# 6. REVISITING THE CONSUMER’S DEMAND FUNCTION

#6-revisiting-demand-function #cfa #cfa-level-1 #economics #microeconomics #reading-14-demand-and-supply-analysis-consumer-demand #study-session-4

We have now come to the reason why we wanted to explore consumer theory in the first place: We want to have a sound theoretical foundation for our use of consumer demand curves. Although we could merely assume that a consumer has a demand curve, we derive a richer understanding of that curve if we start with a more fundamental recognition of the consumer’s preferences and her response to changes in the parameters that constrain her behavior in the marketplace.

## 6.1. Consumer’s Demand Curve from Preferences and Budget Constraints

Recall that to draw a consumer’s demand curve, we appealed to the assumption of “holding all other things constant” and held preferences, income, and the prices of all but one good constant. Graphically, we show such an exercise by representing a given utility function with a set of indifference curves, and then we superimpose a set of budget constraints, each one representing a different price of one of the goods. Exhibit 15 shows the result of this exercise. Notice that we are “stacking” two exhibits vertically to show both the indifference curves and budget constraints and the demand curve below them. In the upper exhibit, we have rotated the budget constraint rightward, indicating successively lower prices of bread, P1B , P2B , P3B , P4B , while holding income constant at I.

Exhibit 15. Deriving a Demand Curve

Note: A demand curve for bread is derived from the indifference curve map and a set of budget constraints representing different prices of bread.

This pair of diagrams deserves careful inspection. Notice first that the vertical axes are not the same. In the upper diagram, we represent the quantity of the other good, wine, whose price is being held constant, along with income. Hence, the budget constraints all have the same vertical intercept. But the price of bread is falling as we observe ever less steep budget constraints with horizontal intercepts moving rightward. Confronted with each respective budget constraint, the consumer finds the tangent point as indicated. This point corresponds to the respective quantities of bread, Q1, Q2, Q3, and Q4. Note also that the horizontal axes of the two diagrams are identical. They measure the quantity of bread purchased. Importantly, the vertical axis in the lower diagram measures the price of bread. As the price of bread falls, this consumer chooses to buy ever greater quantities, as indicated. The price–quantity combinations that result trace out this consumer’s demand curve for bread in the lower diagram. For each tangent point in the upper diagram, there is a corresponding point in the lower diagram, tracing out the demand curve for bread.

## 6.2. Substitution and Income Effects for a Normal Good

The law of demand says that when price falls, quantity demanded rises; however, it doesn’t say why. We can answer that question by delving a little more deeply into consumer theory. When there is a decline in the price of a good that the consumer has been buying, two things happen. The good now becomes relatively less costly as compared to other goods. That is, it becomes more of a bargain than other things the consumer could purchase; thus, more of this good gets substituted for other goods in the consumer’s market basket. Additionally, though, with the decline in that price, the consumer’s real income rises. We’re not saying that the size of the consumer’s paycheck changes; we’re saying that the amount of goods that can be purchased with the same amount of money has increased. If this good is a normal good, then increases in income lead to increased purchases of this good. So the consumer tends to buy more when price falls for both reasons: the substitution effect and the income effect of a change in the price of a good.

A close look at indifference curves and budget constraints can demonstrate how these effects can be separated. Consider Exhibit 16, where we analyze Warren’s response to a decrease in the price of bread. When the price of bread falls, as indicated by the pivoting in budget constraints from BC1 to BC2, Warren buys more bread, increasing his quantity from Qa to Qc. That is the net effect of both the substitution effect and the income effect. We can see the partial impact of each of these effects by engaging in a mental exercise. Part of Warren’s response is because of his increase in real income. We can remove that effect by subtracting some income from him, while leaving the new lower price in place. The dashed budget constraint shows the reduction in income that would be just sufficient to move Warren back to his original indifference curve. Notice that we are moving BC2 inward, parallel to itself until it becomes just tangent to Warren’s original indifference curve at point b. The price decrease was a good thing for Warren. An offsetting bad thing would be an income reduction. If the income reduction is just sufficient to leave Warren as well off but no better off than before the price change, then we have effectively removed the real income effect of the decrease in price. What’s left of his response must be due to the pure substitution effect alone. So, we say that the substitution effect is shown by the move from point a to point b. If his income reduction were then restored, the resulting movement from point b to point c must be the pure income effect.

Exhibit 16. Substitution and Income Effects for a Normal Good

Note: Substitution effect (Qa to Qb) and the income effect (Qb to Qc) of a decrease in the price of a normal good.

An important thing to notice is that the pure substitution effect must always be in the direction of purchasing more when the price falls and purchasing less when the price rises. This is because of the diminishing marginal rate of substitution, or the convexity of the indifference curve. Look again at Exhibit 16. Note that the substitution effect is the result of changing from budget constraint 1 to budget constraint 3—or moving the budget constraint along the original indifference curve, while maintaining tangency. Note that in the process, the budget constraint becomes less steep, just as the marginal rate of substitution decreases. Warren is no better off than before the changes, but his behavior has changed: He now buys more bread and less wine than before the offsetting changes in income and price. This reason for negatively sloped demand curves never changes.

Sellers can sometimes use income and substitution effects to their advantage. Think of something you often buy, perhaps lunch at your favorite café. How much would you be willing to pay for a “lunch club membership card” that would allow you to purchase lunches at, say, half price? If the café could extract from you the maximum amount each month that you would be willing to pay for the half-price option, then it would successfully have removed the income effect from you in the form of a monthly fixed fee. Notice that Exhibit 16 implies that you would end up buying more lunches each month than before you purchased the discount card, even though you would be no better or worse off than before. This is a way that sellers are sometimes able to extract consumer surplus by means of creative pricing schemes. It’s a common practice among big box retailers, sports clubs, and other users of what is called “two-part tariff pricing.”

EXAMPLE 6

# Two-Part Tariff Pricing

Nicole Johnson’s monthly demand for visits to her health club is given by the following equation: Qd = 20 – 4P, where Qd is visits per month and P is euros per visit. The health club’s marginal cost is fixed at €2 per visit.

1. Determine Johnson’s demand curve for health club visits per month.

2. Calculate how many visits Johnson would make per month if the club charged a price per visit equal to its marginal cost.

3. Calculate Johnson’s consumer surplus at the price determined in Question 2.

4. Calculate how much the club could charge Johnson each month for a membership fee.

## Solution to 1:

Qd = 20 – 4P, so when P = 0, Qd = 20. Inverting, P = 5 – 0.25Q, so when Q = 0, P = 5.

## Solution to 2:

Qd = 20 – 4(2) = 12. Johnson would make 12 visits per month at a price of €2 per visit.

## Solution to 3:

Johnson’s consumer surplus can be measured as the area under her demand curve and above the price she pays, for a total of 12 visits: CS = (½)(12)(3) = 18. Johnson would enjoy €18 per month consumer surplus.

## Solution to 4:

The club could extract all of Johnson’s consumer surplus by charging her a monthly membership fee of €18 plus a per-visit price of €2. This is called a two-part tariff because it assesses one price per unit of the item purchased plus a per-month fee (sometimes called an “entry fee”) equal to the buyer’s consumer surplus evaluated at the per-unit price.

## 6.3. Income and Substitution Effects for an Inferior Good

We know that for some consumers and some goods, an increase in income leads to a decrease in the quantity purchased at each price. These goods are called inferior goods, and they have negative income elasticity of demand. When price falls, these goods still exhibit substitution and income effects, but they are in opposite directions. Consider Exhibit 17, in which we see a fairly standard set of indifference curves and budget constraints. But in this case, bread is an inferior good.

Exhibit 17. Income and Substitution Effects for an Inferior Good

Note: The income effect of a price decrease for an inferior good is opposite to the substitution effect, tending to mitigate the change in quantity.

Notice that when the bread’s price falls, as indicated by the shift from budget constraint 1 (BC1) to budget constraint 2 (BC2), the consumer buys more bread, just as we would expect. That is, the consumer’s demand curve is still negatively sloped. When we apply the income adjustment to isolate substitution effect from income effect, we shift the budget constraint back to constraint 3, reducing income sufficiently to place the consumer back on the original indifference curve. As before, the substitution effect is shown as a movement along the original indifference curve from point a to point b. The income effect is, as before, a movement from one indifference curve to the other, as shown by the movement from point b to point c. In this case, however, the income effect partially offsets the substitution effect, causing demand to be less elastic than if the two effects reinforced each other.

We see that for inferior goods, the income effect and the substitution effect are in opposite directions: The decrease in price causes the consumer to buy more, but the income effect tends to mitigate that effect. It’s still true that a decrease in the price of bread represents an increase in real income. But in the case of an inferior good, the increased income causes the consumer to want to buy less of the good, not more. As long as the income effect has a lower magnitude than the substitution effect, the consumer still ends up buying more at the lower price. However, she buys a little less than she would if the good were normal. It is possible, though highly unlikely, for the income effect to have greater magnitude than the substitution effect. We examine that case next.

In the case of savings, the same type of effects can apply. For example, say interest rates rise. Individuals may save more because the reward (price) for saving has risen, and individuals substitute future consumption for present consumption. However, higher interest rates also imply that less saving is required to attain a given future amount of money. If the latter effect (the income effect) dominates, then it is possible to observe higher interest rates resulting in less savings.

## 6.4. Negative Income Effect Larger than Substitution Effect: Giffen Goods

In theory, it is possible for the income effect to be so strong and so negative as to overpower the substitution effect. If that were to occur, then a decrease in price could result in a decrease in quantity demanded and a positively sloped individual demand curve. Let us explore this curiosity in Exhibit 18.

Exhibit 18. Income and Substitution Effects for a Giffen Good

Note: Income and substitution effects of a fall in price for a Giffen good: When price declines, the consumer chooses to buy less of the good.

Once again, we decrease the price of bread as indicated by the pivoting of the budget constraint form BC1 to BC2, and then we move the budget constraint parallel to itself leftward until it just touches the original indifference curve at point b to remove the income effect. What is left is the substitution effect. As always, the substitution effect causes the consumer to substitute more bread for less wine in the basket, as indicated by the movement along the indifference curve from point a to point b. But notice the odd result when we “give back” the income and move from BC3 to BC2. The income effect for this inferior good (from point b to point c) is once again opposite in direction to the substitution effect, as is true for all inferior goods. But in this curious case, its magnitude overwhelms the substitution effect: Point c lies to the left of point a. The consumer actually buys less of the good when its price falls, resulting in a positively sloped demand curve. If we reversed the analysis and increased the price of bread, this consumer would buy more bread when its price rose. Those inferior goods whose income effect is negative and greater in magnitude than the substitution effect are known as Giffen goods. Importantly, all Giffen goods must be inferior, but not all inferior goods are Giffen goods.

This curious result is originally attributed to Robert Giffen, who suspected that Irish peasants might have responded this way to increased prices of staples during the Irish potato famine in the nineteenth century. He reasoned that staples, such as potatoes, comprised a very large portion of the peasants’ total budget. Additionally, potatoes could be a very inferior good, which simply means that when incomes fell, the peasants bought a lot more potatoes; and when incomes rose, they bought a lot fewer. Now because potatoes took up such a large part of total expenditures, any increase in the price of potatoes would result in a very substantial decrease in real income. This combination of strong inferiority coupled with a large amount of the budget spent on potatoes could, in theory, result in the negative income effect not only being opposite in direction to the substitution effect, but in fact overwhelming it.

Although some empirical studies have suggested the existence of Giffen goods, even if they existed they would be extraordinarily rare. Moreover, although they might exist for some small subset of consumers, it is highly unlikely that consumers as a whole would behave this way. So Giffen goods’ role in microeconomics is greater than their role in the empirical world. True, they result in a positively sloped demand curve and they do not violate any of the axioms of consumer choice theory. But any company’s manager who believes that if she raises the price of her product she will sell more of it is very likely to be disappointed.

EXAMPLE 7

# Income and Substitution Effects of a Decrease in Price

Consider the following diagram of budget constraints and indifference curves for a consumer choosing to allocate her budget between books and shoes. Determine whether shoes are normal, inferior, or Giffen goods for this consumer.

## Solution:

When the price of shoes falls, the original budget constraint pivots from BC1 to BC2. The original tangent point was at a and is now at c. We can separate the substitution effect from the income effect by removing enough income to put the consumer back on the old indifference curve at point b. This is shown as a shift in the budget constraint from BC2 to BC3, a parallel shift. The substitution effect is, therefore, from point a to point b, along the original indifference curve. The income effect is from point b to point c, but note that those two points are on the same vertical line. In this case, shoes are on a borderline between normal and inferior. There is zero income effect. (If point c had been to the right of point b, shoes would be normal. And if c had been to the left of point b, they would be inferior. Finally, if c had been to the left of point a, shoes would be a Giffen good.)

## 6.5. Veblen Goods: Another Possibility for a Positively Sloped Demand Curve

Standard choice theory assumes that the consumer can always make comparisons among all pairs of bundles of goods and identify preferences before knowing anything about the prices of those baskets. Then, as we’ve seen, the consumer is constrained by income and prices and makes actual choices of which bundles to purchase with limited income. It is important to note that those preferences are assumed to exist even before knowing the prices at which those goods could be purchased. It is possible, however, that an item’s price tag itself might help determine the consumer’s preferences for it. Thorstein Veblen posited just such a circumstance in his concept of conspicuous consumption. According to this way of thinking, a consumer might derive utility out of being known by others to consume a so-called high status good, such as a luxury automobile or a very expensive piece of jewelry. Importantly, it is the high price itself that partly imparts value to such a good. If that is the case, then a consumer would actually value a good more if it had a higher price. So, it is argued that by increasing the price of a Veblen good, the consumer would be more inclined to purchase it, not less. In the extreme, it could be argued that the consumer’s demand for such a good could be positively sloped, though this need not necessarily follow. In fact, of course, if any seller actually faced a positively sloped demand curve for her product, the rational response would be to increase price because she would sell more at the higher price. Ultimately, at some very high price, demand would necessarily become negatively sloped.

It is important to recognize that, although Veblen goods and Giffen goods share some characteristics, they are in fact quite different. Whether or not they actually exist, Giffen goods are not inconsistent with the fundamental axioms of demand theory. True, they would result in a violation of the law of demand, but that law is not a logical necessity. It is simply recognition that in virtually all observed cases, demand curves are negatively sloped. Giffen goods certainly would not be considered examples of “status goods,” because an increase in income alone would result in a reduced interest in purchasing them. Veblen goods, on the other hand, derive their value from the ostentatious consumption of them as symbols of the purchaser’s high status in society. If they exist, they are certainly not inferior goods. And they do violate the axioms of choice that form the foundation of accepted demand theory.