Utility maximization problem

Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill.Because consumers are modelled as being rational, they seek to extract the most benefit for themselves.However, due to bounded rationality and other biases, consumers sometimes pick bundles that do not necessarily maximize their utility.The utility maximization bundle of the consumer is also not set and can change over time depending on their individual preferences of goods, price changes and increases or decreases in income.For utility maximization there are four basic steps process to derive consumer demand and find the utility maximizing bundle of the consumer given prices, income, and preferences.1) Check if Walras's law is satisfied 2) 'Bang for buck' 3) the budget constraint 4) Check for negativity Walras's law states that if a consumers preferences are complete, monotone and transitive then the optimal demand will lie on the budget line.[1] For a utility representation to exist the preferences of the consumer must be complete and transitive (necessary conditions).[2] Completeness of preferences indicates that all bundles in the consumption set can be compared by the consumer.Transitivity states that individuals preferences are consistent across the bundles.C (A is weakly preferred to C) For a preference relation to be monotone increasing the quantity of both goods should make the consumer strictly better off (increase their utility), and increasing the quantity of one good holding the other quantity constant should not make the consumer worse off (same utility).> 0 Bang for buck is a concept in utility maximization which refers to the consumer's desire 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.The tangency condition is then substituted into this to solve for the optimal amount of the other good.(a set of positive real numbers, the consumer cannot preference negative amount 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 choiceThe consumer will maximise their utility at the kink point in the highest indifference curve that intersects the budget line where x = y.Suppose a consumer finds listening to Australian rock bands AC/DC and Tame Impala perfect substitutes.This means that they are happy to spend all afternoon listening to only AC/DC, or only Tame Impala, or three-quarters AC/DC and one-quarter Tame Impala, or any combination of the two bands in any amount.Therefore, the consumer's optimal choice is determined entirely by the relative prices of listening to the two artists.If the two concert prices are the same, the consumer is completely indifferent and may flip a coin to decide.To see this mathematically, differentiate the utility function to find that the MRS is constant - this is the technical meaning of perfect substitutes.As a result of this, the solution to the consumer's constrained maximization problem will not (generally) be an interior solution, and as such one must check the utility level in the boundary cases (spend entire budget on good x, spend entire budget on good y) to see which is the solution.In this case, any combination of the two goods is a solution to the consumer problem.If consumers reacted to changes in nominal prices and nominal wealth even if relative prices and real wealth remained unchanged, this would be an effect called money illusion.The mathematical first order conditions for a maximum of the consumer problem guarantee that the demand for each good is homogeneous of degree zero jointly in nominal prices and nominal wealth, so there is no money illusion.If either x or y were inferior goods, then demand for these would decrease as income rises (the optimal bundle would be at point B or C).[6] for further information see: Bounded rationality In practice, a consumer may not always pick an optimal bundle.Examples of alternatives to utility maximisation due to bounded rationality are; satisficing, elimination by aspects and the mental accounting heuristic.The relationship between the utility function and Marshallian demand in the utility maximisation problem mirrors the relationship between the expenditure function and Hicksian demand in the expenditure minimisation problem.