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PEAMCQModerate

2026 PEA Q23

A consumer always spends one-fourth of income on XX, and the income elasticity of demand for XX is 55. Is YY inferior?

Reveal answer and solution

Answer

A

Solution

  1. 1

    Let sX=1/4s_X = 1/4 and sY=3/4s_Y = 3/4 be the budget shares. Engel aggregation states

  2. 2

    that the share-weighted sum of income elasticities equals 11:

  3. 3
    sX ηX+sY ηY=1. s_X\, \eta_X + s_Y\, \eta_Y = 1.
  4. 4

    Substituting sX=1/4s_X = 1/4, ηX=5\eta_X = 5, sY=3/4s_Y = 3/4:

  5. 5
    14⋅5+34 ηY=1⟹34 ηY=−14⟹ηY=−13<0. \tfrac{1}{4}\cdot 5 + \tfrac{3}{4}\,\eta_Y = 1 \quad\Longrightarrow\quad \tfrac{3}{4}\,\eta_Y = -\tfrac{1}{4} \quad\Longrightarrow\quad \eta_Y = -\tfrac{1}{3} < 0.
  6. 6

    A negative income elasticity means YY is inferior, irrespective of prices.

Answer structure / marking notes

Engel aggregation follows from differentiating the budget identity with respect to income.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2026/ISI_MSQE_PEA_2026_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.