Assume that the market is currently supplied by ten identical sellers each with variable cost VCq = 3q +14q^2 and the market demand is D = 107 − p Assume that the market demand is sufficiently high th

To find the competitive equilibrium, we need to find the price and quantity at which the market clears, i.e., the quantity supplied equals the quantity demanded.

First, let's find the market supply function. The cost function for each seller is given by VC(q) = 3q + (1/4)(q^2). Since there are ten identical sellers, the market supply function is given by:

QS = 10 * VC(q)

Let's substitute the cost function into the market supply function:

QS = 10 * (3q + (1/4)(q^2)) = 30q + (10/4)(q^2) = 30q + (5/2)(q^2)

Next, let's find the market demand function. The demand function is given by D = 10(7 - p).

Since the market is perfectly competitive, the quantity supplied (QS) must equal the quantity demanded (QD). Therefore, we can set QS equal to QD and solve for the equilibrium quantity (Q):

QS = QD 30q + (5/2)(q^2) = 10(7 - p)

To find the equilibrium price (p), we need to solve the demand function for p. Rearranging the equation, we get:

p = 7 - (1/10)QD

Substituting QD = QS, we get:

p = 7 - (1/10)(30q + (5/2)(q^2))

Now, let's substitute the expression for p back into the equation for QS:

30q + (5/2)(q^2) = 10(7 - (1/10)(30q + (5/2)(q^2)))

Simplifying this equation, we get:

30q + (5/2)(q^2) = 70 - 3q - (1/2)(q^2)

Rearranging terms, we have:

(7/2)(q^2) + 33q - 70 = 0

To solve this quadratic equation, we can use the quadratic formula:

q = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 7/2, b = 33, and c = -70. Substituting these values into the quadratic formula, we get:

q = (-33 ± sqrt(33^2 - 4(7/2)(-70))) / (2 * (7/2)) = (-33 ± sqrt(1089 + 980)) / 7 = (-33 ± sqrt(2069)) / 7

Since the market demand is sufficiently high that no seller shuts down, we can assume that the quantity supplied is positive. Therefore, we can disregard the negative solution and only consider the positive solution:

q = (-33 + sqrt(2069)) / 7

Now, let's substitute this value of q back into the demand function to find the equilibrium price (p):

p = 7 - (1/10)(30q + (5/2)(q^2)) = 7 - (1/10)(30((-33 + sqrt(2069)) / 7) + (5/2)((-33 + sqrt(2069)) / 7)^2)

Evaluating this expression will give us the competitive equilibrium price