any only if; 1) 2) 3) where > 0 2) 'Bang for buck' Bang for buck is a main concept in utility maximization and consists of the consumer wanting to get the best value for their money. If Walras's law has been satisfied, the optimal solution of the consumer lies at the point where the budget line and optimal indifference curve intersect, this is called the tangency condition. To find this point, differentiate the utility function with respect to x and y to find the marginal utilities, then divide by the respective prices of the goods. This can be solved to find the optimal amount of good x or good y. 3) Budget constraint The basic set up of the budget constraint of the consumer is: Due to Walras's law being satisfied: The tangency condition is then substituted into this to solve for the optimal amount of the other good. 4) Check for negativity Negativity must be checked for as the utility maximization problem can give an answer where the optimal demand of a good is negative, which in reality is not possible as this is outside the domain. If the demand for one good is negative, the optimal consumption bundle will be where 0 of this good is consumed and all income is spent on the other good (a corner solution). See figure 1 for an example when the demand for good x is negative. A technical representation Suppose the consumer's consumption set, or the enumeration of all possible consumption bundles that could be selected if there were a budget constraint. The consumption set = (a set of positive real numbers, the consumer cannot preference negative amount of commodities). Suppose also that the price vector (p) of the n commodities is positive, and that the consumer's income is ; then the set of all affordable packages, the budget set is, The consumer would like to buy the best affordable package of commodities. It is assumed that the consumer has an ordinal utility function, called u. It is a real-valued function with domain being the set of all commodity bundles, or Then the consumer's optimal choice is the utility maximizing bundle of all bundles in the budget set if then the consumers optimal demand function is: Finding is the utility maximization problem. If u is continuous and no commodities are free of charge, then exists, but it is not necessarily