Answer to Question #140431 in Microeconomics for Annie

Question #140431
Each firm in a competitive market has a cost function of C =q÷q² +q³. The market has an unlimited number of potential firms, The market demand function is Q = 24- p. Determine the long-run equilibrium price, quantity per firm, mar ket quantity, and number of firms. How do these values change if a tax of $1 per unit is collected from each firm?
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Expert's answer
2020-11-03T11:19:30-0500

C = q + q2 +q3

Q = 24 – p

AC = 1 + q + q2

d(AC)dq=2q+1=0\frac{\mathrm{d}(AC)}{\mathrm{d} q} = 2q + 1 = 0

q = -0.5

d2(AC)dq=2q+1=0=2>0\frac{\mathrm{d^2}(AC)}{\mathrm{d} q} = 2q + 1 = 0 = 2 > 0

At q = -0.5 AC is minimum

min(AC) = 1 + q + q2

min(AC) = 1 + (-0.5) + (-0.5)2

P = 0.75

In the long-run each firm will produce q = 0.5

Total industry production at P = 0.75

Q = 24 – 0.75 = 23.25

Number of firms:

x \times q = Q

n=Qqn = \frac{Q}{q}

n=23.750.5=47n = \frac{23.75}{0.5} = 47

If a tax of $1 per unit is collected from each firm new AC function:

AC = 2 + q + q2

d(AC)dq=2q+1=0\frac{\mathrm{d}(AC)}{\mathrm{d} q} = 2q + 1 = 0

d2(AC)dq=2>0\frac{\mathrm{d^2}(AC)}{\mathrm{d} q} = 2 > 0

At q = -0.5

P = min(AC) = 2 + (-0.5) + (-0.5)2 = 1.75

In the long-run each firm will produce q = 0.5

Total industry production at P = 1.75

Q = 24 – 1.75 = 22.25

Number of firms:

x×q=Qx \times q = Q

n=Qqn = \frac{Q}{q}

n=22.750.5=44.5=45n = \frac{22.75}{0.5} = 44.5 = 45


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