Appendix A: Project Selection
In the main text we discuss four ways to think about public investment efficiency: (1) a fraction ϵ of spending is literally wasted (“corruption”); (2) the costs of the project are higher than they need to be, e.g. because of an inefficiently high use of inputs (“waste”); (3) government may choose projects that yield a greater or lesser flow of capital services for the same investment spending (“poorly designed projects”); and (4) governments may misallocate public investment spending across sectors or categories of investment (“poor investment allocation”). In this appendix we show that all four conceptions of efficiency have similar implications for the public investment/growth relationship.32
To understand public investment efficiency, it is tempting to imagine that all the available public investment projects at a given point in time can be ranked from highest to lowest rate of return. The marginal product of public investment is then the return of the best project available (Figure A-1). In a fully efficient investment process, when an additional dollar is spent, the next best project is chosen. With inefficient project selection, infra-marginal projects are chosen, resulting in a lower overall growth impact.
This notion is static, however, and thus potentially misleading: in general, the rate of return on one project will depend on the size of the capital stock that is already in place. The usual formulation of the public capital stock as the discounted sum of public investment implicitly assumes that all public capital goods are perfect substitutes. In this case, the downward slope of the schedule in Figure A-1 represents not the variety of available projects but simply the fact that capital becomes less productive as it becomes less scarce, as for example in a standard Cobb-Douglas production function. Each of our definitions of inefficiency, however, can be thought of in terms of poor project choice, in different ways.
The first two definitions are identical in terms of the basic equations of section (2), equations (1) and (3): only a fraction ϵ of the spending makes its way into public capital G, though we show in section 3.4 what is done with the 1 – ϵ spending can matter for the general equilibrium outcome. What these definitions mean for project selection is relatively straightforward. If projects differ according to the degree of waste or corruption, then one can think of the schedule in Figure A-1 as measuring the amount of capital produced for given amount of spending. While the figure cannot readily capture the dynamics or even the steady state, the height of the curve would depend on the capital/output ratio, so the selection of more efficient projects would shift the curve down more.
According to efficiency definition (3), different investment projects create capital that yields a greater or lesser flow of public capital services to the economy. So one dollar spent on a “bad project” is one that yields as much public capital as a good project, but the service flow from that project is lower by a factor of ϵ.
Let the infrastructure stock be the sum of spending, discounted for depreciation, denoted Gm and defined as in equation (2). The flow of infrastructure services from this stock depends on how well the particular projects were chosen and is equal to ϵGm. Output then depends on this service flow:
Notice that this is exactly the same as what we get by discounting investment spending by ϵ and putting effective capital G in the production function, as we do in the main text. Thus, all the results from the main text go through, reinterpreted. As before, an inefficient country (one that chooses more bad projects in this sense) does have a lower level of output, but it also has a higher marginal product of service flow (MPSF). If it always tends to choose inefficient projects, the growth impact of subsequent investments will be the same as in the country that has been choosing high-service-flow projects all along. In terms of Figure A-l, again a country could choose infra-marginal projects, but which projects it chooses would influence the overall height of the line through the scarcity of the service flow from public capital.
Finally, the fourth definition of inefficiency gets at the notion that projects differ in a more fundamental sense. In particular, different projects produce different types of public capital that are not perfect substitutes (nor are the service flows from these different stocks capital perfect substitutes). Suppose, for simplicity, there are two types of public capital, G and H. These could be physical infrastructure and human capital, or they could represent roads in the south and roads in the north-the results generalize to any number of types of public capital. In this context, bad project choice is choosing the wrong type of project.
To be concrete:
And there are two associated capital accumulation equations:
where IG is public investment spending on project type G. We can allow ϵ and δ to differ across types.
Now, to model project choice, let θG be the share of total investment spending going to projects of type G. So:
In steady state, we can rewrite equation (A-1) as:
There is an optimal allocation of spending across sectors
We can now show that (1) choosing the wrong projects (i.e. the wrong value of θG) lowers the level of output; and (2) the level of θG does not matter for the growth impact of additional public investment spending. Taking the derivative of equation (A-5) with respect to θG yields:
Equalizing this to zero and solving for θG gives:
This is a maximum, so choosing any other value of θ results in a lower level of output.33 However, the growth impact of additional investment spending does not depend on θG (or on the values of ϵ, for that matter):
In trying to match this definition of efficiency with Figure A-1, the different types could be aligned from highest to lowest marginal product (we have only two in the above equations but there is no reason this could not be generalized to many types). However, an efficient country would over time allocate investment spending to the highest-yielding types, reducing the scarcity of capital in those sectors so that in steady state the curve in the figure would be a horizontal line. An inefficient country would face a downward-sloping curve in steady state.
This fourth conception of “efficiency” as sectoral allocation of spending can be (and indeed in the above equations is) combined with any or all of the other three conceptions of efficiency as captured by ϵ. This yields a fairly rich conception of inefficient project selection: a country may choose the wrong mix of types of projects, and within types, it may choose especially wasteful or corrupt projects or ones where the service flow for a given dollar is relatively low. All of this is consistent with the results in the main text.
Appendix B: Consumption with Corruption
We confirm in a simple steady state analysis that countries with higher investment rates have higher growth independent of the level of efficiency (with corruption). Then we show that consumption rises more in the long run with higher investment in more inefficient (corrupt) countries.
Suppose for simplicity that two countries a and b at an initial steady state have equal output, but that country b is fully efficient while country a is not, and that there is no private capital. Thus:
The equivalence of output in the two countries implies that
Now, consider a new steady state, where the only difference is that I is at a new higher level I1 in both countries.
It is apparent as usual that the change in output is invariant to efficiency:
However, the increase in consumption is not invariant. For any ϵ less than 1, the increase in the inefficient (corrupt) country is bigger than in the efficient country.
The intuition here is that the inefficient country gets more consumption out of an increase in investment spending because it takes less actual realized investment to generate the same output increase, and the rest can be spent on consumption (unlike in the “waste” case).
By assuming that the two countries have the same level of output in the initial steady state, we imply that the inefficient country has a higher level of TFP, and this higher level of TFP allows the country to get the same output (and more consumption) from the same increase in investment spending. Do things change if we assume instead that the two countries initially have the same TFP, and the only difference between the two is in efficiency?
We now have:
Again, there is invariance in output:
When ϵ < 1, consumption growth in the inefficient country a is again higher.34
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We would like to thank Abdul Abiad, Romain Duval, Jason Harris, Thomas Helbling, Richard Hughes, Samah Mazraari, Chris Papageorgiou, Adam Remo, Genevieve Verdier and many IMF colleagues for useful comments. This paper is part of a research project on macroeconomic policy in low-income countries supported by the U.K.’s Department for International Development (DFID). The views expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its management, or to DFID.
In endogenous growth models the outcome depends on the structure of the model. The equilibrium growth rate is higher in countries with efficient public investment in a Barro-type model (Barro, 1990), but not necessarily in the models formulated by Lucas (1988); Manuelli and Jones (1990); Rebelo (1991). The results for growth on the transition path also appear to exhibit considerable variation. In preliminary research, we have found plausible cases in the Lucas model where growth on the transition path is continuously higher in the low-efficiency economy.
Results vary, depending on the sample and the measure of “efficiency.” IMF (2015) use an ‘efficiency frontier’ approach to map cumulative investment spending to a measure of the public capital stock that is itself a combination of a survey-based measure and an index of physical infrastructure. We find a positive but insignificant difference between the growth effect of efficient and inefficient countries according to this measure. (This result, available on request, differs from those in IMF (2015), because we allow for robust standard errors and we align the lag structure with Abiad, Furceri and Topalova (2015).) This insignificant effect nests a positive and significant growth impact of investment spending for a high-efficiency country, using the survey measure, and a negative and significant effect when using the physical infrastructure index. IMF (2014b) find a higher growth impact of public investment in high-efficiency countries in an advanced-country sample. The efficiency measure used is a survey-based index of the ‘quality’ of public infrastructure from the Global Competitiveness Report (GCR, Schwab, 2015). However, this index is meant to capture the effective quantity of infrastructure, not efficiency per se. Abiad, Furceri and Topalova (2015) update IMF (2014b) with a another survey measure of ‘wastefulness of public spending’ from the GCR and obtain a positive and significant effect of efficiency, in a similar sample. Whether this survey question distinguishes the narrow conception of efficiency used in this paper from the broader question of the rate of return to investment spending is impossible to infer from the survey instrument.
We simplify by ignoring private capital and labor here, but this makes no difference to our main point, as we show in section 3.
We suppose that the parameters ψ and δ are the same across the two cases.
From equation (12), this implies that the ratio of the level of output in the two cases (call them ϵh and ϵl) is equal to
In this case, the ratio of the level of output in the two cases is equal to
We follow Pritchett (2000) in switching to continuous time here to simplify the algebra, and we abuse notation using
This can be seen from equations (4) and (5). Of course we are putting various estimation issues aside such as endogeneity.
Some of us have made such a mistake in some of our own calibrations, such as that in Buffie et al. (2012), despite having a brief appendix there on the topic of this paper. Of course, different sources for estimates of rate of return may or may not take into account efficiency. Project-specific analysis of rate of return may not, for example, depending on the nature of the estimate and the project, including whether the cost estimates incorporate the inefficiency. It seems plausible that they would if the inefficiency is related to “waste” or poor project selection, less clear if “corruption”. Moreover, what we call the rate of return is the economy-wide increase in output associated with the project, holding other inputs constant. This may be hard to capture in a project-level analysis of rate of return.
Pritchett (2000) recognizes the broader policy point that the MPK and ϵ effects are likely to offset when he notes (his footnote 19) that
“The case of low efficacy is common in the developing countries reconciles a common paradox. For decades development “experts” have observed the lack of roads, power, schools, and health clinics and assumed that since the stock was so low, the marginal product of public-sector capital must be high and hence “more investment” was the appropriate answer. However, this has left a legacy in the poorest countries of large amounts of public-sector investment (often with official financing) but with little or now public-sector capital to show for it.”
The “paradox” that is reconciled here is that public investment spending should generate very large growth effects, because the MPK is presumably high. But the analogous paradox would be that low-efficiency countries do benefit somewhat from public investment spending, despite wasting so much spending. And the reconciliation to both paradoxes is that it is necessary to take into account both that low efficiency makes the MPK high and that it means that spending generates only a little capital. Similarly, his discussion of the decision about whether to make a particular public investment recognizes the importance of both the MPK and efficiency, but he fails to note that likelihood of a general inverse relationship.
That there is any time variation at all in the difference between ‘efficiency-adjusted’ and unadjusted capital stocks (with time-invariant efficiency measures) in IMF (2014a) is due to assumptions about how the initial capital stock is calculated and in particular whether the same efficiency measure is used for the initial capital stock as is used for subsequent accumulation. If the same efficiency value is used for the initial stock calculation and subsequent investment spending, there will be a constant percentage difference between the adjusted and unadjusted stocks.
Interestingly, the calibrated CES production function in Eden and Kraay (2014) supports this case of higher complementarity between public and private capital. IMF (2014b, p. 78) suggests that complementary may be the more intuitive case: “…infrastructure is an indispensable input in an economy’s production, one that is highly complementary to other, more conventional inputs such as labor and noninfrastructure capital.”
The following results also hold in discrete time. However, continuous time allows to derive simpler analytical expressions, while conveying the same message.
In the long run (t → ∞), the growth rate
The time unit is a year and the discount factor β = 0.94. The initial infrastructure investment is set to be equal to 6 percent of GDP, which is close to the average for LICs in SSA reported by Briceño Garmendia, Smits and Foster (2008). The capital’s share in value added corresponds to α = 0.5 and the depreciation rates are set as δ = 0.05. Lastly the elasticity of output with respect to public capital ip is set to match a rate of return on public capital (net of depreciation) of 25 percent for the low-efficient country which falls in the range of estimates provided by Briceño Garmendia and Foster (2010) for electricity water and sanitation, irrigation, and roads in SSA.
The numerical simulations track the global nonlinear saddle path. The solutions were generated by set of programs written in Matlab and Dynare 4.3.2. See http://www.cepremap.cnrs.fr/dynare.
For these impulse responses, we keep TFP the same across the two cases and let initial income be lower in the low-efficiency case, so the initial public investment/GDP shares are the same. We then compare shocks that correspond to the same percentage increase in investment in the two cases.
Formally, with these private capital adjustment costs, the budget constraint (32) becomes
and the Euler equation (30) changes to
Simulation results are available from the authors upon request.
In the figure, private capital remains higher in the low-adjustment-cost case after 30 years; eventually, though, the two lines converge.
The PIMI and the ICRG measure of “institutions” (a composite indicator of the political, economic and financial risk of a country) have a correlation coefficient of 30 percent on a sample of 50 countries, with ICRG scores averaged over the same 2007-2010 period for which the PIMI is calculated.
In this case, the resource constraint of the economy becomes
since now the government transfers
This is worked out in the steady state in Appendix B, which also shows that the result is not dependent on this particular characterization of the initial steady state.
To underscore, the “increase” here is in time in a given country. In the rest of the paper when we compare low- and high-efficiency countries, these are eternal differences across two cases, the comparison that is more relevant when we compare “low-efficiency” and “high-efficiency” countries using a cross-section indicator of efficiency.
An important corollary of this way of thinking is that efficiency is likely to fall if the investment rate increases, a notion of absorptive capacity limitations discussed briefly above and in Berg et al. (2013).
Further calculations are available upon request.
Buffie et al. (2012) emphasize the interaction of public investment/growth linkages with the fiscal reaction function, absorptive capacity, and other LIC-specific features; Adam and Bevan (2014) explore the role of distortionary taxation and operations-and-maintenance spending in conditioning the growth impact of public investment spending.
There are other definitions of “efficiency” in the related literature. Hulten (1991) defines “efficiency” as the ratio between the amount of investment carried out some time in the past and the amount that would be needed now to provide equal productive capacity. When ϵ is equal to 1, this is related to the depreciation rate. In contrast, Hulten (1996) defines efficiency as the fraction of the capital stock that is available for productive use. This is a useful concept that is related to operations and maintenance expenditure and is also discussed in Adam and Bevan (2014). It is complementary to the concept analyzed in this paper. However, for current purposes it is worth noting that it is indistinguishable from TFP at the macroeconomic level.
It may remain surprising that the values of spending efficiency ϵG and ϵH do not matter for the optimal allocation of spending across types of project, but this is just a reflection of the scarcity/efficiency trade-off emphasized in the main text.
Some algebra using equations (B-9) and (B-10) shows that