Question

Suppose there are \(n\) individuals, each with a linear demand curve for \(Q\) of the form $$Q i=\mathrm{a},+b_{i} P+c j+d i P^{\prime} \quad i=1, n$$ where the parameters \(a, b_{h} c_{h}\) and \(d,\) differ among individuals. Show that at any point, the price elasticity of the market demand curve is independent of \(P^{\prime}\) and the distribution of income. Would this be true if each individual's demand for \(Q\) were instead linear in logarithms? Explain.

Short answer

Explain your answer. Yes, the price elasticity of the market demand curve is independent of \(P^{\prime}\) and the distribution of income when the individual demand for \(Q\) is linear. This is because, in this case, the expression for price elasticity of the market demand curve, \(\eta\), does not involve \(P^{\prime}\) or \(I\). Consequently, the price elasticity does not change with variations in \(P^{\prime}\) or the distribution of income.

Step-by-step solution

1. Derive the market demand curve#

To find the market demand curve, we need to sum up the individual demand curves for all \(n\) individuals: $$Q_{M} = \sum_{i=1}^{n} Q_{i}$$ Substituting the linear demand curve for each individual: $$Q_{M} = \sum_{i=1}^{n} (a_{i} + b_{i} P + c_{i} I + d_{i} P^{\prime})$$

2. Calculate the price elasticity of the market demand curve#

The price elasticity of the market demand curve, denoted as \(\eta\), is defined as the percentage change in quantity demanded, \(Q_{M}\), with respect to the percentage change in price, \(P\): $$\eta = \frac{\frac{\Delta Q_{M}}{Q_{M}}}{\frac{\Delta P}{P}} = \frac{P}{Q_{M}} \cdot \frac{dQ_{M}}{dP}$$ To find the derivative of the market demand curve with respect to price, we need to differentiate \(Q_{M}\) with respect to \(P\): $$\frac{dQ_{M}}{dP} = \sum_{i=1}^{n} b_{i}$$ Therefore, the price elasticity of the market demand curve is: $$\eta = \frac{P}{Q_{M}} \cdot \sum_{i=1}^{n} b_{i}$$

3. Analyze dependence of price elasticity on \(P^{\prime}\) and distribution of income#

Looking at the expression for \(\eta\), we observe that it does not involve \(P^{\prime}\) or \(I\), meaning that the price elasticity of the market demand curve is independent of \(P^{\prime}\) and the distribution of income. Hence, the price elasticity does not change with variations in \(P^{\prime}\) or the distribution of income.

4. Discuss if the same conclusion holds when demand is linear in logarithms#

If the demand for \(Q\) is linear in logarithms, the demand curve for the \(i\)-th individual would look something like this: $$\ln Q_{i} = \alpha_{i} + \beta_{i} \ln P + \gamma_{i} \ln I + \delta_{i} \ln P^{\prime}$$ In this case, the market demand curve, price elasticity, and their dependence on \(P^{\prime}\) and the distribution of income would be different. In order to sum the individual demands, we would need to exponentiate both sides of the equation and sum up the demands in their original form, not their logarithmic form. As a result, the price elasticity of the market demand curve may depend on \(P^{\prime}\) and the distribution of income. In conclusion, when the individual demand for \(Q\) is linear, the price elasticity of the market demand curve is independent of \(P^{\prime}\) and the distribution of income. However, if the demand is linear in logarithms, this conclusion may not hold, and the price elasticity could depend on \(P^{\prime}\) and the distribution of income.

Key concepts

Market Demand Curve

Understanding the market demand curve is essential when studying price elasticity of demand. The market demand curve represents the total quantity of a good or service demanded across all consumers in the market at various price levels. It is derived by horizontally summing all the individual demand curves.

As demonstrated in the exercise, the summation process involves adding the quantities demanded by each individual at given price points to get the market's total quantity demanded. Importantly, the market demand curve slopes downward, reflecting the inverse relationship between price and quantity demanded—one of the foundational principles of economics.

Why does this matter? When it comes to price elasticity, the market demand curve's slope at a specific point tells us how sensitive the quantity demanded is to a price change at that price level. Therefore, analyzing the market demand curve can reveal the overall responsiveness of market demand to price changes, which is pivotal for businesses when setting prices and for policymakers considering tax changes or regulatory interventions.

Individual Demand Curve

Whereas the market demand curve looks at the bigger picture, the individual demand curve zooms in on a single consumer's purchasing behavior. It shows the relationship between the price of a good and the quantity of that good the individual is willing to purchase.

An individual demand curve can also be represented by a linear function, capturing the same inverse relationship between price and quantity. Each person's budget constraints, preferences, and income level cause variations in the individual demand curve, which is represented by different coefficients in the linear equation, as illustrated in the textbook problem.

When working with individual demand curves, whether linear or a more complex form, the key takeaway is to understand how these various components could affect one's purchasing decisions. Recognizing this allows us to better predict how changes in external factors, like a shift in income or a change in price, will influence an individual's demand for a product.

Linear Demand Function

A linear demand function is a simple representation of the demand curve that expresses the relationship between quantity demanded (Q) and price (P) as a straight line. This straight line is typically represented by the equation:
$$Q = a + bP$$
where 'a' is the intercept and 'b' is the slope of the line. The slope 'b' is particularly important as it is negative in most cases, signifying the aforementioned law of demand—that when prices rise, the quantity demanded falls, and vice versa.

In terms of elasticity, which measures responsiveness to price changes, the slope of the demand function is key. A steeper slope indicates less elasticity, as consumers are less responsive to price changes. Conversely, when the slope is flatter, the demand is more elastic since even small price changes can cause large changes in quantity demanded.

Understanding linear demand functions is important not only for creating basic economic models but also for applying these models to real-world scenarios where demand responsiveness translates directly to pricing strategies, market analysis, and economic policy-making.