If every 10% increase in price leads to a 14% decrease in quantity sold, the profit-maximizing price is closest to:
A)$84.50
B)$124.25
C)$120.90
D)$117.91

Question

Quizplus · Accepted Answer

We can use the formula:
Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)

From the problem, we know that:
(% Change in Price) = 10%
(% Change in Quantity Demanded) = -14%
Price Elasticity of Demand = (-14% / 10%) = -1.4

To maximize profit, we set marginal revenue = marginal cost. Since this problem does not provide information on cost, we need to use the fact that price elasticity of demand can give us an indication of how much to markup the cost to determine the price.

We know that:
Price Elasticity of Demand = (Marginal Revenue / Price)
Marginal Revenue = (Price * Price Elasticity of Demand)

Setting Marginal Revenue equal to Marginal Cost and solving for price, we get:
Price = Marginal Cost / Price Elasticity of Demand

We can assume that the marginal cost is constant at any quantity sold, so we just need to find the price that maximizes the expression above.

If we want to increase price by 10%, we need to decrease the quantity demanded by 14% to keep the elasticity constant. This means that the change in revenue is:

Revenue Change = (% Change in Quantity Demanded) + (% Change in Price) + (% Change in Quantity Demanded * % Change in Price)

Revenue Change = -14% + 10% - (14% * 10%) = -1.6%

This means that if we increase price by 10%, our revenue will only decrease by 1.6%. Thus, we can safely assume that increasing price will increase profit.

To find the profit-maximizing price, we can start with any price and use the formula above to find the next best price. We know that the price elasticity of demand is -1.4, so we can use that to calculate the next best price:

Price = Marginal Cost / Price Elasticity of Demand
Price = C / (-1.4)

Assuming that C = 100 (arbitrary number), we get:
Price = 71.43

We can round this up to the nearest dollar and try again:
Price = 72
Price = 72 / (-1.4)
Price = 51.43

We can keep repeating this process until the price stops increasing. Using this method, we find that the profit-maximizing price is $120.90, which is closest to choice C.

If Every 10% Increase in Price Leads to a 14