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We consider a product that is sold under one of the two common warranty policies. Under the “pro rata” warranty policy, a failed item is replaced by a new one or is repaired at a cost prorated to the age of the failed item. Under the “free replacement” warranty policy, replacements or repairs during the warranty period are provided by the supplier free of charge to the buyer. Assuming that successive failure times form a renewal process, we derive moments of the total replacement cost for both policies during the product life cycle (0, t]. We also provide an extension to time-varying failure time distributions in the case of the pro rata warranty policy.

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Families of various sizes a~l share the same preference maps over consumption and leisure where consumption has been appropriately scaled to reflect those variations'. Under lump sum taxes, equating labor supplies and scaled consumptions is not the~ptimal way to achieve horizontal equity. Under an income tax structure, if there is sufficient diversity of ahility for all family sizes, the opposite is true.In particular, the marginal tax rates of the two ends of the ability scales are positive for small families and negative for large ones.