The Demand Function
By now you should understand the factors that affect demand and how to use graphs to
illustrate those influences. The final step in our analysis of the demand side of the market is to
show that all the factors that influence demand may be summarized in what economists refer
Demand Function A function that describes how much of a good will be purchased at
alternative prices of that good and related goods, alternative income levels, and alternative
values of other variables affecting demand.
Then the demand function for good X may be written as
Qxd = f(Px, Py, M, H)
Thus, the demand function explicitly recognizes that the quantity of a good consumed depends
on its price and on demand shifters. Different products will have demand functions of different
forms. One very simple but useful form is the linear representation of the demand function:
Demand is linear if Qxd is a linear function of prices, income, and other variables that influence
demand. The following equation is an example of a linear demand function: Qxd = α0 + αxPx +
αyPy + αM M + αH H
Linear Demand Function
A representation of the demand function in which the demand for a given good is a linear function of prices, income levels, and
other variables influencing demand.
The demand function for good X describes how much X will be purchased at alternative prices of X and
related goods, alternative levels of income, and alternative values of other variables that affect demand.
Formally, let Qxd represent the quantity demanded of good X, Px the price of good X, Py the price of a
related good, M income, and H the value of any other variable that affects demand, such as the level of
advertising, the size of the population, or consumer expectations.
By the law of demand, an increase in Px leads to a decrease in the quantity demanded
of good X. This means that αx < 0. The sign of αy will be positive or negative depending on
whether goods X and Y are substitutes or complements. If αy is a positive number, an increase
in the price of good Y will lead to an increase in the consumption of good X; therefore, good X
is a substitute for good Y. If αy is a negative number, an increase in the price of good Y will lead
to a decrease in the consumption of good X; hence, good X is a complement to good Y.
The sign of αM also can be positive or negative depending on whether X is a normal or an
inferior good. If αM is a positive number, an increase in income (M) will lead to an increase in
the consumption of good X, and good X is a normal good. If αM is a negative number, an
increase in income will lead to a decrease in the consumption of good X, and good X is an
inferior good.
Consumer Surplus
We now show how a manager can use the demand curve to ascertain the value a consumer or
group of consumers receives from a product. The concepts developed in this section are
particularly useful in marketing and other disciplines that emphasize strategies such as value
pricing and price discrimination.
By the law of demand, the amount a consumer is willing to pay for an additional unit of a good
falls as more of the good is consumed. For instance, imagine that the demand curve in Figure
2–5(a) represents your demand for water immediately after participating in a 10K run. Initially,
you are willing to pay a very high price—in this case, $5 per liter—for the first drop of water. As
you consume more water, the amount you are willing to pay for an additional drop declines
from $5.00 to $4.99 and so on as you move down the demand curve. Notice that after you have
consumed an entire liter of water, you are willing to pay only $4 per liter for another drop. Once
you have enjoyed 2 liters of water, you are willing to pay only $3 per liter for another drop.
Figure 2–5 Price per
Consumer liter
Surplus
5
4
3
2
1
0
D
12345
Price
P
x0
Quantity
(liters)
0
Consumer
To find your total value (or benefit) of 2 liters of water, we simply add up the maximum
amount you were willing to pay for each of these drops of water between 0 and 2 liters. This
amount corresponds to the area underneath the demand curve in Figure 2–5(a) up to the
quantity of 2 liters. Since the area of this region is $8, the total value you receive from 2 liters of
water is $8. Fortunately, you don’t have to pay different prices for the different drops of water
you
consume. Instead, you face a per-unit price of, say, $3 per liter and get to buy as many drops
(or even liters) as you want at that price. Given the demand curve in Figure 2–5(a), when the
price is $3 you will choose to purchase 2 liters of water. In this case, your total out-of-pocket
expense for the 2 liters of water is $6. Since you value 2 liters of water at $8 and only have to
pay $6 for it, you are getting $2 in value over and above the amount you have to pay for water.
This “extra” value is known as consumer surplus—the value consumers get from a good but do not
have to pay for. This concept is important to managers because it tells how much extra money
consumers would be willing to pay for a given amount of a purchased product.
More generally, consumer surplus is the area above the price paid for a good but below the demand
curve. For instance, the shaded triangle in Figure 2–5(b) illustrates the consumer surplus of a consumer
who buys Qx0 units at a price of Px0. To see why, recall that each point on the demand curve indicates
the value to the consumer of another unit of the good. The dif
ference between each price on the demand curve and the price Px0 paid represents surplus (the value
the consumer receives but does not have to pay for). When we add up the “surpluses” received for each
unit between 0 and Qx0 (this sum equals the shaded region), we obtain the consumer surplus associated
with purchasing Qx0 units at a price of Px0 each.
Managers can use the notion of consumer surplus to determine the total amount consumers would be
willing to pay for multiunit packages. While this will be discussed in detail in Chapter 11 where we
examine pricing strategies, we illustrate the basic idea in the following problem.
Demonstration Problem”
A typical consumer’s demand for the Happy Beverage Company’s product looks like that in Figure 2–5(a). If the firm
charges a price of $2 per liter, how much revenue will the firm earn and how much consumer surplus will the typical
consumer enjoy? What is the most a consumer would be willing to pay for a bottle containing exactly 3 liters of the
firm’s beverage
ANSWER:
At a price of $2 per liter, a typical consumer will purchase 3 liters of the beverage. Thus, the firm’s revenue is $6 and
the consumer surplus is $4.50 [the area of the consumer surplus triangle is one-half the base times the height, or
.5(3)($5 − $2) = $4.50]. The total value of 3 liters of the firm’s beverage to a typical consumer is thus $6 + $4.50, or
$10.50. This is also the maximum amount a consumer would be willing to pay for a bottle containing exactly 3 liters
of the firm’s beverage. Expressed differently, if the firm sold the product in 3-liter bottles rather than in smaller units,
it could sell each bottle for $10.50 to earn higher revenues and extract all consumer surplus.
SUPPLY
In the previous section we focused on demand, which represents half of the forces that deter mine the
price in a market. The other determinant is market supply. In a competitive market there are many
producers, each producing a similar product. The market supply curve sum marizes the total quantity all
producers are willing and able to produce at alternative prices, holding other factors that affect supply
constant
change in quantity supplied
Changes in the price of a good lead to a change in the quantity supplied of that good. This corresponds to a movement along a
given supply curve.
change in supply Changes in variables other than the price of a good, such as input prices or technological advances, lead to a
change in supply. This corresponds to a shift of the entire supply curve.
While the market supply of a good generally depends on many things, when we graph a sup ply curve,
we hold everything but the price of the good constant. The movement along a supply curve, such as the
one from A to B in Figure 2–6, is called a change in quantity supplied. The fact that the market supply
curve slopes upward reflects the inverse law of supply: As the price of a good rises (falls) and other
things remain constant, the quantity supplied of the good rises (falls). Producers are willing to produce
more output when the price is high than when it is low.
Supply Shifters
Variables that affect the position of the supply curve are called supply shifters, and they include the
prices of inputs, the level of technology, the number of firms in the market, taxes, and producer
expectations. Whenever one or more of these variables change, the position of the entire supply curve
shifts. Such a shift is known as a change in supply. The shift from S0 to S2 in Figure 2–6 is called an
increase in supply since producers sell more output at each given price. The shift from S0 to S1 in Figure
2–6 represents a decrease in supply since producers sell less of the product at each price.
Input Prices
The supply curve reveals how much producers are willing to produce at alternative prices. As production
costs change, the willingness of producers to produce output at a given price changes. In particular, as
the
price of an input rises, producers are willing to produce less output at each given price. This decrease in
supply is depicted as a leftward shift in the supply curve.
Technology or Government Regulations
Technological changes and changes in government regulations also can affect the position of the supply
curve. Changes that make it possible to produce a given output at a lower cost, such as the ones
highlighted in Inside Business 2–2, have the effect of increasing supply. Conversely, natural disasters
that destroy existing technology and government regulations, such as emis
sions standards that have an adverse effect on businesses, shift the supply curve to the left.
Number of Firms
The number of firms in an industry affects the position of the supply curve. As additional firms enter an
industry, more and more output are available at each given price. This is reflected by a rightward shift in
the supply curve. Similarly, as firms leave an industry, fewer units are sold at each price, and the supply
decreases (shifts to the left).
Producer’s expectations
In effect selling of a unit output today and selling of a unit output for tomorrow are substitutes of
production. If firm suddenly expect prices to be higher in the future and the product is not perishable,
producers can hold back output today and sell it later at a higher price. This has the effect of shifting the
supply curve to the left.
You should now understand the difference between supply and quantity supplied and recog nize
the factors that influence the position of the supply curve. The final step in our analysis of
supply is to show that all the factors that influence the supply of a good can be summarized in a
supply function.
The Supply Function
The supply function of a good describes how much of the good will be produced at alter native prices of
the good, alternative prices of inputs, and alternative values of other variables that affect supply.
Formally, let Qxs represent the quantity supplied of a good, Px the price of the of the good, W the price of
the price of technolog
an input (such as the wage rate on labor), Pr
ically related goods, and H the value
of some other variable that affects supply (such as the existing technology, the number of firms in the
market, taxes, or producer expectations). Then the supply function for good X may be written as
Qxs = f(Px, Pr, W, H)
Thus, the supply function explicitly recognizes that the quantity produced in a market depends not only
on the price of the good but also on all the factors that are potential supply shifters. While there are many
different functional forms for different types of products, a particularly useful representation of a supply
function is the linear relationship. Supply is linear if Qxthe good will be produced at alter native prices of
the good, alternative prices of inputs, and alternative values of other variables that affect supply.
Formally, let Qxs represent the quantity supplied of a good, Px the price of the good, W the price of an
the price of technolog
input (such as the wage rate on labor), Pr
ically related goods, and H the value of
some other variable that affects supply (such as the existing technology, the number of firms in the
market, taxes, or producer expectations). Then the supply function for good X may be written as
Qxs = f(Px, Pr, W, H)
Thus, the supply function explicitly recognizes that the quantity produced in a market depends not only
on the price of the good but also on all the factors that are potential supply shifters. While there are many
different functional forms for different types of products, a particularly useful representation of a supply
function is the linear relationship. Supply is linear if Qxs is a linear function of the variables that influence
supply. The following equation is representative of a linear supply function:
Qxs = β0 + βxPx + βrPr + βwW + βH H
s
is a linear function of the variables that influence supply. The following equation is representative of a
linear function
Demonstration Problem
Your research department estimates that the supply function for high definition televisions (HDTVs) is given by
Qxs = 2,000 + 3Px − 4Pt − Pw
where Px is the price of HDTVs, Pt represents the price of a tablet, and Pw is the price of an input used to make
HDTVs. Suppose HDTVs are sold for $400 per unit, tablets are sold for $250 per unit, and the price of an input is
$1,400. How many HDTVs are
produced?
ANSWER:
To find out how many television sets are produced, we insert the given values of prices into the supply function to get
Qxs = 2,000 + 3(400) − 4(250) − 1(1,400)
Adding up the numbers, we find that the total quantity of HDTVs produced is 800.
The information summarized in a supply function can be used to graph a supply curve. Since a supply
curve is the relationship between price and quantity, a representative sup ply curve holds everything but
price constant. This means one may obtain the formula for a supply curve by inserting given values of
the supply shifters into the supply function, butgood, W the price of an input (such as the wage rate on
the price of technolog
labor), Pr
ically related goods, and H the value of some other variable that affects
supply (such as the existing technology, the number of firms in the market, taxes, or producer
expectations). Then the supply function for good X may be written as
Qxs = f(Px, Pr, W, H)
Thus, the supply function explicitly recognizes that the quantity produced in a market depends not only
on the price of the good but also on all the factors that are potential supply shifters. While there are many
different functional forms for different types of products, a particularly useful representation of a supply
function is the linear relationship. Supply is linear if Qxs is a linear function of the variables that influence
supply. The following equation is representative of a linear supply function:
Qxs = β0 + βxPx + βrPr + βwW + βH H leaving Px in the equation to allow for various values. If we do this for
the supply function in Demonstration Problem 2–3 (where Pt = $250 and Pw = 1,400), we get
Qxs = 2,000 + 3Px − 4(250) − 1(1,400)
which simplifies to
Qxs = 3Px − 400 (2−2)
Since we usually graph this relation with the price of the good on the vertical axis, it is useful to
represent Equation 2–2 with price on the left-hand side and everything else on the right-hand
side. This is known as an inverse supply function. For this example, the inverse supply function
is Px = 400 ____
__
3 +1 3 Qxs
which is the equation for the supply curve graphed in Figure 2–9. This curve reveals how much
producers must receive to be willing to produce each additional unit of good X.
Producer Surplus
Just as consumers want price to be as low as possible, producers want price to be as high as
possible. The supply curve reveals the amount producers will be willing to produce at a given
price. Alternatively, it indicates the price firms would have to receive to be willing to produce an
additional unit of a good. For example, the supply curve in Figure 2–9 indicates that a total of
800 units will be produced when the price is $400. Alternatively, if 800 units are produced,
producers will have to receive $400 to be induced to produce another unit of the good.
Producer surplus is the producer analogue to consumer surplus. It is the amount of money producers
receive in excess of the amount necessary to induce them to produce the good. More specifically, note
that producers are willing to sell each unit of output below 800 units at a price less than $400. But if the
price is $400, producers receive an amount equal to $400 for each unit of output below 800, even though
they would be willing to sell those individual units for a lower price.
Geometrically, producer surplus is the area above the supply curve but below the market price of the
good. Thus, the shaded area in Figure 2–9 represents the surplus producers receive by selling 800 units
at a price of $400—an amount above what would be required to produce each unit of the good. The
shaded area, ABC, is the producer surplus when the price is $400. Mathematically, this area is one-half
of 800 times $266.67, or $106,668.
Producer surplus can be a powerful tool for managers. For instance, suppose the manager
of a major fast-food restaurant currently purchases 10,000 pounds of ground beef each week
from a supplier at a price of $1.25 per pound. The producer surplus the meat supplier earns by
selling 10,000 pounds at $1.25 per pound tells the restaurant manager the dollar amount that
the supplier is receiving over and above what it would be willing to accept for meat. In other
words, the meat supplier’s producer surplus is the maximum amount the restaurant could save
in meat costs by bargaining with the supplier over a package deal for 10,000 pounds of meat.
Chapters 6 and 10 will provide details about how managers can negotiate such a bargain.
MARKET EQUILIBRIUM
The equilibrium price in a competitive market is determined by the interactions of all buyers and
sellers in the market. The concepts of market supply and market demand make this notion of
interaction more precise: The price of a good in a competitive market is determined by the
interaction of market supply and market demand for the good.
Since we will focus on the market for a single good, it is convenient to drop subscripts at
this point and let P denote the price of this good and Q the quantity of the good. Figure 2–10
depicts the market supply and demand curves for such a good. To see how the competitive
price is determined, let the price of the good be PL. This price corresponds to point B on the
market demand curve; consumers wish to purchase Q1 units of the good. Similarly, the price of
PL corresponds to point A on the market supply curve; producers are willing to produce only Q0
units at this price. Thus, when the price is PL, there is a shortage of the good; that is, there is
not enough of the good to satisfy all consumers willing to purchase it at that price.
In situations where a shortage exists, there is a natural tendency for the price to rise;
consumers unable to buy the good may offer producers a higher price in an attempt to get the
product. As the price rises from PL to P e in Figure 2–10, producers have an incentive to expand
output from Q0 to Qe. Similarly, as the price rises, consumers are willing to purchase less of the
good. When the price rises to P e, the quantity demanded is Qe. At this price, just enough of the
good is produced to satisfy all consumers willing and able to purchase at that price; quantity
demanded equals quantity supplied.
Thus, the interaction of supply and demand ultimately determines a competitive price, Pe,
such that there is neither a shortage nor a surplus of the good. This price is called the
equilibrium price and the corresponding quantity, Qe, is called the equilibrium quantity for the
competitive market. Once this price and quantity are realized, the market forces of supply and
demand are balanced; there is no tendency for prices either to rise or to fall.
Competitive Market Equilibrium
Equilibrium in a competitive market is determined by the intersection of the market demand and
supply curves. The equilibrium price is the price that equates quantity demanded with quantity
supplied. Mathematically, if Qd(P) and Qs(P) represent the quantity demanded and supplied
when the price is P, the equilibrium price, Pe, is the price such that
Qd(Pe) = Qs(Pe)
The equilibrium quantity is simply Qd(Pe) or, equivalently, Qs(Pe).
INSIDE BUSINESS 2–3
Unpopular Equilibrium Prices
For a recent college graduation, each graduating student was directed to pick up three free
tickets any time between April 9 and April 20. After April 20, remaining tickets were given to
those desiring additional tickets, on a first-come, first-served basis. Once the free tickets were
fully dis tributed, any trades between ticket holders and ticket demanders were left to market
forces—which led to a very unpopular equilibrium price.
With several concurrent events competing for students’ attention during this time, many
students did not claim their three free tickets by the deadline. As graduation approached,
students who did not claim their tickets were willing to pay large sums of money to purchase
tickets for
visiting family members and friends. Since the demand for tickets was great and only a limited
number of tickets were available, some sellers were asking as much as $400 for a ticket!
Several students expressed outrage over the high prices. However, the high prices are merely
a symptom of the high value many students placed on graduation tickets coupled with the
limited supply. Had graduating seniors without tickets better forecasted this market outcome,
they would have picked up their free tickets on time.
PRICE RESTRICTIONS AND MARKET
EQUILIBRIUM
The previous section showed how prices and quantities are determined in a free market. In
some instances, the government places limits on how much prices are allowed to rise or fall,
and these restrictions can affect the market equilibrium. In this section, we examine the impact
of price ceilings and price floors on market allocations.
Price Ceilings
One basic implication of the economic doctrine of scarcity is that there are not enough goods to
satisfy the desires of all consumers at a price of zero. As a consequence, some method must
be used to determine who gets to consume goods and who does not. People who do not get to
consume goods are essentially discriminated against. One way to determine who gets a good
and who does not is to allocate the goods based on hair color: If you have red hair, you get the
good; if you don’t have red hair, you don’t get the good.
The price system uses price to determine who gets a good and who does not. The price
system allocates goods to consumers who are willing and able to pay the most for the goods. If
the competitive equilibrium price of a pair of jeans is $40, consumers willing and able to pay
$40 will purchase the good; consumers unwilling or unable to pay that much for a pair of jeans
will not buy the good.
It is important to keep in mind that it is not the price system that is “unfair” if one cannot
afford to pay the market price for a good; rather, it is unfair that we live in a world of scarcity.
Any method of allocating goods will seem unfair to someone because there are not enough
resources to satisfy everyone’s wants. For example, if jeans were allocated to people on the
48 CHAPTER 2 Market Forces: Demand and Supply
basis of hair color instead of the price system, you would think this allocation rule was unfair
unless you were born with the “right” hair color.
Often individuals who are discriminated against by the price system attempt to persuade
the government to intervene in the market by requiring producers to sell the good at a lower
price. This is only natural, for if we were unable to own a house because we had the wrong hair
color, we most certainly would attempt to get the government to pass a law allowing peo ple
with our hair color to own a house. But then there would be too few houses to go around, and
some other means would have to be used to allocate houses to people.
Price Floors
In contrast to the case of a price ceiling, sometimes the equilibrium competitive price may be
considered too low for sellers. In these instances, individuals may lobby for the government to
legislate a minimum legal price for a good is
called price floor.
Reference:Managerial Economics and Business
Strategy Chap 2 : Market Forces and Supply by
Michael Baye and Jeffrey Prince