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Chapter 6: Problem 7

The accompanying table shows the price and yearly quantity sold of souvenir T-shirts in the town of Crystal Lake according to the average income of the tourists visiting. $$ \begin{array}{c|c|c} & \begin{array}{c} \text { Quantity of T-shirts } \\\ \text { demanded when } \\\ \text { average tourist } \end{array} & \begin{array}{c} \text { Quantity of T-shirts } \\\ \text { demanded when } \end{array} \\\ \text { Price of } & \text { average tourist } \\\ \text { T-shirt } & \text { income is } \$ 20,000 & \text { income is } \$ 30,000 \\ \hline \$ 4 & 3,000 & 5,000 \\\ 5 & 2,400 & 4,200 \\\ 6 & 1,600 & 3,000 \\\ 7 & 800 & 1,800 \end{array} $$ a. Using the midpoint method, calculate the price elasticity of demand when the price of a T-shirt rises from \(\$ 5\) to \(\$ 6\) and the average tourist income is \(\$ 20,000 .\) Also calculate it when the average tourist income is \(\$ 30,000\). b. Using the midpoint method, calculate the income elasticity of demand when the price of a T-shirt is \(\$ 4\) and the average tourist income increases from \(\$ 20,000\) to \(\$ 30,000 .\) Also calculate it when the price is \(\$ 7\)

Short Answer

Expert verified
Answer: The price elasticity of demand when the price of a T-shirt rises from $5 to $6 is -2.2 when the average tourist income is $20,000, and -1.8333 when the average tourist income is $30,000. The income elasticity of demand when the price of a T-shirt is $4 and the income increases from $20,000 to $30,000 is 1.25, and when the price is $7, it is approximately 1.923.

Step by step solution

01

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(5 is 2400 T-shirts, and at \)6 it is 1600 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{1600-2400}{(\frac{1600+2400}{2})} = \frac{-800}{2000} = -0.4\)
02

Calculate the percentage change in price

The price increased from \(5 to \)6. Calculate the percentage change in price using the midpoint method: \(\frac{6-5}{(\frac{6+5}{2})} = \frac{1}{5.5} = 0.1818\)
03

Calculate the price elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in price: \(Elasticity = \frac{-0.4}{0.1818} = -2.2\) So, the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $20,000 is -2.2. Now let's calculate the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $30,000:
04

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(5 is 4200 T-shirts, and at \)6 it is 3000 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{3000-4200}{(\frac{3000+4200}{2})} = \frac{-1200}{3600} = -0.3333\)
05

Calculate the percentage change in price

The price increased from \(5 to \)6. So the percentage change in price is the same as before, which is 0.1818.
06

Calculate the price elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in price: \(Elasticity = \frac{-0.3333}{0.1818} = -1.8333\) So, the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $30,000 is -1.8333. b. Calculate the income elasticity of demand when the price of a T-shirt is \(4 and the average tourist income increases from \)20,000 to $30,000:
07

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(\$20,000\) income is 3000 T-shirts, and at \(\$30,000\) income it is 5000 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{5000-3000}{(\frac{5000+3000}{2})} = \frac{2000}{4000} = 0.5\) Repeat this step for the price of $7: \(\frac{1800-800}{(\frac{1800+800}{2})} = \frac{1000}{1300} \approx 0.7692\)
08

Calculate the percentage change in income

The income increased from \(\$20,000\) to \(\$30,000\). Calculate the percentage change in income using the midpoint method: \(\frac{30000-20000}{(\frac{30000+20000}{2})} = \frac{10000}{25000} = 0.4\)
09

Calculate the income elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in income for both prices: \(Elasticity_{\$4} = \frac{0.5}{0.4} = 1.25\) \(Elasticity_{\$7} = \frac{0.7692}{0.4} \approx 1.923\) So, the income elasticity of demand when the price of a T-shirt is \(\$4\) and the income increases from \(\$20,000\) to \(\$30,000\) is 1.25, and when the price is \(\$7\), it is approximately 1.923.

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Most popular questions from this chapter

The U.S. government is considering reducing the amount of carbon dioxide that firms are allowed to produce by issuing a limited number of tradable allowances for carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) emissions. In an April 25 , 2007, report, the U.S. Congressional Budget Office (CBO) argues that "most of the cost of meeting a cap on \(\mathrm{CO}_{2}\) emissions would be borne by consumers, who would face persistently higher prices for products such as electricity and gasoline \(\ldots\) poorer households would bear a larger burden relative to their income than wealthier households would." What assumption about one of the elasticities you learned about in this chapter has to be true for poorer households to be disproportionately affected?

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