# 3.5. Aggregating the Demand and Supply Functions

#cfa #cfa-level-1 #economics #microeconomics #reading-13-demand-and-supply-analysis-introduction #study-session-4

We have explored the basic concept of demand and supply at the individual household and the individual supplier level. However, markets consist of collections of demanders and suppliers, so we need to understand the process of combining these individual agents’ behavior to arrive at market demand and supply functions.

The process could not be more straightforward: simply add all the buyers together and add all the sellers together. Suppose there are 1,000 identical gasoline buyers in our hypothetical example, and they represent the total market. At, say, a price of $3 per gallon, we find that one household would be willing to purchase 10 gallons per week (when income and price of automobiles are held constant at$50,000 and $20,000, respectively). So, 1,000 identical buyers would be willing to purchase 10,000 gallons collectively. It follows that to aggregate 1,000 buyers’ demand functions, simply multiply each buyer’s quantity demanded by 1,000: Equation (13) Qdx=1,000(8.4−0.4Px+0.06I−0.01Py)=8,400−400Px+60I−10Py where Qdx represents the market quantity demanded. Note that if we hold I and Py at their same respective values of 50 and 20 as before, we can “collapse” the constant terms and write the following Equation 14: Equation (14) Qdx=11,200−400Px Equation 14 is just Equation 3 (an individual household’s demand function) multiplied by 1,000 households (Qdx represents thousands of gallons per week). Again, we can solve for Px to obtain the market inverse demand function: Equation (15) Px = 28 − 0.0025Qx The market demand curve is simply the graph of the market inverse demand function, as shown in Exhibit 5. Exhibit 5. Aggregate Weekly Market Demand for Gasoline as the Quantity Summation of all Households’ Demand Curves It is important to note that the aggregation process sums all individual buyers’ quantities, not the prices they are willing to pay—that is, we multiplied the demand function, not the inverse demand function, by the number of households. Accordingly, the market demand curve has the exact same price intercept as each individual household’s demand curve. If, at a price of$28, a single household would choose to buy zero, then it follows that 1,000 identical households would choose, in aggregate, to buy zero as well. On the other hand, if each household chooses to buy 10 at a price of $3, then 1,000 identical households would choose to buy 10,000, as shown in Exhibit 5. Hence, we say that all individual demand curves horizontally (quantities), not vertically(prices), are added to arrive at the market demand curve. Now that we understand the aggregation of demanders, the aggregation of suppliers is simple: We do exactly the same thing. Suppose, for example, that there are 20 identical sellers with the supply function given by Equation 8. To arrive at the market supply function, we simply multiply by 20 to obtain: Equation (16) Qsx=20(−175+250Px−5W)=−3,500+5,000Px−100W And, if we once again assume W equals$15, we can “collapse” the constant terms, yielding

Equation (17)

Qsx=20[−175+250Px−5(15)]=−5,000+5,000Px

which can be inverted to yield the market inverse supply function:

Equation (18)

Px = 1 + 0.0002Qx

Graphing the market inverse supply function yields the market supply curve in Exhibit 6:

Exhibit 6. Aggregate Market Supply as the Quantity Summation of Individual Sellers’ Supply Curves

We saw from the individual seller’s supply curve in Exhibit 3 that at a price of \$3, an individual seller would willingly offer 500 gallons of gasoline. It follows, as shown in Exhibit 6, that a group of 20 sellers would offer 10,000 gallons per week. Accordingly, at each price, the market quantity supplied is just 20 times as great as the quantity supplied by each seller. We see, as in the case of demand curves, that the market supply curve is simply the horizontal summation of all individual sellers’ supply curves.

EXAMPLE 4

# Aggregating Demand Functions

An individual consumer’s monthly demand for downloadable e-books is given by the equation

Qdeb=2−0.4Peb+0.0005I+0.15Phb

where Qdeb equals the number of e-books demanded each month, Peb is the price of e-books in euros, Iequals the household monthly income, and Phb equals the price of hardbound books, per unit. Assume that household income is €2,300, and the price of hardbound books is €21.40. The market consists of 1,000 identical consumers with this demand function.

1. Determine the market aggregate demand function.

2. Determine the inverse market demand function.

3. Determine the slope of the market demand curve.

## Solution to 1:

Aggregating over the total number of consumers means summing up their demand functions (in the quantity direction). In this case, there are 1,000 consumers with identical individual demand functions, so multiply the entire function by 1,000:

Qeb=1,000(2−0.4Peb+0.0005I+0.15Phb)=2,000−400Peb+0.5I+150Phb

## Solution to 2:

Holding I constant at a value of €2,300 and Phb constant at a value of €21.40, we find

Qeb = 2,000 − 400Peb + 0.5(2300) + 150(21.40) = 6,360 – 400Peb

Now solve for Peb = 15.90 – 0.0025Qeb

## Solution to 3:

The slope of the market demand curve is the coefficient on Qeb in the inverse demand function, which is −0.0025.

EXAMPLE 5

# Aggregating Supply Functions

An individual seller’s monthly supply of downloadable e-books is given by the equation

Qseb=−64.5+37.5Peb−7.5W

where Qseb is number of e-books supplied, Peb is the price of e-books in euros, and W is the wage rate in euros paid by e-book sellers to laborers. Assume that the price of e-books is €10.68 and wage is €10. The supply side of the market consists of a total of eight identical sellers in this competitive market.

1. Determine the market aggregate supply function.

2. Determine the inverse market supply function.

3. Determine the slope of the aggregate market supply curve.

## Solution to 1:

Aggregating supply functions means summing up the quantity supplied by all sellers. In this case, there are eight identical sellers, so multiply the individual seller’s supply function by eight:

Qseb=8(−64.5+37.5Peb−7.5W)=−516+300Peb−60W

## Solution to 2:

Holding W constant at a value of €10, insert that value into the aggregate supply function and then solve for Peb to find the inverse supply function:

Qeb = –1,116 + 300Peb

Inverting, Peb = 3.72 + 0.0033Qeb

## Solution to 3:

The slope of the supply curve is the coefficient on Qeb in the inverse supply function, which is 0.0033.