Utility maximization problem: Difference between revisions

Revision as of 16:14, 7 May 2021

Revision as of 09:23, 5 June 2021

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{{refimprove|date=March 2011}}

Utility maximisationmaximization was first developed by utilitarian philosophers Jeremy Bentham and Josh Stewart Mill. In [[microeconomics]], the '''utility maximisationmaximization problem''' is the problem [[consumer]]s face: "How should I spend my [[money]] in order to maximisemaximize my [[utility]]?" It is a type of [[Optimal decision|optimal decision problem]]. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their [[Preference (economics)|preferences]].

Utility maximisationmaximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are [[Rational choice theory|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 maximisemaximize their utility. The utility maximisationmaximization 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.

==Basic setup==

For utility maximisationmaximization there are four basic steps process to derive consumer demand and find the utility maximisingmaximizing bundle of the consumer given prices, income, and preferences.

1) Check Walras's law is satisfied

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[[Bang for the buck|Bang for buck]] is a main concept in utility maximisationmaximization 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.<ref name=":0">{{Cite book|last=Board|first=Simon|title=Utility maximisationmaximization problem|publisher=Department of economics, UCLA|year=2009|pages=10-17}}</ref> To find this point derive the utility function with respect to x and y to find the marginal utilities then divide by the respective prices of the goods.

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=== 4) Check for negativity ===

[[File:Utility_maximising_bundle_when_demand_is_negative.png|thumb|Figure 1: This represents where the utility maximisingmaximizing bundle is when the demand for one good is negative]]

Negativity must be checked for as the utility maximisationmaximization 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.

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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

:<math>u : \mathbb{R}^n_+ \rightarrow \mathbb{R}_+ \ .</math>

Then the consumer's optimal choice <math>x(p,w)</math> is the utility maximisingmaximizing bundle of all bundles in the budget set if <math>x\in B(p,w)</math> then the consumers optimal demand function is:

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Finding <math>x(p,I)</math> is the '''utility maximisationmaximization problem'''.