We can now use the theory of consumer utility maximization to derive a demand curve for an individual consumer and, by aggregating individual demand curves, we can derive market demand curves.
An Individual Consumer’s Demand Curve
We can use Figure 5.9 to show how an individual consumer’s demand curve is ob- tained. Begin with income of $1,000 and prices of good X and good Y both equal to
$10. The corresponding budget line is given by budget line 1, from 100Y to 100X, in the upper panel of the figure. The consumer maximizes utility where budget line 1 is tangent to indifference curve I, consuming 50 units of X. Thus when income is
$1,000, one point on this consumer’s demand for X is $10 and 50 units of X. This point is illustrated on the price–quantity graph in the lower panel of Figure 5.9.
Following the definition of demand, we hold income and the price of the other good Y constant, while letting the price of X fall from $10 to $8. The new budget line is budget line 2. Because income and the price of Y remain constant, the vertical inter- cept does not change, but because the price of X has fallen, the budget line must pivot outward along the X-axis. The new X-intercept for budget line 2 is 125 (5 $1,000y$8).
With this new budget line, the consumer now maximizes utility where budget line 2 is tangent to indifference curve II, consuming 65 units of X. Thus another point on the demand schedule in the lower panel must be $8 and 65 units of X.
Next, letting the price of X fall again, this time to $5, the new budget line is budget line 3. At the price $5, the consumer chooses 90 units of X, another point on this consumer’s demand curve. Thus we have derived the following demand schedule for good X:
Price Quantity demanded
$10 50
8 65
5 90
Now try Technical Problem 11.
10
Price of X (dollars)
65 90
Quantity of X 8
5
Demand for X 50
0
This schedule, with other points so generated, is graphed as a demand curve in price–quantity space in the lower part of Figure 5.9. This demand curve is downward-sloping. As the price of X falls, the quantity of X the consumer is willing and able to purchase increases, following the rule of demand. Furthermore, we followed the definition of demand, holding income and the price of the other good (goods) constant. Thus an individual’s demand for a good is derived from a series of utility-maximizing points. We used only three such points, but we could easily have used more in order to obtain more points on the demand curve.
Principle The demand curve of an individual for a specific commodity relates utility-maximizing quantities purchased to market prices, holding constant income and the prices of all other goods. The slope of the demand curve illustrates the law of demand: Quantity demanded varies inversely with price.
Market Demand and Marginal Benefit
Managers are typically more interested in market demand for a product than in the demand of an individual consumer. Recall that in Chapter 2 we defined market demand as a list of prices and the corresponding quantity consumers are
market demand A list of prices and the quantities consumers are willing and able to purchase at each price in the list, other things being held constant.
Now try Technical Problem 12.
100
Quantity of Y
65 90100 125 200
Quantity of X III I II
Budget line 1, Px = $10 Budget line 2, Px = $8 Budget line 3, Px = $5
50 0
F I G U R E 5.9 Deriving a Demand Curve
willing and able to purchase at each price in the list, holding constant income, the prices of other goods, tastes, price expectations, and the number of consumers.
When deriving individual demand in this chapter, we pivoted the budget line around the vertical intercept, therefore holding income and the prices of other goods constant. Because the indifference curves remained constant, tasteswere unchanged.
Thus, the discussion here conforms to the conditions of market demand. To obtain the market demand function, we need only to aggregate the individual demand functions of all potential customers in the market. We will now demonstrate this aggregation process, which is frequently called horizontal summation because the aggregation of numerical values takes place along the horizontal axis.
Suppose there are only three individuals in the market for a particular com- modity. In Table 5.1, the quantities demanded by each consumer at each price in column 1 are shown in columns 2, 3, and 4. Column 5 shows the sum of these quantities demanded at each price and is therefore the market demand. Because the demand for each consumer is negatively sloped, market demand is negatively sloped also. Quantity demanded is inversely related to price.
Figure 5.10 shows graphically how a market demand curve can be derived from the individual demand curves. The individual demands of consumers 1, 2, and 3 from Table 5.1 are shown graphically as D1, D2, and D3, respectively. The market de- mand curve DM is simply the sum of the quantities demanded at each price. At $6, consumer 1 demands 3 units. Because the others demand nothing, 3 is the quantity demanded by the market. At every other price, DM is the horizontal summation of the quantities demanded by the three consumers. And if other consumers came into the market, their demand curves would be added to DM to obtain the new market demand.
Relation The market demand curve is the horizontal summation of the demand curves of all consumers in the market. It therefore shows how much all consumers demand at each price over the relevant range of prices.
As explained in Chapter 2, the market demand curve gives demand prices for any quantity demanded of the good. Because the demand price for a specific quantity
Quantity demanded Market
demand
Price Consumer 1 Consumer 2 Consumer 3
$6 3 0 0 3
5 5 1 0 6
4 8 3 1 12
3 10 5 4 19
2 12 7 6 25
1 13 10 8 31
T A B L E 5.1
Aggregating Individual Demands
demanded is the maximum price consumers will pay for that unit of a good, demand price measures, in dollars, the economic value or benefit to consumers of that unit. Thus, for any particular quantity demanded, the price on the vertical axis of the market demand curve measures two things: (1) the maximum price consumers will pay to buy that quantity of the good, and (2) the dollar value of the benefit to buyers of that particular unit of the good. A market demand curve, then, gives the marginal benefit (value) individuals place on the last unit consumed.
In Figure 5.10, we denote the dual roles of market demand by labeling market demand both as “DM” and as “MB.” Consider point A, which shows $4 is the maximum price for which 12 units can be sold in this market of three individual buyers. The marginal benefit for every buyer in this market is $4 for the last unit purchased. Specifically, consumer 1 values the eighth unit he consumes at $4, consumer 2 values the third unit she consumes at $4, and consumer 3 values the first unit he consumes at $4. Clearly, when buyers in this market purchase 12 units at a price of $4, the marginal benefit of the last unit consumed is $4 for everyone in this market. Demand prices along market demand curves, then, measure the marginal benefit or value of the last unit consumed for every individual consumer in society.
Relation The demand prices at various quantities along a market demand curve give the marginal benefit (value) of the last unit consumed for every buyer in the market, and thus market demand can be interpreted as the marginal benefit curve for a good.
F I G U R E 5.10 Derivation of Market Demand
Price and marginal benefit (dollars)
Quantity demanded 1
2 3
4 A
5 6
8 12 16 20 24 28
D2 D1 DM
D3 =MB
0 4 32
Now try Technical Problem 13.