Maximizing Utility

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Learn all about maximizing utility in just a few minutes! Professor Jadrian Wooten of Penn State University walks through how consumers maximize utility when deciding on goods and services to consume.

Consumers maximize utility when the marginal utility per dollar spent on the last unit of each of the goods is equal.

Companies study consumer behavior in order to create more efficient businesses and grow returns. Calculations based on marginal utility (the utility gained by consuming one additional unit of a good) contribute to these decision-making practices. Customer happiness impacts desire for a product and sales, changing how businesses operate.

Understanding marginal utility is important because it is a component of the rational decision-making model. To see how, consider a simple world where there are only two goods available, goods $X$ and $Y$ . ( $X$ and $Y$ also represent the quantities consumed of $X$ and $Y$ , respectively.) An equation is used to examine whether a particular combination of $X$ and $Y$ results in utility maximization—the process of choosing the combination of goods that provides the highest possible level of happiness. To maximize utility, individuals must purchase what is best for their needs and happiness, given the resources they have. Consumers earn money and purchase a new couch and thus are made happier. With utility maximizing, the consumers are happy because they purchased the couch, not because they earned money.

We can determine whether consumption is optimal by looking at two fairly simple conditions. First, consumption is optimal when the marginal utility of the last unit consumed per dollar spent on that last unit is the same for both of the goods. Mathematically, the condition is represented in the optimality equation a formula to calculate the best consumption possibility. Also known as the equimarginal principle.

\frac{\text{MU}_{X}}{\text{P}_{X}}=\frac{\text{MU}_{Y}}{\text{P}_{Y}}

In the formula, MU represents marginal utility and P represents price. (This formula extends in a straightforward manner to accommodate cases where the consumer has more than two goods to choose from.) The second condition is that all of the individual's income is spent on goods

X

and

Y

, assuming

X

and

Y

give positive utility.

The first condition means an individual is behaving optimally when the "bang for the buck" from the last unit of each of the goods is the same. The second condition reflects that if consuming more of the goods makes the consumer happier, it cannot be optimal for the consumer to not spend all of his or her income. Individuals who save most of their money are not behaving optimally because they could be happier if they were using the money on goods instead of saving. Money saved is not optimal for the consumer. A consumer, according to these calculations, would be happier if she spent $5,000 upgrading her car than if she saved the $5,000.

How Consumers Decide What Goods to Consume

When consumption is not optimal, consumers can increase utility by shifting consumption toward the good that has higher marginal utility per dollar spent.

Economists attempt to understand consumer behavior to guess how much of a product will sell. One way to calculate this is the utility-maximizing condition, which assumes that consumers spend every last dollar in ways that optimize utility. The utility maximizing condition can also be used to determine how non-optimal consumption should shift in order to increase utility. The utility maximizing condition is:

\frac{\text{MU}_{X}}{\text{P}_{X}}=\frac{\text{MU}_{Y}}{\text{P}_{Y}}

If consumption is not optimal, either the marginal utility of the last unit of

X

(the

X

term, on the left) is greater than the marginal utility of the last unit of

Y

(the

Y

term, on the right) or the term on the right is greater than the term on the left. For example, for a consumer who values a candy bar more than a piece of fruit of the same price, it would not be optimal consumption for that consumer to spend their last money buying one of each because the marginal utility for them would be greater for the candy bar than the piece of fruit. In either case, utility will increase if consumption is shifted toward the good with the larger term and away from the good with the smaller term. Another way of saying this is that happiness increases when a consumer shifts resources to the good with greater utility.

If this consumption shift toward the good with the higher utility is applied when the optimality equation does not hold, the two sides of the equation will eventually come into balance as long as diminishing marginal utility is present. This is because, as consumption shifts away from a good, marginal utility (and hence marginal utility per dollar spent) increases, and the opposite is true when consumption shifts toward a good. If a consumer finds more pleasure in buying a smartphone than a laptop, the more the consumer purchases, the less the marginal utility will be (it diminishes because they own the product already). Then the marginal utility for the laptop will grow.

Identifying Optimal Consumption

A consumer's optimal consumption can be identified by following the good with the highest marginal utility per dollar spent until all income is used up.

Optimal consumption, individuals consuming in ways to get the best return for consumption, follows the principle of getting the biggest "bang for the buck," or that consumers use utility to get the most good from each last dollar, and can be used to figure out how a consumer can maximize happiness without explicitly doing any sort of optimization calculation. To see how, consider the example with two goods,

X

and

Y

, where the price of

X

is $2, the price of

Y

is $3, and income is $12.

Consumer Behavior and Marginal Utility

Quantity of $X$	MU of $X$	$\frac{\text{MU}_{X}}{\text{P}_{X}}$	Quantity of $Y$	MU of $Y$	$\frac{\text{MU}_{Y}}{\text{P}_{Y}}$
$1$	$20$	$10$	$1$	$21$	$7$
$2$	$16$	$8$	$2$	$18$	$6$
$3$	$12$	$6$	$3$	$15$	$5$
$4$	$8$	$4$	$4$	$12$	$4$

In the table, MU represents marginal utility, the incremental utility associated with consuming one additional unit of a good. The price of $X$ is $2, the price of $Y$ is $3, and income is $12.

Identifying Optimal Consumption

If the consumer is acting optimally, she will choose the good that gives the highest marginal utility per dollar until all money is spent. In this case she will choose the first unit of $X$ because it gives her the most marginal utility. Then she chooses the good that provides the next highest amount of marginal utility, the first second unit of $X$ . The next choice would be to select the first unit of $Y$ . She proceeds in this manner until all her income is used up. She is relying on choosing the product with the highest marginal utility, and marginal utility sinks as consumption of that specific product rises. This is also based on the assumption that the consumer will spend every last dollar on the product that offers the highest marginal utility. When the marginal utilities are the same, she is indifferent between $X$ and $Y$ . Luckily, the consumer can afford both of them, so the particular order of choosing has no impact.

At this point the consumer finds that the optimal way for her to spend all of her $12 is to consume 3 units of $X$ (at $2 each) and 2 units of $Y$ (at $3 each). Note that this consumption does in fact satisfy the condition needed for optimality: the marginal utility per dollar of the last unit of each of the goods is equal.

The Budget Constraint

The budget constraint is a graphical representation of all the ways that the consumer's entire income can be spent.

Consumers want to spend the maximum amount they can, but they are restricted. Many have limited income and only purchase what they can afford. A consumer's budget constraint (BC) is a representation of the choices between two goods that consumers make based on their income. It is the set of all ways that all of the consumer's income can be spent for these two goods. In a simple world with only two goods, called

X

and

Y

in this example, the budget constraint can be represented as a graph showing a line.

The graph depicts the budget constraint sloping downward. The

X

-axis (horizontal) indicates the quantity consumed of movie tickets, the

Y

-axis (vertical) represents the quantity consumed of books. Each increase in the purchase of one good (book or movie ticket) means a decrease in the purchase of the other. The places where the line intersect the axis are where the consumer's total income is spent on a single good; therefore, their income is divided by the price of that good.

Budget	Books Purchased	Movie Tickets Purchased
Books: $18 Movie tickets: $0	3	0
Books: $12 Movie tickets: $6	2	1
Books: $6 Movie tickets: $12	1	2
Books: $0 Movie tickets: $18	0	3

The consumer has $18 to spend, and must decide how to allocate it. Each book purchased decreases the number of movie tickets that can be purchased, and vice versa.

The Budget Constraint

The endpoints of the budget constraint can be found by calculating how much of each of the goods the consumer can afford if all income is spent on that good. For example, if the consumer's income is $18 and the price of a book is $6, then the $Y$ -axis intercept is $18/6=3$ . In other words, the consumer can afford 3 books if she consumes no other good. Note that points in the interior of the triangle formed by the budget constraint line and the axes represent combinations of goods that are affordable but do not exhaust the consumer's entire income.

The budget constraint has two notable features. First, the line slopes downward. This represents the fact that the consumer has to give up some of one good to be able to afford more of another. Second, the budget constraint is a straight line. This is because the slope of the curve is determined by the prices of the goods, which remain constant regardless of the quantities consumed. An individual has a $50 monthly allowance for gas for their car. Their budget limits their consumption, so the individual can only buy up to $50 in gas. The individual can only consume up to what they can afford.

Indifference Curves

Indifference curves are a graphical representation of a consumer's preferences, where each curve represents combinations of goods that provide a particular level of utility.

Consumers make choices based on what will bring them the most happiness. Economists attempt to gauge this in order to predict changes. One way this is represented graphically is through indifference curves. An indifference curve (IC) is graphical representation showing the utility consumers get from consuming combinations of two goods; usually graphed as a curve. Indifference curves can be thought of as the level sets of happiness, where each indifference curve represents a particular level of happiness—that is, utility.

Indifference curves represent consumer choices between goods when each point along the curve has equal utility for the consumer. Consumers can be happy with more books than movie tickets, or they can give up some movie tickets to gain some more books, and be equally happy. Here, IC₁ represents the consumer's ability to be equally happy with either books or movie tickets. If more money is available (and thus the consumer can have more of both books and movie tickets), IC₂ represents their ability to be equally happy with either choice. Moving up again, IC₁ is the curve when quantities of each good are even higher.

Along any one of these curves, the consumer is indifferent among any of the different points on the curve—in other words, the consumer gets the same utility from all of the various combinations of goods

X

and

Y

that the curve represents. If a consumer at a clothing store likes two different sweaters the same amount, the consumer is indifferent: either sweater will bring equal happiness.

Indifference curves have a number of features. First, they slope downward if the goods are actually "goods," or desired by the consumer. This is because a decrease in one good must be offset by an increase in the other in order to maintain the same level of utility, or consumer happiness. If a consumer shopping at a home goods store desires blankets less the more blankets they buy, pillows will likely begin to have a higher utility. Second, indifference curves that are further from the origin represent higher levels of utility, which can be confirmed by the fact that these curves represent greater quantities of both goods. This shows the quality of ordinal values. Higher levels of utility are represented by the direction of the arrow in the diagram. In this feature, the consumer has a higher utility (is happier) with both products, and the location of the curve shifts to the right. Third, indifference curves bend toward the origin when the goods exhibit diminishing marginal utility. Diminishing marginal utility represents a situation where, upon buying an increasing amount of a product, the consumer gains less and less utility from it. Fourth, the curves do not cross each other and do not touch, as a result of the property of ordinal values.

How Consumers Maximize Utility

A consumer's optimal consumption can be found by identifying the point where the budget constraint is tangent to one of the consumer's indifference curves.

Consumers make decisions based on what will bring them the most happiness and what they can afford. Consumers make a variety of choices based on their own budgets and concepts of utility. Utility maximization models the consumer as trying to maximize utility within the boundaries of what is affordable. The rational consumer wants to consume on the highest possible indifference curve that they can afford.

When the budget constraint (BC) is overlaid on the indifference curves (IC), curves within reach become apparent. Here, the consumer can choose from options on IC₁ and IC₂, but all points on IC₃ are out of reach because they are above the consumer's budget. The consumer maximizes utility by purchasing consumption bundle that gives them the highest amount of utility that they can afford. This is shown by the point where the budget constraint touches the utility curve, at

Y^{*}

,

X^{*}

.

Therefore, the optimal consumption point is the one point in optimization where the consumer receives the highest utility. If a consumer at a grocery store is choosing between bananas and apples, he or she needs to purchase the correct amount of each to receive the highest utility given a limited income. The consumer would select the highest amount of bananas and apples that he or she can afford, considering also the respective utility of bananas and apples.

More generally, the optimal consumption point is located where the budget constraint is tangent to one of the indifference curves—in other words, where the budget constraint touches the indifference curve. This one point of intersection is the combination of the two goods that makes the consumer the happiest, the point on the indifference curve with the highest amount of utility that a consumer can afford. The consumer who has a $40 weekly grocery budget will attempt to spend up to $40, as close to it as they can get, while getting the goods they need.

If the consumer's income increases, their budget constraint shifts to the right. This moves them to a new higher indifference curve. Their optimal consumption point is still the point where the consumer receives the highest utility, but it is now on a different higher indifference curve, because the consumer can buy more of both goods.

If the consumer earns more money, their budget constraint (BC₁) shifts to the right (BC₂). They will now increase their spending on the two goods from point A (optimal consumption at lower income) to point B (optimal consumption at higher income), which can be seen as they move to a higher indifference curve (IC₂). They now purchase more movie tickets and books until they spend their entire budget.

<Diminishing Marginal Utility >Suggested Reading