Question

Consider the following cost and benefit functions:

C(X, Y) = 15 (10 X + 2 X²)   with   MC (X) = 15(10 + 4X)                                          

B(X, Y) = 5 (80 X – 2 X² + 40 Y – Y² + 2 X Y) with MB (X) = 5 (80 – 4 X + 2 Y)

For Y = 5, derive the benefit function (hint: simply replace y by its value)

Find the points at which TB (X) = TC (X) if any

Find the value of x for which the net benefit is maximized. (MC = MB).

Calculate the values of the benefit, cost, and net benefit for this value of x.

Answer 1

The following are the cost and benifit functions,

• Cost is

C(X) = 15 (10 X + 2 X²)

Where, Marginal Cost is

MC (X) = 15(10 + 4X)

• Benifit is

B(X, Y) = 5 (80 X – 2 X² + 40 Y – Y² + 2 X Y)

Where, Marginal Benifit is

MB (X) = 5 (80 – 4 X + 2 Y)

Now, let us answer the following questions one by one.

✓ When Y = 5, we get from the Benifit function,

B(X, Y) = 5 (80 X – 2 X² + 40 Y – Y² + 2 X Y)

or, B(X) = 5 (80 X – 2 X² + 40×5 – (5)² + 2×5×X)

or, B(X) = 5.(80X - 2.X^​​​2 + 10.X + 200 - 25)

or, B(X) = 450.X - 10.X^​​​2 + 875

The above is the benifit function for Y = 5.

✓ Now, we have to find where,

TB(X) = TC(X)

or, 450.X - 10.X^​​​2 + 875 = 150.X + 30.X^​​​2

or, 40.X^​​​2 - 300.X - 875 = 0

or, 8.X² - 60.X - 175 = 0

Solving the equation, we get

X = 9.75

We get TB(X) = TC(X) at X = 9.75.

✓ Hence, net benifit is maximized where,

MB(X) = MC(X)

or, 400 - 20.X + 10.Y = 150 + 60.X

Putting Y = 5 we get

400 - 20.X + 10×5 = 150 + 60.X

or, 80.X = 300

or, X = 3.75

Putting X in B(X) and C(X) we get

B(X) = 450×3.75 - 10.(3.75)² + 875

or, B(X) = 2421.875

And, C(X) = 150×3.75+ 30.(3.75)²

or, C(X) = 984.375

Hence net benifit is

NB(X) = B(X) - C(X) = 2421.875 - 984.375

or, NB(X) = 1437.5

• Benifit = 2421.875

• Cost = 984.375

• Net Benifit = 1437.5

Hope the solutions are clear to you my friend.