An automated market maker is a natural and common mechanism to subsidize information acquisition, revelation, and aggregation in a prediction market. The sought-after prediction aggregate is the equilibrium price. However, traders with budget constraints are restricted in their ability to impact the market price on their own. We give a detailed characterization of optimal trades in the presence of budget constraints in a prediction market with a cost-function-based automated market maker. As a concrete application of our characterization, we give sufficient conditions for a property we call budget additivity: two traders with budgets B and B′ and the same beliefs would have a combined impact equal to a single trader with budgetB+B′. That way, even if a single trader cannot move the market much, a crowd of like-minded traders can have the same desired effect. We show that a generalization of the heavily-used logarithmic market scoring rule is budget additive for affinely independent payoffs, but the quadratic market scoring rule is not. Our results may be used both descriptively, to understand if a particular market maker is affected by budget constraints or not, and prescriptively, as a recipe to construct markets.