Human Capital: Theory

1. Human Capital:Theory Lent Term Lecture 2 Dr. Radha Iyengar

2. What is Human Capital? • Part of original conception of inputs in production. Adam Smith said that there were 4 inputs in which we might invest: • Machines or mechanical inputs • Building/infrastructure • land • human capital

3. Education and “General” Human Capital We’re going to study first education (schooling including college/graduate education) This is important because it is: • Expandable and maybe doesn’t depreciate (like physical capital) • Transportable and shareable (not true with “specific capital”)

4. What are we going to study? • Theory • Static Model (Card) • Dynamic Model (Heckman) The goal of theory is to motivate the large body of empirical work • Empirics • Some talk of methods (identification, diff-in-diff, IV) • Reconciling different estimates • Economics of Education (briefly!)

5. A Static Model of Human Capital Acquisition (for details see: David Card, “Causal Effect of Education on Earnings” Handbook of Labor Economics)

6. Basis of Empirical Estimates • Common form of estimation: log( y ) = a + bS + cX + dX2 + e (1) • Usually called a Mincer Regression

8. Some Empirical Facts 1. A simple regression model with a linear schooling term and a low-order polynomial in potential experience explains 20-35% of the variation in observed earnings data, with predictable and precisely-estimated coefficients in almost all applications. 2. Returns to education vary across the population with observables, such as school quality or parent’s education

9. OLS Estimates 1. 10 percent upward bias on OLS estimates of the return to education (based on the most recent, “best” twins studies) 2. Estimates of the return to schooling based on brothers or fraternal twins contain positive ability bias, but less than the corresponding OLS estimate.

10. Does IV Fix the problem? • IV estimates of the return to education based on family background • systematically higher than corresponding OLS estimates • probably have a bigger ability bias than OLS estimates • IV estimates of the return to education based on intervention in the school system • about 20 percent more than the OLS estimates. • return to schooling for these subgroups are especially high, and cannot be generalized to the population.

11. A Static Model of Education and Earnings • Because of its tractability, Card uses a static model that abstracts away from the relationship between completed schooling and earnings over the lifecycle. (we’ll do a dynamic model next). • Two assumptions: • that most people finish schooling and only then enter the labor force (smooth transition). • the effect of schooling independent of experience (Separability above)

12. The basics • Simple Linear regression first introduced by Mincer • Takes the general form of linearity in Schooling, quadratic in experience. Assumptions: • separability of experience and education. • log-earnings are linear in education. • correct measure of schooling is years of education • each year of schooling is the same. (more on this later)

13. Wages or earnings? • Earnings conflates hours and wages • Card reports that about two-thirds of the returns to education are due to the effect of education on earnings—the rest attibutable to the effects on hours/week and week/year. • The specification in (1) explains about 20-30 percent of the variation in earnings data.

14. Why use Semi-Log Specification? • log earnings are approximately normally distributed. • Heckman and Polachek show that the semi-log form is the best in the the Box-Cox class of transformations. (we can talk about this more later in the empirical part)

15. Defining some Terms • Let our utility function U(S, y) = log(y) – h(s) where y is earnings, S is years of schooling, and h(s) is an increasing, convex function. Then, define our discounted present value (DPV) function:

16. Simple relationship between returns and costs • So that we have h(S) = r*S • more generally we could have a convex h(.) function if the marginal cost of each year of schooling increases faster than the foregone earnings for that year—maybe because of credit constraints)

17. Results Optimal schooling is implicitly defined by That is there are two sources of heterogeneity: • Differences in costs (represented by h(S)) • Difference in marginal returns (represented by y’(S)/y(S))

18. Optimal Schooling • a simple specification of these two components (define E(b) = b and E(r)= r and k1, k2> 0) • This gives us the optimal schooling expression:

19. Interpretation of Equilibrium • Individuals do not necessarily know the parameters of their earnings functions when they make their schooling choices. • biinterpretation: individual's best estimate of his/her earnings gain per year of education, as of early adulthood. • One might expect this estimate to vary less across individuals than their realized values of schooling • the distribution of bi may change over time with shifts in labor market conditions, technology, etc. (Skill Premium)

20. Some Assumptions • treat bias known at the beginning of the lifecycle and fixed over time: • assumption probably leads to some overstatement of the role of heterogeneity of biin the determination of schooling and earnings outcomes. • for simplicity, assume jointly symmetric distribution of b and r.

21. Returns to schooling • From our equilibrium expression (4) can get expression for returns to schooling • Even in this simple model there is a distribution of returns unless • Linear indifferent curves with uniform slope • Linear opportunity curves, with uniform slope

22. Within vs. Between Variation • Within: Eq. (4) as a partial equilibrium description of the relative education choices of a cohort of young adults, given their family backgrounds and the institutional environment and economic conditions that prevailed during their late teens and early 20s. • Differences across cohorts in these background factors will lead to further variation in the distribution of marginal returns to education in the population as a whole.

23. Earnings and Schooling Eqn • From equation 3A (FOC), we get • Note that individual heterogeneity affects both the intercept and the slope • Defining αi = ai + a0 • Use this with eqn (4), to define schooling choice in terms of a, b, and r (5)

24. Linear Estimating Function • Define λ0 andψ0as the parametersfrom the linear projection of aiand bi on where is E(Si) (6a) (6b) • That is:

25. OLS estimates of b • Using this notation, we can write the probability limit of the OLS estimate: (7) where the avg. marginal return to schooling in the population is:

26. Homogeneous Returns • Let bi = b and k1 = 0 • Then (5) implies the OLS estimate is not consistent, with upward bias of l0 %. • The bias comes from the correlation of ability to the marginal cost of schooling.

27. Heterogeneous Schooling • Reintroducing a heterogeneous b • we get additional bias terms in due to the self-selection of years of schooling. • The size of this bias depends on the importance of the variation in b in determining the overall variance of schooling outcomes.

28. What did we learn • The linear model appears to fit so well because there is a bias introduced by heterogeneity which is convex and independent of the concavity of the opportunity curve. • More simply put, the concavity from quadratic term in (5) is offset by the convexity from y0 giving an approximately linear relationship.

29. Understanding Observed Linearity-1 • Case 1: gets the standard ability “omitted variable bias” return to schooling • Let ai vary by individual (heterogeneous) • b be fixed across individuals. • Bias comes from the correlation between ability and marginal cost of schooling so σra < 0 which implies that λ0 > 0.

30. Understanding Observed Linearity-2 • Case 2: • ai and bi both vary across individuals. • cross-sectional upward bias because of self selection. • So depending on the relative variance of these components will determine the convexity and concavity.

31. Understanding Observed Linearity-3 • Rewrite (5) and reorganize terms: • This is linear if ψ0≈ 2k1 • The bigger the contribution of bi to the overall variance of schooling, ψ0 is bigger and the more the convexity

32. What about Measurement Error? • The downward bias of measurement error is often thought to offset some if not all of the upward bias in a,b from ability, • only be true if the error is not correlated with level of schooling • Unlikely because individuals with high levels of schooling cannot report positive errors in schooling whereas individuals with very low levels of schooling cannot report negative errors in schooling. • Given this correlation, the measurement error may actually exacerbates the attenuation bias.

33. IV in a Heterogeneous World • even minor difference in mean earnings between the two groups will be exaggerated by the IV procedure. • For example, natural experiments inference are based on small differences between groups of individuals who attended schools at different times, places, etc. However, the uses of these differences might be difficult to generalize.

34. IV-2 • Define a linear relationship between returns to schooling and a set of characteristics, Z, i.e. • So the earnings function can be rewritten as:

35. IV-3 • The big news: In the presence of heterogeneous returns to education the conditions to get an interpretable IV estimator of very strong. • The requirements are that we have individual specific heterogeneity components that are mean independent of the instrument. • The second moment of the return to education is also independent of the instrument • The conditional expectation of the unobserved component of optimal school choice is linear in b.

36. Family Background IV-1 • The strategy: use variables such as parents education, characteristics of parents to control for unobserved ability. • The key idea: if a and S uncorrelated then we get an unbiased estimate, otherwise, we get an upward bias

37. Family Background IV-3 • To illustrate this, consider a linear log earnings function: • linear projection of unobserved ability component on individual schooling and a measure of family background (Fi):

38. Comparing Regressions: Homogeneous Case • In order to compare this to the regression of a on S alone, define: • Using these, we could compare three potential estimators: • OLS from univariate regression of earnings on schooling—bOLS • OLS from bivariate regression of earnings on schooling and family background—bbiv • IV estimator using Fi as an instrument for Si (bIV)

39. Comparing Regressions: Heterogeneous case • introducing heterogeneity across individuals in b, so that • we can relate ψ0 as follows • Assuming , ,

40. Siblings/Twins Models • The key idea behind this strategy: some of the unobserved differences that bias a cross-sectional comparison of education and earnings are based on family characteristics • Key Assumption: within families, these differences should be fixed. • Differencing between schooling levels of individuals will yield consistent results.

41. Defining “Family Effect” • Define “pure family effects” model as the aij=aj and bij=bj • linear projection of a and bi – b on the observed schooling outcomes of the two family members:

42. Estimating with “Family Effects” • Assuming that bi, S1i, S2i have a jointly symmetric distribution • Earning functions are then: • Taking differences, a within family difference in log earnings model:

43. When Family Effects Models Work • With identical twins, it is natural to impose the symmetry conditions so that λ1=λ2=λ,ψ1=ψ2=ψ and • With these assumptions and the pure family effects specification, all biases from ability and schooling are sucked up by the family average schooling component which differences out.

44. When Family Effects Don’t Work • In the case of siblings, or father-son pairs it seems less plausible. • Relax the family effects model as follows:

45. Why doesn’t it work • For a randomly-ordered siblings or fraternal twins, it is natural to assume that the projection coefficients satisfy the symmetry restrictions so that λ11=λ22, λ12=λ21, ψ12=ψ21, ψ11=ψ21 • From this, the earnings model eqn’s are: • From this system, is not identifiable.

46. “Family Effect” or OLS Models? • Without a “pure family effect” and symmetric it is only possible to estimate an upper bound measure of the marginal returns to schooling. • there is no guarantee that this bound is tighter than the bound implied by the cross-sectional OLS estimator. • It is possible that the OLS estimator has a smaller upward bias than the within family estimator.

47. Take-Homes from the Static Model 1 • The OLS estimator has two ability biases, • the intercept • the slope. • The bias in the slope may be relatively small if there is not much heterogeneity. • The necessary conditions for IV estimators to be consistent is strict • many plausible instruments recover only the weighted average of marginal returns of the affected subgroups. • . If the OLS estimator is upward biased, then the IV estimator is likely even more so

48. Take-Homes from the Static Model 2 • If twins or siblings have identical abilities, then a within-family estimator will recover an asymptotically unbiased estimator • otherwise a within-family estimator will be biased • the extent to which depends on the relative importance the variance in schooling attributable to ability differences in families versus the population.

49. Take-Homes from the Static Model 3 • Measurement errors biases are potentially important in interpreting the estimates from different procedures. • OLS estimates are probably downward biased by about 10% • OLS estimates that control for family background may be downward biased by about 15% or more • within-family differenced estimates may be downward-biased by 20-30% with the upper range more likely for identical twins.

50. Empirical Estimates