Question

The local market for widgets has 1,000 identical producers. Each producer has the following short-run cost function:

mcᵢ = 0.05qᵢ + 20 (i=1,000)
You can derive the market supply as: In addition, the market demand for widgets in this market is given by: Q=275,000−2,500P (
a) Calculate the equation for the market supply curve.
(b) What are the market equilibrium price and quantity? How much will each firm produce?
(c) On two graphs side-by-side, graph the market-level and firm-level equilibrium price and quantities.

Answer 1

The market supply curve can be calculated by multiplying the marginal cost of one producer by the number of producers, and it needs to be expressed in terms of total quantity supplied and price. Equilibrium in the market is found where supply equals demand, and the equilibrium quantities for each firm are found at this price.

Step-by-step explanation:

To find the market supply function, we assume each of these producers will supply widgets up to the point where price equals marginal cost (P=MC).

(a) To calculate the equation for the market supply curve, we simply multiply the individual supply (marginal cost in this case) by the number of producers:

• Market Supply = 1,000 * mci = 1,000 * (0.05q + 20). However, this equation needs to be rewritten in terms of Q and P to derive the market supply curve.
• Since marginal cost equals price at supply (0.05qi + 20 = P), we can solve for qi to find the supply of a single producer and then multiply by 1,000 to find total market supply.

(b) Market equilibrium price and quantity are found where market demand equals market supply.

• We use the given market demand equation Q = 275,000 - 2,500P and set it equal to the total market supply we just calculated.
• By solving for P, we find the equilibrium price, and subsequently the equilibrium quantity. Each firm's production can then be found by substituting the equilibrium price into the firm's marginal cost equation mci = P and solving for qi.

(c) We can graph the market-level and firm-level equilibrium by plotting the supply curve and the demand curve in the same graph.

• The intersection point gives us the equilibrium price and quantity for the market. A separate graph for the firm-level will show the quantity supplied by one firm at different prices.

Answer 2

a. The equation is S = 0.05Q + 20,000.

b. The market equilibrium price is $101.58 and the equilibrium quantity is 21325 widgets.

a. Each firm's supply curve is given by Sᵢ = 0.05qᵢ + 20 (for i = 1 to 1,000). So, to get market supply curve, we need to sum up the individual firm supply curves.

The market supply curve of all individual firm supply curves:

S = ΣSᵢ = Σ(0.05qᵢ + 20) for i = 1 to 1,000

S = (0.05Σqᵢ) + (20 * 1,000)

The Σqᵢ is the total quantity supplied by all 1,000 firms which is the market quantity supplied (Q).

So, we have:

S = 0.05Q + 20,000

b. By setting the market supply equal to the market demand and solve for P and Q, we have:

0.05Q + 20,000 = 275,000 - 2,500P

2,500P = 275,000 - 0.05Q - 20,000

2,500P = 255,000 - 0.05Q

P = (255,000 - 0.05Q) / 2,500

We substitute the Q = 275,000 - 2,500P into the equation above:

P = (255,000 - 0.05(275,000 - 2,500P)) / 2,500

P = (255,000 - 13,750 + 125P) / 2,500

2,500P - 125P = 255,000 - 13,750

2,375P = 241,250

P = 241,250/2,375

P = 101.578947368

P = 101.58

The equilibrium quantity is:

Q = 275,000 - 2,500 * 101.47

Q = 21325.