Question

Why does a change in income cause a parallel shift in the budget constraint?

Short answer

A change in income causes a parallel shift in the budget constraint because the prices of the goods remain constant, and only the consumer's ability to purchase the goods changes. The trade-off between the goods does not change, resulting in parallel lines with the same slope representing the new budget constraints.

Step-by-step solution

1. Understanding the Budget Constraint

: A budget constraint represents all the combinations of goods that a consumer can purchase with a given level of income and prices for goods. Mathematically, the budget constraint equation is given by: \(I = p_x X + p_y Y\) Where: - I is the consumer's income - \(p_x\) is the price of good X - \(p_y\) is the price of good Y - X and Y are the quantities of goods X and Y, respectively. The budget constraint line is the graphical representation of this equation on a two-dimensional plane, with the quantity of good X on the x-axis and the quantity of good Y on the y-axis.

2. Changes in Income

: When the consumer's income changes, the budget constraint line will shift. If the income increases, the budget constraint line will shift outward (away from the origin). Conversely, if the income decreases, the budget constraint line will shift inward (toward the origin). To understand this, consider the case when there is an increase in income. Let's denote the new income as \(I'\). The new budget constraint equation would be: \(I' = p_x X + p_y Y\)

3. The Parallel Shift in the Budget Constraint Line

: Now, let's compare the original budget constraint equation (\(I = p_x X + p_y Y\)) to the new budget constraint equation with increased income (\(I' = p_x X + p_y Y\)). Notice that the prices of the goods (i.e., \(p_x\) and \(p_y\)) have not changed in the new budget constraint equation. This means that the slope of the new budget constraint line will remain the same. The slope of the budget constraint line is given by the negative price ratio: \(- \frac{p_x}{p_y}\) Since the slope remains the same for both the original and new budget constraint lines, they will be parallel to each other. This applies to both an increase and a decrease in income.

4. Reasoning

: The reason behind the parallel shift lies in the fact that a change in income does not affect the prices of goods X and Y. When income changes, the consumer's ability to purchase the goods changes as well. However, the relative prices of the goods remain constant, and thus, the trade-off between the goods does not change. This results in the budget constraint line shifting parallel to the initial line. In conclusion, a change in income causes a parallel shift in the budget constraint because the prices of the goods remain constant, and only the consumer's ability to purchase the goods changes. The trade-off between the goods does not change, resulting in parallel lines with the same slope representing the new budget constraints.