Download Section 3.4 Marginal functions in Economics.

Section 3.4
Marginal functions in Economics.
The Cost function C(x) is defined to be the cost of making ‘x’
units of the quantity.
The Revenue function R(x) is defined to be the revenue got by
selling ‘x’ units of the quantity. If the price of a single unit
is p, then R(x)=x.p
The Profit function P(x) is defined to be the profit made in
making and selling ‘x’ units of a quantity.
Q. Let cost of making ‘x’ units of Tektronix 2465 oscilloscope is
C (x) = 2000x + 100 . Let the demand equation for the business model
is p = 2010 − x . Calculate the cost of making x units of the
oscilloscope, the revenue got from selling ‘x’ units of tek 2465.
Finally find the profit made by tektronix when it makes and sells
‘x’ units of tek 2465.
Soln. a. C(x)= 2000x+100
b. R(x)= x.p = x(2010-x)
c. P(x)=R(x)-C(x) = 2010x − x2 − (2000x + 100) = 10x − x2 − 100 .
Marginal Cost is the the cost of making one additional item given
that the production of ‘x’ units has already been made.
Q. Find the marginal cost of making the 201st oscilloscope.
Soln. Cost of making the 201st oscilloscope = C(201)-C(200)=2000.
The marginal revenue is the additional revenue got from selling
one additional unit, given that ‘x’ units have already been sold.
Q. Find the marginal revenue got by selling the 201st
oscilloscope.
Soln. R(201)-R(200) = ?
The marginal profit is the additional profit made in selling one
additional unit of the item.
Q. Find the marginal profit when x=200.
Soln. P(201)-P(200)
The Marginal cost ​function ​defined to be C’(x). It is an
approximation to the marginal cost for a given value of ‘x’.
Marginal revenue ​function​ := R’(x)
Marginal Profit ​function​ := P’(x)
Q. Recall the cost function for making ‘x’ units of tek 2465 was
C(x)=2000x+100, the revenue function was R(x)= 2010x − x2 , and the
profit function P(x) = 10x − x2 − 100 . Find the marginal cost
function, marginal revenue function, and the marginal profit
function for the business model.
Soln. Marginal cost function = 2000
Marginal revenue function = 2010-2x
Marginal profit function = 10-2x
C(x)
x
Revenue function is defined to be R(x) := R(x)
x
P(x)
Profit function is defined to be P (x) := x
Average cost function is defined to be C(x) :=
Average
Average
Q. Recall the cost function for making ‘x’ units of tek 2465 was
C(x)=2000x+100, the revenue function was R(x)= 2010x − x2 , and the
profit function P(x) = 10x − x2 − 100 . Find the average cost
function, average revenue function, and the average profit
function for the business model.
Soln. C(x) =
R(x) =
P (x) =
2000x+100
= 2000 + 100
x
x
2010x−x2
= 2010 − x
x
10x−x2−100
= 10 − x − 100
x
x
The derivative of the average cost function is called the
marginal average cost function. (C(x))′ = [ C(x)
x ]′
Q. Find the marginal average cost function for the business model
in the previous example.
Soln. The marginal average cost function is
100
C (x)′ = [2000 + 100
x ]′ = − x2 .
If the quantity and the price are related by x = f (p) , and f (p) is
a differentiable function of p, then the elasticity of demand of
′(p)
the business model is defined to be E (p) = − p . f
f(p) . The demand is
to be elastic if E(p) > 1, it is said to be unitary if E(p) = 1,
and it is said to be inelastic if E(p) < 1.
Q. Recall that the demand equation for the tek 2465 oscilloscope
is p = 2010 − x . Find the elasticity of demand function E(p). Is
the demand elastic when p = 1900 .
Soln. x = 2010 - p. So f(p) = 2010 - p.
f’(p)= -1.
′(p)
p (−1)
So E (p) = − p . f f(p) = − 2010 − p
1900(−1)
E (1900) = − 2010−1900
= 1900
110 > 1 .
So the demand is elastic when p = 1900.