Question 7 of 26intermediate🔧 ApplyShort Answer3 marks

What happens to the budget set if both the prices as well as the income double?

Correct Answer

When both prices and income double, the budget set remains unchanged. The budget line equation transforms from p1x1+p2x2=Mp_1 x_1 + p_2 x_2 = M to 2p1x1+2p2x2=2M2p_1 x_1 + 2p_2 x_2 = 2M, which when simplified by dividing throughout by 2, gives back the original equation p1x1+p2x2=Mp_1 x_1 + p_2 x_2 = M.

The vertical intercept of the budget line was originally Mp2\frac{M}{p_2}. After doubling both income and prices, it becomes 2M2p2=Mp2\frac{2M}{2p_2} = \frac{M}{p_2}, which remains the same. Similarly, the horizontal intercept changes from Mp1\frac{M}{p_1} to 2M2p1=Mp1\frac{2M}{2p_1} = \frac{M}{p_1}, remaining unchanged.

The slope of the budget line, given by −p1p2-\frac{p_1}{p_2}, also remains unchanged as it becomes −2p12p2=−p1p2-\frac{2p_1}{2p_2} = -\frac{p_1}{p_2}. Since both intercepts and the slope remain the same, the budget line does not shift at all, and the set of available bundles remains exactly the same as before.

Exercise: EXERCISES | Q: 7
For More Understanding

Explanation

This question tests the understanding of how proportional changes in prices and income affect the budget constraint. The key insight is that the budget line depends on the ratio of prices (slope) and the ratio of income to prices (intercepts). When both prices and income double proportionally, all these ratios remain unchanged. The textbook context shows that changes in budget line occur when income changes relative to prices (parallel shifts) or when price ratios change (pivots). In this case, neither happens because the relative purchasing power remains constant.

Solution Steps

  1. Step 1: Write the original budget line equation: p1x1+p2x2=Mp_1 x_1 + p_2 x_2 = M

  2. Step 2: Apply doubling to prices and income: New equation becomes 2p1x1+2p2x2=2M2p_1 x_1 + 2p_2 x_2 = 2M

  3. Step 3: Divide the new equation by 2: p1x1+p2x2=Mp_1 x_1 + p_2 x_2 = M (same as original)

  4. Step 4: Verify intercepts remain same: Vertical intercept =2M2p2=Mp2= \frac{2M}{2p_2} = \frac{M}{p_2}, Horizontal intercept =2M2p1=Mp1= \frac{2M}{2p_1} = \frac{M}{p_1}

  5. Step 5: Verify slope remains same: New slope =−2p12p2=−p1p2= -\frac{2p_1}{2p_2} = -\frac{p_1}{p_2}

  6. Step 6: Conclude that budget set remains unchanged

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