Utility maximization problem: Difference between revisions

Revision as of 02:56, 26 April 2021

Revision as of 20:10, 3 May 2021

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[[Bang for the buck|Bang for buck]] is a main concept in utility maximisation 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 maximisation 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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:<math>u : \mathbb{R}^n_+ \rightarrow \mathbb{R}_+ \ .</math>

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Then the consumer's optimal choice <math>x(p,w)</math> is the utility maximising 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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If ''u'' is continuous and no commodities are free of charge, then <math>x(p,I)</math> exists,<ref>{{Cite book|title=Choice, preference and Utility|publisher=Princeton university press|year=n.d.|pages=14}}</ref>, but it is not necessarily unique. If the preferences of the consumer are complete, transitive and strictly convex then the demand of the consumer contains a unique maximiser for all values of the price and wealth parameters. If this is satisfied then <math>x(p,I)</math> is called the [[Marshallian demand function]]. Otherwise, <math>x(p,I)</math> is set-valued and it is called the [[Marshallian demand correspondence]].

== Utility maximisation of perfect compliments ==

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[[File:Utility_maximisation_with_a_maximum_function.png|thumb|Figure 4: This shows the utility maximising bundles with a maximum function and a budget line.]]

For a maximum function with [[Substitute good|perfect substitutes]], the utility maximising bundle can also not be found using differentiation, therefore intuition is used. The consumer will maximise their utility at the maximum of x or y (whichever commodity there is more of will be the utility). Therefore, utility will be maximised at either x = 0 (spending all income in y) or y= 0 (spending all income in x), depending on the prices of the commodities and which good they can get more of with their set income.<ref>{{Cite book|last=Bun|first=Linh|title=Intermediate Microeconomics|publisher=University of California|year=2012|pages=2}}</ref>. This is intuition, as because the consumer is rational and these goods are perfect substitutes, there is no point them spending money in both goods as their utility is based on the maximum of the two, so they would receive more utility by only spending on one good (figure 4).