Question

An individual is endowed with income of ₹ 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 ₹ 2 and the unit price of $x_2$ is ₹ 3. The utility maximizing bundle is

• A. $x_1=35,\; x_2=20$
• B. $x_1=30,\; x_2=24$
• C. $x_1=35,\; x_2=24$
• D. $x_1=30,\; x_2=20$

Hint:

For interior solutions, set $\text{MRS}=\dfrac{p_1}{p_2}$ and use the budget to solve. For $U=x_2(x_1+1)$, $MU_{x_1}=x_2$ and $MU_{x_2}=x_1+1$.

Answer 1

The Correct Option is C

Step 1: Budget constraint.
$2x_1+3x_2=₹\,142 \Rightarrow x_2=\dfrac{142-2x_1}{3}$.
[2mm] Step 2: Marginal utilities and MRS.
$MU_{x_1}=\dfrac{\partial U}{\partial x_1}=x_2, MU_{x_2}=\dfrac{\partial U}{\partial x_2}=x_1+1$.
For an interior optimum: $\text{MRS}=\dfrac{MU_{x_1}}{MU_{x_2}}=\dfrac{x_2}{x_1+1}=\dfrac{p_1}{p_2}=\dfrac{2}{3} \Rightarrow 3x_2=2(x_1+1)$.
[2mm] Step 3: Solve with the budget.
From $3x_2=2x_1+2$ and $2x_1+3x_2=142$:
$2x_1+(2x_1+2)=142 \Rightarrow 4x_1=140 \Rightarrow x_1=35$.
Then $x_2=\dfrac{2x_1+2}{3}=\dfrac{70+2}{3}=24$.
[1mm] Feasibility ($x_1,x_2\ge 0$) holds, hence the maximizing bundle is \fbox{$(35,\,24)$}.