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Practice Problems

Practice Problems

Utility and Indifference Curves

We know how to represent changes in demand as price or income changes on a graph, but how can we show preferences? What makes buyers happy and how can we measure that happiness? Economists use the term utility when referring to the level of happiness or satisfaction that someone experiences from buying (or selling) goods and services: the more utility, the happier the person. Utility is typically represented on a graph in an indifference curve. An indifference curve represents all of the different combinations of two goods that generate the same level of utility. What this means is that each point on an indifference curve represents a combination of goods. All points on one indifference curve give the person the exact same amount of happiness. For instance, if you give Jim a choice between points A and B on this indifference curve, he won't really mind either way, he is indifferent. One shirt and two hats makes him just as happy as two shirts and one hat, which is why both points are on the same indifference curve.

One of Jim's Indifference Curves for Shirts and Hats

In general, indifference curves bow in towards the origin, rather than being straight lines or outward-bulging curves. The reason for this is that most people do not like extremes: they would rather have a some shirts and some hats than many hats and no shirts. This changing preference results in the traditional inward-curving indifference curves, and illustrates the effects of diminishing returns. In this example, diminishing returns simply means that the first hat Jim gets makes him happier than the second hat, which makes him happier than the third, and so on. His marginal utility--the extra utility he gets with each hat--decreases with the number of hats he gets. After a while, he has had enough of hats, the extra ones don't make him much happier, and he'd rather get a shirt, and might even trade several hats for one shirt. Generally, at the extremes, people are willing to give up many hats to get a few shirts; as the numbers even out, this swapping ratio decreases, and then when they start moving to the other extreme, they want a lot of shirts in exchange for any hats they might give up.

Another example that illustrates the principle of diminishing returns would be the case in which you give Thom a choice between gold and steak. We all know that a bar of gold is worth more than a steak, so only a fool would choose the steak over the gold, right? Thom knows this. If you ask Thom to choose between a bar of gold and a steak, he will probably choose the gold, and be very excited to have a bar of gold. The marginal utility of that first bar of gold is quite high. An hour later, he will choose another bar of gold, and he will still be happy to get another bar of gold; the marginal utility he gets from the second bar of gold might not be quite as high as the marginal utility from the first bar, but it's still higher than the marginal utility he would get from a steak. This will continue, bar after bar, with the added utility of each bar of gold being a little lower than the last. Eventually, Thom will start to get hungry, and if he gets hungry enough, then he will choose the steak over the gold, as the marginal utility from the steak will be higher than the marginal utility from a bar of gold. Thom still knows the relative values of gold and steak, and he knows that he is choosing something that is worth less, but in his situation, he has so much gold that more gold makes very little difference, but a steak can make a large difference, as he is very hungry.

Different indifference curves represent different levels of utility, and in general, more is better: the more goods you have, the happier you are. On the graph, we see this preference for more as an indifference curve that is further away from the origin. Thus, because curve 2 is further out than curve 1, and represents a higher level of utility, any point on curve 2 will be preferable to any point on curve 1, and any point on curve 3 will be preferable to any point on curves 1 or 2.

Indifference Curves

A few more important observations about one person's indifference curves: they can never cross. Why is this true? Think about it this way: if curve 2 is supposed to make you happier than curve 1, but curve two crosses curve 1, then that means that at the point of intersection, you are experiencing two different levels of utility, that is, you are both happy and happier at the same time, which makes no sense. Thus, indifference curves never intersect, but move further away from the origin with increased levels of utility.

A Correct Set of Indifference Curves

An Incorrect Set of Indifference Curves

The indifference curves we have been considering are for normal goods. How can we tell? Because more of either good increases utility. Starting on curve 1 and moving outwards (increasing the number of hats) or upwards (increasing the number of shirts) lands us on curve 2, representing a higher level of utility. Using different types of goods changes what indifference curves look like.

For instance, if one good is a normal good, such as CDs, and the other good is an undesirable good, such as Spam, the indifference curves will look like this, with the second indifference curve being better than the first:

Utility Curves for Normal and Undesirable Goods
As you can see, an increase in number of CDs causes an increase in utility, since we end up on a better indifference curve, but an increase in the amount of Spam results in a decrease in utility.

What if the consumer doesn't care about one of the goods, meaning that getting more or less of that good doesn't make them happier or unhappier? For instance, replace the Spam with expired baseball tickets. Jim likes getting CDs, but really doesn't care how many expired baseball tickets he gets. This makes his indifference curves look like this:

Indifference Curves for Normal and Neutral Goods
Note that increasing or decreasing the number of baseball tickets makes no difference in his indifference curve; only changing the number of CDs moves him to a different indifference curve.

Another instance in which indifference curves behave strangely is in the case of complementary goods. Demand for complementary goods is directly related. In other words, buying one good increases the probability you'll buy the other good, those two goods are complementary. Mittens are an extreme example of complementary goods: if you buy a right mitten, it is almost a sure bet that you'll buy a left mitten. This also means that having extra stray mittens isn't likely to increase your utility. There is virtually no difference in your happiness whether you have one right mitten and one left mitten, or two right mittens and one left mitten. This shows up in the following indifference curves (note that only a simultaneous increase of right and left mittens will result in increased utility).

Indifference Curves for Complementary Goods: Mittens

What if the two goods being evaluated are pretty much the same? Such goods are called substitute goods: the buyer considers them to be interchangeable. One example of perfect substitutes (though some might argue otherwise) could be Coke and Pepsi. If you consider them to be the same thing, then you don't mind which one you get. More is still better, but you don't care what combination of the two you get, which means that when you have a lot of Pepsi, you would not be willing to trade three cans of Pepsi for one can of Coke, eliminating the inward bend of the indifference curves. This results in indifference curves like this:

Indifference Curves for Substitute Goods: Cola

Utility Optimization

While it is impossible to know exactly what goes on inside a buyer's head while they are making a decision, we can assume that a normal person will choose whichever combination will make them as happy as possible, given their choices and their budget. On the graph, this means that they will choose whichever combination lands them on the highest indifference curve possible. We can see this optimization if we draw in the consumer's budget constraint on the same graph as the consumer's indifference curves.

To draw a budget constraint, a line that shows the maximum amount of goods a buyer can purchase with their available funds, you need to know two things: 1) how much money they have, and 2) the prices of the two goods being considered. Once you have both pieces of information, it is simply a matter of finding out the maximum amount of the first good you can buy, without buying any of the second, then finding the maximum amount of the second good you can buy, without buying any of the first. Mark these points on the graph and connect them. To illustrate, suppose Tina has $100. She is deciding how many bottles of wine and how many wine glasses she wants to buy. If wine costs $20 a bottle and glasses cost $5 each, then the most wine she can buy is ($100/$20)=5 bottles. Likewise, she can buy at most ($100/$5)=20 wine glasses. Her budget constraint would look like the darker line, while the filled area includes all of her possible buying decisions, given the amount of money she has. Anything not included in the colored area is out of her budget:

Tina's Budget Constraint

If we know her indifference curves, we can draw her budget constraint in with them on the same graph. After that, it is simply a matter of finding the outermost indifference curve that is tangent to (just barely touches) her budget constraint, and use this tangent point as her optimal combination of wine and glasses. In this case, it is the second indifference curve that optimizes her utility given her budget.

Optimizing Tina's Purchasing Decision
It looks like Tina will buy about 12 wine glasses and 2 bottles of wine. Even though the optimal amount is a little more than 2 bottles, she has to buy either 2 bottles or 3 bottles, and 2 is all she can afford. (When doing such problems, never round up, since that will land you outside of the budget constraints).

Why does it have to be the indifference curve that is tangent to her budget constraint? If it were an indifference curve that crosses her budget constraint, such as the first indifference curve, then we can see that the two points of intersection don't make her as happy as the single tangent point in the previous graph. By picking the outermost curve that still touches her budget constraint, we have maximized her utility. We can't pick a curve any further out, such as the third indifference curve, since she can't afford to buy more than $100 worth of wine and glasses.

Obviously, budget constraints change with changes in income or price. For instance, if Tina now has $125 instead of $100, her new budget constraint will be a parallel shift out from her original budget constraint. The yellow shaded region represents the increase in possible purchases she can make:

A Shift in Tina's Budget Constraint

On the other hand, if Tina still has only $100, but the price of wine changes from $20 a bottle to $10 a bottle, her budget constraint will pivot to reflect this change:

A Pivot in Tina's Budget Constraint

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