Question

Suppose a firm faces the following demand curve: q(p) = 10000 - 800p

Also the varaible cost per unit is $5 and the fixed cost is $10000

1. What price will the firm charge

2. How many units will they produce at that price

3. What is the breakeven prices and quantities?

Answer 1

Facts:

Demand Curve : q(p) = 10000-800p

Variable cost per unit = $5

Fixed Cost = $10000

1. Answer to Sub part 1 - What price the firm will charge:

Profit = Price * Quantity - Cost

= p * q - (5q+10000)

= p(10000-800p)- [5(10000-800p)+10000]

= 10000p-800p² -50000+4000p-10000

= 14000p-800p²-60000

Now differentiate the above profit functionand equate to zero, to get the price the firm should charge to maximize the profit

14000-1600p = 0

1600p= 14000

p= 8.75

2.Answer to Subpart-2 - How many units the firm should produce at that price?

Since demand function, q = 10000-800p

p=8.75 implies, q = 10000- 800(8.75)

q = 3000   

Answer to Question 3 - Breakeven prices and Quantities

The point of sales where the contribution is able to recover fixed cost is Break even point. It means at that point contribution should be equal to fixed cost.

i.e., (p-5)q = 10000

(p-5) (10000-800p) =10000

(p-5) (100-8p) = 100

100p-8p²-500+40p = 100

8p²-140p+600 = 0

2p²-35p+150 = 0

2p²-15p-20p+150 = 0

p (2p-15)-10(2p-15) = 0

(2p-15) (p-10) = 0

Therefore, p-10 = 0 or 2p-15 = 0

which implies p = 10 or p = 7.5 (Breakeven Prices)

Breakeven Quantities

q = 10000-800p implies, q = 10000-800(10) or 10000-800(7.5)

= 10000-8000 or 10000-6000

= 2000 or 4000 units

Final Answer:

Part-1 Price = $8.75

Part-2 Units to produce = 3000 units

Part-3 Breakeven prices =$10 and $7.5

Breakenen quantities = 2000units and 4000units