THEOREM 1.1 Existence of a Real-Valued Function Representing the Preference Relation ~ If the binary relation ~ is complete, transitive, continuous, and strictly monotonic, there exists a continuous real-valued function, u: R" -R, which represents _. THEOREM 1.8 Relations Between Indirect Utility and Expenditure Functions Let v(p. y) and e(p, u) be the indirect utility function and expenditure function for some consumer whose utility function is continuous and strictly increasing. Then for all p >> 0, y 2 0, and u el: 1. e(p, v(p. y)) = y. 2. v(p, e(p, u)) = u. Suppose the binary preference relation is not continuous but holds all other properties mentioned in Theorem 1.1 (Advanced Microeconomic Theory, Jehle and Reny). How would the utility function look like? Can you offer an example of a preference relation to illustrate this utility function? Suppose the binary preference relation is not complete but holds all other properties mentioned in Theorem 1.1 (Advanced Microeconomic Theory, Jehle and Reny). How would the utility function look like? Can you offer an example of a preference relation to illustrate this utility function? Use the following Utility function to derive the corresponding indirect utility function and expenditure function and verify theory 1.8 (Advanced Microeconomic Theory, Jehle and Reny). a. log U(X1, X2) = a log x1 + Blog X2 b. U (X1, X2) = min {a x1, b x2}