Find answers please.

Independent random samples of size n, and n, are taken from the normal populations N(1,of) and N(#2,02 ) respectively. (i) Write down the sampling distributions of X, and X2 and hence determine the sampling distribution of X1 - X2 , the difference between the sample means. (ii) Now assume that of = o} = 02 (a) Express the sampling distribution of X - X, in standard normal form. (b) State the sampling distribution of (m - D)S/ + (12 - 1)S] (c) Using the N(0,1) distribution from (a) and the x- distribution from (b), apply the definition of the / distribution to find the sampling distribution of X - X, when o is unknown.Consider market for masks. Not everyone values the mask equally: marginal private benet is 100 Q and marginal private costs are given by 20 (per Q). Given that wearing masks will help reduce spread of virus, marginal benet to others is 1130 = 50 (per Q). o 3.. Draw the marginal private benet (MPB) marginal social benet (MSB), marginal private cost (MPG), and marginal social cost (MSG) curves on a graph. labeling clearly the axes and intercepts. Pay close attention to what is on the Xiaxis and yiaxis. _ ' o b. What is the equilibrium amount of masks? What is the eicient level? 0 c. What is the deadweight loss (D'WL) caused by the externality? Shade in the area. of the DWL on the graph from part (a). What is the welfare gain from moving to the efcient level of Q? o d. For which good is a market absent in this case? Government solution: Suppose the government steps in and decides to correct the externality by subsi dizing masks. o e. Suggest a Pigouvian tax that would induce the efcient consumption. What is the optimal tax rate per unit, and how much subsidy is needed? Market solution: The Cease theorem states that \"the eicient outcome should occur regardless of which party has the property rights,\" implying that the market can potentially solve externalities by itself (no need for the government to step inl). o f. Interpret the Cease theorem in the context of this question. What are ways in which the Cease theorem could fail? Give some examples in the context of this question. I In this question, we extend the model with search friction discussed in class by adding non-pecuniary preferences over jobs. There are two periods and workers search for a job every period. Workers have linear utility and no saving. If unemployed, workers draw a job offer from a job offer distribution. Each job offer has two components - a wage to and a disutility 1). Assume that w and 'v are independently distributed. The distribution of wages is gm and the distribution of disutilities is g\". If the agent accepts the job offer, her utility at that period is given by: u(w,v)=w'u If the worker accepts a job at period 1, she remains employed for the next period on the same job. If the worker rejects the job offer from period 1, she receives unemployment insurance b at that period and searches for a new job in the next period. The worker has to accept the job offer received in the second period. Utility in the second period is discounted by rate ,8. 1. Suppose the worker receive job offer (110,11) on the rst period. Write down the value of accepting this job offer in period 1, V\"\"\"Pt(w, 'v). . Calculate the expected utility of being unemployed on the next period as function of E(w) and 13(1)). Hint: Use that the utility is linear, u('w, v) = 'w 'v, and that w and v are independently distributed. . Calculate the disutility of rejecting job offer (10,0) on the rst period, Vrew, c). . Calculate the utility of receiving job offer (11:, v) on the rst period, V(w, '0), using your answer to questions 1 to 3. . Cenditional on a disutility of working or, plot the utility of rejecting a job and the utility of accepting a job on a graph with wages on the xaxis and the utility value on the yaxis. Mark the reservation wage 213(1)) on this gure. . Calculate the reservation wage as a function of the disutility, 25(1)). Is the worker willing to accept a lower wage if she nds the job more meaningful? Conditional on a wage w, plot the utility of rejecting a job and the utility of accepting a job on a graph with disutility on the x-axis and the utility value on the y-axis. Mark the reservation disutility 6(a)) on this gure