PROBLEMS ON TOTAL REVENUE MARGINAL REVENUE

Problem 1 :

The total cost function for a company is given by

C(x) = (3/4)x² - 7x + 27

Find the level of output for which MC = AC.

a)  8     b)   6      c)   9      d)  10

Solution :

By finding the derivative of cost function, we will get the marginal cost

By dividing the cost function by output, we get average cost.

C(x) = (3/4)x² - 7x + 27 is given

d(C(x)) = [(3/4)x² - 7x + 27]

= (3/4) 2x - 7(1) + 0

Marginal cost = (3/2)x - 7 -----(1)

Average cost = Cost function / output

=  ((3/4)x² - 7x + 27)/x

= (3/4)x - 7 + (27/x) -----(2)

(1) = (2)

(3/2)x - 7 = (3/4)x - 7 + (27/x)

(3/2)x - (3/4)x - 7 + 7 = 27/x

(6x - 3x)/4 = 27/x

3x/4 = 27/x

3x² = 27(4)

x² = 27(4)/3

x² = 9(4)

x = 6

So, the value of x is 6. Option b is correct.

Use the data, the demand function for a monopolist is given by x = 100 - 4p, where x is the number of product produced and old and p is the price per unit.

Problem 2 :

Find total revenue function

a)  25 - (x²/4)        b)  25 + (x²/4)       c)  25 - (x²/2)

d)  5x - (x²/4)

Solution :

To find the revenue function from the demand function, we have to multiply the number of units sold and price per unit.

x = 100 - 4p

x is number of units produced and p is the price per unit.

Let us solve for p, we get

4p = 100 - x

p = (100 - x)/4

= 25 - (x/4)

Revenue function = [25 - (x/4)] x

= 25x - (x²/4)

Problem 3 :

Find the average revenue function

a)  25 - x/6        b)  25 - x/4     c)  5 - x/4      d)  25 + x/4

Solution :

Average revenue = Total revenue / output

= [25x - (x²/4)] / x

= 25 - (x/4)

So, the answer is option b.

Problem 4 :

Find the marginal revenue function

a)  25 - (x/3)   b)  25 - (x/4)   c)  5 - (x/2)    d)  25 - (x/2)

Solution :

By finding the derivative of total revenue, we get marginal revenue.

Total revenue = 25x - (x²/4)

d(TR)  = d(25x - (x²/4))

= 25(1) - 2x/4

Marginal revenue = 25 - (x/2)

So, option d is correct.

Problem 5 :

The price and quantity at which MR = 0

a)  50, 12.5    b)  70, 12.5   c) 100, 12.5     d)  70, 10

Solution :

Marginal revenue = 25 - (x/2)

25 - (x/2) = 0

25 = x/2

x = 25(2)

x = 50

Number of quantity produced = 50

To find price per unit, we have to apply in the function x = 100 - 4p

50 = 100 - 4p

4p = 100 - 50

4p = 50

p = 50/4

p = 12.5

So, option a is correct.

Use the data, A firm knows that the demand function for one of its products is linear. It also knows that it can sell 1000 units when the price is $4 per unit and it can sell 1500 units when the price is 2 per unit.

Problem 6 :

Find the demand function.

a)  2000 - 250p     b)  2000 - 5p     c)  2000 + 5p

d)  2000 - 25p

Solution :

Since the given function is linear, the function will be in the form

y = mx + b ----(1)

(4, 1000) and (2, 1500)

m = (y₂ - y₁)/(x₂ - x₁)

= (1500 - 1000)/(2 - 4)

= 500/(-2)

= -250

Applying slope (m) = -250 in (1), we get

y = -250x + b

Applying any point here, we get

1000 = -250(4) + b

1000 = -1000 + b

b = 2000

y = -250x + 2000

y = 2000 - 250x

x = p

y = 2000 - 250p

So, option a is correct.

Problem 7 :

Find the total revenue function

a)  (8 - x²)/250     b)  (8x - x²) / 50     c)  (8x - x²)/250

a)  (8x - x²)/25

Solution :

Total function is the product of number of units sold by price per unit.

Revenue = demand x price pre unit

Demand function = y = 2000 - 250x

Dividing by 250, we get

y = (8 - x) / 250

Price per unit = x

y = (8 - x)x / 250

= (8x - x²) / 250

so, option c is correct.

Problem 8 :

Find the average revenue function

a) (8 - x) / 50        b)  (8 - x) / 25    c) (8 + x) / 250

d)  (8 - x)/ 250

Solution :

Average revenue = Total revenue / output

= [(8x - x²) / 250] / x

= (8 - x) / 250

So, option d is correct.

Problem 9 :

Find the marginal revenue function 

a)  8 - x/12    b)  8 - x/25     c) 8 - x/125

d)  8 + x/125

Solution :

By finding derivative of total revenue, we get marginal revenue.

Total revenue = (8x - x²) / 250

d(Total revenue) = d[(8x - x²) / 250]

= 8(1) - 2x/250

= 8 - x/125

So, option c is correct.