Download What are demand and supply?

Question 5: What are demand and supply?
An important application of linear functions is the connection between the price of a
good or service and the quantity of a good or service. If we are interested in the
relationship between the price and the quantity demanded by consumers at that price,
this connection is described by the demand function. The relationship between price of
a good or service and the quantity supplied by manufacturers at that price is described
by the supply function. These functions describe how consumers and suppliers behave
as price is changed.
The consumer’s behavior is given by a table called a demand schedule. This table
reflects the relationship between the price of some good or service and the quantity of
this good or service demanded at this price. The table below is a typical demand
schedule in some market.
Table 5
Average Price of a Gallon of Milk
(dollars per gallon)
Quantity of Milk Sold Per Week
(thousands of gallons)
1.50
125
2.00
115
2.50
105
3.00
95
3.50
85
In one column of the demand schedule are prices of some good. In the other column is
the corresponding numbers of that good sold to consumers. From this demand
schedule for dairies in a competitive market, we can see that as the price per gallon for
milk increases the quantity of milk sold per week decreases. The law of demand states
that if everything else is held constant, as the price of the good increases, the quantity
demanded drops. This pattern is visible in the demand curve shown below.
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The demand function is traditionally graphed with the quantity as the independent
variable and the price as the dependent variable. Because of this, quantity is graphed
on the horizontal axis and price is graphed on the vertical axis. This means that the
information indicating that 125,000 gallons of milk is sold per week at a price of $1.50 per
gallon corresponds to the ordered pair (125, 1.50). The demand function takes an input
of quantity and outputs price,
P  D(Q) .
Demand curves can have many shapes, but the simplest curve is a demand curve
follows a straight line. In the case of the demand function for dairies in a market shown
here, we’ll find a linear demand function of the form
D(Q)  mQ  b .
This is the same form defined earlier, f ( x)  mx  b , with the variable changed to Q from
x and the name changed to D from f.
Figure 11 – The demand D(Q) for milk.
Example 6
Find the Demand Function for Milk
Find the equation of the demand curve graphed in Figure 11.
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Solution The graph above gives us a hint that the demand function is a
linear function. To make sure the ordered pairs all lie along a straight
line, we need to find the slope between adjacent ordered pairs.
Table 6
Adjacent Ordered Pairs
Slope
(125, 1.50) and (115, 2.00)
2.00  1.50
 0.05
115  125
(115, 2.00) and (105, 2.50)
2.50  2.00
 0.05
105  115
(105, 2.50) and (95, 3.00)
3.00  2.50
 0.05
95  105
(95, 3.00) and (85, 3.50)
3.50  3.00
 0.05
85  95
Since the slope between any pair of adjacent points is -0.05, all of the
points lie along a line with slope -0.05.
Insert the slope into the form for a linear demand function to yield
D(Q)  0.05Q  b
All that remains is to find the vertical intercept b in the function. None of
the points are the vertical intercept. To find b, substitute any of the
points into the function. Let’s try the ordered pair (125, 1.50):
D(125)  0.05 125  b  1.50
To find the value of b, simplify the equation to give
6.25  b  1.50
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Adding 6.25 to both sides leads to b equal to 7.75. The function for
demand is
D(Q)  0.05Q  7.75
The supply function also relates the price of a good or service to a quantity of the good
or service. Unlike the demand function which describes the consumer’s willingness and
ability to buy a good or service, the supply function describes the willingness and ability
of a business to supply a good or service at a given price. If all other factors are held
constant, as the price rises the quantity supplied will also increase. A higher price leads
to more profit for a business encouraging higher production.
A supply schedule is a table that shows the relationship between the price of a good or
service and the quantity supplied by a firm.
Table 7
Average Price of a
Gallon of Milk
(dollars)
Quantity of Milk Supplied
Per Week
(thousands of gallons)
0
0
2.40
76
3.00
95
3.60
114
4.20
133
In this supply schedule, an increase in the average price of a gallon of milk corresponds
to an increase in the quantity of milk supplied by dairies. A graph of the data shows a
straight line with a positive slope.
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Figure 12 - Data from the supply schedule and the corresponding supply curve.
Keep in mind that while this is a “typical” supply schedule, supply functions may have
many different shapes. A supply function may be curved concave down or up. It may
even change its concavity.
Figure 13 - Three examples of supply functions. In a, the supply curve is concave down. In b,
the supply curve is concave up. In c, the supply curve changes from concave down to
concave up.
A linear function often models the data adequately and simplifies later calculations. A
linear supply function has the form
S (Q)  mQ  b
where m is the slope of the line and b is the vertical intercept.
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Example 7
Find the Supply Function for Milk
Find the linear supply function S  Q  , where Q is the quantity of milk
supplied, passing through the ordered pairs in Table 7.
Solution Since the function will be a function of Q, interchange the
columns of the table. This change is necessary since the independent
variable is typically listed in the first column.
Quantity of Milk Supplied
Per Week
(thousands of gallons)
Average Price of a
Gallon of Milk
(dollars)
0
0
76
2.40
95
3.00
114
3.60
133
4.20
Let’s check the slope between adjacent points in the Table 7.
Table 8
Adjacent Ordered Pairs
Slope
(0, 0) and (76, 2.40)
2.40  0 2.4

 0.032
76  0
76
(76, 2.40) and (95, 3.00)
3.00  2.40 0.6

 0.032
95  76
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(95, 3.00) and (114, 3.60)
3.60  3.00 0.6

 0.032
114  95
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(114, 3.60) and (132, 4.20)
4.20  3.60 0.6

 0.032
132  114
19
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The fractions
2.4
76
and
0.6
19
simplify to
3
95
. Since all of the slopes are the
same at about 0.032, we know the points lie along a straight line.
Unlike the demand function, we know that the vertical intercept is (0, 0)
so the equation of the line is
S (Q) 
3
Q
95
Notice that the fraction has been used for the slope. If the rounded
decimal had been used instead of the exact fraction, the line would go
very close to the data in the supply schedule, but not through exactly.
Decimals are acceptable when they are exact decimals and not
rounded or where approximations are acceptable.
Example 8
Find the Surplus at a Fixed Price
At a price of $3.36 per gallon, dairies will be willing to supply more milk
than consumers are willing to buy. Find the surplus amount of milk at
this higher price.
Solution To find the amount of the surplus, we must find the difference
between the quantity supplied and the quantity demanded. These
quantities are found by setting the outputs of the supply and demand
functions equal to $3.36.
To find the quantity supplied, set S(Q) equal to 3.36 and solve for the
quantity Q :
3
Q  3.36
95
Q  3.36 
95
 106.4
3
Multiply by reciprocal of 3/95
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At a price of $3.36 per gallon, the dairies would be willing to supply
106.4 thousand gallons or 106,400 gallons of milk per week.
To find the quantity demanded by consumers, set D(Q) equal to 3.36
and solve for Q:
0.05Q  7.75  3.36
0.05Q  4.39
Q  87.8
Subtract 7.75 from both sides
Divide both sides by -0.05
The demand at a price level of $3.36 per gallon is 87,800 gallons of milk
per week.
The surplus of milk is the difference between the two quantities or
Surplus  106, 400  87,800  18,600 gallons per week
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