Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.Homework: Factor Markets: With Emphasis on the Labor Market (Ch 13) 9. How price changes impact the labor market Consider the labor market for the fast-food industry, which consists mainly of high school and college students. Assume that all fast-food restaurants are profit maximizing. The following calculator shows the market demand curve (blue curve) and market supply curve (orange curve) for student workers, who are responsible for making cheeseburgers. At any time in this problem, you can click the Reset to Initial Values button to return the elements in the calculator to their original positions. You will not be graded on any changes to the calculator; it's just here to help you answer the following questions. Tool tip: You can directly change the values in the boxes with the white background by clicking in the box and typing. The graph and any related values will change accordingly. Graph Input Tool (?) LABOR MARKET CALCULATOR Wage rate 14 Labor demanded Labor supplied Thousands of Thousand's of 70 workers) workers WAGE RATE Price of a cheeseburger 0 10 2 040 50 60 70 10 90 QUANTITY OF LABOR (Thousands of workers) When the price of a cheeseburger is $4, the equilibrium wage in the fast-food labor market is $ per hour. Suppose that the demand for cheeseburgers increases enough so that the price of a cheeseburger rises to $8. Ordinarily, this would result in a new equilibrium employment level and wage in the labor market for young people who work in fast food restaurants. However, restaurants claim they can only afford to pay the initial equilibrium wage. In this labor market, If the price of cheeseburgers increases, but restaurants continue to pay the equilibrium wage that prevailed before the increase in demand for cheeseburgers, there will be a labor Z of workers.1. Suppose that the market for hotel rooms (one-night stay) has the following supply and demand schedules: Price of a Quantity demanded Quantity supplied room (thousands) (thousands) $150 100 10 $160 90 15 $170 80 20 $180 70 25 $190 60 30 $200 50 35 $210 40 40 $220 30 45 $230 20 50 $240 10 55 $250 0 60 a) Using this information, draw the demand curve and the supply curve for hotel rooms. b) What is the equilibrium price and quantity for rooms? c) Assume the government levies a tax of $30 per night. What is the price that consumers will pay for a room now? d) What is the price that hotels will receive when someone stays for a night? e) Illustrate the effect of this tax in your diagram from part a. f) Calculate the government revenue raised by this tax. diagram. g) Is the market efficient. with the $30 excise tax? Explain. Illustrate your answer with a(a) First consider an economy where the government budget is required to be balanced in recession. The economy is given by: Czcu+c1(YT) T=t+t1Y I=I_ G=T With{61{1,{t1{1 i. Derive an expression for Equilibrium output. ii. Suppose for that for some reason demand were to fall, so that en were to decrease. We are interested in by how much output would fall due to this decline in ca. Derive an expression for the decrease in y do to a decrease in Ca. (b) Now ippose that the economy is the same as above except that G = G, La. that G is constant, and the government budget is not required to be balanced in recession. i. Derive an expression for equilibrium output. ii. Derive an expression for the decrease in 3,; due to a decrease in [30. (c) Interpretation: i. Based on the expressions you derived in aii and bii, would the same decline in g cause a larger decline in the economy with a balanced budget requirement, or the economy in which the government was not required to run a balanced budget? ii. In economy a, the govermnent's budget remains balanced after the decline in output, i.e. that (T G) = 0. Suppose that in economy I: the government's budget was balanced before the decline in output. What happens to the government's budget after the decline in gutput; is it still balanced, in surplus, or decrt? 1.e (T G) 2: [1? iii. How do you think the condition of the government's budget af fects how the economy responds to a decline in demand in the two eoonomies above? How does the answer to part cii concern- ing the state of the government's budget relate to your answer in part ci concerning which economy would experience a. larger decline in output given the same fall in ca