Question

When prices are \(\left(p_{1}, p_{2}\right)=(1,2)\) a consumer demands \(\left(x_{1}, x_{2}\right)=(1,2)\) and when prices are \(\left(q_{1}, q_{2}\right)=(2,1)\) the consumer demands \(\left(y_{1}, y_{2}\right)=(2,1)\) Is this behavior consistent with the model of maximizing behavior?

Short answer

The behavior is consistent with utility maximization.

Step-by-step solution

Understand the Utility Maximization Problem

The utility maximization problem involves a consumer choosing quantities of goods \(x_1\) and \(x_2\) to maximize their utility given a budget constraint. For this problem, we must check if the observed demand behavior is consistent with utility maximization given different price scenarios.

Analyze Budget Constraints

For the first price scenario, the consumer faces prices \((p_1, p_2) = (1, 2)\) and buys \((x_1, x_2) = (1, 2)\), giving a budget constraint \(1 \cdot 1 + 2 \cdot 2 = 5\). For the second price scenario, prices are \((q_1, q_2) = (2, 1)\) and the consumer buys \((y_1, y_2) = (2, 1)\), leading to a budget constraint \(2 \cdot 2 + 1 \cdot 1 = 5\). Both scenarios indicate a consistent budget of 5.

Check Consistency with Utility Maximization

To be consistent with utility maximization, the consumer's demand should follow the weak axiom of revealed preference (WARP). Given \(\left(p_1, p_2\right)\) with \(\left(x_1, x_2\right)\) and \(\left(q_1, q_2\right)\) with \(\left(y_1, y_2\right)\), under WARP the first scenario bundle should not be affordable in the second price scenario if \(\left(y_1, y_2\right)\) is chosen.

Apply the Weak Axiom of Revealed Preference (WARP)

Calculate the cost of the first bundle \((1, 2)\) under the second prices \((2, 1)\): \(2 \cdot 1 + 1 \cdot 2 = 4\). Since 4 is less than 5 (the budget constraint with \(\left(q_1, q_2\right)\)), the consumer could afford \((1, 2)\) but chooses \((2, 1)\). This does not violate WARP.

Conclude on Consistency

Since the observed consumer behavior does not violate the weak axiom of revealed preference, it is consistent with utility maximizing behavior. The consumer's choices in both scenarios are rational and consistent with the principles of utility maximization.

Key concepts

Revealed Preference

Revealed Preference is a fundamental concept in economics that helps us understand a consumer's preferences based on their purchasing decisions. Instead of relying on what consumers say they like, economists observe what choices they make when faced with different scenarios of prices and quantities. By analyzing these choices, we can infer which bundles of goods are preferred over others.

• Revealed Preference assumes that if a consumer selects a particular bundle of goods over another, given their budget constraint, then the chosen bundle offers them more satisfaction or utility.
• This assumption allows economists to explore if the observed behavior aligns with rational utility maximization, which is the idea that consumers will always try to get the most satisfaction for the limited budget they have.

By examining the combinations of goods that consumers choose when prices change, we can deduce certain aspects of their preferences, like whether they are consistent and rational in their decision-making.

Budget Constraint

A budget constraint represents the choices a consumer can afford given their income and the prices of goods. It creates a boundary line whereby different combinations of goods can be purchased with a particular budget.

• The formula for a budget constraint is typically expressed as: \(p_1x_1 + p_2x_2 = ext{income}\), where \(p_1\) and \(p_2\) are the prices of goods \(x_1\) and \(x_2\), respectively.
• It showcases the trade-offs that consumers need to make between the different goods due to limited resources.

In our example, when the prices are set at \((1, 2)\), the consumer's budget is 5, and they choose the bundle \((1, 2)\). When prices change to \((2, 1)\), the budget remains 5, and the chosen bundle becomes \((2, 1)\). This shows that with every price change, the consumer re-evaluates to maximize utility without exceeding their budget.

Weak Axiom of Revealed Preference

The Weak Axiom of Revealed Preference (WARP) is a rule used to analyze if a consumer's choices are consistent with utility maximization. According to WARP, if a consumer chooses one bundle of goods over another in a given budget, and if the rejected bundle costs less at different prices, the consumer's choice must reflect a stronger preference for the chosen bundle.

• This means if in one scenario, bundle \((x_1, x_2)\) was preferred over \((y_1, y_2)\), then there should not be any scenario where \((y_1, y_2)\) becomes cheaper, prompting a switch to \((y_1, y_2)\).
• WARP helps in determining if consumers are being consistent in their preference ordering, effectively reflecting their rational and utility-maximization behavior.

In the example provided, the cost of the first bundle \((1, 2)\) when recalculated under the second price scenario \((2, 1)\), is only 4 units. Since choosing \((2, 1)\) while \((1, 2)\) was affordable (5 units), it indicates a consistent preference aligning with the principles of WARP, showing rational utility-maximizing behavior without violation.