Yesterday's post outlined how to construct an indifference curve. I left the question of why indifference curves are curves, and not straight lines, unanswered. So, I want to cover that off in this post.
As I noted in yesterday's post, indifference curves are the way that we represent the decision-maker's preferences in the constrained optimisation model. We'll stick with using the consumer choice model as our illustrative model. For the consumer, an indifference curve connects all of the bundles of goods that provide the consumer with the same amount of utility (satisfaction, or happiness).
To understand why indifference curves are curves, we first need to recognise that consumers' choices are subject to diminishing marginal utility. Marginal utility is the additional utility that the consumer receives from consuming one more unit of the good. Marginal utility decreases as the consumer consumes more of a good because of satiation (which, taken literally, means that the consumer gets full as they eat more). Consider the example of a consumer that is hungry and has a pizza in front of them. They eat a slice. It gives them some marginal utility (satisfaction). They eat another slice. The second slice gives them some more marginal utility, but not as much as the first slice (because they aren't as hungry anymore). They eat another slice. The third slice gives them some more marginal utility, but not as much as the first or second slice. And so on.
Now, how does that relate to the indifference curve? The slope of the indifference curve is known as the marginal rate of substitution. It is the quantity of one good (Good Y) that the consumer is willing to give up to get one more unit of the other good (Good X), and be just as satisfied (they would have the same utility afterwards). This marginal rate of substitution (MRS) is equal to the ratio of the marginal utilities, [-MUX/MUY], for reasons that I won't go into here (because it requires calculus - but if you are interested, scroll down towards the end of this blog post). The MRS is a negative number, because the indifference curve is downward sloping.
It turns out that it is diminishing marginal utility that leads the indifference curve to be a curve. Consider the indifference curve I0 shown in the diagram below. At the top of the curve, at Bundle A, the consumer has not much of Good X, but lots of Good Y. The marginal utility of Good X will be high (because the consumer doesn't have much, and diminishing marginal utility means that the marginal utility will be high when the consumer doesn't have much of a good). The marginal utility of Good Y will be low (because the consumer has a lot, and diminishing marginal utility means that the marginal utility will be low when the consumer has a lot of a good). So, the ratio [-MUX/MUY] will be a big number in absolute terms (because we are dividing a big number by a small number). The marginal rate of substitution is high, which means that the slope of the indifference curve is steep.
At the bottom of the curve, at Bundle B, the consumer has lots of Good X, but not much of Good Y. The marginal utility of Good X will be low (as explained above, but in reverse). The marginal utility of Good Y will be high. So, the ratio [-MUX/MUY] will be a small number in absolute terms (because we are dividing a small number by a big number). The marginal rate of substitution is low, which means that the slope of the indifference curve is flat.
Now think about moving along the indifference curve. It is a curve, and not a straight line, because as we move along the indifference curve, the marginal utility of Good X decreases, and the marginal utility of Good Y increases. So, the ratio [-MUX/MUY] becomes a smaller number (in absolute terms), which means that the marginal rate of substitution is smaller, and the indifference curve must therefore be a bit flatter. We start at bundles of goods (like Bundle A) where the indifference curve is steep, and end at bundles of goods (like Bundle B) where the indifference curve is flat.
However, having established that diminishing marginal utility leads the indifference curve to be a curve and not a straight line, it turns out that the indifference curve is not always a curve. There are some special cases. One special case is perfect substitutes (which I have blogged about before here). Perfect substitutes are goods that, in the eyes of the consumer, are identical. The consumer is indifferent between them. One example is red M&Ms and blue M&Ms. When goods are perfect substitutes, the indifference curves are straight lines, as shown in the diagram below for red M&Ms and blue M&Ms. That's because, for every red M&M the consumer gives up, they would be willing to accept one blue M&M and remain just as satisfied (same utility). In other words, the marginal rate of substitution is constant (and equal to one, in this case). Because the marginal rate of substitution is constant, the indifference curve must always have the same slope, so it is a straight line.
Another special case is perfect complements. Perfect complements are goods that must be consumed together, in a specific ratio. One example is left shoes and right shoes. If the consumer has one pair of shoes, and is given more left shoes, their utility remains the same - they are no better off, because they still only have one pair of shoes. Similarly, if they are given more right shoes, their utility remains the same, because they still only have one pair of shoes. That leads to indifference curves that are right angles, as shown in the diagram below. The consumer's utility is determined by the number of complete pairs of shoes that they have.
Perfect substitutes and perfect complements are opposite extremes of what indifference curves may look like. In most cases, indifference curves are somewhere between the two extremes. And, if you think about it, in-between a right angle and a straight line is a curve, which is how we usually draw indifference curves.
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