Given the demand function q=173-5p, determine the price where demand has unit elasticity.
Since the p-q curve has negative slope (which is normal), unit elasticity means -1.
Elasticity = (ΔQ/Q) / (ΔP/P)
=(ΔQ/ΔP) / (P/Q)
=(dQ/dP)*(P/Q)
=-1
dQ/dP = d(173-5P)/dP = -5
Therefore
-5*(P/Q) = -1
-5P = -1(173-5P)
Solve for P to get P=17.3
To find the price at which demand has unit elasticity, we need to find the price (p) when the absolute value of the demand elasticity (E) equals 1.
The elasticity of demand (E) is given by the formula:
E = (dq/dp) * (p/q)
Where dq/dp represents the derivative of demand with respect to price.
Let's calculate the derivative dq/dp:
dq/dp = 0 - 5
Since the derivative is a constant (-5), we can directly substitute the value into the formula for elasticity:
E = (-5) * (p/q)
Now, we can substitute the demand function (q = 173 - 5p) into the elasticity formula:
1 = (-5) * (p / (173 - 5p))
Simplify the equation:
1 = -5p / (173 - 5p)
Cross multiply to get:
173 - 5p = -5p
Bring the terms involving p on one side:
173 = -5p + 5p
173 = 0
Since the equation simplifies to 173 = 0, this means there is no solution. It implies that demand does not have a unit elasticity under the given demand function q = 173 - 5p.
To determine the price at which demand has unit elasticity, we need to find the price at which the absolute value of the price elasticity of demand is equal to 1.
The price elasticity of demand is given by the formula:
E = (dq/dp) * (p/q)
Where:
E is the price elasticity of demand
dq/dp is the derivative of the demand function with respect to price
p is the price
q is the quantity demanded
First, let's find dq/dp, the derivative of the demand function with respect to price:
dq/dp = -5
Now, let's substitute this derivative back into the price elasticity of demand formula:
E = (-5) * (p/q)
Since we want to find the price at which the absolute value of the price elasticity of demand is equal to 1, we can set the absolute value of E equal to 1 and solve for p:
|E| = 1
|-5*(p/q)| = 1
5*(p/q) = 1
p/q = 1/5
p = q/5
Now, let's substitute the demand function q=173-5p into the equation p=q/5 and solve for p:
p = (173-5p)/5
5p = 173-5p
10p = 173
p = 173/10
p = 17.3
Therefore, the price at which demand has unit elasticity is $17.3.