# Marginal Utility

Marginal utility is the change in wellbeing (or change in total utility) that a person experiences if they consume one additional unit of a good or service.

One of the most basic ideas in economics is that the consumption of goods and services changes our level of satisfaction or happiness (also referred to as utility). Often, we measure this change in satisfaction in monetary terms. If, for example, we want to know how much the wellbeing of a person changes if we give them a book (for free), we can simply ask this person how much he or she is willing to pay for the book. Suppose we ask a student (let´s call her Ann) what the highest price she would be willing to pay for a particular book was and her answer was € 15. This means that Ann’s increase in happiness when we give her the book is the same as the increase in happiness she would experience if we gave her € 15.

In the special case when consumers consume just one unit of a good, the marginal utility is the same as the maximum price the consumer is willing to pay for that good. However, with respect to most goods and services we do not consume just one unit, but several units. For example, the number of cups of coffee we drink each day, the number of text messages we send, the number of megabytes we stream, the liters of gasoline that we fuel our car with, and so forth. Each additional unit that we consume (e.g. each additional cup of coffee we drink) changes our wellbeing. This change in wellbeing from one unit to the next is what we call marginal utility and for many goods and services it is not constant from one unit to the next.

## Law of diminishing marginal utility

Consuming additional units of a good or service increases our wellbeing, but the change is not necessarily the same for each unit. For instance, let´s say we consume three cups of coffee. The first cup we enjoy a lot, it wakes us up in the morning and gives us a lot of energy to help start the day. The second cup, we might also enjoy, but probably a bit less than the first one, and then the third cup might not increase our overall wellbeing at all - for many this is a daily disappointment. This property, that the marginal utility decreases with the number of units we consume, is called the law of diminishing marginal utility. For many goods and services, the law of diminishing marginal utility holds, although in some cases the change in utility might decline faster than in others. For certain types of goods, the change in utility might be almost the same for each unit or not decrease at all, think e.g. about fresh uncontaminated air. Marginal utility can also be negative if our wellbeing decreases after some point. The 5th cup of coffee, for example, might actually make us anxious and feel unwell. It often does…

## How to calculate marginal utility

As explained above, marginal utility is the change in total utility if we consume one more unit. Suppose we want to calculate the marginal utility that the 3rd cup of coffee gives to an individual who gains the following total utility from consuming this (see below) number of coffees:

 Nr of cups of coffee 0 1 2 3 4 5 Total utility (measured in monetary units) 0 5 7 8 8 5

As we see from the table, the first cup of coffee increases the total utility of our individual by the same amount as if we gave them € 5. It is the difference in total utility between consuming 1 cup and zero cups of coffee:

Marginal Utility (MU) of nth unit Total utility (n units) Total utility (1 units)

MU of 1st cup of coffee Total utility 1 cup Total utility 0 cups = 5 - 0.

We can compute the marginal utility of the other cups of coffees as follows:

 Nr of cups of coffee 0 1 2 3 4 5 Marginal utility 0 5 2 1 0 -3

Let´s say the price per cup is € 2. How many cups will a rational individual consume? The change in total utility of the first cup is equivalent to € 5, while the price is € 2. Thus, the marginal utility exceeds the price of the good and a rational utility maximizing person will buy the first cup at a price of € 2. What about the second cup? The increase in utility is the same as the price of the cup. This means that our individual is indifferent, they are unsure whether to buy the second cup of coffee and or keep the money. A rational consumer will buy or consume a unit of a good if the marginal utility of this unit exceeds the price he or she has to pay for this unit.

In many applications we assume that goods are perfectly divisible and we will have a particular utility function given. In these cases, the marginal utility is simply the derivative of total utility. Let’s consider an example: suppose we have an individual with utility function u(q) = ln⁡ (q), where q denotes the number of units the person consumes of a particular good. For this utility function, marginal utility MU ∂u/∂q 1/q. This is the marginal utility of the qth unit. If we want to know what the marginal utility of a specific unit is, for instance the 3rd unit, we simply compute the value of MU for q = 3 and obtain MU(3) 1/3.

## Marginal utility and the law of demand

The marginal utility is the change in wellbeing of consuming one more unit and is equivalent to the maximum willingness to pay for this additional unit. The law of diminishing marginal utility tells us that the change in wellbeing decreases as we consume more and more units. The willingness, therefore, to pay for one additional unit decreases as we consume more units. This implies that there is a negative relationship between the quantity consumers want to consume and the price they are willing to pay for that quantity. Put differently, the higher the price, the lower the quantity demanded. This is what we call the law of demand.