Intermediate Microeconomics Problem Set 2 Part 1: California Dreamin’ California is facing a serious drought, but policy-makers in the state are unwilling to alter the prices at which water is sold for fear that it will unduly harm the poorest citizens.
Intermediate Microeconomics Problem Set 2 Part 1: California Dreamin’ California is facing a serious drought, but policy-makers in the state are unwilling to alter the prices at which water is sold for fear that it will unduly harm the poorest citizens. Some have suggested that the price be raised and the revenues returned to each citizen in the form of a tax rebate. You will use the economic methods you have learned to evaluate such a policy. The Model Let’s imagine there are only three people living in California, but that each of them have preferences described by the following utility function: 𝑢(𝑥1, 𝑥2 ) = 𝑥1 1/10 𝑥2 9/10 In that function, 𝑥1 is the quantity of water consumed, and 𝑥2 is the quantity of the composite good consumed (where the ‘composite good’ represents ‘all other goods’). The State of California will impose a quantity tax on water (𝑡) but will rebate the revenue to each consumer as a lump-sum. The total revenue generated by the tax (𝑅) will be: 𝑅 = 𝑋 1 𝑡 Where 𝑋 1 is the total market demand for water, and 𝑡 is the quantity tax on water. This revenue will be split between the three consumers equally, so that the rebate (𝑟) will be: 𝑟 = 1 3 𝑋 1 𝑡 The Details The three people in our model all have the same preferences, but each has a different income. One of them has an income of $100, another an income of $1,000, and the third has an income of $10,000. The price of water1 is $1 and the price of the composite good is $1. The State of California must reduce the total consumption of water to 300 units. Use this information to answer the questions on the next page. 1 The interpretation here is that we are measuring units of water in terms of what $1 can currently buy. So when we say that a consumer is buying 10 ‘units’ of water this means however much water $10 currently gets you. This is a common and easy simplification to make. But, just to be clear, nothing in this footnote is important to getting the right answer to the problems below. The Questions 1. Derive each of the consumer’s demand functions for water without any taxes or rebates: 𝑥1(𝑝) 2. Derive the total market demand for water without any taxes or rebates: 𝑋 1 (𝑝) 3. Determine the total amount of water consumed in this market. 4. Calculate the quantities consumed for each individual consumer without any taxes or rebates. 5. Calculate the level of utility consumed for each individual consumer at these quantities. 6. Repeat questions 1 and 2, but add in the tax and rebate. 7. Derive a formula for revenue that is a function of only the tax: 𝑅(𝑡) 8. Derive a formula for the rebate that is a function of only the tax: 𝑟(𝑡) 9. Determine what the tax must be to reduce total consumption of water to 300 units. 10. What is the rebate at the tax determined in previous question? 11. Calculate the quantities consumed for each individual consumer with that tax and rebate. 12. Calculate the level of utility consumed for each individual consumer at these quantities. 13. Which consumers have been made better off and which have been made worse off. 14. How much would the wealthiest consumer we willing to pay to get rid of the tax (aka calculate their equivalent variation)? 15. Would the other two consumers be willing to accept the wealthiest consumer’s offer (aka calculate their compensating variation)? 16. Suppose that the State placed the same tax on water, but did not offer a rebate. Calculate the income and substitution effects for the poorest consumer. And remember…
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