Z Treeintro
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Z-Tree is a software program that allows researchers to easily develop and carry out economic experiments by facilitating communication between experimenters and subjects. Key features include saving experimental data, controlling display times, and calculating profits. Researchers write experimental code in Z-Tree, which subjects then participate in by connecting through the Z-Leaf program. The document provides an example of a simple public goods experiment that could be conducted using Z-Tree.
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The optimal quantity, , here satisfies:
Notice the usual tradeoff in the monopoly problem:
Higher quantity entails higher amount sold, but lower price.
The monopolist at optimum equates marginal revenue to marginal cost.
Back to Duopoly
Now suppose that there are two firms in the market.
How does the amount supplied change?
To solve for the supply decisions of each firm, we use iterated dominance/rationalizability.
Iterated Dominance/Rationalizability
We first look for the set :
The pure strategies that are not strictly dominated in the original game.
To compute this set, we first look at pure strategies that are best responses to a pure strategy of the opponent firm.
Best Responses
Suppose that you are firm 1 and that firm 2 produces .
Then what is firm 2’s best response?
To solve this, we need to solve the following problem:
The optimal quantity then is given by:
Best Responses
Suppose that you are firm 1 and that firm 2 produces .
Then what is firm 2’s best response?
Similarly, if firm 1 produces , the firm 2’s best response is:
Best Responses in Cournot Duopoly
Higher the quantity of the opponent, the less I want to produce:
Intuitively, higher quantity of the opponent decreases the price lower marginal revenue.
Best Responses in Cournot Duopoly
Now using the above, we know that the quantities are all in .
These pure strategies are all best responses to some pure strategy of firm 2.
What about the quantities ?
We know that such pure strategies are not best responses against any pure strategy.
But we do not know whether they are best responses against a mixed belief.
We now show that they are not, and in fact they are strictly dominated.
Strictly Dominated Strategies in Cournot Duopoly
Claim: For any is strictly dominated by .
To prove this, we need to show that for all ,
To see this, note that if we take the derivative of the left hand side with respect to ...
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