Question

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Calculating Elasticities Show your work for full points. Given Qd = 100-5(Pq) + 1 (PZ) + 0.5 (inc) Where Qd-quantity of Q demanded Pq price of Q Pz- price of another (cross) good Z and nc income levels. Using Calculus, calculate the own price elasticity of demand and determine if the demand for Q is elastic or inelastic if Qd-80 and Pq-20. (3 points) Using Calculus, calculate the cross price elasticity of demand for Z and determine if it is a complement or substitute if Q-80 and Pz-5. (3 points) Using Calculus, calculate the income elasticity of demand for Z and determine if it is an inferior or normal good if Q-80 and Inc #150. (3 points)

Answer 1

Q_d = 100 - 5P_q + P_z + 0.5I Own price elasticity measures how much (%) the quantity demanded change for a change (%) in its price. e_p = dQ/Q/dP/P Where e_p is the own price elasticity of demand. Using calculus, e_p = partial differential Q/partial differential P/Q/P partial differential Q/partial differential P = -5 Q/P = 80/20 = 4 implies e_p = -5/4 = -1.25 (Please note, we usually ignore the negative sign and take the absolute value) Absolute value of Own Price Elasticity of Demand > 1. Hence the demand is elastic when Q = 80 and P = 20. Cross price elasticity measures how much (%) the quantity demanded change for a change (%) in price of a different commodity. e_cp = dQ/Q/dP_z/P_z Where P_z represent the price of a different commodity Using calculus, e_cp = partial differential Q/partial differential P_z/Q/P_z partial differential Q/partial differential P_z = 1 Q/P_z = 80/5 = 16 implies e_cp = 1/16 = 0.0625 > 0 The cross price elasticity is greater than

1) Own price elasticity measures how much (%) the quantity demanded change for a change (%) in its price.

Where e_p is the own price elasticity of demand.

Using calculus,

  

(Please note, we usually ignore the negative sign and take the absolute value)

Absolute value of Own Price Elasticity of Demand > 1. Hence the demand is elastic when Q=80 and P = 20.

2) Cross price elasticity measures how much (%) the quantity demanded change for a change (%) in price of a different commodity.

Where P_z represent the price of a different commodity

Using calculus,

The cross price elasticity is greater than 0. This means, as the price of other good goes up, demand Q increases. This implies the goods are substitutes.

3) Income elasticity measures how much (%) the quantity demanded change for a change (%) in Income.

Where e_I represent the income elasticity

Using calculus,

Income elasticity is positive. That means as the income increases, demand for the good also increases. Therefore, it is a normal good.

(Please note that, the variables used, like I for income, may not be exactly as per the text book you follow. In case of any difficulty, please do comment and I will revert)