A Math Appendix

A.1 Exponential functions

If we’re working with the Cobb-Douglas production function specifically, we end up using exponential functional forms, and so it can be handy to remember some basic rules of exponents:

Exponential functions are a compact way of writing products. For example: \[ 10 \times 10 \times 10 = 10^3 \]

We can also have fractional exponents. You may recall that the square root function is

\[ \sqrt(x) = x^{\frac{1}{2}} \]

More generally

\[\sqrt[n]{x} = x^{1/n}\].

Some important rules that you could derive from the definition:

• \(x^{-n} = \frac{1}{x^n}\)
• \(x^a \times x^{b} = x^{a+b}\)
• Combining the two previous rules, \(\frac{x^{c}}{x^d} = x^{c-d}\)
• \(x^0 = 1\) in general (to be safe, say this is true for \(x\neq 0\).)

Remember that \(x^a + x^b \neq x^{a+b}\). We add the powers across multiplication, not addition.

An important thing to remember is that something growing at a constant growth grows exponentially over time. That is, if the stating value is \(x_0\), \(t\) time periods have passed, and the variable grows at a rate \(g\) every time period:

\[x_t = x_0 \times (1+g) \times (1+g) \times \ldots \times (1+g) = x_0 (1+g)^t\]

A.2 Logarithmic Functions

Logarithmic functions are related to exponential functions, so we’ll often use them together. The relationship is that: if \(x^b = z\), then

\[\log_x (z) = log_x (x^b) = b \times log_x(x) = b \times 1 = b\]

Other useful properties of logs for our purposes are the following (illustrated using a log of base \(a\); think about \(a = 10\) if that’s helpful, but this is true for any log.)

\[a^{\log_a x} = x\]

\[\log_a(x^y) = y \times log_a(x)\]

\[\log_a (xy) = \log_a x + \log_a y\]

Be careful: \(\log_a (x + y) \neq \log_a x + \log_a y\)

We will also often use logarithms to make plots of data easier to view. Taking our exponential growth example, notice that if we were to plot \(x_t\) against time, it would increase exponentially: the slope would be increasing over time. But if we plot \(\log_a(x_t)\), the slope is constant and equal to the growth rate. Moreover, plotting on a logarithmic scale (or a ratio scale) means that equal spaces on a vertical axis are proportional changes in the variable rather than constant changes.

We’ll also often use \(\log_{10}\), which is handy for plotting. Going from \(log_{10}(x) = z\) to \(\log_{10}(y) = z+1\) implies that \(y = 10 x\).