Perfectly Competitive Market Equilibrium: Solving for Price and Quantity

To find the competitive equilibrium, we need to find the equilibrium price and quantity.

First, let's find the equilibrium quantity by setting the market demand equal to the total supply:

D = 10(7 - p) = 10q

Solving for q, we get:

7 - p = q

Now, let's substitute the variable cost function into the equation to find the equilibrium price:

VC(q) = 3q + (1/4)(q^2) = 3(7 - p) + (1/4)(7 - p)^2

Expanding and simplifying the equation, we get:

21 - 3p + (1/4)(49 - 14p + p^2) = 0

Multiplying through by 4 to get rid of the fraction, we have:

84 - 12p + 49 - 14p + p^2 = 0

Rearranging the equation, we get:

p^2 - 26p + 133 = 0

Using the quadratic formula, we can solve for p:

p = (-(-26) ﾱ sqrt((-26)^2 - 4(1)(133))) / (2(1))

p = (26 ﾱ sqrt(676 - 532)) / 2

p = (26 ﾱ sqrt(144)) / 2

p = (26 ﾱ 12) / 2

p = 19 or p = 7

Since the market demand is sufficiently high that no seller shuts down, the equilibrium price should be set at the higher value of p, which is p = 19.

Substituting p = 19 back into the equation 7 - p = q, we get:

7 - 19 = q

q = -12

However, negative quantity doesn't make sense in this context, so we discard this solution.

Therefore, the competitive equilibrium is a price of p = 19 and a quantity of q = 0.