https://file.aiquickdraw.com/mathbot/file/1725258849027kemns20h.jpg

Redraw the graph in slide 4 in your answer (including all its parts) and illustrate the effect of a drop in the consumer’s income, assuming that the optimal bundle is interior. Clearly indicate which budget lines and optimal bundles correspond to the new and old income. Is x an inferior good for a utility function whose indifference curves have the properties illustrated in the figure? Is y an inferior good for a utility function whose indifference curves have the properties ilustrated in the figure? (5 marks)

### Explanation of the Graph and the Effect of Income Drop

The graph above illustrates the effect of a drop in the consumer's income on their choice of optimal bundles:

- **Indifference Curves \( U_1 \) and \( U_2 \)**: These represent different levels of utility, with \( U_2 \) being higher than \( U_1 \). The consumer prefers higher indifference curves as they represent higher utility.
  
- **Old Budget Line (Green Dashed Line)**: This represents the consumer's budget constraint before the income drop. The line is tangent to the higher indifference curve \( U_2 \) at the point marked as the **Old Optimal Bundle** (green dot).

- **New Budget Line (Red Dashed Line)**: After the income drop, the budget line shifts inward, becoming parallel to the old one but at a lower level, reflecting reduced income. The new budget line is tangent to the lower indifference curve \( U_1 \) at the point marked as the **New Optimal Bundle** (red dot).

### Inferior Goods Analysis

- **Is \( x \) an Inferior Good?**  
  For \( x \) to be an inferior good, the quantity consumed should decrease as income increases. Here, when income decreases (the budget line shifts inward), the quantity of \( x \) consumed decreases from the Old Optimal Bundle to the New Optimal Bundle. Thus, \( x \) is not an inferior good in this scenario; it behaves as a normal good.

- **Is \( y \) an Inferior Good?**  
  Similarly, for \( y \) to be an inferior good, the quantity consumed should decrease as income increases. However, the quantity of \( y \) also decreases as income decreases, indicating that \( y \) is not an inferior good either.

In conclusion, both \( x \) and \( y \) behave as normal goods under the conditions illustrated in the graph.

### Questions for Further Exploration

1. How would the graph change if the prices of \( x \) and \( y \) also changed along with income?
2. What happens if the indifference curves are linear rather than curved?
3. How would you graphically represent a situation where \( x \) is an inferior good?
4. What effect does a change in the price of \( x \) have on the budget line?
5. How does the slope of the indifference curve relate to the marginal rate of substitution?

**Tip:** Always check if the utility function is well-behaved (i.e., convex indifference curves) before analyzing the effect of income changes on optimal bundles.

Explore the effect of a drop in consumer income on optimal bundles and the classification of goods as inferior or normal based on indifference curves and budget constraints. Detailed graphical analysis included.

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