Question

The utility function of a consumer from consumption of $x_1$ and $x_2$ is given by \[ u(x_1, x_2) = x_1 + 2\sqrt{x_2}. \] At the current prices and income, the consumer’s optimal consumption bundle is given by $(x_1 = 10, x_2 = 10)$. The consumer’s optimal choice of $x_2$, if his income increases by 50% but prices remain unchanged, is \_\_\_\_\_\_\_\_\_\_\_. (in integer)

Hint:

For utility functions that are quasi-linear in one good, income changes affect only the non-linear good’s consumption if marginal utilities remain constant in ratio.

Answer 1

Correct Answer: 10

Step 1: Derive the optimal condition.
The marginal utilities are: \[ MU_1 = 1, \quad MU_2 = \frac{1}{\sqrt{x_2}}. \] At optimum, \[ \frac{MU_1}{p_1} = \frac{MU_2}{p_2} \Rightarrow \frac{1}{p_1} = \frac{1/\sqrt{x_2}}{p_2} \Rightarrow \sqrt{x_2} = \frac{p_2}{p_1}. \]
Step 2: Use the given initial condition.
Given optimal bundle $(x_1, x_2) = (10, 10)$, let prices be $(p_1, p_2)$ and income be $m$. From budget constraint: \[ p_1x_1 + p_2x_2 = m \Rightarrow 10p_1 + 10p_2 = m. \]
Step 3: Substitute $\sqrt{x_2 = \frac{p_2}{p_1}$ at optimum.}
From $x_2 = 10$, \[ \sqrt{10} = \frac{p_2}{p_1} \Rightarrow p_2 = p_1\sqrt{10}. \] Then, \[ m = 10p_1 + 10p_1\sqrt{10} = 10p_1(1 + \sqrt{10}). \]
Step 4: When income increases by 50%.
New income $m' = 1.5m = 15p_1(1 + \sqrt{10})$. At new optimum (same prices), \[ p_1x_1' + p_2x_2' = 1.5m. \] Substituting $p_2 = p_1\sqrt{10}$ and $\sqrt{x_2'} = \frac{p_2}{p_1} = \sqrt{10}$ gives same ratio of marginal utilities. Budget expands proportionally → $x_1'$ and $x_2'$ scale by 1.5.
Step 5: Compute new $x_2$.
\[ x_2' = 1.5 \times 10 = 15. \] Thus, new $x_2 = 15$.
Step 6: Conclusion.
\[ \boxed{x_2' = 15.} \]