In: Economics
Consider the following cost functions:C(Q)=F+cQ,C(Q)=F+Q12, andC(Q)=F+aQ2, whereFrepresents fixed cost. Draw the curves corresponding to average costand marginal cost. Discuss whether any of these production technologies generates anatural monopoly.
DIAGRAM - MATHEMATICAL EXPLANATION FOLLOWS AFTER THE PICTURE
CURVE 1 :
C = F + cQ ; assuming c to be a constant
Marginal Cost : dC/dQ = c
Average Cost : C/ Q = F/Q + c
dAC/dQ = -F/Q2 < 0 i.e. downward sloping
d2AC/dQ2 = F/Q3 >0 i.e. convex
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CURVE 2 :
C = F + 12Q
Marginal Cost : dC/dQ = 12
Average Cost : C/ Q = F/Q + 12
dAC/dQ = -F/Q2 < 0 i.e. downward sloping
d2AC/dQ2 = F/Q3 >0 i.e. convex
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CURVE 3 :
C = F + 2aQ ; assuming a to be a constant
Marginal Cost : dC/dQ = 2a
Average Cost : C/ Q = F/Q + 2a
dAC/dQ = -F/Q2 < 0 i.e. downward sloping
d2AC/dQ2 = F/Q3 >0 i.e. convex
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A natural monopoly has a flat linear MC curve and a downward sloping convex AC curve. So, yes all the cost functions generate natural monopoly.