A consumer spends a fixed weekly income I on Good X (plotted on the horizontal axis) and G… — JAMB UTME

A consumer spends a fixed weekly income $I$ on Good $X$ (plotted on the horizontal axis) and Good $Y$ (plotted on the vertical axis) with unit prices $P_x$ and $P_y$ . Suppose the government imposes a $20\%$ ad valorem tax on Good $X$ while simultaneously providing the consumer with a lump-sum income subsidy equal to $20\%$ of their initial total income $I$ , leaving $P_y$ unchanged. Which of the following correctly describes the structural transformation of the consumer's budget line?

The vertical intercept increases by $20\%$ , the horizontal intercept remains unchanged, and the budget line pivots counter-clockwise around the fixed horizontal intercept to become steeper.Answer
B
The budget line shifts outward in a parallel direction because both nominal income and the price of Good $X$ increased by the same proportion.
C
The horizontal intercept decreases by $20\%$ while the vertical intercept remains unchanged because the lump-sum grant exactly neutralizes the price increase of Good $X$ .
D
The slope of the budget line remains constant because the relative price ratio $P_x/P_y$ is offset by the $20\%$ increase in purchasing power.

Answer

The vertical intercept increases by 20%, the horizontal intercept remains unchanged, and the budget line pivots counter-clockwise around the fixed horizontal intercept to become steeper.

The initial budget line has a horizontal intercept at

I/P_x

, vertical intercept at

I/P_y

, and slope of magnitude

P_x/P_y

. Following the policy changes, the new income is

1.20I

and the new price of

X

1.20P_x

. The new horizontal intercept is

(1.20I)/(1.20P_x) = I/P_x

, which is unchanged. The new vertical intercept is

(1.20I)/P_y

, representing a

20\%

increase. The slope magnitude increases to

(1.20P_x)/P_y = 1.20(P_x/P_y)

, making the budget line steeper. Graphically, anchoring the budget line at the same point on the horizontal axis while pushing its vertical endpoint upward corresponds to a counter-clockwise pivot around the fixed horizontal intercept.

Step-by-Step Solution

Express the initial budget equation and its key boundary intercepts and slope.

Initial equation:

P_x X + P_y Y = I

. Horizontal intercept (

X

-axis) =

\frac{I}{P_x}

, Vertical intercept (

Y

-axis) =

\frac{I}{P_y}

, Slope =

-\frac{P_x}{P_y}

Establishing initial parameter values allows direct comparison with post-tax and post-subsidy values.

Determine the updated price of Good

X

and updated nominal income.

P_x' = (1 + 0.20) P_x = 1.20 P_x

;

I' = (1 + 0.20) I = 1.20 I

;

P_y' = P_y

The

20\%

tax inflates

P_x

by a factor of

1.20

, and the

20\%

income subsidy inflates

I

by a factor of

1.20

Calculate the new horizontal intercept, vertical intercept, and slope.

New Horizontal Intercept =

\frac{I'}{P_x'} = \frac{1.20 I}{1.20 P_x} = \frac{I}{P_x}

(Unchanged).
New Vertical Intercept =

\frac{I'}{P_y'} = \frac{1.20 I}{P_y} = 1.20 \left(\frac{I}{P_y}\right)

(

20\%

increase).
New Slope =

-\frac{P_x'}{P_y'} = -\frac{1.20 P_x}{P_y} = 1.20 \left(-\frac{P_x}{P_y}\right)

(

20\%

magnitude increase, steeper).

Comparing the relative scaling factors reveals which intercept shifts and how the opportunity cost ratio changes.

Synthesize the geometrical movement of the budget line.

Since the

X

-intercept is fixed while the

Y

-intercept shifts outward, the budget line pivots counter-clockwise around the

X

-intercept, making it steeper.

A fixed horizontal intercept combined with a higher vertical intercept results in a counter-clockwise rotation.

Key Concept

Budget Line Intercepts and Slope Transformation under Asymmetric Price and Income Changes

Estimated Time:2m 0s

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Step-by-Step Solution

Key Concept

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