^d}Disagreement between two or more sources.

^{^e}Only one source (Labor Force Survey for Cote d’Ivoire and Namibia; Population Census for Guyana).

^{^f}Data cross-checked for a different year.

^{^g}Unemployment rate changed more than 10% across two years in the Labor Force Survey and the Census.

Table 1:

Unemployment rates in LDCs

Country	Year	National	Rural	Urban
Algeria	2013	9.8	8.1	10.6
Bahamas	2013	15.8	−	15.8^a
Bangladesh	2010	4.5	4.0	6.5
Belize	2013	11.7^d	11.8	11.5
Botswana	2010	17.9	−	9.8^b
Brazil	2009	8.4^d	3.4	9.3
Cameroon	2010	3.8	1.4	8.1
Colombia	2014	9.1	5.7	10.0
Costa Rica	2014Q2	8.6^d	8.8	8.5
Cyprus	2014	16.1	17.1	15.7
Cote d’Ivoire	2015	6.9^e	3.0	13.4, 7.7^c
Dominican Rep.	2013	15.0^d	14.2	15.4
Egypt	2013	13.2	10.7	16.5
Gabon	2010	20.4	19.0	21.0
Gambia	2012	29.8	31.1	28.4
Guyana	2002	−	−	11.8^b
India	2014	4.9^d	5.5	4.7
Indonesia	2012	6.1^f	4.7	7.7
Iran	2013	10.4	7.0	11.8
Jamaica	2013	15.2	−	−
Jordon	2012	12.2	14.2	11.8
Kyrgyzstan	2012	8.4^f	7.9	9.5
Malawi	2013	6.6	6.0	11.5
Malaysia	2014	2.9	3.2	3.1
Mauritania	2012	10.1	−	−
Mauritius	2014	7.8	−	−
Mexico	2014	4.8	2.8	5.3
Morocco	2014	9.7	4.2	15.0
Mozambique	2012	22.6^d	−	−
Namibia	2014	28.1^e	30.2	26.2
Nigeria	2015Q3	9.9^d,f	9.0	12.1
Philippines	2014	6.8	−	9.3^b
Senegal	2015	13.4^g	10.1	17.2
South Africa	2012	24.7	28.0	24.3
Tajikistan	2009	11.5	9.6	16.8
Thailand	2015	.9	1.0	.9
Tunisia	2012Q4	16.7^f	−	20.4^b
Turkey	2013	9.7	6.1	11.5
Uruguay	2014	6.6	4.7	6.7, 7.0^c
Zambia	2012	7.8	3.3	14.2

^{^a}Unemployment rate for New Providence Island (where the capital city is located).

^{^b}Unemployment rate for the capital city.

^{^c}Unemployment rates in the capital city and other urban centers, respectively.

^{^d}Disagreement between two or more sources.

^{^e}Only one source (Labor Force Survey for Cote d’Ivoire and Namibia; Population Census for Guyana).

^{^f}Data cross-checked for a different year.

^{^g}Unemployment rate changed more than 10% across two years in the Labor Force Survey and the Census.

3 Inputs

Before deriving any welfare results, we need to locate the steady-state equilibrium, delineate the transition paths for P3 and OI, and define the social welfare function. These tasks occupy the next three sections. To obtain exact closed-form solutions, we treat the increase in infrastructure investment as a small, differential change.

3.1 The Long-Run Solution

Suppose policy makers opt for P3. At the new steady state,

\begin{matrix} qk = ρ + δ & (14) \end{matrix}

and

\begin{matrix} c = q - r_{p} (z - z_{o}) - δ (z + k) . & (15) \end{matrix}

Equations (9), (12), (14), and (15) can be solved for k, L, w, and c as a function of z. Straight forward algebra delivers

\begin{matrix} dc = pdk + (R_{p} - r_{p}) dz + wdL, & (16) \end{matrix}

\begin{matrix} dw = b_{2} wdL, & (17) \end{matrix}

\begin{matrix} dL = \frac{(R_{p} + δ) (1 - α)}{w f_{1}} dz + \frac{(ρ + δ) (1 - α)}{{wf}_{1}} dk, & (18) \end{matrix}

and

\begin{matrix} dk = \frac{R_{p} + δ}{ρ + δ} (\frac{α}{1 - α}) dz + \frac{k}{L} dL, & (19) \end{matrix}

where

f_{1} \equiv α + b_{2} L = α + b_{2} (1 - u),

and R_p = q_z − δ is the return on infrastructure (net of depreciation).

Figure 1 depicts the solutions for the capital stock and employment in equations (18) and (19). Since k and L are gross complements in production, the kk and LL schedules slope upward: growth in employment increases the equilibrium capital stock and vice versa.¹⁶

An increase in the supply of infrastructure acts like a positive productivity shock, shifting kk vertically upward and LL horizontally to the right. In the long run,

\begin{matrix} dk = \frac{R_{p} + δ}{ρ + δ} (\frac{α}{1 - α}) \underset{\overset{⏟}{Multiplier}}{\frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)}} dz, & (20) \end{matrix}

and

\begin{matrix} dL = \frac{R_{p} + δ}{w f_{1}} (1 - α) \underset{\overset{⏟}{Multiplier}}{[1 + (\frac{α}{1 - α}) \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)}]} dz . & (21) \end{matrix}

The corresponding solutions for OI substitute the return R for R_p. Thus

\begin{matrix} {\begin{matrix} dk | \end{matrix}}_{P 3} - {dk |}_{OI} = \frac{R_{p} - R}{ρ + δ} (\frac{α}{1 - α}) \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)} dz, & (22) \end{matrix}

and

\begin{matrix} {\begin{matrix} dL | \end{matrix}}_{P 3} - {dL |}_{OI} = \frac{R_{p} - R}{w f_{1}} (1 - α) [1 + (\frac{α}{1 - α}) \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)}] dz . & (23) \end{matrix}

These solutions underpin many of the results in the paper. Two points merit emphasis. First, the positive direct effects on capital accumulation and employment depend on the return on infrastructure. Given the consensus view that R_p exceeds R, this strengthens the case for P3. Second, feedback effects between changes in the capital stock and employment greatly magnify the direct effects. Empirical estimates of b₂ cluster between 0.5 and 1.5. For values of b₂ in this range, the feedback multiplier is on the order of 1.5 − 3 for k and 2 − 4 for L. Point D in Figure 1 is far north of point B and far east of point C.

Return now to equation (16). Because the multiplier effects on capital accumulation and employment are so large, partial and general equilibrium analysis often reach different conclusions about whether P3 or OI pays the highest return. Substituting the solutions for k and L into (16) gives

\begin{matrix} {\begin{matrix} \frac{dc}{dc} | \end{matrix}}_{P 3} = R_{p} - r_{p} + (R_{p} + δ) F, & (24) \end{matrix}

where

F \equiv [ρ + \frac{(ρ + δ) (1 - α)}{α + b_{2} (1 - u)}] [\frac{α}{(ρ + δ) (1 - α)}] \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)} + \frac{1 - α}{α + b_{2} (1 - u)} > 0.

The solution for OI replaces R_p and r_p with R and r:

\begin{matrix} {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{OI} = R - r + (R + δ) F . & (25) \end{matrix}

Therefore

\begin{matrix} {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{P 3} - {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{OI} = \underset{\overset{⏟}{Direct Rerurn Gap}}{R_{p} - r_{p} -} (R - r) + (R_{p} - R) F . & (26) \end{matrix}

In the case of full employment (b₂ →∞), F becomes $\frac{ρα}{(ρ + δ) (1 - α)}$ and consequently

\begin{matrix} \begin{matrix} {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{P 3} - {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{OI} = \end{matrix} \underset{\overset{⏟}{Direct Return Gap}}{R_{p} - r_{p} - (R - r)} + (R_{p} - R) \frac{ρα}{(ρ + δ) (1 - α)} . & (27) \end{matrix}

Figure 2 plots the solutions in equations (26) and (27). The Direct Return Gap (DRG) is R_p − r_p − (R − r)¹⁷. The Total Return Gap (TRG) mirrors the full solution and includes the term (R_p − R)F. Everything depends on the size of this term relative to the DRG. To quantify the two competing effects, we computed solutions for

α = .30- .50, α = .05, r = .06, ρ = .10, R = .16 − .25, r_p = .15, and b₂ = .5 - 1.5, ∞.

The income share of capital, the depreciation rate, and the external borrowing rate take ordinary values.¹⁸ For the private time preference rate, we chose the highish value 10% based on the estimates in Isham and Kaufmann (1999) and Dalgaard and Hansen (2005) of the return to private capital in LDCs. Estimates of the return on infrastructure in LDCs are all over the map, but the weight of the evidence in both micro and macro studies points to high average returns of 15 − 30%—see Buffie et al. (2012). Accordingly, we set the return at 16% for OI and 16 − 25% for P3. Finally, r_p is a pricey 15% because it reflects all of the costs associated with P3. In addition to the return paid to the private partner–many points above r for reasons discussed in the introduction—it includes annuitized payments for the very high transaction costs incurred in P3s.¹⁹ Unfortunately, the paucity of data on transaction costs and actual payments to the private partner means there is a lot of uncertainty about the right value for r_b. Our choice of 15% is a crude, semi-educated guess. It is also, we admit, a debating tactic. Since R_p tops out at 25%, the DRG ranges from zero to -9%. The case for P3 rests entirely therefore with general equilibrium effects; by assumption, OI enjoys a big advantage in the comparison of direct returns.

Table 2 confirms that general equilibrium effects are potent, especially when increases in employment reinforce the gains from capital accumulation. In the column at the far right, where R_p = .25 and both direct returns equal 10%, the consumption return for P3 is 16 − 25% above the return for OI with full employment and 31 − 39% higher when involuntary unemployment exists. Alternatively, consider the breakeven points in Figure 2. The schedule for full employment breaks into the first quadrant at ${\bar{R}}_{p} = .222$ . This is neither exciting nor insignificant. But when infrastructure investment reduces unemployment, ${\bar{R}}_{p}$ drops to .179 − .196. Strikingly, in five of nine runs where P3 pays a direct return of only 4%, it increases real income more than OI. Although these are not welfare results, they foreshadow the conclusion that P3 may dominate OI even when the DRG is very large.

Table 2:

Percentage difference in the increase in steady-state consumption for P3 vs. OI.¹

^¹R = .16, r = .06, r_p = .15, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Solution with full employment.

Table 2:

Percentage difference in the increase in steady-state consumption for P3 vs. OI.¹

α = .50
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−8.8	−3.6	1.7	7.0	12.3	17.5	22.8	28.1	33.4	38.6
1		−14.3	−8.7	−3.1	2.5	8.1	13.7	19.3	24.9	30.5	36.0
1.5		−18.0	−12.2	−6.4	−.6	5.2	11.0	16.8	22.7	28.5	34.3
FE²		−37.5	−30.6	−23.6	−23.6	−9.7	−2.8	4.2	11.1	18.1	25.0

α = .40
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−10.4	−5.0	.4	5.7	11.1	16.5	21.8	27.2	32.6	37.9
1		−17.0	−11.2	−5.5	.3	6.0	11.8	17.5	23.3	29.0	34.8
1.5		−21.5	−15.5	−9.5	−3.5	2.5	8.5	14.6	20.6	26.6	32.6
FE		−46.6	−39.1	−31.6	−24.1	−16.7	−9.2	−1.7	5.7	13.2	20.7

α = .30
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−11.8	−6.4	−.9	4.5	10.0	15.4	20.9	26.3	31.8	37.2
1		−19.6	−13.7	−7.8	−1.9	4.0	9.9	15.8	21.7	27.6	33.5
1.5		−25.0	−18.8	−12.6	−6.3	−.1	6.1	12.3	18.5	24..7	31.0
FE		−56.2	−48.2	−40.2	−32.1	−24.1	−.16.1	−8.0	0	8.0	16.1

^¹R = .16, r = .06, r_p = .15, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Solution with full employment.

Table 2:

Percentage difference in the increase in steady-state consumption for P3 vs. OI.¹

α = .50
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−8.8	−3.6	1.7	7.0	12.3	17.5	22.8	28.1	33.4	38.6
1		−14.3	−8.7	−3.1	2.5	8.1	13.7	19.3	24.9	30.5	36.0
1.5		−18.0	−12.2	−6.4	−.6	5.2	11.0	16.8	22.7	28.5	34.3
FE²		−37.5	−30.6	−23.6	−23.6	−9.7	−2.8	4.2	11.1	18.1	25.0

α = .40
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−10.4	−5.0	.4	5.7	11.1	16.5	21.8	27.2	32.6	37.9
1		−17.0	−11.2	−5.5	.3	6.0	11.8	17.5	23.3	29.0	34.8
1.5		−21.5	−15.5	−9.5	−3.5	2.5	8.5	14.6	20.6	26.6	32.6
FE		−46.6	−39.1	−31.6	−24.1	−16.7	−9.2	−1.7	5.7	13.2	20.7

α = .30
	R_p
b₂		.16	.17	.18	.19	.20	.21	.22	.23	.24	.25
.5		−11.8	−6.4	−.9	4.5	10.0	15.4	20.9	26.3	31.8	37.2
1		−19.6	−13.7	−7.8	−1.9	4.0	9.9	15.8	21.7	27.6	33.5
1.5		−25.0	−18.8	−12.6	−6.3	−.1	6.1	12.3	18.5	24..7	31.0
FE		−56.2	−48.2	−40.2	−32.1	−24.1	−.16.1	−8.0	0	8.0	16.1

^¹R = .16, r = .06, r_p = .15, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Solution with full employment.

3.2 The Transition Path

The transition path is governed by equations (3), (4), (7), and (13), along with equation (18), which links the path of employment to the paths of the capital stock and the stock of infrastructure. Linearizing (7) and (13) yields

\begin{matrix} \dot{c} = τc (q_{kk} dk + q_{kzL} dL) & (28) \end{matrix}

and

\begin{matrix} \dot{k} = pdk + (R_{p} - r_{p}) dz + wdL - dc . & (29) \end{matrix}

Substitute for dL from (18) and note that the second derivatives of the production function satisfy

q_{kk} = \frac{α (α - 1)}{k^{2}} = \frac{α - 1}{k} (ρ + δ), q_{kz} = \frac{αψaq}{kz} = \frac{α}{k} (R_{p} + δ), and q_{kL} = - q_{kk} = - q_{kk} \frac{k}{L} .

After collecting terms,

\begin{matrix} \dot{c} = τ (c / k) (ρ / δ) (α - 1) f_{2} (k - \bar{k}) + τ (c / k) α (R_{p} + δ) f_{3} (z - \bar{z}) & (30) \end{matrix}

and

\begin{matrix} \dot{k} = f_{4} (k - \bar{k}) + f_{5} (z - \bar{z}) - (c - \bar{c}), & (31) \end{matrix}

where

\begin{matrix} f_{2} \equiv 1 - \frac{α}{f_{1}} > 0, & f_{3} \equiv 1 + \frac{1 - α}{f_{1}} > 0, \end{matrix}

f_{4} \equiv ρ + \frac{(ρ + δ) (1 - α)}{f_{1}} > 0, and f_{5} \equiv R_{p} - r_{p} + \frac{(R_{p} + δ) (1 - α)}{f_{1}} > 0.

Until Section 4.3, we ignore differences in implementation and the speed of construction. Infrastructure investment is constant at $\bar{ι_{z}}$ and

\begin{matrix} \dot{z} = - δ (z - \bar{z}) & (32) \end{matrix}

for both P3 and OI.

Equations (30)-(32) form a self-contained system of three differential equations in c, k, and z:

\begin{matrix} [{\dot{k}}_{\dot{z}}^{\dot{c}}] = [{- 1}_{0}^{0} {f_{4}}_{0}^{τ (c / k) (ρ + δ) (α - 1) f_{2}} {f_{5}}_{- δ}^{τ (c / k) α (R_{p} + δ) f_{3}}] [{k - \bar{k}}_{z - \bar{z}}^{c - \bar{c}}] . & (33) \end{matrix}

The stationary equilibrium is a saddle point with two state variables, k and z, and two negative eigenvalues:

λ_{1} = - δ and λ_{2} = \frac{f_{4} - \sqrt{f_{4}^{2} + 4 τ (1 - α) (ρ + δ) (c / k) f_{2}}}{2} .

3.3 The Welfare Criterion

We measure social welfare as²⁰

\begin{matrix} SW = \int_{o}^{\infty} \frac{c^{1 - 1 / τ}}{1 - 1 / τ} e^{- ρ * t} dt, with ρ^{*} \leq ρ . & (34) \end{matrix}

SW allows the benevolent social planner to set ρ* below the private time preference rate ρ. Should (s)he?

The position of policy makers is clear. In both developed and less developed countries, the social time preference rate used to calculate the cost-benefit ratio for public sector projects is usually much lower than the private rate. HM Treasury (2003) recommends, for example, ρ* = 2 − 3.5%.

Theory cannot tell us whether 3% is a sensible number for ρ*, the view of HM Treasury notwithstanding. It does, however, provide cogent arguments for ρ* < ρ. In Sen’s (1967) isolation paradox, private saving is suboptimal because individuals would be willing to enter into a social contract that required everyone to save more. Feldstein (1964) and Baumol (1965) reach the same conclusion more quickly by appealing to the notion that economic development is partly a public good; if the premise is granted, then the social time preference rate “must be administratively determined as a matter of public policy [because] the market cannot express the ‘collective’ demand for investment to benefit the future” (Feldstein, 1964, pp. 362, 365).

4 Welfare Comparisons

All of the machinery is now in place to rank the welfare gains for P3 vs. OI. We start with the simplest case.

4.1 Full Employment

We derive the solutions for c and SW in Appendix A. The welfare gain, measured in units of consumption, is

\begin{matrix} {\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ} (\bar{z} - z_{o})} | \end{matrix}}_{P 3} = \int_{o}^{\infty} (c - c_{o}) e^{- ρ * t} dt = (\frac{R_{p} - r_{p}}{ρ + δ}) \frac{δ}{ρ^{*}} (1 + H_{p}), & (35) \end{matrix}

for P3, where ρ* ≤ ρ and

H_{p} \equiv \frac{ρ - ρ^{*}}{ρ^{*} + δ} Λ + \frac{(R_{p} + δ) α (ρ - ρ^{*})}{(ρ^{*} - λ_{2}) (1 - α) (ρ + δ)} Γ,

Λ \equiv 1 + \frac{ρ^{*} (λ_{2} + δ)}{J (ρ^{*} - λ_{2})} with Λ > 0,

Γ \equiv \frac{τ (c / k) (1 - α) (λ_{2} + δ)}{J} - λ_{2} \frac{ρ^{*} + δ}{ρ^{*}} with Γ > 0,

and

J \equiv τ (c / k) (1 - α) - δ .

Similarly,

\begin{matrix} {\begin{matrix} \frac{SW - SW}{c^{- 1 / τ} (z^{*} - z_{o})} | \end{matrix}}_{OI} = (\frac{R - r}{ρ + δ}) \frac{δ}{ρ^{*}} (1 + H), & (35') \end{matrix}

with

H \equiv \frac{ρ - ρ^{*}}{ρ^{*} + δ} Λ + \frac{(R + δ) α (ρ - ρ^{*})}{(ρ^{*} - λ_{2}) (1 - α) (ρ + δ)} Γ

for OI. It can be shown with some effort that the terms involving (ρ − ρ*) in H_p and H are positive (see Appendix A).

The welfare solutions look forbidding, but they have the expected form and agree with intuition. When ρ* = ρ, then H_p = H = 0. In this case, the impact on private capital accumulation is irrelevant. The welfare gain is simply the capitalized value of the stream of returns generated by increases in the supply of infrastructure:²¹

\begin{matrix} {\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ}} | \end{matrix}}_{P 3, ρ^{*} = ρ} = \int_{o}^{\infty} (R_{p} - r_{p}) (z - z_{o}) e^{- pt} dt = (\frac{R_{p} - r_{p}}{ρ + δ}) \frac{δ}{ρ} (\bar{z} - z_{o}) & (36) \end{matrix}

and

\begin{matrix} {\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ}} | \end{matrix}}_{OI, ρ^{*} = ρ} = \int_{o}^{\infty} (R - r) (z - z_{o}) e^{- pt} dt = (\frac{R - r}{ρ + δ}) \frac{δ}{ρ} (\bar{z} - z_{o}) & (36') \end{matrix}

Discounting at ρ* < ρ changes the welfare calculation in two ways. First, the lower discount rate increases the present value welfare gain from increases in the stock of infrastructure. Second, since the private capital stock is below its socially optimal level, crowding in private investment generates an additional welfare gain. The impact of this gain on the welfare comparison depends on R_p vs. R as the increase in the service flow determines the stimulus to private investment. Thus P3 may increase welfare more than OI despite a lower direct return. From (35) and (35′),

{SW |}_{P 3} > {SW |}_{OI}

if and only if

\begin{matrix} \frac{(R_{p} - R) α (ρ - ρ *)}{\underset{\overset{⏟}{LHS}}{(ρ * - λ_{2}) (1 - α) (ρ - δ) G}} Γ > & \underset{\overset{⏟}{Direct Return Gap (DRG)}}{R - r - (R_{p} - r_{p})}, & \begin{matrix} (37) \end{matrix} \end{matrix}

where

G \equiv \frac{δ}{ρ^{*} (ρ + δ)} (1 + \frac{ρ - ρ^{*}}{ρ^{*} + δ} Λ) with G > 0 if ρ^{*} < ρ .

The term on the left side (LHS) of (37) is small but significant. When ρ* = .02 − .06, it lowers the breakeven value of R_p in Table 3 from .25 to .228 − .239. P3 wins close races where the DRG is 1 − 2 points.

Table 3:

Ratio of welfare gain in P3 vs. OI in the model with full employment.¹

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

Table 3:

Ratio of welfare gain in P3 vs. OI in the model with full employment.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.47	.39	.30	.20	.10
.17	.55	.47	.39	.30	.20
.18	.63	.56	.48	.39	.30
.19	.70	.64	.57	.49	.40
.20	.78	.72	.65	.58	.50
.21	.86	.81	.74	.67	.60
.22	.94	.89	.83	.77	.70
.23	1.02	.97	.92	.86	.80
.24	1.10	1.06	1.01	.96	.90
.25	1.18	1.14	1.10	1.05	1.0
$\bar{R_{p}}$	.228	.233	.239	.245	.250

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

Table 3:

Ratio of welfare gain in P3 vs. OI in the model with full employment.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.47	.39	.30	.20	.10
.17	.55	.47	.39	.30	.20
.18	.63	.56	.48	.39	.30
.19	.70	.64	.57	.49	.40
.20	.78	.72	.65	.58	.50
.21	.86	.81	.74	.67	.60
.22	.94	.89	.83	.77	.70
.23	1.02	.97	.92	.86	.80
.24	1.10	1.06	1.01	.96	.90
.25	1.18	1.14	1.10	1.05	1.0
$\bar{R_{p}}$	.228	.233	.239	.245	.250

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

4.2 Unemployment

There is no need to repeat the algebraic derivations in the previous section. Obviously, the term multiplying R_p − R in equation (37) is much larger and the case for P3 much stronger when infrastructure investment reduces unemployment. P3 produces an additional welfare gain of 27 − 34% in the runs where direct returns are equal (R_p = .25 in Table 4), and the breakeven value of R_p decreases to .17 − .22 with ρ* between .04 and .1 (Figure 3). Policy makers can rest assured that P3 is the right choice when the DRG is three points or less. In economies with highly rigid wages (b₂ = .1 − .5), it may be the right choice even when the direct return is 7 − 8 points lower than for OI.

Table 4:

Ratio of welfare gain in P3 vs. OI in the model with open employment and b₂ = 1.¹

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

Table 4:

Ratio of welfare gain in P3 vs. OI in the model with open employment and b₂ = 1.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.81	.77	.74	.70	.66
.17	.86	.83	.80	.77	.73
.18	.92	.90	.86	.83	.80
.19	.98	.96	.93	.90	.87
.20	1.04	1.02	.99	.96	.93
.21	1.10	1.08	1.05	1.03	1.0
.22	1.16	1.14	1.12	1.09	1.07
.23	1.22	1.20	1.18	1.16	1.13
.24	1.28	1.26	1.24	1.22	1.20
.25	1.34	1.32	1.30	1.29	1.27
$\bar{R_{p}}$	.193	.197	.202	.206	.210

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

Table 4:

Ratio of welfare gain in P3 vs. OI in the model with open employment and b₂ = 1.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.81	.77	.74	.70	.66
.17	.86	.83	.80	.77	.73
.18	.92	.90	.86	.83	.80
.19	.98	.96	.93	.90	.87
.20	1.04	1.02	.99	.96	.93
.21	1.10	1.08	1.05	1.03	1.0
.22	1.16	1.14	1.12	1.09	1.07
.23	1.22	1.20	1.18	1.16	1.13
.24	1.28	1.26	1.24	1.22	1.20
.25	1.34	1.32	1.30	1.29	1.27
$\bar{R_{p}}$	.193	.197	.202	.206	.210

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The direct return in the P3 equals the direct return on own investment when R_P = .25.

Table 5:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when there is full employment and P3 builds infrastructure faster than OI.¹

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

Table 5:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when there is full employment and P3 builds infrastructure faster than OI.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.53	.53	.52	.51	.51
.17	.62	.64	.66	.67	.69
.18	.71	.75	.79	.83	.88
.19	.80	.86	.92	.99	1.07
.20	.90	.97	1.06	1.15	1.26
.21	.99	1.08	1.19	1.31	1.45
.22	1.08	1.19	1.32	1.47	1.63
.23	1.17	1.30	1.46	1.63	1.82
.24	1.26	1.41	1.59	1.79	2.01
.25	1.35	1.52	1.72	1.95	2.20
$\bar{R_{p}}$	.212	.203	.196	.190	.186
DRG²	.038	.047	.054	.06	.064

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

Table 5:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when there is full employment and P3 builds infrastructure faster than OI.¹

	ρ*
R_p	.02	.04	.06	.08	.10
.16	.53	.53	.52	.51	.51
.17	.62	.64	.66	.67	.69
.18	.71	.75	.79	.83	.88
.19	.80	.86	.92	.99	1.07
.20	.90	.97	1.06	1.15	1.26
.21	.99	1.08	1.19	1.31	1.45
.22	1.08	1.19	1.32	1.47	1.63
.23	1.17	1.30	1.46	1.63	1.82
.24	1.26	1.41	1.59	1.79	2.01
.25	1.35	1.52	1.72	1.95	2.20
$\bar{R_{p}}$	.212	.203	.196	.190	.186
DRG²	.038	.047	.054	.06	.064

^¹R = .16, r = .06, r_p = .15, α = .40, τ = .50, δ = .05, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

Table 6:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when b₂ = 1 in the model with open employment and P3 builds infrastructure faster than OI.¹

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

Table 6:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when b₂ = 1 in the model with open employment and P3 builds infrastructure faster than OI.¹

		ρ*
R_p		.02	.04	.06	.08	.10
.16		.91	1	1.10	1.20	1.30
.17		.98	1.08	1.20	1.32	1.43
.18		1.05	1.17	1.29	1.43	1.56
.19		1.12	1.25	1.39	1.54	1.69
.20		1.18	1.33	1.49	1.66	1.82
.21		1.25	1.41	1.59	1.77	1.95
.22		1.32	1.49	1.68	1.88	2.08
.23		1.39	1.57	1.78	2.0	2.21
.24		1.46	1.66	1.88	2.11	2.34
.25		1.53	1.74	1.97	2.22	2.47

$R_{p} = \bar{R_{P}}$ (DRG)²
	ρ*
b₂		.02	.04	.06	.08	.10
.5		.160 (.09)	.145 (.105)	.133 (.117)	.126 (.124)	.119 (.131)
1		.173 (.077)	.160 (.09)	.150 (.10)	.142 (.108)	.137 (.113)
1.5		.180 (.07)	.168 (.082)	.159 (.091)	.152 (.098)	.147 (.103)
2		.186 (.064)	.174 (.076)	.165 (.085)	.158 (.092)	.153 (.097)
3		.192 (.058)	.181 (.069)	.173 (.077)	.166 (.084)	.161 (.089)
4		.196 (.054)	.185 (.065)	.177 (.073)	.171 (.079)	.166 (.084)

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

Table 6:

Ratio of welfare gain in P3 vs. OI and the breakeven value of R_P when b₂ = 1 in the model with open employment and P3 builds infrastructure faster than OI.¹

		ρ*
R_p		.02	.04	.06	.08	.10
.16		.91	1	1.10	1.20	1.30
.17		.98	1.08	1.20	1.32	1.43
.18		1.05	1.17	1.29	1.43	1.56
.19		1.12	1.25	1.39	1.54	1.69
.20		1.18	1.33	1.49	1.66	1.82
.21		1.25	1.41	1.59	1.77	1.95
.22		1.32	1.49	1.68	1.88	2.08
.23		1.39	1.57	1.78	2.0	2.21
.24		1.46	1.66	1.88	2.11	2.34
.25		1.53	1.74	1.97	2.22	2.47

$R_{p} = \bar{R_{P}}$ (DRG)²
	ρ*
b₂		.02	.04	.06	.08	.10
.5		.160 (.09)	.145 (.105)	.133 (.117)	.126 (.124)	.119 (.131)
1		.173 (.077)	.160 (.09)	.150 (.10)	.142 (.108)	.137 (.113)
1.5		.180 (.07)	.168 (.082)	.159 (.091)	.152 (.098)	.147 (.103)
2		.186 (.064)	.174 (.076)	.165 (.085)	.158 (.092)	.153 (.097)
3		.192 (.058)	.181 (.069)	.173 (.077)	.166 (.084)	.161 (.089)
4		.196 (.054)	.185 (.065)	.177 (.073)	.171 (.079)	.166 (.084)

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, ρ = .10, s=.20, and n=1.6. The direct return in the P3 equals the direct return on own investment when R_P = .25.

^²Direct return gap for R_p = $\bar{R_{p}}$ .

4.3 The Importance of Speed

When P3 builds infrastructure faster than OI, i_z enters the core dynamic system as a state variable. The new system is

\begin{matrix} [\begin{matrix} \dot{c} \\ \dot{k} \\ \dot{z} \\ {di}_{z} / dt \end{matrix}] = [\begin{matrix} 0 & τ (c / k) (ρ + δ) (α - 1) f_{2} & τ (c / k) α (R_{p} + δ) f_{3} & 0 \\ - 1 & f_{4} & f_{5} & 0 \\ 0 & 0 & - δ & 1 \\ 0 & 0 & 0 & - s \end{matrix}] [\begin{matrix} c - \bar{c} \\ k - \bar{k} \\ z - \bar{z} \\ i_{z} - \bar{ι_{z}} \end{matrix}] . & (38) \end{matrix}

Let $n \equiv [i_{z, o +} - i_{z, o}] / ({\bar{ι}}_{z} - i_{z, o})$ , where i_z,o is the post-jump value of i_z at t = o. n and s jointly determine the path of i_z and thus the path of z:²²

z - z_{o} = [1 - e^{δt} + (n - 1) δ \frac{e^{- st} - e^{- δt}}{δ - s}] (\bar{z} - z_{o}) .

When the government invests on its own, n = 1 and z traverses 50% of ground to its new steady-state level in t₁ = − ln(.50)/δ years. In the comparison run for P3, we set s = .20 and chose n so that z reaches the 50% benchmark in 25% less time.²³ Figure 4 shows the two paths for z. In an infinite horizon model where the choice between P3 and OI is repeated in a sequence of projects, the gap between the paths widens for 15 years before starting to shrink.

Infrastructure deficits are large in low-income countries (LICs), middle-income countries (MICs) and most emerging economies (EMEs). Consequently, speed matters. It matters a lot in the full-employment economy, the case least favorable to P3. It matters even more in countries with unemployed workers because shorter construction periods for infrastructure increase labor demand directly (q_Lz > 0)and via faster growth in the private capital stock. Time is money, as they say in the business world.

The numbers in Tables 5 and 6 are surprising, nevertheless. Compare these tables with Tables 3 and 4. When direct returns are equal, the welfare-gain-advantage of P3 soars from 0 − 34% in Tables 3 and 4 to 35 − 147% in Tables 5 and 6. The breakeven value of R_p decreases to .186 − .212 in the full-employment case and to .119 − .180 in the runs with efficiency wages and b₂ = .5 − 1.5. It should be emphasized in connection with this last result that the distinction between perfect and extremely high wage flexibility is important. A value of 1.5 for b₂ is at the upper end of empirical estimates. Yet, in runs where b₂ = 1.5 − 4, ρ* ≥ .04, and the DRG is 7 − 8 percentage points (R_p = .17 − .18), P3 beats OI 62 − 88% of the time. This simplifies life for some policy makers. Arguably, there is no need for a public sector comparator or complex general equilibrium analysis in countries where involuntary unemployment exists: if P3 passes the benefit-cost test by a small margin, based on its direct return only, then it is virtually certain to dominate OI.²⁴

5 The Fiscal Challenge

Lump-sum taxes adjust to pay the private partner in P3 and foreign bond holders in OI. Since r_p >> r, it seems obvious that the fiscal challenge is greater for P3 than for OI. This is not necessarily the case, however. P3s are much more expensive than OI, but they also increase the supply of infrastructure services and real output more than OI, generating more revenue at existing tax rates. Let Y denote net national income–e.g., Y = q − r_p(z − z_o) − δ(k + z) for P3—and define $\bar{χ} \equiv χ / (R + δ)$ , the ratio of the user fee to the monetary value of infrastructure services produced by OI. Solving the government budget constraints (2) and (2^′) for $ν \equiv T / Y$ then gives

\begin{matrix} {\begin{matrix} (ν - ν_{o}) | \end{matrix}}_{P 3} = [r_{p} + δ - (R_{p} + δ) Ω - ν (R_{p} - r_{p})] \frac{z - z_{o}}{Y} - h \frac{c - c_{o}}{Y} - νΦ \frac{k - k_{o}}{Y} & (39) \end{matrix}

and

\begin{matrix} {\begin{matrix} (ν - ν_{o}) | \end{matrix}}_{OI} = [r + δ - (R + δ) Ω - ν (R - r)] \frac{z - z_{o}}{Y} - \frac{h}{Y} (c - c_{o}) - νΦ \frac{k - k_{o}}{Y}, & (39') \end{matrix}

where

Ω \equiv \bar{χ} + ν (1 - α) / f_{1} > 0 and Φ \equiv ρ + (ρ + δ) (1 - α) / f_{1} > 0.

When the speed of construction is the same for P3 and OI,

\begin{matrix} {\begin{matrix} (ν - ν_{o}) | \end{matrix}}_{P 3} - {(ν - ν_{o}) |}_{OI} = (r_{p} - r) (1 + ν) \frac{z - z_{o}}{Y} - (R_{p} - R) (Ω + ν) \frac{z - z_{o}}{Y} & (40) \end{matrix} - h ({\frac{c - c_{o}}{Y} |}_{P 3} - {\frac{c - c_{o}}{Y} |}_{OI}) - νΦ ({\frac{k - k_{o}}{Y} |}_{P 3} - {\frac{k - k_{o}}{Y} |}_{OI}) .

The existing literature worries, rightly, that the first term is positive and large. But the difference in endogenous revenue—the sum of the last three terms—carries the opposite sign and is also potentially large.

We track $CFS \equiv \frac{(ν - ν_{o})}{({\bar{ι}}_{z} - i_{z, o}) / Y}$ , the ratio of the increase in υ to to the increase in infrastructure investment measured as a percentage of National Net Income (NNI). Translating into plain English, CFS is the “coefficient of fiscal stress.” CFS(t₁) = .75 means, for example, that the government has to collect an additional .75% of NNI in lump-sum taxes at time t₁ to finance a permanent increase in infrastructure investment equal to 1% of NNI.²⁵

Figure 5 shows the path of υ for a variety of cases in which r_p = .10 − .15 and P3 ties or wins the welfare comparison.²⁶ The diversity of outcomes is extreme.²⁷ Sometimes the P3 line is always above the OI line, sometimes it is always below; sometimes the two lines cross; sometimes the lines stay in the first quadrant, sometimes they end up in the fourth quadrant. Clearly, there is no general presumption that P3 creates more fiscal stress than OI. Much depends on the productivity differential (R_p − R), the structure of taxes, and the labor market:

When R_p = .20, r_p = .10, (i.e., the DRG = 0) and user fees cover only depreciation costs $(\bar{χ} = .25)$ , the gap between the P3 and OI lines is positive and large for several decades (Figures 5a and 5e). But the gap shrinks considerably when R_p = .25, and in the model with unemployment the sign changes to the advantage of P3 at year 13 (Figures 5b and 5f).
Because P3 builds better, more productive infrastructure than OI, modest user fees that cover O+M costs plus 30−50% of interest/return payments make a big difference to the fiscal outcome. In the runs where R_p = .25, r_p = .10, and $\bar{χ} = .40$ , P3 pays for itself immediately or within a few years (Figures 5d and 5g) and delivers a much larger fiscal dividend than OI. The comparison still favors OI when R_p = .20 (Figures 5c and 5h), but the choice in Figure 5h is between good and better—fiscal adjustment is easy and stress-free for both P3 and OI.

The import of these results is that policy makers should not pre-judge how fiscal concerns affect the case for P3 vs. OI. P3 wins the welfare comparison decisively when R_p = .25 or when R_p = .20 and involuntary unemployment exists. In half of the runs, it is also the choice that minimizes fiscal stress, provided a little help is available from user fees. Indeed, Figure 5 suggests that welfare-maximizing P3s may ease fiscal constraints and improve the prospects for debt sustainability. The Ministry of Finance and the Ministry of Public Works should agree as often as not.

6 Extensions

The benchmark model assumes that lump-sum taxes adjust in the government budget constraint and that social welfare depends solely on consumption of the representative agent. We relax both assumptions below. The first proves innocuous, but the second is potentially important.

6.1 Distortionary Taxes Adjust

Distortionary taxes may bear part or all of the cost of financing public investment. To investigate this scenario, treat c as a CES aggregate of two consumer goods and assume that only c₂ is subject to the tax h. The exact consumer price index is

\begin{matrix} P = {[κ {(P_{1}^{*})}^{1 - ε} + (1 - κ) P_{2}^{1 - ε}]}^{1 / (1 - ε)}, & (41) \end{matrix}

where

\begin{matrix} P_{2} = P_{2}^{*} (1 + h), & (42) \end{matrix}

$P_{i}^{*}$ is the world market price of good i; к is a CES distribution parameter; and ∊ is the elasticity of substitution between c₁ and c₂. We choose units so that $P_{i}^{*} = 1$ and $P_{2}^{*} = 1 / (1 + h_{o})$ . Thus P₁ = P₂ = P = 1 at the initial equilibrium.

The public, private, and national budget constraints are

\begin{matrix} h P_{2}^{*} c_{2} = r_{p} (z - z_{o}) + δz - χ [a (z - z_{o}) + z_{o}] - T, & (43) \end{matrix}

\begin{matrix} \dot{k} = q - T - Pc - δk - χ [a (z - z_{o}) + z_{o},] & (44) \end{matrix}

and

\begin{matrix} \dot{k} = q - r_{p} (z - z_{o}) - Pc - δ (z + k) + h P_{2}^{*} c_{2} . & (45) \end{matrix}

Demand for good 2 is

\begin{matrix} c_{2} = (1 - κ) {(\frac{P_{2}}{P})}^{- ε} c & (46) \end{matrix}

and h adjusts endogenously in (43) to satisfy the government budget constraint.

Efficiency gains/losses from the tax distortion show up in the term $h P_{2}^{*} c_{2}$ in the national budget constraint. Digging into the details, linearize equation (45) and solve (43) and (46) for c₂ and h. This yields

\begin{matrix} \dot{k} = f_{4} dk + f_{5} dz - dc + h P_{2}^{*} {dc}_{2}, & (47) \end{matrix}

\begin{matrix} \frac{{dc}_{2}}{c_{2}} = \frac{dc}{c} - ε (1 - γ) \frac{dh}{1 + h}, & (48) \end{matrix}

and

\begin{matrix} \frac{dh}{1 + h} = \frac{r_{p} + δ - \bar{χ} (R_{p} + δ)}{g_{0}} dz - \frac{h}{{cg}_{0}} dc, & (49) \end{matrix}

where

g_{0} \equiv 1 + h - εh (1 - γ),

differentials refer to deviations from the steady state and γ ≡ P_2c2/P_c, the consumption share of good 2.

Consistent with the results in Section 5, the impact on the fiscal budget and the consumption tax can go either way. Regardless of whether h rises or falls, however, the welfare effects are merely “little triangles.” Consider the solution for steady-state consumption. From (47)-(49), and the solution for dk in (20),

\begin{matrix} {\begin{matrix} \frac{dc}{dz} | \end{matrix}}_{P 3} = & {(1 - hγ / g_{0})}^{- 1} {[ρ + \frac{(ρ + δ) (1 - α)}{f_{1}}] a_{2} + R_{p} - r_{p} & (50) \\ + \frac{(R_{p} + δ) (1 - α)}{f_{1}} - \frac{hɛ (1 - γ)}{g_{0}} [r_{p} + δ - \bar{χ} (R_{p} + δ)]} . \end{matrix}

where

a_{2} \equiv \frac{R_{p + δ}}{ρ + δ} (\frac{α}{1 - α}) \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)} .

The term $r_{p} + δ - \bar{χ} (R_{p} + δ)$ picks up the direct effect on the budget—see (49). It is negative for plausible values of r_p, R_p, and $\bar{χ}$ . But since ∊ 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, $hε (1 - γ) / g_{0}$ , is extremely small. Moreover, the small negative term in the numerator is counterbalanced by a multiplier slightly larger than unity (reflecting the gain from increases in c that translate into increases in c₂). Consumption may increase more or less than in the case where lump-sum taxes adjust, but the difference is pocket change. And 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 r for R_p and r_p in (50). Hence

{\frac{dc}{dz} |}_{P 3} - {\frac{dc}{dz} |}_{OI} > 0

if and only if

\begin{matrix} \underset{\overset{⏟}{Solution in the Lump-Sum Tax Case in (26)}}{\begin{matrix} R_{p} - r_{p} - (R - r) + (R_{p} - R) F \end{matrix}} & + \frac{hε (1 - γ)}{g_{0}} [\bar{χ} (R_{p} - R) - (r_{p} - r)] > 0. & (51) \end{matrix}

Barring an exceptionally large productivity gap, (R_p − R)/(r_p − r) $< 1 / \bar{χ}$ and the comparison of consumption gains is more favorable to OI than before. To repeat, however, the little triangles are truly little. The term involving h functions essentially as a tiebreaker: for R_p = .23, R = .16, r_p = .15, r = .06, δ = .05, ∊ = γ = .5, b₂ = 1, α = .40, $\bar{χ}$ = .25, and h = .15, it equals all of −.00261 vs. 123 for the component that mirrors lump-sum tax solution.

At the risk of beating a dead horse, we went to the trouble of deriving the transition path (see Appendix C) and computing exact welfare comparisons for lump-sum vs. distortionary tax adjustment. Unsurprisingly, the numbers in Table 7 are always very close. In many cases, the difference amounts to rounding error.

Table 7:

Ratio of welfare gain in P3 vs. OI for distortionary and lump-sum taxation when b₂ = 1 in the model with open unemployment.¹

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The solution with lump-sum taxation is shown in parentheses.

Table 7:

Ratio of welfare gain in P3 vs. OI for distortionary and lump-sum taxation when b₂ = 1 in the model with open unemployment.¹

	$\bar{χ}$ = 0
	r_p = .10			r_p = .15
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.10 (1.12)	1.02 (1.04)	.94 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.01 (1.05)	.95 (1.0)
.22	1.26 (1.27)	1.25 (1.26)	1.25 (1.25)	1.14 (1.16)	1.07 (1.62)	1.02 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.14 (1.18)	1.09 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.16 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.27 (1.30)	1.24 (1.27)

	$\bar{χ}$ = 2.5
	r_p = .10			r_p = .15
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.11 (1.12)	1.02 (1.04)	.95 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.01 (1.05)	.96 (1.0)
.22	1.26 (1.27)	1.25 (1.26)	1.25 (1.25)	1.14 (1.16)	1.08 (1.62)	1.03 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.14 (1.18)	1.10 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.17 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.28 (1.30)	1.24 (1.27)

	$\bar{χ}$ = .40
	r_p .10			r_p .15
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.11 (1.12)	1.02 (1.04)	.94 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.02 (1.05)	.96 (1.0)
.22	1.26 (1.27)	1.26 (1.26)	1.25 (1.25)	1.14 (1.16)	1.08 (1.62)	1.03 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.15 (1.18)	1.10 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.17 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.28 (1.30)	1.24 (1.27)

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The solution with lump-sum taxation is shown in parentheses.

Table 7:

Ratio of welfare gain in P3 vs. OI for distortionary and lump-sum taxation when b₂ = 1 in the model with open unemployment.¹

	$\bar{χ}$ = 0
	r_p = .10			r_p = .15
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.10 (1.12)	1.02 (1.04)	.94 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.01 (1.05)	.95 (1.0)
.22	1.26 (1.27)	1.25 (1.26)	1.25 (1.25)	1.14 (1.16)	1.07 (1.62)	1.02 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.14 (1.18)	1.09 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.16 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.27 (1.30)	1.24 (1.27)

	$\bar{χ}$ = 2.5
	r_p = .10			r_p = .15
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.11 (1.12)	1.02 (1.04)	.95 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.01 (1.05)	.96 (1.0)
.22	1.26 (1.27)	1.25 (1.26)	1.25 (1.25)	1.14 (1.16)	1.08 (1.62)	1.03 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.14 (1.18)	1.10 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.17 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.28 (1.30)	1.24 (1.27)

	$\bar{χ}$ = .40
	r_p .10			r_p .15
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.20	1.14 (1.15)	1.12 (1.14)	1.11 (1.12)	1.02 (1.04)	.94 (.99)	.89 (.93)
.21	1.20 (1.21)	1.19 (1.20)	1.18 (1.19)	1.08 (1.10)	1.02 (1.05)	.96 (1.0)
.22	1.26 (1.27)	1.26 (1.26)	1.25 (1.25)	1.14 (1.16)	1.08 (1.62)	1.03 (1.07)
.23	1.32 (1.33)	1.32 (1.32)	1.32 (1.32)	1.20 (1.22)	1.15 (1.18)	1.10 (1.13)
.24	1.39 (1.39)	1.39 (1.39)	1.39 (1.39)	1.26 (1.28)	1.21 (1.24)	1.17 (1.20)
.25	1.45 (1.44)	1.45 (1.45)	1.46 (1.46)	1.32 (1.34)	1.28 (1.30)	1.24 (1.27)

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The solution with lump-sum taxation is shown in parentheses.

6.2 Incorporating Distributional Concerns

Policy makers obsess about creating more and better-paying jobs. Short of building a model with heterogeneous agents, the most natural way to incorporate the gains from more/better jobs is to enter real wage income in the social welfare function. In this section,

\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ}} = \int_{o}^{\infty} [c - c_{o} + f (wL - w_{o} L_{o})] e^{- ρ^{* t}} dt . & (52) \end{matrix}

In principle, f is the answer to the problem: when real wage income decreases by one dollar, social welfare is unchanged if aggregate consumption increases by _____ dollars. Although policy makers will not reveal their answer, Benthamite utilitarian calculations shed some light on reasonable values for f. In out model, a utilitarian welfare function implies f = .47 − 2.05 when τ = .35 − 1 and other parameters take their usual values.²⁸

Our earlier results predict the qualitative and quantitative changes in the welfare arithmetic reported in Tables 8 and 9. Naturally, since P3 increases labor demand real wages more than OI, it benefits much more from the distributional externality. For ρ* = .06 and values of f that we consider (too) conservative (.5) or reasonable (1, 1.5), the breakeven value of R_p drops from .202 to .169 − .185 in the runs with efficiency wages and from .239 to .189 − .214 for projects in regions with full employment. 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.

7 Concluding Remarks

There is a clean division of labor between microeconomic and macroeconomic analysis of P3s. The job of microeconomics is to secure the best possible deal for the LDC by structuring the P3 contract so as to maximize the incentives for delivery of high-quality services while minimizing transaction costs for the government and rents paid to the private partner. In the existing literature, the story ends here. The final step in the evaluation process compares the net return in the optimal P3 contract, calculated as simply the return on infrastructure net of total costs for the government, to the net return on OI by the public sector.

But the partial equilibrium comparison of net returns should not be the final step. In a world that is irredeemably second-best, general equilibrium macroeconomic analysis is required for an accurate comparison of social returns to P3 and OI. Typically P3s cost more but produce higher-quality infrastructure and boast a better on-time completion record than OI. Because they score better on efficiency and speed, P3s are comparatively more effective in reducing underinvestment in private capital, underinvestment in infrastructure, unemployment and poverty. How much should policy makers be willing to pay for these benefits? Our two-word answer is “a lot.” The 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 current paper is part of a larger project the IMF has initiated to improve the analysis of P3s in LDCs. Future research in the project will focus on construction of policy tools that are theory-based but flexible and user-friendly. We plan specifically to integrate P3s into the DIG (Debt, Investment, and Growth) model developed by Buffie et al. (2012). The P3-DIG model promises to cover all the bases: policy makers will be able to specify the exact terms in annuity or DBOT (design, build, operate, and transfer) contracts, to rank the welfare gains for P3 vs. OI, and to track the paths of key macroeconomic variables (sectoral wages, employment, and output, private investment, the real exchange rate, external debt and the current account deficit, etc.).

Table 8:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with full employment.¹

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, and ρ = .10. The solution for f = 0 reproduces the solution in the benchmark model where social welfare depends only on consumption of the representative agent.

Table 8:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with full employment.¹

	f = 0			f = .5
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.47	.30	.10	.68	.63	.55
.17	.55	.39	.20	.75	.70	.63
.18	.63	.48	.30	.82	.77	.70
.19	.70	.57	.40	.88	.84	.77
.20	.78	.65	.50	.95	.91	.85
.21	.86	.74	.60	1.01	.98	.92
.22	.94	.83	.70	1.08	1.04	.99
.23	1.02	.92	.80	1.15	1.11	1.07
.24	1.10	1.01	.90	1.21	1.18	1.14
.25	1.18	1.10	1.0	1.28	1.25	1.21
R_p	.228	.239	.250	.208	.214	.221

	f = 1			f = 1.5
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.78	.76	.73	.84	.84	.82
.17	.84	.83	.79	.89	.90	.88
.18	.90	.89	.85	.95	.95	.94
.19	.96	.95	.92	1.01	1.0	1.0
.20	1.02	1.01	.98	1.06	1.07	1.05
.21	1.08	1.07	1.04	1.12	1.12	1.11
.22	1.14	1.13	1.11	1.18	1.18	1.17
.23	1.20	1.19	1.17	1.24	1.24	1.23
.24	1.26	1.26	1.24	1.29	1.29	1.29
.25	1.32	1.32	1.30	1.35	1.35	1.34
R_p	.196	.198	.203	.189	.189	.191

Table 8:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with full employment.¹

	f = 0			f = .5
R_P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.47	.30	.10	.68	.63	.55
.17	.55	.39	.20	.75	.70	.63
.18	.63	.48	.30	.82	.77	.70
.19	.70	.57	.40	.88	.84	.77
.20	.78	.65	.50	.95	.91	.85
.21	.86	.74	.60	1.01	.98	.92
.22	.94	.83	.70	1.08	1.04	.99
.23	1.02	.92	.80	1.15	1.11	1.07
.24	1.10	1.01	.90	1.21	1.18	1.14
.25	1.18	1.10	1.0	1.28	1.25	1.21
R_p	.228	.239	.250	.208	.214	.221

	f = 1			f = 1.5
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.78	.76	.73	.84	.84	.82
.17	.84	.83	.79	.89	.90	.88
.18	.90	.89	.85	.95	.95	.94
.19	.96	.95	.92	1.01	1.0	1.0
.20	1.02	1.01	.98	1.06	1.07	1.05
.21	1.08	1.07	1.04	1.12	1.12	1.11
.22	1.14	1.13	1.11	1.18	1.18	1.17
.23	1.20	1.19	1.17	1.24	1.24	1.23
.24	1.26	1.26	1.24	1.29	1.29	1.29
.25	1.32	1.32	1.30	1.35	1.35	1.34
R_p	.196	.198	.203	.189	.189	.191

Table 9:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with open employment.¹

^¹R = .16, r* = .06, r_p = .15, α= .40, τ = .50, δ = .05, u = .10, b₂, and ρ = .10. The solution for f = 0 reproduces the solution in the benchmark model where social welfare depends only on consumption of the representative agent.

Table 9:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with open employment.¹

	f = 0			f = .5
R^P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.81	.74	.66	.88	.86	.83
.17	.86	.80	.73	.93	.91	.88
.18	.92	.86	.80	.99	.97	.94
.19	.98	.93	.87	1.04	1.03	1.0
.20	1.04	.99	.93	1.10	1.08	1.06
.21	1.10	1.05	1.0	1.15	1.14	1.11
.22	1.16	1.12	1.07	1.21	1.19	1.17
.23	1.22	1.18	1.13	1.26	1.25	1.23
.24	1.28	1.24	1.20	1.32	1.31	1.29
.25	1.34	1.30	1.27	1.37	1.36	1.35
$\bar{R_{p}}$	.193	.202	.210	.182	.185	.190

	f = 1			f = 1.5
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.92	.92	.90	.94	.95	.95
.17	.97	.97	.96	.99	1	1
.18	1.02	1.02	1.01	1.04	1.05	1.05
.19	1.07	1.08	1.06	1.09	1.11	1.10
.20	1.13	1.13	1.12	1.14	1.16	1.15
.21	1.18	1.18	1.17	1.20	1.21	1.20
.22	1.23	1.23	1.22	1.25	1.26	1.25
.23	1.28	1.29	1.28	1.30	1.31	1.30
.24	1.34	1.34	1.33	1.35	1.36	1.36
.25	1.39	1.39	1.38	1.40	1.41	1.41
$\bar{R_{p}}$	.176	.176	.178	.172	.169	.170

Table 9:

Ratio of welfare gain in P3 vs. OI when real wage income enters the social welfare function in the model with open employment.¹

	f = 0			f = .5
R^P	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.81	.74	.66	.88	.86	.83
.17	.86	.80	.73	.93	.91	.88
.18	.92	.86	.80	.99	.97	.94
.19	.98	.93	.87	1.04	1.03	1.0
.20	1.04	.99	.93	1.10	1.08	1.06
.21	1.10	1.05	1.0	1.15	1.14	1.11
.22	1.16	1.12	1.07	1.21	1.19	1.17
.23	1.22	1.18	1.13	1.26	1.25	1.23
.24	1.28	1.24	1.20	1.32	1.31	1.29
.25	1.34	1.30	1.27	1.37	1.36	1.35
$\bar{R_{p}}$	.193	.202	.210	.182	.185	.190

	f = 1			f = 1.5
R_p	ρ* = .02	ρ* = .06	ρ* = .10	ρ* = .02	ρ* = .06	ρ* = .10
.16	.92	.92	.90	.94	.95	.95
.17	.97	.97	.96	.99	1	1
.18	1.02	1.02	1.01	1.04	1.05	1.05
.19	1.07	1.08	1.06	1.09	1.11	1.10
.20	1.13	1.13	1.12	1.14	1.16	1.15
.21	1.18	1.18	1.17	1.20	1.21	1.20
.22	1.23	1.23	1.22	1.25	1.26	1.25
.23	1.28	1.29	1.28	1.30	1.31	1.30
.24	1.34	1.34	1.33	1.35	1.36	1.36
.25	1.39	1.39	1.38	1.40	1.41	1.41
$\bar{R_{p}}$	.176	.176	.178	.172	.169	.170

A Appendix A

The core dynamic system in the benchmark model is given in equation (³³). On the convergent saddle path,

\begin{matrix} c - c_{o} = \bar{c} - c_{o} + [(f_{4} - λ_{2}) (X_{21} - a_{2}) e^{λ 2 t} - X_{11} e^{- δt}] (\bar{z} - z_{o}), & (A 1) \end{matrix}

and

\begin{matrix} k - k_{o} = [a_{2} (1 - e^{λ 2 t}) + X_{21} (e^{- δt})] (\bar{z} - z_{o}), & (A 2) \end{matrix}

where

\begin{matrix} X_{11} & = & \frac{τ (c / k) [(1 - α) f_{5} f_{2} + α (R_{p} + δ) f_{3}^{2}]}{V}, \\ X_{21} & = & \frac{(R_{p} + δ) [τα (c / k) f_{3} + δ (1 - α) / f_{1}] + δ (R_{p} - r_{p})}{(ρ + δ) V}, \\ V & \equiv & τ (c / k) (1 - α) f_{2} - δ f_{3}, \\ a_{2} & \equiv & \frac{\bar{k} - k_{o}}{\bar{z} - z_{o}} = \frac{R_{p} + δ}{ρ + δ} (\frac{α}{1 - α}) \frac{1 + b_{2} (1 - u)}{b_{2} (1 - u)} . \end{matrix}

The Welfare Solution Under Full Employment

In the full-employment case, f₄ ≡ ρ and a₂ ≡ (R_p + δ)α/(1 − α)(ρ + δ). Recall that ρ* ≤ ρ. Thus

\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ}} & = & \int_{o}^{\infty} (c - c_{o}) e^{- ρ * t} dt, \\ \Rightarrow & \frac{SW - {SW}_{o}}{c^{- 1 / τ} (\bar{z} - z_{o})} & = & \int_{o}^{\infty} (ρ \frac{R_{p} + δ}{ρ + δ} \frac{α}{1 - α} + R_{p} - r_{p}) e^{- ρ * t} dt \\ - \int_{o}^{\infty} X_{11} e^{- {(ρ * + δ)}^{t}} dt + \int_{o}^{\infty} (ρ - λ_{2}) (X_{21} - \frac{R_{p} + δ}{ρ + δ} \frac{α}{1 - α}) e^{(λ_{2} - ρ *) t} dt, \\ \Rightarrow & \frac{SW - {SW}_{o}}{c^{- 1 / τ} (\bar{z} - z_{o})} & = & \frac{λ_{2 (ρ * - ρ)}}{ρ * (ρ * - λ_{2})} \frac{R_{p} + δ}{ρ + δ} \frac{α}{1 - α} + \frac{R_{p} - r_{p}}{ρ *} + \frac{ρ - λ_{2}}{ρ * - λ_{2}} X_{21} - \frac{X_{11}}{ρ * + δ} . \end{matrix}

Substitute for X₁₁ and X₂₁. After rearranging terms, the solution may be written as

\begin{matrix} {\begin{matrix} \frac{SW - {SW}_{o}}{c^{- 1 / τ} (\bar{z} - z_{o})} | \end{matrix}}_{P 3} = \int_{o}^{\infty} (c - c_{o}) e^{- ρ * t} dt = (\frac{R_{p} - r_{p}}{ρ + δ}) \frac{δ}{ρ *} (1 + H_{p}), & (A 3) \end{matrix}

for P3, where

H_{p} \equiv \frac{ρ - ρ *}{ρ * + δ} Λ + \frac{(R_{p} + δ) α (ρ - ρ *)}{(ρ * - λ_{2}) (1 - α) (ρ + δ)} Γ,

Λ \equiv 1 + \frac{ρ * (λ_{2} + δ)}{J (ρ * - λ_{2})} with Λ > 0,

Γ \equiv \frac{τ (c / k) (1 - α) (λ_{2} + δ)}{J} - λ_{2} \frac{ρ * + δ}{ρ *} with Γ > 0,

and

J \equiv τ (c / k) (1 - α) - δ .

To show that Λ > 0 and Γ > 0, note first that λ₂ + δ and J are opposite in sign. Hence

\begin{matrix} \frac{λ_{2} + δ}{J} > - 1. & (A 4) \end{matrix}

is sufficient for $Λ \equiv 1 + \frac{ρ * (λ_{2} + δ)}{J (ρ * - λ_{2})}$ to be positive. When J > 0, the condition holds if

λ_{2} + δ + J > 0,

\Rightarrow λ_{2} + N > 0,

where N ≡ τ(c/k)(1 − α). Substituting for γ₂ produces

\begin{matrix} \frac{ρ - \sqrt{ρ^{2} + 4 (ρ + δ) N}}{2} + N & > & 0, \\ \Rightarrow ρ + 2 N & > & \sqrt{\begin{matrix} ρ^{2} + 4 (ρ + δ) N \end{matrix}}, \\ \Rightarrow N J & > & 0, \end{matrix}

which holds as J > 0 in the case under examination.

Suppose next that J < 0. In this case, we require

λ₂ + δ + J > 0.

The same algebra delivers

N J < 0,

which also holds.

Turn now Γ which affects H_p in (A3). Since J = N – δ, the bracketed term multiplying δ − δ*. is positive if

\frac{N (λ_{2} + δ)}{N - δ} - λ_{2} > 0,

or, equivalently,

\begin{matrix} λ_{2} + N ≷ 0 as J = N - δ ≷ 0. & (A 5) \end{matrix}

But we already know that λ₂ + N and J take the same sign. Thus the condition in (A5) holds.

The solution for OI takes the same form as (A3), with R and r replacing R_p and r_p.

B Appendix B

When P3 builds infrastructure faster than OI, the core dynamic system is

\begin{matrix} [\begin{matrix} \dot{c} \\ \dot{k} \\ \dot{z} \\ {di}_{z} / dt \end{matrix}] = [\begin{matrix} 0 & τ (c / k) (ρ + δ) (α - 1) f_{2} & τ (c / k) α (R_{p} + δ) f_{3} & 0 \\ - 1 & f_{4} & f_{5} & 0 \\ 0 & 0 & - δ & 1 \\ 0 & 0 & 0 & - s \end{matrix}] [\begin{matrix} c - \bar{c} \\ k - \bar{k} \\ z - \bar{z} \\ i_{z} - {\bar{ι}}_{z} \end{matrix}] . & (B 1) \end{matrix}

The saddle point solutions for c, k, and z are

\begin{matrix} [\begin{matrix} \begin{matrix} c - c_{o} \\ k - k_{o} \\ z - z_{o} \end{matrix} \end{matrix}] = [\begin{matrix} \begin{matrix} \bar{c} - c_{o} \\ \bar{k} - k_{o} \\ \bar{z} - z_{o} \end{matrix} \end{matrix}] + [\begin{matrix} X_{11} & f_{4} - λ_{2} & X_{13} \\ X_{12} & 1 & X_{23} \\ 1 & 0 & 1 / (δ - s) \end{matrix}] [\begin{matrix} \begin{matrix} h_{1 e^{- δt}} \\ h_{2} e^{λ_{2} t} \\ h_{3} e^{- st} \end{matrix} \end{matrix}], & (B 2) \end{matrix}

where

\begin{matrix} X_{13} & = & \frac{τ (c / k)}{(δ - s) V_{1}} [1 - α f_{2} f_{5} + α (R_{p} + δ) f_{3} \frac{f_{4} + s}{ρ + δ}], \\ X_{23} & = & \frac{τ (c / k) α (R_{p} + δ) f_{3} + s f_{5}}{(δ - s) (ρ + δ) V_{1}} \\ V_{1} & \equiv & τ (c / k) (1 - α) f_{2} - s \frac{f_{4} + s}{ρ + δ}, \end{matrix}

and h₁ - h₃ are determined by the initial jump in i_z and the initial values of k and z:

\begin{matrix} h_{1} & = & [(1 - n) δ / (δ - s) - 1] (\bar{z} - z_{o}), \\ h_{2} & = & {X_{21} - a_{2} + (n - 1) δ [X_{21} / (δ - s) + X_{23}] (\bar{z} - z_{o}), \\ h_{3} & = & (n - 1) δ (\bar{z} - z_{o}), \\ n & = & \frac{i_{z, o +} - i_{z, o}}{\bar{ι_{z}} - i_{z, o}} \end{matrix}

C Appendix C

In the model where the distortionary consumption tax adjusts to satisfy the government budget constraint,

\begin{matrix} \dot{c} = τc (qk - ρ - δ) - τcγ \frac{\dot{h}}{1 + h} & (C 1) \end{matrix}

replaces equation (⁷). Interpret the differentials in (49) as time derivatives and substitute for $\dot{h}$ . After collecting terms, we have

\begin{matrix} (1 - τhγ / g_{0}) \dot{c} = τ c (qk - ρ - δ) - τcγ g_{1} (i_{z} - δz), & (C 2) \end{matrix}

where

\begin{matrix} g_{0} \equiv 1 + h - ε h (1 - γ) & and & g_{1} \equiv \frac{r_{p} + δ - \bar{χ} (R_{p} + δ)}{P_{2}^{*} c_{2} g_{0}} . \end{matrix}

The linearized version of (C2) is

\begin{matrix} \dot{c} = \frac{τ (c / k) (α - 1) (ρ + δ) f_{2}}{g_{3}} (k - \bar{k}) + \frac{τ (c / k) (α - 1) (R_{p} + δ) f_{3} - g_{4}}{g_{3}} (z - \bar{z}), & (C 3) \end{matrix}

where

g_{3} \equiv 1 - \frac{τhγ}{g_{0}} and g_{4} \equiv \frac{τδ (1 + h)}{g_{0}} [r_{p} + δ - \bar{χ} (R_{p} + δ)] .

Turn next to equation (⁴⁷). Using equations (⁴⁸) and (⁴⁹) to solve for c₂ and substituting it into equation (⁴⁷) leads to

\begin{matrix} \dot{k} = f_{4} (k - \bar{k}) + f_{6} (z - \bar{z}) - f_{7} (c - \bar{c}), & (C 4) \end{matrix}

where

f_{6} = f_{5} - \frac{hε (1 - γ) [r_{p} + δ - \bar{χ} (R_{p} + δ)]}{g_{0}} and f_{7} = 1 - \frac{hγ}{g_{0}} .

Equations (C3), (C4), and

\begin{matrix} \dot{z} = - δ (z - \bar{z}) & (C 5) \end{matrix}

comprise the core dynamic system. The solution for the path of consumption is

\begin{matrix} c - c_{o} = \bar{c} - c_{o} + [\frac{f_{4} - λ_{2}}{f_{7}} (X_{21} - a_{2}) e^{λ 2 t} - X_{11 e^{- δt}}] (\bar{z} - z_{o}), & (C 6) \end{matrix}

but now

\begin{matrix} λ_{2} & = & \frac{f_{4} - \sqrt{f_{4}^{2} + 4 τ (1 - α) (ρ + δ) (c / k) f_{2} f_{7} / g_{3}}}{2} \\ X_{11} & = & \frac{f_{3}}{g_{3} V} [τ (c / k) (1 - α) f_{6} f_{2} / f_{3} + τ (c / k) α (R_{p} + δ) f_{3} + g_{4}], \\ X_{21} & = & \frac{δ f_{6} + [τα (c / k) (R_{p} + δ) f_{3} + g_{4}] f_{7} / g_{3}}{{(ρ + δ)}^{V}} \\ V & = & τ (c / k) (1 - α) f_{2} f_{7} / g_{3} - δ f_{3} . \end{matrix}

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Sen, A., 1967, “Isolation, Assurance and the Social Rate of Discount.” Quarterly Journal of Economics 81, 112–124.
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Abstract

1 Introduction

2 The Model

2.1 The Choice: P3 or OI?

2.2 The Private Sector

2.3 The National Budget Constraint

2.4 The Case for Efficiency Wages

3 Inputs

3.1 The Long-Run Solution

3.2 The Transition Path

3.3 The Welfare Criterion

4 Welfare Comparisons

4.1 Full Employment

4.2 Unemployment

4.3 The Importance of Speed

5 The Fiscal Challenge

6 Extensions

6.1 Distortionary Taxes Adjust

6.2 Incorporating Distributional Concerns

7 Concluding Remarks

A Appendix A

The Welfare Solution Under Full Employment

B Appendix B

C Appendix C

References