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Note: These explanations are in the process of being adapted from my textbook.
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Shifts in Demand Curves

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The demand curve shows the quantity demanded of a good as a function of its price. However, we’ve shown that the quantity demanded is a function not only of the good’s own price, but also a function of income and the prices of other goods. These are sometimes known as demand shifters, because when they change, the entire demand curve (may) shift. The direction of that shift depends on the nature of the demand function.

Responses to changes in the prices of other goods: Complements and substitutes

The way the demand curve shifts in response to the price of another good depends on the relationship between those two goods:

Let’s look at each of these in turn, by examining the effect of a price increase in one flavor of jelly on the quantity demanded of peanut butter, strawberry jam, and t-shirts.


If two goods are complements, an increase in the price of either good will result in a decrease in the quantity bought of both goods. For example, if you enjoy sandwiches made with peanut butter (good 1) and grape jelly (good 2), an increase in the price of jelly increases the price of making a PB&J sandwich; so you might have fewer such sandwiches and do something else for lunch.

If peanut butter and jelly are perfect complements with the utility function $u(x_1,x_2) = \min{x_1,x_2}$, we could illustrate the effect of this price change in the following graph:

– Graph to be added –

Note, however, that goods might be complements without being perfect complements. That is, we can observe complementary behavior even if you don’t only consume goods in a strict ratio. For example, you might have other uses for peanut butter than just making PB&J sandwiches!

One useful family of utility functions is a CES utility function, of the form \(u(x_1,x_2) = (ax_1^r + bx_2^r)^{1 \over r}\) In this case the $r$ parameter measures how complementary or substitutable goods are. As $r$ approaches $-\infty$, the preferences approach the “perfect complements” or Leontief functional form we’ve become familiar with. However, for any negative value of $r$, the goods are complementary. For example, the graph below shows the same price change for $r = -1$:

– Graph to be added –

Even though these goods aren’t perfect complements, an increase in the price of good 2 still causes the consumer to buy less good 1 and less good 2, and causes the demand curve for good 1 to shift in (to the left).


If two goods are substitutes, an increase in the price of one good will result in a decrease in the quantity bought of that good, and an increase in the quantity of the other. For example, if you view strawberry jam (good 1) and grape jelly (good 2) as substitutes, then an increase in the price of grape jelly will cause you to use more of the relatively cheaper strawberry jam in recipes which could use either.

For a CES function, a value of $r$ between 0 and 1 represents goods which are substitutes but not perfect substitutes. (When $r = 1$, you can check to see that they’re perfect substitutes.) For example, with $r = {1 \over 2}$, we can illustrate the effect of this price change in the following graph:

– Graph to be added –

As you can see, the case of perfect substitutes is an extreme example of this kind of behavior: when the price of one good increases beyond a certain threshold, the optimal bundle jumps from buying only one good to buying only the other. But that really is an extreme case: in general, changes in the prices of imperfect substitutes like strawberry jam and grape jelly will affect the demand for the other good, but not necessarily drive it down to zero. After all, sometimes you crave grape jelly, even if it’s pricier!

Independent Goods

Finally, let’s think about goods like t-shirts (good 1) and grape jelly (good 2), which have no obvious connection. For such goods, we would not expect a change in grape jelly to affect the quantity of t-shirts bought at all:

The utility function we’ve seen that exhibits this behavior is Cobb-Douglas, in which the consumer will spend a given fraction of their income on each good. For example, with the Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$, the consumer’s demand for good 1 is $x_1^\star(p_1,p_2,m) = m/2p_1$, which doesn’t depend at all on $p_2$.

– Graph to be added –

In the next chapter, we’ll go into much more detail about what makes goods complements, substitutes, or independent goods.

Responses to changes in income: Normal and inferior goods

– coming soon –

Copyright (c) Christopher Makler /