All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

Let's return once again to our list of equations.

• y = 4x + 3
• y = 4x^1/2 + 3
• y = 4x - 2z +3
• y = 10/x
• y = 2^x
• lny = 4lnx
• y = 3 + 4x²
• lnQ = -2lnP +1.5lnY

Not all relationships can be captured by linear equations, but you probably know that from your algebra and trigonometry courses. You probably also remember that nonlinear equations were often more difficult to deal with, and that has not changed. You can, however, make your life a bit easier if you realize that as long as all of the parameters have specified values, you can use spreadsheets to help you visualize the nature of the nonlinearities. For example, let's look at the seventh equation, a simple quadratic.

y = 4x² + 3

We can set up a simple spreadsheet in which we put in values for x extending from 1 to 31 and program excel so that the value for y appearing in the second column is specified by the equation. In this nonlinear case the rate of change varies with the initial value of x. If we are at x = 1 and move to x = 2, y increases by 12. The rate of change Dy/Dx = (19-7)/(2-1) = 12. If we begin at x = 30, however, and move to x = 31, the rate of change (Dy/Dx) = (3887-3603)/(31-30) = 284/1. It should come as no surprise that if we plot these points we have what looks like a parabola.

x	y
0	3
1	7
2	19
3	39
...	...
30	3603
31	3887

Equation 6 ]lny = 4lnx] is also a nonlinear equation. It is called a log linear equation and economists tend to like this specification of a relationship between two variables. Fortunately, you do not need to know anything about logs at this time. If you ever have a problem where you run into a problem involving log, you can use excel which has logarithmic functions. What is important know is the interpretation of the coefficient. In the linear equation the coefficient of x was equal to Dy/Dx. In the log linear case the coefficient of lnx equals %Dy/%Dx.

%Dy/%Dx = 4

As we saw with the linear equations, we can interpret this relationship as one equation with three unknowns. Using the 'laws' of algebra, we can rewrite this equation in the following two ways - each providing us with a different variable on the left hand side of the = sign. If you know the value of the coefficient, which is 4 in equation 6, then you rewrite the equation two ways:

1 %Dy = %Dx* 4

2 %Dx = %Dy/4

The first equation can be used to answer questions such as; how much will y increase if x increases by 3 percent? if x decreases by 4 percent? In the first case y would increase by 12 percent when y increased by 3 percent. If on the other hand you happened to know that y increased by 12 percent, you would know that x must have increased by 3 percent.

The reason why this formulation is so important is that the ratio of percentage changes has a special significance in economics. From your microeconomics you will recall the concept of elasticity. The price elasticity of demand (e_p) is defined as the percentage change in demand (Qd) divided by the percentage change in price(p) (e_p = %DQd/%DP). Similarly the income elasticity of demand is defined as the percentage change in demand divided by the percentage change in income (Y) (e_y = %DQd/%DY).

Equation 5 (y = 2^x) is also an equation which you are likely to run into repeatedly. This is an exponential function which is at the heart of present and future value analysis which in turn is at the center of the large majority of financial decisions. First, however, let's look at a couple of important equations that specify change.

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