Utility maximization problem: Difference between revisions

Revision as of 01:42, 23 June 2022

Latest revision as of 15:50, 15 June 2024

(12 intermediate revisions by 11 users not shown)

Line 6:

}}

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==

Line 19:

=== 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 ====

Line 37:

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>

Line 49:

=== 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, differentiate 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>

Line 77:

<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>

Line 87:

:<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>

Line 95:

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 ==

Line 115:

== 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 ==

Line 123:

* 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 ==

Line 132:

==See also==

*[[Welfare maximization]]

*[[Profit maximization]]

*[[Choice modelling]]

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