1 mark, −0.333MicroeconomicsBudget constraint and consumer equilibriumDemand under complements, substitutes and quasilinear utilityModerateabout 3 min

GATE 2023 Question 27: Budget constraint and consumer equilibrium

An individual is endowed with income of Rs. 142 and has the utility function $U(x_1,x_2) = x_2(x_1+1)$ , where $x_1 \ge 0, x_2 \ge 0$ . The unit price of $x_1$ is Rs. 2 and the unit price of $x_2$ is Rs. 3. The utility maximizing bundle is

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Answer and worked solution

Answer: (C)

Budget constraint: $2x_1 + 3x_2 = 142$ .

$MU_1 = \dfrac{\partial U}{\partial x_1} = x_2$ , $\quad MU_2 = \dfrac{\partial U}{\partial x_2} = x_1+1$ .

Tangency condition: $\dfrac{MU_1}{MU_2} = \dfrac{p_1}{p_2} \Rightarrow \dfrac{x_2}{x_1+1} = \dfrac{2}{3} \Rightarrow 3x_2 = 2(x_1+1) = 2x_1+2 \Rightarrow x_2 = \dfrac{2x_1+2}{3}$ .

Substitute into the budget constraint:

$2x_1 + 3\left(\frac{2x_1+2}{3}\right) = 142 \Rightarrow 2x_1 + 2x_1+2 = 142 \Rightarrow 4x_1 = 140 \Rightarrow x_1 = 35$

$x_2 = \frac{2(35)+2}{3} = \frac{72}{3} = 24$

Therefore the utility-maximizing bundle is $x_1=35, x_2=24$ , option (C).

The trap: (A)

x_1=35, x_2=20

Students sometimes treat this as an ordinary Cobb-Douglas problem and apply the standard constant-expenditure-share shortcut, ignoring the '+1' shift inside the utility function — this shifted marginal utility changes the tangency condition and produces a different, incorrect $x_2$ value (option A keeps the correct $x_1=35$ but gets $x_2$ wrong).

Plain-text version

An individual is endowed with income of Rs. 142 and has the utility function U(x₁,x₂) = x₂(x₁+1), where x₁ ≥ 0, x₂ ≥ 0. The unit price of x₁ is Rs. 2 and the unit price of x₂ is Rs. 3. The utility maximizing bundle is (A) x₁=35, x₂=20 (B) x₁=30, x₂=24 (C) x₁=35, x₂=24 (D) x₁=30, x₂=20

Also written as: An individual is endowed with income of Rs. 142 and has the utility function U(x_1,x_2) = x_2(x_1+1), where x_1 >= 0, x_2 >= 0. The unit price of x_1 is Rs. 2 and the unit price of x_2 is Rs. 3. The utility maximizing bundle is

Budget constraint and consumer equilibrium has come up 99 times in 7 exams since 2004.

What this question tests

This question is on Demand under complements, substitutes and quasilinear utility, part of Budget constraint and consumer equilibrium in Microeconomics.

Deriving demand when preferences are L-shaped, linear or quasilinear, and what that implies for income and price responses.

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