Determine elasticity along a linear demand curve (geometric measure) and relate elasticity to total expenditure.
Deriving e_D = -bp/q and the geometric measure DA/DB.
You might assume that a straight-line demand curve has a constant elasticity because its slope never changes. But slope and elasticity are entirely different concepts!
A linear demand curve is written as . Even though the slope () is constant, the price elasticity of demand () actually changes at every single point along this line.
Mapping values from 0 to infinity.
Price elasticity of demand varies at different points on a linear demand curve, ranging from infinity at the price intercept to zero at the quantity intercept.
Identifying elasticity at the exact midpoint of a linear demand curve.
What is the price elasticity of demand exactly at the midpoint of a linear demand curve?
Linking Price * Quantity changes to the elasticity classification.
Ever wonder what happens to your total spending when a store changes its prices? It depends entirely on your buying habits.
Total Expenditure () on a good is equal to the demand for the good () multiplied by its price ().
Because price and quantity almost always move in opposite directions, the final effect on your total expenditure depends on how responsive your demand is to that price change.
Approximation formula for change in expenditure.
Let's mathematically prove how total expenditure on a good responds when its price changes. We start by comparing the "before" and "after" states.
Initial State:
New State:
Applying the expenditure rules to a specific scenario.
The problem asks how expenditure will be affected if the price elasticity of demand for a good is and there is a increase in its price. First, we determine the nature of the elasticity by taking its absolute value, which gives us . Since this value is less than , the demand for this good is considered price . By definition, this means the percentage decline in quantity is than the percentage increase in price. Specifically, the demand will only go down by . Because the percentage decline in quantity is smaller than the increase in the price, the total expenditure on the good will go .