Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 7
pollev.com/chrismakler
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
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...
The total revenue is the price times quantity (area of the rectangle)
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 \(p\): \(dq \times p\)
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
Special simpler case: constant marginal cost, no fixed costs.
Simple case: linear demand, constant MC, no fixed costs
Baseline Example: Monopoly
14
2
units
$/unit
14
P
Q
Baseline Example: Monopoly
14
2
units
$/unit
14
P
Q
Profit
Baseline Example: Monopoly
14
8
2
6
Q
P
36
How will firm 2 react to firm 1's quantity?
2
P
"Firm 2's Residual Demand Curve"
Firm 2's "reaction function"
How will firm 1 choose, knowing firm 2's reaction?
Firm 2's "reaction function"
Firm 2's strategy: whatever \(q_1\) firm 1 produces, produce \(6 -{1 \over 2} q_1\)
Firm 1's strategy: produce 6 units of output
Given what the other firm is doing,
does either firm have any incentive
to change its strategy?
In equilibrium, firm 2 produces 3 units.
Why don't we just say that its strategy is "produce 3 units of output"?