Assume that you are determining how much time to study and how much time to party...

 

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Assume that you are determining how much time to study and how much time to party while you are in university, and that your preferences at are described by the following utility function: U₁(s, p) = = sp, where s is the time you dedicate to studying, and p the one you dedicate to partying. To keep things simple, assume that you have one unit of time while you are at university, and that you split it between studying and partying. You do not spend this unit of time doing anything else. However, assume that you also care about the future. In the future, you will work and earn an income which you will spend on goods. To keep things simple, assume that you will spend all your future income on a single good, call it x, and assume it will have a price p > 0. In the future, your preferences over good x will be given by U₂(x) = x. Importantly, your income in the future will depend on how much you study at university. In other words, your income will be a function of s, denoted by I(s). For simplicity, assume I(s) = s. Finally, assume that increasing your utility today in u units is equivalent to increasing it in du units in the future, where > 0. In other words, your overall utility function is given by: U(u₁, U₂) = ₁ + du₂, where u, denotes your utility in the present (which depends on how much you study and how much you party in university), and u₂ denotes your utility in the future (which depends on how much of good a you consume). (a) (5 points) Write down your utility maximization problem. (Hint: your problem has three decision variables: s, p, and r, and two budget con- straints, one for the present and one for the future.) (b) (10 points) Simplify the problem as follows: solve for x from your future budget constraint and replace this value in the objective function. Since we've gotten rid of x, your objective function should be a utility function over bundles (s, p). Plot the indifference curves for this utility function on the (s, p) plane. What is the MRS of these preferences? Assume that you are determining how much time to study and how much time to party while you are in university, and that your preferences at are described by the following utility function: U₁(s, p) = = sp, where s is the time you dedicate to studying, and p the one you dedicate to partying. To keep things simple, assume that you have one unit of time while you are at university, and that you split it between studying and partying. You do not spend this unit of time doing anything else. However, assume that you also care about the future. In the future, you will work and earn an income which you will spend on goods. To keep things simple, assume that you will spend all your future income on a single good, call it x, and assume it will have a price p > 0. In the future, your preferences over good x will be given by U₂(x) = x. Importantly, your income in the future will depend on how much you study at university. In other words, your income will be a function of s, denoted by I(s). For simplicity, assume I(s) = s. Finally, assume that increasing your utility today in u units is equivalent to increasing it in du units in the future, where > 0. In other words, your overall utility function is given by: U(u₁, U₂) = ₁ + du₂, where u, denotes your utility in the present (which depends on how much you study and how much you party in university), and u₂ denotes your utility in the future (which depends on how much of good a you consume). (a) (5 points) Write down your utility maximization problem. (Hint: your problem has three decision variables: s, p, and r, and two budget con- straints, one for the present and one for the future.) (b) (10 points) Simplify the problem as follows: solve for x from your future budget constraint and replace this value in the objective function. Since we've gotten rid of x, your objective function should be a utility function over bundles (s, p). Plot the indifference curves for this utility function on the (s, p) plane. What is the MRS of these preferences?

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