Ten competitive sawmills currently supply lumber to a market whose demand q, dep

Question

Ten competitive sawmills currently supply lumber to a market whose demand q,

depends on lumber price, p, as follows:

                            q = 3550 - 350p.

The cost function of each mill is identical:

                   Ci = 5qi + 0.05qi2 + 80                                  i = 1........10

(d) Technical development changes the cost function of new mills to:

                         Cj = 5qj + 0.1qj2 + 22.5

Ninety new competitive mills enter the market. What becomes the new short-run equilibrium market price and output? Assume the old mills still operate and that the entry and exit of mills do not affect individual mill cost functions.

(e) What will be the long run market equilibrium price and output? How many mills of what type - new or old - will survive?

Explanation / Answer

MC = dC/dq

= 5 + .1q = P or q = 10P - 50

The industry supply curve is the horizontal sum of the 10 firm's individual supply curves.
Q = 10q = 100P - 500

The equilibrium price and quantity can be found by equating supply and demand equation
Q = 3550 - 350P = 100P - 500

Q = 4050 = 450P or P = 9

Q = 3550 - 350(9) = 400

Now for each firm P = MR = MC

So equating the equations

9 = 5 + 0.1q which gives us q = 40

Now the new cost curve is applicable for 90 firms

Cj = 5qj + 0.1qj2 + 22.5

So again MC = dC/dq = 5 + .2q = P or q = 5P - 25

Total supply will be supply curve of 10 old firms and 90 new firms

Q = 100P - 500 + 450P - 2250 = 550P - 2750

Again equating demand and supply equation

Q = 3550 - 350P = 550P - 2750 or 900P = 6300 or P = 7

Q = 3550 - 350(7) = 1100

For each firm P = MR = MC

For old firms

7 = 5 + .1qi so qi = 20

profiti = Pqi - C = 7(20) - [5(20) + .05(20)2 + 80] = -60

For new firms

7 = 5 + .2qj so qj = 10
profitj = Pqj - C = 7(10) - [5(10) + .1(10)2 + 22.5] = -12.50

The lowest market price at which a firm would operate in the long run is equal to the minimum level of long run average cost
ACi = C/qi = 5 + .05q + 80/q
min ACi = dACi/dqi = 0.05 - 80/q2 so q = 40

ACi = 5 + .05(40) + 80/(40) = 9

ACN = C/qN = 5 + .1q + 22.5/q

min ACj = dACN/dqN = .1 - 22.5/q2 so that  q = 15

ACj = 5 + .1(15) + 22.5/(15) = 8

In long run equilibrium, any firm that remains in the industry must be making zero economic profit. Therefore, the long run equilibrium price must be $8.

At this price,

Q = 3550 - 350P = 3550 - 350(8) = 750 units.

Now at price of $8, older firms are making losses and all of them will have to be shut down

For newer entrants, at $8 there will be 50 (750/15) of the new firms which would remain in the industry with each firm producing 15 units of output.

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