2. Consider the two-good model of the utility maximization program subject to a budget constraint. The utility function U of a hypothetical rational consumer and his/her budget constraint are given, respectively, by: U = x1x2, (U) B = p1x1 + p2x2, (B) where xi = the consumer’s demand for consumption good i (i = 1, 2), pi = the price of consumption good i (i = 1, 2), and B = the (exogenously given) budget of the consumer. In this maximization program, assume the following data: B = 240, p1 = 10, p2 = 2. (a) Using the Lagrangian function L, derive the first-order (necessary) conditions for a (local) maximum of the utility function. (b) Compute the optimal values of all choice variables, i.e., x*1 , x*2, and λ* , in the program, where λ signifies the Lagrange multiplier. (c) Using the information of the bordered Hessian matrix H¯ , verify the second order (sufficient) condition for a (local) maximum of the utility function.
2. Consider the two-good model of the utility maximization program subject to a budget constraint. The utility function U of a hypothetical rational consumer and his/her budget constraint are given, respectively, by:
U = x1x2, (U)
B = p1x1 + p2x2, (B)
where xi = the consumer’s demand for consumption good i (i = 1, 2), pi = the price of consumption good i (i = 1, 2), and B = the (exogenously given) budget of the consumer. In this maximization program, assume the following data: B = 240, p1 = 10, p2 = 2.
(a) Using the Lagrangian function L, derive the first-order (necessary) conditions for a (local) maximum of the utility function.
(b) Compute the optimal values of all choice variables, i.e., x*1 , x*2, and λ* , in the program, where λ signifies the Lagrange multiplier.
(c) Using the information of the bordered Hessian matrix H¯ , verify the second order (sufficient) condition for a (local) maximum of the utility function.
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