metadata: authors: 0: author: Devadas,Sharmila
1: author: Pennings,Steven Michael
count: World
docna: 0: docna: Assessing the Effect of Public Capital on
            Growth : An Extension of the World Bank Long-Term Growth Model
docty: Policy Research Working Paper
subtopic: Hydrology,Energy Policies & Economics,Infrastructure Regulation,Infrastructure Economics,Infrastructure and Law,Infrastructure Finance,Labor & Employment Law,Macroeconomic Management
teratopic: Energy,Water Resources,Infrastructure Economics and Finance,Macroeconomics and Economic Growth,Law and Development
abstracts: cdata!: To analyze the effect of an increase in
            the quantity or quality of public investment on growth, this
            paper extends the World Bank’s Long-Term Growth Model
            (LTGM), by separating the total capital stock into public
            and private portions, with the former adjusted for its
            quality. The paper presents the Long-Term Growth Model
            Public Capital Extension (LTGM-PC) and accompanying freely
            downloadable Excel-based tool. It also constructs a new
            Infrastructure Efficiency Index (IEI), by combining quality
            indicators for power, roads, and water as a cardinal measure
            of the quality of public capital in each country. In the
            model, public investment generates a larger boost to growth
            if existing stocks of public capital are low, or if public
            capital is particularly important in the production
            function. Through the lens of the model and utilizing
            newly-collated cross-country data, the paper presents three
            stylized facts and some related policy implications. First,
            the measured public capital stock is roughly constant as a
            share of gross domestic product (GDP) across income groups,
            which implies that the returns to new public investment, and
            its effect on growth, are roughly constant across
            development levels.  Second, developing countries are
            relatively short of private capital, which means that
            private investment provides the largest boost to growth in
            low-income countries. Third, low-income countries have the
            lowest quality of public capital and the lowest efficient
            public capital stock as a share of gross domestic product.
            Although this does not affect the returns to public
            investment, it means that improving the efficiency of public
            investment has a sizable effect on growth in low-income
            countries. Quantitatively, a permanent 1ppt GDP increase in
            public investment boosts growth by around 0.1-0.2ppts over
            the following few years (depending on the parameters), with
            the effect declining over time.
repnme: repnme: Assessing the Effect of Public Capital on
            Growth : An Extension of the World Bank Long-Term Growth Model
display_title: Assessing the Effect of Public
            Capital on Growth : An Extension of the World Bank Long-Term
            Growth Model
listing_relative_url: /research/2018/10/30482900/assessing-effect-public-capital-growth-extension-world-bank-long-term-growth-model
topicv3: Infrastructure and Growth m1326244 1083,Current Account m1326278 1135,Water m1326522 137,Power and Electricity Sector m1327094 515,Working Age Population m1504053 2749
publishtoextweb_dt: 2018-10-09T00:00:00Z
historic_topic: Energy,Water Resources,Infrastructure Economics and Finance,Macroeconomics and Economic Growth,Law and Development
no_of_pages: 38
author_keys: 0: author_key: 1463367
1: author_key: 1324256
model_extractions: WPS8604 Policy Research Working Paper 8604 Assessing the Effect of Public Capital on Growth An Extension of the World Bank Long-Term Growth Model _Sharmila Devadas Steven Pennings_ Development Economics Development Research Group October 2018
Policy Research Working Paper 8604

# **Abstract**

To analyze the effect of an increase in the quantity or quality of public investment on growth, this paper extends the World Bank’s Long-Term Growth Model (LTGM), by separating the total capital stock into public and private portions, with the former adjusted for its quality. The paper presents the Long-Term Growth Model Public Capital Extension (LTGM-PC) and accompanying freely downloadable Excel-based tool. It also constructs a new Infrastructure Efficiency Index (IEI), by combining quality indicators for power, roads, and water as a cardinal measure of the quality of public capital in each country. In the model, public investment generates a larger boost to growth if existing stocks of public capital are low, or if public capital is particularly important in the production function. Through the lens of the model and utilizing newly-collated cross-country data, the paper presents three stylized facts and some related policy implications. First, the measured public capital stock is roughly constant as a share of gross domestic product (GDP) across income groups, which implies that the returns to new public investment, and its effect on growth, are roughly constant across development levels. Second, developing countries are relatively short of private capital, which means that private investment provides the largest boost to growth in low-income countries. Third, low-income countries have the lowest quality of public capital and the lowest efficient public capital stock as a share of gross domestic product. Although this does not affect the returns to public investment, it means that improving the efficiency of public investment has a sizable effect on growth in low-income countries. Quantitatively, a permanent 1ppt GDP increase in public investment boosts growth by around 0.1–0.2ppts over the following few years (depending on the parameters), with the effect declining over time.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/research. The authors may be contacted at spennings@worldbank.org and sdevadas1@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Assessing the Effect of Public Capital on Growth:**

**An Extension of the World Bank Long‐Term Growth Model**

Sharmila Devadas and Steven Pennings<sup>1</sup> Development Research Group, World Bank JEL Classification: O40, O57, H41 Keywords: Long‐term growth, infrastructure, public capital, public investment efficiency

> 1 The views expressed here are the authors’, and do not necessarily reflect those of the World Bank, its Executive Directors, or the countries they represent. The authors are grateful to Norman V. Loayza for guidance, Luis Serven for helpful comments and Jorge Luis Guzman for research assistance. The authors also thank the Global Facility for Growth for Development Trust Fund, supported by the Korean Government, for financial support. The companion LTGM‐PC spreadsheet and appendices are available at www.worldbank.org/LTGM..
# **1. Introduction**

Inadequate infrastructure, especially public infrastructure, is often viewed as a key impediment to economic growth and development in low‐ and middle‐income countries. While increasing infrastructure investment has been a part of national development strategies for decades, its perceived importance has gained prominence with the rapid development of China and its infrastructure‐led growth strategy; as well increased infrastructure‐specific finance through new bilateral lending, the Asian Infrastructure Bank, and the Belt and Road Initiative.

Despite the importance of public infrastructure investment, there is wide disagreement about the size and significance of its effect on growth in developing countries (Calderon and Serven 2014). On one hand, the needs are clearly great – close to 700 million people do not have access to safe drinking water and 1.2 billion are without electricity – and so one should expect a sizable impact.<sup>2</sup> Several papers have estimated large returns to infrastructure investment ‐‐ most frequently cited, Aschauer (1989). But as infrastructure investment is endogenous – for example, growth for other reasons might generate public revenues which allows the construction of infrastructure ‐ many of those empirical studies lack causal validity and estimated impacts are implausibly large. Many other papers have found insignificant or negative impacts (Bom and Ligthart 2014), possibly because public investment in developing countries often fails to generate productive capital due to corruption and the presence of “white elephants” (Pritchett 2000).

Perhaps less appreciated is that there is a great deal of confusion in the empirical and policy discussion about the dynamics and mechanisms through which public infrastructure investment would affect growth. For example, empirical studies (and policy reports) are often vague about whether it is the level of infrastructure that affects growth, or whether infrastructure investment (and hence changes in infrastructure levels) affects growth. Likewise, empirical studies often have difficulty estimating _when_ the boost to growth might occur (whether the size of the effect will increase or decrease over time) and what country‐level factors determine the impact on growth (as different studies are for particular countries or reflect a cross‐country average). All these aspects are crucial for evaluating the effectiveness of a country’s public investment‐led growth plans.

This paper makes contributions in two areas to try to address these gaps. First, we develop a model of the effect of public investment on long‐term growth – called the Long‐Term Growth Model Public Capital Extension (LTGM‐PC) – that is simple enough to be solved in an Excel spreadsheet without macros (which is provided as a companion to this paper on the website www.worldbank.org/LTGM).<sup>3</sup> Unlike coefficients estimated in most empirical studies, the LTGM‐PC allows for the effect of extra public investment to vary across countries and over time within the same country. In the model, the effect of an increase in public investment (or the quality of that investment) and the full dynamic growth path depend on country‐specific factors such as the scarcity of public capital (relative to GDP) and some crowding in of private investment. The model also allows for the fact that the public capital stock might be of low‐quality construction, which is a practical concern in many developing countries.

More technically, our model builds on the celebrated Solow‐Swan growth model and another World Bank Excel‐based tool known as the Long‐Term Growth Model (LTGM) (Loayza and Pennings 2018, and

> 2 <u>http://www.worldbank.org/en/news/feature/2016/04/16/spending‐more‐and‐better‐essential‐to‐tackling‐the‐infrastructure‐gap</u>

> 3 The relevant data and parameters for all countries are already pre‐loaded into the LTGM‐PC spreadsheet.

2
Hevia and Loayza 2012), which we refer to as the Standard LTGM. However, in the Solow‐Swan model (and Standard LTGM), capital is simply an aggregate, and so those models cannot simulate the specific effect of an increase in public investment. In contrast, in the LTGM‐PC, total capital is split into public and private portions. The LTGM‐PC retains many other realistic growth drivers and features of the Standard LTGM, including other growth fundamentals (human capital, TFP, demographics, labor market participation by gender), and also implications for poverty rates. Section 2 presents the model and Section 3 describes how it is implemented in the Excel‐based tool.

Despite being theoretical, the paper draws extensively on the empirical literature to guide the choice of parameters. The most important parameter is the elasticity of output to public capital, φ , which we call the _usefulness_ of public capital. In Section 5 we review the evidence from two meta‐analyses and other literature, which suggests an elasticity of φ =0.17 for essential infrastructure and φ =0.10 for generic public capital like buildings (though users can also specify its value). We also calculate the country‐specific scarcity of public capital using a new public capital database from the IMF’s Fiscal Affairs Department.

However, we could not find a suitable measure of the fraction of public capital that is of high quality (we use “efficiency” and “quality” interchangeably).<sup>4</sup> So in Section 4 we develop a new cardinal Infrastructure Efficiency Index (IEI) to quantify the extent to which public capital is of high quality in different countries. The IEI is based on estimates of the fraction of roads that are in poor condition, water that never reaches its final customers, and electricity that is lost through transmission and distribution.

Our second contribution is to document how the quantity and quality of public capital vary across countries with different levels of development, and how this affects the impact of new public and private investment on growth (Section 6). This analysis is conducted through the lens of the LTGM‐PC and utilizes the cross‐country data on the IEI and public capital stocks collected for the Excel‐based tool.

Surprisingly, we find that the effect of an extra 1ppt of GDP of public investment on growth is roughly constant across different levels of development.<sup>5</sup> This puts us at odds with optimistic commentators claiming that sizable “infrastructure gaps” mean a larger growth dividend from public investment in low‐income countries. But it also puts us at odds with pessimistic commentators who claim that the low efficiency of public investment in developing countries ‐ due to corruption and mismanagement for example ‐ means that such projects have little effect on growth.<sup>6</sup> Overall a 1ppt increase in public investment as a share of GDP increases growth by 0.1‐0.2ppts in our model, depending on the calibration. As public investment is typically around 5% of GDP and usually less than 10% of GDP, higher public investment alone cannot turn a slow‐growing country into a tiger economy.

Instead, developing countries are short of _private_ capital, both relative to GDP and in absolute terms. Private capital as a share of GDP in low‐income countries is only 2/3 of that in middle‐income countries, and almost half that in high‐income countries. By our calculations, this means the return to

> 4 Other indices like the World Economic Forum’s infrastructure quality index or the IMF Public Investment Efficiency Index (PIE‐ X) include survey‐based scores or distance to the frontier analysis, which means that a quality or efficiency score does not reflect the cardinal or absolute fraction of public capital operating as it should (see Section 4). The literature uses the terms quality and efficiency interchangeably as well.

> 5 This result follows from measured public capital as a share of GDP being roughly constant across countries with different levels of development (which is possibly overstated in low‐income countries with weak governance; Keefer and Knack 2007).

> 6 As in Berg et al. (2015), the level of efficiency in the LTGM‐PC has no effect on the return to new public investment because the low quality of new public investment is exactly offset by greater need for public capital due to the poor quality of past public investment. See Sections 2 and 6.5.

3
private capital is highest in low‐income countries, relative to both advanced countries and also relative to the return on public capital. This stems from the relatively low levels of private investment in low‐income countries (whereas public investment in low‐income countries is actually larger as a share of GDP).

However, low‐income countries also have the most inefficient public investment – with an infrastructure efficiency index 1/5 lower than middle‐income countries and 1/3 lower than high‐income countries. Even though low‐income countries might not be short of _measured_ public capital – as public investment is likely overstated in many low‐income countries with poor institutions (Keefer and Knack 2007) ‐ low‐income countries are likely short of _efficient_ public capital that is actually useful in production. This means that in low‐income countries (i) the marginal product of efficient public capital ‐ if it could be installed ‐ is extremely high and (ii) there is substantial room for low‐income countries to boost growth through increases in efficiency. As high efficiency only affects output through new investment, countries with high existing rates of public investment (and low existing efficiency) have the most to gain. However, efficiency is extremely difficult to increase quickly, and so in practical terms the return to public investment will still be similar across different levels of development (as claimed above).

# _1.1 Definitions and related literature_

In this paper, we generally equate public capital with infrastructure for simplicity, though we recognize that not all public capital is infrastructure, and not all infrastructure is public. Public capital in the literature is defined as _core_ infrastructure made up of transport (roads, railways, airports) and utilities (water supply and sanitation, energy, ICT); but also hospitals, education buildings, other public buildings and public physical assets (Agenor 2013, Bom and Ligthart 2014).<sup>7</sup> Although the public sector dominates the provision of infrastructure in low‐ and middle‐income countries, in high‐income countries the private sector plays an increasingly important role, including in hybrid categories like Public‐Private Partnerships, or PPPs (IMF 2015). In the literature, infrastructure is generally thought to increase the productivity of private factors much like TFP (see for instance Romp de Haan (2007), Serven (2010) and Straub (2008)) – an approach we take here.

The closest modeling project to ours is the IMF’s Debt, Investment and Growth (DIG) Model (Buffie et al. 2012). While the DIG model seeks (in part) to estimate the effect of infrastructure on growth, it also aims to provide more analysis on the fiscal side on how public infrastructure might be financed. The DIG model also accounts for traded and non‐traded goods and optimizing consumers (among other things). While the fiscal analysis and other features of the DIG model are missing from the LTMG‐PC, the cost of those extra features is in complexity and transparency: for example, the DIG model cannot be solved in a standard Excel spreadsheet. Our default calibration of the usefulness of public capital φ� 0.17 , is the same as that used in the the DIG model. The LTGM‐PC and the DIG model in turn build on an earlier generation of models involving public capital, such as Baxter and King (1993) and Barro and Sala‐i‐ Martin (1992) and more recent models like Leeper et al. (2010).

> 7 IMF (2015): public capital is the accumulated value of public investment over time, which is the principal input into the production of public infrastructure, comprising _economic_ infrastructure (transport and utilities) and _social_ infrastructure (for example, public schools, hospitals and prisons).

4
# **2. A model of long‐term growth with public capital**

In this section, we provide an overview of the model structure (Sections 2.1‐2.3) and some intuition on growth drivers (Section 2.4). Section 3 describes how these model equations enter the LTGM‐ PC Excel‐based tool which enables users to run policy simulations.

# _2.1 The production function_

To analyze the effects of public capital on growth, we adapt the Standard LTGM by splitting aggregate capital stock into public and private portions. We assume a Cobb‐Douglas specification, where the two capital stocks have unitary elasticity of substitution, in contrast to being perfect substitutes in the Standard LTGM.<sup>8</sup> Based on the models in Eicher and Turnovsky (2000) and Agenor (2013), we first consider the following production function at time, t : Each firm takes technology (TFP), A� and public services S� as given, that is, these are externalities to the firm. K�� is the private capital stock, h�L� is effective labor, which can be further decomposed into h� , human capital per worker and L� , the number of workers. 1 �β and β are private capital and labor income shares. Next, we consider the following specification for public services S� : G� is the efficient physical public capital stock – the public capital that is actually used in production. ζ captures whether public capital is subject to congestion (or not) – discussed further below. φ is the _usefulness_ of public capital (more technically the elasticity of output to efficient public capital).

Due to corruption, mismanagement or pork‐barreling, only a fraction θ� �1 of measured public capital is useful for production. The measured capital stock K��� is what is recorded in international statistical databases, constructed using the perpetual inventory method. θ� is the average _efficiency/quality_ of the public capital stock. Equations (1), (2A) and (2B) can be written in a more conventional production function as:

# <u>Congestion ( ζ ∈�0,1� )</u>

In principle, the congestion parameter in Equations (2A) and (3) can take values between: ζ�1 (full congestion) and ζ�0 (no congestion). As long as ζ�0 , it is the _ratio_ of public capital to private capital that provides public services, rather than the absolute amount of public capital (Barro and Sala‐i‐ Martin 1992).<sup>9</sup> When there is a large amount of private capital relative to public capital, the public capital

> 8 See Appendix A1.5 for a general idea of how the Standard LTGM differs from the LTGM‐PC.

> 9 Congestion can also be measured in terms of both private capital and labor supply (see for instance Glomm and Ravikumar (1997)) or aggregate output, but these do not result in substantial changes to the analysis (Eicher and Turnovsky 2000). Aside from absolute congestion, there can also be relative congestion, in which case, congestion increases only if aggregate usage increases relative to individual usage (see Eicher and Turnovsky (2000) for further details on how relative and absolute congestion affect growth analysis).

5
becomes “congested” and its benefits diminish. The intuition for this is a road network: when there are too many cars on the road, it becomes jammed, reducing its capacity to add to output.

In the Excel‐based tool, we only allow for two cases, ζ∈�0,1�, for simplicity. In our main “congestion” specification, ζ�1. This means that K��� must grow faster than K�� to have a positive effect on output. In this scenario, there is decreasing returns to scale to private inputs (private capital and effective labor), and constant returns to scale to all inputs. In the appendix (available on the website www.worldbank.org/LTGM) and in some parts of the paper we take the alternative assumption that ζ� 0: public capital is a pure public good. When ζ�0 , there are constant returns to scale to private inputs but increasing returns to scale to all inputs, though as we assume φ��1 �β��1 , endogenous growth through capital accumulation is ruled out. ζ�0 is a polar case – in reality, almost all public goods are characterized by some degree of congestion.

# <u>The</u> _<u>efficiency/quality</u>_ <u>of public capital ( θ ∈�0,1��</u>

θ� ∈ [0, 1], reflecting that “a dollar’s worth of public investment spending often does not create a dollar’s worth of public capital” (Pritchett 1996) ‐ K��� units of capital act like G� �θ�K��� units, and it is only the latter that is useful for increasing output. That is, productive capital is sometimes not created at all; or supposedly productive capital is created but subject to implementation weaknesses and/or operational inefficiencies such that the cost is higher than the minimum required to build the capital.

More concretely, a low θ most closely resembles poor construction quality which impedes efficient operation of the public capital project. A good example of low quality/efficiency is a corrupt road construction project where the construction firm reduces the thickness of pavement to save money (and pays kickbacks to politicians/bureaucrats). The road surface then deteriorates much more quickly than it should if it were properly constructed, resulting in reduced travel speeds and capacity. This example closely relates to how we measure θ in practice based on the fraction of unpaved roads (or electricity/water transmission losses).

If θ mostly reflects construction quality, readers might wonder about other aspects of the public investment management process, such as poor project selection, excessive public investment in politically sensitive regions, or large vanity projects with little economic value. Unfortunately, it is close to impossible to assess the scale of these problems quantitatively across countries and so they are excluded from our Infrastructure Efficiency Index (IEI) (and from θ ), which is discussed in Section 4. To the extent that vanity projects are a different class of public investment (even less essential than other public buildings), it could be argued that they are less _useful_ for producing output and hence have a lower φ. But we would generally prefer adjusting down θ ‐ below that implied by the IEI ‐ which allows for potential improvement in the efficiency/quality of public investment in the future (and is closer to Pritchett’s original formulation).

# <u>The</u> _<u>usefulness</u>_ <u>of public capital for production ( φ∈�0,1� )</u>

The elasticity of output with respect to the public capital stock measures the _usefulness_ of public capital for production, assuming that the project is of maximum quality/efficiency ( θ�1 ). Essential infrastructure like roads, ports, power and water, tend to have a higher usefulness than less essential forms of public capital, like public buildings, even if these different types are constructed properly. The calibration of values for this parameter is discussed in Section 5.2.

6
<!-- Start of picture text --> t<br><!-- End of picture text --> 6 ~~Y;~~ = AlAc(Le)S_ emgp CRE -~B-ESOP t P Gm t t t—Or tt t t t—Or tt t t t ~~Mean~~ <u>Vt</u> ~~[es~~ Me ~~es~~ Qc ~~Neva]~~ Ne OOF ~~Peas)~~ At ~~[Oe~~ Or # ~~eee~~ kgm ~~fi~~ kP e] ~~as~~ hy }" t+1 t+1s: 1+ Gy,t+1 = [d+ Tea) O99] (1 + Gatsi)(1 + 9oe+1) (1 + 9m ¢44)° (1 + Ie) (A + Tne+1) t+1 x,tt1 14+Tiya =(1+ Jort+1)(1 + Got41)(1 + 9nt41) 1+ Tey (¢ = 1) PC Po ~~Yt~~ _ ~~Ye~~ t t M bt ¢t 1+ 9¥f41 = (1+ Iyt+i(4 + Jot+1)(1 + Ju,t+1) 14+ Gye. = (1+ yeas) + 9nt+1) (1 — 6%) KeTM G G t om = (1-8) Kg" + If

> om om = (1- ~~89)~~ + ~~1&~~ <u>/y,</u> 1+ g,Gm ~~KAT Lesa~~ _ ~~__§~~ 1-5¢) ~~em4t/%t~~ G ~~<u>jy,</u>~~ kote. KgTM Le (14+.90,t41)(14+90,t41)(14+9N,¢41)
The stock of efficiency‐adjusted public capital (which is actually used in production) evolves based on the previous period’s efficiency‐adjusted undepreciated stock and efficiency‐adjusted new investment θ��I�� .

Readers will note that Equation (13A) is the same Equation (1) in Berg et al. (2015), with the � efficiency of new investment being θ� rather than ε . Consequently, all of Berg et al.’s results on the effects of efficiency also go through here (discussed further below). Equation (13A) is also equivalent to equation (2) in Pritchett (2000), who refers to γ as the efficiency of public investment.<sup>10</sup> Here one can interpret 1/ θ�� as the dollar cost of providing an extra dollar of usable public capital. Hence, corruption or other rent‐seeking which reduce the quality of public investment effectively increase the cost of a given increase in the productive capital stock as found empirically by Olken (2007) and Collier et al. (2015).

θ� is the average efficiency of existing public capital (rather than the efficiency of new investment). Substituting G� �θ�K��� into Equation 13A and rearranging as 13B, one can see the θ��� evolves as a weighted average of the quality of existing public capital θ� , and the quality of new investment � θ� .

As such, the quality/efficiency of the stock of public capital only changes when the quality of new � 11 investment projects is different from that of the existing public capital stock: θ� �θ�. Using equation (13B), the growth in quality which enters equation (6) can be written as follows: The quantity of private capital follows the same accumulation process as public capital. But with δ<sup>�</sup> as the private capital depreciation rate, and I�� as private investment. The growth rate of private capital per worker is as follows:

> 10 The measured public capital stock K��� here is G�� in Berg et al (2015). Pritchett (2000) refers to K�� in his Equation (2) as the efficient capital stock.

> 11 The treatment of “new” investment versus maintenance expenditure requires some clarification. For instance, Buffie et al. (2012), in their macroeconomic model of public investment effects, consider infrastructure investment as encompassing net investment, as well as operations and maintenance; and treat the depreciation rate as exogenous. Kalaitzidakis and Kalyvitis (2004), in their infrastructure‐led growth model, specify the accumulation of public capital as a function of new investment, and the depreciation rate depends on maintenance expenditure. Our model is more in line with Buffie et al. (2012), in that

> depreciation is exogenous. Conceptually, I�� in our model could include spending on major repairs, which along with new investment helps offset the capital decumulation effects of depreciation. But practically, we note that maintenance spending is typically subsumed under public consumption data and hence is hard to gauge. (From a national accounts perspective, the SNA (1993) notes “ordinary maintenance and repairs to keep fixed assets in good working order are intermediate consumption. However, major improvements, additions or extensions to fixed assets which improve their performance, increase their capacity or prolong their expected working lives count as gross fixed capital formation. In practice, it is not easy to draw the line...Some analysts...would favor a more “gross” method...all such activities are treated as gross fixed capital formation.”) User concern about insufficient maintenance spending could thus be reflected as higher depreciation rates.  Developing countries tend to spend less on operations and maintenance, which could imply higher depreciation rates than developed countries (Devarajan, Swaroop and Zou (1996), in regression analysis for a sample of developing countries, find public capital expenditure and economic growth to be negatively correlated but that current expenditure has positive effects, illustrating how capital expenditure may have been excessive, while current expenditure insufficient). However, developed countries are more likely to hold a higher share of more sophisticated assets that are subject to faster depreciation, making the net implication for depreciation rates not readily obvious (Arslanalp et al. 2010).

8
1+ ~~(1-8")+igre~~ IP /y. Ik? t+1 ~~ood~~ <u>(14+.90,t+1)(1+90,t+1)(1+9N,t41)</u> Ind+g)=@9 yeet ~~fot~~ = Gatsi t B(Got+1 F Gutt1 + Snt+1) -(- B)(9nt+1) +¢ lov tt t _ 3 +(1-B_—p-~ 66) ~~(eeeM~~ —_ 5”)sp <u>Og ees</u> _ <u>?</u> N aie /y, a,KeTM/y, © ~~_~~ O:KETM/¥, t=O— 2mG tN t tN_=% p/(KETM 4) (KeTMG 4)/¢ <u>dgyts1</u> _ <u>1-B-Co</u> arp /¥, —KP/Y¥e t <u>le /Ye KETM/Y,</u> _ 65 Kom —=KevtG ooG 1 — <mark>|p.</mark> Get _ ~~[at 1f/%~~ <u>yam</u> -—6 c| __ ~~[o-8] 17Gm/%~~ + ~~leGm [Ye~~ -—6 G 0, KETM/Y a IKeTM/Y KeTM/Y

~~[2%~~ 6, 1KETM/y, ~~lf /Ye~~ Kem tN_=% t
forces in the production function. First, lower quality/efficiency naturally means that there is a smaller increase in efficient public capital for each extra 1ppt of public investment. Second, in economies with lower efficiency, the stock of efficient public capital is scarcer, and hence has higher marginal product.<sup>12</sup> From equation (16), TFP growth, g�,��� has the largest direct effect on growth. The effect of most other factors depends on the labor share, β�1 . The larger is β , the lower is the effect of private capital accumulation on growth. For both public and private capital accumulation, holding all else constant, the same level of investment‐to‐output becomes less efficient as the capital‐to‐output ratio rises.

# **3.           Implementing public capital in the Long‐Term Growth Model (LTGM‐PC)**

||Sub‐model 1|Sub‐model 2|Sub‐model 3|
|---|---|---|---|
|Public Investment/GDP<br>Private Investment/GDP|**Input**<br>**Input**|Output<br>**Input**|**Input**<br>Output|
|Growth Target|Output|**Input**|Output|
|National Savings/GDP|Output|Output|**Input**|

In practical terms, using the LTGM‐PC involves choosing the path for several inputs in the future (exogenous variables), and then the LTGM‐PC calculates the future implied path of the outputs (endogenous variables). The LTGM‐PC has three sub‐models (1‐3) where the endogenous and exogenous variables in the model are switched. Other growth drivers – growth in TFP (A), human capital (h), labor participation rate ρ , working age‐population ratio ( ω ), and population ( N� respectively are always exogenous, as in the Standard LTGM. The LTGM‐PC also allows for output growth to affect poverty rates, as in the Standard LTGM. More technically, the LTGM‐PC has three “state” variables, which are predetermined at any point and change slowly over time: the public and private capital stocks (usually expressed as a ratio to GDP) and the average efficiency of installed public capital.

In this paper, we mostly use **Sub‐model 1** where future paths of public and private investment (as a share of GDP) are exogenous and the path of GDP (or GDP per capita) is endogenous. Alternatively, this can be reversed and **Sub‐model 2** can calculate the required public investment ratio to achieve a specified growth target (given an exogenous private investment share). In both Sub‐models 1 and 2, savings rates are calculated as a residual for an assumed path of the current account balance to GDP ratio. In **Sub‐ model 3** , the user instead specifies national savings rates and public investment rates as exogenous, with the model calculating implied private investment and growth rates.

> 12 Countries with different efficiency levels will have the same marginal product of public investment given the same parameters and initial conditions. The marginal product of public investment depends on the marginal product of efficient public capital and the translation of public investment to efficient capital and can be expressed as follows (using Equation (3) and (13)): ∂Y���⁄∂I�� ��∂Y���⁄∂θ���K������ �∂θ���K�����⁄∂I�����φY���⁄θ���K������θ�� Note that θ��� and θ�� cancel out if efficiency is unchanged.

10
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# **4** . **Evidence on the efficiency/quality of the public capital stock,** θ

This section develops a new Infrastructure Efficiency Index (IEI) to measure the proportion of public capital spending that delivers useful public capital/infrastructure services. For example, power lines and power plants might not deliver electricity to households and businesses, dams and pipelines might not be able to deliver water due to leaks, and roads may be in poor condition (such as being unpaved) (World Bank 1994). The IEI combines these measures into a single index for all countries.

# _4.1 A new Infrastructure Efficiency Index (IEI)_

An infrastructure efficiency index for measuring θ� in the model above should have several key features: (i) informative about many countries; (ii) simple and transparent in its construction; and (iii) cardinal rather than conveying a relative rank/score. A cardinal index is needed because a doubling of θ� in our model _doubles_ the efficient public infrastructure stock, whereas a doubling in score or rank could mean the increase in the efficient public infrastructure stock is smaller or greater than double. While there are other infrastructure efficiency indices in the literature (surveyed below in Section 5.2), none of them have all three features. Our new Infrastructure Efficiency Index (IEI) does, following a similar methodology as Rioja (2003)<sup>15</sup> but for as many countries as possible subject to data availability.

Specifically, we construct the index using more recent data for three indicators – (i) electricity transmission and distribution losses (% of output); (ii) water losses (% of provision); and (iii) paved roads (% of total roads). Ideally, we would like to have used the percentage of paved roads in good condition as a subcomponent of the IEI. However, this statistic is no longer available. As such, we follow Calderon and Serven (2010) who use paved roads (% of total roads) as an indicator of the quality of road networks in their analysis of the quantity and quality of infrastructure services in Latin America. We nevertheless recognize that unpaved roads are not always undesirable or inefficient and may depend on country‐ specific geographic features. Electricity losses reflect inefficiency at the transmission and distribution stages ‐ which is what our index intends to measure ‐ but also electricity delivered but not paid for which can be attributable to theft and unmetered supply. For some countries the second type of non‐technical losses can be large – however, this can still to some extent be seen as related to dysfunctional infrastructure in terms of construction and management (or operational inefficiency) (Jimenez et al. 2014). We also do not include telecommunications in the composite index since fixed‐telephone line faults may no longer be relevant due to the rising importance of mobile telephony, and data on the quality of mobile phone service are not as extensive.

In calculating the IEI, we want to include the latest data, but also recognize that infrastructure losses for a single year can be very noisy. As such we take the average of the index for the latest available year and the post 2000 average, using a weighted average of water losses, power losses and paved roads:

> 15 Rioja (2003) uses the physical infrastructure losses reported in World Bank (1994), weighted by corresponding infrastructure stock shares (from Ingram and Fay (1994)), to proxy the efficiency parameter for public capital stock in seven Latin American countries and five industrialized countries. The infrastructure loss indicators comprise electricity power transmission and distribution losses (% of output), faults per 100 main telephone lines per year, percentage of paved roads not in good condition, and water losses (% of total provision).  Calderon and Serven (2004, 2010) construct an infrastructure quality index based on the first principal component of electricity losses, the share of paved roads, and the waiting time for telephone line installation.

13
> [EI = ~~21~~ (Sie1 n n,avg(2000-—latest) + Yae1 n n,latest available value (starting 2000) )

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Further details on the sources of data, IEI summary statistics as well as discussion on robustness checks for the index are provided in Appendix 2.

- _4.2 IEI versus other measures of public investment efficiency_

In this section, we briefly discuss some of the other measures of public investment efficiency that are available. These serve as useful checks against the IEI, but do also face some limitations for use in our model – primarily that they indicate relative performance rather than being a cardinal measure.

Afonso, Schuknecht and Tanzi (2005, 2010) and various IMF papers, starting with Albino‐War et al. (2014) take the approach of distance to frontier, where inefficiency is measured relative to best performing peer countries. In the case of the papers by Afonso, Schuknecht and Tanzi (2005, 2010) which respectively cover 23 industrial countries (2005 paper), and 23 emerging and new EU member states (2010 paper), the output measure is a composite of public sector performance based on a series of quantitative and qualitative socio‐economic indicators, while the input measure is public sector spending (i.e. more than just public investment).<sup>16</sup> For our model analysis purposes, we find that the results from these two papers may not be suitable because: one, the outcome variable encompasses broad, indirect macroeconomic outcomes; and two, efficiency is compared within a small group of countries – this may be particularly worrisome in the case of the emerging market/new EU member sample.<sup>17</sup> On the other hand, the IMF public investment efficiency indicator (PIE‐X) covers more than 100 countries. The output variables are directly related to infrastructure ‐ a quantity index (physical infrastructure coverage and provision of social services<sup>18</sup> ) and a survey‐based quality index<sup>19</sup> respectively, as well as a hybrid of the two; while the input variable is public capital stock per capita.<sup>20</sup> While individual country efficiency scores have not been published, group averages of the quantity indicator suggest that advanced economies are 70% efficient (infrastructure output could be increased by 30% for the same amount of public capital input), emerging markets are 60% efficient, while low‐income developing countries are at about 45% efficiency. Nevertheless, these are still _relative_ performance indicators rather than cardinal indicators of quality: a score of 70% does not mean that 30% of infrastructure stock is not productive but rather the economy is operating 30% below the best performer in its peer group.

Aside from the above measures of inefficiency, there is also the IMF Public Investment Management Index (PIMI) (see Dabla‐Norris et al. 2012), a purely qualitative indicator based on scores for individual country performance in terms of the investment process (project appraisal, selection, implementation and evaluation). While it provides information on relative performance across 71 developing countries and shows positive correlation with GDP per capita and indicators of governance quality, it is not a cardinal indicator of the proportion of public capital that is productive, and only

> 16 Qualitative indicators of corruption, red tape, judiciary efficiency, public infrastructure quality; and quantitative indicators ‐ shadow economy size, secondary school enrollment, education achievement, infant mortality, life expectancy, as well as broad macroeconomic outcomes – income distribution, growth performance and stability, inflation and unemployment.

> 17 Sinha (2017) uses the emerging market/new EU member state average efficiency as a reference point for public investment efficiency in Bangladesh.

> 18 Length of road network, electricity production, access to water, number of secondary teachers and number of hospital beds.

> 19 Based on the World Economic Forum Global Competitiveness Report survey responses on the quality of key infrastructure services.

> 20 With GDP per capita as an auxiliary input variable.

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measures the quality of input process. Despite this, Gupta et al. (2014) normalize the index on a 0‐1 scale and use it as a measure of efficiency‐adjusted public capital effects on growth based on a sample of 52 countries. They find that upper middle‐income countries have on average 57% efficient public capital stock against 46% for lower middle‐income countries, and 38% for low‐income countries.<sup>21</sup> Table 2 summarizes the PIE‐X and PIMI and how they compare against the IEI. While caution should be exercised in comparing outcomes in absolute terms across different methodologies, one crucial takeaway is that all methodologies point to a gap between high‐income countries and low‐income countries. Thus, there appears to be substantial room for improvement in efficiency in low‐income countries which could lead to a better growth performance.

**<u>Table 2: Indicators of Efficient Public Capital Stock ‐ Different Methodologies</u>**

|**Infrastructure**<br>**Index**<br>**(this pa**|<br>**Efficiency**<br>**(IEI)**<br>**per)**|<br>**E**|<br>**Public Inv**<br>**fficiency Indi**|<br>**estment**<br>**cator (PIE‐X)**||<br>**Efficiency‐Adj**<br>**Capital, usi**<br>**Investment M**<br>**Index (**|**usted Public**<br>**ng Public**<br>**anagement**<br>**PIMI)**|
|---|---|---|---|---|---|---|---|
||**Average**||**Hybrid,**<br>**average**|**Physical,**<br>**average**|**Survey‐**<br>**based,**<br>**average**||**Average**|
|High Income<br>(21)|0.844|All countries<br>(132)|0.730|0.570|0.800|Upper Middle<br>Income<br>(14)|0.572|
|Upper Middle<br>Income<br>(25)|0.767|Advanced<br>(26)|0.870|0.700|0.900|Lower Middle<br>Income<br>(22)|0.456|
|Lower Middle<br>Income<br>(27)|0.738|Emerging<br>Markets<br>(62)|0.730|0.600|0.800|Low Income<br>(14)|0.382|
|Low Income<br>(10)|0.576|Low Income<br>Developing<br>Countries<br>(44)|0.600|0.450|0.750|Notes:<br>The capital accum<br>process is based<br>public investmen|ulation<br>on efficient<br>t. The portion|
|||||||of public investm|ent deemed|
|Notes:<br>Mean of simple an<br>average of output<br>delivery (for elect<br>water) and paved<br>percentage of tot|d weighted<br>not lost in<br>ricity and<br>roads as<br>al roads.|Notes:<br>Based on efficie<br>stock per capita<br>and/or quality a<br>mean of the phy<br>Source: IMF (20|ncy frontier an<br>as input and in<br>s output. The<br>sical and surve<br>15).|alysis with publi<br>frastructure qu<br>hybrid indicator<br>y‐based indicat|c capital<br>antity<br>is a simple<br>ors.|<br>as efficient is bas<br>invariant PIMI sc<br>1 and 4) that hav<br>normalized (4 to<br>Source: Gupta et|<br>ed on a time‐<br>ore (between<br>e been<br>1).<br>al. (2014).|

> 21 We group Gupta et al. (2014)’s individual country calculations according to the World Bank income classification scheme.

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# **5. Model calibration**

# _5.1 Setting up the baseline and scenarios_

To make empirical comparisons _between_ the Standard LTGM and the public capital extension (LTGM‐PC) as well as _within_ the public capital extension (public versus private investment, increasing quantity vs quality of public investment), we run simulations using both models for a sample of 147 countries. To solve each model, we need to first input baseline parameters, initial conditions and paths of variables for individual countries. Data sources and calculations are detailed in Table 3.

**<u>Table 3: Baseline Parameters, Initial Conditions and Paths of Variables</u>**

|<br>**Constant Parameters**|<br>**Model(s)**|<br>**Source/Calculation**|
|---|---|---|
|Labor shareβ|Std LTGM<br>LTGM‐PC|Latest individual countryβ, based on Penn World Tables (PWT) version 8.1,<br>or the Global Trade Analysis Project (GTAP) database. Labor shares below<br>0.45 or above 0.7 were trimmed to those values to remove outliers.|
|Elasticity of output to public<br>capitalφ|LTGM‐PC|0.170 (default) and 0.100 for all countries. See Section 5.2 for discussion of<br>this calibration.|
|Congestionζ|LTGM‐PC|1(congestion)or 0(purepublicgood)is assumed for all countries.|
|Efficient public capital stockθ|LTGM‐PC|Infrastructure Efficiency Index (IEI) – see Section 4. Unchanged efficiency is<br>assumed for all countries.|
|Aggregate depreciation rateδ|Std LTGM|Latest value from PWT 8.1.|
|Public capital and private capital<br>depreciation ratesδ<sup>�</sup>, δ<sup>�</sup>|LTGM‐PC|δ<sup>�</sup>= 2% for all countries, which is the average depreciation rate for<br>structures in the PWT 8.1.δ<sup>�</sup>, individual country‐specific, is the residual<br>from the weighted average calculation.|
|**Initial Conditions**|||
|Initial capital‐to‐output ratio<br>��<br>��|Std LTGM|Individual country values are based on the most recent data in PWT 8.1.<br>(Capital stock at constant 2005 national prices (2005 US$ mil.)/Real GDP at<br>constant 2005 national prices (2005 US$ mil.)). K/Y below 1.5 or above 3.5<br>were trimmed to those values to remove outliers.|
|Initial measured public capital‐<br>to‐output and private capital‐to‐<br>output ratios<br>��<br>��<br>��<br>~~,~~<br>��<sup>�</sup><br>��|LTGM‐PC|Individual country values are derived by applying 2015 shares of public and<br>private capital calculated by the IMF Fiscal Affairs Dept. to PWT data on<br>��<br>��<br>~~.~~<br>IMF data:http://www.imf.org/external/np/fad/publicinvestment/.|
|**Projected Paths**|||
|TFP growthg�|Std LTGM<br>LTGM‐PC|Individual country average value over 2001‐2010 based on PWT 8.1 (or<br>applying PWT methodology with GTAP labor shares). Missing TFP in PWT<br>calculated by Barrot (2016) using GTAP labor share. Negative TFP growth<br>adjusted upto zero.|
|Human capital growth rateg�|Std LTGM<br>LTGM‐PC|Individual country average value over 2001‐2010 from PWT 8.1, based on<br>years of schoolingand returns to education.|
|Growth in labor participation<br>rateg�|Std LTGM<br>LTGM‐PC|International Labor Organization (ILO) individual country data for 2014,<br>available from the World Development Indicators(WDI)Database.|
|Population growthg�|Std LTGM<br>LTGM‐PC|World Bank compilation of data sourced from the United Nations<br>Population Division and countrycensuses. Projections for 2016 to 2050.|
|Growth in the working age‐to‐<br>population rateg�|Std LTGM<br>LTGM‐PC|World Bank estimates based on United Nations Population Division data on<br>age and sex distribution ofpopulation. Projections for 2016 to 2050.|
|Investment‐to‐output ratio<br>�<br>�|Std LTGM|Individual country data is based on most recent data in WDI (Gross fixed<br>capital formation (constant 2010 US$)/GDP (constant 2010 US$)).|
|Public investment‐to‐output and<br>private investment‐to‐output<br>ratios<br>�<sup>�</sup><br>�<br>,<br>�<sup>�</sup><br>�|LTGM‐PC|Individual country values are derived by applying the 2015 shares of public<br>and private investment calculated by the IMF FAD to WDI data on<br>�<br>�<br>~~.~~IMF<br>data:http://www.imf.org/external/np/fad/publicinvestment/|
|The sample of countries is guided by t<br>affected countries based on the media<br>countries was reduced to 147 to exclu|he availability o<br>n value of avail<br>de outliers that|f human capital data. For missing data in other variables, these are interpolated for the<br>able data for the corresponding country groups by income. An initial sample of 151<br>appeared in the top graphs of Graph 2(The Bahamas,Egypt,Iran and Uzbekistan).|

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# _5.2 Calibration of select parameters_

# <u>5.2.1.  The usefulness of public capital φ (elasticity of output with respect to public capital)</u>

The effect of public investment on growth is most sensitive to the elasticity of output to public capital, φ , which we calibrate carefully in this subsection based on two meta‐analysis studies ‐‐ Bom and Ligthart (2014) and Nunez‐Serrano and Velazquez (2016), and a recent paper by Calderon et al. (2015). Meta‐analyses are necessary due to the controversy in the literature on this parameter with a range of studies with different samples, definitions and methodologies. Our default value is φ�0.17 , which is the upper bound across the two meta‐analyses and should be applied to “essential infrastructure” or “productive capital” only and is the same as in the IMF’s DIG model (Buffie et al. 2012). As such, our default calibration should be viewed as being relatively optimistic about the growth effects of public capital. An alternative value (which could be viewed as a lower bound) is φ�0.10 for all public capital. This latter value is slightly larger than the estimates in Calderon et al. (2015) of φ�0.07 �0.10 .

Bom and Ligthart (2014) look at 68 empirical studies<sup>22</sup> which cover the period 1983 – 2008. These studies are based on the production function approach and measure public capital in monetary and stock terms. Bom and Ligthart’s meta‐analysis indicates an average elasticity of output to public capital of 0.106 (not reported), which is higher in the long run and for core public capital, but lower for aggregate public capital/national government level (compared to the regional/local government level).

Nunez‐Serrano and Velazquez (2016) consider 145 papers<sup>23</sup> which cover the period 1983 – 2011. The empirical studies scrutinized are predominantly those that take a production function approach (85% of the sample) and include studies that use non‐monetary (17%) and flow (7%) measures of public capital. They find an average elasticity of output to public capital of 0.132, which is higher in the long run, 0.161. Though with less statistical significance, there is also indication that the elasticity value is higher with productive public capital. The distinction between monetary and non‐monetary measures of public capital does not have a discernible effect. A summary of select results from the respective meta‐analyses of Bom and Ligthart (2014) and Nunez‐Serrano and Velazquez (2016) is presented in Table 4.

**<u>Table 4: Elasticity of Output to Public Capital,</u>** φ **(“Usefulness”)**

|Bom and Ligthart(2014):*||
|---|---|
|Core infrastructure<sup>a</sup>|0.170|
|Totalpublic capital|0.122|
|Nunez‐Serrano and Velazquez(2016):*||
|Productive capital<sup>b</sup>|0.175|
|Totalpublic capital|0.161|
|Calderon et al. (2015) (infrastructure):*|0.070 to<br>0.100|
|aRefers to roads, railways, airports and utilities.<br>bRefers to capital aimed at health, education, housing and community se<br>installations, communication and transport infrastructure. Values are comp<br>results reported in Table 3, Section B.a. of Nunez‐Serrano and Velazquez (201<br>*Longrun estimates. Note that most studies control for the total orprivate capi|rvices, energy<br>uted based on<br>6).<br>tal stock.|

> 22 Of the 68 studies, 5 cover country groups (of which one is exclusively developing countries) and 63 are country‐specific (all advanced economies).

> 23 Of the 145 studies, 26 cover country groups (of which five are exclusively developing countries) and 119 are country‐specific (of which five are developing countries).

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Neither of the two meta‐analyses contains a discussion of potential differences in φ between developed and developing countries. However, Calderon, et al. (2015),<sup>24</sup> using a relatively extensive cross‐country sample, find that the long‐run elasticity of infrastructure does not seem to vary with countries’ per capita income levels, infrastructure endowment or population size.

# <u>5.2.2.  Initial capital‐to‐output ratios and depreciation rates</u>

Our default data source for aggregate data on capital stocks and depreciation rates is the PWT 8.1, which is also used in the Standard LTGM. Unfortunately, PWT 8.1 does not include the split into public and private capital so we rely on data from the IMF for the relative shares of public and private capital stocks (see IMF 2015 and <u>http://www.imf.org/external/np/fad/publicinvestment/).</u><sup>25</sup> Given different computation methodologies, aggregate K/Y ratios differ between the PWT 8.1 and the IMF, and are on average lower in the latter case.<sup>26</sup> We assume that public capital is mostly structures, and so apply the 2% structures depreciation rate from PWT 8.1. The private capital depreciation rate is a residual determined by the country‐specific PWT 8.1 depreciation rate for aggregate K: δ�δ<sup>��</sup> �<sup>��</sup> + δ<sup>��</sup> �<sup>�</sup> ~~.~~

# = = <u>5.2.3.  Congestion (ζ 1) vs pure public good (ζ 0) and actual vs measured TFP growth</u>

Users of the LTGM‐PC must choose either a calibration with congestion (ζ=1) or without it (pure public good ζ=0). Unless users strongly feel that public capital in their country is not congested, we recommend using the congestion calibration. We use this ourselves as the default for the results in this paper and in the Excel‐based tool. The reason is that without congestion, _actual_ TFP in the model ( A� in equation 1) tends to depart from _measured_ TFP using standard growth accounting exercises. In any practical application of the LTGM‐PC, the actual TFP growth rate is one of the most important assumptions ‐ and also the most difficult to calibrate. When using the congestion specification, the actual TFP growth rate is similar to what one would get applying a standard growth accounting exercise. In contrast, without congestion the measured TFP growth rate is above the actual TFP growth rate, and so cannot be used to guide the calibration of the actual TFP growth rates without some kind of adjustment.

To see this, for each country we used the LTGM‐PC to generate a growth path by assuming that actual TFP growth A� was the same as that recorded in PWT 8.1 (over a 10‐year average).<sup>27</sup> Then we performed a standard growth accounting exercise on the generated growth rates (given growth rates of other growth fundamentals like human capital and the total capital stock), to generate _measured_ TFP growth. We then compared the measured TFP growth to the inputted actual TFP growth.

> 24 In the study, infrastructure is measured as a composite of several physical infrastructure indicators (as opposed to the monetary value of capital stock). A panel data set is used, comprising 88 countries that cut across different income levels and infrastructure endowments.

> 25 We use the following data from the IMF to calculate the shares: general government capital stock and private capital stock in billions of constant 2011 international dollars.

> 26 Both sources of data use variations of the perpetual inventory method to estimate aggregate capital stocks and do not take account of the destruction/damage of capital from wars/conflicts (which is naturally difficult to measure). Wars/conflicts also reduce output, perhaps faster than they destroy capital, and so the aggregate K/Y ratio may rise as a country enters a war/conflict. In the reconstruction period, it would not be surprising if measured public capital it extremely high because it fails to take account of the public capital destroyed in the conflict.

> 27 To be clear, the exercise is not dependent on the source of assumed actual TFP growth, but just that its distribution across countries is reasonable.

19
A. = gy -B(gnt+ 91) -A-B)gx K Aactual —= Gy _ B(Gn + 91) — P(90 + 9xem) — (1 — B - BP) OKP (1 —B)gx gees = gqcrual (99 + Gxam)+(1-B-Sh)gKP = @=0 K © G_om © Oye # 0 meas = gactual > l=1 go = 0.17 <!-- Start of picture text --> e e e 8<br>° ry e 3 °?, °<br>e ° of, e °<br>* 2<br><!-- End of picture text -->
# **6. Results: Stylized facts and the effect of an increase in investment on growth**

- _6.1 Stylized facts on the infrastructure gap and the return to public/private investment_

It is often argued that there is a large public infrastructure gap in developing countries, with current public infrastructure falling far short of what is needed. From a human development perspective this is definitely true, based on figures like 700 million people without safe drinking water and 1.2 billion people without electricity (quoted in the Introduction). But does this public infrastructure gap mean that the return to new public investment in developing countries is much higher than that in developed countries? Put another way, are developing countries particularly short of public capital _relative to their level of development?_ What about if we adjust for the lower quality of public capital in developing countries? And are they short of public infrastructure relative to, say, private capital?

In this section we answer these questions through the lens of our calibrated model for representative countries at various stages of development based on the World Bank classification:<sup>29</sup>

- Low income (LI) ‐ GNI PC<$1,000

- Lower Middle Income (LMI):  $1,000< GNI PC<$4,000

- Upper Middle Income (UMI):  $4,000< GNI PC<$12,000

- High Income (HI): GNI PC>$12,000.

Parameters for each “representative country” are the within‐group medians (Table 5)<sup>30</sup> , taken from an overall sample of 108 countries with complete (non‐interpolated) data. A caveat here is that the sample of LI countries with complete data is quite small (only 12 countries), and so there is a chance that results for that group might change with better data. We report results using the default congestion setting (ζ=1) for essential infrastructure ( φ =0.17) – with robustness to other parameters reported in Appendix 3. In addition to answering the questions above, this also provides a guide to how the LTGM‐PC might be used in specific countries.

We find no evidence that _measured_ public capital is particularly scarce for LI or LMI countries relative to GDP. In fact, public capital as a share of output is relatively constant across various levels of development at around K<sup>Gm</sup> /Y =0.92 (±0.05), with LI countries having the highest K<sup>Gm</sup> /Y and LMI countries being in the middle of the group (Table 5 Panel A). If anything, it is HI countries that are relatively short of public capital, as their ratio of K<sup>Gm</sup> /Y =0.86 is the lowest.

How do we square this with the narrative of infrastructure gaps above? The first answer is that developing countries are short of productive public capital ( G�/Y�� , rather than measured public capital �K���/Y�� – an issue we revisit in Section 6.4. As argued in Keefer and Knack (2007), developing countries with poor institutions tend to have higher rates of public investment, which the authors argue are inflated to provide rents and kick‐backs.

The second answer is that developing countries have a shortage of public capital relative to their development aspirations, but not relative to their current development level. That is, people in low‐ income countries have many unmet needs, with public infrastructure capital being in just as short supply as everything else.

> 29 Cutoffs are expressed in GNI per capita (to 2 significant figures), calculated using the World Bank Atlas method.

> 30 Replication files for the main Tables and Figures are available from the authors upon request.

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As the “usefulness” of public capital does not vary with income (Calderon et al. 2015), the stability of the public capital‐to‐output ratio across income levels also means that the marginal product of measured public capital ( MPK<sup>��</sup> ) is not relatively larger in developing countries. As we will see in the next section, this is the effect of an expansion of public investment on growth in the short‐run with unchanged efficiency. Specifically, we find that with our relatively generous calibration of usefulness ( φ =0.17), that the MPK<sup>��</sup> is around 18.5% (±1%), which varies inversely with the public capital‐to‐output ratio (Table 5, Panel B). After subtracting the depreciation rate of 2% (constant across countries), this yields a return to new public investment of around 16.5% (±1%) which also does not vary systematically across levels of development. The relatively high absolute returns stem from the high assumed usefulness of public capital in our default calibration. If instead we assumed φ =0.1 (for public buildings), then the return to public investment falls to around 9% (±1%), though the ranking across income groups is unchanged.

While some might interpret the lack of a higher return to public investment in LI countries as negative, we are more sanguine. It also means that development banks need not refrain lending for infrastructure projects in countries with poor implementation capacity ‐ as they are often encouraged to do ‐ because that low capacity also means the projects are even more in need.<sup>31</sup> In contrast, it seems that LI countries have a shortage of private capital for their level of development, and the scarcity of private capital falls with per capita income. Specifically, K<sup>P</sup> /Y is around 1.25 for LI countries, around 1.80 for middle‐income countries and 2.25 for HI countries (80% higher than that of LI countries). Measured public capital is also the largest share of total capital in LI countries (44%) and the lowest share in HI countries (28%). This reflects the fact that private financial markets are typically underdeveloped in LI countries and so the private sector finds it difficult to raise funds for investment. In some countries insecure property rights also reduce their incentive to invest in the first place.

A consequence is that the marginal product of private capital is the highest in LI countries (25%), which is double that in HI countries (12%), with middle‐income countries in between. Note that the MPK<sup>�</sup> does not vary exactly inversely with the K<sup>P</sup> /Y across income groups, because of cross‐country variation in the capital share 1‐β.<sup>32</sup> After subtracting depreciation (around 5%), the return to private investment is the highest in LI countries (around 20%), lower in middle‐income countries (13‐17%) and lowest in HI countries (7%). Interestingly, the return to private capital for LI countries is actually higher than for public investment (20% versus 16%), suggesting that if savings are scarce, governments need to be careful that public investment does not crowd out private investment.

As today’s capital stocks reflect past investment, one might expect that public investment would make up a larger share of total investment in LI countries ‐ which is exactly what we find. Public investment is 37% of total investment in LI countries, double the share in HI countries (18%), with middle

> 31 For example, Keefer and Knack (2007) argue that their “results therefore signal the need for donor agencies to exercise particular caution in supporting public investment in countries with a weak institutional environment.”

> 32 The return to private capital is even higher with φ =0.1 (though its ranking across groups is unchanged), as the penalty for reducing the congestion of public capital is lower.

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income countries in between (23%).<sup>33</sup> Translated to a share of GDP, public investment spending increases steadily with lower income per capita: 4% of GDP in HI countries, 5% in UMI countries, 6% in LMI countries and 7% of GDP in LI countries. Keefer and Knack (2007) argue that this is likely due to poor quality governance in developing countries, rather than the level of income per se. Consistent with earlier results, private investment is particularly lacking in LI countries – around 13% of GDP versus 17‐18% of GDP in the other three income groups.

**<u>Table 5:</u> Median Values of Baseline Parameters and Paths of Variables by Income Group**

|**Parameter/Variable**|**Note**|**HI**|**UMI**|**LMI**|**LI**|
|---|---|---|---|---|---|
||**A.**|**Capital and**|**Investmen**|**t ‐ Group Me**|**dians***|
|Labor shareβ|(1)|0.561|0.450|0.503|0.520|
|Public capital share of total capitalK<sup>Gm</sup>/K||0.277|0.356|0.330|0.440|
|Capital‐to‐output ratio – TotalK0/Y0<br>(Measured) – PublicK0<br>Gm/Y0 <br>– PrivateK0<br>P/Y0|(2)<br>(3)<br>(3)|3.110<br>0.863<br>2.247|2.706<br>0.965<br>1.741|2.722<br>0.900<br>1.822|2.208<br>0.972<br>1.235|
|Depreciation rate – Totalδ<br>– Publicδ<sup>G</sup><br>– Privateδ<sup>P</sup>|(4)|0.040<br>0.020<br>0.047|0.038<br>0.020<br>0.048|0.042<br>0.020<br>0.053|0.035<br>0.020<br>0.047|
|Public investment share of total investmentI<sup>G</sup>/I|(7)|0.177|0.219|0.243|0.369|
|Investment‐to‐output ratio – TotalI/Y<br>– PublicI<sup>G</sup>/Y|(5)<br>(6)|0.223<br>0.039|0.223<br>0.049|0.236<br>0.057|0.201<br>0.074|
|– PrivateI<sup>P</sup>/Y|(6)|0.183|0.174|0.179|0.127|
|_No of countries_||_48_|_29_|_19_|_12_|
|_. _<br>|**B. T**|<br>**he return t**|<br>**o investmen**|<br>**t(withφ=0.**|<br>**17; ζ=1)***|
|Marginal product of measured public capital<br>_MP_K<sup>��</sup>�φ/�K�<br>��Y�<br>⁄<br>�||0.197|0.176|0.189|0.175|
|Return to_public_investment(MPK<sup>Gm</sup>‐δ<sup>G</sup>)||0.177|0.156|0.169|0.155|
|Marginal product of_private_capital<br>MPK<sup>�</sup>��1 �β�φ�/�K�<br>�Y�<br>⁄<br>�||0.120|0.218|0.180|0.251|
|Return to_private_investment(MPK<sup>P</sup>‐δ<sup>P</sup>)||0.072|0.170|0.127|0.204|
|For a general description of the sources of data, see Table 3. The sa<br>individual countries of PWT 8.1 data for labor share, capital‐to‐out<br>A3.1 for median values of other key variables used in the simulatio<br>Countries are classified according to the 2018 World Bank classifica<br>* Multiply by 100 (except values in ppts) to obtain parameter/varia<br>Note:<br>_(1) _Forβ,individual country values below 0.45 were increased to<br>_(2) _ForK�Y�<br>⁄<br>,individual country values below 1.5 were increase<br>_(3) _K�<br>��Y�<br>⁄<br>andK�<br>�Y�<br>⁄<br>are derived based on the median values o<br>_(4) _δ<sup>�</sup>is derived based on the median values ofδ,δ<sup>�</sup>andK<sup>��</sup>K<br>⁄ <br>_(5) _Median of 15‐year averages across countries over 2001‐2015.<br>Qatar, for which the same variable is tabulated from the IMF F<br>_(6) _I<sup>�</sup>Y<br>⁄andI<sup>�</sup>Y<br>⁄are derived based on the median values ofI<br>⁄|mple size for<br>put ratio, hum<br>ns.<br>tion of count<br>ble values in<br>0.45 and thos<br>d to 1.5 and t<br>fK�Y�<br>⁄<br>and<br><br> .<br> I/Yis gross fi<br>AD Investme<br>Y<br>andI<sup>�</sup>I<br>⁄ .|all variables i<br>an capital gro<br>ries by incom<br>percentage sh<br>e above 0.70<br>hose above 3.<br>K<sup>��</sup>K<br>⁄ .<br>xed capital fo<br>nt and Capital|s guided by the<br>wth and TFP gr<br>e for the 2017 c<br>are or growth t<br>reduced to 0.7<br>5 reduced to 3.<br>rmation (% of G<br>Stock Databas|consistent avail<br>owth. See Appe<br>alendar year.<br>erms (%).<br>0.<br>5.<br>DP) from WDI,<br>e.|ability for<br>ndix 3, Table<br>except for|

> 33 These figures are average investment rates over 2001‐15. In steady state K�/Y�I/Y/�g� �δ�� for _j=G,P_ . If the countries in Table 5 were in steady state, headline GDP growth rates would need to be ≈3% for HI countries, ≈3.8% for UMI countries, ≈4.8% for LMI and ≈5.6% for LI countries, which are fairly close to what we observe.

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# _6.2. The effect of an increase in public investment on growth_

Graph 3 shows the effect of a permanent 1ppt GDP increase in public investment on growth. In the year following the shock, growth increases by the marginal product of measured public capital (equation 16A with θ�� �θ� ), which as argued above is around 0.17‐0.19ppts and is surprisingly similar across countries with different levels of development (blue dotted line with φ�0.17 , congestion specification ( ζ�1 )).

**<u>Graph 3: Incremental Output Growth from a 1 ppt. Increase in Public Investment</u>**

**in the LTGM‐PC (Congestion** , ζ�1 **)**

<!-- Start of picture text --> HI UMI<br>% %<br>0.25 0.25<br>0.20 0.20<br>0.15 0.15<br>0.10 0.10<br>0.05 0.05<br>0.00 0.00<br>Standard LTGM Congestion, φ = 0.17 Standard LTGM Congestion, φ = 0.17<br>Congestion, φ = 0.10 Congestion, φ = 0.10<br>LMI LI<br>% %<br>0.25 0.25<br>0.20 0.20<br>0.15 0.15<br>0.10 0.10<br>0.05 0.05<br>0.00 0.00<br>Standard LTGM Congestion, φ = 0.17 Standard LTGM Congestion, φ = 0.17<br>Congestion, φ = 0.10 Congestion, φ = 0.10<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br><!-- End of picture text --> The second thing to note from Graph 3 is that the increment to growth falls over time. This is because as public capital accumulates, K<sup>��</sup> ⁄Y increases, which reduces the marginal product of measured public capital. This is intuitive: one would expect an increase in public investment to become less effective in boosting growth over time as “infrastructure needs” are met. There is some heterogeneity across income groups: the boost to growth is slightly less persistent in LI countries. This could be because these countries have a lower capital share ( 1 �β ), and so the marginal product of private capital which is “crowded in” declines more quickly (in addition to marginally higher K<sup>��</sup> ⁄Y which means the marginal product of public capital dwindles faster over time).

Finally, the effect of an increase in public investment in the LTGM‐PC (with φ�0.17 and ζ�1 ) is on average similar to the effect of a same‐sized increase in the aggregate investment in the Standard LTGM (red solid line) for all but HI countries. Specifically, the effect in the LTGM‐PC is very similar to the 24
Standard LTGM for middle‐income countries, slightly lower for LI countries, and higher for HI countries. The latter is because HI countries tend to have the lowest share of total capital owned by the public sector. The effect of public investment on growth is naturally much smaller in the LTGM‐PC when φ =0.10 (dashed gray line), but also much smaller than the comparable effect in the Standard LTGM (except in HI countries). Greater consistency with the Standard LTGM is one reason we prefer the φ�0.17 calibration over the φ =0.10 calibration.<sup>34</sup>

# _6. 3 The effect of private investment on growth_

Graph 4 reports the effect of a 1ppt of GDP expansion in _private_ investment on growth, which is largest for LI countries. In the first period following the shock, growth increases by the MPK<sup>�</sup> (Equation 16B) which as argued above is twice as large for LI countries as HI countries (25ppts versus 12ppts). As before, the increase in growth falls over time as private capital accumulates, raising K<sup>�</sup> ⁄Y and lowering the marginal product of additional private capital. Nonetheless the effect of higher private investment on growth is quite persistent, verifying our claim above that developing countries are particularly short of private capital.

**<u>Graph 4: Incremental Output Growth from a 1 ppt. Increase in Private Investment</u>**

**in the LTGM‐PC (Congestion** , ζ�1 **)**

<!-- Start of picture text --> HI UMI<br>% %<br>0.35 0.35<br>0.30 0.30<br>0.25 0.25<br>0.20 0.20<br>0.15 0.15<br>0.10 0.10<br>0.05 0.05<br>0.00 0.00<br>Standard LTGM Congestion, φ = 0.17 Standard LTGM Congestion, φ = 0.17<br>Congestion, φ = 0.10 Congestion, φ = 0.10<br>LMI LI<br>% %<br>0.35 0.35<br>0.30 0.30<br>0.25 0.25<br>0.20 0.20<br>0.15 0.15<br>0.10 0.10<br>0.05 0.05<br>0.00 0.00<br>Standard LTGM Congestion, φ = 0.17 Standard LTGM Congestion, φ = 0.17<br>Congestion, φ = 0.10 Congestion, φ = 0.10<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br><!-- End of picture text --> 34 With the pure public good setting (Appendix 3, Graph A3.1), the immediate effect of higher public investment is broadly similar to that of the congestion setting, but is more persistent and higher than the congestion setting (and hence the Standard LTGM).

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1—-B-$@ = 0.17 (1-B-¢o)/o > P Gm tt PGm t t ~~st~~ <u>a t+</u> ~~et~~ <u>a itt</u> = ~~E~~ l-p - ce) - 6 ~~(—)~~ | Private Public

¢=017 f=1 P Gm P Gm ~=017 ¢=1 P Gm a—B-co/ ~~E)~~ > o/ ~~(©)~~ ¢ = 0.10 $ = 0.10
<!-- Start of picture text --> g = 0.17 f=1<br>eo. ome oeomngpe Bore me ee 4<br>2, ace ae eo ee °<br>° 3, e eod ‘ a Ce e<br>wD<br>e eee eo ><br>*| a-p-<#)- 6( 45) |<br><!-- End of picture text -->

<!-- Start of picture text --> @ = 0.17).<br>@ = 0.17 f=1<br>° ° e e ors oe<br>e . °. — e<br><!-- End of picture text -->
# _6. 5 The efficiency of public investment_

# <u>6.5.1  Stylized facts</u>

One of the reasons that public capital might have less effect on growth in developing countries is that it is inefficient. However, as already noted in Section 2, the _level_ of efficiency θ does not affect the marginal product of measured public capital and the effect of additional public investment on growth. For low efficiency countries, only a fraction of new investment might become productive capital, but this is offset by the greater need for public capital given that past investment was also inefficient.<sup>39</sup> However this does not mean that _increasing_ public investment efficiency cannot have a large effect on growth.

The potential for higher growth through greater efficiency of public investment depends on current efficiency being low. In the first row of Table 6 we calculate the median of the Infrastructure Efficiency Index (IEI) for each income group (see Section 4 for details on the construction of the IEI). As expected, based on the full set of computed IEI, HI countries have the highest median efficiency, with around 87% of roads being paved or water/electricity reaching their final destination. For middle‐income countries, efficiency is about 74%, or 1/7<sup>th</sup> lower. Efficiency is the lowest for LI countries, where only 59% of roads are paved or water/electricity reach their final destination; which is about 1/3 lower than efficiency in HI countries.

**<u>Table 6:</u> Efficiency – Income Group Medians (with φ=0.17; ζ=1)**

|**Parameter/Variable**|**HI**|<br>**UMI**|**LMI**|**LI**|
|---|---|---|---|---|
|Efficient public capital stockθ|0.870|0.753|0.730|0.590|
|_No. of countries**_|_21_|_25_|_27_|_10_|
|Efficient public capital‐to‐output ratioG�/Y��θK�<br>��/Y�|0.751|0.726|0.656|0.574|
|Marginal product of e_fficient_public capital_MP_K�<br>G= ~~�~~<br>�<br>����<br>����<br>�<br>�|0.226|0.234|0.259|0.296|
|Marginal product of efficiencyMPe�φ�<br>��<br>���<br>�<br>����<br>����<br>⁄<br>~~�~~|0.009|0.011|0.015|0.022|
|μ=I<sup>�</sup>/Y/θ � pptsincrease inI<sup>�</sup>/Yequivalent to 1pptsincrease<br>inθ<sup>�</sup>after oneyear(also equivalent by2040)|0.045|0.065|0.079|0.125|
|For a general description of the sources of data, see Table 3.<br>Countries are classified according to the 2018 World Bank classification of countries<br>* Multiply by 100 (except values in ppts) to obtain parameter/variable values in per<br>** Number of countries for efficiencyθis based on IEI data.<br>Calculations forθK�<br>��/Y�,_MP_K�<br>�,MPeandμ, are formed by combining the data in<br>(not on a constantgroupof countries).|by income fo<br>centage shar<br>Table 5 with|r the 2017 ca<br>e or growth te<br>the efficiency|lendar year.<br>rms (%).<br> θfor each inco|me group|

One can also use the IEI to calculate the _efficient public capital‐to‐output ratio_ θK���/Y� . Because the public capital‐to‐output ratio ( KGm0 /Y0 ) is roughly constant across levels of development, but efficiency θ increases with development, the combined efficient public capital‐to‐output ratio also increases with development. Specifically, the efficient public capital‐to‐output ratio is around 0.75 in HI countries, 0.73 in UMI countries, 0.66 in LMI countries and 0.57 in LI countries. This suggests that LI countries do not have a shortage of measured public capital (as argued above), but rather a shortage of _efficient_ public capital.

> 39 Berg et al. (2015) show that for the Cobb‐Douglas production function (as used here), these two are exactly offsetting. However, even with the CES production function the two factors are mostly offsetting unless public and private capital are extremely complementary (their Figure 3).

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How should we interpret the efficient public capital to output ratio θK���/Y� for policy? Equation (26) is the _marginal product of efficient public capital_ ( MPK�� ) _‐_ the effect of an extra ppt of GDP of efficient investment θ���I��⁄Y�� on growth (the derivative of equation 16 with respect to θ�� I��⁄Y� ). One can see that the MPK�� is inversely proportional to the efficient public capital‐to‐output ratio θK���/Y� . As such, a low efficient public capital‐to‐output ratio means that the return to an extra _efficient_ unit of public investment is high.

Because LI countries have the lowest G�/Y� �θK���/Y� they also have the highest MPK�� = 30%, which is almost _double_ the regular marginal product of public capital from Section 6.1. In contrast, middle‐income countries have a MPK�� of about 25%, while HI countries, a lower MPK�� of 23%.40 The high MPK�� means that if a typical LI country (with low efficiency) were somehow able invest efficiently, the returns for growth would be very high. But this is a hypothetical scenario. As Berg et al. (2015) and others point out, quickly increasing public investment implementation capacity is difficult, largely because public investment capacity has deep determinants ‐ like poor institutional quality and a lack of relevant bureaucratic human capital.

# <u>6.5.2  Increasing efficiency (“investing in in vesting”)</u>

However, it can still be that the efforts to improve efficiency are highly cost effective, even if they only lead to a slow increase in the average quality of the public capital stock. Here we consider the effect on growth of a 5‐ppt increase in the efficiency of new public investment from 2017 onwards (Graph 7), which is roughly a sixth of the gap between the efficiency of low and high‐income countries (first row Table 6). We also assume public investment for each income group is at its group median from Table 5 panel A (3.9% for HI countries, through to 7.4% for LI countries). Average efficiency reflects characteristics of the installed public capital stock, and so changes only take place slowly. The effect on growth is strongest for the representative LI country given its low‐quality stock, where increased efficiency boosts growth by 0.11ppts.<sup>41</sup> For middle‐income and HI countries, the average efficiency levels are much higher, and so the boost to growth is much smaller – around 0.05ppts for UMI and HI countries (with φ�0.17 ), and 0.075ppts for LMI countries.<sup>42</sup> For LI countries, “catch up” is easier because practices are so far from the frontier.

What determines the increase in growth from an extra unit of efficiency in the short run? Taking the derivative of Equation 16 with respect to θ�� , produces the _marginal product of efficiency_ ( _MPe_ ), which is � the boost to growth from a 1ppt increase in the efficiency of new public investment ( θ� � :

> 40 The large absolute size of the MPK�� stems from the generous calibration of usefulness ( φ =0.17). Instead, with φ =0.10, the MPK�<sup>�</sup> falls to 17% for LI countries, and 14‐15% in middle‐income and 13% HI countries.

> 41 Low base efficiency is important because it is the percentage (not percentage point) increase in efficiency that determines the effect on growth. A fixed 5 ppts increase in efficiency is a larger proportion of a low base.

> 42 As φ is the elasticity of output with respect to _efficient_ public capital, the effect of an increase in the efficiency is much lower with φ�0.10 than with φ�0.17.

29
In Table 6, one can see that the _MPe_ is highest in LI countries (0.022), which is more than double the rate in HI (0.009). UMI and LMI countries are in‐between (0.011 and 0.015 respectively). As such a 5ppt � immediate increase in θ� for LI countries will raise growth by 0.022 � 5ppts=11ppts, which is similar to the boost to growth in the first period in Graph 7. For HI countries, in contrast, a 5ppt increase in efficiency would raise growth by a much lower 5ppts (0.009 x 5ppts). The _MPe_ is inversely proportional to the _efficient public capital‐to‐output ratio_ , is increasing in the usefulness of public capital ( φ ) and also increasing in the public investment to output ratio ( I��⁄Y� ).

The last somewhat surprising result is because an increase in efficiency θ�� only affects new investment. Intuitively, in countries with low public investment rates, an increase in the efficiency of new public investment will only have a small short‐run effect on the average efficiency of installed capital – and hence on growth ‐ because the new efficient public capital is only a small fraction of the total capital stock. In these countries, a permanent increase in the efficiency of public investment will still boost output, but it will just take more time for these gains to materialize.

**<u>Graph 7: Incremental Output Growth from a 5‐ppt. Increase in the Efficiency of Public Investment in the LTGM‐PC (Congestion</u>** , ζ�1 **)**

<!-- Start of picture text --> HI UMI<br>% %<br>0.15 0.15<br>Initial θ: 0.87 Initial θ: 0.75<br>Scenario, θ N in 2017 and onwards: 0.92 Scenario, θ N in 2017 and onwards: 0.80<br>θ in 2035: 0.90 θ in 2035: 0.78<br>0.10 0.10<br>0.05 0.05<br>0.00 0.00<br>Congestion, φ = 0.17 Congestion, φ = 0.10 Congestion, φ = 0.17 Congestion, φ = 0.10<br>LMI LI<br>% %<br>0.15 0.15<br>Initial θ: 0.73 Initial θ: 0.59<br>Scenario, θ N in 2017 and onwards: 0.78 Scenario, θ N in 2017 and onwards: 0.64<br>0.10 θ in 2035: 0.76 0.10 θ in 2035: 0.63<br>0.05 0.05<br>0.00 0.00<br>Congestion, φ = 0.17 Congestion, φ = 0.10 Congestion, φ = 0.17 Congestion, φ = 0.10<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br>2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049<br><!-- End of picture text --> 30
# <u>6.5.3  Which countries should invest in an increase in the quantity versus quality of public investment?</u>

One can compare the effect on growth of an extra unit of public investment (Equation (16A) at constant efficiency), and the return to an extra unit of efficiency (Equation (27)). As the two are in different units, it is not appropriate to assess which marginal product is larger (as we did with private and public investment). Instead, we calculate the size of the increase in the public investment rate ( μ�� ppts of GDP) equivalent to a 1ppt increase in efficiency. The larger the value of μ�� , the more effective an increase in investment quality is at boosting short‐run growth (relative to boosting the quantity of investment). Setting MPe�μ�� �MPK<sup>��</sup> (from Equation (27) and (16A)): A nice feature of this comparison is that it does not depend on measured public capital scarcity ( K���⁄Y� ), or the usefulness of public capital ( φ ), which cancel out in Equation (28). That is, greater scarcity and usefulness increase MPe and MPK<sup>��</sup> proportionately and so do not affect the relative effectiveness of quality versus quantity (though they do affect the aggregate size of both marginal products). Rearranging implies: � ������ μ�� �� (29) Equation (29) suggests that increases in investment quality are particularly effective in (i) countries with a high rate of public investment (because only new investment is affected by the improved investment management processes), and (ii) countries with low existing quality of public capital (so the � improvement in quality is larger in percentage (not percentage point) terms). For LI countries, μ�� 0.13 ppts of GDP ‐ that is a 1ppts increase in the quality of public investment has the equivalent effect on short‐run growth as a 0.13ppts increase in quantity of public investment. Given that improving investment processes could be almost free (if feasible), saving 0.13ppts of GDP on public investment expenditure for the same short‐run growth outcome is good policy. Of course, improvements in quality are not as powerful elsewhere. For middle income countries, an extra unit of efficiency is worth μ�� � 0.07ppts of public investment, and for HI countries (with the lowest I��⁄Y� and highest quality θ ) μ�� � 0.05ppts.

For individual countries with available data on IEI and public investment, Graph 8 plots the size of recent public investment‐to‐output ratios (15‐year average, 2001‐2015) on the Y‐axis, and existing quality (as reflected by the IEI) on the X‐axis. The further a country is towards the top right of the figure, the more effective a 1ppt increase in investment _quality_ is relative to greater investment _quantity_ . Lines represent the locus of points for μ� 0.05ppts, 0.1ppts or 0.2 ppts. Specifically, many LI countries have public investment‐to‐output ratios that are greater than 5% and efficiency levels lower than 0.6 and so fit on the upper RHS with the most to gain. Most HI countries are on the lower LHS with high efficiency and low public investment rates, suggesting limited gains from higher efficiency. Outliers are China (CHN) and Malaysia (MYS) which appear to have relatively high efficiency levels ( � 0.85) but can still make sizable gains given relatively high public investment ratios of 20% and 11% respectively. China and Mozambique benefit the most overall, where a 1ppt higher efficiency is equivalent to ≈0.25ppts GDP increase in public investment.

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**<u>Graph 8: Quantity versus Quality of Public Investment</u>**

<!-- Start of picture text --> Who benefits the most from higher public investment efficiency?<br>LI<br>MORE gain from a 1ppt<br>CHN increase in efficiency LMI<br>UMI<br>HI<br>MOZ<br>NERVEN μ=0.05ppts<br>MYS<br>BOLMNG μ=0.10ppts<br>ECU<br>IND μ=0.20ppts<br>BHR NAM<br>MUSJORNZLTUN MDAKGZ SENGAB BEN<br>KORKAZHRV TUR CMR TZA<br>SGP NORBGRUSAPOL LKHUN AME PANXPRYLTUPE R URYKEN HND TGO<br>SVK CRI<br>DNK UKREGYCYP AUS<br>IDNSRB ZAF CIV<br>ARG<br>GBR CHL<br>BEL<br>LESS gain from a 1ppt BRA<br>increase in efficiency ZWE<br>1 .9 .8 .7 .6 .5 .4<br>Efficiency (IEI) [reversed order]<br>Note: μ=Ig/Y/IEI is ppts Ig/Y equivalent to a 1ppt increase in public investment efficiency<br>.25<br>.2<br>.15<br>.1<br>.05<br>0<br>Public Investment-to-Output, I(G)/Y [log scale]<br><!-- End of picture text --> Our calculations so far have involved the _short‐run_ increase in investment equivalent to a 1ppt increase in efficiency ( μ�� ). Instead, one might be interested in the permanent increase in investment that generates the same increase in GDPPC by 2040 as a 1ppt increase in efficiency – what we call μ��. Using numerical simulations we find that the values of μ�� and μ�� are almost identical. This is because the increase in efficiency (and equivalently sized increase in I<sup>�</sup> /Y ) is small, which means that any non‐ linearities are second order.

# **7. Conclusion**

In this paper, we develop a new model of public investment and growth ‐ the Long‐Term Growth Model Public Capital Extension (LTGM‐PC) ‐ which is designed to capture the effect of increases in public infrastructure investment quantity or quality on growth, while at the same time being simple enough to solve in a spreadsheet without macros (the Excel‐based tool is provided as a companion to this paper at the website www.worldbank.org/LTGM). Relative to the Standard Long‐Term Growth Model (LTGM), our extension allows public and private capital to enter the production function separately and for public capital to be of poor quality such that only a fraction can be used in production.

The effects of public and private investment on growth in our model vary substantially across countries depending on whether the country is relatively short of public or private capital ‐ but on average are similar to the effect of aggregate investment in the Standard LTGM. We show analytically and numerically that the effect of public investment on growth is higher when the public capital‐to‐output ratio is lower ‐ that is when public capital is scarce. Conversely, in countries where public capital is abundant _relative_ to other factors – even if it is scarce in absolute terms – public investment will have a 32
smaller effect on growth. The growth impact is also larger when public investment is more _useful_ , such as when it is in the form of essential infrastructure (public investment in other areas will have a lower return).

In contrast with several popular narratives, we find the growth impact of an increase in public investment is very similar across different levels of development. For a _typical_ low‐ or middle‐income country with our default parameters, a permanent 1ppt of GDP increase in public investment in essential infrastructure tends to boost growth by around 0.18ppts in the short term, but the boost to growth falls slowly over time as public capital accumulates. Other, less useful types of public investment (like public buildings) have a boost to growth of around 0.1ppts. In contrast, a permanent 1ppt of GDP increase in private investment leads to a slightly higher short‐term boost to growth of about 0.22ppts, although the effect tapers off faster over time.

Model simulations also show that there can be substantial growth dividends from improvements in the _quality_ of new public investment. Our new Infrastructure Efficiency Index (IEI) suggests a public capital efficiency loss of about 30 ppts for low‐income countries, and 10‐15 ppts for middle‐income countries (relative to the efficiency of high‐income countries). For countries with poor quality public capital and a large public investment share of GDP – such as many low‐income countries – an increase in the quality of public investment can be just as effective as a modest increase in quantity of public investment. For example, for the typical low‐income country a 1ppt increase in efficiency boosts growth by the same amount as a 0.13ppt of GDP increase in public investment. Despite this, the _level_ of efficiency has no effect on the marginal product of public capital because the low quality of new public investment is exactly offset by greater need for public capital due to the poor quality of past public investment (as in Berg et al. 2015).

In closing, it is worth mentioning a few caveats to our model and stylized facts. In order to keep the LTGM‐PC as simple as possible, we abstract from the effects of the financing of public investment via distortionary taxation. In many cases this will act as a drag on growth, and so our growth impact should be seen as an upper bound in that context (unless public investment can be financed by reducing unproductive expenditure elsewhere). We also abstract from endogenous private investment and return‐ seeking international capital flows. These factors might lead to a larger crowd‐in of private investment, but they could also amplify any negative impacts of distortionary taxation. Finally, our stylized facts depend on available data, and the quality of that data. While we have data on many high‐ and middle‐ income countries, the sample size for low‐income countries is small, which might increase the volatility of our estimates.

# **Appendices are available online at www.worldbank.org/LTGM**

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