Fundamentals of Pricing
Pushan Dutt
Professor of Economics
An Airline’s Pricing “Dilemma”
I am a struggling airline (Aren’t they all?)
I currently price tickets at P = $300, and expect to sell Q = 330,
which is less than my capacity
Should I lower prices to attract one more passenger?
Question: How much is an additional passenger worth?
Total Revenue
Total revenue is price times quantity
TR = P*Q
Recognize that quantity depends on price
TR = P*Q (P)
Alternatively:
TR = P (Q)*Q
where P (Q) is the inverse demand curve
P
Total
Revenue
= 300*330 = 99,000
Example:
Q (P) = 480 – 0.5P (the demand curve)
P (Q) = 960 – 2Q (the inverse demand curve)
TR
= P*Q = 960Q – 2Q2
300
330
Q
Marginal Revenue
So how much is an additional passenger worth?
Inverse demand curve: P = 960 – 2Q
To sell 331 seats, set price 960 – 2(331) = $298
Must lower price from $300 to $298 to sell an additional seat
So is the additional passenger worth $298?
No!
TR with 330 seats sold = $300*330 = $99,000
But... TR with 331 seats sold = $298*331 = $98,638
Selling an extra seat decreases my total revenue by $362!
Marginal revenue of an extra seat = – $362 < 0
What’s going on?
Inexorable Volume-Margin Trade-Off
Margins you sacrifice
Old price
New price
Demand
Volume-Margin
Tradeoff
Extra unit brings:
P in revenue
Revenue gain from
selling an additional
unit (more volume)
Q Q+1
…but costs
(dP/dQ) in price cut
applied to Q units or
(dP/dQ)*Q
Marginal Revenue (MR)
Marginal Revenue:
Contribution of extra unit to revenue
dTR/dQ = MR = P(Q) + (dP/dQ)Q
The marginal unit is worth: the price, plus the adjustment term
which is a negative number (dP/dQ)*Q
$
Example:
P
MR
▪ P = 960 – 2*Q
▪ dP/dQ =
▪ MR =
P > MR
Demand
Q
Marginal Q
Revenue
How About Costs?
Simple Cost structure: c(Q) = FC + SC +k*Q
▪ FC is fixed cost
▪ SC is sunk cost – ignore this
$
ac (Q)
▪ k*Q is variable cost
ac (Q) = k + FC/Q
avc (Q) = k
mc(Q) = k
k
Q
The Fundamental of Profit Maximization:
Optimal Pricing Decision
$
MR > MC → increase Q
MR < MC → decrease Q
MR = MC → optimal Q = Q*
P*
Marginal
Cost = k
Inverse demand
P(Q)
Q*
Q
MR (Q)
Total Revenues at Optimal Price
$
P*
Marginal
Cost = k
Revenue
Inverse demand
P(Q)
Q*
Q
Total Variable Costs at Optimal Price
$
P*
Marginal
Cost = k
Variable
Costs
Inverse demand
P(Q)
Q*
Q
Variable Profits at Optimal Price
$
P*
Variable
Profits
Marginal
Cost = k
Inverse demand
P(Q)
Q*
Q
Total Profits at Optimal Price
$
P*
Profits
Average Costs
Marginal
Cost = k
Inverse demand
P(Q)
Q*
Q
Key Takeaways
Marginal Revenue
▪ Captures volume-margin trade off
Marginal costs
▪ COGS from financial statements
▪ Be aware of assumption of linear cost function
Profit Maximization: How much to produce; What price to charge?
▪ Produce where marginal revenues equals marginal costs
▪ Determine price from demand curve