Utility maximization problem

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Utility maximization problem

← Previous revision Revision as of 06:35, 21 April 2026
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Note that if the consumer gives up MRS_{i,j} units of j to obtain one more unit of i his utility remain unchanged.
Note that if the consumer gives up MRS_{i,j} units of j to obtain one more unit of i his utility remain unchanged.

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== Reaction to changes in prices ==
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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.

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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 homogeneous, 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 cheaper. The same goes if the price of one good increases, consumers will buy less of that good and more of the other.{{Cite book|title=Utility Maximization and Demand|publisher=University of Minnesota library|year=2011|pages=chapter 7.2}}

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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 (cheaper to consume the same bundle, they can therefore consume more of their desired combination of goods).

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== Reaction to changes in income ==
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[[File:Optimal_bundle_reaction_to_changes_in_income.png|thumb|232x232px|Figure 3: This figure shows how the optimal bundle of a consumer changes when their income is increased.]]
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If the consumers income is increased their budget line is shifted outwards and 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).{{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/|access-date=22 April 2021|website=Press books}}


== Solving the consumer problem ==
== Solving the consumer problem ==
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However, x_2^{*} may be negative. In this case, x_1^{*} = \frac {I}{p_1} and x_2^{*}=0 .
However, x_2^{*} may be negative. In this case, x_1^{*} = \frac {I}{p_1} and x_2^{*}=0 .

⚫
== 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 homogeneous, 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 cheaper. The same goes if the price of one good increases, consumers will buy less of that good and more of the other.{{Cite book|title=Utility Maximization and Demand|publisher=University of Minnesota library|year=2011|pages=chapter 7.2}}

⚫
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 (cheaper to consume the same bundle, they can therefore consume more of their desired combination of goods).

⚫
== Reaction to changes in income ==
⚫
[[File:Optimal_bundle_reaction_to_changes_in_income.png|thumb|232x232px|Figure 3: This figure 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 and 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).{{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/|access-date=22 April 2021|website=Press books}}


== Do consumers maximize utility? ==
== Do consumers maximize utility? ==
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