Answer to Question #140431 in Microeconomics for Annie

Question

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?

Answer 1

C = q + q² +q³

Q = 24 – p

AC = 1 + q + q²

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

q = -0.5

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

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

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

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

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

AC = 2 + q + q²

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

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

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:

$x \times q = Q$

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

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