Question

Consider a monopoly with constant returns to scale starting in the sort run with TC=9+1/4Q^2, and marginal cost MC= 1/2Q, facing a market demand curve of P=12-1/4Q.

1, Graph and calculate the short run consumer surplus, profit, and deadweight loss to welfare.

2, Graph and calculate the long run consumer surplus, profit, and deadweight loss to welfare.

Answer 1

Answer 2

To graph and calculate the short-run and long-run consumer surplus, profit, and deadweight loss for the monopoly, we'll follow these steps:

Step 1: Find the monopoly's profit-maximizing output in the short run and the long run. Step 2: Calculate the price at the profit-maximizing output. Step 3: Calculate the short-run and long-run consumer surplus. Step 4: Calculate the monopoly's profit in the short run and the long run. Step 5: Calculate the deadweight loss to welfare in both the short run and the long run.

Let's start with the short run:

1. Short Run: a) Find the profit-maximizing output: In the short run, the monopoly will maximize its profit where marginal cost (MC) equals marginal revenue (MR).

MR = d(TR)/dQ = P + Q(dP/dQ) MR = (12 - 1/4Q) + Q(-1/2*1/4) MR = 12 - 1/4Q - 1/8Q MR = 12 - 3/8Q

MC = 1/2Q

Setting MR = MC: 12 - 3/8Q = 1/2Q

Solving for Q: 12 = 11/8Q Q = (12 * 8) / 11 Q ≈ 10.91

b) Calculate the price at the profit-maximizing output: Using the demand curve P = 12 - 1/4Q:

P = 12 - 1/4 * 10.91 P ≈ 9.72

c) Calculate the short-run consumer surplus: Short-run consumer surplus = 1/2 * (P - MC) * Q Short-run consumer surplus = 1/2 * (9.72 - (1/2 * 10.91)) * 10.91 Short-run consumer surplus ≈ 26.85

d) Calculate the monopoly's profit: Profit = TR - TC TR = P * Q = 9.72 * 10.91 TC = 9 + 1/4 * 10.91^2 = 9 + 1/4 * 118.88 ≈ 37.22 Profit ≈ 9.72 * 10.91 - 37.22 ≈ 64.29

e) Calculate the deadweight loss to welfare: Deadweight loss = 1/2 * (P - MC) * (Q - Qc) Where Qc is the competitive equilibrium quantity.

In a perfectly competitive market, the quantity is where MC = P: 1/2Q = 12 - 1/4Q 1/2Q + 1/4Q = 12 3/4Q = 12 Qc = (12 * 4) / 3 Qc = 16

Deadweight loss ≈ 1/2 * (9.72 - (1/2 * 10.91)) * (10.91 - 16) ≈ -10.59 (negative because it represents a gain in welfare)

Next, let's move on to the long run:

2. Long Run: In the long run, the monopoly will adjust its output level to achieve maximum profit given the market demand.

a) Find the long-run profit-maximizing output: In the long run, the monopoly will produce where MC = MR = P, which occurs under perfect competition.

MC = 1/2Q P = 12 - 1/4Q

Setting MC = P: 1/2Q = 12 - 1/4Q 1/2Q + 1/4Q = 12 3/4Q = 12 Q = (12 * 4) / 3 Q ≈ 16

b) Calculate the price at the long-run profit-maximizing output: Using the demand curve P = 12 - 1/4Q:

P = 12 - 1/4 * 16 P = 12 - 4 P = 8

c) Calculate the long-run consumer surplus: Long-run consumer surplus = 1/2 * (P - MC) * Q Long-run consumer surplus = 1/2 * (8 - (1/2 * 16)) * 16 Long-run consumer surplus = 32

d) Calculate the monopoly's profit: In the long run, in a monopolistically competitive market, the profit approaches zero. Therefore, the monopoly's profit in the long run is close to zero.

e) Calculate the deadweight loss to welfare: There is no deadweight loss in the long run as the market reaches an equilibrium where P = MC, which is efficient.

To summarize:

Short-run:

• Consumer Surplus: ≈ 26.85

• Profit: ≈ 64.29

• Deadweight Loss: ≈ -10.59 (gain in welfare)

Long-run:

• Consumer Surplus: 32

• Profit: Close to zero (approaching competitive equilibrium)

• Deadweight Loss: 0 (no deadweight loss in the long run)

Answer 3

To analyze the short-run and long-run outcomes of the monopoly, we need to understand the difference between the two scenarios. In the short run, the monopoly is operating with a fixed level of inputs and cannot adjust its scale of production. In the long run, however, the monopoly can change its scale of production and adjust its inputs.

1. Short Run Analysis:

First, let's find the monopoly's output quantity and price in the short run:

Given: Total cost (TC) = 9 + 1/4 * Q^2 Marginal cost (MC) = 1/2 * Q Demand curve (P) = 12 - 1/4 * Q

To find the monopoly's output quantity, set MC equal to the demand curve's slope:

1/2 * Q = 1/4 Q = 1/2 * 4 Q = 2

Now, let's find the price corresponding to the output quantity:

P = 12 - 1/4 * Q P = 12 - 1/4 * 2 P = 12 - 1/2 P = 11.5

Now we can calculate short-run consumer surplus, profit, and deadweight loss:

a) Short-run consumer surplus: Consumer surplus (CS) is the area above the market price and below the demand curve up to the quantity sold (Q).

CS = (1/2) * (P - MC) * Q CS = (1/2) * (11.5 - 1/2 * 2) * 2 CS = (1/2) * (11.5 - 1) * 2 CS = (1/2) * 10.5 * 2 CS = 10.5

b) Short-run profit: Profit (π) is the difference between total revenue (TR) and total cost (TC).

TR = P * Q TR = 11.5 * 2 TR = 23

TC = 9 + 1/4 * Q^2 TC = 9 + 1/4 * 2^2 TC = 9 + 1/4 * 4 TC = 10

π = TR - TC π = 23 - 10 π = 13

c) Short-run deadweight loss (DWL): DWL occurs in a monopoly due to the reduction in consumer surplus below the perfectly competitive level.

DWL = (1/2) * (Pmonopoly - Pcompetitive) * (Qmonopoly - Qcompetitive) DWL = (1/2) * (11.5 - Pcompetitive) * (2 - Qcompetitive)

In the short run, there is no deadweight loss because the monopoly price and quantity are the same as in a perfectly competitive market (where MC equals demand). Thus, Pcompetitive = MC = 1/2 * 2 = 1, and Qcompetitive = 2.

DWL = (1/2) * (11.5 - 1) * (2 - 2) DWL = 0

2. Long Run Analysis:

In the long run, the monopoly can adjust its scale of production to maximize profit. Since it's a natural monopoly with constant returns to scale, the long-run equilibrium output occurs at the point where marginal cost (MC) equals the minimum average total cost (ATC).

ATC = TC / Q ATC = (9 + 1/4 * Q^2) / Q ATC = 9/Q + 1/4 * Q

To find the output quantity that minimizes ATC, we take the derivative of ATC with respect to Q and set it to zero:

d(ATC)/dQ = -9/Q^2 + 1/4 0 = -9/Q^2 + 1/4 9/Q^2 = 1/4 Q^2 = 9 * 4 Q = √(36) Q = 6

Now, we can find the corresponding price:

P = 12 - 1/4 * Q P = 12 - 1/4 * 6 P = 12 - 1.5 P = 10.5

Now we can calculate long-run consumer surplus, profit, and deadweight loss:

a) Long-run consumer surplus: Consumer surplus (CS) in the long run is the area above the market price and below the demand curve up to the quantity sold (Q).

CS = (1/2) * (P - MC) * Q CS = (1/2) * (10.5 - 1/2 * 6) * 6 CS = (1/2) * (10.5 - 3) * 6 CS = (1/2) * 7.5 * 6 CS = 22.5

b) Long-run profit: In the long run, the monopoly adjusts its scale to maximize profit, which means it produces at the minimum ATC. Thus, there is no economic profit in the long run; it's at the break-even point.

c) Long-run deadweight loss (DWL): Similar to the short run, there is no deadweight loss in the long run for this natural monopoly since it produces at the efficient scale.

In conclusion, in the short run, the monopoly generates consumer surplus of 10.5, earns a profit of 13, and has no deadweight loss. In the long run, the consumer surplus increases to 22.5, the profit becomes zero, and there is no deadweight loss