Problem 1: Non-differentiable utility with multiple constraints There are two goods, coffee and mineral water, available in arbitrary non- negative quantities (so the consumption set is R²). A consumer has pref- erences over consumption bundles that are represented by the following utility function: u(c, m) = min{c, m}+c+m, where e is the quantity of coffee (in grams) and m is the quantity of mineral water (in liters). The consumer has wealth in Dirhams of w> 0. The price of coffee is p > 0 (in grams/Dirham) and the price of mineral water q> 0 (in liters/Dirham). (a) In an appropriate diagram, illustrate the consumers map of indifference curves. Make sure you label the diagram clearly, and include as part of your answer any calculations about the slopes of the indifference curves. (b) Formulate and solve the consumer's utility maximization problem. Your final answer should describe the consumer's demand for coffee and mineral water as a function of the wealth w and the prices p and q, as well as the consumer's indirect utility function. Now assume that w = 100, p= 10 and q = 10. In addition to the monetary budget constraint, the consumer has a time constraint. The consumer has only 100 minutes available for the consumption of both goods. It takes 5 minutes/liter to consume water and and 20 minutes/gram to consume coffee. (e) In an appropriate diagram, illustrate the consumer's constraint set. Your diagram should illustrate the monetary budget constraint, the time constraint, and indicate the consumer's overall constraint set (i.e., the set of all consumption bundles the consumer can consume given their limited income and the prices of the goods, as well as the consumers limited time and the time it takes to consume each of the goods).
Problem 1: Non-differentiable utility with multiple constraints There are two goods, coffee and mineral water, available in arbitrary non- negative quantities (so the consumption set is R²). A consumer has pref- erences over consumption bundles that are represented by the following utility function: u(c, m) = min{c, m}+c+m, where e is the quantity of coffee (in grams) and m is the quantity of mineral water (in liters). The consumer has wealth in Dirhams of w> 0. The price of coffee is p > 0 (in grams/Dirham) and the price of mineral water q> 0 (in liters/Dirham). (a) In an appropriate diagram, illustrate the consumers map of indifference curves. Make sure you label the diagram clearly, and include as part of your answer any calculations about the slopes of the indifference curves. (b) Formulate and solve the consumer's utility maximization problem. Your final answer should describe the consumer's demand for coffee and mineral water as a function of the wealth w and the prices p and q, as well as the consumer's indirect utility function. Now assume that w = 100, p= 10 and q = 10. In addition to the monetary budget constraint, the consumer has a time constraint. The consumer has only 100 minutes available for the consumption of both goods. It takes 5 minutes/liter to consume water and and 20 minutes/gram to consume coffee. (e) In an appropriate diagram, illustrate the consumer's constraint set. Your diagram should illustrate the monetary budget constraint, the time constraint, and indicate the consumer's overall constraint set (i.e., the set of all consumption bundles the consumer can consume given their limited income and the prices of the goods, as well as the consumers limited time and the time it takes to consume each of the goods).
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.13P
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Transcribed Image Text:Problem 1: Non-differentiable utility with multiple constraints
There are two goods, coffee and mineral water, available in arbitrary non-
negative quantities (so the consumption set is R). A consumer has pref-
erences over consumption bundles that are represented by the following
utility function:
u(c, m) = min{c,m} +c+m,
where c is the quantity of coffee (in grams) and m is the quantity of mineral
water (in liters).
The consumer has wealth in Dirhams of w > 0. The price of coffee is p >0
(in grams/Dirham) and the price of mineral water q >0 (in liters/Dirham).
(a) In an appropriate diagram, illustrate the consumers map of indifference
curves. Make sure you label the diagram clearly, and include as part
of your answer any calculations about the slopes of the indifference
curves.
(b) Formulate and solve the consumer's utility maximization problem.
Your final answer should describe the consumer's demand for coffee
and mineral water as a function of the wealth w and the prices p and
q, as well as the consumer's indirect utility function.
Now assume that w 100, p= 10 and q = 10. In addition to the monetary
budget constraint, the consumer has a time constraint. The consumer has
only 100 minutes available for the consumption of both goods. It takes
5 minutes/liter to consume water and and 20 minutes/gram to consume
coffee.
(e) In an appropriate diagram, illustrate the consumer's constraint set.
Your diagram should illustrate the monetary budget constraint, the
time constraint, and indicate the consumer's overall constraint set
(i.e., the set of all consumptic
bundles the consumer can consume
given their limited income and the prices of the goods, as well as the
consumers limited time and the time it takes to comsume each of the
goods).
(d) Find the optimal consumption bundle and illustrate the solution in an
appropriate diagram.
The government wants the consumer to consume an equal quantity of coffee
and water. To achieve this policy objective, the government implements
a tax of 7> 0 Dirhams/gram for coffee. This tax increases the price of
coffee for the consumer from p Dirhams/gram top+T Dirhams/gram.
(e) Assume the same prices and wealth as in parts (c) and (d). What
is the minimum tax 7 required to guarantee that the consumer will
consume coffee and mineral water in equal quantities? Compute r and
illustrate your answer in an appropriate diagram.
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