Question

A luxury is defined as a good for which the income elasticity of demand is greater than 1 Show that for a two-good economy, both goods cannot be luxuries. (Hint: What happens if both goods are luxuries and income is increased by 10 percent?)

Short answer

Answer: In a two-good economy, both goods cannot be luxuries because if both goods have an income elasticity of demand greater than 1, the total quantity demanded for both goods would increase by more than the total income when income increases. However, this is not possible, as consumers have a budget constraint and cannot spend beyond their increased income, creating a contradiction. Therefore, both goods cannot be luxuries in a two-good economy.

Step-by-step solution

Definition of Income Elasticity of Demand (IED)

The income elasticity of demand (IED) is a measure of how sensitive the quantity demanded of a good is to a change in income. It is calculated as the percentage change in quantity demanded divided by the percentage change in income. Mathematically, it's expressed as: IED = $\frac{\mathrm{\%}ChangeinQuantityDemanded}{\mathrm{\%}ChangeinIncome}$ In the given problem, a luxury good is defined as a good with an IED greater than 1.

Examine the effects of a 10% increase in income

Let's say we have a two-good economy where good A and good B are both luxuries. If income increases by 10%, we can analyze the effect on the quantity demanded for each good.

Calculate the new quantity demanded for Good A

Since good A is a luxury, its IED is greater than 1. Let's call the IED for good A as $IE{D}_A>1$. When income increases by 10%, the new quantity demanded for good A (let's call it $Q_A^{\ast }$) can be calculated by: $Q_A^{\ast }=Q_A+(\mathrm{\%}ChangeinIncome)\times (IE{D}_A)\times Q_A$ Plugging in the values, we get: $Q_A^{\ast }=Q_A+(0.1)\times (IE{D}_A)\times Q_A$ Since $IE{D}_A>1$, the new quantity demanded for good A will also be more than 10% higher ($Q_A^{\ast }>Q_A+0.1\times Q_A$).

Calculate the new quantity demanded for Good B

Similarly, for good B, we'll call the IED for good B as $IE{D}_B>1$. When income increases by 10%, the new quantity demanded for good B (let's call it $Q_B^{\ast }$) can be calculated by: $Q_B^{\ast }=Q_B+(\mathrm{\%}ChangeinIncome)\times (IE{D}_B)\times Q_B$ Plugging in the values, we get: $Q_B^{\ast }=Q_B+(0.1)\times (IE{D}_B)\times Q_B$ Since $IE{D}_B>1$, the new quantity demanded for good B will also be more than 10% higher ($Q_B^{\ast }>Q_B+0.1\times Q_B$).

Analyze the results

From Steps 3 and 4, we can see that when income increases by 10%, the quantity demanded for both goods A and B also increase by more than 10%. However, if both goods are luxuries and the total quantity demanded for both goods increases by more than the total income, it means that the consumers are spending beyond their 10% increased income, which is not possible in a two-good economy, as consumers have a budget constraint. Therefore, it is not possible for both goods in a two-good economy to be luxuries.

Key concepts

Luxury Good

In the context of economics, a luxury good is an item that experiences greater demand as people's income increases, disproportionately compared with other goods. This is quantitatively expressed through the income elasticity of demand (IED), which measures the sensitivity of the quantity demanded to changes in income. For a luxury good, the IED is greater than 1, meaning that if a person's income rises by a certain percentage, the quantity demanded for the luxury good rises by an even higher percentage.

For instance, consider an exotic sports car, which is typically regarded as a luxury good. If a consumer's income increases by 10%, they may increase their consumption of sports cars by 15%, indicating an IED greater than 1. This trend reflects that as people earn more, they are likely to allocate a larger portion of their income to purchasing goods that are not merely necessities, but also provide higher satisfaction or status.

However, it's crucial to recognize that the categorization of a good as a luxury can change over time and across different income groups. What may be considered a luxury to one demographic may be a necessity to another, especially as income levels and societal norms evolve.

Two-Good Economy

A two-good economy is a simplified conceptual model used to study economic behaviors involving two distinct goods. This type of model is beneficial to understand the trade-offs and choices an individual or an entire economy makes when faced with limited resources. The two-good model assumes that consumers can only choose between two goods, which makes the analysis more manageable and the insights more apparent.

In such an economy, if both goods are considered to be luxuries, an issue arises when income levels increase. Each good's demand would increase at a rate higher than the income growth due to their higher income elasticity of demand. This could lead to an unrealistic outcome where consumers are apparently spending more than the additional income received, disregarding any budget constraints. This paradox is often used in educational scenarios to illustrate the importance of trade-offs and the concept of budget constraints in economic theory. The oversimplification of considering only two goods helps in highlighting how various economic principles interact in a controlled environment.

Budget Constraint

The budget constraint represents the combinations of goods and services that a consumer can purchase with their limited income. It's a fundamental concept in economics that encapsulates the trade-off between different choices that adhere to financial limitations. In other words, it's the frontier of the consumer's purchasing capability, illustrating that there's a ceiling to what can be bought.

Graphically, a budget constraint is often shown as a straight line on a two-dimensional graph where each axis represents the quantity of one good. The slope of this line indicates the rate at which one good can be substituted for another while keeping the total spending unchanged. An important implication of the budget constraint is that consumers can't spend more than they earn in a particular period; this ensures that economic models are grounded in realism.

In the exercise's scenario, if a consumer's income increases by 10%, the budget constraint shifts outward, allowing for more consumption. However, if both goods in the two-good economy are luxuries with an IED greater than 1, the outward shift of the budget constraint is insufficient to cover the proportionally larger increase in the consumption of both goods. This discrepancy serves to underscore that in a realistic economic model, preferences, income levels, and budget limitations must all be considered to accurately predict consumer behavior and demand patterns.