Price elasticity of demand and slope of the demand curve are two important concepts in economics. Elasticity considers relative, or percent, changes. Slopes consider absolute unit changes.

Despite their differences, slope and elasticity are not entirely unrelated concepts, and it is possible to figure out how they relate to each other mathematically.

### The Slope of the Demand Curve

The demand curve is drawn with the price on the vertical axis and quantity demanded (either by an individual or by an entire market) on the horizontal axis. Mathematically, the slope of a curve is represented by rise over run or the change in the variable on the vertical axis divided by the change in the variable on the horizontal axis.

Therefore, the slope of the demand curve represents the change in price divided by change in quantity, and it can be thought of as answering the question "by how much does an item's price need to change for customers to demand one more unit of it?"

### Responsiveness of Elasticity

Elasticity, on the other hand, aims to quantify the responsiveness of demand and supply to changes in price, income, or other determinants of demand. Therefore, the price elasticity of demand answers the question "by how much does the quantity demanded of an item change in response to a change in price?" The calculation for this requires changes in quantity to be divided by changes in price rather than the other way around.

### Formula for Price Elasticity of Demand Using Relative Changes

A percent change is just an absolute change (i.e. final minus initial) divided by the initial value. Thus, a percent change in quantity demanded is just the absolute change in quantity demanded divided by quantity demanded. Similarly, a percent change in price is just the absolute change in price divided by price.

Simple arithmetic then tells us that price elasticity of demand is equal to the absolute change in quantity demanded divided by the absolute change in price, all times the ratio of price to quantity.

The first term in that expression is just the reciprocal of the slope of the demand curve, so the price elasticity of demand is equal to the reciprocal of the slope of the demand curve times the ratio of price to quantity. Technically, if price elasticity of demand is represented by an absolute value, then it is equal to the absolute value of the quantity defined here.

This comparison highlights the fact that it's important to specify the range of prices over which elasticity is calculated. Elasticity is not constant even when the slope of the demand curve is constant and represented by straight lines. It is possible, however, for a demand curve to have constant price elasticity of demand, but these types of demand curves will not be straight lines and will thus not have constant slopes.

### Price Elasticity of Supply and the Slope of the Supply Curve

Using similar logic, the price elasticity of supply is equal to the reciprocal of the slope of the supply curve times the ratio of price to quantity supplied. In this case, however, there is no complication regarding arithmetic sign, since both the slope of the supply curve and the price elasticity of supply are greater than or equal to zero.

Other elasticities, such as the income elasticity of demand, don't have straightforward relationships with the slopes of the supply and demand curves. If one were to graph the relationship between price and income (with price on the vertical axis and income on the horizontal axis), however, an analogous relationship would exist between the income elasticity of demand and the slope of that graph.