2) a) Originally Ramesh faces the budget line 𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚. Then the price of the good 1 doubles, the price of good 2 becomes 5 times larger, and income becomes 3 times larger. Write down the equation for the new budget line in terms of the original prices and income.b) Emy was consuming 150 units of X and 50 units of Y. The price of X rose from 5 to 7. The price of Y remained at 3. How much would Emy’s income have to rise so that she can still afford the 150 units of X and 50 units of Y?

### Part (a): New Budget Line Equation

Originally, Ramesh's budget line is given by:

\[
p_1 x_1 + p_2 x_2 = m
\]

where:
- \( p_1 \) is the price of good 1,
- \( p_2 \) is the price of good 2,
- \( x_1 \) is the quantity of good 1,
- \( x_2 \) is the quantity of good 2,
- \( m \) is the income.

Now, the changes are:
- The price of good 1 doubles: \( p_1' = 2p_1 \),
- The price of good 2 becomes 5 times larger: \( p_2' = 5p_2 \),
- The income becomes 3 times larger: \( m' = 3m \).

The new budget line equation can be written as:

\[
2p_1 x_1 + 5p_2 x_2 = 3m
\]

This is the equation for the new budget line in terms of the original prices \( p_1 \) and \( p_2 \), and the original income \( m \).

### Part (b): Income Adjustment for Emy

Emy was initially consuming 150 units of good \( X \) and 50 units of good \( Y \). The prices were:
- Price of \( X \) = \( 5 \),
- Price of \( Y \) = \( 3 \).

Her original expenditure was:

\[
\text{Original Expenditure} = 150 \times 5 + 50 \times 3 = 750 + 150 = 900
\]

Now, the price of \( X \) rises from 5 to 7, while the price of \( Y \) remains at 3. To afford the same quantities (150 units of \( X \) and 50 units of \( Y \)), her new expenditure would be:

\[
\text{New Expenditure} = 150 \times 7 + 50 \times 3 = 1050 + 150 = 1200
\]

To maintain the same consumption levels, Emy's income would have to rise from 900 to 1200.

**Required Income Increase:**

\[
\text{Income Increase} = 1200 - 900 = 300
\]

So, Emy's income needs to rise by 300 units to afford the same quantities of \( X \) and \( Y \) after the price change.

Would you like more details, have any questions, or need further clarification?

### Related Questions:

1. How do you derive the equation of a budget line?
2. What happens to a budget line if only income changes?
3. How do changes in the prices of both goods affect the slope of the budget line?
4. Can you explain the concept of a budget set?
5. What is the effect of a proportional change in all prices and income on the budget line?
6. How would the budget line change if income decreased instead of increasing?
7. What is the importance of the slope of the budget line in consumer theory?
8. How does a change in only one price affect the budget line?

**Tip:** When analyzing budget lines, always pay attention to how changes in prices or income affect both the slope and the intercepts of the line.

Explore how changes in prices and income affect budget lines in consumer theory. Learn how to derive new budget line equations when prices and income change. Includes a detailed example of adjusting income to maintain consumption levels.

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