Cost and profit functions

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Paul Verheyen

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Paul Verheyen 4 years, 7 months ago

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Paul Verheyen 5 years, 5 months ago

Published this.

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Paul Verheyen 5 years, 5 months ago

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Paul Verheyen 5 years, 5 months ago

Created this as a copy of Kosten- en winstfuncties versie 1.

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Question statement

Consider a firm with cost function $C: \mathbb{R}^+ \to \mathbb{R} : q \mapsto C(q) = \simplify[]{{a}q^3-{b}q^2+{c}*q+{d}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
s	integer	6
t	integer	3
grensq	integer	12
grensy	number	272
b	integer	54
c	number	327
d	integer	20
v	integer	3
prijs	number	84
q1	integer	3
q2	integer	9
startupprijs	number	84
bovenaan	number	0
onderaan	number	-364
topCy	number	1352
qstup	rational	9

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Determine the marginal cost function:

$ MC(q) = $ Unnamed gap

Advice

a) en b) From the expression \[ C(q) = \simplify[]{{a}*q^3-{b}*q^2+{c}*q+{d}}\] we get the marginal cost

\[ MC(q) = \frac {dC}{dq} (q) =\simplify[]{{3*a}*q^2-{2*b}*q+{c}}\] and the average cost

\[ AC(q) = \frac {C(q)}{q} = \simplify[]{{a}*q^2-{b}*q+{c}+{d}/q} .\]

c) The profit equals the difference of the revenues and the costs:

\[ W(q) = R(q) - C(q) = \var{prijs} \cdot q - \left( \simplify[]{{a}*q^3-{b}*q^2+{c}*q+{d}} \right) = \simplify[]{{-a}*q^3+{b}*q^2+{prijs-c}*q-{d}}\]

d) In order to maximize the profit, the derivative of $ W $ with respect to $ q $ must be equal to 0:

\[ \frac {dW}{dq} (q) = \simplify[]{{-3*a}*q^2+{2*b}*q+{prijs-c}} . \]

This first order derivative equals 0 at the values $ q_1 = \var{q1} $ and $ q_2 = \var{q2} $.

Since

\[ \frac {d^2 W}{dq^2} (q) = \simplify[]{{-6*a}*q+{2*b}} \]

it follows

\[ \frac {d^2 W}{dq^2} (\var{q1}) = \var{-6*a*q1+2*b} > 0 \]

and

\[ \frac {d^2 W}{dq^2} (\var{q2}) = \var{-6*a*q2+2*b} < 0 . \]

This implies that the pofit is maximal at $ q = \var{q2} $.

e) $ W( \var{q2}) = \var{-a*q2^3+b*q2^2+(prijs-c)*q2-d}$

Graph of the profit function:

{ShowWinst()}

f) The startup/shutdown price is the minimal value of the average variable cost function:

\[ p_{startup/shutdown} = min \, AVC ,\]

where $ AVC(q) $ equals the total variable cost divided by $ q $:

\[ AVC(q) = \frac {TVC(q)} {q} .\]

In this problem

\[ AVC(q) = \simplify[]{{a}*q^2-{b}*q+{c}} \]

has as derivative

\[ \frac {dAVC}{dq} (q) = \simplify[]{{2*a}*q-{b}} .\]

This derivative is equal to 0 at

\[ q = \simplify[all,FractionNumbers]{{b}/{2*a}} .\]

This value indeed leads to a minimum of $ AVC $, with

\[ AVC \left( \simplify[all,FractionNumbers]{{b}/{2*a}} \right) = \simplify[all,FractionNumbers]{{a*(b/(2*a))^2-b*(b/(2*a))+c}} ,\]

such that rounded up to 2 decimals $ p_{startup/shutdown} $ is given by

\[ \simplify[]{{a*(b/(2*a))^2-b*(b/(2*a))+c}} .\]

Graphs of MC , AC and AVC :

{ShowKosten()}

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Paul Verheyen 4 years, 7 months ago

Paul Verheyen 5 years, 5 months ago

Paul Verheyen 5 years, 5 months ago

Paul Verheyen 5 years, 5 months ago

Consider a firm with cost function $C: \mathbb{R}^+ \to \mathbb{R} : q \mapsto C(q) = \simplify[]{{a}q^3-{b}q^2+{c}*q+{d}}$

Error in variable testing condition

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Generated value: `integer`

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Cost and profit functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Paul Verheyen 4 years, 7 months ago

Paul Verheyen 5 years, 5 months ago

Paul Verheyen 5 years, 5 months ago

Paul Verheyen 5 years, 5 months ago

Consider a firm with cost function C:R+→R:q↦C(q)={{a}q3−{b}q2+{c}q+{d}} C: \mathbb{R}^+ \to \mathbb{R} : q \mapsto C(q) = \simplify[]{{a}*q^3-{b}*q^2+{c}*q+{d}}

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Consider a firm with cost function $C: \mathbb{R}^+ \to \mathbb{R} : q \mapsto C(q) = \simplify[]{{a}q^3-{b}q^2+{c}*q+{d}}$

Generated value: `integer`