Answer the following questions.
Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted X, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted Y, is regarded as the response, outcome, or dependent variable. Suppose that we are given n-i.i.d observations { (x;, y;)}"_, from the assumed simple linear regression model Y = BIX + Bo + . Answer the following questions on simple linear regression. 5-a. Denote 1 and Bo as the point estimators of B, and Bo, respectively, that are obtained through the least squares method. Show, step by step, that the two point estimators are unbiased. Derive the least squares estimator of of and determine whether it is unbiased or not. Show your work step by step. 5-b. Calculate _'_1(yi - Bix; - Bo) (Bix, + Bo). Determine whether the point (X, Y) is on the line Y = 1X + Bo. Explain your reasoning mathematically. 5-c. Using the maximum likelihood estimation (MLE) technique, derive a point estimator for the coefficient B1 and the intercept Bo, respectively. Determine whether the point estimators that you obtained via MLE are unbiased or not. Justify your conclusion mathematically. 5-d. Calculate the variance of the four estimators from Questions 5-a and 5-c, respectively. Show your work step by step. 5-e. Suppose that we are using the simple linear regression model Y = B1 X + Bo + 1 while the true model is Y = 1X1 + B2X2 + Bo + 82 where Bo, B1, and B2 are constants. We assume that the distributions of &, and e2 are both N(0,02), i.e., normal distribution with variance o?. We further assume that the two noise variables are uncorrelated. Find the least squares estimator of B, in this case and determine whether the point estimator that you obtain is biased or not. If it is biased, calculate the bias.Question 1 There are 10,000 identical individuals in the market for commodity X, each with a demand function given by Qodx = 12 - 2Px, and 1000 identical producers of commodity X, each with a function given by Qsx = 20 Px. where Qox is an individual's quantity demanded, Qsx is a single producer's quantity supplied, and Px is the price of the commodity. (a) Find the market demand function (QDx) and the market supply function (QSx) for commodity X (b) Determine the market demand schedule and the market supply schedule of commodity X (for whole dollar prices) and from them find the equilibrium price and the equilibrium quantity. (c) Plot, on one set of axes, the market demand curve and the market supply curve for commodity X and show the equilibrium point (d) Obtain the equilibrium price and the equilibrium quantity mathematically. (e) Explain why the equilibrium condition is considered stable. (f) Determine the elasticity of demand for commodity X at the equilibrium point. Question 2 Suppose that from the condition of equilibrium in Question 1, there is an increase in consumers' incomes (ceteris paribus) so that a new market demand curve is given by QDx = 140,000 - 20,000Px. (a) Derive the new market demand schedule (b) Show the new market demand curve on the graph used in Question 1(c) (c) State the new equilibrium price and equilibrium quantity for commodity X (d) Determine the Income Elasticity at the original equilibrium price and at the new equilibrium price. (Assume that the increase in income is 106). (e) In the light of your answer to 2 (d), comment on the nature of commodity X.MICRO PLASTIC PROJECT ECONOMIE INF. Initial Cost of techology $ 450D, OOD Installation Lost (design | Castruction) $11 1000, DOD Annual cost For Opratug / maintenance $ 3BoiDDD Duch bul Cost Annual savings from $ 6SDIDOD replacing old techolog Annal benefits to communty $ 1500, DDD Residual value of benefits aft Pict horizon $ SOOD DOD Pjet horizon 5 yrs Useful life of techology Infinite Inhest rate + 2.44% Capatalize Cost NPHI ? ! ! A Find Capartialize Cost ( P ) = 1! knowing that P = Al . ) FIND B Draw Cash flow diagram it Calculate Net presity worth ( NPHI = ?Problem 2 (30 pts) Explain your answers! The market demand function for computer games is qD (p) = BOOSp and the market supply function is qs(p) = 4}). (a) Find the equations of (express price in terms of quantity) and plot the demand and supply curves corresponding to the given market demand and supply functions. Find the market equilibrium price and quantity p* and if". (b) Find the producer's surplus and consumer's surplus at the equilibrium. (c) What is the price elasticity of demand at the equilibrium point? ((1) Derive the equation of the marginal revenue curve and compute the marginal revenue at the equilibrium point in (a). (e) Given the demand function, at what price would total revenue be maximized? (f) Suppose a quantity tax of $5 per game is imposed on the computer game suppliers. What is the new equilibrium price paid by the consumers? What is the new equilibrium price received by producers? How much is the tax revenue? How much is the deadweight loss of the tax? Use a graph to explain your answers. Problem 3 (15 pts) Explain your answers! Suppose the production function of a rm is given by f (K ,L) = SK 1/4111\(3) You and a classmate are assigned a project on which you will receive one combined grade. You each want to receive a good grade, but you also want to avoid hard work. In particular, here is the situation: If you both work hard, you get an A, which gives you each 40 units of happiness .If only one of you works hard, you both get a B, which gives you each 30 units of happiness. .If neither of you works hard, you both get a D, which gives each of you 10 units of happiness. . Working hard costs 25 units of happiness. (a) Fill in the following payoff matrix: Your decision Work Shirk Classmate's Work decision Shirk (b) What is the likely outcome? Explain your answer. T (c) If you get this classmate as your partner on a series of projects throughout the year, rather than only once, how might that change the outcome you predicted in part (b)
