Question

The utility function of a consumer in a two-good world is given by

Hint:

For perfect complements utility functions, optimal consumption occurs where the quantities of goods are equal. Substitute this condition into the budget equation to derive the Engel curve.

Answer 1

Correct Answer: 3

Step 1: Read the utility.
The utility is $u(x_1,x_2)=\min\{x_1,x_2\}$, the perfect complements case. So at the best choice the two goods are used in equal amounts, $x_1=x_2=x$.

Step 2: Write the budget.
With $p_1=1$ and $p_2=2$ and income $M$,
\[ x_1+2x_2=M \]

Step 3: Use the equal amounts.
Putting $x_1=x_2=x$,
\[ x+2x=M\;\Rightarrow\;3x=M \]

Step 4: Get the demand for good 2.
So $x_2=\dfrac{M}{3}$.

Step 5: Flip it for the Engel curve.
With income on the vertical axis, $M=3x_2$. This is a straight line through the origin with slope $3$.
\[ \boxed{3} \]