Unit (or unitary) price easticity of demand (PED) is defined as 'unit' or 'unitary' when the change in demand as a result of a change in price is 'proportionate'.
This means that a given % change in price leads to the same % change in demand, with the co-efficient of PED equaling 'one' (hence the name 'unitary').
If a producer reduces the price of a product from $200 to $190, and over a given period of time demand increases from 500 to 525, PED will be:
When PED is unitary, any given 'small' change in price will keep total revenue (TR) = (price x quantity, PxQ) the same. This means that when PED = 1, marginal revenue (MR) must be at zero. This is because any change in price (either up or down) will not reduce total revenue - hence the change in total revenue must be zero.
A demand curve with a ‘unit’ PED value over its whole length is called a rectangular hyperbola. This means that at all points, price times quantity is the same value. As price times quantity equals total revenue, total revenue (TR) is equal at all points.
In the above diagram, any change in price (P to P1, or P to P2) leads to the same total revenue as at the original price. Hence, the areas (P,A,Q,0,- P1,B,Q1,0, and P2,C,Q2,0) are identical.
In some sense, unit PED is a highly theoretical concept given that the probability that any given percentage change in price will lead to an equal and opposite percentage change in quantity demanded is very low. However, it is still a useful concept when used by organisations (especially in the public sector) to try to engineer a situation of constant revenue.
For example, if a coffee producing country such as Brazil wishes to stabilise the income (revenue) going to coffee producers, and assuming it can set the domestic price, it could attempt to set a price each year which tried to stabilise income.
If the government or organisation predicted that the demand for coffee was likely to rise by 5% next year, it could engineer a price reduction of 5% so that revenue remained constant. Of course, this policy would be designed to stabilise income (rather than increase income, which would happen if coffee prices are allowed to move upwards.)
The logic behind this can be best understood when appreciating that income instability can be a major problem for commodity producers in developing countries. However, this strategy can only work effectively when the particular government establishes tight control of the market through the establishment of a 'marketing board'. See marketing boards in Canada.