Jen, Eric, and Kurt are all buyers of chain saws.Jen's demand function is Q_j  520  13P, Eric's demand function is Q_e  40  P, and Kurt's demand function is Q_k  200  5P.Together, these three constitute the entire demand for chainsaws.At what price will the price elasticity of market demand be 21?

A) $19
B) $20
C) $25
D) $15
E) none of the above.

Question

Faith Ngaira · Accepted Answer

Final Answer : B, Explanation :  The elasticity of demand for a market is calculated using the formula:E = (% change in quantity demanded)/(% change in price)We're given the demand functions for Jen, Eric, and Kurt and we need to find the market demand function. We can get the market demand function by summing up the individual demand functions:Q = Q j  + Q e  + Q k Plugging in the demand functions, we get:Q = 520 - 13P + 40 - P + 200 - 5PQ = 760 - 19PNow we need to find the price at which the elasticity of demand is 21.We know that:E = 21We also know that:E = (% change in quantity demanded)/(% change in price)We can rearrange this equation to get:(% change in quantity demanded) = E*(% change in price)Using the midpoint formula, we can calculate the percentage changes in quantity demanded and price:% change in quantity demanded = [(Q2 - Q1)/((Q2 + Q1)/2)]*100% change in price = [(P2 - P1)/((P2 + P1)/2)]*100Let Q1 = 760 and P1 = 20 (initial quantity demanded and price)Let Q2 = 760*1.21 and solve for P2 (final quantity demanded after a 21% increase in quantity)% change in quantity demanded = [(760*1.21 - 760)/((760*1.21 + 760)/2)]*100 = 10%% change in price = [(P2 - 20)/((P2 + 20)/2)]*100Using the equation:(% change in quantity demanded) = E*(% change in price)We get:10 = 21*(% change in price)% change in price = 10/21Now we can solve for P2:(P2 - 20)/((P2 + 20)/2)*100 = 10/2121(P2 - 20) = 10(P2 + 20)21P2 - 420 = 10P2 + 20011P2 = 620P2 = 56.36Rounding down to the nearest dollar, the price at which the elasticity of demand is 21 is $56.Plugging this price into the market demand function, we get:Q = 760 - 19*56Q = 644Therefore, the final answer is B ($20).

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