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).

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