Question

Suppose that the market demand for beef is given by \(Q^{D}=200-6 P+2 Y\), where \(P\) is the price of meat per \(\mathrm{kg}\) and \(Y\) is the consumers' income. Suppose that consumers' income is \(£ 100\). If the price of beef decreases from \(£ 10\) to \(£ 8\) per \(\mathrm{kg}\), find the corresponding elasticity of demand. Now suppose that the price is fixed to \(£ 8\) while consumers' income increases from \(£ 100\) to \(£ 150\); find the corresponding income elasticity of demand. Is beef a normal good?

Short answer

The price elasticity of demand is approximately -0.156, and the income elasticity is approximately 0.621. Beef is a normal good.

Step-by-step solution

Finding Initial and Final Quantities for Price Elasticity of Demand

First, calculate the initial quantity demanded when the price is £10 and the income is £100. Plug these values into the demand equation:\[Q^D = 200 - 6(10) + 2(100) = 200 - 60 + 200 = 340\]Now calculate the final quantity when the price decreases to £8:\[Q^D = 200 - 6(8) + 2(100) = 200 - 48 + 200 = 352\]

Calculate Price Elasticity of Demand

Using the formula for elasticity: \(E_d = \frac{\Delta Q}{\Delta P} \times \frac{P_1 + P_2}{Q_1 + Q_2}\), where \(\Delta Q = 352 - 340 = 12\) and \(\Delta P = 8 - 10 = -2\).\[E_d = \frac{12}{-2} \times \frac{10 + 8}{340 + 352} = -6 \times \frac{18}{692} \approx -0.156\]

Finding Initial and Final Quantities for Income Elasticity of Demand

First, find the initial quantity demanded when the price is £8 and income is £100:\[Q^D = 200 - 6(8) + 2(100) = 352\]Now calculate the final quantity when income increases to £150:\[Q^D = 200 - 6(8) + 2(150) = 200 - 48 + 300 = 452\]

Calculate Income Elasticity of Demand

Using the formula for income elasticity: \(E_i = \frac{\Delta Q}{\Delta Y} \times \frac{Y_1 + Y_2}{Q_1 + Q_2}\), where \(\Delta Q = 452 - 352 = 100\) and \(\Delta Y = 150 - 100 = 50\).\[E_i = \frac{100}{50} \times \frac{100 + 150}{352 + 452} = 2 \times \frac{250}{804} \approx 0.621\]

Assessing the Nature of the Good

Since the income elasticity of demand \(E_i \) is positive and greater than zero, it indicates that as income rises, the quantity demanded also increases. Therefore, we can conclude that beef is a normal good.

Key concepts

Price Elasticity

Price elasticity of demand measures how much the quantity demanded of a good responds to a change in its price. In simple terms, it tells us if consumers will buy more or less of a product when its price changes.
To calculate the price elasticity of demand, you use this formula:

• Change in quantity demanded (\(\Delta Q\))
• Change in price (\(\Delta P\))
• Average price (\(\frac{P_1 + P_2}{2}\))
• Average quantity (\(\frac{Q_1 + Q_2}{2}\))

In the exercise, when the price of beef decreases from £10 to £8, we found that the elasticity is approximately -0.156.
This tells us that the demand for beef is inelastic. A price change doesn't lead to a large change in demand. Why? Possibly because people need beef regardless of price changes. Understanding this can help businesses decide how much to charge.

Income Elasticity

Income elasticity of demand measures how the quantity demanded of a good changes as consumer income changes. It's like asking, "If someone earns more, will they buy more of this product?"
To compute income elasticity, you'll need:

• Change in quantity demanded (\(\Delta Q\))
• Change in income (\(\Delta Y\))
• Average income (\(\frac{Y_1 + Y_2}{2}\))
• Average quantity (\(\frac{Q_1 + Q_2}{2}\))

In the scenario given, when consumers' income rose from £100 to £150, the elasticity was approximately 0.621.
This positive elasticity shows that as people's income increases, they tend to purchase more beef. Such insight is crucial for businesses to predict sales trends based on economic cycles.

Normal Goods

Normal goods are products that see an increase in demand as consumer incomes rise. It's the type of product that people buy more of when they have more money.
The concept of normal goods stems from the understanding of consumer behavior related to purchasing power. People typically upgrade or buy more when they earn more. In everyday life, this is why people might choose more steaks when they get a raise!
In our problem, beef was determined to be a normal good because the computed income elasticity was positive. When testing products, a positive elasticity generally indicates that a product is a normal good. Knowing if a product is normal helps in categorizing it correctly in economic models and making better business decisions.