Theorem 6 (Envelope Theorem)
Consider P1 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
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.
Problem Set 3: Due on May 28
Advanced Microeconomics I (Spring, 1st, 2013) 1. Question 1 (5 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.
A practical guide. Chichester, West Sussex: Wiley-Blackwell.
• Seliger, H. W., & Shohamy, E. (1989). Second language research methods. Oxford: Oxford
Constant Absolute Risk Aversion | 絶対的リスク回避度一定 Def We say that preference relation % exhibits invariance to wealth if (x + p 1 ) % (x + p 2 ) is true or false independent of x.
Thm If u is a vNM continuous utility function representing preferences that are monotonic and exhibit both risk aversion and invariance to wealth, then u must be exponential,
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.
f (x, a) s.t. g(x, a) = 0.
where x is a vector of choice variables, and a := (a 1 , ..., a m ) is a
vector of parameters that may enter the objective function, the constraint, or both. Suppose that for each vector a, the solution is unique and denoted by x(a).