If the supply function for a commodity is p = q2 + 6q + 16 and the demand function is p = −7q2 + 2q + 436, find the equilibrium quantity and equilibrium price.


Answer:

equilibrium quantity = 7

equilibrium price = 107

Explanation:

Data provided in the question:

supply function, p = q² + 6q + 16  ........(1)

demand function is p = −7q² + 2q + 436

Now at equilibrium

Demand = Supply

Thus,

q² + 6q + 16 = −7q² + 2q + 436

or

q² + 6q + 16 + 7q² - 2q - 436 = 0

or

8q² + 4q - 420 = 0

or

2q² + q - 105 = 0

on solving for the roots of q

using the Quadratic Formula where

a = 2, b = 1, and c = -105

[ x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }]

a = 2, b = 1, and c = -105

Thus,

[ q = \frac{ -1 \pm \sqrt{1^2 - 4(2)(-105)}}{ 2(2) }]

[ q = \frac{ -1 \pm \sqrt{1 - -840}}{ 4 }]

[ q = \frac{ -1 \pm \sqrt{841}}{ 4 }]

The discriminant ( b² - 4ac > 0)

so, there are two real roots.


Therefore,

q = [\frac{ -1 \pm 29}{ 4 }]

[ q = \frac{ 28 }{ 4 } \; \; \; q = -\frac{ 30 }{ 4 }]

[ q = 7 \; \; \; q = -\frac{ 15}{ 2 }]

since,

Quantity cannot be negative

Thus,

q = 7

therefore, substituting q in (1)

p = 7² + 6(7) + 16

or

p = 49 + 42 + 16

or

p = 107


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