Question

You need to determine the profit-maximizing price for a video game console. Currently, you charge $180 and sell 2 million consoles per year. It costs $150 to produce a console, and the price elasticity of demand for consoles is 3. What price should you charge for consoles?

Now assume that, on average, a purchaser of your video game console buys 10 video games, and you earn $10 profit on each video game. What is the correct price for consoles?

Answer 1

Let the demand curve be represented by the linear equation -

mPc

where,

Q is the quantity demanded

P is the selling price

m is the slope of the curve

c is the intercept of the line (a constant)

It is given that price elasticity of demand is 3. Using this information -

we know that price elasticity of demand is given by -

/Op dP/P d ed

Now, through this, we find the slope of our curve, i.e. dQ/dP

а 180 3 dP 2m

(Note: here 2m is 2 million)

dQ 1m . dP 30

now, we have to find c in the curve equation, thus we substitute for values of Q, P and m

Im 2m 180. c 30

C= 8m

thus, our demand curve equation is -

8 30

which has P in dollars and Q in millions

Now,

P = 240 - 30CQ

Revenue (R) from Q consoles sold would be = price x quantity

R P Q= (240 30Q) Q240Q 30Q2

Cost (C) per console = 150

cost of Q consoles -

C = 150Q

Profit (p) from selling Q consoles = R - C

p30Q2240Q 150Q=30Q90Q

To maximize profit, we differentiate p with respect to Q and equate it to 0

аp -60Q90 0 аQ

$Q=1.5$

The above is the profit maximizing quantity to be sold.

substituting for Q in demand curve equation, we get the price to be charged for the consoles -

$P=-30Q+240=-30*1.5+240=195$