Motivated by applications such as voluntary carbon markets and educational testing, we consider a market for goods with varying but hidden levels of quality in the presence of a third-party certifier. The certifier can provide informative signals about the quality of products, and can charge for this service. Sellers choose both the quality of the product they produce and a certification. Prices are then determined in a competitive market. Under a single-crossing condition, we show that the levels of certification chosen by producers are uniquely determined at equilibrium. We then show how to reduce a revenue-maximizing certifier's problem to a monopolistic pricing problem with non-linear valuations, and design an FPTAS for computing the optimal slate of certificates and their prices. In general, both the welfare-optimal and revenue-optimal slate of certificates can be arbitrarily large.

翻译：受自愿碳市场和教育测试等应用场景的启发，我们考虑存在第三方认证机构时，商品质量可变化但不可观测的市场。认证机构可以提供关于产品质量的增量信息，并对此项服务收费。卖方同时决定其生产产品的质量水平和是否进行认证。随后由竞争市场决定价格。在单一交叉条件下，我们证明生产者选择的认证水平在均衡状态下具有唯一确定性。进而，我们将追求收入最大化的认证机构问题转化为具有非线性估值的垄断定价问题，并设计了求解最优认证方案及其定价的全多项式时间近似方案（FPTAS）。通常情况下，社会福利最优与收益最优的认证方案均可能具有任意大的规模。