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 consumer 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.
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 in where . That is, every element in is a nonnegative orthant in . A consumption bundle takes the following form: where . 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 there is a price vector where and . That is a price of a good is a positive number.
Furthermore, assume that the consumer's income is . The budget set, or the set of all possible consumption bundles is:
.
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: .
The budget set is homogenous of degree 0 in prices and income: since every that exhibits that exhibits that (we can delete the 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, which is assumed to be a closed and convex set. Every element in is a nonnegative orthant in . That is, every consumption bundle take the following form: where .
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 and in , either or or both. That is, the consumer prefers over , or he prefers over , or he is indifferent between and .
Note that,
If holds but not then we can learn that . That is, the consumer prefers over . If and hold, then we can learn that . That is, the consumer is indifferent between and .
Reflexive For all in , .
The consumer is indifferent between a consumption bundle and the same consumption bundle (a very trivial assumption).
Transitivity For all , if and , then .
Namely, if the consumer weakly prefers over and over , then he weakly prefers over .
Continuity Suppose that and then
were is a ball with radius around the bundle, and .
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 over the consumption bundle . However, the consumer prefers over were and . 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 is monotone if and only if
means that for all with at least one case for which . means that for all .
A preference relation is strictly monotone if any increase of good makes the consumer better off:
.
Convexity
The assumption of convexity states that the consumer prefers "average" bundles over extreme ones.
More formally: suppose that and where . Then, .
In simple words, suppose that the consumer is indifferent between and , and is a bundle that is a weighted average of and with weights and respectively, then is no worse than or .
If the preference relation exhibits strict convexity than . 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:
The consumer's optimal choice is the utility maximizing bundle of all bundles in the budget set.
.
is set-valued and it is called the Marshallian demand correspondence.
If is continuous and no commodities are free of charge, then exists, but it is not necessarily unique. 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 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. 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. If this is satisfied then 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:
By differentiating with respect to we obtain the first order conditions:
.
From the first order conditions we obtain that for each two goods , the Marginal Rate of Substitution is equal to the price ratio between these goods:
.
By differentiating with respect to we obtain the budget constraint: .
The first order condition, , 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).
Suppose that . This implies that the consumer values an extra unit of more than the amount of he must give up to buy it. Hence, substituting toward good raises utility.
Note that if the consumer gives up units of to obtain one more unit of his utility remain unchanged.
Reaction to changes in prices The budget set is homogenous of degree zero, therefore, if the both prices and income increase by the same rate, the budget set does not change, and hence the optimal solution to the consumer's problem remain unchanged. That is, the demand for all goods is homogenous of degree 0 in income and prices.
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.
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
Suppose we have two goods: x and y. 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).
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.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).
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 where . Note that . The MRS decreases when decreases and 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.
The first order condition:
Inserting to the budget constraint: