Question

. Suppose consumer 1 gets utility ϕ 1 (h) if she can listen to music, where h≥0 is the noise level. Consumer 2 gets disutility ϕ

2 (h) if consumer 1 enjoys music with noise level h. Suppose that ϕ
1 (h) is twice differentiable, increasing and concave and that ϕ
2 (h) is twice differentiable, decreasing and concave.
a) What level of h ∗ maximizes the sum of the two individuals' utilities? What level of
h^ maximizes consumer 1's utility, what level h
~maximizes consumer 2's utility?
b) Suppose that the utility of consumer i with music at level h and receiving a transfer T is ϕ
i (h)+T(−T if the consumer has to pay T). Suppose that consumer 1 has the right to choose the level that maximizes his payoff,
h^ . However, consumer 2 can make a take-it-or-leave-it offer to consumer 1 to pay the amount T to consumer 1 to reduce h to some level. What level of h will be chosen if consumer 2 makes the optimal offer to consumer 1 ?

Answer 1

To determine the optimal noise level h* providing maximum sum of utilities for both consumers, one must calculate the first and second derivatives of their combined utility function. Consumer 1's utility is maximized at a certain noise level h^, while Consumer 2's at h~. For part b), an optimal transfer T from Consumer 2 to Consumer 1 would be proposed, affecting Consumer 1's choice of h to maximize their combined utility given the transfer.

Step-by-step explanation:

The question posed involves a scenario where two consumers have differing preferences in regards to the noise level, h, associated with playing music. Consumer 1 experiences utility, denoted as φ1(h), from the noise, while Consumer 2 experiences disutility, denoted as φ2(h). The utilities are twice differentiable functions, with Consumer 1's utility being increasing and concave, and Consumer 2's disutility being decreasing and concave.

To find the optimal level of noise, h*, that maximizes the sum of utilities of both consumers, we would set up the following equation:

Maximize: U = φ1(h) - φ2(h)

By taking the first derivative of U with respect to h and setting it to zero, and then checking the second derivative for a maximum, one could find h*. Consumer 1's utility alone is maximized at h^, where the derivative of φ1(h) is zero, while Consumer 2's utility is maximized at h~, which would be a level of no noise where φ2(h) is at its highest.

In part b), introducing the concept of a transfer T to the utility functions involves a game-theoretic scenario where Consumer 1 has the right to choose the noise level, but Consumer 2 can offer a payment to change this level. An ideal level of h will be chosen based on maximizing Consumer 1's overall utility, taking into account the offered transfer T.

To deduce the optimal offer T and the resulting noise level h, one would apply the utility maximization principle, taking both the transfer and the changed noise level into account. A transfer that makes Consumer 1 indifferent between keeping h at his preferred level or reducing the noise level to Consumer 2's preference would be the solution.

Answer 2

Income changes can lead to different allocations of spending based on personal preferences and whether goods are normal or inferior. The utility-maximizing choice aligns with the highest indifference curve achievable within a given budget constraint, adhering to the rule that the ratios of marginal utility to price for each good should be equal at the optimal choice.

Step-by-step explanation:

When individuals seek the highest level of utility, they aim to reach the highest possible indifference curve within the confines of their budget constraints. This constraint represents the possible trade-offs between goods they can afford, given their income and prices of goods. In the scenario where Manuel and Natasha's income rises to $60, their budget constraints shift outward, allowing for a higher consumption of goods.

Their new utility-maximizing choices are determined by their preferences for each good. Manuel's preference for movies leads him to allocate most of his additional income to movies, while Natasha, with a preference for yogurt, allocates more income to purchasing yogurt. This demonstrates how income changes can influence consumption choices depending on individual preferences and whether goods are considered normal or inferior.

Utilizing the indifference curve approach, we observe varied responses based on personal preferences. The rule for maximizing utility is that the ratios of the marginal utility to the price of each good should be equal at the optimal choice, indicating that there is no better way to allocate the budget that would result in a higher utility level.