There are two goods, coffee and mineral water, available in arbitrary nonnegative quantities (so the consumption set is R2+). A consumer has preferences 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). 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. 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 τ > 0 Dirhams/gram for coffee. This tax increases the price of coffee for the consumer from p Dirhams/gram to p + τ Dirhams/gram. What is the minimum tax τ required to guarantee that the consumer will consume coffee and mineral water in equal quantities? Compute τ and illustrate your answer in an appropriate diagram.
There are two goods, coffee and mineral water, available in arbitrary nonnegative quantities (so the consumption set is R2+). A consumer has preferences 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).
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.
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 τ > 0 Dirhams/gram for coffee. This tax increases the price of coffee for the consumer from p Dirhams/gram to p + τ Dirhams/gram.
What is the minimum tax τ required to guarantee that the consumer will consume coffee and mineral water in equal quantities? Compute τ and illustrate your answer in an appropriate diagram.
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