I don't know how to solve question d, e and f

Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model. Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model. f. Calculate the present value of the consumer's expected total con- sumption, i.e. E (T"c*(ler at). as well as the present value of the consumer's expected total wage payments. We must of course have that the former is equal to the latter plus the initial savings A, but it is not! The mistake that we make is that we haven't accounted for the fact that if the consumer dies, then she has not used her remaining savings: the bank obtains them and thereby makes an extra profit. Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model. Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model. Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model. f. Calculate the present value of the consumer's expected total con- sumption, i.e. E (T"c*(ler at). as well as the present value of the consumer's expected total wage payments. We must of course have that the former is equal to the latter plus the initial savings A, but it is not! The mistake that we make is that we haven't accounted for the fact that if the consumer dies, then she has not used her remaining savings: the bank obtains them and thereby makes an extra profit. Exercises: 1. (40 points) Consider a consumer who wants to maximize her lifetime utility. At any point in time the utility she enjoys from at that point in time consuming an amount c is U(C), where U : R++ + R is a strictly increasing C2 function. So, U gives us the consumer's instanteneous utility. Her rate of time preference, which captures her level of impa- tience, is p > 0. Of course, the consumer does not know what her lifetime exactly entails: from her point of view her time of death T is a random variable. Suppose that the distribution of T is given by a cu- mulative distribution function F with density f = F'. The support of f is [0,T], where I can be finite or infinite.? So, her expected lifetime utility associated with a consumption path {c(t)}t>0, equals El vct)e=* dt). (1) where E is the expectation operator. a. Note that (1) is a double integral: You are integrating with respect to T and with respect to t. Rewrite (1) in such a way that you arrive at a single integral only involving integration with respect to t. So far we have not specified how the consumer is going to pay for her consumption. Let s(t) be her savings at time t, w(t) her wage at time t, and c(t) her consumption at time t. Then it must be that s'(t) = rs(t) + w(t) c(t), (2) where r > 0 is the instanteneous interest rate charged by the bank. We measure consumption by the amount of money spent on it. So, this c is just the amount of money spent buying stuff. Of course, T = p is not realistic. However, we will see that with an infinite T it is much easier to develop a tractable model