Problem%20Set%203%20Solutions - Economics 3010 Fall 2009...

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 3 1. (Choice: Schoolwork vs. Facebook) [ This question refers to empirical analysis from a recent research paper: Goolsbee, Austan, and Jonathan Guryan (2005), “The impact of Internet subsidies in public schools,” University of Chicago mimeo. (In addition to being an economics professor at the University of Chicago, Austan GooIsbee is one of Barack Obama’s main economic advisors.) If you are interested, feel free to read more details: http://faculty.chicagogsb.edu/jonathan.guryan/research/GuryanGoolsbeeERATE.pdf ] Recently, a number of economics research papers have focused on evaluating the causal effect of technology (such as new computers and Internet access) on education. For example, Goolsbee and Guryan (2005) study the effect of the E‐Rate program on California public schools. In an effort to close the “digital divide” between wealthier and poorer households, the U.S. government began subsidizing Internet access in public schools and libraries as part of the Telecommunications Act of 1996. The E‐ Rate program, which began in 1998, provided $2.25 billion in subsidies – which is enormous relative to the total public school spending on computers in 1999 of $3.3 billion. Goolsbee and Guryan (2005) find that although school districts responded strongly to these incentives (especially lower‐income and higher‐minority school districts), dramatically increasing their Internet access over the next few years, there was essentially no effect on the students’ academic achievement. (a) Some proponents of increased technology spending in schools have responded to these and other, similar findings by arguing that it will take time for technology to have positive effects on academic achievement. Why might that be so? There are many possibilities. For example, it may take teachers time to learn how to use the computers effectively in their teaching. (b) In this problem, we’re going to explore another possible explanation for the finding of no effect on academic achievement. Suppose that students do two things with Internet access: schoolwork and social networking (email, Facebook, etc.). Suppose each student has 5 hours per day to divide between these two computer activities. Prior to the introduction of additional technology, each hour spent doing schoolwork increases the student’s grade by 10 percentage points (out of 100 total); the student will get 0 percentage points if he/she spends no time doing schoolwork on the computer. Each hour spent social networking increases the number of Facebook friends by 5. However, the student will have 100 Facebook friends even if he/she spends no time doing social networking (because everybody has at least 100 Facebook friends these days). Draw the student’s budget constraint for grade (in percentage points) and number of Facebook friends, and write down the budget constraint algebraically. We draw the budget line as follows. First, we imagine how many facebook friends the student can have if she devotes all of her time to online networking. The answer is 5 times 5 plus the initial starting point of 100, which is 125. This is the y –intercept. Let’s now figure out what her facebook and grade pairing would be if she devotes all her time to studying. No matter what, she has 100 friends, and her grade will be 5 times 10, which is 50. To get the algebraic expression, we take the slope and y‐intercept and plug it into the equation for a line. Facebook Friends = 125 – 0.5*grade, for 0 < = grade = < 50. (c) Now suppose that the introduction of additional technology does increase the productivity of time spent doing schoolwork so that now, each hour spent doing schoolwork increases the student’s grade by 20 percentage points. Draw the student’s new budget constraint for grade and number of Facebook friends, and write down the budget constraint algebraically. We take the same approach used in part b to find the budget line with additional technology. Following those steps, we get the graph drawn below. The equation is Facebook Friends = 125 – 0.25*grade, for 0< = grade = < 100. (d) If the student spent the same amount of time doing schoolwork after the introduction of new technology as before, would the student’s grade go up? If the student spent less time doing schoolwork after the introduction of new technology, is it possible that the student’s grade would go down? What must happen to the number of Facebook friends in that case? Draw indifference curves for the old and new budget constraint illustrating the case where the student’s grade stays exactly the same despite the increased productivity of time spent doing schoolwork. If the student spent the same amount of time studying after the introduction of new technology, then her grade would go up. Intuitively, she’s doing the same amount of work with better equipment, so she does better in class. We can show this graphically by holding facebook friends constant, and seeing what the new grade will be. If she spent 3 hours studying before and after the introduction of technology, then she spent 2 hours networking in both cases. Therefore, her facebook friends will be 110 in each situation. As we can see from the graph, 110 friends in the new budget line gets us a higher grade. (Note: These curves are meant to look smooth.) If the student spent less time studying after the introduction of new technology, it is possible for her grade to go down. To do this, she would have to spend quite a bit less time studying, since the technology is now more productive. The picture below shows how this is possible. As we can see, the number of facebook friends goes up. (e) If the student’s preferences represent his/her well‐being (in addition to representing how he/she makes decisions), can we say whether the new technology makes the student better off or worse off? The introduction of the new technology expands the choice set. Therefore, the student is at least as well off with the new technology. She will be strictly better off if her original optimal bundle was not (0,125). (f) There is not enough information in the existing research papers to test the “Facebook substitution hypothesis” outlined in this question. What kind of data would you need to test it? (Note: There are many possible answers to this question. I’d like you to describe one testable prediction of the “Facebook substitution hypothesis,” and explain what you would need to look for in real‐world data from public schools to test it.) To begin, we need to know how much time students spend studying and social networking. This answer will give us the total time endowment of the students. To find their optimal points, we need to obtain the students’ final grades and the number of facebook friends they have. Once we have this information for one time period, we can compare it to values at an earlier time period (i.e. one with older technology). After studying the students’ chosen points, we can find if they substitute away from studying in the presence of new technology. To obtain this information, surveys could be handed out to each student asking for their GPA, number of facebook friends, and how many hours they spend on each task. This survey would have to be conducted in two time periods to see what happens after new technology is introduced. Note: it is also possible to learn about the facebook substitution hypothesis in an experimental setting. We could divide people into two groups‐ one with inferior technology, and one with better technology. Each person in each group could then be assigned the same task to perform in a specified time period. When they are not working, they can be allowed to chat with others through an online chat room. So that there is incentive to do a decent job on the task, money could be handed out for higher performance. At the end, the time allocations, quality, and time chatting can be compared for each group. 2. (Choice: Cost‐of‐living adjustments) [You can find solutions to some parts of this question in ch. 7.9. Feel free to use what you learn from that part of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] Wages and government transfer programs are often “indexed” to keep up with inflation. The idea is that if prices go up, income should go up enough to offset the price increase. To take one example, Social Security is the major government program that provides income to elderly Americans, and Social Security payments are indexed to the price level. (a) Write down Grandma’s budget constraint for two goods that have prices p1 and p2, and suppose that Grandma’s sole income is her Social Security check. Illustrate on a diagram the budget constraint and indifference curve that is tangent at the optimal point, (x1*, x2*). Suppose that there is a perfectly‐balanced inflation; prices of both goods increase by the same proportion. Show algebraically that increasing Social Security payments by the same proportion as the inflation leaves Grandma on exactly the same budget constraint as before. What is her new optimal point? Is she better off or worse off than before? Original budget constraint (B1): p1 x1 + p2 x2 = m Now, denote the inflation by π. When everything increases by (1+ π), The budget constraint (B2) is: p1 (1+ π) x1 + p2 (1+ π) x2 = m (1+ π), which is exactly same as (B1) Hence, there will be no change in her choice of bundle. x2 (I1) * x2 (B1) * x1 x1 (b) Now suppose (as is almost always the case in real life) the inflation is not perfectly balanced; prices of both goods increase to p1’> p1 and p2’> p2, but by different proportions. Suppose the government increases Grandma’s Social Security payment by exactly the amount necessary so that she could afford her old consumption bundle, (x1*, x2*). Draw the new budget constraint (which must go through (x1*, x2*)). Show that the new optimal point, (x1*’, x2*’), will not be the same as the old optimal point, (x1*, x2*). Show that Grandma will be on a higher indifference curve. x2 (I1) (I2) * x2 * x2 ' (B1) * x1 * x1 ' (B2) x1 x2 Note: If x1 and x2 are complete complements then the optimal bundle does not change. In this problem, since it was given that the budget constraint and indifference curve that is tangent at the optimal point, we rule out that the indifference curve is L shaped. (I1) * x2 (B1) * x1 * x1 ' (B2) x1 (c) Now suppose there is another inflation that is not perfectly balanced. Although prices p1’’> p1’ and p2’’> p2’ are now even higher, this new inflation brings the price ratio, p1’’/ p2’’, back to the old price ratio, p1/p2. Suppose the government increases Grandma’s Social Security payment by exactly the amount necessary so that she can still afford (x1*’, x2*’). Draw the new budget constraint (which must go through (x1*’, x2*’)), and show that Grandma will yet again be on a higher indifference curve at her new optimum. Show that the government is paying more to Grandma than it would be if prices had increased in a perfectly‐balanced manner from (p1, p2) to (p1’’, p2’’). The new budget constraint that goes through (x1*’, x2*’) is labeled as (B3). Under this budget constraint, Grandma can again achieve higher utility (I3) then before by choosing (x1*”, x2*”). x2 (I3) (I2) (I1) * x2 " * x2 * x2 ' (B3) (B1) * x1 * x1 " * x1 ' (B2) x1 Show that the government is paying more to Grandma than it would be if prices had increased in a perfectly‐balanced manner from (p1, p2) to (p1’’, p2’’). If the government paid in a balanced manner, p1 (1+ π”) x1 + p2 (1+ π”) x2 = m (1+ π”), the budget line is same as (B1). Hence, the increase in the social security is m π”. As we can see from the graph, the slope of (B3) is the same as (B1), but the vertical intercept is ~ ~ higher, which means p1 (1+ π”) x1 + p2 (1+ π”) x2 = m (1+ π ) (B3) where π > π”. Hence, the ~ increase in the social security in this case is m π which is greater than m π”. (d) How does the analysis in this question help explain why the U.S. Social Security system has inadvertently become more generous over time? As seen so far, this type of cost of living adjustment enables the consumers to choose a better bundle than before and the government pays more than it would if the adjustment was made in a perfectly balanced manner. 3. (Choice and Demand: The case of Cobb‐Douglas utility) [You can find solutions to some parts of this question in the appendix to ch. 5. Feel free to use what you learn from that part of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] (a) Consider a Cobb‐Douglas utility function, u(x1, x2) = x1α x21‐α, where 0 < α < 1 is a constant. Write down the Lagrangian for maximizing this utility function with respect to the standard budget constraint, p1 x1 + p2 x2 ≤ m. L = x1α x21‐α – λ( p1 x1 + p2 x2 – m) (b) Prove that the consumer’s demand functions are given by x1*( p1, p2, m) = αm/p1 x2*( p1, p2, m) = (1 – α)m/p2. (1) ∂L/∂x1 = αx1α‐1 x21‐α – λp1 = 0 ∂L/∂x2 = (1 – α)x1α x2‐α – λp2 = 0 (2) ∂L/∂λ = p1 x1 + p2 x2 – m = 0 (3) (1)/(2) leads to {(1 – α)x1P1}/ α = p2 x2 (4) By putting (4) into (3), we get x1*( p1, p2, m) = αm/p1, x2*( p1, p2, m) = (1 – α)m/p2 (c) Show that Good 1 is a normal good for this consumer. Is it a luxury, a necessity, or neither? A normal good is one for which an increase in income leads to an increase in demand. Hence, by ∂x1/∂m = α/p1 > 0, with given range of 0 < α < 1 and assuming that p1 > 0, Good 1 is indeed a normal good. If an income elasticity of demand is greater than 1, than the good is a luxury good. Let’s denote x1*( p1, p2, m) = αm/p1 ∂x1 ε x * ,m = 1 ∂m * x1 m = αm ∂x1 m = = 1 (unit elastic) ∂m Q1 p1 αm p1 Hence, Good 1 is neither a luxury nor a necessity good. (d) Take the (natural) log of the consumer’s demand function for Good 1. Show that the consumer’s demand function can be expressed in the log‐linear form: ln x1*( p1, p2, m) = ln(α) + ln(m) – ln(p1) lnx1*( p1, p2, m) = ln (αm/p1) = ln(αm) – ln(p1) = ln(α) + ln(m) – ln(p1) (e) What is the own‐price elasticity of demand for Good 1 (the elasticity of quantity demanded of Good 1 with respect to p1)? Is the consumer’s demand function for Good 1 elastic, inelastic, or unit‐ elastic? What is the cross‐price elasticity of demand for Good 1 (the elasticity of quantity demanded of Good 1 with respect to p2)? Is the consumer’s demand function for Good 1 with respect to the price of Good 2 elastic, inelastic, or unit‐elastic? What is the elasticity of demand for Good 1 with respect to income (the “income elasticity of demand for Good 1”)? We can compute the elastic ties by ∂x1 p1 . But, when the demand function is express in the ∂p1 x1 logarithmic form as in (d), the coefficients of the log terms in the right hand side represent the elasticity of demand corresponding the term, i.e. Own price elasticity of demand for Good 1 ( ε x * , p ) = ‐ 1 (coefficient of ln(p1)) 1 1 Cross price elasticity of demand for Good 1 ( ε x * , p ) = 0 (because there is no ln(p2) in the 1 2 equation) Income elasticity of demand for Good 1 ( ε x * ,m ) = 1 (coefficient of ln(m)) 1 This is because, with for the case of the own price elasticity of demand for Good 1 as an example, From d ln( x1 ) 1 dx = , d ln( x1 ) = 1 (a) dx1 x1 x1 From d ln( p1 ) 1 dp = , d ln( p1 ) = 1 (b) dp1 p1 p1 Recall the own elasticity of demand is the ratio of the right side terms in (a) and (b), which should be equivalent to the ratio of the left hand side, i.e. d ln( x1 ) measures the own price d ln( p1 ) elasticity of demand, which is exactly the coefficient of ln(p1) in the log‐linear equation ln x1*( p1, p2, m) = ln(α) + ln(m) – ln(p1) ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell University (Engineering School).

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