Question 9.1.1a

In lecture, Professor Mark explained discrete compounding interest. Interest can also be compounded continuously. Here we explain the difference.

Professor Mark calculated future value as jQuery22405720561620500767_1596069437086=?(1+?) ?, where ? is the principal, ? is the interest rate,and ? is the term of the contract (often in years). This formula can be generalized to ??=?(1+?/?) ??, where ? is the number of compounding periods per year (in lecture, this was 1). That is, after every compounding period, more interest accrues on both the principle and the previous accrued interest. This is discrete compounding.

Supposethat ? becomes large. For example, the interest rate could be 10% per year, but compounded each minute. Future value rises as ? increases, but it rises at a diminishing rate. It turns out that as ? goes to infinity, future value is described by an exponential function,??=????, where ? is the base of the natural logarithm. This is continuous compounding.

You are planning to invest in fine wine. Each case costs $100 (at time 0), and you know from experience that the (future) value of a case of wine held for ? years is 100? for ?1. (Suppose that the value is 100 for 0<?<1. Why do you think we make this assumption?) Wine is thus increasing in value over time. One case of wine is available to purchase, and the annual interest rate you can earn by keeping money in the bank is 10 percent.

Should you buy the case of wine?

A) You are indifferent between buying the case of wine and not

B) No, you should not buy the case of wine

C) Yes, you should buy the case of wine

How many years should you hold the case of wine before selling it? (Consider only integer years. Suppose that interest is compounded discretely (annually), not continuously.)

How much money will you receive for the case of wine at the time of its sale?

What is the net present value of that investment?

Question 9.1.1b

Let's consider the problem that is the continuous-time analog to the above discrete-time problem. This is perhaps a more realistic problem.

Suppose interest is compounded continuously instead of discretely. You may again choose how many years to hold the case of wine before selling it - but now suppose you can sell at any time, not just at the integer years.

How many years should you hold the case of wine before selling it?

How much money will you receive for the case of wine at the time of its sale?

What is the net present value of that investment?

Question 9.1.2

Suppose we are still in the continuous-time model.

Immediately after you purchase the case of wine, still at time ?=0, a fellow wine enthusiast offers to buy your case of wine for $130. (If you sell your case, you won't have an opportunity to buy another for yourself.)

Should you accept this offer?

A) You are indifferent between selling the case of wine and not

B) No, you should not sell the case of wine

C) Yes, you should sell the case of wine

Question 9.1.3

How would your answers qualitatively change if the interest rate were only 5 percent? Would you still buy the wine? Would you keep it for longer or shorter?

A)

You would buy the case of wine, but keep it for a longer time B)You would buy the case of wine, but keep it for a shorter time

C)You are indifferent between buying the case of wine and not

D)You would buy the case of wine, and keep it for the same length of time

E)You would not buy the case of wine

F)You would buy the case of wine, whereas before you would not have bought it

Question 9.2.1

Suppose that there are only 10 individuals in the economy each with the following utility function over present and future consumption: ?(?1,?2)=?1+23?2, where ?1 is consumption today, and ?2 is consumption tomorrow. Consumption tomorrow is less valued because people are impatient and prefer consuming now rather than later. Buying 1 unit of consumption today costs $1 today and buying 1 unit of consumption tomorrow costs $1 tomorrow. All individuals have income of $10 dollars today and no income tomorrow (because they will be retired) but they can save at the market interest rate ?0.

What is the price today of one unit of consumption tomorrow?

To verify that you have the correct answer, calculate the price today of one unit of consumption tomorrow if ?=0.05.

?=

Question 9.2.2

Write an expression for an individual's budget constraint in terms of today's and tomorrow's consumption expenditure.

To verify that you have the correct answer, calculate the maximum you can consume today if you consume 6.24 tomorrow and the interest rate is 0.07.

?1=

Question 9.2.3

Draw the indifference curve. (This is for your understanding. Your drawing will not be graded.)

How much of his or her income will an individual consume today given that the interest rate is 0.3?

A. Less than half of it

B. Exactly half of it

C. The individual is indifferent between consuming today and saving

D. More than half of it

E. All of it

F. None of it

How much of his or her income will an individual consume today given that the interest rate is 0.5?

A. Less than half of it

B. The individual is indifferent between consuming today and saving

C. None of it

D. All of it

E. More than half of it

F. Exactly half of it

How much of his or her income will an individual consume today given that the interest rate is 0.7?

A. Less than half of it

B. All of it

C. None of it

D. More than half of it

E. Exactly half of it

F. The individual is indifferent between consuming today and saving

Question 9.2.4

Suppose that in this economy all the funds for capital come from savings by the 10 individuals. Firms' demand for capital is given by ??=100100?.

What is the market supply for funds if the interest rate is 30%?

??=

What is the market supply for funds if the interest rate is 70%?

??=

What is the equilibrium interest rate that clears the capital market?

?=

What is aggregate consumption in each period at that interest rate?

?1=

?2=