Please see attached photos as questions:

oblem 1 (30 marks total; with 10 marks for each item) 1, a utility maximizer, make a delicious 4 kg cake and plan to eat it in three consecutive days: today , tomorrow (t = 1), and the next day (t = 2). Your roommate will eat half of whatever part of the ou leave in the fridge during the night from today to tomorrow. However, your roommate will travel ow morning and will not eat more than this one time; the rest of your cake will be safe during the rom tomorrow (t = 1) to the next day (t = 2). Your instantaneous satisfaction for eating ct kg of v(ct) = Inc. You discount time with an intertemporal discount factor S = 1/2 per day. Your utility discounted flow of future distantaneous utilities. You do not have the ingredients to make more cake; more cake is impossible. 1) Write the cake consumption problem in an organized, mathematical way. State the optimization ns that you will need to solve. What restrictions each variable have? Solve the cake consumption n. What is the optimal consumption plan? Explain the economic intuition of your results. 2) Suppose that you overcook the cake and have to trash half of it. From the very start you have only w. Because of this minor cooking accident, you become nervous and anxious, so your discount factor es to 8 = 1/4. Find the optimal consumption plan. Explain the economic intuition of your results. re this consumption plan with the previous one (your answer for item 1.1). 3) You have 4 kg of cake again, and o = 1/2. Suppose your roommate has their trip cancelled. This that whatever cake left in the fridge from t = 1 to t = 2 will be halved during the night (and, as the cake in the fridge is also halved from t = 0 to t = 1). You know this from the moment you start the cake. Solve the cake consumption problem. What is the optimal consumption plan? Compare nsumption plan with the original consumption plan in item 1.1. each item, justify all steps of your reasoning clearly and precisely, using math formulas, hers as fractions, if necessary. Do not use