Editing Utility maximization problem (section)

==Basic setup==
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

=== 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 ====
For a utility representation to exist the preferences of the consumer must be complete and transitive (necessary conditions).<ref>{{Cite book|last=Salcedo|first=Bruno|title=Utility representations|publisher=Cornell university|year=2017|pages=18–19}}</ref>

===== Complete =====
Completeness of preferences indicates that all bundles in the consumption set can be compared by the consumer. For example, if the consumer has 3 bundles A,B and C then;

A <math>\succcurlyeq</math> B, A <math>\succcurlyeq</math> C, B <math>\succcurlyeq</math> A, B <math>\succcurlyeq</math>C, C <math>\succcurlyeq</math>B, C <math>\succcurlyeq</math>A, A <math>\succcurlyeq</math>A, B <math>\succcurlyeq</math>B, C <math>\succcurlyeq</math>C.  Therefore, the consumer has complete preferences as they can compare every bundle.

===== Transitive =====
Transitivity states that individuals preferences are consistent across the bundles.

therefore, if the consumer weakly prefers A over B (A <math>\succcurlyeq</math> B) and B <math>\succcurlyeq</math>C this means that A <math>\succcurlyeq</math> C (A is weakly preferred to C)

===== Monotone =====
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 and only if;

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

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

3) <math>(x+\epsilon, y+\epsilon)\succ(x,y)</math>

where  <math>\epsilon</math> > 0

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

[[Bang for the buck|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.<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>

This can be solved to find the optimal amount of good x or good y.

=== 3) Budget constraint ===
The basic set up of the [[budget constraint]] of the consumer is: <math> p_xx + p_yy \leq I</math>

Due to Walras's law being satisfied:  <math> p_xx + p_yy = I</math>

The tangency condition is then substituted into this to solve for the optimal amount of the other good.

=== 4) Check for negativity ===
[[File:Utility_maximising_bundle_when_demand_is_negative.png|thumb|Figure 1: This represents where the utility maximizing bundle is when the demand for one good is negative]]
Negativity must be checked for as the utility maximization 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.