5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2,...

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5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2, where x1, x2 0 denote the consumed amounts of good 1 and 2 (the only two goods available to the consumer), and t > 0 is a free parameter. The wealth of the consumer is equal to 4 and the price of each good is 1. Due to medical reasons, the consumer is not allowed to consume more than 2 units of good 2 for every consumed unit of good 1. The objective of the consumer it to maximise their utility given the budget and the medical constraint. (a) Write down the consumer optimisation problem. Sketch the set of all feasible consumption bundles (x1, x2) in a two-dimensional graph and determine if it is closed, or convex, or compact. Precisely motivate your answer. (b) [4 marks] Determine whether the utility function of the consumer is continuous, or con- cave, or strictly concave. What can you say about the number of optimal bun- dles? Precisely motivate your answer. [3 marks] (c) Write down the Lagrange function corresponding to the optimisation problem of the consumer. Write down the Kuhn-Tucker conditions for this problem. You may neglect the non-negativity constraints. [4 marks] (d) By examining each possible combination of binding constraints, derive all so- lutions to the Kuhn-Tucker conditions for any strictly positive value of the pa- rameter t. In each case, clearly state the corresponding values of x1, x2, and the Lagrange multipliers. Determine which solutions to the Kuhn-Tucker con- ditions are solutions to the consumer optimisation problem. [6 marks] (e) Suppose that a recent medical research concluded that a marginal increase in consumption of good 2 for every unit of good 1 causes no health issues. This is to say that now the consumer may consume 2 units of good 2 for every 1 + units of good 1, where is an infinitesimally small number. How would this affect the utility of the agent in the optimum? [3 marks] 5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2, where x1, x2 0 denote the consumed amounts of good 1 and 2 (the only two goods available to the consumer), and t > 0 is a free parameter. The wealth of the consumer is equal to 4 and the price of each good is 1. Due to medical reasons, the consumer is not allowed to consume more than 2 units of good 2 for every consumed unit of good 1. The objective of the consumer it to maximise their utility given the budget and the medical constraint. (a) Write down the consumer optimisation problem. Sketch the set of all feasible consumption bundles (x1, x2) in a two-dimensional graph and determine if it is closed, or convex, or compact. Precisely motivate your answer. (b) [4 marks] Determine whether the utility function of the consumer is continuous, or con- cave, or strictly concave. What can you say about the number of optimal bun- dles? Precisely motivate your answer. [3 marks] (c) Write down the Lagrange function corresponding to the optimisation problem of the consumer. Write down the Kuhn-Tucker conditions for this problem. You may neglect the non-negativity constraints. [4 marks] (d) By examining each possible combination of binding constraints, derive all so- lutions to the Kuhn-Tucker conditions for any strictly positive value of the pa- rameter t. In each case, clearly state the corresponding values of x1, x2, and the Lagrange multipliers. Determine which solutions to the Kuhn-Tucker con- ditions are solutions to the consumer optimisation problem. [6 marks] (e) Suppose that a recent medical research concluded that a marginal increase in consumption of good 2 for every unit of good 1 causes no health issues. This is to say that now the consumer may consume 2 units of good 2 for every 1 + units of good 1, where is an infinitesimally small number. How would this affect the utility of the agent in the optimum? [3 marks]