1. Continuous Growth and the Natural Exponential Function
Chapter 12

2. 12.1 Introduction
When discounting is continuous rather than discrete, it is convenient to use natural exponential functions.
An inverse to the natural exponential function is the natural logarithmic function.
In Chapter 13 we will see that differentiating both types of function is very easy, which makes these functions even more attractive.

3. 12.2 Limitations of Discrete Compound Growth
Recap. If we deposit a USD at an annual interest rate r added n times a year for the period of x years, our deposit in the end of that period will grow to the value of
Note. For the same nominal interest rate r the effective interest rate will increase as n grows larger.
What happens to our deposit when n is infinitely large?
In other words, we want to find
Note. This limit is not equal to infinity since, as n grows larger, the small-period interest rate r/n tends to zero.
Many economic variables grow virtually every second (e.g. stock prices) so that n can be taken to be infinity without sacrificing correspondence with reality.

4. 12.3 Continuous Growth: the Simplest Case
Continuous growth is understood in terms of changes happening in very small steps: stock prices, foreign exchange, height of plants and humans, body weight etc.
Example. Suppose you invest $1 for the period of one year at the nominal interest rate of 100%. However, there is a range of possibilities as to when to add interest.
Case 1: Interest added just once. Your deposit grows to , i.e. your money just doubles.
Case 2: Interest added twice. Your deposit grows to
Case 3: n=10 implies
...
Case n= implies
A simple EXCEL simulation can help visualize the process.

5. The Number e Definition. The irrational number e is defined as
Note 1. Number e is the value of your deposit in the previous example when interest is added an infinite number of times during this one year, i.e. virtually every single instant.
Note 2. Since number e is irrational, we can never write “all of it”, but approximately it is equal to e=

6. Graphical Treatment of e
In all cases we have plots of
in the end of the first year, interest given twice a year.
in the end of the first year, interest given ten times a year.
in the end of the first year, interest given one-hundred times a year.

7. 12.4 Continuous Growth: the General Case
Recap. The number e is the value of your account if you invest $1 for one year at an annual 100% nominal interest rate.
Generalization. Suppose you invest $a at a nominal annual interest rate r for x years. In this case the value of your account is going to be
In the limit,

8. Continuous Growth
When y grows continuously from the level of a at an annual r during x years, its value at the end of year x is equal to

9. 12.5 The Graph of Simple case:
This is just an exponential function with base e, which we call the natural base.
Note 1. Since the base e is irrational, the graph of can never be drawn precisely.
Note 2. Since x measures time, x=0 means the beginning of observed time period and so is rather arbitrary.

10. The Graph of Case 1: r>1 Same intercept with , but the graph of
is steeper.
Case 1: r<1
Same intercept with ,
but the graph of
is flatter.

11. Negative r Consider the graph of
This graph is a mirror image of the graph of

12. The Graph of
Adding parameter a>0 just changes the scale and the intercept of all the graphs we considered before.
However, if a<0, not only the scale is affected, but the graph itself “flips about” the horizontal axis:

13. 12.6 Natural Logarithms
Definition. The function inverse to is called natural logarithmic function, and is denoted as
Note. Natural logarithms are so common in economics and in math that they enjoy a special notation, namely,

14. The Graph of

15. 12.7 Rules for Natural Logs
Natural logs work exact same way as do logs to any other base:

16. 12.9 Continuous Growth Applications
Consider a continuous growth case, After one year, or x=1, you get back , where a is your original investment.
The interest alone is thus
The proportionate return is thus
Rule. In case of continuous growth given by , the effective growth rate is
Example Suppose a variable is growing continuously at a rate r=10% per year. What is the effective interest rate annually?
Directly applying the formula gets us the proportionate return:
This return is about 0.5 percentage points higher than the nominal return of 10%.

17. Example GDP Growth
Suppose GDP grows at 4% per year. What is its level after 5 years? What is the cumulative percentage growth over 5 years?
Method 1. Assuming our GDP is growing in jumps, i.e. discretely, we use the formula
The cumulative percentage growth is then
Method 2. If the GDP grows continuously, we apply
The cumulative factor in method 2 is compared to in method 1, which is what we expected: cumulative growth is always higher in the continuous compared to the discrete case.

18. Continuous vs Discrete Growth
Continuous growth formula is applicable when changes occur very often.
Discrete growth formula should be used when changes happen in discrete jumps, e.g. when GDP is measured annually or quarterly (but not every minute, for instance.)

19. Example 12.3
If GDP is growing at 5% annually, how long does it take to double?
Method 1: assuming discrete growth,
Since GDP doubles in x years, , which means we need to solve
To solve this equations, we log both sides of it:
The GDP of an economy growing 5% annually will double in a little over 14 years.

20. Example 12.3: Continuous Growth
Method 2:
Taking natural logs of both sides of this equation, one obtains
Since
GDP will double in less than 14 years, i.e. faster compared to the case of discrete growth.

21. The Rule of 69
Question: How long does it take a variable to double if it grows at r annually?
The answer to this question is given by a solution to the equation
We are assuming continuous growth here. This equation boils down to
Another name for the rule is “the rule of 70” since, if we use r in percentages, we obtain the following:

22. Example 12.4
UK GDP measured in 2000 prices rose from $256,501 million in 1948 to $988,338 million in 2002, where $ stands for “pounds.” What was the average annual growth rate?
Method 1: discrete growth.
Plugging in the values for GDP levels in 1948 and 2002, we obtain:
Take natural logs of both sides of this equation:

23. Example 12.4: Continuous Growth
Method 2: Continuous growth.
Taking natural logs of both sides, one obtains:
As expected, in case of a continuous growth formula we get a lower annual growth rate since continuously growing variables grow to a higher level during the same period of time compared to their discrete analogues if the nominal growth rate is the same.

24. Continuous Discounting and Present Value
Recap: To learn the present value of a single payment a received after x years in the future given the annual interest rate r, we use the following formula:
The rationale for the present value formula: suppose I have USD today. I can deposit this amount with a bank at the interest rate r. After x years my initial deposit will grow to:

25. An Example
The present value of the same amount of money decreases over time.

26. Half-Yearly Payments
Assume now that discounting occurs twice a year.

27. Continuous Discounting
With continuous discounting, we discount n times a year:

28. Continuous Discounting

29. Semi-Log Scale
If a variable y is growing at a constant proportionate rate over time, the graph of log(y) against time is a straight line.
Example Consider an economic variable y that doubles every year as displayed by Table 12.6:
Proportionate growth from year 0 to year 1:
Proportionate growth from year 1 to year 2:

30. Semi-Log Scale
Part (a) of the graph tells us that y is growing, but we have no idea whether the proportionate growth rate is constant, increasing, or decreasing. We know it’s constant.
Part (b): 3rd row of Table 12.6 says that log(y) is increasing by the same absolute amount, which reflects constant proportionate growth rate. As a result, the graph of log(y) is a straight line.
Part (c) is better than part (b) because we have values of y marked on the vertical axis so we don’t have to guess what value of y corresponds to e.g

31. Example 12.7
Variable y grows over a ten-year period at a non-constant growth rate, see row 3. In fact, it is growing at a decreasing rate. The semi-log graph has a concave shape.
Natural Scale
Semi-log Scale

32. Example: Real Data
In pane (a) the slope of the GDP line becomes steeper after 1984, but we don’t know whether it is due to the increase in proportionate growth rate
On a semi-log scale the growth rate appears almost constant everywhere, including the period after 1984