Problem 1: Roberts' New Way Vacuum Cleaner Company
Problem 1roberts New Way Vacuum Cleaner Company Is A Newly Started S
Problem 1: Robert’s New Way Vacuum Cleaner Company is a newly started small business that produces vacuum cleaners and belongs to a monopolistically competitive market. Its demand curve for the product is expressed as Q = 5000 – 25P where Q is the number of vacuum cleaners per year and P is in dollars. Cost estimation processes have determined that the firm’s cost function is represented by TC = 1500 + 20Q + 0.02Q^2. Show all of your calculations and processes. Describe your answer for each question in complete sentences, whenever it is necessary.
What are the profit-maximizing price and output levels? Explain them and calculate algebraically for equilibrium P (price) and Q (output). Then, plot the MC (marginal cost), D (demand), and MR (marginal revenue) curves graphically and illustrate the equilibrium point. How much economic profit do you expect that Robert’s company will make in the first year? Do you expect this economic profit level to continue in subsequent years? Why or why not?
Paper For Above instruction
Introduction
In a monopolistically competitive market, firms face a downward-sloping demand curve, which allows them a degree of pricing power. Robert’s New Way Vacuum Cleaner Company, as a new entrant in such a market, aims to determine its profit-maximizing output and price level based on its demand and cost functions. This analysis is critical for understanding the firm's short-term profitability and sustainability in a competitive environment characterized by product differentiation and free entry and exit.
Demand and Cost Functions
The demand function is given by Q = 5000 – 25P, which indicates that as price increases, quantity demanded decreases, conforming to the law of demand. The total cost (TC) function is TC = 1500 + 20Q + 0.02Q^2, reflecting fixed costs, linear variable costs, and increasing marginal costs associated with higher production levels (Varian, 2014). To find the profit-maximizing output and price, we first need to derive the marginal revenue (MR) and marginal cost (MC) functions.
Derivation of Revenue and Marginal Functions
Rearranging demand to express P as a function of Q:
P = (5000 – Q) / 25 = 200 – 0.04Q
Total revenue (TR) as a function of Q:
TR = P × Q = (200 – 0.04Q)Q = 200Q – 0.04Q^2
Marginal revenue (MR), the derivative of TR with respect to Q:
MR = d(TR)/dQ = 200 – 0.08Q
Marginal cost (MC), the derivative of total cost:
MC = d(TC)/dQ = 20 + 0.04Q
Profit-Maximization Condition and Equilibrium Calculation
Profit maximization occurs where MR = MC:
200 – 0.08Q = 20 + 0.04Q
Solving for Q:
200 – 20 = 0.08Q + 0.04Q
180 = 0.12Q
Q = 180 / 0.12 = 1500 units
Substituting Q back into the demand function to find the optimal price:
P = 200 – 0.04 × 1500 = 200 – 60 = $140
Graphical Illustration
Graphically, the demand curve slopes downward from P = 200 at Q = 0, while the MR curve lies below the demand curve with twice its slope, starting at the same intercept. The MC curve is upward-sloping, intersecting MR at Q = 1500, which determines the equilibrium output. The intersection points illustrate the firm’s optimal output and price (P = $140).
Economic Profit Calculation
Total revenue at equilibrium:
TR = P × Q = 140 × 1500 = $210,000
Total cost at equilibrium:
TC = 1500 + 20(1500) + 0.02(1500)^2 = 1500 + 30,000 + 45,000 = $76,500
Economic profit:
π = TR – TC = 210,000 – 76,500 = $133,500
This significant profit indicates that Robert’s company is likely to gain a competitive advantage initially. However, in the long term, the free entry characteristic of monopolistic competition will attract new entrants, reducing the company's profits, potentially to zero as more firms enter and market supply increases (Pindyck & Rubinfeld, 2018).
Conclusion
The profit-maximizing output for Robert’s New Way Vacuum Cleaner Company is approximately 1500 units at a price of $140 per vacuum cleaner. The expected economic profit in the first year is substantial, around $133,500, signaling strong short-term profitability. Nevertheless, in a monopolistically competitive market, the profit level tends to diminish over subsequent years due to the entry of new firms, which erodes the firm’s market share and pushes profits toward normal levels in the long run. Maintaining competitive advantage will require innovation, branding, or cost efficiencies.
References
- Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach (9th ed.). W. W. Norton & Company.
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