Optimisation: Find quantity to maximise profits

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From users who are members of MASH Bath: Question Bank :

Ben McGovern

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2 years ago

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Ed Southwood 2 months, 4 weeks ago

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Ed Southwood 2 years ago

Created this as a copy of Optimisation - find quantity to maximise profits.

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Question statement

A demand function for a generic good, where $P$ is the price and $q$ is the quantity is: $P = \var{p} - q$ . The Marginal Cost is given by, $MC = \simplify{{x}q+{y}}$ , and fixed cost is, $FC = \var{z}$ . Find the quantity $q$ to maximise profits.

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			Name	Type	Generated Value

p	integer	236
y	integer	20
x	integer	5
z	integer	40
ans	rational	18

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Generated value: `integer`

236

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$q$ = Unnamed gap

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The total profit can be found by calculating the total revenue (price times quantity) subtract the total cost (marginal cost times quantity add fixed costs).

As an equation this looks like:

$\begin{split} \pi &= TR - TC \\ &= Pq - (MC \times q + FC) \end{split}$

Using the equations in this question we have:

$\pi = \simplify{({p} -q)q - ({x}q+{y})q - {z}}$

Collecting like terms:

$\pi = \simplify[!noLeadingMinus]{-{x+1}q^2 + {p-y}q +{z}}$

Differentiating with respect to $q$ gives:

$\frac{d\pi}{dq} = \var{-2*(x+1)}q + \var{p-y}$

Using the first order condition $\frac{d\pi}{dq} = 0$ we have:

$\var{-2*(x+1)}q + \var{p-y}= 0$

Solving to find $q$ :

$q = \var{-(p-y)/(-2*(x+1))}$

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Optimisation: Find quantity to maximise profits

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Taxonomy: mathcentre

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Taxonomy: Context

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Ed Southwood 2 months, 4 weeks ago

Ben McGovern 2 years ago

Ed Southwood 2 years ago

Error in variable testing condition

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Generated value: `integer`

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Optimisation: Find quantity to maximise profits

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ed Southwood 2 months, 4 weeks ago

Ben McGovern 2 years ago

Ed Southwood 2 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`