# A firm has the following demand and cost functions: Q=500-20P TC=18Q + 6Q² - 0.10Q³ a) Is the firm operating in a perfectly competitive market?  b) Is the firm operating in the short run?...

A firm has the following demand and cost functions:

Q=500-20P

TC=18Q + 6Q² - 0.10Q³

a) Is the firm operating in a perfectly competitive market?

b) Is the firm operating in the short run? Give your reason

c) Derive the profit function for this firm

d) Determine the profit maximizing price and quantity

mathsworkmusic | Certified Educator

We are given that the firm has the demand and cost functions:

Q=500-20P

TC=18Q + 6Q² - 0.10Q³

a) Is the firm operating in a perfectly competitive market?

'A perfectly competitive market is a hypothetical market where competition is at its greatest possible level.  Neo-classical economists argued that perfect competition would produce the best possible outcomes for consumers, and society' (see weblink 1 below)

In a 'perfectly competitive market' the demand function for an individual firm is constant with price P. This is known as perfectly elastic demand. As the demand function is not constant here with P, the firm is not operating in a perfectly competitive market. (see weblink 2 below).

b) Is the firm operating in the short run? Give your reason

In the short run price elasticity demand (PED) is usually less than 1 so that we have price inelastic demand. PED is the % change in quantity demanded / % change in price. We write

PED = % change in Q/ % change in P

If this is <1 and the demand is price inelastic then the change in demand is small compared to change in price.

Since, from the demand function, whenever price P increases demand reduces, we have that the PED is negative, so definitely <1. Therefore the firm is likely to be operating in the short run. Reasons for the demand being price inelastic could be that the goods

1. Have few or no close substitutes, e.g. petrol
3. They are addictive, e.g. sugar
4. They cost a small % of income or are bought infrequently/luxury

c) Derive the profit function for this firm

Profit = Revenue - Total Cost = R - TC

where

Revenue = Price x Quantity Demand

R = PQ

So that

Profit = PQ - TC

To find P rearrange the demand function:

P = (500 - Q)/20

so that

Profit = 25Q - Q^2/20 - (18Q + 6Q^2 - 0.1Q^3)

= 7Q + 5.95Q^2 -0.1Q^3   (1)

d) Determine the profit maximizing price and quantity

The quantity demanded Q0 when the profit is maximised is obtained by differentiating (1) with respect to Q and setting to zero (the gradient is zero at a maximum or minimum):

7 + 11.9Q0 - 0.3Q0^2 = 0

Q0 = (-11.9 - sqrt(11.9^2 - 4(7)(-0.3)))/(-0.6) = (-11.1 - 12.24786)/(-0.6) =

-23.34786/-0.6 = 38.9131

Therefore profit is maximised when the demand Q0 = 39. Substituting this into the inverse demand function gives a corresponding price P0 of

P0 = (500 - 39)/20 = \$23.05

Profit maximising price P0 = \$23.05 and profit maximising quantity Q0 = 39.