Question

In a two-good world, the utility function of a consumer is given by

Hint:

Compensating Variation measures the extra income required after a price increase to keep the consumer at the original utility level.

Answer 1

Correct Answer: 41.4

Step 1: Identify the initial prices and income.
Initially, the prices are
\[ p_1=1,\quad p_2=1 \] and income is
\[ M=100 \]
The utility function is
\[ u(x_1,x_2)=x_1x_2 \]
This is a Cobb-Douglas utility function with equal exponents.
For such a utility function, the consumer spends equal fractions of income on both goods.

Step 2: Find the initial optimal consumption bundle.
Since income is equally divided, expenditure on each good is
\[ \frac{100}{2}=50 \]
Because both prices are equal to \(1\),
\[ x_1=50 \] and
\[ x_2=50 \]
Thus, the initial utility level is
\[ u_0=50\times50 \] \[ u_0=2500 \]

Step 3: Write the new prices after price change.
After the price increase,
\[ p_1=1 \] and
\[ p_2=2 \]
We now calculate the minimum expenditure required to achieve the original utility level
\[ u_0=2500 \]
This minimum expenditure gives the expenditure function and helps determine compensating variation.

Step 4: Use the expenditure minimization condition.
For the Cobb-Douglas utility function
\[ u=x_1x_2 \] the expenditure function is
\[ e(p_1,p_2,u)=2\sqrt{up_1p_2} \]
Substitute
\[ u=2500,\quad p_1=1,\quad p_2=2 \]
\[ e=2\sqrt{2500\times1\times2} \]
\[ e=2\sqrt{5000} \]
\[ e=2(70.7107) \] \[ e=141.4214 \]

Step 5: Calculate Compensating Variation.
Compensating Variation is the additional income needed after the price rise to restore the consumer to the original utility level.
Thus,
\[ CV=e-M \]
\[ CV=141.4214-100 \]
\[ CV=41.4214 \]
Rounded off to one decimal place,
\[ CV=41.4 \]

Step 6: Final conclusion.
Hence, the compensating variation associated with the price increase is
\[ \boxed{41.4} \]