The required supply function is p = 3q - 300 and the required demand function is p = -3q + 1110. The equilibrium quantity is 235 and the equilibrium price is $405.
a) Find an equation for the supply function's price p as a function of q as follows.
Let p = mq + b be the required linear supply function.
From the given information, the supply function passes through points (125, 75) and (150,150).
Find the slope m using the points (125, 75) and (150,150) as shown below.
m = (p2-p1)/(q2-q1)
m = (150-75)/ (150-125) = 75/25 = 3
Now find the value of b using the point (125, 75) and slope m = 3 as follows.
p = mq + b
75 = 3(125) + b
b = - 375 + 75 = -300
Therefore, the required supply function is p = 3q - 300.
b) Find an equation for the demand function's price p as a function of q as follows.
Let p = mq + b be the required linear supply function.
From the given information, the demand function passes through points (355, 45) and (330,120).
Find the slope m using the points (355, 45) and (330,120) as shown below.
m = (p2-p1)/(q2-q1)
m = (120-45)/ (330-355) = 75/-25 = -3
Now find the value of b using the point (355, 45) and slope m = -3 as follows.
p = mq + b
45 = -3(355) + b
b = 1065 + 45 = 1110
Therefore, the required demand function is p = -3q + 1110.
Find the equilibrium quantity and equilibrium price as follows.
In order to find the equilibrium quantity, equate the demand function and supply function and solve the equation solve for q as shown below as p is common.
3q - 300 = -3q + 1110
=> 6q = 1410
=> q = 235
Thus, the equilibrium quantity is 235.
Find the equilibrium price by substituting q = 235 in p = 3q -300.
p = 3 x 235 - 300
p = 705 -300 = 405
Hence, the equilibrium price is $405.
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