In microeconomic theory, the utility maximization problem formalizes how a consumer allocates limited resources across different goods and services. The consumer is assumed to have well-defined preferences over all feasible bundles of goods and to be able to rank these bundles according to the level of utility they provide. Given a budget constraint determined by income and prices, the consumer chooses the most preferred bundle that is affordable. The utility maximization problem yields a systematic analysis of consumer demand and how it changes in response to changes in income or prices.
The utility maximization problem was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. It is formulated as follows: find the consumption bundle that maximizes the consumer's utility subject to his budget constraint.
Consumption bundle
A consumption bundle is an element <math>x</math> in <math>X</math> <math>(x\in X)</math> where <math>x\in R_+^k</math>. That is, every element <math>x </math> in <math>X </math> is a nonnegative orthant in <math>R^{k} </math>. A consumption bundle takes the following form: <math>x=(x_1, x_2,...,x_k)</math> where <math>x_i\geq0 </math> <math>\forall i=1,..,k</math>. In simple words, the consumer cannot consume a negative amount of good.
The budget constraint
The consumer maximizes his utility subject to his budget constraint. 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.
Assume their is a price vector <math>p</math> where <math>p=(p_1,...,p_k)</math> and <math>p_i>0 \forall i=1,..,k</math>. That is a price of a good is a positive number.
Furthermore, assume that the consumer's income is <math>I</math>. The budget set, or the set of all possible consumption bundles is:
<math>B(p, I) = \{x \in \mathbb{R}^k_+ | \mathbb{\Sigma}^k_{i=1} p_i x_i \leq I\} </math>.
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.
In a two good world, the basic set up of the consumer's budget constraint is: <math> p_1x_1+p_2x_2 \leq I</math>.
The budget set is homogenous of degree 0 in prices and income: <math>B(p, I) = B(\lambda p, \lambda I) </math> since every <math>x </math> that exhibits that <math>{\Sigma}^k_{i=1} \lambda p_i x_i \leq \lambda I\ </math> exhibits that <math>= {\Sigma}^k_{i=1} p_i x_i \leq I\ </math> (we can delete the <math>\lambda </math> from both sides). That is, if the prices and income increase by the same rate, the budget set does not change.
Preferences
The consumer preferences are defined over the of all possible bundles, that is, over<math>X</math>, which is assumed to be a closed and convex set. Every element <math>x </math> in <math>X </math> <math>(x\in X)</math> is a nonnegative orthant in <math>R^{k} </math>. That is, every consumption bundle take the following form: <math>x=(x_1, x_2,...,x_k)</math> where <math>x_i\geq0 </math> <math>\forall i=1,..,k</math>.
We want the consumer's preferences to create a well-defined order over the consumption bundles. Therefore, the some properties must be satisfied:
Completeness
Completeness of preferences indicates that all bundles in the consumption set can be compared by the consumer. For all <math>x </math> and <math>y</math> in <math>X </math>, either <math>x\succeq y</math> or <math>y\succeq x</math> or both. That is, the consumer prefers <math>x </math> over <math>y </math>, or he prefers <math>y </math> over <math>x </math>, or he is indifferent between <math>x </math> and <math>y </math>.
Note that,
- If <math>x\succeq y</math> holds but not <math>y\succeq x </math> then we can learn that <math>x\succ y </math>. That is, the consumer prefers <math>x </math> over <math>y </math>.
- If <math>x\succeq y</math> and <math>y\succeq x </math> hold, then we can learn that <math>x\thicksim y </math>. That is, the consumer is indifferent between <math>x </math> and <math>y </math>.
Reflexive
For all <math>x </math> in <math>X </math>, <math>x\succeq x</math>.
The consumer is indifferent between a consumption bundle and the same consumption bundle (a very trivial assumption).
Transitivity
For all <math>x, y , z \in X </math>, if <math>x\succeq y</math> and <math>y\succeq z</math>, then <math>x\succeq z </math>.
Namely, if the consumer weakly prefers <math>x </math> over <math>y </math> and <math>y </math> over <math>z </math>, then he weakly prefers <math>x </math> over <math>z </math>.
Continuity
Suppose that <math>x \succ y</math> and <math>x'\in B(x,\epsilon), y'\in B(y,\epsilon) </math> then <math>x' \succ y' </math>
were <math>B </math> is a ball with radius <math>\epsilon </math> around the bundle, <math> \epsilon>0 </math> and <math>\epsilon\rightarrow 0</math>.
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 <math> x=(8,10)</math> over the consumption bundle <math> y=(8,5)</math>. However, the consumer prefers <math> y'=(8+\epsilon,5)</math> over <math> x'=(8,10+\epsilon)</math> were <math> \epsilon>0 </math> and <math>\epsilon\rightarrow 0</math>. Consequently, there is no utility function that represents the lexicographic preference relation.
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.
The preference relation <math>\succcurlyeq</math> is monotone if and only if
- <math>x>y \Rightarrow x\succeq y</math>
- <math>x \gg y \Rightarrow x \succ y</math>
- <math>x>y </math> means that <math>x_i\geqslant y_i </math> for all <math>i=1,..,k</math> with at least one case for which <math>x_i > y_i</math>.
- <math>x \gg y</math> means that <math>x_i > y_i</math> for all <math>i=1,..,k</math>.
A preference relation is strictly monotone if any increase of good makes the consumer better off:
<math>x>y \Rightarrow x\succ y</math>.
Convexity
The assumption of convexity states that the consumer prefers "average" bundles over extreme ones.
More formally: suppose that <math> x\thicksim y</math> and <math>z=\alpha x + (1-\alpha) y</math> where <math>0 < \alpha <1</math>. Then, <math>z \succeq x, y </math>.
In simple words, suppose that the consumer is indifferent between <math>x</math> and <math>y</math>, and <math>z</math> is a bundle that is a weighted average of <math>x</math> and <math>y</math> with weights <math>\alpha </math> and <math>(1-\alpha)</math> respectively, then <math>z</math> is no worse than <math>x</math> or <math>y</math>.
If the preference relation exhibits strict convexity than <math>z\succ x,y</math>. That is, the consumer strictly prefers the average bundle.
The consumer problem
The consumer chooses a bundle to maximize his utility subject to the budget constraint and the non-negativity condition.
More formally:
<math>\max_{x_1,..,x_k}\; u(x_1,..,x_k)</math>
<math>s.t. \sum p_i x_i \leq I</math>
<math>x_i \geq 0
</math> <math>\forall i=1,..,k
</math>
The consumer's optimal choice <math>x(p,I)</math> is the utility maximizing bundle of all bundles in the budget set.
<math>x(p, I) = \{x \in B(p,I)| U(x) \geq U(y) \forall y \in B(p,I)\}</math>.
<math>x(p,I)</math> is set-valued and it is called the Marshallian demand correspondence.
- 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). If this is satisfied then <math>x(p,I)</math> 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:
<math>\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)</math>
By differentiating <math>\mathcal{L} </math> with respect to <math>x_i </math> <math>(i=1,..,k) </math> we obtain the first order conditions:
<math>u_i(\cdot)-\lambda p_i =0 </math> <math>\forall i=1,..,k </math>.
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).
If the consumer's preference relation exhibits monotonicity than the consumer spends his entire resources, as mentioned earlier. This means that at least one good is a normal good. Why? if he does not change the consumed amount of the goods (the goods are neutral), or if he lowers the consumed amount of some goods (some goods are inferior), than he cannot spend his entire income. That is, he must increase the consumed amount of at least one good.
Solving the consumer problem
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.thumb|284x284px|Figure 1: This figure shows the optimal amounts of goods x and y that maximize utility given a budget constraint.2) Check that the utility function is quasi-concave. That is, the second order condition for maximum holds. 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).
Behavioral economist also challenge the theory of the rational consumer who maximizes utility subject to his budget constraint. 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.
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. 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) or change of states over time (in the case of a state dependent utility function).
Bounded rationality
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.
- 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.
- 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.
Related concepts
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 as 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.
See also
- Welfare maximization
- Profit maximization
- Choice modelling
- Expenditure minimization problem
- Optimal decision
- Substitution effect
- Utility function
- Law of demand
- Marginal utility