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)
Consider the case that M ≻ m. By I and C, there must be a single number v(s) ∈ [0, 1] such that
v(s) ◦ M ⊕ (1 − v(s)) ◦ m ∼ [s]
where [s] is a certain lottery with prize s, i.e., [s] = 1 ◦ s. In particular, v(M ) = 1 and v(m) = 0. I implies that
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: