Suppose That Drexapparel Gets A Patent For A New Type Of T-S

Suppose that Drexapparel gets a patent for a new type of t-shirt fabric

Evaluate the market behavior of Drexapparel as a monopolist with a patent, given the demand function Q = 30 – p and cost function 50 + 0.5Q^2. Determine the inverse demand function, revenue, marginal revenue, profit-maximizing quantity and price, graphical representation, economic profits, consumer surplus, producer surplus, deadweight loss, market power index, impact of government-imposed price ceiling, and the optimal price ceiling to eliminate deadweight loss.

Paper For Above instruction

In this paper, we analyze the monopolistic behavior of Drexapparel after obtaining a patent for a novel stain-resistant t-shirt fabric. Our focus centers on the derivation of key economic metrics such as demand functions, revenue, marginal revenue, profit maximization, and the effects of government interventions on market efficiency and welfare. This comprehensive analysis reveals the strategic considerations a monopolist faces and the implications of policy measures like price ceilings.

Inverse Demand Function and Revenue Calculations

The given demand function Q = 30 – p allows us to express the price as a function of quantity, which forms the inverse demand function. Rearranging, p = 30 – Q. This inverse demand function indicates that as quantity increases, the price consumers are willing to pay decreases linearly, reflecting typical demand behavior for a monopolist.

The total revenue (TR) for Drexapparel is computed as TR = p × Q. Substituting p from the inverse demand function yields TR = (30 – Q) × Q = 30Q – Q². The revenue function explicitly expresses total revenue based on quantity sold.

The marginal revenue (MR) is derived by differentiating TR with respect to Q: MR = d(TR)/dQ = 30 – 2Q. Marginal revenue decreases twice as fast as the demand curve's slope and is crucial for determining profit-maximizing output levels.

Profit-Maximizing Quantity and Price

The cost function is given as C(Q) = 50 + 0.5Q², with the marginal cost (MC) obtained by differentiating: MC = dC/dQ = Q. The profit maximization condition equates MR and MC: 30 – 2Q = Q, which simplifies to 30 = 3Q, yielding Q* = 10 (thousand shirts).

Substituting Q back into the inverse demand function provides the optimal price: p = 30 – 10 = $20 per shirt.

Graphical Representation and Profit Visualization

Graphing the demand curve (p = 30 – Q), the MR curve (MR = 30 – 2Q), and the MC curve (MC = Q), visually illustrates the profit-maximizing point at Q = 10 and p = 20. The area between the price and average total cost (ATC) over this quantity represents economic profit, which can be computed subsequently.

Economic Profits

The total revenue at profit maximization is TR = (20)(10) = $200. The total cost at this quantity is C(10) = 50 + 0.5(10)² = 50 + 50 = $100. Therefore, the economic profit is π = TR – C = 200 – 100 = $100, indicating a profitable monopolistic operation.

Consumer Surplus and Producer Surplus

The consumer surplus is the area of the triangle between the demand curve and the price line, up to Q = 10. The maximum price consumers are willing to pay (at Q=0) is $30, and the price they actually pay is $20, with a quantity of 10 thousand shirts. Thus, consumer surplus (CS) = (1/2) × base × height = (1/2) × 10 × (30 – 20) = 50.

The producer surplus, in this case, is equivalent to the economic profit since fixed costs are incorporated in total costs, and the monopoly captures the entire surplus. The producer surplus is therefore $100.

Deadweight Loss and Market Power

The deadweight loss (DWL) arises because the monopolist produces less than the socially optimal quantity where MR = D = MC = Q. The socially optimal quantity occurs when p = MC: 30 – Q = Q, which yields Q = 15. Comparing this with the monopoly quantity (Q=10), the DWL area is a triangle with base (15 – 10) = 5 and height (price at Q=15) p = 15, thus DWL = (1/2) × 5 × 5 = 12.5.

Lerner Index and Demand Elasticity

The Lerner Index, reflecting market power, is computed as (P – MC)/P = (20 – 10)/20 = 0.5. The elasticity of demand (ε) faced by Drexapparel can be derived from the inverse demand function, where ε = (dQ/dp) × (p/Q). Here, dQ/dp = –1, so ε = –(p/Q) = –(20/10) = –2. This indicates elastic demand, consistent with the monopolist’s pricing decisions.

Impact of Price Ceiling

If the government imposes a price ceiling of $18, the new equilibrium quantity can be computed by setting p = 18 in the demand function: Q = 30 – 18 = 12. Since the price ceiling is below the monopolist’s optimal price, the monopolist will sell 12 thousand shirts at $18 each.

The total revenue becomes TR = 18 × 12 = $216, and total cost at Q=12 is C(12) = 50 + 0.5(12)² = 50 + 72 = $122. The economic profit now is 216 – 122 = $94. The deadweight loss arises because the monopolist reduces output to 12 instead of the socially optimal 15, leading to DWL calculated as (1/2) × (15 – 12) × (30 – 18) = (1/2) × 3 × 12 = 18.

Optimal Price Ceiling to Eliminate Deadweight Loss

To eliminate DWL, the price ceiling must equate to the marginal cost at the socially optimal quantity, i.e., p = MC at Q=15. From the inverse demand function: p = 30 – Q = 30 – 15 = $15. Imposing a price ceiling of $15 ensures the market produces Q=15, aligning private and social optimality, thus eliminating deadweight loss.

Conclusion

This comprehensive analysis demonstrates the critical interplay between demand, costs, and market power in shaping monopoly outcomes. Policy measures like price ceilings can significantly influence welfare, either reducing or eliminating deadweight loss, depending on their alignment with marginal cost. Understanding these dynamics is vital for informed economic policymaking and strategic business operations.

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