Chapter 16, Problem 6E

A.

To determine

To Calculate: The profit maximizing level of price and output.

Answer to Problem 6E

The detailed solution with explanation is provided in the next section.

Explanation of Solution

  $\begin{array}{l}\text{Total Revenue = Price }\times \text{ Quantity}\\ \text{TR=P}\text{.Q}\\ \text{TR=(800-20Q)Q}\\ {\text{TR=800Q-20Q}}^2\\ MR=\frac{d}{dQ}(TR)\\ MR=800-40Q\end{array}$

  $\begin{array}{l}TC=300+500Q+10Q^2\\ MC=\frac{d}{dQ}(TC)\\ MC=500+20Q\end{array}$

Under monopoly, the profit maximization condition will be;

  $\begin{array}{l}MR=MC\\ MR=800-4Q\\ MC=500+20Q\\ 60Q=300\\ Q=5\end{array}$

Substituting this value of Q in demand equation, the price of,

  $\begin{array}{l}P=800-2Q\\ P=800-20(5)\\ P=800-100\\ P=700\end{array}$

Thus, the profit maximizing levels of price and output under monopoly are 700 and 5 respectively.

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

B.

To determine

To Calculate: The profits at the derived price and output levels in the above part.

Answer to Problem 6E

After necessary calculations, the profit is derived as 450.

Explanation of Solution

  $\begin{array}{l}\mathrm{Pr}ofit=TR-TC\\ TR=800Q-20Q^2\\ $ TR=800(5)-20{(5)}^2$TR=4000-500\\ TR=3500\\ TC=300+500Q+10Q^2\\ $ TC=300+500(5)+10{(5)}^2$TC=300+2500+250\\ TC=3050\\ \mathrm{Pr}ofit=3500-3050\\ \mathrm{Pr}ofit=450\end{array}$

Thus, the profit obtained at the given levels of price and output are 450.

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

C.

To determine

To Calculate: The profit maximizing levels of price and output based on the given conditions.

Answer to Problem 6E

After necessary calculations, the price and output levels in the competitive market are 620 and 6 respectively.

Explanation of Solution

  $\begin{array}{l}TR=P.Q\\ TR=620Q\\ MR=\frac{d}{dQ}(TR)\\ MR=620\\ TC=300+500Q+10Q^2\\ MC=\frac{d}{dQ}(TC)\\ MC=500+20Q\\ MC=500+20Q\\ 620=500+20Q\\ Q=6\end{array}$

Thus, the price and output in the competitive market is 620 and 6 respectively.

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

D.

To determine

To Calculate: Given the derived answer in the previous part, find out the total profits.

Answer to Problem 6E

After necessary calculations, the total profit under the competitive market is 450.

Explanation of Solution

  $\begin{array}{l}\mathrm{Pr}ofit=TR-TC\\ TR=620Q\\ TR=620(6)\\ TR=3720\\ TC=300+500Q+10Q^2\\ $ TC=300+500(6)+10{(6)}^2$TC=300+3000+360\\ TC=3660\\ \mathrm{Pr}ofit=3720-3660\\ \mathrm{Pr}ofit=60\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

E.

To determine

To Calculate:

Suppose that the monopoly situation as derived in earlier parts prevails, describe the profits of the industry.

Answer to Problem 6E

After necessary calculations, the total profit under the said monopoly after taking investments into consideration is 150.

Explanation of Solution

  $\text{Return on investment =}\frac{200000000\times 15}{100}=300million$

The Total Cost function is given as,

  $\begin{array}{l}TC=300+500Q+10Q^2\\ TC=300+300+500Q+10Q^2\\ TC=600+500Q+10Q^2\\ \text{With the derived quantity level of 5 for the monopoly,}\\ $ TC=600+500(5)+10{(5)}^2$TC=600+2500+250\\ TC=3350\\ TR=800Q-20Q^2\\ $ TR=800(5)-20{(5)}^2$TR=4000-500\\ TR=3500\\ \mathrm{Pr}ofit=TR-TC\\ \mathrm{Pr}ofit=3500-3350\\ \mathrm{Pr}ofit=150\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

F.

To determine

To Calculate:

Supposing the given condition, find out whether the industry is operating under equilibrium conditions, and explain with reasons.

Answer to Problem 6E

After necessary calculations and observations, it can be seen that it satisfies equilibrium condition.

Explanation of Solution

  $\begin{array}{l}\text{Marginal Cost function is,}\\ MC=500+20Q\\ MC=500+20(6)\\ MC=620\end{array}$

The equilibrium condition in perfect competition is P=MR=MC, equaling to $620. And, hence, it satisfies the equilibrium condition.

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

G.

To determine

To Calculate:

On the given monopoly situation, find out the impact of this change on price, output and total profits.

Answer to Problem 6E

With the given pollution control standards, since the total cost increases, the total profit will reduce from $450 to $80 million, and total quantity from 5 to 4 millions of pounds of plastic. This effect has increased the price from $700 to $720 million.

Explanation of Solution

  $\begin{array}{l}\text{Marginal Cost function is,}\\ MC=500+20Q\\ MC=500+20(6)\\ MC=620\end{array}$

The equilibrium condition in perfect competition is P=MR=MC, equaling to $620. And, hence, it satisfies the equilibrium condition.

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

H.

To determine

To Calculate:

The impact of the new standards on the firms in the region and in the rest of the industry.

Answer to Problem 6E

After necessary calculations, it can be seen that the profits distributed for the firms in the region will be 6.2 and for the rest of the industry will be 0.68.

Explanation of Solution

Price under competitive form will be, P=620.

  $\begin{array}{l}TR=P.Q\\ TR=620Q\\ MR=\frac{d}{dQ}(TR)\\ MR=620\\ TC=400+560Q+10Q^2\\ MC=\frac{d}{dQ}(TC)\\ MC=560+20Q\end{array}$

Under the competitive market, the profit maximization condition is MR=MC;

  $\begin{array}{l}MR=620\\ MC=560+20Q\\ 620=560+20Q\\ Q=3\end{array}$

Thus, in the competitive market, the profit maximizing levels of price and output are 620 and 3 respectively.

  $\begin{array}{l}\mathrm{Pr}ofit=TR-TC\\ TR=620Q\\ TR=620(3)\\ TR=1860\\ TC=400+560Q+10Q^2\\ $ TC=400+560(3)+10{(3)}^2$TC=400+1680+90\\ TC=2170\\ \mathrm{Pr}ofit=2170-1860\\ \mathrm{Pr}ofit=310\end{array}$

Thus, the profit will be distributed as,

  $Firm\sin theregion=\frac{310}{50}=6.2$

And, for the rest of the industry, the profit will be distributed as 0.68

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.