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Provide an example of any utility function with decreasing, hammock-shaped indifference curves displaying the main feature shown in the figure of slide 4. Justify in detail why the utility function you chose satifies all these characteristics.

The image you provided shows a graph with indifference curves (ICs) that are vertically parallel, which is a characteristic of specific types of utility functions. The figure also indicates that the indifference curves are downward sloping, which suggests that the utility function in question exhibits substitutability between the two goods (represented on the x and y axes).

### Example of a Utility Function:
One utility function that could result in decreasing, hammock-shaped indifference curves that are vertically parallel is:

\[ U(x, y) = -\sqrt{(x - a)^2 + (y - b)^2} \]

Here, \( a \) and \( b \) are constants, and \( x \) and \( y \) represent the quantities of two goods. The negative square root ensures that the utility decreases as you move away from the point \( (a, b) \).

### Justification:
1. **Hammock Shape**: The utility function is designed to decrease as you move away from a specific point \( (a, b) \). This means the indifference curves will be hammock-shaped, reflecting that utility is maximized at \( (a, b) \), and decreases as you move in any direction away from this point. The shape of the indifference curves resembles a hammock, which is concave toward the origin.

2. **Decreasing Indifference Curves**: The utility function is a negative value of the square root, which means that as you increase the distance from \( (a, b) \), the utility decreases. The indifference curves, therefore, are downward sloping as moving in either the \( x \) or \( y \) direction away from \( (a, b) \) results in lower utility, which corresponds to lower levels of indifference curves.

3. **Vertically Parallel Indifference Curves**: Vertically parallel indifference curves imply that for any given \( x \), the utility function changes only depending on \( y \). The function \( U(x, y) \) depends on the distance from \( (a, b) \), and as you move vertically along the \( y \)-axis with \( x \) constant, the indifference curves will remain parallel. This characteristic aligns with the vertical parallelism shown in the figure.

### Conclusion:
The utility function \( U(x, y) = -\sqrt{(x - a)^2 + (y - b)^2} \) exhibits the hammock-shaped, vertically parallel indifference curves as depicted in the figure. The shape of these curves, the fact that they are vertically parallel, and their decreasing nature as utility decreases with distance from the point \( (a, b) \), all satisfy the characteristics observed in the provided figure.

Would you like more details or have any specific questions? Here are five related questions you might find interesting:

1. How do changes in the constants \( a \) and \( b \) affect the position of the hammock-shaped indifference curves?
2. Can this utility function be extended to more than two goods, and how would the indifference curves look in higher dimensions?
3. How does the concept of substitutability between goods relate to the shape and slope of indifference curves in general?
4. Are there real-world examples of goods whose utility functions might produce hammock-shaped indifference curves?
5. What mathematical properties are necessary for a utility function to have convex (rather than concave) indifference curves?

**Tip**: When analyzing utility functions and their corresponding indifference curves, always consider how changes in quantities of goods influence overall utility. This helps in understanding consumer preferences and behaviors.

Explore an example of a utility function with decreasing, hammock-shaped indifference curves. Understand how the chosen utility function, U(x, y) = -\sqrt{(x - a)^2 + (y - b)^2}, exhibits characteristics such as hammock shape, decreasing nature, and vertically parallel indifference curves.

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