Vaccine rollout is different than allocating other (initially) scarce goods because it involves overcoming vaccine hesitancy. Here's a meta-analysis which concludes that advertising was helpful and cost effective.
Athey, Susan, Kristen Grabarz, Michael Luca, and Nils Wernerfelt. "Digital public health interventions at scale: The impact of social media advertising on beliefs and outcomes related to COVID vaccines." Proceedings of the National Academy of Sciences 120, no. 5 (2023): e2208110120.
Abstract: Public health organizations increasingly use social media advertising campaigns in pursuit of public health goals. In this paper, we evaluate the impact of about $40 million of social media advertisements that were run and experimentally tested on Facebook and Instagram, aimed at increasing COVID-19 vaccination rates in the first year of the vaccine roll-out. The 819 randomized experiments in our sample were run by 174 different public health organizations and collectively reached 2.1 billion individuals in 15 languages. We find that these campaigns are, on average, effective at influencing self-reported beliefs—shifting opinions close to 1% at baseline with a cost per influenced person of about $3.41. Combining this result with an estimate of the relationship between survey outcomes and vaccination rates derived from observational data yields an estimated cost per additional vaccination of about $5.68. There is further evidence that campaigns are especially effective at influencing users’ knowledge of how to get vaccines. Our results represent, to the best of our knowledge, the largest set of online public health interventions analyzed to date.
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