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:

Question

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

Answer 1

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