the optimality of . Continuity The maximum theorem implies that if: The utility function is continuous with respect to , The correspondence is non-empty, compact-valued, and continuous with respect to , then is an upper-semicontinuous correspondence. Moreover, if is unique, then it is a continuous function of and . Combining with the previous subsection, if the consumer has strictly convex preferences, then the Marshallian demand is unique and continuous. In contrast, if the preferences are not convex, then the Marshallian demand may be non-unique and non-continuous. Homogeneity The optimal Marshallian demand correspondence of a continuous utility function is a homogeneous function with degree zero. This means that for every constant This is intuitively clear. Suppose and are measured in dollars. When , and are exactly the same quantities measured in cents. When prices and wealth go up by a factor a, the purchasing pattern of an economic agent remains constant. Obviously, expressing in different unit of measurement for prices and income should not affect the demand. Demand curve Marshall's theory exploits that demand curve represents individual's diminishing marginal values of the good. The theory insists that the consumer's purchasing decision is dependent on the gainable utility of a goods or services compared to the price since the additional utility that the consumer gain must be at least as great as the price. The following suggestion proposes that the price demanded is equal to the maximum price that the consumer would pay for an extra unit of good or service. Hence, the utility is held constant along the demand curve. When the marginal utility of income is constant, or its value is the same across individuals within a market demand curve, generating net benefits of purchased units, or consumer surplus is possible through adding up of demand prices. Examples In the following examples, there are two commodities, 1 and 2. 1. The utility function has the Cobb–Douglas form: The constrained optimization leads to the Marshallian demand function: 2. The utility function is a CES utility function: Then In both cases, the preferences are strictly convex, the demand is unique and the demand function is continuous. 3. The utility function has the linear form: The utility function is only weakly convex, and indeed the demand is not unique: when , the consumer may divide his income in arbitrary ratios between product types 1 and 2 and get the same utility. 4. The