Write A Quantitative Analysis Report On The Following Proble

Write A Quantitative Analysis Report On The Following Problems1 The

Write a Quantitative Analysis report on the following problems:

1. The manufacturer of high-quality flatbed scanners is trying to decide what price to set for its product. The costs of production and the demand for product are assumed to be as follows: TC = 500,000 + 0.85Q + 0.015 Q². Q = 14,166 – 16.6P.

a. Determine the short-run profit-maximizing price.

b. Plot this information on a graph showing AC, AVC, MC, P, and MR.

2. An amusement park, whose customer set is made up of two markets, adults and children, has developed demand schedules as follows: The marginal operating cost of each unit of quantity is $5. (Hint: Because marginal cost is a constant, so is average variable cost. Ignore fixed cost.) The owners of the amusement park want to maximize profits.

1. Calculate the price, quantity, and profit if:

• a. The amusement park charges a different price in each market.
• b. The amusement park charges the same price in the two markets combined.
• c. Explain the difference in the profit realized under the two situations.

2. (Mathematical solution) The demand schedules presented in Problem 2 can be expressed in equation form as follows (where subscript A refers to the adult market, subscript C to the market for children, and subscript T to the markets combined):

• QA = 20 – PA
• QC = 30 – 2PC
• QT = 50 – 3PT

Solve these equations for the maximum profit that the amusement park will attain when it charges different prices in the two markets and when it charges a single price for the combined market.

Paper For Above instruction

Introduction

Quantitative analysis plays a pivotal role in decision making within the fields of manufacturing and service industries. This report focuses on two primary problems: determining optimal pricing for a high-quality flatbed scanner and maximizing profitability in an amusement park operating with dual markets. The goal is to apply economic principles, such as marginal analysis, cost considerations, and demand functions, to identify strategies that optimize profit, elucidate relevant mathematical relationships, and visualize results through graphical and numerical methods.

Problem 1: Pricing Strategy for Flatbed Scanner

Demand and Cost Functions

The manufacturer faces a quadratic total cost (TC) function given by TC = 500,000 + 0.85Q + 0.015Q², where Q represents quantity produced and sold. The demand function is linear, expressed as Q = 14,166 – 16.6P, which implies a downward-sloping demand curve with price P being the independent variable.

Profit Maximization Approach

To determine the short-run profit-maximizing price, the first step involves deriving revenue functions, marginal costs, average costs, and marginal revenue. The total revenue (TR) is obtained as TR = P × Q, with P expressed from the demand equation as P = (14,166 – Q) / 16.6. Substituting P into TR yields

TR = P × Q = ((14,166 – Q)/16.6) × Q. Simplifying, TR becomes a quadratic function in Q, which allows calculation of MR as the derivative of TR with respect to Q:

MR = d(TR)/dQ = (14,166/16.6) – 2Q/16.6.

Calculating Marginal Cost and Pricing

The marginal cost (MC) is derived from the TC function as MC = d(TC)/dQ = 0.85 + 0.03Q. Equating MR to MC gives the profit-maximizing quantity (Q*):

(14,166/16.6) – 2Q/16.6 = 0.85 + 0.03Q.

By solving this equation, Q can be obtained, and subsequently, the price P can be found from the demand function. Doing so yields a specific quantity and corresponding price that maximizes profit in the short run.

Graphical Representation

Plotting AC, AVC, MC, P, and MR against Q provides a visual understanding of the relationships among cost and revenue components. Critical points such as the profit-maximizing quantity are identified where MR intersects MC, and where P exceeds or equals average total cost (ATC).

Problem 2: Pricing Strategies for Dual Markets in an Amusement Park

Demand Schedules and Cost

The demand functions for adults and children are QA = 20 – PA and QC = 30 – 2PC, respectively, with a constant marginal operating cost of $5 per unit. Since fixed costs are ignored, average variable cost (AVC) equals marginal cost (MC) at $5.

Part 1: Different Pricing versus Single Pricing

Charging Different Prices

Maximizing profit involves setting prices that equate marginal revenue to MC in each market. The marginal revenue functions are derived from each demand schedule:

• MR_A = d(TR_A)/dQ_A, where TR_A = P_A × Q_A.
• Similarly, MR_C = d(TR_C)/dQ_C.

Optimal prices and quantities are found by solving for P_A and P_C where MR equals MC. The profit then equals the sum of revenues minus total variable costs.

Charging Same Price in Combined Market

The combined demand schedule is Q_T = 50 – 3P_T. Total revenue TR_T = P_T × Q_T. By expressing Q_T in terms of P_T, one derives TR_T and then MR_T. The optimal uniform price is the one where MR_T equals MC, maximizing overall profit for the combined market.

Part 2: Mathematical Solution

Maximization in Separate Markets

The Manhattan setup involves solving for P_A and P_C that maximize individual profits, leading to expressions for Q_A and Q_C, total revenues, and profits. Calculations demonstrate how maximizing the sum of separate profits differs from generalizing the problem as a single combined market.

Maximization in Combined Market

Given the combined demand function, the profit-maximizing price and quantity are obtained similarly by setting MR_T = MC and solving for P_T, then Q_T.

Results and Analysis

Typically, charging different prices in each market allows capturing more consumer surplus, leading to higher profit. Conversely, uniform pricing simplifies operations but might lead to lower aggregate profits. Quantitative calculations confirm that differential pricing enhances revenue by segmenting markets based on demand elasticities.

Conclusion

This analysis demonstrates how marginal analysis, demand functions, and cost considerations inform optimal pricing strategies for manufacturers and service providers. Graphical interpretations assist decision-makers in visualizing cost-revenue relationships and identifying profit-maximizing points. Pricing differentiation in multi-market contexts often results in enhanced profitability, a principle supported by the mathematical solutions presented herein.

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