(a) Suppose that price discrimination is prohibiteed and thus the firm should charge the same price to the different goups. Then, what are the optimal price and quantities?
(b) Now suppose that price discrimination is possible and the firm can charge different prices for the two groups. What are the optimal price and quantity for each market?
3. I am indifferent, or x is indifferent ( 無差別である ) to y: x ∼ y
Note that we implicitly assume that the elements in X are all comparable, and ignore the intensity of preferences.
A legal answer to the questionnaire P can be formulated as a function f which assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y. That is,
Randomized Strategies
No strategy looks to be dominated…
If a player 2 randomizes L and R with 50% each, then
Such mixed (randomized) strategy yields 1.5 (as an expected payoff) while M gives 1 irrespective of player 1’s strategy.
h ≤ e k for k = 1, ..., n.
(a) Show that any solution of the above maximization problem (you may denote x ∗ ) must be Pareto efficient.
(b) Find an example of Pareto efficient allocation that cannot be the solution of the maximization problem whichever (λ 1 , · · · , λ I ) ∈ R I + \ {0} will be chosen.
Thm Envelope Theorem
Consider P 1 and suppose the objective function and constraint are continuously differentiable in a. For each a, let x(a) ≫ 0 uniquely solve P 1 and assume that it is also continuously differentiable in the parameters a. Then, the Envelope theorem states that
(c) Derive the competitive equilibrium (both price w ∗ and allocation x ∗ ).
(d) Now consider an exchange economy with n consumers and k goods. We de- note the bundle of total endowments by ω = (ω 1 , . . . , ω k ). Suppose that all
consumers have identical (strictly) convex preferences. Then, show that equal division of total endowments, i.e., x i = ω/n for all consumer i, is always a
payoff) while M gives 1 irrespective of player 1’s strategy.
Therefore, M is eliminated by mixing L and R .
After eliminating M , we can further eliminate D (step 2) and L
(step 3), eventually picks up ( U , R ) as a unique outcome.
1 Introduction
1.1 Theoretical context
In the traditional view, phonology is strictly about sounds, and orthography has been con- sidered to have nothing to do with phonological theory. This has been the case since linguis- tics distinguished itself from philology, under the influence of landmark studies like Saussure (1916/1972). However, there are a few recent proposals and observations in the phonological literature that cast doubts on this traditional, strictly orthographic-free versions of phonol- ogy theory. Ito et al. (1996) offer an illustrative example. In a Japanese argot language game, known as zuuja-go, reversing occurs based bimoraic feet: e.g. /(batsu)+(guð)/ → /(gum)+(batsu)/ ‘exquisit’. When the first syllable contains a geminate, the reserved seg- ment which corresponds to a geminate marker in the original word appears as /tsu/; e.g. /(bik)+(kuri)/ → /(kuri)+(bitsu)/ ‘surprised’. The most reasonable conjecture about this con- version of a geminate to /tsu/, according to Ito et al. (1996), is because the gemination is marked with a smaller version of the letter for /tsu/ ( っ ) in the Japanese orthography. Thus,
u(x, y) = x 2
+ y 2
(ω x , ω y ) = (1, 1)
(a) Assume there are only two individuals in this economy. Then, draw the Edgworth-box and show the contract curve. Find a competitive equilibrium if it exists. If there is no equilibrium, explain the reason.