Question

A recent report by the U.S. Centers for Disease Control and Prevention (CDC), published in the CDC's Morbidity and Mortality Weekly Report, studied the effect of an increase in the price of beer on the incidence of new cases of sexually transmitted disease in young adults. In particular, the researchers analyzed the responsiveness of gonorrhea cases to a tax-induced increase in the price of beer. The report concluded that "the.... analysis suggested that a beer tax increase of \(\$ 0.20\) per six-pack could reduce overall gonorrhea rates by \(8.9 \% . "\) Assume that a sixpack costs \(\$ 5.90\) before the price increase. Use the midpoint method to determine the percent increase in the price of a six-pack, and then calculate the cross-price elasticity of demand between beer and incidence of gonorrhea. According to your estimate of this cross-price elasticity of demand, are beer and gonorrhea complements or substitutes?

Short answer

Answer: The cross-price elasticity of demand between beer and the incidence of gonorrhea is -2.67, and they are complements since the calculated value is negative.

Step-by-step solution

Determine the New Price of a Six-Pack with the Tax Increase

To determine the new price of a six-pack after the tax increase of \(0.20\), we need to add the tax increase to the original price. The original price is \(5.90\), so the new price is \(5.90 + 0.20 = \$ 6.10\).

Calculate the Percent Increase in the Price of a Six-Pack using the Midpoint Method

The midpoint method is given by the formula: `Percent Increase = (New Price - Old Price) / Midpoint × 100` Here, the midpoint is `((Old Price + New Price) / 2)`. So we have: Midpoint = (5.90 + 6.10) / 2 = \(6.00\) Percent Increase = (6.10 - 5.90) / 6.00 × 100 ≈ 3.33 %

Calculate the Cross-Price Elasticity of Demand Between Beer and Incidence of Gonorrhea

The cross-price elasticity of demand is given by the formula: `Cross-Price Elasticity = (Percentage Change in Quantity Demanded of Good 2) / (Percentage Change in Price of Good 1)` In this case, the percentage change in the price of beer is 3.33%, and the percentage change in the incidence of gonorrhea is -8.9%. So we have: Cross-Price Elasticity = (-8.9) / 3.33 ≈ -2.67

Determine if Beer and Gonorrhea are Complements or Substitutes

The cross-price elasticity of demand tells us the relationship between two goods. If the cross-price elasticity is positive, the two goods are considered substitutes. If it's negative, the goods are considered complements. Since the cross-price elasticity we calculated in Step 3 is -2.67, which is negative, we can conclude that beer and gonorrhea are complements. This means that an increase in the price of beer leads to a reduction in the incidence of gonorrhea.

Key concepts

Midpoint Method

The midpoint method is a useful way of calculating percentage changes that provides a more accurate estimate than the simple percentage change method. It's particularly handy because it gives the same result regardless of which direction the price change was initiated from. In our problem, we're calculating the percentage increase in the price of a six-pack of beer from \(5.90 to \)6.10 after a tax increase.

The formula for the midpoint method is:

• Percent Increase = \(\frac{\text{New Price - Old Price}}{\text{Midpoint}} \times 100\)

Here, the midpoint is the average of the old and new prices, which is \((5.90 + 6.10) / 2 = 6.00\). Using the formula, we plug the values into our equation:

• Percent Increase = \(\frac{6.10 - 5.90}{6.00} \times 100 \approx 3.33\%\)

This method allows us to understand the relative size of the price increase in relation to the midpoint, ensuring that differences felt proportionately at different levels are taken into account.

Complements and Substitutes

In economics, the relationship between two goods can be examined through cross-price elasticity of demand. This economic concept helps us understand whether two goods are complements or substitutes.

Here's how it works:

• If the cross-price elasticity is positive, it means that as the price of one good increases, the quantity demanded of the other good increases as well, indicating they are substitutes.
• If the cross-price elasticity is negative, an increase in the price of one good decreases the quantity demanded of the other, suggesting the goods are complements.

For the problem at hand, we calculated a cross-price elasticity of -2.67 between beer and incidence of gonorrhea. Since this value is negative, it implies that beer and gonorrhea are complementary goods. When the price of beer increases, the occurrence of gonorrhea decreases, suggesting a linked consumption or behavioral pattern.

Beer Tax Effect

The effect of a beer tax demonstrates the practical application of cross-price elasticity. In the given exercise, the imposition of a $0.20 tax per six-pack induces a 3.33% price increase. This leads to an observed 8.9% decrease in gonorrhea cases. Such an incidence proves how taxes can influence consumer behavior and public health outcomes.

The relationship identified here proves insightful for policy makers:

• A small tax increase can lead to significant behavioral changes, highlighted by the considerable drop in gonorrhea cases.
• This negative cross-price elasticity suggests that public health can be indirectly affected by economic measures targeting seemingly unrelated goods.

By understanding and leveraging the beer tax effect, governments can strategize to achieve desired public health outcomes, showing the intricate link between economic policy and social welfare.