Download A brief summary of Sec 3.4

A brief summary of Sec 3.4
October 5, 2012
1
Now I give a brief summary of Sec 3.4: Some Financial Models.
Suppose that the cost function, or the total cost is given by C(x), which describes the
relationship between the number of units produced and the cost incurred. For example,
if the cost functions is C(x) = x2 + 1, then the cost when producing 1 unit is 2 dollars,
and the cost when producing 2 units is 5 dollars, and so on.
Now we examine the quantity C ′ (x). We know, by sec 3.7 (the differentials), the following relationship holds true:
C(x + 1) − C(x) = ∆C ≈ dC = C ′ (x)[(x + 1) − x] = C ′ (x)
|
{z
}
Def inition of dif f erential
us C ′ (x) gives the approximation of additional cost incurred when producing one
more unit (Note that we have on the le hand side: C(x + 1) − C(x)) provided that the
production is already on the level x. We call C ′ (x) the Marginal Cost Function whose
economic meaning is given above.
Take C(x) = x2 + 1 for example: e cost to produce 20 units is 401(that is , 202 + 1)
dollars while the cost to produce 21 units is 442 dollars. us the actual cost (the exact
value) in producing the 21st unit is
442 − 401 = 41 dollars
If we use the marginal cost function:
C ′ (x) = 2x
to approximation the actual cost, we have:
C ′ (20) = 2 · 20 = 40 dollars
2
which is fairly close to the real cost.
Our next concept is the Average Cost Function, which is de ned by
C̄(x) =
C(x)
x
It is clear that the average cost function gives the cost on each individual unit. Similarly,
we can de ne the Marginal Average Cost Function which is simply the derivative of
C̄(x), i.e.
[
C(x) ′
]
x
is the marginal average cost function. It gives an approximation of the average cost incurred when producing one more unit provided the production is already on the level x.
Take C(x) = x2 + 1 for example: the average cost is
C̄(x) =
C(x)
1
=x+
x
x
and the marginal average cost is
C̄ ′ (x) = 1 −
1
x2
Next, we know that if the relationship between the unit price and the number of units
produced is given by
p = p(x)
then the Revenue Function (i.e. the total income)is given by
R(x) = x · p(x)
and the Pro t Function is given by
P (x) = R(x) − C(x) = x · p(x) − C(x)
3
where C(x) is the cost function.
Similarly we can de ne the Marginal Revenue Function and Marginal Pro t Function
as the derivatives of R(x) and P (x), i.e.
R′ (x) = [xp(x)]′ = p(x) + xp′ (x)
and
P ′ (x) = R′ (x) − C ′ (x) = p(x) + xp′ (x) − C ′ (x)
are marginal revenue and marginal pro t, respectively.
erefore, when we talk about 'marginal', we mean the derivative. Please remember that
the marginal 'blablabla' is an Approximation of the actual 'blablabla' when producing
one more unit provided the production is on certain level.
For example, C(x) = x2 + 1 and p(x) = x3 + x2 where p(x) is the unit price function,
then we have:
R(x) = x · p(x) = x4 + x3
P (x) = R(x) − C(x) = x4 + x3 − x2 − 1
us the marginal revenue function is
R′ (x) = 4x3 + 3x2
while the marginal pro t function is
P ′ (x) = 4x3 + 3x2 − 2x
e above is all you need to know for Sec 3.4 and I hope that clarify things. You can
skip reading the subsection: Elasticity of Demand. ey are not in midterm. However,
you can read it for extra knowledge and, if possible, for fun.
4