Question

Suppose the supply function for a certain item is given by S(q)= (q+6)2 and the demand funtion is given by D(q)= (1000)/(q+6).
A. Find the point at which supply and demand are in equilibrium?
B. Find the consumer's surplus?
C. find the producer's surplus?

Answer 1

Answer: The equilibrium point is where; Quantity supplied = 100 and Quantity demanded = 100

Step-by-step explanation: The equilibrium point on a demand and supply graph is the point at which demand equals supply. Better put, it is the point where the demand curve intersects the supply curve.

The supply function is given as

S(q) = (q + 6)^2

The demand function is given as

D(q) = 1000/(q + 6)

The equilibrium point therefore would be derived as

(q + 6)^2 = 1000/(q + 6)

Cross multiply and you have

(q + 6)^2 x (q + 6) = 1000

(q + 6 )^3 = 1000

Add the cube root sign to both sides of the equation

q + 6 = 10

Subtract 6 from both sides of the equation

q = 4

Therefore when q = 4, supply would be

S(q) = (4 + 6)^2

S(q) = 10^2

S(q) = 100

Also when q = 4, demand would be

D(q) = 1000/(4 + 6)

D(q) = 1000/10

D(q) = 100

Hence at the point of equilibrium the quantity demanded and quantity supplied would be 100 units.

Answer 2

A. The point at which supply and demand are in equilibrium is $q=4$.

B. The consumer's surplus is 178.16 .

C. The producer's surplus is 66.6 .

Given,

The supply function for a certain item is,

$S(q)= (q+6)^2$

The demand function is,

$D(q)= \dfrac{1000}{ (q+6)}$

Now we know that the supply and demand are in equilibrium where the supply and demand functions are equal.

So for equilibrium,

$S(q)= D(q)$

$(q+6)^2=\dfrac{1000}{q+6}$

$(q+6)^3=1000$

$q+6=\sqrt[3]{1000}$

$q+6=10$

$q=4$

Hence the point is $q=4$, at this point supply and demand are in equilibrium.

At equilibrium the supply is $(4+6)^2=100$ and demand is also 100.

so, $(q^*,p^*)$ is $(4,100)$

Now, the consumer's surplus will be,

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=\int\limits^4_0 {\dfrac{1000}{q+6} } \, dq -4\times 100$

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000[log10-log6]-400$

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000[1-0.778]-400$

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000\times 0.22184-400$

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=221.84-400$

$\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=178.16$

Now, the producer's surplus will be,

$p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=400-\int\limits^{4}_0(q+6)^2dq$

$p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=400-\frac{1}{3} [1000-0]$

$p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=\dfrac{200}{3}$

$p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=66.66$

For more details on Demand and Supply function follow the link:

https://brainly.com/question/2254422