Is the supply and demand elasticity equal at the equilibrium?

I am working through homeowrk problems and we were asked to calculate the supply and demand at the equilibruim using either the point or arc method. I chose to use the point method and will provide a picture of the graph within this post. THe demand equation is 100-2p=Q and supply is -20+2p=Q. Here are my calculations [note the equilibrium occurs at (40,30)]

Demand elasticity: point method = Slope* (P/Q)

• = -2(30/40)
• = -2(3/4) Elasticity = -1.5 = 1.5 Therefore the demand elasticity is 1.5.

Supply elasticity: point method = 2(30/40) =2(3/4) Elasticity = 1.5 Therefore the supply elasticity is 1.5

It is not true that at the equilibrium point the elasticity of demand is equal to the elasticity of supply.

This is a coincidence in your particular example, depending on the fact that the slope of the demand function and the slope of the supply function are equal in absolute value: they are $$-2$$ and $$2$$.

Consider the formula of the point elasticity you wrote:

$$\epsilon=|Slope|\times \frac {P}{Q}$$.

The value of the elasticity depends on both $$|Slope|$$, the absolute value of the slope of the functions, and on the ratio $$\frac {P}{Q}$$.

$$\frac {P}{Q}$$ at the equilibrium point is of course equal for demand and supply, because the equilibrium point is a point that belongs to both functions.

But nothing ensures that the slope of demand and supply functions are equal (in absolute value).

Consider the example in the graph below, where it is clear that the slopes of demand and supply, $$-b$$ and $$a$$, are different in absolute value, and $$a>b$$:

Therefore, the point elasticity of supply $$\epsilon_S$$ is

$$\epsilon_S= a\times \frac {30}{40}$$

and the point elasticity of demand is

$$\epsilon_D= b\times \frac {30}{40}$$.

They are evidently different and $$\epsilon_S>\epsilon_D$$.

supply & demand elasticities are independent of each other and don't depend on the equilibrium.

No, the supply and demand elasticity are not always equal at the equilibrium.

Here is an example of a market where the elasticity of demand and supply are not equal at the equilibrium. Suppose the demand curve for a good is given by the equation Qd = 20 - 2P and the supply curve for the good is given by the equation Qs = 2P. The equilibrium price for the good is P = 5 and the equilibrium quantity is Q = 10. The elasticity of demand at the equilibrium is -1 and the elasticity of supply at the equilibrium is 1. Therefore, the elasticity of demand is not equal to the elasticity of supply at the equilibrium.