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...

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 ModelLet'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^1t

Where X^1is 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=1/3X^1t

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 water1is $1 and the price of the composite good is $1. The State of California must reduce the total consumption of water to 300 units.

1. Derive each of the consumer's demand functions for water with the taxes and rebates

2. Derive the total market demand for water with the taxes and rebates

3. Derive a formula for revenue that is a function of only the tax: R(t)

4. Derive a formula for the rebate that is a function of only the tax: r(t)

5. Determine what the tax must be to reduce total consumption of water to 300 units.

6. What is the rebate at the tax determined in previous question?

7. Calculate the quantities consumed for each individual consumer with that tax and rebate.

8. Calculate the level of utility consumed for each individual consumer at these quantities.

9. Which consumers have been made better off and which have been made worse off.

10. How much would the wealthiest consumer we willing to pay to get rid of the tax (aka calculate their equivalent variation)?

11. Would the other two consumers be willing to accept the wealthiest consumer's offer (aka calculate their compensating variation)?

12. 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.

CARLIFONIA STATEQuestion 1Analysis of the questionUtility U(X1 X2) = X1 /10 X29/10U(X1 X2) = 0.1X10.1X29Total revenue (r ) = XtX = Total market demand for waterT = Tax on waterDivision of...