3 Application (20 points) Consider a possible policy application. Suppose we are interested in peoples' choices of how much to drive. We consider a simple model where a person consumers two goods with the utility function U(x1, x2) = x²+x² with budget constraint P₁x1 + x2 = 1. Good 1 x is the kilometers that this person drives and good 2 r2 is money spent on all other goods. Notice that the price of good 2 is set to 1, or p2 = 1. 1. (1 point) Set up the utility maximization problem. 2. (1 point) Set up the Lagrangian. 3. (5 points) Find the first-order conditions for the utility maximization problem. 4. (5 points) Solve the demand functions for goods r₁ and 2 as a function of price P₁, income I, and the parameter 8. (don't worry about the second derivative.) = 5. (1 point) Suppose p₁ = 10, I = 1100, and 8 1/2. Find the optimal con- sumption bundle ₁ and 2. How much does the consumer spend on driving (px)? 3 6. Suppose the government would like to discourage driving. To do so, the gov- ernment imposes a tax at rate t on spending on miles driven. So now, the price paid by the consumer for good 1 is (1+t)p₁. (a) (1 point) Write down the new budget constraint for the consumer given this tax, and write the corresponding demand functions (b) (1 point) Write down the new demand functions of r₁ and 12. (c) (2 points) Let t = 1 (a huge tax that doubles the cost of good 1). Suppose you would like to estimate the tax revenue raised by this policy, the tax revenue is given by tp₁₁ where ₁ is quantity of good 1 consumed, and P₁₁ is the spending on driving. Suppose you estimated this revenue by using the value for p₁₂ in part 5, what would your estimate be? (d) (3 points) Solve for the optimal consumption bundle with the tax. What is the tax revenue raised by the government provided p₁ = 10, I = 1100, and 1/2. (hint, you can use the solution in part 6b and plug in the appropriate t, pi, I, and B. Explain the difference between 6c and 6d. =