Question

Q1) A company produces a special new type of TV. The company has fixed costs of ​$471,000​, and it costs ​$1100 to produce each TV. The company projects that if it charges a price of​$2500 for the​ TV, it will be able to sell 750 TVs. If the company wants to sell 800 ​TVs, however, it must lower the price to ​$2200. Assume a linear demand.

What is the marginal profit if 250 TVs are produced?

It is ​$_____ per item.

​(Round answer to nearest​ dollar.)

Q2) A company produces a special new type of TV. The company has fixed costs of ​$450,000​, and it costs ​$1000 to produce each TV. The company projects that if it charges a price of $2300 for the​ TV, it will be able to sell 800 TVs. If the company wants to sell 850 ​TVs, however, it must lower the price to $2000. Assume a linear demand.

What are the​ company's profits if marginal profit is​ $0?

The profit will ​$_____.

​(Round answer to nearest​ cent.)

Please help solve these two questions, thanks!

Answer 1

Solution to Q1

First step here is to find the linear demand function which is in the form y = mx + c

Here Price is y and m is the slope of demand, x is quantity demanded and c is a constant.

Find the linear demand function, first calculate m with the formula: m = (y₂  - y₁ )/(x₂ - x₁ )

Consider the demand pairs,(2500,750) and (2200,800) Substitute the values to get (800-750)/(2200-2500) which equals -0.16666667. Hence m is -0.166667

To find c, subsitute m in one of the pairs. 750 = -0.166667 (2500) +c solving this we get c = 1166.67

Hence the linear demand function is P = -0.166667Q + c

At level of 250 TV sets, Price will be as follows: 250 = -0,166667(P) + 1166.67 = Approx. 5500

To find the marginal profit we need to find the Marginal cost and Marginal revenue. Marginal cost is given as 1100. Marginal revenue for 250th unit can be found by subtracting total revenue at 249 unit and 250 unit. Using demand function above, for 249 units (249 = -0.166667p+ 1166.67) price is 5506. Total revenue at 250 units is 1375000 (250*5500) and for 249 units is 1370994. Difference is 4006 which is Marginal revenue.

Marginal profit is the difference between Marginal revenue and Marginal cost. Hence Marginal profit is 4006 - 1100 = 2906

Solution to Q2

First step here is to find the linear demand function which is in the form y = mx + c

Here Price is y and m is the slope of demand, x is quantity demanded and c is a constant.

Find the linear demand function, first calculate m with the formula: m = (y2  - y1 )/(x2 - x1 )

Consider the demand pairs,(2300,800) and (2000,850) Substitute the values to get (850 -800)/(2300-2000) which equals -0.16666667. Hence m is -0.166667

To find c, subsitute m in one of the pairs. 850 = -0.166667 (2000) +c solving this we get c = 1183.3333

Hence the linear demand function is P = -0.166667Q + 1183.3333

To find the company's profit when marginal profit is zero, we need to find the Marginal cost and Marginal revenue. Marginal cost is given as 1000.

The demand function is the Average Revenue function, Total revenue can be found by multiplying q to AR.

TR = -0.16666672Q²+ 1183.3333Q

MR = -0.333334Q + 1183.33 This is found by taking the differential of TR.

MC = 1000, as given.

To find the level of output where Marginal profit is zero, we have to equate MC and MR

1000 = -0.333334Q+ 1183.33

Q = 183.33/0.333334

Q = 550

At 550 units, Total revenue will be 701249.98 when substituted in equation for TR

Similarly, Total cost at 550 units will be 450,000(Fixed cost) + (550*1000) Variable cost = 1000000

Total profits will be -298750.02 Hence this will be a loss to the firm