Question

A price-taking profit-maximising business, which operates at its optimal plant
size, has a TC curve give by TC = q³ − 2q² + 2q + 0.576 where TC
is measured in 1000s of £ per week and q is the rate of output measured in
1000s of units per week.
a) If the market price of output is £1.00 show that the firm will produce 1000
units per week and make a loss. Explain why the business is, at this price,
indifferent between producing and closing down. Draw a diagram to show
this.
b) If the market price of output is £3.28 what will each firm produce and what
profit is made?
c) Assuming that all firms in the industry have identical cost structures and that
there is freedom of entry, show that the long-run weekly rate of output will fall
to 1200 units.
d) What is the price of output when the industry is in equilibrium?

Answer 1

Answer on Question #67416 – Math – Calculus

A price-taking profit-maximising business, which operates at its optimal plant size, has a $TC$ curve given by $TC = q^3 - 2q^2 + 2q + 0.576$ where $TC$ is measured in 1000s of $\mathcal{E}$ per week and $q$ is the rate of output measured in 1000s of units per week.

Question

a) If the market price of output is £1.00 show that the firm will produce 1000 units per week and make a loss. Explain why the business is, at this price, indifferent between producing and closing down. Draw a diagram to show this.

Solution

$MR$ - marginal revenue; $MC$ - marginal cost; $TR$ total revenue; $p$ - price.

Since this is profit-maximising business, then

$$MR = MC$$ $$TR = pq$$ $$MR = \frac{d}{dq}(pq) = p$$ $$MC = \frac{d(TC)}{dq} = 3q^2 - 4q + 2$$ $$p = 3q^2 - 4q + 2$$ $$1 = 3q^2 - 4q + 2$$ $$q = \frac{4 \pm 2}{6}$$ $$q_1 = 1; \quad q_2 = \frac{1}{3}$$

Total Revenue:

$$TR = 1 \cdot q = 1000 \text{ £ per week}$$

Then profit is

$$TR - TC = 1000 - (1 - 2 + 2 + 0.576) \cdot 1000 = -576 \text{ £ per week} - \text{the loss}$$

Profit at this price is increasing, but on the other hand, it is negative. So the business is indifferent between producing and closing down.

Question

b) If the market price of output is £3.28 what will each firm produce and what profit is made?

Solution

$$p = 3q^2 - 4q + 2 = 3.28$$ $$q = \frac{4 \pm \sqrt{16 + 3 \cdot 1.28}}{6}$$ $$q = 1409 \text{ units per week}$$

Profit:

$$\begin{array}{l} TR - TC = 3.28 \cdot 1409 - (1.409^3 - 2 \cdot 1.409^2 + 2 \cdot 1.409 + 0.576) \cdot 1000 \\ = 2400.82 \text{ £ per week} \end{array}$$

Question

c) Assuming that all firms in the industry have identical cost structures and that there is freedom of

entry, show that the long-run weekly rate of output will fall to 1200 units.

Solution

$$MC = \frac{TC}{q}$$ $$3q^2 - 4q + 2 = q^2 - 2q + 2 + \frac{0.576}{q}$$ $$2q^2 - 2q - \frac{0.576}{q} = 0$$ $$2q^3 - 2q^2 - 0.576 = 0$$ $$q = 1.2 \text{ (1200 units)}$$

Question

d) What is the price of output when the industry is in equilibrium?

Solution

$$q = 1.2$$ $$p = 3q^2 - 4q + 2$$ $$p = 3 \cdot 1.2^2 - 4 \cdot 1.2 + 2 = 1.52 \text{ £}$$

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