Chapter 10, Problem 8E

a)

To determine

To Calculate: The level of output that should be produced to maximize short-run profits.

Answer to Problem 8E

Profit maximization output is $44.8806$

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

  $P=1760-12Q$

  $\text{Total Revenue(TR)=P}\times \text{Q}$

  $\text{=}\left(1760-12Q\right)Q$

  $=1760Q-12Q^2$

Total revenue function is differentiated to know the Marginal Revenue (MR)

  $MR=\frac{d}{dQ}\left(1760Q-12Q^2\right)\mathrm{.....}\text{(1)}$

  $\text{=1760}-\text{24Q}$

Total Cost (TC)

  $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

On differentiating the total cost function

  $MC=\frac{d}{dQ}\left(\frac{1}{3}{Q}^3-15Q^2+5Q+24,000\right)\mathrm{.....}\text{(2)}$

  ${\text{=Q}}^{\text{2}}-30\text{Q+5}$

Following is the condition for profit maximization

  $\text{MR=MC}$

  $\text{1,760}-{\text{24Q=Q}}^{\text{2}}-30\text{Q+5}$

  $1,760-5={\text{Q}}^{\text{2}}-30\text{Q}+\text{24Q}$

  $1,745={\text{Q}}^{\text{2}}-6Q$

  ${\text{Q}}^{\text{2}}-6Q-1,745$

On solving the quadratic equation

  $Q=\frac{-b+\sqrt{b^2-4ac}}{2a}$

  $=\frac{6+\sqrt{36+6980}}{2}$

  $=44.8806$

Economics Concept Introduction

Introduction: Suppose if the firm in the competitive market try to increase the profits. If the firm is in short-run, there is more possible to the firm’s economic profits will be positive, negative or zero. When the firm on the perfectly competitive market might be generate a profit for the short-run.

b)

To determine

To Calculate: The kind of price should be charged.

Answer to Problem 8E

Price to be charged is 1,221.44

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

Substituting the value of Q in demand function:

  $P=1760-12Q$

  $=1760-12\left(44.8806\right)$

  $=1760-538.56$

  $=1,221.44$

Economics Concept Introduction

Introduction: One of the main object in the economic condition is Price. Price is the most important value for the goods and services when transaction occurs. Price is any kind of specific good or service through the relationship between the demand and supply.

c)

To determine

To Compute: The total profits at this given price-output level

Answer to Problem 8E

Total Profit at this given price-output level is $30,674.42

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

Total Revenue $=P\times Q$

  $=1221.44\times 44.88$

  $=54818.22$

Substituting the value of Q in total cost function:

  $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

  $=\frac{1}{3}{\left(44.88\right)}^3-15{\left(44.88\right)}^2+5\left(44.88\right)+24,000$

  $=30132.6-30213.2+224.4+24,000$

  $=2,4143.8$

  $\text{Total Profit}\left(\pi \right)=TR-TC$

  $=54,818.22-24,143.8$

  $=\$30,674.42$

Economics Concept Introduction

Introduction: One of the common measure of the success is the total profit. This is always equal to the net revenue will remain once all the costs have been deducted. Total profit the base income of the business through this tax will be computed and determines the payment to dividend for the shareholders.

d)

To determine

To Compute: The point price elasticity of demand at the profit-maximizing level of output.

Answer to Problem 8E

The point price elasticity of demand at the profit-maximizing level of output is $-2.267$

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

Following is the demand function given:

  $P=1760-12Q$

  $12Q=1,760-P$

  $Q=\frac{1,760}{12}-\frac{1}{12}P$

Point of Elasticity of Demand $=\frac{dQ}{dP}\times \frac{P}{Q}$

Differentiating the demand function with respect to price

  $\frac{dQ}{dP}=-\frac{1}{12}$

Substituting the values in the formula given

Point of Elasticity of Demand $=\frac{dQ}{dP}\times \frac{P}{Q}$

  $=-\frac{1}{12}\times \frac{1,221.44}{44.88}$

  $=-2.267$

Economics Concept Introduction

Introduction: Profit-Maximizing is one of the short-run and also long-run process. In that, a firm will describe the level of the price, input and output range to yield the best higher profit. Suppose if the firm need to reach the highest equilibrium, there is having some changes in the level of the output to maximize the product.

e)

To determine

To describe: The level of fixed costs is the firm experiencing on its bed production.

Answer to Problem 8E

The level of fixed costs does not vary with the firm experiencing on its bed production.

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

The poster Company having the demand for its canopy bed to the given estimated demand function and also the cost analysis department having the given estimated total cost function for the poster bed.

Based on the given estimated demand function and estimated total cost function, the fixed cost is 24,000.

Hence, the fixed cost is 24,000 that does not vary with change in the level of output.

Economics Concept Introduction

Introduction: Fixed cost is define as the cost that will not change to the increase or decrease of the quantity of the goods and services to be produce or sale. Fixed cost is the main expenses of the firm that will be paid by the company.

f)

To determine

To describe: The impact of a $5,000 increase in the level of fixed costs on the price charged, output produced, and profit generated.

Answer to Problem 8E

The impact of a $5,000 increase in the level of fixed costs on the price and output will not be affected.

Explanation of Solution

Given:

Estimated demand function for the bed $P=1760-12Q$

Total cost function for the poster bed $TC=\frac{1}{3}{Q}^3-15Q^2+5Q+24,000$

Explain:

The poster Company having the demand for its canopy bed to the given estimated demand function and also the cost analysis department having the given estimated total cost function for the poster bed.

Price and output would not be affected by the increase in the fixed cost. Fixed cost does not increase with change in output. But Profit would be declined by the $5,000.

Economics Concept Introduction

Introduction: Fixed cost is the opportunity cost occurred on the short-run production.it will not base by the quantity of the product. Insurance premium, payment of the loan are the examples of the fixed cost.