Very elastic supply + very inelastic demand

Question

Suppose we have a very elastic supply curve and a very inelastic demand curve. Further suppose there is a decrease in input prices which shifts the supply curve outward. By taking an extreme representation (vertical demand curve and horizontal supply curve), I can see that it will be mainly price that is affected: the equilibrium price decreases while equilibrium quantity hardly changes.

How can I show this more intuitively or mathematically?

Edit: Just thought about this. I don't think the elasticity of the supply curve matters. Only the elasticity of the curve that isn't shifting does. Is this the answer?

Answer 1

Consider the inverse functions

$$Q^d = A-bP$$

$$Q^s = C + dP$$

to have the situation described by the OP, we will have that $b$ is "very small" while $d$ is "very large".

Equilibrium price and quantity are

$$P^* = \frac {A-C}{d+b} = \frac {A}{d+b}-\frac {C}{d+b}$$

$$Q^* = A- b\frac {A-C}{d+b} = A- b\frac {A}{d+b}+b\frac {C}{d+b}$$

The change described by the OP amounts to $\Delta C > 0$. Then

$$\Delta P^*|_{\Delta C} = -\frac {1}{d+b}\Delta C $$

while

$$\Delta Q^*|_{\Delta C} = \frac {b}{d+b}\Delta C$$

Comparing the two in absolute terms, we get

$$\frac {|\Delta P^*|}{\Delta Q^*} = \frac 1 b$$

... from which we conclude:

1) To have equilibrium price be relatively more affected than equilibrium quantity when the supply curve shifts outwards, we need the marginal effect of price on quantity demanded ($b$) to be smaller than unity

2) The above does not depend on how elastic or inelastic is the supply curve.

Answer 2

I'd say the best option is to draw the graph, with actual values and compare the percentage change in the quantity demanded to the percentage change in price. Then use the PED formula to calculate the elasticity. Since the only thing being considered here is the elasticity of demand, the elasticity of the supply curve is irrelevant.