Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 12
The profit from \(q\) units of output
PROFIT
REVENUE
COST
is the revenue from selling them
minus the cost of producing them.
We will assume that the firm sells all units of the good for the same price, \(p\). (No "price discrimination")
The revenue from \(q\) units of output
REVENUE
PRICE
QUANTITY
is the price at which each unit it sold
times the quantity (# of units sold).
The price the firm can charge may depend on the number of units it wants to sell: inverse demand \(p(q)\)
Demand curve:
quantity as a function of price
Inverse demand curve:
price as a function of quantity
QUANTITY
PRICE
Demand
Inverse Demand
If the firm wants to sell \(q\) units, it sells all units at the same price \(p(q)\)
Since all units are sold for \(p\), the average revenue per unit is just \(p\).
By the product rule...
let's delve into this...
Demand
Inverse Demand
Revenue
The total revenue is the price times quantity (area of the rectangle)
Note: \(MR < 0\) if
The total revenue is the price times quantity (area of the rectangle)
If the firm wants to sell \(dq\) more units, it needs to drop its price by \(dp\)
Revenue loss from lower price on existing sales of \(q\): \(dp \times q\)
Revenue gain from additional sales at \(p\): \(dq \times p\)
(multiply first term by \(p/p\))
(simplify)
(since \(\epsilon < 0\))
Notes
Elastic demand: \(MR > 0\)
Inelastic demand: \(MR < 0\)
In general: the more elastic demand is, the less one needs to lower ones price to sell more goods, so the closer \(MR\) is to \(p\).
Demand curve:
quantity as a function of price
Inverse demand curve:
price as a function of quantity
QUANTITY
PRICE
Special case: perfect substitutes
Demand curve:
quantity as a function of price
Inverse demand curve:
price as a function of quantity
QUANTITY
PRICE
Special case: perfect substitutes
For a small firm, it probably looks like this...
(multiply first term by \(p/p\))
(simplify)
(since \(\epsilon < 0\))
Note
Perfectly elastic demand: \(MR = p\)
Optimize by taking derivative and setting equal to zero:
Profit is total revenue minus total costs:
"Marginal revenue equals marginal cost"
Example:
What is the profit-maximizing value of \(q\)?
What is the profit-maximizing value of \(q\)?
Multiply right-hand side by \(q/q\):
Profit is total revenue minus total costs:
"Profit per unit times number of units"
AVERAGE PROFIT
CHECK YOUR UNDERSTANDING
Find the profit-maximizing quantity.
Recall our elasticity representation of marginal revenue:
Let's combine it with this
profit maximization condition:
Really useful if MC and elasticity are both constant!
Inverse elasticity pricing rule:
Find the optimal price and quantity if a firm has the cost function $$c(q) = 200 + 4q$$ and faces the demand curve $$D(p) = 6400p^{-2}$$
Inverse elasticity pricing rule:
Fraction of price that's markup over marginal cost
(Lerner Index)
What if \(|\epsilon| \rightarrow \infty\)?
Optimize by taking derivative and setting equal to zero:
Profit is total revenue minus total costs:
For a price taker (or competitive firm), revenue equals price times quantity:
"Price equals
marginal cost"
Price
MC
\(q\)
$/unit
P = MR
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