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：在自愿碳市场和教育测试等应用的推动下,我们考虑在第三方验证人在场的情况下,为质量不同但隐藏的商品建立一个市场。验证人可以提供关于产品质量的信息信号,并可为此服务收取费用。销售人既选择其生产的产品质量,又选择认证。然后价格在竞争性市场中确定。在单一交叉条件下,我们表明生产者选择的认证水平在平衡时决定得独特。然后我们表明如何减少收入最大化验证人对非线性估值的垄断定价问题造成的收入最大化验证人问题,并设计一种FPTAS,用于计算最佳证书日期及其价格。一般来说,福利最优和收入最优的证书日期都是任意的。