Question

The prices are \(\left(p_{1}, p_{2}\right)=(2,3)\), and the consumer is currently consuming \(\left(x_{1}, x_{2}\right)=(4,4) .\) Now the prices change to \(\left(q_{1}, q_{2}\right)=(2,4) .\) Could the consumer be better off under these new prices?

Short answer

The consumer cannot afford the same bundle, but might find a better one within the budget.

Step-by-step solution

Calculate Initial Budget

First, determine the consumer's initial budget. Multiply the initial prices by the consumed quantities: \( p_1 \cdot x_1 + p_2 \cdot x_2 = 2 \cdot 4 + 3 \cdot 4 = 8 + 12 = 20 \). The budget is 20 units.

Evaluate New Price Scenario

Calculate how much the same bundle would cost under the new prices: \( q_1 \cdot x_1 + q_2 \cdot x_2 = 2 \cdot 4 + 4 \cdot 4 = 8 + 16 = 24 \). The new price for the original bundle is 24 units.

Compare Bundles

Since the initial budget is only 20 units and to consume with new prices would cost 24 units, the consumer cannot afford the initial consumption bundle with their current budget. However, if other combinations at or below the new budgetary cost exist that provide the same utility or better, the consumer could potentially be as well off or better off.

Analyze Utility

For the consumer to be better off, they would need to find a new bundle that provides higher utility than the initial bundle within the budget constraint. Without information on the utility function, conclusions about the consumer being better off cannot be confidently drawn.

Key concepts

Budget Constraint

In the world of consumer choice theory, a budget constraint represents the range of options available to a consumer based on their income and the prices of goods. It essentially delineates what combinations of goods a consumer can afford. For instance, in this exercise, the consumer originally has a budget of 20 units. This budget limits their purchasing power, determining what they can buy given the initial prices for goods 1 and 2, which were 2 and 3 units, respectively.

A budget constraint can be visualized as a line on a graph, showing the trade-offs between two goods. On one side of the line, all combinations of these goods are affordable, while on the other side, they are not. The slope of this line is determined by the ratio of the prices of the two goods. Understanding this concept is crucial because it sets the boundary within which consumers must make their choices based on preferences and prices.

Price Change

Changes in the prices of goods can significantly affect a consumer's budget constraint. When prices change, the line representing their budget constraint shifts. It could either expand or contract based on whether prices increase or decrease. For example, in this exercise, the price of good 2 increased from 3 to 4 while the price of good 1 remained the same.

This price increase affects the affordability of the same bundle of goods the consumer was initially buying. Specifically, what used to cost 20 units now costs 24 units, making it unaffordable under the same budget. Such shifts in prices force the consumer to re-evaluate their purchases, potentially opting for different quantities or different goods altogether. This is a dynamic aspect of consumer choice theory, highlighting how consumers must constantly adapt their choices to maximize satisfaction within their financial limits.

Utility Analysis

Utility in economics refers to the satisfaction or pleasure a consumer derives from consuming goods and services. Utility analysis is used to understand and measure how consumers make decisions to maximize their happiness or satisfaction given their budget constraints. In our exercise, even with a restricted budget due to a price increase, the consumer needs to find a mix of goods that maximizes their utility.

Unfortunately, without specific information about the utility function, we cannot precisely determine if the consumer can be better off with these new prices. However, the goal remains to identify a combination of goods that yields the highest possible utility that fits within the new budget constraint. Utility functions often assume that more of a good is better than less, but the specific tastes and preferences of the consumer will ultimately shape their purchasing choices. This analysis underscores the importance of understanding individual preferences and the impact they have on consumption decisions.