Exercises - Math Problem Example

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Exercises

1 A firm manufactures and sells q units of a product at price = £(575 – ½ q)
which has unit costs of £(q2 – 25q) and fixed costs of £45,000.

(a) Write down expressions for: revenue, profit and average cost
in terms of output(q) of the firm. [1 mark]

Revenue = (575 – ½ q ) q
= 575q – ½ q2

Profit = Revenue – Total Cost

= 575q – ½ q2 - [(q2 – 25q )q +45,000]

= 575q – ½ q2 – q3 + 25q2 - 45,000

= – q3 + 24.5q2 + 575q - 45,000

Average Cost = Total Cost / q

= q2 – 25q + 45,000/q


(b) Find expressions for: marginal revenue, marginal cost, marginal
profit and marginal average costs in terms of output (q). [2 marks]


Marginal Revenue, Marginal Cost, Marginal Profit and Marginal Average Costs is the derivative of Revenue, Cost, Profit, Average Costs . Since the derivative of f(x) = xn is nxn-1, we have:

Marginal Revenue = 575 – q

Marginal Cost = 3q2 -50q

Marginal Profit = -3q2 + 49q +575

Marginal Average Cost = 2q – 25 -45,000/q2 (since 1/q = q-1)

(c) Find the output levels of the firm that and confirm that the output levels found do indeed maximise or minimise these functions [ 1 mark]


(i) Maximise revenue

















• This is the graph of Revenue = 575q – ½ q2 , we can see that it is maximised at q = 575.


(ii) Minimise costs

To minimise costs, set marginal costs to 0

q = 50 / 3 or approx 17 units ...Show more

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