Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 12
If you're interested in this stuff..
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
What is firm 2's best response function?
2
P
"Firm 2's Residual Demand Curve"
Firm 2's "best response function"
Cournot Model:
Both firms choose simultaneously and independently
Firm 1's
best response function
Firm 2's
best response function
In equilibrium, each firm is
correct in its beliefs (so \(q_1 = \hat q_1\) and \(q_2 = \hat q_2\) ),
so each firm's quantity is a best response to the other firm's quantity.
Another way of thinking about this:
if everyone knows everything about this model (and everyone knows that everyone knows everything about this model), what do each of the firms know about the other firm's beliefs?
Each firm knows the other
will never produce more than 6.
Because \(6 - {1 \over 2}6 = 3\),
this means each firm knows the other
will never produce less than 3.
Because \(6 - {1 \over 2}3 = 4.5\),
this means each firm knows the other
will never produce more than 4.5.
The only set of quantities that survives this is (4,4).
Profits in Cournot Equilibrium
Each firm is producing 4 units, so the market price is \(14 - 4 - 4 = 6\).
Each unit costs $2, so each firm is making
$4 of profit on 4 units = $16.
Remember our monopoly: it produced 6 units,
sold them at a price of 8, and earned a total profit of 36.
If each of these two firms produced 3 units, they could earn 18...
so why don't they?
Next Thursday, we'll look at collusion between firms.
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Indy
Indy is located \(x\) blocks from store A, and \(y\) blocks from store B.
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Indy
Indy is located \(x\) blocks from store A, and \(y\) blocks from store B.
A's market share
Suppose he is indifferent between buying from the two stores.
B's market share
Then everyone to his left goes to A, and everyone to his right goes to B.
🤓
Indy
Indy is located \(x\) blocks from store A, and \(y\) blocks from store B.
A's market share:
Suppose he is indifferent between buying from the two stores.
B's market share:
Cost of buying from store A
To buy from a store, Indy incurs cost \(c\) per block, plus the price charged by the store.
Cost of buying from store B
If he is indifferent, these two must be equal:
A's market share:
B's market share:
Cost of buying from store A
Cost of buying from store B
If he is indifferent, these two must be equal:
Substitute in \(y = l - (a + b + x)\) and solve for \(x\):
🤓
Indy
A's market share
B's market share
A's market share
B's market share
A's revenue (payoff function)
B's revenue (payoff function)
A's revenue (payoff function)
B's revenue (payoff function)
Take derivatives, set equal to zero to find best response functions:
Solve to find Nash equilibrium:
Profits:
Equilibrium prices
Market shares:
What can we say about how these firms feel about the parameters \(a,b,c,\) and \(l\)? Why?
STRATEGIES ARE
QUANTITIES
STRATEGIES ARE
PRICES
In each case: find the best response functions,
by taking the derivative of the profit function with respect to one's own strategy,
holding the other player's strategy as given;
then find strategies which are best responses to each other