Final answer:
The current elasticity of demand for the aluminum doors is inelastic at approximately -0.58. The current value of the manufacturer's marginal revenue function suggests that reducing the price by $15 would lead to a decrease in total revenue, with a marginal revenue of approximately -$45 per additional door sold.
Step-by-step explanation:
To estimate the current elasticity of demand for the aluminum doors, we can use the midpoint formula, which is defined as the percentage change in quantity demanded divided by the percentage change in price. We have the original quantity (Q1) of 400 doors at a price (P1) of $70, and the new quantity (Q2) of 460 doors at a price (P2) of $55. Using the formula, the elasticity of demand (Ed) equals:
Ed = ((Q2 - Q1) / ((Q2 + Q1) / 2)) / ((P2 - P1) / ((P2 + P1) / 2))
Ed = ((460 - 400) / (460 + 400) / 2)) / (($55 - $70) / ($55 + $70) / 2))
Ed = (60 / 430) / (-15 / 62.5)
Ed = 0.1395 / -0.24 = -0.58125
The negative sign indicates the inverse relationship between price and quantity demanded. Since the absolute value of Ed is less than 1, demand is inelastic at the current price range.
To estimate the current value of the manufacturer's marginal-revenue (MR) function, we need to know the change in total revenue when an additional unit is sold. Assuming a linear demand curve, a decrease in price results in an increase in quantity demanded. Because we are only given two points, we can't provide an exact MR function, but we can give an approximation based on the given information:
MR ≈ ΔTR / ΔQ
MR ≈ (($55 - $70) * 400 + $55 * 60) / 60
MR ≈ (($-15 * 400) + $3300) / 60
MR ≈ (-$6000 + $3300) / 60
MR ≈ -$2700 / 60
MR ≈ -$45
This negative MR indicates that reducing the price will decrease total revenue, which is consistent with inelastic demand.