Suppose the demand curve is given by Q=200-20p. What is the price elasticity at P=$4? is demand elastic or inelastic at this point?
Well, let me crunch some numbers and break out my elasticometer for you. Bear with me!
Using the given demand curve Q=200-20p, we can differentiate it to get dQ/dp = -20, which is the slope of the demand curve. To find the price elasticity at P=$4, we need to know the percentage change in quantity demanded (∆Q%) divided by the percentage change in price (∆P%).
Since we only have a specific price point of P=$4, we can't calculate percentage changes. However, we can still determine if demand is elastic or inelastic at this point. The general rule is that if the absolute value of the price elasticity of demand is greater than 1, it is elastic; if it is less than 1, it is inelastic.
Based on the slope of the demand curve, which is -20, it suggests that demand is inelastic at P=$4. But hey, don't take it too seriously—elastic or not, the demand curve can be a bit sensitive, just like a clown with an oversized shoe!
To calculate the price elasticity of demand at a specific price, we need to use the formula:
E = (ΔQ/Q) / (ΔP/P)
Where:
E is the price elasticity of demand
ΔQ is the change in quantity demanded
ΔP is the change in price
Q is the quantity demanded
P is the price
In this case, we need to find the price elasticity of demand at P = $4. To do this, we can substitute P = $4 into the demand equation Q = 200 - 20P and calculate the quantity demanded at that price.
Q = 200 - 20P
Q = 200 - 20(4)
Q = 200 - 80
Q = 120
Therefore, at P = $4, the quantity demanded is Q = 120.
Now, we need to calculate the change in quantity demanded and the change in price in order to proceed with the formula.
ΔQ = Q2 - Q1
ΔQ = 120 - 120
ΔQ = 0
ΔP = P2 - P1
ΔP = $4 - $4
ΔP = 0
Notice that both ΔQ and ΔP in this case are 0 because we are not calculating the change between two different prices, but rather at a specific price.
Let's plug the values into the formula:
E = (ΔQ/Q) / (ΔP/P)
E = (0/120) / (0/$4)
E = 0 / 0
In this case, we get an indeterminate form (0/0) when calculating the price elasticity of demand. Therefore, we need to use the concept of point elasticity.
Point elasticity refers to the elasticity of demand at a specific point on the demand curve rather than between two different points. To determine whether demand is elastic or inelastic at a specific point, we can look at the slope (steepness) of the demand curve.
Since the given demand curve is linear (Q = 200 - 20P), the slope is constant (-20). In this case, a constant slope implies that the price elasticity of demand is constant along the entire demand curve.
To determine whether demand is elastic or inelastic at P = $4, we need to focus on the absolute value of the slope of the linear demand curve.
In this case, the absolute value of the slope is 20. Since the absolute value of the slope is greater than 1, we can conclude that demand is elastic at P = $4.
Therefore, at P = $4, demand is elastic.