D(x) is the price, in dollars per unit, that consumers are willing to pay for x

Question

D(x) is the price, in dollars per unit, that consumers are willing to pay for x units of an item, and S(x) is the price, in dollars per unit, that producers are willing to accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.

(a) What are the coordinates of the equilibrium point?

________ (Type an ordered pair)

(b) Find the consumer surplus at the equilibrium point

$________    (Type an integer or decimal)

(c) Find the producer surplus at the equilibrium point

$ _________     (Type an integer or decimal)

Explanation / Answer

D(x) = (x - 6)^2

S(x) = x^2 + 4x + 4

a) Equi point :

D = S

(x - 6)^2 = x^2 + 4x + 4

x^2 - 12x + 36 = x^2 + 4x + 4

-12x + 36 = 4x + 4

-16x = -32

x = -32/-16

x = 2

When x = 2 , D = (2 - 6)^2 = (-4)^2 = 16

So, the equilibrium point is (2 , 16) -----> ANSWER

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b) Consumer surplus :

First find the demand when quantity = 0

(0 - 6)^2

(-6)^2

36

Equilibrium point, x = 2

So, base of the right triangle would be : 2

The value of price when D = S was 16

So, the height of the triangle is 36 - 16 = 20

And area of the triangle = 0.5 * base * hright

A = 0.5 * 2 * 20

A = 20

So, the consumer surplus = 20 ----> ANSWER

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c) Producer Surplus :

Find the supply when quantity = 0

S(0) = 0^2 + 4(0)   + 4 ---> 4

Now, base of the right triangle is again 2

And the height = 16 - 4 = 12

So, area = 0.5 * 2 * 12

= 12

So, producer surplus = 12 --------> ANSWER

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