Question

Originally the consumer faces the budget line \(p_{1} x_{1}+p_{2} x_{2}=m .\) Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income.

Short answer

The new budget line is \(2p_1 x_1 + 8p_2 x_2 = 4m\).

Step-by-step solution

Understand the original budget line

The original budget line is given by the equation \(p_1 x_1 + p_2 x_2 = m\), where \(p_1\) is the price of good 1, \(p_2\) is the price of good 2, and \(m\) is the consumer's income.

Identify changes in prices and income

The problem states that the price of good 1 doubles, meaning it becomes \(2p_1\). The price of good 2 becomes 8 times larger, meaning it becomes \(8p_2\). The income becomes 4 times larger, meaning it becomes \(4m\).

Write down the new budget line equation

Substitute the new prices and income into the budget line equation: the new budget line is \(2p_1 x_1 + 8p_2 x_2 = 4m\). This represents the consumer's new budget constraint after the changes in prices and income.

Key concepts

Understanding Price Changes in Microeconomics

Price changes are a key concept in microeconomics that impact consumers' budget constraints. When the price of a good changes, it directly affects how much of that good a consumer can purchase within their budget. To illustrate, in the problem above, the price of good 1 doubles, meaning consumers will need twice as much money to buy the same amount. On the other hand, the price of good 2 becomes 8 times higher. This makes it significantly more expensive for consumers, further limiting their purchasing power for that particular good.

Price changes are not just limited to individual goods. In this example, both goods see a price increase. The adjustment in prices forces consumers to reevaluate their spending patterns, as the most they could initially buy has now shifted.

• Price of good 1 changes from \(p_1\) to \(2p_1\).
• Price of good 2 changes from \(p_2\) to \(8p_2\).

These changes can squeeze a consumer’s budget, influencing them to allocate their income differently compared to the past.

Examining the Income Effect

The income effect is an important phenomenon in microeconomics that describes how changes in a consumer's income influence their purchasing decisions. When a consumer's income increases or decreases, their ability to buy goods and services also changes.

In the given problem, the consumer’s income multiplies by fourfold. This means their purchasing power has increased dramatically. Instead of sticking to the original constraint, they now have more financial flexibility. However, because the prices of goods have also changed, the real impact on consumption depends on the combination of price changes and the income effect.

• Original income: \(m\).
• New income post-change: \(4m\).

While the increased income suggests greater purchasing ability, the steep price changes dampen this potential. This scenario exemplifies how income changes often need to be seen in conjunction with price shifts to understand the full economic impact.

Microeconomics: Budget Constraints and Consumer Behavior

In microeconomics, understanding budget constraints is vital to analyzing consumer behavior. A consumer's budget constraint outlines the combinations of goods and services they can purchase, given their income and the prices of goods. This important tool helps to predict how consumers will react to changes in economic circumstances.

In the exercise, the budget constraint is presented through the formula \(p_1 x_1 + p_2 x_2 = m\). With the price and income changes, the updated constraint becomes \(2p_1 x_1 + 8p_2 x_2 = 4m\). This scenario forces consumers to adjust their consumption choices based on the new terms.

• Original constraint: a balanced perspective of consumer ability.
• Updated constraint: reflects price inflation and income growth dynamically.

These shifts are crucial in microeconomic studies as they provide insights into how consumers prioritize different goods when faced with financial pressures and how they alter consumption patterns accordingly.