The demand for your college newspaper is 2200 copies each week if the paper is given away free of charge and drops to 1100 each week if the charge is 10/copy. However, the university is only prepared to supply 600 copies each week free of charge but will supply 1600 each week at 20/copy.
(a) Write down the associated linear demand function
Write down the associated linear supply function.
To write down the associated linear demand function, we need to consider the relationship between the demand for the newspaper and its price. We have two data points:
When the newspaper is given away free, the demand is 2200 copies per week.
When the charge is 10 cents per copy, the demand drops to 1100 copies per week.
Let's assume the demand function is represented by D(p), where p is the price per copy. We can write the demand function equation in the form of a linear equation:
D(p) = mp + b
To find the slope (m) and the y-intercept (b), we will use the two given data points:
When p = 0 (free of charge), D(p) = 2200.
When p = 10 cents, D(p) = 1100.
Using the equation D(p) = mp + b, we can plug in the values for the two data points to form a system of equations:
2200 = m(0) + b
1100 = m(10) + b
Simplifying these equations, we get:
2200 = b
1100 = 10m + b
Substituting b = 2200 into the second equation:
1100 = 10m + 2200
Now, we can solve for m:
10m = -1100
Dividing both sides by 10:
m = -110
Therefore, the associated linear demand function is:
D(p) = -110p + 2200
Now, let's write down the associated linear supply function.
Similarly, we have two data points for the supply:
When the newspaper is given away free, the supply is 600 copies per week.
When the charge is 20 cents per copy, the supply increases to 1600 copies per week.
Again, let's assume the supply function is represented by S(p), where p is the price per copy. We can write the supply function equation in the form of a linear equation:
S(p) = mp + b
Using the two given data points, we can set up a system of equations:
When p = 0 (free of charge), S(p) = 600.
When p = 20 cents, S(p) = 1600.
Again, we can solve for m and b by substituting the values:
600 = b
1600 = 20m + b
Substituting b = 600 into the second equation:
1600 = 20m + 600
Now, we can solve for m:
20m = 1000
Dividing both sides by 20:
m = 50
Therefore, the associated linear supply function is:
S(p) = 50p + 600