# Optimal Price and Quantity Calculations

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A monopoly is considering selling several units of a homogeneous product as a single package. A typical consumer's demand for the product is Qd = 130 - 0.25P, and the marginal cost of production is $160

As a manager of a chain of movie theaters that are monopolies in their respective markets, you have noticed much higher demand on weekends than during the week. You therefore conducted a study that has revealed two different demand curves at your movie theaters. On weekends, the inverse demand function is P = 20 - 0.001Q; on weekdays, it is P = 15 - 0.002Q. You acquire legal rights from movie producers to show their films at a cost of $25,000 per movie, plus a $2.50 "royalty" for each moviegoer entering your theaters (the average moviegoer in your market watches a movie only once).

BAA is a private company that operates some of the largest airports in the United Kingdom, including Heathrow and Gatwick. Suppose that BAA recently commissioned your consulting team to prepare a report on traffic congestion at Heathrow. Your report indicates that Heathrow is more likely to experience significant congestion between July and September than any other time of the year.

Based on your estimates, demand is Q1d = 600 - 0.25P, where Q1d is quantity demanded for runway time slots between July and September. Demand during the remaining nine months of the year is Q2d = 220 - 0.1P, where Q2d is quantity demanded for runway time slots.

The additional cost BAA incurs each time one of the 80 different airlines utilizes the runway is £1,100 provided 80 or fewer airplanes use the runway on a given day. When more than 80 airplanes use Heathrow's runways, the additional cost incurred by BAA is £6 billion (the cost of building an additional runway and terminal). BAA currently charges airlines a uniform fee of £1,712.50 each time the runway is utilized.

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#### Solution Preview

Question 1:

First, set the demand function in the form of the inverse demand function to later generate the MR = MC condition to find optimal Q* and optimal P*.

Qd = 130 - 0.25P

P = 520 - 4Q

Then, find marginal revenue (MR) by deriving total revenue with respect to Q.

TR = P*Q

TR = 520Q - 4Q^2

MR = 520 - 8Q

Then, equate MR to MC to find P* and Q*.

MR = MC

520 - 8Q = 160

8Q = 520 - 160

8Q = 360

Q* = 45

i) Therefore, the optimal number of units is 45.

Input Q* in the inverse demand function to find P* optimal price.

P = 520 - 4Q

P = 520 - 4 (45)

P* = 340

ii) Therefore, the firm should charge $340 for this package.

Question 2:

First, the total cost function must be calculated before calculating the MR=MC condition, hence optimal P* and Q*, for each demand curve (demand ...

#### Solution Summary

This solution looks at three example problems and solves for their optimal price and optimal quantity of units, based on the competitive environment. Conditions such as MR = MC are applied in these cases.

Pricing Calculations for Optimal Prices

CASE 4: Pricing Light Bulb Market

Case No. Chapters Case Title

4 8 Pricing: Acme Lamp Company

You are the marketing manager for Acme Lamp Company. Acme specializes in the manufacture of lamps (light bulbs) for industrial applications. You are in charge of launching Acme's new LED-12 light emitting diode (LED) lamp. The LED-12 uses an array of 12 high-intensity LEDs to replace a standard medium-base incandescent lamp. As part of the launch plan, you must select a price. You have the following data:

Attribute Data Description

Investment $20,000 Money invested to develop product

Fixed Cost $10,000 Overhead costs not changing with quantity produced

Variable Cost $10 Labor and material costs to produce each unit

Unit Sales 5,000/ year Quantity of units forecast to sell at $20 per unit

Unit Sales, Max 10,000/ year Constraint on production; Maximum production quantity

% Markup 20% Desired return on sales

Target ROI 20% Target return on investment for new projects

LED-12: Life 24 months Long life due to rugged LED design

Existing lamps: Price $1 Price of existing lamps: Incandescent and CFL

Existing lamps: Life 3 months Shortened life due to severe conditions in industrial plant

Existing lamps: Labor $20/ lamp to replace Labor cost to replace existing lamp

Price elasticity 1 % change in demand given a % change in price

1. Calculate the target price using Markup/ Cost-Plus pricing.

Pricing Calculations Results

Unit Cost

Markup Price

2. Calculate the target price using Target Return pricing.

Pricing Calculations Results

Unit Cost

Target-Return Price

3. Calculate the target price using Value-In-Use pricing. Assume industrial plant uses 100 lamps.

Pricing Calculations Results

Current cost

Value In Use Price

4. Calculate the target price using the Optimal Price Analysis: http://www.ccdconsultants.com/calculators/optimal-price-analysis.html

Pricing Calculations Results

Optimal Price