Consider a market with demand curve 𝐷 = 25 − 3𝑝 and supply curve 𝑆 = −4 + 4𝑝. The unit of price is dollar. (a) Find the competitive equilibrium. (b) Compute consumer surplus and producer surplus at the equilibrium. (c) The government decides to impose a tax of $2 per unit. Compute the competitive equilibrium with the tax. (d) Compute consumer surplus, producer surplus and tax revenue at the new equilibrium.
Question
Consider a market with demand curve 𝐷 = 25 − 3𝑝 and supply curve 𝑆 = −4 + 4𝑝. The unit of price is dollar. (a) Find the competitive equilibrium. (b) Compute consumer surplus and producer surplus at the equilibrium. (c) The government decides to impose a tax of $2 per unit. Compute the competitive equilibrium with the tax. (d) Compute consumer surplus, producer surplus and tax revenue at the new equilibrium.
Solution
Sure, let's break this down:
(a) To find the competitive equilibrium, we need to set the demand equal to the supply and solve for the price (p).
25 - 3p = -4 + 4p
Solving for p, we get:
7p = 29
p = 29/7 = $4.14 (approximately)
Substitute p = 4.14 into either the demand or supply equation to find the equilibrium quantity (Q):
Q = 25 - 3*4.14 = 12.58 (approximately)
So, the competitive equilibrium is at price $4.14 and quantity 12.58.
(b) Consumer surplus is the area between the demand curve and the price line up to the equilibrium quantity. It's given by the formula 0.5*(Q)*(Pmax - P), where Pmax is the maximum price consumers are willing to pay (where Q=0).
Pmax = 25/3 = $8.33 (approximately)
Consumer surplus = 0.512.58(8.33 - 4.14) = $26.47 (approximately)
Producer surplus is the area between the price line and the supply curve up to the equilibrium quantity. It's given by the formula 0.5*(Q)*(P - Pmin), where Pmin is the minimum price suppliers are willing to accept (where Q=0).
Pmin = 4/4 = $1
Producer surplus = 0.512.58(4.14 - 1) = $19.73 (approximately)
(c) With a tax of 2. The new supply curve is S = -2 + 4p. Setting this equal to the demand curve:
25 - 3p = -2 + 4p
Solving for p, we get:
7p = 27
p = 27/7 = $3.86 (approximately)
Substitute p = 3.86 into the demand equation to find the new equilibrium quantity:
Q = 25 - 3*3.86 = 13.42 (approximately)
So, the competitive equilibrium with the tax is at price $3.86 and quantity 13.42.
(d) Compute consumer surplus, producer surplus and tax revenue at the new equilibrium:
Consumer surplus = 0.513.42(8.33 - 3.86) = $29.97 (approximately)
Producer surplus = 0.513.42(3.86 - 1) = $19.23 (approximately)
Tax revenue is the tax per unit times the quantity sold, which is 26.84 (approximately).
Similar Questions
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