Question

Market inverse demand is 𝑝 = 10 − 2𝑄𝑑. There are two competitive firms in the market. Firm A’s total cost of production is 𝑇𝐶 (𝑞 ) = 𝑞a^2 and firm B’s total cost of production is 𝑇𝐶 (𝑞 ) = 2𝑞b^2. Suppose that the government imposes a per-unit sales tax of 𝑡 < 10 dollars on only firm B. Then:

1. Find the competitive equilibrium {𝑝*, 𝑞a *, 𝑞b *}

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the competitive equilibrium, we need to find the quantities and prices at which the supply and demand curves intersect.

Given the inverse demand equation p = 10 - 2Qd, we can set this equal to the sum of the quantities supplied by firm A and firm B, Qa and Qb, respectively:
10 - 2Qd = Qa + Qb

Now let's find the quantities supplied by firm A and firm B. Firm A minimizes its total cost of production, so we need to find the quantity, Qa, that minimizes its cost function TC(q) = qa^2:
TC(qa) = qa^2
Marginal cost (MC) = dTC/qa = 2qa
Equating marginal cost to the price, p, gives us the quantity supplied by firm A:
2qa = p
qa = p/2

Similarly, firm B minimizes its total cost of production, so we need to find the quantity, Qb, that minimizes its cost function TC(q) = 2qb^2:
TC(qb) = 2qb^2
Marginal cost (MC) = dTC/qb = 4qb
Equating marginal cost to the price, p, plus the per-unit sales tax, t, gives us the quantity supplied by firm B:
4qb = p + t
qb = (p + t)/4

Now we can substitute these quantities into the inverse demand equation to find the equilibrium price:
10 - 2Qd = Qa + Qb
10 - 2Qd = (p/2) + ((p + t)/4)
10 - 2Qd = (2p + p + t)/4
40 - 8Qd = 3p + t
t = 40 - 8Qd - 3p

Next, let's find the equilibrium quantities by substituting the equilibrium price, p*, into the quantity supplied equations for firm A and firm B:
qa* = p*/2
qb* = (p* + t)/4

Finally, we can solve for the equilibrium price, p*, by substituting the equilibrium quantities into the inverse demand equation:
10 - 2Qd = qa* + qb*
10 - 2Qd = p*/2 + (p* + t)/4
40 - 8Qd = 2p* + p* + t
40 - 8Qd = 3p* + t
3p* = 40 - 8Qd - t
p* = (40 - 8Qd - t)/3

So, the competitive equilibrium is given by the prices and quantities:
p* = (40 - 8Qd - t)/3
qa* = p*/2
qb* = (p* + t)/4

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the competitive equilibrium {𝑝*, 𝑞a *, 𝑞b *}, we need to determine the quantity produced by each firm and the market price after the tax is imposed.

First, we need to find the quantity demanded and quantity supplied in the market without the tax. In a competitive equilibrium, the quantity demanded by consumers is equal to the quantity supplied by the firms.

Quantity demanded: 𝑄𝑑 = (10 - 𝑝) / 2
Quantity supplied by Firm A: 𝑄𝑎 = 𝑞𝑎 *
Quantity supplied by Firm B: 𝑄𝑏 = 𝑞𝑏 *

Setting the quantity demanded equal to the total quantity supplied, we have:
(10 - 𝑝) / 2 = 𝑞𝑎 * + 𝑞𝑏 *

Next, we need to calculate the total cost of production for each firm.

Total cost of production for Firm A: 𝑇𝐶𝑎(𝑞) = 𝑞𝑎^2
Total cost of production for Firm B: 𝑇𝐶𝑏(𝑞) = 2𝑞𝑏^2

Now, let's consider the effects of the tax on Firm B. With a per-unit sales tax of 𝑡, the new total cost of production for Firm B will be:

𝑇𝐶𝑏_𝑡(𝑞) = 2𝑞𝑏^2 + 𝑡𝑞𝑏

Now, we can set up the profit maximization problem for each firm:

Profit for Firm A: 𝜋𝐴 = 𝑝(𝑄𝑎) - 𝑇𝐶𝑎(𝑄𝑎)
Profit for Firm B: 𝜋𝐵 = 𝑝(𝑄𝑏) - 𝑇𝐶𝑏_𝑡(𝑄𝑏)

Finally, we can solve for the competitive equilibrium by simultaneously solving the profit maximization problem for each firm using the market demand and the new total cost of production for Firm B.

By solving these equations, we can find the values of 𝑝*, 𝑞𝑎 *, and 𝑞𝑏 * that constitute the competitive equilibrium.

Answer 3

[AI-generated answer — Explain Bot]

To find the competitive equilibrium {𝑝*, 𝑞a*, 𝑞b*}, we need to consider the conditions for perfect competition:

1. Equilibrium price (𝑝*): In perfect competition, the equilibrium price is determined by the market demand and supply. In this case, the market inverse demand is given by 𝑝 = 10 − 2𝑄𝑑.

To find the equilibrium price, we set the market demand equal to the total quantity supplied by both firms (𝑞a* + 𝑞b*):
10 − 2(𝑞a* + 𝑞b*) = 𝑞a* + 𝑞b*

Simplifying the equation:
10 − 2𝑞a* − 2𝑞b* = 𝑞a* + 𝑞b*
10 − 3𝑞a* − 3𝑞b* = 0

2. Equilibrium quantity for firm A (𝑞a*): The equilibrium quantity for firm A is determined by the profit maximization condition. Firm A maximizes its profit when marginal cost (MC) equals marginal revenue (MR) which is equal to the equilibrium price (𝑝*).

The marginal cost for firm A is the derivative of its total cost function, 𝑇𝐶(𝑞) = 𝑞a^2, with respect to quantity (𝑞a):
MCa = d(TC)/d(𝑞a) = 2𝑞a

Setting MCa equal to 𝑝* and solving for 𝑞a*:
2𝑞a* = 10 − 3𝑞a*
5𝑞a* = 10
𝑞a* = 2

3. Equilibrium quantity for firm B (𝑞b*): Similar to firm A, the equilibrium quantity for firm B is determined by the profit maximization condition. Firm B maximizes its profit when MC equals MR, which is equal to the equilibrium price (𝑝*).

The marginal cost for firm B is the derivative of its total cost function, 𝑇𝐶(𝑞) = 2𝑞b^2, with respect to quantity (𝑞b):
MCb = d(TC)/d(𝑞b) = 4𝑞b

Setting MCb equal to 𝑝* after accounting for the per-unit tax (𝑡):
4𝑞b + 𝑡 = 10 − 3𝑞b
7𝑞b = 10 − 𝑡
𝑞b* = (10 − 𝑡) / 7

Now we have found the competitive equilibrium {𝑝*, 𝑞a*, 𝑞b*}:
- Equilibrium price (𝑝*): 10 − 3𝑞a* − 3𝑞b* = 0
- Equilibrium quantity for firm A (𝑞a*): 2
- Equilibrium quantity for firm B (𝑞b*): (10 − 𝑡) / 7