1. In an economy there are two types of jobs: H-type jobs have a high level of some disamenity z (z = 1), while L-type jobs have a low level of the disamenity (z = 0). Everyone prefers working in the low-disamenity jobs, but the degree of this preference varies across people. In particular, the preference (or reservation price) is distributed uniformly from $6 to $11. Formally, workers’ preferences are given by the following utility function:
Ui(w,z) = w - θiz
θi ˜ U(6,11)
Thus, if the wage in H-type jobs is $6 more than the wage in L-type jobs, nobody will choose to work in the H-type jobs. If the wage in H-type jobs is $8 more than the wage in L-type jobs, 40 percent (or two-fifths) of the worker population will choose to work in the H-type jobs. Labor supply is perfectly inelastic, but firms compete for labor. There are a total of 25,000 workers to be distributed between the two types of jobs. Demand for labor in both types of jobs is described by the following inverse labor demand functions:
L-type jobs: wL = 50 – 0.0024EL.
H-type jobs: wH = 50 – 0.0004EH.
a) Solve for the labor market equilibrium by finding the number of workers employed in both types of jobs, the wage paid in both types of jobs, and the equilibrium wage differential. (Hint: write first the inverse demand for workers in H-type jobs as a function of the wage differential. Use the fact that EH + EL = 25000. Also, can you show that the supply of workers to H-type jobs is Δw = 6 + 0.0002 EH?).
Now assume that 6,000 immigrants suddenly flow to this economy. All the immigrants have identical preferences equal to:
UM = w - θMZ
θM = 6
That is, all immigrants are as tolerant of the disamenity as the most tolerant of the native workers.
b) Solve for the new labor market equilibrium. How many workers are employed in both types of jobs? What is the wage paid in both types of jobs? What is the new equilibrium wage differential? What is the allocation of native workers between H-type and L-type jobs? (Hint: remember that now there are a total of 31,000 workers in the economy. Use this number to derive the inverse demand function for workers in H-type jobs as a function of the wage differential).
c) What happens to the utility of workers who:
i. Work in L-type jobs both before and after the immigration shock.
ii. Worked in H-type jobs before the immigration shock, but move to L-type jobs after the immigration shock.
iii. Work in H-type jobs both before and after the immigration shock.
d) A researcher finds that the share of immigrants in a locality is negatively correlated with the fraction of natives that have unpleasant work schedules (e.g., working at nights or weekends). He therefore concludes that immigration may be beneficial to native workers because it pushes them out of “bad jobs.” Evaluate the researcher’s claim in light of the results of parts (a)-(c).
2. In the country of Utopia, individuals live two periods (period 0 and period 1), and must decide between two alternative income paths.
- “No College”: work full time in both periods, and receive an income of 200 in both periods (Y0,NC = Y1,NC = 200). One can save any amount at interest rate r = 0, but cannot borrow at all (i.e., the interest rate for borrowing is effectively infinite).
- “College”: In period 0, go to college, work part time and earn 120 (Y0,C = 120). In period 1, work full time and earn 300 (Y1,C = 300). One can save any amount at interest rate r = 0. One can also take on a student loan of up to 40: that is, it’s possible to borrow up to 40 at interest rate 0 (but it’s not possible to borrow more than 40 – the interest rate for any borrowing beyond 40 is effectively infinite).
Assume that attending college does not yield any direct utility, and that the amount of leisure is identical under both income paths. Assume also initially that tuition is zero.
Under the current circumstances, some individuals in Utopia choose to attend college, and others choose not to attend college.
a) Draw the budget constraints under the two alternative paths. Make sure to label your graph carefully.
b) Assume that the period 1 income if one attends college (Y1,C) increases, while all the other parameters remain constant. What will happen to the fraction of people who attend college? What will happen to the amount of borrowing in the economy? Use a graph to
explain your answer.
c) Assume that tuition increases, meaning that the net period 0 income (Y0,C) if one attends college decreases. What will happen to the fraction of people who attend college? What will happen to the amount of borrowing in the economy? Use a graph to explain your answer.
d) In recent years, there has been an explosion in the amount of student loans, and also an increase in college enrollment. In light of your answers to parts b) and c), what factor can explain both trends: an increase in the economic returns to college, or an increase in tuition?
3. Consider the following version of the human capital model. The income generating function is of the form:
Y(Si) = exp[biSi - (Si2/2)]
where bi > 0, Si is the schooling level chosen by individual i and Y is income. The costs of education are determined by the following function:
g(Si) = c + riSi
where c > 0 and ri > 0. Individuals choose S to maximize utility: U(S)=log[Y(S)] – g(S).
a) What is the optimal level of schooling S* that individuals choose in this model?
b) Given the optimal choice of S, what is the return to schooling β* = Y'(S)/Y(S) in this economy?
c) Now, interpret the variable S so that individual attends college if S > 0 and only gets the compulsory education if S ≤ 0. Assume that parameters r and b are distributed in the population in the following way:
bi
ri 0 1
0 20% 30%
1 30% 20%
How large a fraction of the population attends college in this economy? Suppose that the government introduces a policy that reduces ri to 1/2 for those whose ri was previously 1. Howlarge a fraction of the population will attend college as a result of this reform?
4. (Borjas, Problem 6-8) Suppose there are two types of persons: high-ability and low-ability. A particular diploma costs a high-ability person $8,000 and costs a low-ability person $20,000. Firms wish to use education as a screening device where they intend to pay $25,000 to workers without a diploma and $K to those with a diploma. In what range must K be to make this an effective screening device?
5. In this exercise, you will be asked to evaluate alternative instrumental variables for the returns to schooling. Download the 2005 March CPS (follow the instructions given in Problem Set 2). You will need to download the following variables: REGION, AGE, SEX, RACE, MARST, EDUC, CLASSWKR, STATEFIP, INCWAGE, in addition to the basic technical variables that are included in every extract (YEAR, SERIAL, HWTSUPP, MONTH, etc.). You will also need to attach to each individual the educational level of his/her spouse. You can easily do this using the “attach characteristics” option on the IPUMS website.
Downloading and cleaning the data:
1) Transform the EDUC variable into a continuous variable based on the following scheme:
None or preschool: 0 years of schooling
Grades 1,2,3 or 4: 4 years of schooling
Grades 5 or 6: 6 years of schooling
Grades 7 or 8: 8 years of schooling
Grade 9: 9 years of schooling
Grade 10: 10 years of schooling
Grade 11: 11 years of schooling
12th grade, no diploma: 12 years of schooling
High school diploma or equivalent: 12 years of schooling
Some college but no degree: 13 years of schooling
Associate’s degree, any type: 14 years of schooling
Bachelor’s degree: 16 years of schooling
Master’s degree: 18 years of schooling
Professional school degree: 18 years of schooling
Doctorate degree: 20 years of schooling
2) Calculate a person’s potential experience as AGE – Years of Schooling – 6. Drop any observations with negative values of experience.
3) Calculate the fraction of workers in a state with at least a B.A. degree using the full sample (all individuals aged 25 and up).
4) Restrict the sample to married men, aged 25-54, employed as salaried workers (use the CLASSWKR variable).
5) Drop all observations with zero yearly earnings.
6) Among those with positive earnings, drop those with earnings below the 1st percentile and above the 99th percentile.
Questions:
a) Run a regression of log yearly earnings on years of schooling, experience, experience squared, dummies for non-hispanic blacks and for Hispanics, and regional dummies. What is the estimated return to schooling? Why is this estimate potentially biased? Comment briefly on the size and significance of the other coefficients.
A researcher is concerned that the estimated return to schooling from part a) suffers from omitted variable bias, and is considering three potential instruments for years of schooling:
i. The wife’s years of schooling
ii. The fraction of workers in one’s state of residence that have a college degree.
iii. A dummy variable indicating whether the state’s FIPS code is an even number or not.
b) For each one of the three potential instruments, say whether you think they satisfy the necessary conditions needed for an instrument to be valid.
c) For each one of the three instruments, estimate the first stage regression, i.e., the regression of years of schooling on the instrument and all the other explanatory variables. Using the test command, calculate the first-stage F-statistic for the null hypothesis that the coefficient on the instrument is equal to zero. An instrument is said to be “weak” if the F-statistic is lower than 10. Which one of the three instruments used is a weak instrument?
d) Re-estimate the regression in part (a) by 2SLS, using each one of the instruments in turn. Comment on the results.
e) Re-estimate the regression in part (a) by OLS, but add in turn each one of the instruments as additional control variables. Comment on the results.
6. In this exercise, you will reanalyze the data set used by Ashenfelter and Krueger in their study on the returns to schooling using data on identical twins. Download the data sets twins_long.dta from the course website. The data includes 680 twins (this is a slightly larger sample than the one used by Ashenfelter and Krueger in their final regressions).
Answer the following questions.
a) Estimate an OLS regression of log wages on years of schooling (educ), pooling together the data from both twins. What is the percentage change in wages associated with an additional year of schooling?
b) Now add to the regression controls for age, age squared, a female dummy, a racial dummy, and a marital status dummy. What happens to the estimated returns to schooling?
c) Add to the regression in part (b) controls for mother and father’s years of schooling. What happens to the estimated returns to schooling? Is this what you would have expected? Explain.
d) Repeat the regression in part (c), but now instrument own (self-reported) years of schooling with the twin’s report of years of schooling (educ_twinsreport). What is the rationale for running this regression? What are the estimated returns to schooling? How does it compare to the estimate in part (c)? Is this what you would have expected?
e) Estimate the regression in part (c) using first differences (i.e.., regress the difference in log wages between twin 1 and 2 on the difference in years of schooling, the difference in age, etc.). What is the rationale for running this regression? What happens to the estimated returns to schooling relative to the estimate in part (c)? Is this what you would have expected? Is this the same result obtained by Ashenfelter and Krueger?