Question

Hand made bicycle frames are produced by a number of identically sized firms.Total monthly costs for a typical firm are
C(q)=q^3-20q^2 +100q+8000
q is the number of frames produced per month
Demand for hand made bicycle frames is given by
Qd=2500-3p
Qd is qty demanded per month P is price per frame.
a) Determine longrun equilibrium in this market.
b) what is the relationship between the AC (average cost) and MC (marginal cost )at this point
c) what is the output level (q*) at this point
d) What is the AC and MC at the longrun equilibrium and what is the price
c) what will be the qty demanded at the longrun​ equilibrium
d) Calculate the equilibrium number of firms and show that the supply will precisely balance what is demanded at the price calculated above

Answer 1

Cost function is C = 8000 +100q – 20q^2 + q^3

Marginal cost is MC = derivative of Cost function = 100 – 40q + 3q^2

Average total cost function = C/q = 8000/q + 100 – 20q + q^2

Long run price has LRAC = LMC

8000/q + 100 – 20q + q^2= 100 – 40q + 3q^2

8000/q + 20q – 2q^2 = 0

4000 + 10q^2 – q^3 = 0

This gives q = 20. Hence long run price is LMC =100 – 40*20 + 3*(20^2) = 500

Long run Q = 2500 – 3*500 = 1000 units.

Long run n = 1000/20 = 50 firms

a) Determine longrun equilibrium in this market.

Long run has price = 500 and quantity = 1000 units

b) what is the relationship between the AC (average cost) and MC (marginal cost )at this point

AC = MC = 500 where each firm produces 20 units

AC = 8000/20 + 100 – 20*20 + 20^2 = 500

MC = 100 – 40*20 + 3*(20^2) = 500

c) what is the output level (q*) at this point

This is q = 20 for each firm

d) What is the AC and MC at the longrun equilibrium and what is the price

AC = MC = 500 and the price is minimum of AC. This is true because long run price is determined at minimum efficient scale

c) what will be the qty demanded at the longrun? equilibrium

Long run quantity = 2500 – 3*500 = 1000 units.

d) Calculate the equilibrium number of firms and show that the supply will precisely balance what is demanded at the price calculated above

Long run n = 1000/20 = 50 firms

When each firm supplies 20 units and there are 50 firms, total quantity is 1000 units which is what was demanded by consumers.