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Question Four {10 Marks} Consider a competitive industry with several identical firms
Question Four {10 Marks} Consider a competitive industry with several identical firms. The long run average cost of a firm producing Q units of output is given by: C = 50 – 4Q + Q2 Suppose the market demand is given by: Qp = 246 – P; where P denotes market price. Determine the following: Answer i. The quantity at which the LRAC is minimised. {2 Marks} ii. The value of LRAC at its minimum point. {2 Marks} iii. The market demand when LRMC ELRAC (minimum point). {2 Marks} iv. The number of firms in the industry in the long run equilibrium.
Expert Solution
(i)
Long run average cost of a firm producing Q units of output is given by: LRAC = 50 - 4Q + Q2.
LRAC = 50 - 4Q + Q2.
Differentiate LRAC w.r.t Q
=> dLRAC / dQ = -4 + 2Q = 0
=> 2Q = 4
=> Q = (4/2)
=> Q = 2
At Q=2, the LRAC minimised
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(ii) LRAC = 50 - 4Q + Q2
Put Q = 2
=> LRAC = 50 - 4(2) + (2)2.
=> LRAC = 50 - 8 + 4
=> LRAC = 46
The minimum value of LRAC is 46
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(iii) At minimum point of LRAC, LRMC = LRAC
Hence, LRMC = LRAC = 46
A perfectly competitive firm produces at P = LRMC = LRAC
P = LRMC
=> P = 46
Hence, the long run equilibrium price is 46
Qd = 246 - P
=> Qd = 246 - 46
=> Qd = 200
The market demand is 200 when LRMC = LRAC.
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(iv) Each firm produces Q=2 in the long run.
The market demand in the long run is 200
Number of firms in the long run = (Market demand / Each firm production)
=> Number of firms in the long run = (200 / 2)
=> Number of firms in the long run = 100
There are 100 firms in the industry in the long run equilibrium point.
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