Answer all... i need answer plz

Question 1 {20 points} Consider the standard neoclassical growth model in discrete time. There is a large number of identical households normalised to 1. Each household wants to maximise life-time diseounted utility \"Hamel = Edited}, 13 e (D, 1). t- Each household has an initial capital to at time (I, and one unit of productive time in each period that can be devoted to work. Final output is produced using capital and labor, according to a CBS production function F. This technology is owned by rms (whose measure ow not really matter because of the CRS assumption). Output can he oonsurned (q) or invested (it). Households own the capital [so they make the investment decision)1 and they rent it out to rms. Let r5 E (0, 1) denote the depreciation rate of capital Households own the rms, i.e., they are claimants to the rms' prots, but thae prots will be zero in equilibrium. The function a is twice continuously differentiable and bounded, with n'(c} :- Ill, u"(o] c: D, a'm) = so, and a'(oo) = 0. Regarding the production technology, we will introduoe the useful function at} E F(a:, 1)+{1 Elm, V1: 5 11.... The function f is twice continuously di'erentiable with fits} } {1, Fits] 4: (I, ll} = I], 1'10] = on, and f'[oo] = l 6. In this model the government taxes households' investment at the constant rate 1- E [i], 1]. The government returns all the tax revenues, T, to the households in the form of lump-sum transfers. Throughout this question focus on recursive oompetitive equilibrium {RUE}. a} Write down the problem of the household recursively.1 Carefully distinguish between aggregate and individual state variables. Then, dene a ROE. b) Write down the dynamic equation that the aggregate capital stock follows in this economy. c) Now focus on steady-states. Describe the steady-state equilibrium value of the aggregate capital stock in this economy, and denote it by K* (7). d) Describe the value of K* when + = 0 and when r = 1. Here firms face a static problem. I am not asking you to explicitly spell it out, but it will be critical for correctly defining a RCE. 2 Hint: Obtain the Euler equation for the typical household and impose the RCE conditions. I ecommend you express the equilibrium condition(s) in terms of the function f, rather than F. This will make life easier in the forthcoming parts.01. The ratio of the 90'\" percentile wage to the 10'\" percentile wage is a common measure of earnings inequality. a) Use the diagram below to roughlyr show (draw) how this measure has changed over time. Log earnings ratio :9153 [957' 1.971 1995 1979 I959 193'? 1991 1995 1999' EDS b} Researchers have described a number of possible explanations for the eann'ngs inequality patterns that have been observed over the past half eentm'y. The two leading explanations are 1} changes in technology, and 2) changes in institutions. Briey summarise these taro explanations, being sure to explain why theyr might contribute to changes in inequality. c) Some researchers have tried to disentangle the relative importance of technology vs. institutions by foeusin g on changes in \"within group\" earnings inequality\". What do we mean by \"within group\" inequalityr in this context. and hon:r can changes in within group inequality provide clues about the factors that are contributing to changes in earnings inequality over time? cl) Lendeua (Z) maesgates how changes in within group inequality have been affected by changes in the composition of wodters by estimating the following equation: Vt = E; 5": thU} where Vtis the total earnings variance in year t, EU) is the earnings variance in year t for group j, and arm is the share of the workforce in groupj. Since we would expect VrUj to be bigger for older, more educated workers, and the US population has been getting both older and more educated, part of the change in overall inequality ever time might result from the act that groups with bigger earnings variance are contributing more weight to the ouerall mequality meastne. What does Leanieux nd when he investigates this possibility and how Q2. Suppose that you are interested in estimating the effect of class-size on children's test scores. You have a dataset that includes teed scores for all public school children in state A, along with information about each child's race. gender, and whether hefshe is eligible for the eeireduced price school lunch program. The dataset also includes student, school and grade identifiers. It contains test score information covering a period from 1995-2005. a} h} '11} One approach to estimating the effect of classeise on children's test scores would be to run an 0L5 regression. How would you implement this estimation stt'ategy'? Include a regression equation and a detailed description. What critical assumptions are needed in order for 01.5 to yield unbiased estimates of the causal immct of interest? What are some weaknesses associated with this strategy? An alternative estimation approach that has been used by Caroline Hoxby {and others) is based on class-size variation across cohorts. Describe how you wrould implement this strategy, including a regression equation. What critical assumptions are needed in order for this strategy to yield unbiased estimates of the causal impact of interest? What are some strengths and wealmcsses associated with mis strategy? _ A third estimation approach, which has been used by Hanoshck, Rivkin and Kain, uses variation within cohorts over time. Using a regression equation. describe how you would implement this strategy. What critical assumptions are needed in order for this strategy to yield unbiased estimates of it! effect of classeizc'? What are some strengths and weaknesses associated with this strategy? A class-size redaction policy was implemented in California in the late 1990s. "the policy gave schools strong fmancial incentives to keep grades [(-3 puplmacher ratios below 20. The policy was put into place very suddenly, and so it was not anticipated. How could you use yam- data and the policy change to esrnate the effect of classsize on student achievement