Scientists have developed a vaccine called See-Ya-Corona. The government needs to encourage people to get vaccinated. Consider a representative citizen named Citizen Korona. Any income spent on health care is exempt from income tax, while income spent on other goods and services is subject to tax. To further incentivize citizens to invest in health care, the government will match any health care spending. That is, for every dollar a citizen spends on health care, the government will spend an additional dollar on the Citizen Korona's behalf. Let x1 denote total spending on the vaccine and other health care expenses (including government matching) and x2 denote Citizen Korona's spending on other goods and services. You can assume Citizen Korona has an income of 50000 and a marginal income tax rate of 20 percent. Citizen Korona has Cobb-Douglas preferences with = 23 and = 2.

1. (a)Find the first-order necessary condition(s) for utility maximization.
2. (b)How much should Citizen Korona spend on health care?
3. To successfully fight COVID-19, the government needs total spending on health care to be at least 15,000. For parts (c) and (d), you will compare two policies for achieving this spending level. You can assume the government wants to maximize net revenue (i.e., tax revenue minus spending). Unless otherwise specified, everything else in the problem remains the same.
4. (c)The government will now contribute r dollars for every dollar spent on health care by Citizen Korona. Find the optimal value of r that achieves the government's desired spending level on health care.
5. (d)The government eliminates the health care spending matching program and the tax exemption for health care spending. It will instead send Citizen Korona a direct payment P that can only be used for health care. How large does P need to be to achieve the 15000 spending level on health care? Does the government prefer this policy or the policy in (c)? Justify your answer.