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Price For Maximizing Profit by Ted Mitchell

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Learning Goal Finding the Price that Maximizes the Profit is not necessarily the same as finding the Price that Maximizes Revenue

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Revenue - total Variable Cost - Fixed Cost = Profit Revenue – Total COGS – Total Fixed = Z PQ - VQ - F = Z Z = PQ - VQ - F And Quantity Sold is a function of Selling Price

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Expand The Basic Profit Equation Z = PQ - VQ - F substitute Q = ƒ(P) = a - bP Z = P(a-bP) - V(a-bP) - F Z = aP - bP 2 - aV + bPV - F

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Consider the Fixed Costs

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Z = aP - bP 2 - aV + bPV - F Consider the Fixed Costs Consider the Variable Costs

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Z = aP - bP 2 - aV + bPV - F Consider the Fixed Costs Consider the Variable Costs Consider the Revenue

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Revenue looks like R = aP - bP 2 With zero costs Revenue = Profit Revenue = Profit Price 0

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Revenue = Profit Price 0 Subtract Fixed Costs from Revenue R - F = aP - bP 2 - F P*

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Revenue = Profit Price 0 P* Subtract Variable Costs from Revenue R - VQ - F = aP - bP 2 - aV + bPV - F P*

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With NO variable cost per unit, V = 0 And Only fixed or period costs, F > 0 If you find the price that maximizes revenue, then you have found the price that maximizes profit. Price that maximizes revenue is Pr* = a/2b

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Revenue = Profit Price 0 P* Subtract Variable Costs from Revenue R - VQ - F = aP - bP 2 - aV + bPV - F P* Breakeven Points

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Slope of Revenue Curve is Zero Revenue Price 0 R P

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Slope of Profit Curve is Zero Profit Price 0 Z P

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Example Exam Question The Demand is estimated by market research to be Q = 5,000 – 500P The variable cost per unit is, V = $2 The fixed cost for the period is, F = $7,000 What is the selling price that will maximize the Profit? First build the Profit Equation The Revenue is R = P(a-bP 2 ) = P(1,500-500P) The Profit is Z = R – VQ – F Z = P(1,500-500P) – 2(1,500-500P) – 1,000 Z = 1,500P – 500P2 -3,000-1000P – 1,000

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Example Exam Question The Demand is estimated by market research to be Q = 5,000 – 500P The variable cost per unit is, V = $2 The fixed cost for the period is, F = $7,000 What is the selling price that will maximize the Profit? First build the Profit Equation, Z The Revenue is R = P(a-bP 2 ) = P(5,000-500P) The Profit is Z = R – VQ – F Z = P(5,000-500P) – 2(5,000-500P) – 7,000 Z = 5,000P – 500P 2 -10,000+1000P – 7,000

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Example Exam Question The Demand is estimated by market research to be Q = 5,000 – 500P The variable cost per unit is, V = $2 The fixed cost for the period is, F = $7,000 What is the selling price that will maximize the Profit? Second: Find the first derivative wrt P, Z = 5,000P – 500P 2 -10,000 + 1,000P – 7,000 dZ/dP = 5,000 –2(500)P +1,000, set dZ/dP= 0 5,000 –2(500)P +1,000 = 0, solve for P –2(500)P = -5,000 -1,000 = P = 6,000/1,000 = $6

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The optimal Price for maximizing profit in the example is Pz* = $6

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Price 0 $ $5 $6 Profit Revenue

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There is a General Solution for Finding Optimal Price for Max Profit 1) Establish the Profit equation Z = aP - bP 2 - aV + bPV – F 2) Find the first derivative of the profit equation dZ/dP = a – 2bP – bV 3) Set the first derivative equal to zero dZ/dP = a – 2bP – bV = 0 4) Solve for the optimal price P = a/2b + bV/2b = a/2b + V/2b

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The Price That Maximizes Profit Consider Market Potential Consider The Customer’s Sensitivity to Price Changes Consider Your Variable Costs

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The Price That Maximizes Profit P = (Price that maximizes revenue) + (Half of the Variable Cost)

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The Price That Maximizes Profit Is always equal to or higher than the price that maximizes sales revenue!

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Quantity Price Maximum Revenue Pr* = $5 Pz* = $6 Revenue for Maximum Profit

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Quantity Price Maximum Revenue Pr* = a/2b a/2 Pz* = a/2b + V/2 Revenue for Maximum Profit

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The Price That Maximizes Profit Says if you get an increase in your variable costs pass half of it on to the customer.

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The Price That Maximizes Profit Note: It Says Do NOT change your price just because you get an increase in your fixed costs!

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Any Questions?

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