Suppose that the total cost of producing pizzas for the typical firm in a local town is given by: C(q)=2q+2q2. In turn, marginal cost is given by: MC = 2 + 4q. (If you know calculus, you should be able to derive this expression for marginal cost.)

a. Show that the competitive supply behavior of the typical pizza firm is described by: q = P − 1 .

behavior of the typical pizza firm is described by: q = P/4 − 1/2 .

b. If there are 100 firms in the industry, each acting as a perfect competitor, show that the market supply curve is, in inverse form, given by: P = 2 + Q/25.

a. use the information that an MC marginal cost curve is a portion of the supply curve so we use the integral of the MC curve as the supply curve in a competitive market where companies are price takers.

b. multiply the Marginal cost function slope by 1/100 now that the competitors are limited to 100 firms.

Explanation:

firstly we know that at MC(marginal cost curve) intersect with AC (average cost curve) is the minimum point that a firm can start selling to break even on quantity produced or supplied and we also know that from that point above the company can be profitable and supply to the market properly without suffering that is why the cost of producing 1 more unit is equal to the cost of production and the portion of the Marginal cost curve above the intersection point can be given the assumption that it is a supply curve.

Therefore if the price at the intersection is price P then P=MC

this is when the marginal benefit of the company is equal to the marginal cost. then now we have the total cost function which is C(q) = 2q + 2q2

and the marginal cost function which is MC= 2+ 4q which is derived from the total cost curve so if we simplify the total cost curve and factorise it further to C(q) = 2q(1+q) we can see that the part in brackets is the cost of producing 1 more unit in the company which represents the marginal cost curve, therefore P=MC so P = 1+q then we solve for q which in turn is the quantity supplied when producing 1 more unit of pizza where MC(Marginal cost curve) and AC( average cost curve) meet

then therefore the competitive supply of that quantity is q=P-1.

then we can equate the marginal cost function which is MC= 2+ 4q

which above we mentioned that is equal to the price P, so P= 2+ 4q

then we solve for q the quantity supplied at price P to get the supply function.

P-2= 4q, then we divide the whole equation by 4 to solve for q.

P/4 - 2/4 =q , we then simplify and rearrange.

q = P/4 - 1/2.

b. if we have 100 firms in the industry then we would multiply the slope for the marginal cost function by 1/100 because now the price in the market and the quantity will be determined by these 100 firms in market supply so MC= 2 + 4q now to adjust the quantity supplied to a competitive market of 100 firms Q,

MC = 2 + 4q(1/100)

MC = 2 + (4/100)Q then simplify the fraction

MC =  2 + (1/25)Q which can be written as

MC = 2 + Q/25 then we know for supply function MC=P so

therefore P= 2 + Q/25.