Utility maximization problem: Difference between revisions

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{{For|a less technical introduction|Utility}}

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{{More footnotes needed|date=August 2010}}

{{more citations needed|date=March 2011}}

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Utility maximization was first developed by utilitarian philosophers [[Jeremy Bentham]] and [[John StewartStuart Mill]]. In [[microeconomics]], the '''utility maximization problem''' is the problem [[consumer]]s face: "How should I spend my [[money]] in order to maximize 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 [[Natural borrowing limit|constraint on total spending]] (income), the prices of the goods and their [[Preference (economics)|preferences]].

Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are modelled as being [[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 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.

==Basic setup==

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=== 1) Walras's Law ===

[[Walras's law]] states that if a consumers preferences are complete, monotone and transitive then the optimal demand will lie on the [[Budget constraint|budget line]].<ref>{{Cite book|last=Levin|first=Jonothan|title=Consumer theory|publisher=Stanford university|year=2004|pages=4–6}}</ref>

==== Preferences of the consumer ====

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For a preference relation to be [[Monotone preferences|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).

The preference <math>\succcurlyeq</math> is monotone if anyand only if;

1)<math>(x+\epsilon, y)\succcurlyeq(x,y)</math>

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=== 2) 'Bang for buck' ===

[[Bang for the buck|Bang for buck]] is a main concept in utility maximization andwhich consistsrefers ofto the consumer's wantingdesire 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 maximization problem|publisher=Department of economics, UCLA|year=2009|pages=10–17}}</ref> To find this point, derivedifferentiate the utility function with respect to x and y to find the marginal utilities, then divide by the respective prices of the goods.

<math> MU_x/p_x = MU_y/p_y</math>

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<math>p \in \mathbb{R}^n_+ \ ,</math>

and that the consumer's income is <math>wI</math>; then the set of all affordable packages, the [[budget set]] is,

<math>B(p, I) = \{x \in \mathbb{R}^n_+ | \mathbb{\Sigma}^n_{i=1} p_i x_i \leq I\} \ ,</math>

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

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

<math>x(p, I) = \{x \in B(p,I)| U(x) \geq U(y) \forall y \in B(p,I)\}</math>

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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 complimentscomplements ==

U = min {x, y}

[[File:Utility_maximisation_of_a_minimum_function.png|thumb|Figure 3: This shows the utility maximisation problem with a minimum utility function.]]

For a minimum function with goods that are [[Complementary good|perfect complimentscomplements]], the same steps cannot be taken to find the utility maximising bundle as it is a non differentiable function. Therefore, intuition must be used. The consumer will maximise their utility at the kink point in the highest indifference curve that intersects the budget line where x = y.<ref name=":0" /> This is intuition, as the consumer is rational there is no point the consumer consuming more of one good and not the other good as their utility is taken at the minimum of the two ( they have no gain in utility from this and would be wasting their income). See figure 3.

== Utility maximisation of perfect substitutes ==

U = max {x, + y}

For a utility function with [[Substitute good|perfect substitutes]], the utility maximising bundle can be found by differentiation or simply by inspection. Suppose a consumer finds listening to [[Australia|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 attending a Tame Impala concert is cheaper than attending the AC/DC concert, the consumer chooses to attend the Tame Impala concert, and vice versa. 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 [[marginal rate of substitution|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. The special case is when the (constant) MRS equals the price ratio (for example, both goods have the same price, and same coefficients in the utility function). In this case, any combination of the two goods is a solution to the consumer problem.

[[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).

== Reaction to changes in prices ==

For a given level of real wealth, only relative prices matter to consumers, not absolute prices. 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.

When the prices of goods change, the optimal consumption of these goods will depend on the substitution and income effects. The [[substitution effect]] says that if the demand for both goods is homogenoushomogeneous, when the price of one good decreases (holding the price of the other good constant) the consumer will consume more of this good and less of the other as it becomes relatively cheeper. The same goes if the price of one good increases, consumers will buy less of that good and more of the other.<ref name=":1">{{Cite book|title=Utility Maximization and Demand|publisher=University of Minnesota library|year=2011|pages=chapter 7.2}}</ref>

The income effect occurs when the change in prices of goods cause a change in income. If the price of one good rises, then income is decreased (more costly than before to consume the same bundle), the same goes if the price of a good falls, income is increased (cheeper to consume the same bundle, they can therefore consume more of their desired combination of goods).<ref name=":1" />

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== Reaction to changes in income ==

[[File:Optimal_bundle_reaction_to_changes_in_income.png|thumb|232x232px|Figure 5: This shows how the optimal bundle of a consumer changes when their income is increased.]]

If the consumers income is increased their budget line is shifted outwards andsand they now have more income to spend on either good x, good y, or both depending on their [[Preference (economics)|preferences]] for each good. if both goods x and y were [[normal good]]s then consumption of both goods would increase and the optimal bundle would move from A to C (see figure 5). If either x or y were [[inferior good]]s, then demand for these would decrease as income rises (the optimal bundle would be at point B or C).<ref>{{Cite web|last=Rice University|date=n.d.|title=How changes in income and prices affect consumption choices|url=https://opentextbc.ca/principlesofeconomics/chapter/6-2-how-changes-in-income-and-prices-affect-consumption-choices/|url-status=live|access-date=22 April 2021|website=Press books}}</ref>

== Bounded rationality ==

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* The [[satisficing]] heuristic is when a consumer defines an aspiration level and looks until they find an option that satisfies this, they will deem this option good enough and stop looking.<ref>{{Cite book|last=Wheeler|first=Gregory|title=bounded rationality|publisher=Stanford Encyclopedia of Philosophy|year=2018}}</ref>

* [[Heuristics in judgment and decision-making|Elimination by aspects]] is defining a level for each aspect of a product they want and eliminating all other options that don'tdo not meet this requirement e.g. price under $100, colour etc. until there is only one product left which is assumed to be the product the consumer will choose.<ref>{{Cite web|date=2018|title=Elimination-By-Aspects Model|url=https://www.monash.edu/business/marketing/marketing-dictionary/e/elimination-by-aspects-model|url-status=live|access-date=20 April 2021|website=Monash University}}</ref>

* The [[mental accounting]] heuristic: In this strategy it is seen that people often assign subjective values to their money depending on their preferences for different things. A person will develop mental accounts for different expenses, allocate their budget within these, then try to maximise their utility within each account.<ref>{{Cite web|date=2021|title=Why do we think less about some purchases than others?|url=https://thedecisionlab.com/biases/mental-accounting/|url-status=live|access-date=20 April 2021|website=The decision lab}}</ref>

== Related concepts ==

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==See also==

*[[Welfare maximization]]

*[[Profit maximization]]

*[[Choice modelling]]

*[[Expenditure minimization problem|Expenditure minimisation problem]]