Analyzing the impact of public capital on private capital productivity in a panel of African nations

Ezzahid, Elhadj; Rafik, Hamid Article Analyzing the impact of public capital on private capital productivity in a panel of African nations Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Ezzahid, Elhadj; Rafik, Hamid (2024) : Analyzing the impact of public capital on private capital productivity in a panel of African nations, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 12, Iss. 5, pp. 1-18, https://doi.org/10.3390/economies12050118 This Version is available at: https://hdl.handle.net/10419/329044 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Ezzahid, Elhadj, and Hamid Rafik. 2024. Analyzing the Impact of Public Capital on Private Capital Productivity in a Panel of African Nations. Economies 12: 118. https:// doi.org/10.3390/economies12050118 Academic Editor: Bruce Morley Received: 28 March 2024 Revised: 2 May 2024 Accepted: 7 May 2024 Published: 14 May 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). economies Article Analyzing the Impact of Public Capital on Private Capital Productivity in a Panel of African Nations Elhadj Ezzahid and Hamid Rafik * Department of Economics, Faculty of Law, Economic and Social Sciences Agdal, Mohammed V University, Rabat 6430, Morocco; [email protected] *Correspondence: [email protected] Abstract: This research contributes to the ongoing discourse concerning the efficiency of public capital and its influence on the productivity of private capital and total factor productivity within African economies. Employing the standard production approach, we include public capital as a distinct input to assess its specific impact on output growth and the enhancement of total factor productivity. We argue that public capital, predominantly manifesting through infrastructure, constitutes an indispensable element for fostering growth. Fundamental to the productivity of private capital is its reliance on a sufficient stock of public infrastructure for operational efficiency. Our empirical analysis reveals that public capital exhibits a substantive long-term influence on output growth and the productivity of private capital. However, in the short term, the discernible impact of public capital is less pronounced. Moreover, while public capital emerges as a noticeable factor in output growth, its influence on total factor productivity remains relatively subdued. Keywords: public capital; economic growth; total factor productivity; Africa 1. Introduction Assessing the extent to which public infrastructures, such as roads, highways, electricity networks, airports, railways, and other facilities designed for public use can boost output growth and productivity is of paramount importance to policy makers. Central to the objective of this paper is the sign and magnitude of the elasticity of output per unit of private capital with respect to public capital and the contribution of the latter to total factor productivity (henceforth, TFP).1 Undeniably, public infrastructures play a pivotal role in supporting the private sector, given that the production of goods and services typically necessitates the transportation of essential inputs such as energy, water, and other vital components—a task unfeasible for the private sector to accomplish on its own. 2 Hence, a logical presupposition arises regarding the indispensability of public infrastructure to foster sustained economic growth. Nevertheless, government-provided infrastructure can potentially create distortions if it starts to displace private investment. Barro (1990)’s endogenous growth model posits that as the proportion of government spending in the gross domestic product (GDP) expands, the economy’s growth rate also increases until it attains an optimal point, beyond which it begins to decline. Thus, the provisioning of infrastructure by the government in African nations might not be optimal due to inadequate public investment. For private capital to be productive, it needs an efficient and diversified infrastructure. That is, the productivity of private investment is contingent upon the existence of a robust stock of public capital surpassing a specific threshold. 3 Thus, the productivity of public capital is expected upon reaching this threshold, serving as an essential requirement for the operation of the private sector. It is crucial to study the effect of public capital on private capital in African nations for several reasons. First, public capital can enhance the productivity and growth of the Economies 2024,12, 118. https://doi.org/10.3390/economies12050118 https://www.mdpi.com/journal/economies Economies 2024,12, 118 2 of 18 private sector by providing essential infrastructure and services that the private sector cannot or will not provide on its own. Second, public capital can also affect the allocation and efficiency of private capital by creating positive or negative externalities, depending on the quality and quantity of public investment. Third, public capital can have implications for the fiscal and macroeconomic stability, as it involves public spending, borrowing, and taxation decisions. Therefore, studying the effect of public capital on private capital can help policy-makers design optimal public investment strategies that can maximize the benefits and minimize the costs of public capital for the private sector and the economy as a whole. This study focuses on Africa for several reasons. First, infrastructure in Africa significantly lags behind that of most other developing nations (Foster and Briceño-Garmendia 2010). Second, infrastructure is a major catalyst for economic growth, productivity enhancement, and a facilitator of trade (Donaubauer et al. 2018;Aschauer 1989). This work aims to shed light on the contribution of public capital to the productivity of private capital in African economies, using the standard production function approach and data from 1960 to 2017. The rest of this paper is organized as follows. Section 2deals reviewis the literature on the nexus between public capital and growth. Section 3outlines the channels through which public infrastructure affects output and productivity. Section 4 details the empirical approach. Section 4presents the model. Section 5discusses the data and stylized facts. Section 6presents the results. Finally, Section 7concludes. 2. Public Capital and Growth The potential role of public capital in enhancing economic growth and productivity poses a challenging relationship to measure empirically. This stems from the difficulties in estimating the stock of public capital, requiring assumptions about its initial value, depreciation rate, and the price deflator of public investment (Everhart and Sumlinski 2001). While it is the case that the stock of public capital is the sum of public investments made each year minus the depreciation rate, studying the impact of public capital instead of investment is more relevant due to the fact that it takes a long time for public projects to have a positive return manifested through the enhanced productivity of the private sector. While there is abundant research studying empirically public capital in relation to output growth and private capital, there is, however, none for the case of African economies. Thus, further research is imperative to assess the influence of public capital on growth. Such task is not only of great importance from an academic perspective, but it can also guide policy making by shifting attention on the likely equal importance of public capital as, in recent years, there has been increased interest in the primordial role of the private sector. 4 Estimates of capital stock, both public and private, are predicated on assumptions related to initial stock levels and depreciation rates, potentially leading to inaccuracies. Consequently, empirical studies could exhibit biases in determining the extent of public capital’s contribution to growth, particularly in countries with unsatisfactory statistical systems. Government investment might not effectively translate into intended infrastructure like roads, bridges, or public buildings due to inefficiencies. Inefficient government investments might be costly due to corruption and the rent-seeking behavior of politicians and bureaucrats (Keefer and Knack 2007). Indeed, the aggregate data used by experts to estimate public capital stock might not accurately reflect the real physical capital available. Significant infrastructure gaps are observed in African countries. Increases in factor inputs at the initial stages of development are assumed to generate high returns due to high marginal productivity. Accordingly, we should expect to find elasticities of output with respect to public capital to be relatively higher in African economies compared to advanced and middle-income economies. The main rationale for this assumption is the observed gaps of public infrastructures in Africa, especially in sub-Saharan countries. The empirical literature presents a varied spectrum regarding the impact of public capital on economic growth. Some studies find a negative contribution, while others highlight its significance in fostering growth (Ligthart and Suárez 2011). In these studies, public Economies 2024,12, 118 3 of 18 capital is typically integrated as a factor input alongside labor and private capital within production functions. 5 Bom and Ligthart’s (2014) meta-regression analysis, examining 578 estimates from 68 studies between 1983 and 2008, revealed short-run output elasticities of about 0.083, long-run elasticities around 0.122, with an average output elasticity of approximately 0.106, signifying its substantial contribution to growth. In contrast, Aschauer (1989), in a seminal paper which aimed to investigate the 1970s productivity decline in the U.S, finds a remarkable 0.39% increase in output per unit of private capital after a 1% rise in the public–private capital ratio. This elasticity starkly contrasts with the average 0.106 found in Bom and Ligthart’s meta-analysis. However, critiques from authors like Munnell (1990) challenged the credibility of Aschauer’s results, considering them disproportionately high relative to common findings in the literature. Ouédraogo et al. (2019) estimated the impact of public investment on private investment in sub-Saharan African countries, revealing an overall positive effect of public investment on its private counterpart. Similarly, Henderson and Kumbhakar (2006) explored a panel encompassing 48 states, 6 estimating a positive and significant return on public capital. Ligthart (2000) estimated the effect of public capital on output growth in Portugal. Employing the production function approach and the unrestricted vector autoregressive model (VAR) with yearly data spanning from 1965 to 1995, the study unveiled public capital’s crucial role as a determinant of long-term growth. The elasticity of output concerning public capital in this analysis aligned with prior literature, standing at 0.19. Arslanalp et al. (2010) examined the impact of public capital across OECD (Organisation for Economic Co-operation and Development) and developing countries, using data from 1960 to 2001. Their findings indicated a positive elasticity of output with respect to public capital. Notably, for non-OECD nations, the elasticity was relatively higher, with a public capital coefficient at 0.13, while developing countries exhibited an insignificant coefficient of public capital. Bleaney et al. (2001) uses a panel data model to investigate the effects of distortionary taxes, that is, taxes that cause inefficiencies in the economy by discouraging consumption and saving, and government expenditures for OECD countries. Their results aligned with the predictions of Barro’s endogenous growth model. Likewise, Ramirez (2002) analyzed the correlation between public spending on infrastructure and labor productivity in Mexico. Through cointegration analysis, their study concluded that government spending on infrastructure, rather than overall government investment, significantly influenced the growth of labor productivity. Ram (1996) conducted an estimation in which output is a function of the shares of private and public investments in GDP, along with the annual growth rate of exports. The findings revealed that public investment exhibited notably higher productivity compared to its private counterpart. Examining the role of public capital accumulation in output growth for 28 developing countries between 1981 and 1991, Dessus and Herrera (2000) employed a simultaneous equations model. Their conclusion pointed towards the positive impact of public capital accumulation on long-term growth. However, they highlighted an indirect crowding-out effect of private capital accumulation resulting from public capital formation. In this regard, Holtz-Eakin (1992) argued that it is counter-intuitive to suggest that the public provision of roads, bridges, water, sewage facilities, and other components of infrastructure do not affect production directly. However, the study acknowledged that certain public projects might not be subject to cost–benefit constraints and could survive despite being unproductive. Consequently, the study emphasized the limitations of using aggregate data to convincingly establish public capital as the sole driver of improvements in private productivity. Compared to the cited literature (see Table 1), this study is an empirical contribution regarding the effect of public capital, firstly on output in the short- and long-term; secondly, on the productivity of private capital; thirdly, on the total factor productivity in the context Economies 2024,12, 118 4 of 18 of African economies. In this context, this study seeks to fill the gap regarding empirical studies assessing the elasticity of output with respect to public capital. Table 1. A summary of the literature on the efficiency of public capital. Study Focus Key Findings Ram (1996)Output estimation based on private and public investments Public investment has a higher productivity compared to private investment. Dessus and Herrera (2000) Role of public capital in output growth Positive impact of public capital accumulation on long-term growth, yet noted indirect crowding-out effect on private capital. Holtz-Eakin (1992)Public infrastructure’s impact on production Disputes counter-intuitive idea of no direct impact but acknowledges some unproductive public projects due to cost–benefit constraints. Ligthart (2000)Effect of public capital on output growth in Portugal Public capital found as a crucial determinant of long-term growth, showing a certain elasticity of output concerning public capital. Arslanalp et al. (2010) Impact of public capital across OECD and developing countries Positive elasticity of output concerning public capital, with relatively higher impact for non-OECD nations, while developing countries exhibited an insignificant coefficient. Bleaney et al. (2001)Effects of distortionary taxes and government expenditures Aligns with predictions of Barro’s endogenous growth model, exploring the impact on OECD countries. Henderson and Kumbhakar (2006)Return on public capital for a panel of 48 states Shows a positive and significant return on public capital. Ramirez (2002)Correlation between public spending on infrastructure and growth Government spending on infrastructure, rather than overall government investment, significantly influences the growth of labor productivity. Aschauer (1989)1970s productivity decline and public–private capital ratio Indicates a substantial increase in output per unit of private capital with a rise in the public–private capital ratio, a finding challenged due to disproportionately high results. Source: authors. The Dynamics of Public Capital, Private Capital, and Output Exogenous growth models place significant emphasis on investment for transitional dynamics and achieving steady-state growth. However, earlier models like that of Solow (1956) did not differentiate between the impacts of private and public capital on these dynamics. Private capital notably enhances output directly by bolstering the economy’s productive resources and indirectly by fostering technological advancements, thereby enhancing overall efficiency. Conversely, public capital, largely represented by infrastructure, does not directly impact production. Instead, its role lies in enhancing the efficacy of private capital and labor productivity, facilitating the adoption of new technologies and modes of production. Yet, due to the absence of a direct pricing mechanism for government services, which are primarily financed through taxes or borrowing, the allocation of public capital might not always be optimally managed. Economies 2024,12, 118 5 of 18 In a neoclassical framework of economic growth, positive technological shocks are presumed to be the primary catalyst for long-term growth. Easterly and Levine (2001) found empirical evidence that corroborates this assertion, suggesting that TFP stands as a significant driver of economic growth. Without the productivity gains, the growth of per capita income would have been substantially lower in industrialized economies. Additionally, within Solow’s growth model, the accumulation of physical capital contributes to per capita output growth only until the steady state is reached. Subsequently, sustained growth is predominantly derived from exogenous technological changes.7 In light of what precedes, we believe that the absence of robust public infrastructure hinders the private sector from propelling productivity, even in the presence of other growth catalysts such as technology, physical and human capital, and well-established institutions. For instance, if we consider a set stock of private inputs—where kis a broad concept of capital, covering both human and physical capital—alongside institutions ensuring property rights, law enforcement, and contract enforcement, the private sector’s inability to access crucial elements like roads, energy, or water directly impedes output generation. To illustrate, assuming Cobb–Douglass technology, y=A·kαg1−α , with low levels of g, output will be low despite sufficient endowments of k. Therefore, it follows that public infrastructure is necessary to spur the emergence of growth in the private sector. Boosting the stock of public capital necessitates public investment which can be funded through taxes or borrowing. However, taxes, once they exceed a certain threshold, can introduce distortions, 8 hindering growth rather than facilitating it. Although external borrowing can be effective, it is not a sustainable strategy when the social rate of return to public investment falls below the cost of borrowing. Despite public capital being a major input in the production function besides private inputs, it does not necessarily contribute significantly to growth. One of the reasons why public investment does not contribute to the growth of private production is its crowding out effect of private investment (see, Sundararajan and Thakur 1980;Higgins and Link 1981, among others). This was one of the chief counter arguments to public spending since it was first advocated by Keynes. Financing public spending through taxes reduces the resources available to the private sector, resulting in decreased private investment and, consequently, reduced overall growth.9 Alternatively, when public spending is funded through borrowing, the heightened demand for loanable funds raises interest rates, making it costlier for the private sector to invest, thereby suppressing private production. Inefficient public investment further complicates the financing of public projects through taxes or external borrowing, as investor trust correlates closely with the economy’s fiscal position. Proponents of deficit spending contend that judicious public borrowing, invested in projects stimulating private investment, can indirectly enable new channels of production and directly bolster aggregate demand through classical Keynesian channels. Barro’s seminal work in 1990 delineates two categories of public expenditures. The first pertains to productive investments that significantly contribute to long-term growth, while the second category involves expenditures deemed irrelevant to growth, such as spending on social services. Tax reductions on investment stimulate growth by incentivizing the private sector to invest, much like public spending aimed at enforcing property rights and maintaining law and order, which, in turn, enhances the returns on private capital. Conversely, distortionary taxes impede private investment, thereby diminishing longterm growth. Building upon this, the Barro and Sala-i-Martin (1992) model assumes n producers generating output y(where the aggregate output is represented as Y = ny) using the Cobb–Douglas technology: y=Ak1−αgα, where Ais technology, krepresents a broad concept of capital 10 per producer (k = K/n), g represents the government-provided services per producer (g = G/n); Gand Krepresents aggregate public and private capitals. Government finances its services through a propor- Economies 2024,12, 118 6 of 18 tional tax on output τ and levies a lump-sum tax L. 11 The government is subject to the following constraint: ng +C=L+τny, where Crepresents government consumption, i.e., non-investment goods that are not included in the production function. They showed that the growth rate ∅is as follows: ∅=λ(1−τ)(1−α)A1 1−α(g y)α/(1−α)−µ, where λ and µ are parameters in the utility function. This equation illustrates that distortionary taxes diminish the growth rate, whereas government-provided services denoted by ‘g’ augment growth. Non-distortionary taxes, specifically L, exhibit no impact on growth. It is noteworthy that government consumption does not affect the growth rate. However, within the model, there is the assumption that the government maintains a balanced budget, thereby avoiding surpluses or deficits. Bleaney et al. (2001) conducted a test on the aforementioned model, yielding robust results that affirm its assertions—highlighting the enduring impact of fiscal policy on economic growth. 3. The Channels through Which Public Infrastructure Affect Output and Productivity There are many channels through which public investment can affect output and TFP (see Figure 1). One type of public investment might not be enough to encourage growth because the private sector needs many infrastructures in order to function efficiently. The following figure is a list of the main components of public capital and how they contribute to improving output and TFP. Economies 2024, 12, x FOR PEER REVIEW 6 of 19  +  =  +  , where C represents government consumption, i.e., non-investment goods that are not included in the production function. They showed that the growth rate ∅ is as follows: ∅ =  ( 1 −  ) ( 1 −  )      (   )  / (    ) −  , where  and  are parameters in the utility function. This equation illustrates that distortionary taxes diminish the growth rate, whereas government-provided services denoted by ‘g’ augment growth. Non-distortionary taxes, specifically L, exhibit no impact on growth. It is noteworthy that government consumption does not affect the growth rate. However, within the model, there is the assumption that the government maintains a balanced budget, thereby avoiding surpluses or deficits. Bleaney et al. (2001) conducted a test on the aforementioned model, yielding robust results that affirm its assertions—highlighting the enduring impact of fiscal policy on economic growth. 3. The Channels through Which Public Infrastructure Affect Output and Productivity There are many channels through which public investment can affect output and TFP (see Figure 1). One type of public investment might not be enough to encourage growth because the private sector needs many infrastructures in order to function efficiently. The following figure is a list of the main components of public capital and how they contribute to improving output and TFP. Roads play a primordial role in bolstering private sector productivity by significantly reducing transaction costs. Agrawal et al. (2017) conducted a study utilizing data on highways, railroads, and routes, aiming to unveil the impact of inter-state transportation on regional innovation. Their empirical findings revealed that a 10% increase in the stock of highways correlates with a 1.7% surge in registered patents. Moreover, they highlighted the underestimated influence of transportation infrastructure on economic growth. Interconnecting regions through roads yields a dual benefit—it allows regions to leverage their specialized production by trading surplus goods and fosters interdependence, enabling regional specialization. Figure 1. The channels through which public capital influences output and productivity. Source: authors. Figure 1. The channels through which public capital influences output and productivity. Source: authors. Roads play a primordial role in bolstering private sector productivity by significantly reducing transaction costs. Agrawal et al. (2017) conducted a study utilizing data on highways, railroads, and routes, aiming to unveil the impact of inter-state transportation on regional innovation. Their empirical findings revealed that a 10% increase in the stock of highways correlates with a 1.7% surge in registered patents. Moreover, they highlighted the underestimated influence of transportation infrastructure on economic Economies 2024,12, 118 7 of 18 growth. Interconnecting regions through roads yields a dual benefit—it allows regions to leverage their specialized production by trading surplus goods and fosters interdependence, enabling regional specialization. In the context of the African continent, the expansion of road networks is poised to significantly enhance continental and global trade. The expansion and creation of transportation channels across African countries will likely streamline the movement of goods, fostering increased trade links. Importantly, transportation infrastructure not only eases the movement of goods but also facilitates the mobility of factors of production (Buys et al. 2006;Arvis et al. 2007). From a microeconomic perspective, a firm endowed with specific factor inputs experiences enhanced productivity due to reduced transaction costs stemming from either the construction of new public roads or the enhancement of existing ones. This improvement enables the firm to generate more output using the same input levels, effectively elevating the total factor productivity. An essential characteristic of publicly provided infrastructures is their broad-reaching benefit to numerous private agents. This reasoning extends to various public infrastructures that facilitate transportation. Supporting this assertion, in the case of manufacturing establishments in India during the period 1998–2012, Kailthya and Kambhampati (2022) found that a 1% increase in road density correlates with a 0.25% rise in TFP. Considering the aforementioned points, the inefficiency of public investment in an environment marked by a limited stock of transportation infrastructure does not stem from the inherent ineffectiveness of public capital itself. Instead, it often relates to the challenges in implementing and managing public projects, issues like corruption, and the elevated costs of public investment. Dzhumashev (2014) illustrates that, at lower levels of the government’s share in GDP, corruption significantly hampers the efficiency of public spending, consequently impacting overall growth. Electricity networks stand as a vital component of infrastructure, acting as the lifeblood of economic production in developed economies. In fact, electricity infrastructure is indispensable for technology-intensive machinery and high-end devices, acknowledged in literature as pivotal drivers of productivity. Lowering energy costs for established businesses can substantially reduce their overall expenses and bolster their output. Extending electricity networks to previously unelectrified rural areas offers households the capacity to operate electric appliances for household tasks, have access to illumination during night hours, facilitate tasks like well-digging and installing water pumps, ultimately leading to increased agricultural production. Several studies demonstrate the positive impact of energy consumption on economic growth (e.g., Stern et al. 2019;Banerjee et al. 2017;Saidi and Hammami 2015). While the private sector can supply electricity, the initial phases of establishing largescale energy infrastructures in impoverished nations often demand substantial investments, making them more feasible by the government. Conversely, large-scale infrastructures that might not yield immediate or medium-term profits might not attract private sector involvement. Investments in healthcare and education play a pivotal role in enhancing the quality of human capital, a well-explored concept in the literature. Alongside these, the provision of essential infrastructures is indispensable for fostering skilled human capital, especially for firms engaged in technology-intensive goods. Tsaurai and Ndou (2019) conducted an analysis on the influence of infrastructure and human capital development on economic growth in transitional economies. Employing dynamic panel and generalized methods of moments estimators (GMMs), the study spans 25 transitional economies from 1995 to 2015. The findings underscore the importance of the interaction between infrastructure and human capital development in bolstering economic growth in transitional economies. This study recommends policies focusing on augmenting human capital development to amplify the capacity of infrastructure development in influencing economic growth. Economies 2024,12, 118 8 of 18 Finally, Canning and Bennathan (2000) explore the impacts of infrastructure and human capital on economic development utilizing cross-country data spanning from 1950 to 1992. Their research estimates a production function encompassing infrastructure and human capital as inputs while controlling for geographical, institutional, and openness factors. The study affirms the significant and positive effects of both infrastructure and human capital on output per worker. Furthermore, it highlights that, in countries with higher levels of human capital, the marginal product of infrastructure is proportionally higher. While each public infrastructure component independently influences output, the collective execution of multiple public projects—such as roads, highways, electricity networks, telecommunication infrastructures, dams, and sewer facilities—yields a cumulative and amplified effect on overall growth. A synchronized increase in these varied infrastructure components significantly contributes to economic development. Conversely, an isolated surge in one aspect, like public investment in roads without a concurrent development of other public facilities, is unlikely to manifest noticeable impacts on output growth. This understanding, although challenging to empirically test, strongly advocates for a comprehensive expansion of various public infrastructure components throughout the African continent. 4. Model To estimate the elasticity of output with respect to the public capital stock, we use a standard Cobb–Douglas production function, such as that in Aschauer (1989): Yit =F(Nit,Kit,Git)=AitNeN it KeK it GeG it (1) where Y it is output, A it is the level of productivity, N it is labor, K it is private capital, and G it is the public capital stock. eN,eK , and eG are the elasticity of output with respect to labor, private and public capital, respectively. The subscripts iand trepresent the country and time variables, respectively. It is possible to include other kinds of capital such as human and institutional capital, but we assume that they indirectly affect the growth rate via their impact on the productivity measure Ait. Following the work of Aschauer (1989), we introduce natural logarithms which yield the following equation: yit =ait +eN·nit +eK·kit +eG·git, (2) This equation does not assume any restrictions on the parameters, assuming that competitive markets private inputs will be paid their marginal products. It is assumed that government services make private inputs more productive and allow for increasing returns over all inputs. It is assumed that there are constant returns to scale over private inputs but increasing returns to scale over all inputs, i.e., eN+eK= 1. Thus, we obtain the following equation: yit =ait +eN·nit +(1−eN)·kit +eG·git (3) Simplifying yields the following: yit =ait +eN·nit +kit −eN·kit +eG·git (4) Therefore: yit −kit =ait +eN·(nit −kit)+eG·git (5) Additionally, productivity is given by: pit =yit −sN·nit −sk·kit =ait +eG·git (6) Equation (5) expresses output per unit of capital as a function of productivity, the labor–capital ratio, and the stock of public capital, while Equation (6) expresses total factor productivity as a function of technical progress and the flow of government services assuming it is related to the stock of public capital, where sNand skare factor shares. Economies 2024,12, 118 15 of 18 Table 6. Output per unit of capital (y−k). Equation Estimation Method Constant Time n−k g −k g k (13) aFixed effects −5.85 *** (−8.55) 0.003 *** (9.87) 0.481 *** (38.63) 0.133 *** (8.58) (14) bFixed effects −15.172 *** (−5.68) 0.008 *** (6.00) 0.323 *** (7.07) 0.128 *** (8.30) −0.296 *** (−6.19) a F(3;2118) = 744.85 ***; R 2 : within = 0.50, between = 0.50, overall = 0.48. b F(4;2217) = 564.92 ***; R 2 : within = 0.50, between = 0.47, overall = 0.45. *** significant at 5% acceptance threshold. Equation (14) (Table 6) facilitates separate estimates for the coefficients associated with public and private capital. Specifically, a 1% increase in public capital is associated with an approximate 0.13% rise in private capital’s productivity. This result underscores the pivotal role of public capital in augmenting the productivity of the private sector. Equation (15) (Table 7) shows that a 1% increase in public capital causes only a 0.053% increase in total factor productivity. Upon including private inputs, the response of total factor productivity to a 1% increase in public capital rises to 0.128% (Equation (16), Table 7), which indicates the presence of serial correlation among explanatory variables. Equation (17) (Table 7) relates TFP to public capital and the combination of private inputs. We observe that the combination of private inputs carries a negative sign, which is in accordance with the assumption of constant returns to scale. Equation (18) (Table 7) exhibits the strong relationship between public capital per unit of the combination of private inputs and TFP. A 1% increase in the public capital per combination of private inputs raises total factor productivity by 0.13%. Table 7. Total factor productivity (p). Equation Estimation Method Constant Time g k n i g −i (15 aFixed effects 3.671 *** (3.73) −0.001 *** (−2.65) 0.053 *** (4.53) – – – – (16) bFixed effects −15.172 *** (−5.68) 0.008 *** (6.00) 0.128 *** (8.30) − 0.104 *** (−6.72) − 0.328 *** (7.19) – – (17) cFixed effects −7.190 (−4.52) 0.004 (5.03) 0.142 (9.46) – – −0.266 (−9.22) – (18) dFixed effects 0.820 (1.25) −0.000 (−0.06) – – – – 0.13 (8.65) a F(2;2219) = 11.78 ***; R 2 : within = 0.01, between = 0.01, overall = 0.00. b F(4;2217)= 30.96 ***; R2: within = 0.05 , between = 0.36, overall = 0.28. c F(3;2218) = 36.48 ***; R 2 : within = 0.04, between = 0.32, overall = 0.25. dF(2;2219) = 38.96 *** ; R 2 : within = 0.03, between = 0.31, overall = 0.25. *** significant at 5% acceptance threshold. The results of our empirical testing show that public capital is a significant contributor to the private sector’s productivity. However, its direct impact on TFP is small. Nonetheless, upon including private inputs in the specification, this impact increases significantly due to the strong correlation between public capital and private inputs. Moreover, our results are supportive of the assumption of constant returns to scale. Public capital seems to have a significant impact on output growth in the long run. However, in the short run, its impact on growth is unnoticeable. 7. Conclusions This study is a contribution to the debate over the effectiveness of public capital and its contribution to the growth of output and TFP. Indeed, this study advances the empirical literature on public capital’s impact on output and private capital productivity both in the short and long term, as well as on total factor productivity in the context of African economies. We address a significant gap in the literature by estimating the elasticity of Economies 2024,12, 118 16 of 18 output with respect to public capital for the case of African economies, highlighting its substantial long-term influence on output growth and the productivity of private capital. Although the short-term effects are less marked, and the impact on total factor productivity is subdued, our findings underscore the critical role of public capital in economic growth. It is important to note that economic growth in Africa is not matching what we should expect from a continent that is rich with natural resources and, on the other hand, has an abundant supply, though unskilled, of labor. Using the standard production function approach, we estimated the elasticity of output with respect to public capital and tested the assumption of constant returns to scale. Our results show that public capital is an important contributor to output in the long run. We also estimated the response of private capital’s productivity to changes in public capital. Our empirical model shows that public capital is a significant factor in enhancing private capital’s productivity. However, we were unable to find a strong association between public capital and total factor productivity. The study of the dynamics of economic growth assigns a primordial role to institutional quality which, through its role in spurring a climate that favors private incentives and lowering transaction costs, is the main driver of economic growth (Acemoglu et al. 2005; North 1991). Therefore, forthcoming studies dealing with the same issue should control for the impact of institutions on the efficiency of both public and private investments. The magnitude of the impact of public capital on output in African economies is in accordance with the preceding literature that uses panel data. Consequently, our model is in favor of more public investment in infrastructure to boost the productivity of the private sector. As it is well documented in the literature, factor inputs alone are not enough for growth and productivity enhancement (Easterly and Levine 2001). It is important to bear in mind that one of the caveats in our dataset is the degree of its accuracy, since the statistical system in African countries is far from being satisfactory. Author Contributions: Conceptualization, E.E. and H.R.; methodology, E.E. and H.R.; software, H.R.; validation, E.E.; formal analysis, H.R.; investigation, H.R.; resources, H.R.; data curation, H.R.; writing—original draft preparation, H.R.; writing—review and editing, H.R. and E.E.; visualization, H.R.; supervision, E.E.; project administration, E.E.; All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data used in this study can be accessible at: https://www.rug.nl/ ggdc/productivity/pwt/pwt-releases/pwt9.1?lang=en and https://data.imf.org/?sk=1ce8a55f-cfa7 -4bc0-bce2-256ee65ac0e4 (accessed on 2 May2024). Conflicts of Interest: The authors declare no conflict of interest. Notes 1From a public policy perspective, the higher the elasticity, the more fruitful it is to invest in public facilities. 2We assume that the private sector is unable to provide infrastructure on its own. 3 According to Pereira and Andraz (2013), the output elasticity of public capital is 0.093 when the public capital stock is below the threshold level of 72.4% of GDP and 0.193 when it is over the threshold. 4 The Washington Consensus has placed emphasis on privatization and the transfer of state-owned enterprises and other forms of public property to the private sector. 5 Typically, a log-linearized Cobb–Douglas function is used either in levels or in first differences. When estimating the elasticity of output with respect to public capital in logarithmic levels, a concern arises about potential spurious regression due to the upward sloping trend of factor inputs. Conversely, employing estimation in first logarithmic differences addresses this issue, although it compromises the long-term relationship between output and the public capital stock. 6Specifically, 48 American states. Economies 2024,12, 118 17 of 18 7 Within the well-known neoclassical Solow (1956) growth model, an uptick in the saving rate initially elevates the capital stock to its steady state level. Beyond this point, additional investment primarily serves to offset the per capita depreciation of capital. Once the economy reaches its steady state, sustained growth predominantly arises from exogenous technological shocks. 8Taxes diminish the pool of wealth accessible to the private sector, thereby exerting a negative influence on private incentives. 9 This rationale holds when factor accumulation is considered the primary growth driver. 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