Question

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

Short answer

Yes, the behavior is consistent with utility maximization.

Step-by-step solution

Define the Problem

We have two scenarios of prices and demanded bundles: (1) \((p_1, p_2) = (2, 1)\) and demand \((x_1, x_2) = (1, 2)\); (2) \((q_1, q_2) = (1, 2)\) and demand \((y_1, y_2) = (2, 1)\). We need to check if this is consistent with utility maximization.

Determine Income in Each Scenario

First, determine the consumer's income for each price-demand scenario.For the first scenario: \[ I = p_1 x_1 + p_2 x_2 = 2 \times 1 + 1 \times 2 = 4 \]For the second scenario:\[ J = q_1 y_1 + q_2 y_2 = 1 \times 2 + 2 \times 1 = 4 \] The income in both cases is the same, \(4\).

Check Revealed Preference Condition for Scenario 1

If the consumer maximizes utility, then the chosen bundle under one price should be at least as good as any other, given budget constraints.For the first scenario, bundle \((x_1, x_2) = (1, 2)\) should be preferred or equal to \((y_1, y_2) = (2, 1)\):\[ p_1 y_1 + p_2 y_2 = 2 \times 2 + 1 \times 1 = 5 \] The cost of bundle \((y_1, y_2)\) is \(5\), which exceeds the budget \(4\). So, bundle \((y_1, y_2)\) is not affordable under prices \((p_1, p_2)\).

Check Revealed Preference Condition for Scenario 2

Similarly, for the second scenario, bundle \((y_1, y_2) = (2, 1)\) should be preferred or equal to \((x_1, x_2) = (1, 2)\):\[ q_1 x_1 + q_2 x_2 = 1 \times 1 + 2 \times 2 = 5 \] The cost of bundle \((x_1, x_2)\) is \(5\), which also exceeds the available budget \(4\) in this scenario. Hence, bundle \((x_1, x_2)\) is not affordable under prices \((q_1, q_2)\).

Conclusion

The revealed preference conditions show that for each price scenario, the non-chosen bundle exceeds the budget constraint, meaning that each selected bundle is truly maximized given the budget. Hence, the consumer's behavior is consistent with utility maximization.

Key concepts

Revealed Preference

Revealed Preference is a concept used in economics to infer a consumer's preferences based on their purchasing decisions rather than their stated preferences.
This idea assumes that if a consumer chooses a particular bundle over another when both are affordable, the chosen bundle is preferred.
In this exercise, we have two scenarios where different bundles are purchased at different prices. According to the revealed preference theory:

• The bundle chosen in each price scenario must be at least as good as, or better than, other possible bundles, given the consumer’s income.
• The bundles that the consumer did not choose must either cost more than their income allows or must not provide as much satisfaction, showing that the selected bundle is optimizing their utility.

By confirming that the unchosen bundles exceed the consumer’s budget in this case, we see that the chosen bundles are consistent with maximizing utility.

Consumer Demand

Consumer Demand refers to how much of a product or bundle of products consumers are willing to purchase at various price levels.
Essentially, it’s the relationship between a consumer's wants, their financial limitations, and the prices of goods.
In our given exercise, we have:

• In scenario one, at prices \((p_1, p_2) = (2, 1)\), the demand is \((1, 2)\).
• In scenario two, at prices \((q_1, q_2) = (1, 2)\), the demand shifts to \((2, 1)\).

This shift in demand between scenarios indicates that consumers adjust their purchase decisions based on price changes to maximize satisfaction.
Hence, consumer demand is an essential factor for understanding how consumers respond to price fluctuations and budget constraints, ultimately impacting market outcomes and business strategies.

Budget Constraint

A Budget Constraint represents the limitations on the consumer’s buying capabilities due to their income and the prices they face.
It essentially defines the boundary of goods they can afford.
In both scenarios considered:

• The consumer had a fixed income of \(4\).
• The cost of each chosen bundle matched or was well within their budget limit.
• Unchosen bundles in each scenario were outside the budget constraint, exceeding the affordable cost of \(4\).

This budget constraint effectively shapes the set of possible choices for the consumer.
Understanding this concept allows us to see why certain bundles are chosen and others are not, thus reflecting utility maximization behavior through the feasible purchase options determined by income and goods' prices.