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.

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