Utility maximization problem

In microeconomics, a consumer is defined as an individual or a household consisting of one or more individuals. The consumer is the basic decision-making unit that determines which goods and services are purchased and in what quantities. Each day, millions of such choices are made, shaping the allocation of the trillions of dollars worth of goods and services produced annually in the world economy.^[2]

The utility maximization problem was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill.^[3] It is formulated as follows: find the consumption bundle that maximizes the consumer's utility subject to his budget constraint.^[4]

A consumption bundle is an element ${\displaystyle x}$ in ${\displaystyle X}$ ${\displaystyle (x\in X)}$ where ${\displaystyle x\in R_{+}^{k}}$. That is, every element ${\displaystyle x}$ in ${\displaystyle X}$ is a nonnegative orthant in ${\displaystyle R^{k}}$. A consumption bundle takes the following form: ${\displaystyle x=(x_{1},x_{2},...,x_{k})}$ where ${\displaystyle x_{i}\geq 0}$ ${\displaystyle \forall i=1,..,k}$. In simple words, the consumer cannot consume a negative amount of good.^[5]

The consumer maximizes his utility subject to his budget constraint.^[6] The budget constraint is the most simple and intuitive constraint faced by a consumer. The consumer may face a time constraint (the act of consuming takes time), a constraint of both time and money, an intertemporal budget constraint and many more. The economic problem originates from scarcity, therefore, when formulating and economic problem we will usually see some formulation of a constraint.^[7]

Assume their is a price vector ${\displaystyle p}$ where ${\displaystyle p=(p_{1},...,p_{k})}$ and ${\displaystyle p_{i}>0\forall i=1,..,k}$. That is a price of a good is a positive number.^[8]

Furthermore, assume that the consumer's income is ${\displaystyle I}$. The budget set, or the set of all possible consumption bundles is:

${\displaystyle B(p,I)=\{x\in \mathbb {R} _{+}^{k}|\mathbb {\Sigma } _{i=1}^{k}p_{i}x_{i}\leq I\}\ }$.^[9][8]

In simple words, the consumer can choose a consumption bundle whose cost does not exceed his income.

In general, the set of all possible consumption bundles is assumed to be a closed and convex.^[9]

In a two good world, the basic set up of the consumer's budget constraint is: ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq I}$.^[10]

The consumer preferences are defined over the of all possible bundles, that is, over${\displaystyle X}$, which is assumed to be a closed and convex set. Every element ${\displaystyle x}$ in ${\displaystyle X}$ ${\displaystyle (x\in X)}$ is a nonnegative orthant in ${\displaystyle R^{k}}$. That is, every consumption bundle take the following form: ${\displaystyle x=(x_{1},x_{2},...,x_{k})}$ where ${\displaystyle x_{i}\geq 0}$ ${\displaystyle \forall i=1,..,k}$.^[11]

We want the consumer's preferences to create an well-defined order over the consumption bundles. Therefore, the some properties must be satisfied:^[12]

Completeness

Completeness of preferences indicates that all bundles in the consumption set can be compared by the consumer. For all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$, either ${\displaystyle x\succeq y}$ or ${\displaystyle y\succeq x}$ or both. That is, the consumer prefers ${\displaystyle x}$ over ${\displaystyle y}$, or he prefers ${\displaystyle y}$ over ${\displaystyle x}$, or he is indifferent between ${\displaystyle x}$ and ${\displaystyle y}$.

Note that,

• If ${\displaystyle x\succeq y}$ holds but not ${\displaystyle y\succeq x}$ then we can learn that ${\displaystyle x\succ y}$. That is, the consumer prefers ${\displaystyle x}$ over ${\displaystyle y}$.

• If ${\displaystyle x\succeq y}$ and ${\displaystyle y\succeq x}$ hold, then we can learn that ${\displaystyle x\thicksim y}$. That is, the consumer is indifferent between ${\displaystyle x}$ and ${\displaystyle y}$.

Reflexive

For all ${\displaystyle x}$ in ${\displaystyle X}$, ${\displaystyle x\succeq x}$.

The consumer is indifferent between a consumption bundle and the same consumption bundle (a very trivial assumption).

Transitivity

For all ${\displaystyle x,y,z\in X}$, if ${\displaystyle x\succeq y}$ and ${\displaystyle y\succeq z}$, then ${\displaystyle x\succeq z}$.

Namely, if the consumer weakly prefers ${\displaystyle x}$ over ${\displaystyle y}$ and ${\displaystyle y}$ over ${\displaystyle z}$, then he weakly prefers ${\displaystyle x}$ over ${\displaystyle z}$.

Continuity

Suppose that ${\displaystyle x\succ y}$ and ${\displaystyle x'\in B(x,\epsilon ),y'\in B(y,\epsilon )}$ then ${\displaystyle x'\succ y'}$

were ${\displaystyle B}$ is a ball with radius ${\displaystyle \epsilon }$ around the bundle, ${\displaystyle \epsilon >0}$ and ${\displaystyle \epsilon \rightarrow 0}$.^[13]

This assumption means that if the consumer prefers one bundle over the other, an infinitesimal change in the bundles will not change the preference relation. That is, the preferences are well established.

The four assumptions ensure that the consumer's preferences are well-defined and consistent. Moreover, if the four assumption hold, then the consumer's preferences can be represented by a continuous utility function.

• We will show that the lexicographic preference relation does not exhibits continuity: The consumer prefers the consumption bundle ${\displaystyle x=(8,10)}$ over the consumption bundle ${\displaystyle y=(8,5)}$. However, the consumer prefers ${\displaystyle y'=(8+\epsilon ,5)}$ over ${\displaystyle x'=(8,10+\epsilon )}$ were ${\displaystyle \epsilon >0}$ and ${\displaystyle \epsilon \rightarrow 0}$. Consequently, there is no utility function that represents the lexicographic preference relation.^[14]

Monotonicity

The monotonicity assumption emphasizes that the goods are "good" and not "bad". That is, more of a good cannot make the consumers worse off. For a preference relation to be 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.^[15]

The preference relation ${\displaystyle \succcurlyeq }$ is monotone if and only if

1. ${\displaystyle x>y\Rightarrow x\succeq y}$
2. ${\displaystyle x\gg y\Rightarrow x\succ y}$

• ${\displaystyle x>y}$ means that ${\displaystyle x_{i}\geqslant y_{i}}$ for all ${\displaystyle i=1,..,k}$ with at least one case for which ${\displaystyle x_{i}>y_{i}}$.
• ${\displaystyle x\gg y}$ means that ${\displaystyle x_{i}>y_{i}}$ for all ${\displaystyle i=1,..,k}$.

A preference relation is strictly monotone if any increase of good makes the consumer better off:

${\displaystyle x>y\Rightarrow x\succ y}$.^[15]

Convexity

The assumption of convexity states that the consumer prefers "average" bundles over extreme ones.^[16]

More formally: suppose that ${\displaystyle x\thicksim y}$ and ${\displaystyle z=\alpha x+(1-\alpha )y}$ where ${\displaystyle 0<\alpha <1}$. Then, ${\displaystyle z\succeq x,y}$.

In simple words, suppose that the consumer is indifferent between ${\displaystyle x}$ and ${\displaystyle y}$, and ${\displaystyle z}$ is a bundle that is a weighted average of ${\displaystyle x}$ and ${\displaystyle y}$ with weights ${\displaystyle \alpha }$ and ${\displaystyle (1-\alpha )}$ correspondently, then ${\displaystyle z}$ is no worse than ${\displaystyle x}$ or ${\displaystyle y}$.

If the preference relation exhibits strict convexity than ${\displaystyle z\succ x,y}$. That is, the consumer strictly prefers the average bundle.

The consumer chooses a bundle to maximize his utility subject to the budget constraint and the non-negativity condition.

More formally:

${\displaystyle \max _{x_{1},..,x_{k}}\;u(x_{1},..,x_{k})}$

${\displaystyle s.t.\sum p_{i}x_{i}\leq I}$

${\displaystyle x_{i}\geq 0}$ ${\displaystyle \forall i=1,..,k}$

The consumer's optimal choice ${\displaystyle x(p,I)}$ is the utility maximizing bundle of all bundles in the budget set.

${\displaystyle x(p,I)=\{x\in B(p,I)|U(x)\geq U(y)\forall y\in B(p,I)\}}$.

${\displaystyle x(p,I)}$ is set-valued and it is called the Marshallian demand correspondence.^[17]

• If ${\displaystyle u}$ is continuous and no commodities are free of charge, then ${\displaystyle x(p,I)}$ exists^[18], but it is not necessarily unique.^[17]

• If the utility function also exhibits monotonicity then at the optimum, the consumer spends all his resources. The intuition for this result is straightforward: as long as the consumer has money he can by more goods and increase his utility (due to monotonicity).^[17][19]
• If the consumer's preferences are complete, reflexive, transitive, monotone, and strictly convex then the solution to the consumer problem is unique. Suppose that there are two solutions on the budget set. A bundle that is an average of the two solutions is preferable (due to strict convexity) and is on the budget set. That is, the two bundles were not optimal.^[17][18]
• If in addition to the previous, the utility function exhibits the Inada condition than the solution to the consumer problem in an internal one. That is, the consumer chooses to consume a positive amount from each good.^[20] If this is satisfied then ${\displaystyle x(p,I)}$ is called the Marshallian demand function. Otherwise,

Assuming an internal solution (the consumer consumes a positive amount from each good), the solution to the consumer problem is achieved using the Lagrange multiplier:

${\displaystyle {\mathcal {L}}\left(x_{1},\ldots ,x_{n},\lambda \right)=u\left(x_{1},\ldots ,x_{k}\right)+{\lambda }({I}-{\sum \limits _{i=1}^{k}}p_{i}x_{i})}$

By differentiating ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle x_{i}}$ ${\displaystyle (i=1,..,k)}$ we obtain the first order conditions:

${\displaystyle u_{i}(\cdot )-\lambda p_{i}=0}$ ${\displaystyle \forall i=1,..,k}$.^[17]

From the first order conditions we obtain that for each two goods ${\displaystyle i,j}$ , the Marginal Rate of Substitution is equal to the price ratio between these goods:^[17][21]

${\displaystyle MRS_{i,j}={\tfrac {u_{i}}{u_{j}}}={\tfrac {p_{i}}{p_{j}}}}$ ${\displaystyle \forall i,j\in (1,..,k)}$.

By differentiating ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle \lambda }$ we obtain the budget constraint: ${\displaystyle {I}-{\sum \limits _{i=1}^{k}}p_{i}x_{i}=0}$.

The first order condition, ${\displaystyle MRS_{i,j}={\tfrac {p_{i}}{p_{j}}}}$, implies that at the optimum the maximal price the consumer is wiling to pay for a good, the "subjective" value of a good (in terms of another good) equals the "objective" price of that good (in terms of the other good).^[22]

Suppose that ${\displaystyle MRS_{i,j}>{\tfrac {p_{i}}{p_{j}}}}$. This implies that the consumer values an extra unit of ${\displaystyle i}$ more than the amount of ${\displaystyle j}$ he must give up to buy it. Hence, substituting toward good ${\displaystyle i}$ raises utility.^[17]

Note that if the consumer gives up ${\displaystyle MRS_{i,j}}$ units of ${\displaystyle j}$ to obtain one more unit of ${\displaystyle i}$ his utility remain unchanged.

Assume that there are only two goods. For utility maximization there are five basic steps process to derive consumer's. optimal bundle and find the utility maximizing bundle of the consumer given prices, income, and preferences.

1) Check that utility function is monotone. That is, at the optimum, the consumer spends all of his income.

2) Check that the utility function is quasi-concave. That is, the second order condition for maximum holds.^[23][24] In the two goods example the second-order condition implies that the utility function should be convex. In this case, optimal bundle lies in the tangency point between the utility function (See Figure 1).

3) Apply the first order condition to extract one of the variables.

4) Insert into the budget constraint to find the solution.

5) Check for negativity. 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).

Examples

Exponential utility function

Assume that ${\displaystyle u(x_{1},x_{2})=x_{1}^{\alpha }x_{2}}$ where ${\displaystyle \alpha <1}$. Note that ${\displaystyle MRS_{1,2}={\frac {\alpha x_{2}}{x_{1}}}}$. The MRS decreases when ${\displaystyle x_{2}}$ decreases and ${\displaystyle x_{1}}$ increases. That is, the utility function is convex. Therefore, the tangency point between the utility function and the budget set is the solution to the consumer's maximization problem.

${\displaystyle \max _{x_{1},x_{2}}\;x_{1}^{\alpha }x_{2}}$

${\displaystyle s.t.p_{1}x_{1}+p_{2}x_{2}\leq I}$

${\displaystyle x_{1},x_{2}\geqslant 0}$

The first order condition:

${\displaystyle MRS_{1,2}={\frac {\alpha x_{2}}{x_{1}}}={\frac {p_{1}}{p_{2}}}\Rightarrow \alpha x_{2}={p_{1}x_{1}}}$

Inserting to the budget constraint:

${\displaystyle \alpha p_{2}x_{2}+p_{2}x_{2}=I\Rightarrow p_{2}x_{2}(1+\alpha )=I\Rightarrow x_{2}^{*}={\frac {I}{p_{2}(1+\alpha )}}\Rightarrow x_{1}^{*}={\frac {\alpha I}{p_{1}(1+\alpha )}}}$

It can be easily seen that both goods are normal, that is, the demanded amount from each good increases with income. The demanded amount from each good decreases with each own price. Moreover, the good are not substitutes nor complements. Namely, the demand for one good does not change with a change in the other good's price.

Utility maximization of perfect complements

${\displaystyle u(x,y)=min\{x,y\}}$

For a minimum function with goods that are perfect complements, the same steps cannot be taken to find the utility maximizing bundle as it is a non differentiable function. Therefore, intuition must be used. The consumer will maximize their utility at the kink point in the highest indifference curve that intersects the budget line where x = y.^[23] 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 2.

Utility maximization of perfect substitutes

${\displaystyle u(x,y)=x+y}$

For a utility function with perfect substitutes, the utility maximizing bundle can be found by differentiation or simply by inspection. Suppose a consumer finds listening to 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 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.

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.^[25]

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).^[25]

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 preferences for each good. if both goods x and y were normal goods 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 goods, then demand for these would decrease as income rises (the optimal bundle would be at point B or C).^[26]

Consumers are assumed However, due to bounded rationality, which prevents individuals from examining all the possible bundles and acquiring all the available information, for example due to lack of time, consumers sometimes pick bundles that do not necessarily maximize their utility.^[27]

Behavioral economist also challenge the theory of the rational consumer who maximizes utility subject to his budget constraint.^[27] For example, Daniel Kahneman and Amos Tversky conducted experiments that showed that people act irrationally.^[28][29] However, Robert Aumann challenges view and claims that rational acts should be distinguished from rational rules. The first refers to short-term utility maximization, while the latter refers to adhering to rules or habits that promote long-term utility maximization. When people face an unfamiliar situation they choose the action that maximizes their utility in most situation similar to the new one. This action does not necessarily maximize their utility in the new situation they face, but due to lack of time to study the new situation they resort to an action that "works" most of the time.^[30]

Dynamic utility maximization

The utility maximization bundle of the consumer is also not set and can change over time. For example, in the overlapping generation model the prices of the production factors (the price of labor - wage and the price of capital - the interest rate) change over time and so does the decision of the consumer.^[31] Consumer can modify their decisions due to a change of preference over time (for example in an optimal choice of consumption bundle over time under hyperbolic discounting)^[32] or change of states over time (in the case of a state dependent utility function).^[33]

for further information see: Bounded rationality

In practice, a consumer may not always pick an optimal bundle. For example, it may require too much thought or too much time. Bounded rationality is a theory that explains this behaviour. Examples of alternatives to utility maximization due to bounded rationality are; satisficing, elimination by aspects and the mental accounting heuristic.

• 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.^[34]
• Elimination by aspects is defining a level for each aspect of a product they want and eliminating all other options that do 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.^[35]
• 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 maximize their utility within each account.^[36]

The relationship between the utility function and Marshallian demand in the utility maximization problem mirrors the relationship between the expenditure function and Hicksian demand in the expenditure minimization problem. In expenditure minimization the utility level is given and well as the prices of goods, the role of the consumer is to find a minimum level of expenditure required to reach this utility level.

The utilitarian social choice rule is a rule that says that society should choose the alternative that maximizes the sum of utilities. While utility-maximization is done by individuals, utility-sum maximization is done by society.

• Welfare maximization
• Profit maximization
• Choice modelling
• Expenditure minimization problem
• Optimal decision
• Substitution effect
• Utility function
• Law of demand
• Marginal utility