**(25%) This question applies the concept of indirect utility to study migration and voting decisions.**

Suppose there are 99 households in a country. The households consume two goods in their leisure time: books (good 1) and guns (good 2). Household i's preference is represented by the utility function:

\[ U(X_1, X_2) = (x_1)^{1-\theta} (x_2)^{\theta} \]

For example, the preference of household i = 30 is represented by:

\[ U_{30}(x_1, x_2) = (x_1)^{100} (x_2)^{100} \]

**(a)**
i. What percentage of household i's expenditure is spent on books?

ii. Show that household i's indirect utility function, the maximal level of utility attainable, is proportional to:

\[ m (p_1)^{-100} (p_2)^{100} \]

where \( p_1 \) and \( p_2 \) denote the prices of the two goods, and \( m \) denotes the after-tax income level.

The country is composed of two municipalities: the Blue state and the Red state. The two states have different fiscal policies:

- Red state enacts a flat income tax rate of 10%.
- Blue state enacts an ad valorem subsidy of 25% on books and a flat income tax rate of 20%.

The states are identical in every other aspect. The good prices \( p_1 \) and \( p_2 \) are the same, and households receive the same before-tax income level \( M_0 \) in both states. Households can settle in any state without incurring moving costs.

**(b)**
Find out who will live in the Red state and who will live in the Blue state using the indirect utility function.

**Hint:** Consider how the tax and subsidy in each state affect a household's budget line. Modify the indirect utility function accordingly.

**(c)**
Is there any systematic difference in the consumption between the residents in the two states? Why do households with similar consumption preferences segregate into different states? Explain your answer in fewer than 180 words.

**(d)**
The governor of the Blue state plans to replace the flat income tax with an ad valorem gun tax \( T_2 \). The state will hold a referendum. Two-thirds of the votes are needed for the new policy to pass. Residents will vote for the policy that benefits them.

Assume prices and income levels remain unchanged if the gun tax is enacted. As a political consultant to the governor, advise on the gun tax rate \( T_2 \).

i. Using the indirect utility function, explain why if household i = 60 votes for the new policy, all households i = 60, 61, ..., 99 will also vote for it.

**Hint:** Substituting numerical values into the indirect utility function may help.

ii. What is the highest possible gun tax rate \( T_2 \) the governor may propose to win the referendum?

**Economics Survey**

An economist conducted a survey to study the incomes of fresh graduates from a university in Hong Kong. He randomly selected 12 graduates and obtained their monthly salary data (in dollars) as follows:

15,400, 16,100, 18,100, 17,000, 16,500, 15,800, 16,900, 17,400, 17,800, 14,500, 15,900, 16,600.

Assume the salaries follow a normal distribution with an unknown mean \(\mu\) and a known standard deviation \(\sigma\).

**(a) (1 point)** Calculate the sample mean, \(\bar{x}_{12}\).

**(b) (1 point)** If the width of a 95% confidence interval for \(\mu\) is 1,075, find \(\sigma\), rounded to the nearest whole number.

**(c) (1 point)** To narrow down the 95% confidence interval, the economist took another random sample of size 13. The mean of this second sample, \(\bar{x}_{13}\), is $16,550. Using the combined information of the two samples, calculate the sample mean of the combined 25 data, \(\bar{x}_{25}\).

**(d) (5 points)** Test, at the 0.05 significance level, whether this academic year's average salary for fresh graduates is greater than last year's average salary of $16,200, using all information provided in part (c). Clearly state the hypotheses, test statistic, critical value, and conclusion.

**(e) (1 point)** Calculate the p-value of the test.