Which rationing rule does a single consumer follow?

We will investigate the amount of residual demand in a market consisting of only one consumer and two producers. Since there is only one consumer, we cannot really speak about a rationing rule, but we can ask ourselves whether a known rationing rule reflects the consumer’s utility maximizing behavior. We will show that, if the consumer has a Cobb-Douglas utility function, then the amount purchased by the consumer from the high-price firm lies between the values determined according to the efficient rationing rule and the random rationing rule. We will show further, that if the consumer has a quasilinear utility function, then in the economically interesting case his residual demand function will be equal to the residual demand function under efficient rationing