Final answer:
The elasticity of demand at a price of $20 must be computed using the demand curve and the derivative of that curve with respect to quantity demanded. Depending on the value determined, demand can be classified as elastic, inelastic or unit elastic, which will guide whether the company should raise or lower the price for maximum revenue. For maximum revenue, set marginal revenue equal to zero and solve for the price.
Step-by-step explanation:
Determining Elasticity of Demand
To find the elasticity of demand when the price is $20 using the demand curve p = 540 - 18√q, we first need to find the corresponding quantity demanded (q). Plugging in p = 20 into the demand equation yields:
20 = 540 - 18√q
√q = (540 - 20)/18
√q = 520/18
q = (520/18)²
q = 841.1111...
Now, the elasticity of demand (E) can be calculated using the formula E = (p/q) * (dq/dp), where dp/dq is the derivative of the demand curve with respect to q. The derivative dp/dq is
-18/(2√q) which is -9/(√q).
So, we compute:
E = (20 / 841.1111) * (-9/√q) at the quantity demanded found above.
Plugging in the values gives us the elasticity at q = 841.1111...
To answer the subsequent questions, if E is greater than 1, demand is considered elastic, if E is less than 1, demand is inelastic, and if E is equal to 1, the demand is unit elastic. Depending on whether demand is elastic or inelastic, the firm should lower or raise prices to increase revenue. Specifically, if demand is elastic, the firm should lower the price, and if demand is inelastic, the firm should raise the price. When demand is unit elastic, changing the price will not affect the revenue.
For part d, we would set the marginal revenue equal to zero and solve for p to find the maximum revenue point. The marginal revenue is the derivative of the revenue function R(p) which is equal to p*q.