• Assume you have $1 million now, and you have just retired from your job. This would be your initial wealth, W₀.
• You expect to live for 20 years, and you want to have the same level of consumption for each of the 20 years, after adjusting for inflation. That is, you want to keep the same purchasing power of consumption for all 20 years. In the equation, this means that C_t will be constant in real dollars over the rest of your lifetime.
• You also wish to leave the purchasing power equivalent of $100,000 today to your kids at the end of the 20 years as a bequest (or to pay them to take care of you). In real dollars, of course, since there is no inflation in real dollars, you will need to have $100,000 to give them at the end of the 20 years.
• You expect real interest rates to stay at 5% per year. So, your savings account will earn 5% in real dollars over the next 20 years.
• The Omar example in the textbook shows Omars human capital through labour earnings, but our assignment problem does not have any labour earnings. You have already earned your retirement savings of $1 million.
• The assignment problems will ask you to apply the formula to calculate the level of consumption that fits the intertemporal budget constraint, given assumptions about initial wealth and a planned bequest.

5.2 Intertemporal Budget Constraint with Nominal Cash Flows (7 marks)

Using the same information as stated in 5.1 and adding the assumption that inflation is expected to be 3% per year over the next 20 years, translate all of the values (i.e., the bequest, the initial wealth, and the consumption, into nominal dollars (i.e., actual dollars rather than purchasing power).

The trickiest part is the fact that the consumption amount will no longer be constant, because inflation will require that you spend 3% more each year to maintain the same consumption. So, if you spent $1 last year to buy a chocolate bar, the same chocolate bar will cost you $1.03 at the end of year 1, $1.0609 at the end of year 2 (i.e., 1.0609 = 1.03 x 1.03 = 1.03^2), and so on.

a. The present value of your bequest will be the same as in part 5.1a, because the present value of the nominal amount must be the same as the real amount at time 0. But in 20 years, the real amount of $100,000 will not be the same as the nominal amount. Calculate the nominal amount you will need to have 20 years from now to satisfy the bequest for your kids. (2 marks)

Hint: One way to visualize how to do this is to realize that $100,000 of purchasing power is no different than the price of 100,000 $1 chocolate bars today. All you need to do is determine how much youd have to pay to buy the same chocolate bars 20 years in the future.

b. Calculate the actual amount of consumption, in nominal dollars, for each of the 20 years, using the stated assumptions. This requires you to calculate 20 numbers, so you might want to use an Excel spreadsheet, such as the one provided with this assignment (5.2 template). (4 marks)

Note: If you want to calculate the result algebraically, you could use the formula for the present value of a growing annuity. You can find the formula in the lesson notes at the end of Note 7 in Lesson 4. That formula shows the value of a $1 annuity, which grows to a value of (1+g) at the end of the first period, and then grows at a rate of g every year after that first period.

In your assignment solution, tell us the amount of consumption at the end of the first year and at the end of the 20^th year. You can cut and paste the table from your Excel spreadsheet, or you can show your calculations algebraically.

c. Many financial planners use real rates to plan for their clients retirements. Given your recent experience with questions 5.1 and 5.2, why might they use real rates rather than nominal rates? (1 mark)