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).

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