Consider the following cost and benefit functions: C(X, Y) = 150 X + 30 X2. MC (X) = 150 + 60 X B(X, Y) = 400 X – 10 X2 + 200 Y – 5 Y2 + 10 X Y MB (X) = 400 – 20 X + 10 Y . For Y = 5 • Derive the benefit function • Find the value of x for which the net benefit is maximized. • Calculate the values of the benefit, cost and net benefit for this value of x. • Graph the marginal benefit and marginal cost and show this equilibrium graphically. • Graph the benefit and cost functions and show the net benefit maximizing value for x.

Consider the following cost and benefit functions:

C(X, Y) = 150 X + 30 X2.
MC (X) = 150 + 60 X B(X, Y) = 400 X – 10 X2 + 200 Y – 5 Y2 + 10 X Y
MB (X) = 400 – 20 X + 10 Y .
For Y = 5

• Derive the benefit function
• Find the value of x for which the net benefit is maximized.
• Calculate the values of the benefit, cost and net benefit for this value of x.

Benefit Function: In order to find out the benefit function ,we must integrate our Marginal benefit function.

MB(X) = 400 – 20X +10Y

MB(X)=400-20X +10*5 =400-20X +50 = 450 -20X.

Now integrate both the sides

Total benefit = 450X -20 (X^2 /2 )

T.B(x) = 450-10X^2………….This is our total benfit function.

2. Now come to Net benefit.

Net benefit = Total benefit – Total Cost

NB =450X -10X^2 – 150 X – 30 X^2.

In order to find out X that maximizes our net benefiT ,we need to differenctiate NB with respect to X.So let us differentiate it.

$\delta$NB/$\delta$X = 450 -20X -150 – 60X =0

=300-80X=0

X =300/80

X= 3.75.

Other way to find out Maximum value of X is to Equate MB(X)=MC(X) and solve for X.

3. Benefit = 450-10X^2. = 450 – 10 *3.75^2 =450-140.625 =309.375

Cost = 150X +30X^2 = 150 + 30 *3.75^2=571.875

Net benefit = 450X -10X^2 – 150 X – 30 X^2 = 450*3.75-10*3.75^2 -150*3.75 -30*3.75^2

=1687.5-140.625-562.5-421.875 =562.5

Asked on February 14, 2018 in