(四)質傳中之 Sherwood number 與熱傳中之 Nusselt number 相似，其定義分別為何？(4 分)

1. Suppose the market supply and demand for milk can be expressed as:$$Q^S = 3P$$$$Q^D = 100 - 2P,$$If the government imposes a $5 unit tax to be collected from sellers, how much tax revenue is collected? What fraction is paid by sellers? (10%)

a. Paul's utility function is \( U = 2x + 3y \). (5%)

b. Paul's utility function is \( U = \min \{x, 3y\} \). (5%)

3. James has a monopoly on a board game called "Fly High" and faces a demand curve given by \( Q_T = 100 - P \), where \( Q_T \) equals the combined total number of games produced per hour in the company's two factories (\( Q_T = q_1 + q_2 \)). If factory 1 has a marginal cost function given by \( MC_1 = q_1 - 5 \) and factory 2 has a marginal cost function given by \( MC_2 = 0.5q_2 - 5 \), how much total output will James choose to produce and how will he divide this output between its two factories in order to maximize profits? (10%)

4. XYZ company has a Cobb-Douglas production function, \( Q = L^{0.5}K^{0.5} \). Use \( w \) and \( r \) to denote the price of the two inputs, \( L \) and \( K \), respectively. Given a required output level \( \bar{Q} \), compute XYZ's conditional factor demand, \( L(w, r; \bar{Q}) \) and \( K(w, r; \bar{Q}) \). (10%)

a. What is the optimal level of mosquito control? (5%)

b. Now, suppose mosquito control is a public good. Once it is produced everyone can benefit from it. What would be the optimal level of this activity? (5%)

a. Using Year 1 as the base year, what is the growth rate of real GDP from Year 1 to Year 2? (6%)

b. Based on the GDP deflator, what is the inflation rate from Year 1 to Year 2? (6%)

7. Comment on the following statement and show what would happen in the long run if such advice were followed by the Central Bank:"The increase in the stock market has increased people's wealth. As a result, their consumption has increased, increasing aggregate demand and output. So the Central Bank needs to increase the money supply, since with higher income, people's demand for real money balances will be higher." (12%)

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