Question

One of the key concepts we have discussed this semester is the concept of “Elasticity”. Generally speaking, elasticity is simply a measure of how responsive one variable is to a change in another variable. While we have focused on the “own price elasticity of demand”, the “income elasticity of demand” and the “cross-price elasticity of demand” this concept can be extended to a number of different situations.

1. (10 pts) The demand curve for a product is given by Q_x=1,000-2P_x+0.02P_z,  where P_z = $400.

1. What is the own price elasticity of demand when P_x  = $154? Is demand elastic or inelastic at this price? What would happen to the firm’s revenue if it decided to charge a price below $154?
   
2. What is the own price elasticity of demand when P_x= $354? Is demand elastic or inelastic at this price? What would happen to the firm’s revenue if it decided to charge a price above $354?

3. What is the cross-price elasticity of demand between good X and good Z when P_x= $154? Are goods X and Z substitutes or complements?

2. (10 pts) Cross-Price Elasticity of Demand characterizes how the demand for one good, X, changes in response to a price change for good Y. The responsiveness tells you a number of things, including whether or not the two goods are substitutes or complements.

1. How can you tell if goods are substitutes or compliments based on the value of the cross-price elasticity of demand?

2. In a 90’s paper, some economists estimated the cross-price elasticities listed in the table below. Based on these estimates, characterize whether the goods are substitutes or compliments.

Goods/Services	Cross-Price Elasticity
Transportation vs Recreation	-0.05
Food vs Recreation	0.15
Clothing vs Food	-0.18

3. Now let’s assume you opened a little food shop. While you are reading the WSJ, you see an article where economists and businesses are anticipating a 18% increase in the price of recreation. Using the formula for cross-price elasticity and the estimated cross-price elasticity for food & recreation, calculate the anticipated % change in demand for food arising from an 18% increase in the price of recreation.

Answer 1

Qx = 992 – 2Px (when Pz=400)

a. Own price elasticity of demand when Px: Qx= 154 = 992- (2*154) = 684.
Price elasticity = (-2*154) / 684 = -0.45
Demand is inelastic since the elasticity is between -1 and 0.

The revenue of the firm will reduce if a price below $154 is charged since the demand is inelastic at this price.

b. Own price elasticity of demand when Px is 354 Qx = 992- (2*354) = 284
Price elasticity = (-2*354) / 284 = -2.49
Demand is elastic at this point.
The revenue of the firm will increase if a price above $354 is charged.

c.  
Cross price elasticity = Percentage Change in Price of X / Percentage Change in Quantity of Z​
= 0.02 * 400 / 684

= 0.01

X and Z are substitute goods.

d. if the cross price elasticiities are positive, they are substitutes, if they are negative they are complimentary

e. Transport vs recreation are complimentary
food vs recreation are substitutes.
clothing vs food are complimentary

f. change in demand for food = 0.18 / 0.15 = 1.2
120 % increase in the demand for food