Mathematical Economics
Dr. hab. David Ramsey
e-mail: [email protected]
home page: www.ioz.pwr.edu.pl/pracownicy/ramsey
Office 5.16, B-4
Office hours: Monday 16-18, Thursday 14-16
28 września 2016
1 / 40
Outline of Course
1. Interest rates, Inflation and Discounting
2. Valuation of Income Streams. Life Insurance.
3. Theory of Supply and Demand.
4. Direct and Indirect Taxation
5. Introduction to Decision Theory
6. Introduction to Game Theory.
7. Applications of Game Theory to Economics.
2 / 40
1.1 The Present and Future Value of Money
Even when there is no inflation, it is rational to prefer obtaining
$100 today rather than obtaining $100 in e.g. a month’s time.
This due to the following factors
1. The possibility of utilizing the money now rather
than later.
2. Uncertainty. Will I really get the money in a month’s
time?
The face value of money, here $100, (which does not take into
account when the money is obtained) is called the nominal value.
3 / 40
Discounting the Value of Money over Time
The simplest method of discounting the value of money over time
is exponential (or geometric) discounting.
The present value of the promise of obtaining x at t time units in
the future is given by V (x; t), where
V (x; t) = xαt ,
α is the discount factor, 0 < α < 1.
One may say that an individual is indifferent between receiving the
present value (xαt ) now and receiving x at t time units in the
future.
4 / 40
Example 1.1
Suppose the discount factor is 0.9 and time is measured in years.
Calculate the present value of $100 to be obtained in t years where
t = 1, 2, 3.
5 / 40
Example 1.1
6 / 40
The Discount Rate for Different Units of Time
The discount rate is normally given as an annual rate, α.
We may wish to calculate the monthly or daily rate.
Suppose that there are k periods per year (e.g. when a period is
one month k = 12), then the discount rate per period is given by
αp , where
√
αp = k α
and α is the annual discount rate.
7 / 40
The Discount Rate for Different Units of Time
Suppose the annual discount rate is 0.9. Then the monthly
discount rate is given by
√
12
0.9 ≈ 0.99126.
Note that the annual discount rate (i.e. for 12 months) is
αp12 = 0.9.
8 / 40
Advantages of Exponential Discounting
In mathematical terms, exponential discounting shows the property
of time consistency, i.e.
If an inidividual is indifferent between obtaining x1 now and
obtaining x2 at t time units in the future, then
that individual is indifferent between obtaining x1 at k time units
in the future and obtaining x2 at k + t time units in the future.
9 / 40
Advantages of Exponential Discounting
For example, if an individual is indifferent between receiving $90
today and receiving $100 in a year’s time.
Under exponential discounting,
that individual is indifferent between receiving $90 in 5 years’ time
and receiving $100 in 6 years’ time.
Note that the phrase ”indifferent between... and” in the above
condition can be replaced by ”prefers...to” or ”does not prefer...to”
as appropriate.
10 / 40
Disadvantages of Exponential Discounting
In reality, individuals often place a higher stress on immediate
payments, e.g it is quite possible that an individual
a) prefers obtaining $90 today to obtaining $100 in a year’s time,
but
b) does not prefer obtaining $90 in five years’ time to obtaining
$100 in six years’ time.
Exponential discounting cannot describe such preferences.
11 / 40
Quasi-hyperbolic Discounting
Under quasi-hyperbolic discounting, the present value of obtaining
x at t time units in the future, where t > 0, is given by
V (x; t) = xβαt ,
where 0 < α < 1, 0 < β ¬ 1.
The parameter β describes how strongly an individual demands
immediate payment.
The larger β, the more patient an individual is, i.e. less demanding
of immediate payment.
β = 1 corresponds to exponential discounting.
12 / 40
1.2 Simple and Compound Interest
When investing money, we expect that the nominal value of our
investment increases over time.
Suppose simple interest is applied to an investment of nominal
value x at an annual rate of 100R%.
Then the nominal value of the investment increases by Rx per time
unit.
i.e. interest is always calculated on the nominal value of the initial
investment.
13 / 40
Simple Interest
It follows that the nominal value of an initial investment x at a
simple interest rate of 100R% after t time units is given by
I (x; t) = x(1 + Rt)
Note that if simple interest is added to an investment of $100 at
5% per annum, then the nominal value of the investment increases
by $5 each year.
14 / 40
Compound Interest
In practice, compound interest is applied to investments.
Suppose compound interest is applied to an investment of nominal
value x at an annual rate of 100R%.
Then the nominal value of the investment is multiplied by (1 + R)
per time unit.
i.e. the interest is always calculated on the present nominal value
of the investment.
15 / 40
Compound Interest
It follows that the nominal value of an initial investment x at a
simple interest rate of 100R% after t time units is given by
I (x; t) = x(1 + R)t
16 / 40
Example 1.2
Suppose $1000 is invested.
Calculate the nominal value of the investment after 3 years,
when interest is
a) simple at annual rate 5%,
b) compound at annual rate 3%
17 / 40
Example 1.2
18 / 40
Example 1.2
19 / 40
1.3 Practical Aspects of Interest Rates - Capitalisation
The formulas used above assume that interest is added
continuously, i.e. t can take any real, non-negative value.
In practice, interest is only added at the end of each period, e.g. at
the end of a day or month.
Hence, when interest is not added continuously, the above
formulas are only appropriate at the end of a period.
The annual interest rate is quoted (sometimes with the monthly or
the daily rate).
It is assumed that there are 12 months and 360 days in a year, i.e.
30 days per month.
20 / 40
Capitalisation of Simple Interest
Suppose the annual rate is 100R% and interest is added k times a
year (there are k periods per year).
In this case, the interest rate per period, Rp , is simply Rp =
R
k.
21 / 40
Capitalisation of Simple Interest
Hence, for example, simple interest of 6% per annum is equivalent
to simple interest of 0.5% per month.
If interest is added each month, then e.g. after 60 days (2 months)
the value of the investment has increased by 2 × 0.5 = 1%
compared to the initial investment.
The value of this investment will then remain at the same value
until 90 days (3 months) after the initial investment, when the
value of the initial investment will have increased by 1.5%.
22 / 40
Capitalisation of Compound Interest
Suppose the annual rate is 100R% and compound interest is added
k times are year.
In this case, the interest rate per period, Rp , satisfies
√
√
k
k
(1 + Rp )k = (1 + R) ⇒ (1 + Rp ) = 1 + R ⇒ Rp = 1 + R − 1
23 / 40
Capitalisation of Compound Interest
For example, if the annual rate of interest is 6% and interest is
added monthly, i.e. k = 12, then the rate of interest per month is
given by
√
12
Rp = 1.06 − 1 ≈ 0.004868 = 0.4868%.
It should be noted that when compound interest is applied k times
per year, then Rp < Rk .
24 / 40
Capitalisation of Compound Interest
On the other hand, when monthly interest is compounded, the
annual interest rate is greater than 12 times the monthly interest
rate.
From above, we have (1 + Rp )k = (1 + R), where Rp is the
interest rate per period, k the number of interest payments per
year and R the annual interest rate.
Given a monthly rate of 0.5%, we have
(1 + R) = (1.005)12 = 1.06168.
Hence, the annual rate of interest is 6.168%.
25 / 40
Example 1.3
a) Suppose simple interest is paid at a rate of 7.2% per annum.
Calculate the interest rate per period when interest is added
i) monthly
ii) daily.
b) Suppose $200 is invested. Calculate the value of this investment
after 75 days when interest is added
i) monthly
ii) daily.
c) Carry out the corresponding calculations for the case when
compound interest is paid.
26 / 40
Example 1.3
Note: The number of capitalisations is equal to the number of
complete periods.
e.g. 75 days is 2.5 months, i.e. 2 complete months.
Thus when capitalisation is carried out monthly (or quarterly),
when calculating the value of an investment, this number is always
rounded downwards to the complete number of months (quarters).
75 days is less than 1 quarter (3 months = 90 days). Hence, if
capitalisation is carried out quarterly, then after 75 days the value
of the investment is still $200 (no capitalisation has occurred).
27 / 40
Example 1.3
28 / 40
Example 1.3
29 / 40
Example 1.3
30 / 40
Example 1.3
31 / 40
Time to reach a given nominal value
We now consider the smallest amount of time required for the
nominal value of an investment to reach a given value k.
In the case of simple interest, we have to solve the linear equation
x(1 + Rt) = k to find t.
In the case of compound interest, we have to solve the equation
x(1 + R)t = k ⇒ (1 + R)t =
k
x
This is done by taking logs on both sides.
32 / 40
Time to reach a given nominal value
It should be noted that when capitalisation is not continuous, the
appropriate nominal value is only achieved at the end of the
appropriate period.
Hence, in this case the number of periods should be rounded
upwards.
33 / 40
Example 1.4
Calculate the time required for the value of an investment to
double when compound interest is added at a rate of 7.2% per
annum and interest is added
a) continuously
b) daily,
c) monthly.
34 / 40
Example 1.4
35 / 40
Example 1.4
36 / 40
Example 1.4
37 / 40
Example 1.4
38 / 40
1.4 The Return Rate from Fixed Capital
Suppose that the annual income from renting a flat of value x (or
using some other form of fixed capital of value x) is I .
Then the return rate, R, is given by
I
x
Multiplying by 100, we obtain the percentage return rate.
R=
39 / 40
The Return Rate from Fixed Capital
For example, given the value of a flat is $400 000 and the income
from its rental is $1 500 per month, then the annual return rate is
R=
1 500 × 12
= 0.045.
400 000
Hence, the return rate is 4.5%.
Note: I use the same notation for return rate and interest rate,
since they both measure the rate of return on a certain investment.
40 / 40