A function f : D (⊂ R n ) → R is called
1 continuous at a point x 0 if, for all ε > 0, there exists δ > 0
such that d(x, x 0 ) < δ implies that d(f (x), f (x 0 )) < ε.
2 continuous if it is continuous at every point in its domain. 3 uniformly continuous if, for all ε > 0, there exists δ > 0 such
vNM Utility Function (1)
Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function.
where x is a vector of choice variables, and a := (a 1 , ..., a m ) is a vector of
parameters ( パラメータ ) that may enter the objective function and constraint.
Suppose that for each vector a, the solution is unique and denoted by x(a).
◮ A maximum-value function, denoted by M (a), is defined as follows:
“Soon after Nash ’s work, game-theoretic models began to be used in economic theory and political science,. and psychologists began studying how human subjects behave in experimental [r]
with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n +1
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)
with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+1
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)
with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+1
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)
5. Production Economy (25 points)
Consider an economy with two firms and two consumers. Firm 1 is entirely owned by consumer 1; it produces good A from input X via the production function a = 2x. Firm 2 is entirely owned by consumer 2; it produces good B from input X via the production function b = 3x. Each consumer owns 10 units of X. Consumers’ preferences are given by the following utility functions:
Three Firms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.
Klemperer (2002), “How (not) to Run Auctions: The European 3G Telecom Auctions,” European Economic Review. Milgrom (2004) Putting Auction Theory to Work Cambridge U Press[r]
. Consumers buy at most one unit and have utility function u (s|θ) = θs if they consume one unit of quality s and 0 if they do not consume. The monopolist decides on the quality and price that it is going to produce. Con- sumers observe qualities and prices and decide which quality to buy if at all.
Price Discrimination (1)
A monopoly firm may further raise profit by charging different prices across consumers. This exercise is called price
discrimination. The traditional classification of the forms of price discrimination listed as follows is due to Pigou (1920):