PLEASE ANSWER THIS!
Consider the problem of a monopolist that sells its product on two different markets m, with m=1,2. Each market has an aggregate demand function given by 1200−α_m*p_m, where p_m denotes the price in market m, and α_m=m measures the responsivity of demand to prices in market m.
The monopolist's cost function is given by c(q)=12q2, where q denotes the total amount produced for all markets.
The monopolist is owned by a foreign company, so none of the monopolist's profits are received by the consumers in these markets.
The law allows the monopolist to charge different pricees in different markets, but does not allow any other forms of price discrimination.
(i)What is the equilibrium level of production in market 2?
(ii)What is total consumer surplus in the economy (i.e., taking both markets into account)?
(iii)Suppose that the government behind market 1 introduces a tax of $100 per unit on the monopolist's sales in its market (paid by the firm), and that the tax revenue is given back to consumers in market 1 using lump-sum transfers. Suppose also that no such tax is introduced by the government behind market 2.
What is the new equilibrium level of production in market 2?
(iv)What is the the new total level of consumer surplus in the economy (including the tax revenues)?
To answer the questions, we need to analyze the monopolist's profit maximization problem and the effects of the tax on the equilibrium level of production and total consumer surplus.
(i) To determine the equilibrium level of production in market 2, we need to find the quantity that maximizes the monopolist's profit.
Profit function:
π(q_1, q_2) = (1200 - α_1 * p_1) * q_1 + (1200 - α_2 * p_2) * q_2 - c(q_1 + q_2)
where q_1 is the quantity produced in market 1, q_2 is the quantity produced in market 2, p_1 is the price in market 1, p_2 is the price in market 2, and c(q) is the cost function.
Differentiating the profit function with respect to q_2 and setting it equal to zero yields:
∂π/∂q_2 = 1200 - α_2 * p_2 - 24 * (q_1 + q_2) = 0
Solving for q_2:
1200 - α_2 * p_2 - 24 * (q_1 + q_2) = 0
1200 - α_2 * p_2 = 24 * (q_1 + q_2)
q_2 = (1200 - α_2 * p_2) / 24 - q_1
(ii) To calculate the total consumer surplus in the economy, we sum the consumer surpluses in both markets:
Consumer Surplus in Market 1:
CS_1 = ∫[0, q_1] (1200 - α_1 * p_1 - p_1) * dq_1
= ∫[0, q_1] (1200 - (α_1 + 1) * p_1) * dq_1
= (1200 - (α_1 + 1) * p_1) * q_1
Consumer Surplus in Market 2:
CS_2 = ∫[0, q_2] (1200 - α_2 * p_2 - p_2) * dq_2
= ∫[0, q_2] (1200 - (α_2 + 1) * p_2) * dq_2
= (1200 - (α_2 + 1) * p_2) * q_2
Total Consumer Surplus:
CS = CS_1 + CS_2
= (1200 - (α_1 + 1) * p_1) * q_1 + (1200 - (α_2 + 1) * p_2) * q_2
(iii) With the introduction of a tax of $100 per unit on the monopolist's sales in market 1, we need to adjust the profit function and find the new equilibrium level of production in market 2.
Modified profit function:
π(q_1, q_2) = (1200 - α_1 * (p_1 + 100)) * q_1 + (1200 - α_2 * p_2) * q_2 - c(q_1 + q_2)
Deriving the new equilibrium level of production in market 2, we follow the same steps as in (i).
(iv) With the tax, the new total level of consumer surplus in the economy includes the tax revenues. Therefore, we need to calculate the consumer surplus and add the tax revenue from market 1:
New Consumer Surplus in Market 1:
CS_1_new = (1200 - (α_1 + 1) * (p_1 + 100)) * q_1
New Total Consumer Surplus:
CS_new = CS_1_new + CS_2 + Tax Revenue
= (1200 - (α_1 + 1) * (p_1 + 100)) * q_1 + (1200 - (α_2 + 1) * p_2) * q_2 + (100 * q_1)
To find the equilibrium level of production in market 2, we need to determine the monopolist's profit-maximizing quantity.
Step 1: Find the monopolist's marginal revenue function.
The monopolist's marginal revenue (MR) is the change in total revenue resulting from selling one additional unit of the product. Since the monopolist faces downward-sloping demand curves in both markets, the marginal revenue is given by the derivative of the total revenue function.
For market 2, the demand function is given by D_2 = 1200 - α_2 * p_2, where p_2 is the price in market 2. To find the marginal revenue, we differentiate the total revenue function with respect to quantity (q_2) and solve for MR_2:
MR_2 = d(TR_2)/d(q_2)
Step 2: Determine the monopolist's profit-maximizing quantity.
To maximize profit, the monopolist chooses the quantity level where marginal revenue equals marginal cost. Since the cost function is given as c(q) = 12q^2, the monopolist's marginal cost (MC) is the derivative of the cost function with respect to quantity.
MC = d(c(q))/d(q)
Set MR_2 = MC and solve for q_2 to find the equilibrium level of production in market 2.
To calculate total consumer surplus in the economy, we need to find the area between the demand curve and the price line for each market.
Step 1: Find the equilibrium price for each market.
The equilibrium price is where the demand curve intersects the marginal cost curve. Set MR_2 = MC and solve for p_2 to get the equilibrium price in market 2.
Step 2: Calculate consumer surplus for each market.
Consumer surplus is the difference between what consumers are willing to pay for a product (based on their demand) and what they actually pay. To calculate consumer surplus, we need to determine the area between the demand curve and the equilibrium price line in each market.
For market 2, consumer surplus (CS_2) is the integral of the demand function (1200 - α_2 * p_2) with respect to price (p_2), evaluated from 0 to the equilibrium price.
CS_2 = ∫[0 to p_2] (1200 - α_2 * p_2) dp_2
The new equilibrium level of production in market 2 after the introduction of the tax can be found by considering the tax as an additional cost to the monopolist. So the new marginal cost is MC + Tax = MC + 100.
We repeat the steps to find the new equilibrium level of production in market 2, but this time using the modified marginal cost.
Finally, to find the new total level of consumer surplus in the economy including the tax revenues, we need to calculate the consumer surplus in each market (taking into account the new equilibrium prices) and sum them up. Additionally, we add the tax revenue received by the government in market 1 to the total consumer surplus.
I hope this helps you work through the problem and find the answers step by step!