How to Work with Growth Rates in Economics

Growth Rates in Economics

When working with various economic models, case in point being the Extended Solow Model (a topic of several recent articles, see here), it’s important to have a solid base in working with growth rates of ratios, products, and powers.

Specifically, growth rates are any rate describing how a variable increases over time. When working with GDP, productivity, price levels, and other economic variables, we’ll often see growth rates of this sort.

We’ll explore these two core rules that will cover most economic problems you’ll deal with:

1. The growth rate of a product of variables growing at a constant rate is equivalent to the sum of the growth rates. Conversely, the growth rate of a ratio of two variables growing at a constant rate is equivalent to the growth rate of the numerator minus the growth rate of the denominator.
2. The growth rate of a variable raised to a constant equals the constant multiplied by the growth rate of the variable.

To break down these rules, we’ll use Z, Y, and Z to represent three variables that are increasing over time. “a” is a constant parameter, meaning that it remains the same regardless of changes in the variable it influences—it is exogenous, and used to scale or adjust variable behavior in a model, regardless. Finally, gᵢ is the growth rate (increase) of ᵢ — so, the growth rate of a variable “ₚ” describing GDP is gₚ.

Growth Rates of Products

Say I give you the inflation rate and real GDP, respectively sitting at 4% and 2%, and ask you to compute the growth rate of nominal GDP. How would you do it?

That’s what we’ll figure out with this rule. Let’s say that Z = Y*X, or that Z is the product of Y and X. For example, we know that nominal GDP is the product of the price level and real GDP, such that $Y = P*Y. Given this setup, we can say:

• gz = gᵧ + gₓ

Or, in our example:

• gᵧ = gₚ + gᵧ

So, basically, if we’re multiplying the bases, then we just add the growth rates. In the case of our nominal GDP setup, we’re saying that the growth rate of nominal GDP is equivalent to the growth rate of the price level (inflation rate) added to the growth…