# **2.2 Bils, Klenow and Ruane (2020) framework**

HK have attracted large attention partly because their study relies on model assumptions that allow for straightforward implementation of its framework across countries. With increasing attention, however, has also come stronger scrutiny of the robustness of their distortion metrics. A key concern of the HK framework is that what is perceived as misallocation could be confounded by measurement error, as acknowledged in the original paper. BKR develop a new methodology adapted from HK to address the concerns of measurement error in revenues and intermediate inputs.

BKR argue that, in a standard monopolistic competitive context, a firm’s _true_ revenues and costs respond in the same proportion to productivity shocks. In the absence of measurement error, therefore, revenue growth would move in the same proportion as inputs cost growth across all firms. In contrast, measurement error dampens the response of _measured_ revenues and costs to productivity shocks. Then, in the presence of measurement error, revenue growth would be less responsive to input cost growth. Moreover, assuming that measurement error is additive and orthogonal to the true marginal products, BKR show that measurement error dampens the response of revenue growth to input cost growth more than proportionally for firms with higher average revenue products (TFPR). This allows them to back out the measurement error and, most importantly, quantify the extent to which measured misallocation reflects true misallocation. BKR propose estimating: BKR propose estimating: (T (T where and ss are firm-specific log of revenue and inputs cost (labor, capital, and intermediates), ss ss ss ss ss ss ∆RR = ΨΨ ∆II + ΦΦ ff(l PPRR )) − ΨΨ(1 − λλ) gg(l PPRR )∆II ) + DD + ξξ l (T ) is the Tornqvist average for current and previous year and are polynomials, ss ss and RR are sector-year fixed effects.II<sup>7</sup> The parameter of interest is which is defined as the ratio of ss 2 the variances of true distortions and revenue productivity, i.e. ssPPRR , ff(∙)2 <u>ln τ</u> ~~,~~ gg(∙) where is the firmDD λλ ln T σσ level distortion. If the response of measured revenue with respect to inputs cost varies with the λλ≡ σσ PPTT ττ level of TFPR, then is different from one and the presence of measurement error is implied. Indeed, the presence of additive measurement error in either or both revenue and inputs costs can explain observing a marginal change in inputs but not a corresponding change in revenue. Note λλ that is a country-specific value. We can then estimate revenue productivity explicitly accounting for measurement error, λλ T ) where ln (T ).<sup>8</sup> � Using the estimated ıı T ss for all firms, and following the HK framework, BKR estimate ss ss PPRR) �= exp (λλ̂ ln ( TPPRR PPRR ) + εε εε ~NN(0, (λλ̂ −λλ�)σσ<sup>2</sup> allocative efficiency. This measure accounts for measurement error. Analogously to the HK ıı framework, the estimated allocative efficiency is used to back out potential TFP gains from moving PPRR toward the efficient allocation.

> 7 The subscript t for time is omitted, to simplify the notation.

> 8 The circumflex sign indicates _measured_ values.

8