# Assuming a fall in price, does consumer surplus fall more with elastic or inelastic price elasticity of demand

I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.

But what about the change in consumer surplus. Some sources i've seen say the change is larger for elastic P.E.D with other sources saying the opposite, all with minimal explanation of the reason.

Would any of you be able to give a definitive answer with a reason behind it because nothing I seem to

Let $$D$$ denote the demand function. The consumer surplus when the price is $$p$$ is:

$$CS(p)=\int_p^\infty D(x)dx$$

The derivative (using the Leibniz integral rule) is

$$CS'(p)=-D(p)$$

So the (absolute) change in consumer surplus (with a marginal change in price) is equal to the quantity demanded at the initial price. It follow that the absolute change in consumer surplus cannot be related to the price elasticity of demand (without further assumptions).

The proportional change in consumer surplus (the elasticity of consumer surplus with respect to price) is:

$$\frac{pCS'(p)}{CS(p)}=-\frac{pD(p)}{CS(p)} \tag{1}$$

So the proportional change in consumer surplus is equal to the negative of the ratio of expenditure to consumer surplus (and cannot be related directly to the price elasticity of demand).

This first paragraph of your question is incorrect (at least without further assumptions about demand and about the supply side):

I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.

The magnitude of consumer surplus depends on more than just the elasticity. In particular, it depends on the price (and the equilibrium price depends on the cost function(s) and market structure).

One case where we can say that consumer surplus decreases with an increase in price elasticity of demand (at the initial equilibrium price) is when:

• There is perfect competition
• Demand is linear
• Demand becomes more elastic by rotating it anticlockwise about the initial equilibrium price (so that the demand becomes flatter and so more elastic at the initial equilibrium price)

(That consumer surplus rises in this case can easily be verified diagrammatically.)

In particular, when demand is perfectly elastic consumer surplus will be zero and as demand becomes steeper it grows without bound (consumer surplus is undefined when demand is perfectly inelastic).

The consumer surplus in price inelastic is much greater than the consumer surplus of price elastic demand.

When the price falls, attributed to a increase in supply (rightward shift in the supply curve), there is an increase in quantity demanded (movement along the demand curve) due to the lower price.

This leads to an increase in consumer surplus in both price inelastic and elastic demand situations. However, the incremental consumer surplus from the fall in price in price inelastic demand is greater than the incremental consumer surplus in price elastic demand.

You can find this out by graphing/drawing hypothetical demand and supply curves.