Question

Imagine that you want to launch a product to the market. Estimate two combinations of prices and quantities, to determine the demand curve. Determine the demand equation and the revenue equation (R). Assume the equation for variable costs (VC) and fixed costs (FC). Whit this determine the profit equation (R - VC - FC). How many units will you produce to maximize profits?

Answer 1

Let suppose two combinations of price and quantity are -

P = $5 , Qd = 45 units

P = $10 , Qd = 20 units

Slope of the demand curve = ∆Qd/∆P = -25/5 = -5.

At P = 10 , Q = 20 , the equation of demand curve is

P - 10 = (∆Q/∆P)[Q - 20]

Or, P - 10 = -5*(Q - 20)

Or, P - 10 = - 5Q + 100

Or, 5Q = 100 + 10 - P

Or, 5Q = 110 - P

Or, Q = (110/5) - (1/5)P

Or, Q = 22 - 0.2P ...... This is the demand function.

P = 110 - 5Q ..... Inverse demand function.

Revenue function = TR = P*Q

Revenue (R) = (110 - 5Q)*Q = 110Q - 5Q^2 ..Revenue function

Variable cost equation is = 10Q + 5Q^2

Fixed Cost = 50.

Profit equation (π) = R - VC - FC

π = 110Q - 5Q^2 - (10Q + 5Q^2) - 50

π = 110Q - 5Q^2 - 10Q - 5Q^2 - 50

π = 100Q - 10Q^2 - 50 ...... This is profit equation.

At profit maximising level F.O.C of profit (π) function w.r.t Q = 0.

Therefore, dπ/dQ = d/dQ(100Q - 10Q^2 - 50) = 0

Therefore, 100 - 20Q = 0

Or, 20Q = 100

Or, Q* = 100/20 = 5.

Therefore, at Profit maximising level there should produce 5 units of output. (Answer).