Essays on quantity discounts
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This dissertation consists of five chapters: The first one reviews the literature, the next three are the core essays, and the final chapter provides concluding remarks. Quantity-discount pricing such as "buy more, save more" or "buy one and get second one at half price" is ubiquitous in the business world. Unlike two-part tariffs, quantity discounts charge buyers declining tier prices for the same good as the purchase amount increases without charging a fixed or entry fee. The first core essay explores the optimal multi-tier quantity-discount pricing scheme. In a model in which a monopolist offers an n-tier quantity-discount schedule for its product, this essay finds that all individual tiers' marginal costs (other than the last tier's) are irrelevant in determining the optimum tier structure; an increase in the equilibrium total output is associated with an increase in each tier's own and its cumulative outputs; and the amount of increase in each tier's cumulative output rises monotonically from the first tier. Furthermore, if demand is strictly convex (concave), the equilibrium tiers' outputs are monotonically increasing (decreasing) starting from the first tier. When a government policy affects a firm's cost function and thus its optimal tier prices, should the government impose the same or different tax/subsidy rates among tiers? The second essay explores the tax/subsidy policies and the optimal quantity discounts for a monopoly. It covers four tax/subsidy schemes: uniform and non-uniform systems in both specific and ad valorem regimes. Under general demand and total cost functions, this essay examines the effects of an exogenous change in a tier's tax on the equilibrium tier sizes among the four schemes. It shows that, for the social optimum, it is necessary that the last tier's consumer price be equal to the marginal cost evaluated at total output. Furthermore, the individual optimal rates in all four schemes are indeterminate, but they must be constrained by each scheme's respective optimal policy locus. To solve for the explicit policy structure, this essay further applies linear demand and quadratic total cost functions and shows that the optimal policy calls for using specific rate subsidies to all tiers under the uniform and no-uniform regimes, but using ad valorem rate subsidies under the uniform regime. The essay further explores which of the four schemes is the most cost effective for the government. The final core essay produces new theorems on tridiagonal matrices that are crucial in proving the various results in the other core essays. One of the key theorems shows that in an n × n tridiagonal matrix T that has positive dominant diagonals and negative super- and sub-diagonals, there exists a strictly positive vector x that satisfies the system Tx = b, where b is non-negative and b is not a zero vector. Furthermore, if T is symmetric, the the components of x can be ranked under certain conditions.