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You, a utility maximizer, make a delicious 4 kg cake and plan to eat it in three consecutive days: today (t = 0), tomorrow (t = 1), and the next day (t = 2). Your roommate will eat half of whatever part of the cake you leave in the fridge during the night from today to tomorrow. However, your roommate will travel tomorrow morning and will not eat more than this one time; the rest of your cake will be safe during the night from tomorrow (t = 1) to the next day (t = 2). Your instantaneous satisfaction for eating ct kg of cake is v(ct) = 1n ct. You discount time with an intertemporal discount factor 6 = 1 / 2 per day. Your utility is the discounted ow of future instantaneous utilities. You do not have the ingredients to make more cake; making more cake is impossible. (1.1) Write the cake consumption problem in an organized, mathematical way. State the optimization problems that you will need to solve. What restrictions each variable have? Solve the cake consumption problem. What is the optimal consumption plan? Explain the economic intuition of your results. (1.2) Suppose that you overcook the cake and have to trash half of it. From the very start you have only 2 kg now. Because of this minor cooking accident, you become nervous and anxious, so your discount factor decreases to 6 = 1 / 4. Find the optimal consumption plan. Explain the economic intuition of your results. Compare this consumption plan with the previous one (your answer for item 1.1). (1.3) You have 4 kg of cake again, and 6 = 1/2. Suppose your roommate has their trip cancelled. This means that whatever cake left in the fridge from t = 1 to t = 2 will be halved during the night (and, as before, the cake in the fridge is also halved from t = O to t = 1). You know this from the moment you start making the cake. Solve the cake consumption problem. What is the optimal consumption plan? Compare this consumption plan with the original consumption plan in item 1.1. In each item, justify all steps of your reasoning clearly and precisely, using math formulas, theorems and symbols as needed. Leave numbers as fractions, if necessary. Do not use decimals