1. Consider the following constrained optimization problem: you have to decide about purchasing , and x2...

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1. Consider the following constrained optimization problem: you have to decide about purchasing ₁, and x2 units of goods with prices p₁, p2 (per unit). Utility of these goods in the production (output) is described by the utility function: 1-a u(x₁, x₂) = axix 1/2 - where a € (0, 1) and a > 0 is a given constant. 2 You are given the fixed utility U that must be achieved by the purchased amounts (i.e. u(x₁, x₂) = U) while the total cost p₁x₁ + p2x2 is minimized. a) Formulate the optimization problem to be solved. [2] b) Find the solution of the problem. [5] c) How will the optimal cost of the purchase change if the total utility U is de- creased by 2%. [3] 1. Consider the following constrained optimization problem: you have to decide about purchasing ₁, and x2 units of goods with prices p₁, p2 (per unit). Utility of these goods in the production (output) is described by the utility function: 1-a u(x₁, x₂) = axix 1/2 - where a € (0, 1) and a > 0 is a given constant. 2 You are given the fixed utility U that must be achieved by the purchased amounts (i.e. u(x₁, x₂) = U) while the total cost p₁x₁ + p2x2 is minimized. a) Formulate the optimization problem to be solved. [2] b) Find the solution of the problem. [5] c) How will the optimal cost of the purchase change if the total utility U is de- creased by 2%. [3]