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2022 PEA Q2

For the linear utility u(x1,x2)=3x1+2x2u(x_1,x_2) = 3x_1 + 2x_2 with budget 2x1+3x2≤M2x_1 + 3x_2 \leq M, x1,x2≥0x_1, x_2 \geq 0, the Lagrangian is

L(x1,x2;λ)=3x1+2x2+λ[M−2x1−3x2]. L(x_1,x_2;\lambda) = 3x_1 + 2x_2 + \lambda[M - 2x_1 - 3x_2].

Determine the equilibrium (x1∗,x2∗,λ∗)(x_1^*, x_2^*, \lambda^*).

Reveal answer and solution

Answer

A

Solution

  1. 1

    Compare the ratios of marginal utility to price:

  2. 2
    MU1p1=32,MU2p2=23. \frac{MU_1}{p_1} = \frac{3}{2}, \qquad \frac{MU_2}{p_2} = \frac{2}{3}.
  3. 3

    Since 32>23\tfrac{3}{2} > \tfrac{2}{3}, the consumer spends entire income on good 1, so

  4. 4

    x1∗=M/2, x2∗=0x_1^* = M/2,\ x_2^* = 0. The Lagrange multiplier equals the marginal utility of income:

  5. 5

    λ∗=MU1/p1=3/2\lambda^* = MU_1/p_1 = 3/2.

Answer structure / marking notes

λ∗\lambda^* equals the maximum of MUi/piMU_i/p_i, not the smaller ratio.

Content note

Imported from public/resources/isi/msqe/solutions/pea/2022/ISI_MSQE_PEA_2022_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.