We present a novel bilevel programming formulation that aims to solve a resource allocation problem for request-response systems. Our formulation is motivated by potential inefficiencies in the allocation of computational resources to incoming user requests in such systems. In our experience, systems often operate with a surplus of resources despite potentially incurring unjustifiable cost. Our work attempts to optimize the tradeoff between the financial cost of resources and the opportunity cost of unfulfilled user demand. Our bilevel formulation consists of an \textit{upper} problem which has a constraint value appearing in the \textit{lower} problem. We derive efficient methods for finding global solutions to the upper problem in two settings; first with logarithmic utility functions, and then with a particular type of sigmoidal utility function. A solution to the model we describe (1) determines the optimal number of total resources to allocate and (2) determines the optimal distribution of such resources across the set of user requests.