Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
We conduct the following power analysis on our ability to detect a difference in credence markups between any two countries. We assume that prices within a country are generated by the following DGP: Price_{i,s} = β0 + β1CRE + v_{s} + ϵ_{i,s}, where v_{s} is a store-level random effect. To obtain values for the σ’s for v_{s} ∼ N (0, σ_{s}) and ϵ_{i,s} ∼ N (0, σ_{i}), we conduct a regression on data from a previous repair experiment on a different product that also conducted two visits per store. We multiply those previous prices by 2.5 to reflect the increased costs and cost range of the current type of repair, and obtain estimates of σ_{s} = 35 and σ_{s} = 27 and β0 = 115. Based on this transformed data, we also assume a minimum price of 75 (10% quantile). We set the maximum price to 350 Euros based on our experimental design.
Given this DGP, we can generate prices for each country assuming different β1’s, calculate the credence markup (assuming CRE ≥ ORD), and estimate the regression: reg Markup Country, vce(robust). We assume Country 1 has β1 = 10, and simulate the smallest possible β1 for Country 2 so that we can detect
the effect with 80% power at the 5% level. Using 1000 simulations, this value is β1 = 41.1, which implies an absolute difference in markups of 31.1 Euros, or for the relative difference in markups as defined of 21.7 units (e.g. we could detect the difference between a country with an average markup of 110 and another country with a markup of 131.7). This seems reasonable given we are mainly interested in substantial differences between countries. For context, Hall et al. (2019) report an equivalent markup of 144.4 in Turkey.