Basic Mathematical Notes for Macroeconomics

 

Percent Change (from one period to another)
 
 

Change in macroeconomic variables over time is commonly expressed as the percent change from one period to the next period (e.g., percent change in GDP from the 1st quarter of 1999 to the 2nd quarter of 1999).  Since the change is more often positive than negative, the percent change is also referred to as the growth rate.  The following shows the calculation:
From the Bureau of Economic Analysis (BEA), the government agency that tracks the U.S. national income accounting data, the real GDP data for the 1st and 2nd quarters of 1999 are :
 
 
Period I 99 II 99
GDP 8,717.6 8,758.3
 
Therefore, the percent change from I 99 to II 99 is:
 
[(8,758.3 - 8,717.6)/8,717.6 ] x 100%  =  0.47%
However, it is customary to express growth in an annual rate (i.e., 4 quarters) so that the the annual percent change is:
 
0.47% x 4  = 1.86% =~ 1.9
Of course, if you calculate the percent change from one quarter to the same quarter in the consecutive year (e.g., I 99 to I 00), then you do not need to multiply the figure by 4.
 
 

 
 
 
 

First Differences
 
 

It is customary to approximate the growth rate of a time series in terms of first differences of its log levels.  The following provides an explanation. 
For a variable Z that takes on the value Zt in period t the growth rate between period t and t-1 is defined as

       g = (Zt+1_ Zt)/Zt

such that 

       Zt+1/Zt= 1 + g

If we take logs of both sides of the expression, then we can see that

          log(Zt+1) _ (logZt) = log(1 + g)
 
 

Since g is usually much smaller than 1, we can take advantage of the Taylor Series approximation, which takes the following form:

      log (1 + g) = g _ (g2/2) + (g3/3) _ (g4/4) + . . .

Hence, based on the first-order Taylor Series approximation, we can simply rewrite the expression for the first-differences as:

      log (1 + g) = g
 

Therefore, the growth rate (or percentage change) of the time series Z can be approximately measured by the first-difference of its log levels as:

          log(Zt+1) _ (logZt) =  g
 

 
 
 

Rule of 70
 
 

A simple way to calculate the number of periods for any number to double at the compound growth rate is to divide 70 by the percent increase per period.  For example, if a population grows at 7% per year, then it will take about 10 years for the population to double (i.e., 70/7=10).
To show how the rule of 70 is derived, let us first write out the compound growth formula for a given number Z that grows at the rate of g per period continuously for T periods:

       Z(1+g)T

For Z to double, we can apply the above formula such that:

       Z(1+g)T = 2Z

Since Z appears on both sides of the equation, they cancel out.  We can then solve for the number of periods to double, i.e., T, by taking natural logs of both sides.  The above expression becomes:

          T*log(1+g) = log2,                      or
          T = 0.69 / log(1+g)
 

Recall that based on the first-order Taylor Series expansion, we can approximate log (1+g) by g in this situation.  Therefore, we can roughly write out the last equation as:

           T =~ 0.70 / g

Since g is in percentage term, we can simply rewrite the "rule of 70" as:

           T =~ 70 / g%
 
 

Compounding/Continuous Growth
 
 

A variable P which is compounded or grows at the rate of r for a given number of periods t will have a value at the end of that time given by the exponential function

 
S = P(1+r)t. 

 
If P grows m times (continuously) at the rate r for t periods, then 

 
S = P limm-->oo[1 + (1/m)]mrt = P(2.71828) = Pert

 
Application: 100e0.03t
A function of 100e0.03t means that a variable begins with the value of 100 at the beginning and grows continuously over time (t) at 3% (.03) at each period.
 

Money Multiplier
 
 

Solving the money multiplier, which is the number of times that an increase (or decrease) in the money supply, such as bank deposits, potentially results in the total increase (decrease) in the money supply.  
 
 
Let M be the total change money supply, D be the initial change in bank deposits, and r be the required reserve ratio such that M = (1/r)D where the money multiplier is 1/r. 

Proof:
 

M = D + (1-r)1D + (1-r)2D + (1-r)3D + (1-r)4D + ... (1)
(1-r)M = (1-r)1D + (1-r)2D + (1-r)3D + (1-r)4D + ... (2)=(1-r)*(1)
M - (1-r)M = D (1)-(2)
(1 - 1 + r)M = D
M = (1/r)D
 

Created by Jim Lee.