HW #1 Solutions. 1. a. In order to Önd the utility maximizing bundle, we must Önd where MRS = MUc MUh = pc ph and where I = pc c + ph h: First, lets Önd the MRS. MUc = dU dc = 1 c : MUh = 1: So MRS=1 c : Now lets Önd pc ph ; which in this case is 1 1 : So from MRS=Pc ph ; we get 1 c = 1; so c=1. In order the Önd h, plug optimal c intro the budget constraint. 5 = 1 c + 1 h; so 5 = 1 + 1 h; so 4 = h: b. If income increases to 10, the comsumption of c is unchanged as 1 c still = 1 : However, now we get 10 = 1 c+ 1 h; so 10 = 1 + 1 h; so 9 = h: As income went up, so did expenditure on h, so h is a normal good. Consumption of c was unchanged, so c is neither a normal or an inferior good. c. Now the price of c increases to 2 (assuming incomes stays at 10, but same results if we examine income=5). Now we have MRS= pc ph ; so 1 c = 2 1 ; so c is 1/2. We Önd the e§ect on by using 10 = 2 c + 1 h; which can be written as 10 = 2 1 2 + 1 h; so 9 = h: As consumption of h did not change, the goods do not appear to substitutes or complements. For your own beneÖt also try to see what happens when the price of h changes. 2. a. Education could e§ect health for several reasons. 1) More education leads to higher incomes which may lead to higher demand for health if health is a normal good. 2) More edcation may lead to more information on the beneÖts of health 3) More education may lead individuals to have lower discount rates which would lead to increased returns on future utility increasing the return for health. 4) More education may improve the ability of individuals to process information/follow doctors orders 5) More educaiton may place individuals in safer jobs/jobs with higher medical care 6) More education may reduce fertility (family size) which may increase the amount of spending per family availble for medical care. 7) Iíd be willing to hear others? b. The key assumptions for the estimator to be unbiased is E(ui jeduci) = 0: We also need for the model to be linear in the parameters. c. This will likely be violated. There are many factors which are unobserved (such a parental income or risk aversion) which likely a§ect both your educational pursuits and your health directly. d. We could use an instrumental variables approach. That approach will isolate the part of the variation in education for which it is true that E(ui jeduci) = 0: To do so we need an instrumental (like education laws, or perhaps college openings and closings) which a§ect both the amount of variation in education, but are unrelated to the unobserved factors which determine your health. 1 https://www.coursehero.com/file/9345653/HW1Solutions/ This study resource was shared via CourseHero.co 3. a. Yes, most likely. Because winning the lottery is determined by a random draw, we expect the assumption E(ui jlotteryi) to be zero always. We would need to examine only a sample of lottery players because the choice to play the lottery is not random. There might be a slight bias, just because playing the lottery multiple times increases your odds of winning, but this is something we could control for, and previous research has suggested this does not create much bias due to the very low frequency nature of wins (playing multiple times increases your odds from 1 100;000;000 to 10 100;000;000 ) which are both incredibly small probabilities. b. Because the lottery variable is an indictor variable (also known as a categorical or dummy variable), a unit change in the regressor is the di§erence between winning and losing the lottery. So an interpretation would be that winning the lottery increases daily exercise time by ^ 1 hours. If ^ 1 = 1:3; winning the lottery increases daily exercise time by 1.3 hours. If the estimated standard error on 1 is .48, we can test the null hypothesis that 1 = 0; by 1:30 :48 which is greater than 1.96, so we can reject the null, and say that with at least probability .95, 1 does not equal 0. Another approach is to construct a 95 percent conÖdence interval and conÖrm the interval does not include zero. c. In this the OLS estimator for 1 can be shown to reduce yT yC ; given the regressor of interest is an indicator variable (see notes for a mathematically involved proof). An easier logic is to look at the regression set up yi = 0 + 1 lotteryi + ui ; where yi is our measure of health. E(yi jlottery = 1); or the average level of health when someone wins the lottery (which is yT ) is 0 + 1 because E(ui jlottery = 1) = 0: Likewise, E(yi jlotteryi = 0); or the average level of health for lottery losers (which is yC ) is 0 because E(ui jlotteryi = 0) = 0: To Önd 1 ; subtract E(yi jlottery = 1) E(yi jlotteryi = 0) = yT yC : 4. a. Yes as individuals get older, the number of days of poor health increase. The estimate of 1 is .065 b. The estimate of 2 is -2.21. This does correspond with Grossmanís prediction. I think the estimate will be biased because there are other factors related to health (such as risk aversion, income, parental education) that are also correlated with education. This violates the strict-exogeneity assumption. c. Using height as an instrument, the estimate of 2 is -8.49. Although height is a predictor of college attainment, its also correlated with gender, which is not included in the model and therefore is an important unobservable. In addition, height might be correlated with parental education or early childhood health, and therefore belongs in the main equation and is not excludeable. 2 https://www.coursehero.com/file/9345653/HW1Solutions/ This study resource was shared via CourseHero.com Powered by TCPDF (www.tcpdf.
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HW #1 Solutions. 1. a. In order to Önd the utility maximizing bundle, we must Önd where MRS…