Mr. Smith has the following demand equation for a certain product: Q = 30 - 2P.
a. At price of $7, what is point elasticity?
b. Between prices of 5$ and 6$, what is the arc elasticity?
c. If the market is mark up of 100 individuals with demand curve identical to Mr. Smith's, what will be the point and arc elasticity for the conditions specified in parts a and b?
Q= 30-2(7)= 16 E p = -2(7/16)= -0.88 Raising the price by 1%, the quantity demanded goes down by 0.88%
a. To find the point elasticity at a price of $7, we first need to find the quantity demanded at that price. We can substitute 7 for P in the demand equation and solve for Q:
Q = 30 - 2P
Q = 30 - 2(7)
Q = 30 - 14
Q = 16
So, at a price of $7, the quantity demanded is 16.
To find the point elasticity, we can use the formula:
E = (dQ/dP) * (P/Q)
Where dQ/dP represents the derivative of the demand equation with respect to price.
dQ/dP = -2
E = (-2) * (7 / 16)
E = -14 / 16
E = -0.875
So, the point elasticity at a price of $7 is -0.875.
b. To find the arc elasticity between prices of $5 and $6, we need to find the quantities demanded at each of these prices. We can substitute these prices into the demand equation:
For P = $5:
Q = 30 - 2(5)
Q = 30 - 10
Q = 20
For P = $6:
Q = 30 - 2(6)
Q = 30 - 12
Q = 18
Now, we can use the formula for arc elasticity:
E = ((Q2 - Q1) / (Q1 + Q2)) / ((P2 - P1) / (P1 + P2))
E = ((18 - 20) / (20 + 18)) / ((6 - 5) / (5 + 6))
E = (-2 / 38) / (1 / 11)
E = -2/38 * 11/1
E = -22/19
So, the arc elasticity between prices of $5 and $6 is approximately -1.157.
c. If the market consists of 100 individuals with demand curves identical to Mr. Smith's, we can multiply the point and arc elasticities by 100 to find the total elasticities for the market.
For part a, the price is $7 and the point elasticity is -0.875. Multiply the point elasticity by 100:
Point elasticity for the market = -0.875 * 100 = -87.5
For part b, the prices are $5 and $6 and the arc elasticity is approximately -1.157. Multiply the arc elasticity by 100:
Arc elasticity for the market = -1.157 * 100 = -115.7
So, for the market of 100 individuals, the point elasticity is -87.5 and the arc elasticity is -115.7.
To find the elasticity, we need to differentiate between point elasticity and arc elasticity.
a. **Point elasticity** measures the elasticity of demand at a specific point on the demand curve. It is calculated using the formula:
Point elasticity = (dQ/dP) * (P/Q)
Given the demand equation Q = 30 - 2P, we can differentiate with respect to P to find dQ/dP:
dQ/dP = -2
And we can substitute the given price P = $7 into the equation:
Q = 30 - 2P
Q = 30 - 2(7)
Q = 30 - 14
Q = 16
Now, we can substitute these values into the formula for point elasticity:
Point elasticity = (-2) * ($7/16)
Point elasticity = -14/16 (-0.875)
Therefore, the point elasticity at a price of $7 is -0.875.
b. **Arc elasticity** measures the elasticity of demand between two points on the demand curve. It is calculated using the formula:
Arc elasticity = [(Q2 - Q1) / (Q1 + Q2)] / [(P2 - P1) / (P1 + P2)]
Given the demand equation Q = 30 - 2P, we can substitute the given prices P1 = $5 and P2 = $6:
Q1 = 30 - 2(5) = 30 - 10 = 20
Q2 = 30 - 2(6) = 30 - 12 = 18
Now, we can substitute these values into the formula for arc elasticity:
Arc elasticity = [(18 - 20) / (20 + 18)] / [(6 - 5) / (5 + 6)]
Arc elasticity = [-2/38] / [1/11]
Arc elasticity = -22/38 (-0.579)
Therefore, the arc elasticity between prices of $5 and $6 is approximately -0.579.
c. If the market is made up of 100 individuals with demand curves identical to Mr. Smith's, the point elasticity and arc elasticity from parts a and b would remain the same. The elasticity measures are not affected by the number of individuals in the market, as long as they all have identical demand curves.