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: Understand the nature of the utility function.
The given utility function is
\[ u(x_1,x_2)=\min\{x_1,x_2\} \] This is a perfect complements utility function.
For perfect complements, the consumer always consumes the two goods in fixed proportion.
Therefore, at utility maximization, the optimal bundle satisfies
\[ x_1=x_2 \]
Let
\[ x_1=x_2=x \]

Step 2: Write the budget constraint.
The prices are given as
\[ p_1=1 \] and
\[ p_2=2 \]
Let income be denoted by
\[ M \]
The budget equation is
\[ p_1x_1+p_2x_2=M \]
Substituting the prices,
\[ 1\cdot x_1+2\cdot x_2=M \]
Since
\[ x_1=x_2=x \] we get
\[ x+2x=M \] \[ 3x=M \]

Step 3: Find the demand function for \(X_2\).
From
\[ 3x=M \] we obtain
\[ x=\frac{M}{3} \]
Since
\[ x_2=x \] therefore,
\[ x_2=\frac{M}{3} \]
This is the Engel curve for good \(X_2\).

Step 4: Express income as a function of \(x_2\).
The question states that income is measured on the vertical axis.
Thus, we express income \(M\) in terms of \(x_2\).
From
\[ x_2=\frac{M}{3} \] multiplying both sides by \(3\),
\[ M=3x_2 \]

Step 5: Find the slope of the Engel curve.
The Engel curve equation is
\[ M=3x_2 \]
Comparing with the straight-line form
\[ y=mx \] the slope is
\[ 3 \]

Step 6: Final conclusion.
Hence, the slope of the Engel curve for \(X_2\), when income is measured on the vertical axis, is
\[ \boxed{3} \]