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GPEM, Universitat Pompeu Fabra, Academic year 1998/99
1. A TU game (N,v) is called convex if for all coalitions S and T, v(S)+v(T)-v(S∩T)≤v(S∪T).
2. A TU game (N,v) is called simple if for every coalition S, v(S) is either 0 or 1, and v(N)=1. A coalition S with v(S)=1 is called a winning coalition. A player in a simple game is called a veto player if it belongs to all winning coalitions.
3. A game (N,v) is a weighted simple majority game if there exist non-negative numbers (qi)i∈N, qi≥0, ∑i∈Nqi=1 such that for all coalitions S, ∑i∈Nqi≠1/2, and v(S)=1 iff ∑i∈Nqi>1/2, and v(S)=0 otherwise.
4. Show that a TU market may not have a price equilibrium when all players have zero quantity of a given input. Hint: Consider Cobb-Douglas production functions.
5. A production function f:Rk+→R+ is called distributive if it is non-decreasing and for any input vectors x1,...,xm, and non negative numbers a1,...,am such that xi≤∑aixi for all i=1,...,m, it is satisfied that ∑aif(xi)≤f(∑aixi).