Human Capital – SchoolingA. Two basic notions1. There are mutable attributes of individuals that affect theirlabor market productivitya. General physical fitness and well-being – important forsome types of manual labor; affected by nutrition, restand exerciseb. Manual skills (e.g., typing) – important for other typesof manual labor; affected by practice and trainingc. Knowledge – affected by schooling and training2. People choose the levels of these attributes rationallya. People recognize that changing these attributesinvolves near-term costs and long-term benefitsb. People optimally choose the level of these attributesbalancing these costs and benefitsc. Becker’s insight – this optimization problem isanalogous to a firm’s physical capital investmentproblem, hence the term human capitald. Though widely accepted now, this view was initiallycontroversialIt may seem odd now, but I hesitated a while beforedeciding to call my book Human Capital – and evenhedged the risk by using a long subtitle. In the earlydays, many people were criticizing this term and theunderlying analysis because they believed it treatedpeople like slaves or machines.Gary Becker, Human Capital 3rd Edition, p. 16VefStfS e t fSerCfSerrtSrtSrtSrS===−+⎛⎝⎜⎞⎠⎟=−∞−∞−∞−∫∫()d() d ()()Max ( )SrSVfSer=−B. Simple formal model of skill acquisition (schooling)1. Assumptions and notationa. will work in a continuous time frameworkb. let S denote years of schoolingc. let Y = f(S) denote per-period earnings; for simplicity,assume that labor supply is fixed and that earningsdepend only on schoolingd. assume that lifetimes are infinitee. assume that there are no direct costs of schooling butalso that there are no earnings while in schoolf. people seek to maximize their lifetime earnings, V2. Wealth maximization problema. present discounted value of life-time earningsassociated with a given level of schooling isb. individual’s optimization problem isdd() ()VSfSerfSerSrS=′−=−−0′=fSrfS()()c. first-order conditiond. optimal level of schooling satisfies1) first term represents the marginal discounted life-time return to an additional unit of schooling(recall that the present discounted value of aninfinite, constant pay-off stream of B is B/r)2) second term represents the earnings that areforegone while an addition unit of schooling ispursued, i.e., the marginal opportunity coste. assume fO(S) < 0,can depict thesolution graphicallyf. solution is anexample of anoptimal stoppingrule (problem isanalogous to the“when to sell thewine” and “when tocut the tree” problems)g. example: let f(S) = e" + $S ! (S²1) fN(S) = ($ ! 2(S) e" + $S ! (S²2) then the solution is S* = $ / 2( ! r / 2(()Max ,SUVS()()∂∂∂∂UVSSUVSVVS,,dd+=0C. Extensions to the simple model1. What accounts for the variation in schooling attainment?a. in the simple model, variation in schooling choicesmust reflect either1) differences in the returns to schooling or2) differences in borrowing costs (imperfect capitalmarkets)b. can consider some extensions to the model thatintroduce other types of heterogeneity2. Consumption value of schoolinga. one criticism of the simple model is that it assumesthat people attend school only to increase their wealth;schooling (knowledge) is not valued for its own sakeb. assume that instead of being only concerned withlifetime wealth (consumption), people havepreferences defined over both wealth and schoolingsuch that U = U(V, S)c. optimization problem isd. first-order conditiondiffers from earlier solution in which we simply setdV/dS = 0Max ( ) d dSrtSrtSVfSetKet=−−∞−∫∫0′=+fSrfS K()()e. graphically, the solution isshown at the right1) SW* – wealthmaximizing level ofschooling2) SU* – utilitymaximizing level ofschoolingf. assuming that wealth andschooling are goods (enter utility function positively),optimal schooling attainment is higher and optimallifetime wealth is lower than in the simple modelwhat happens if there is disutility associated withschooling (e.g., disutility from boring professors)?3. Direct costs of schoolinga. assumptions1) return to the wealth maximization framework2) make the additional assumption that students mustpay a per-period cost of K while they are in schoolb. the optimization problem now becomesc. the first-order condition can be expressedindicates that people balance1) marginal lifetime earnings benefit with2) the opportunity costs and direct costs of anadditional unit of schoolingd. education subsidies1) elementary and secondary education is publiclyprovided; in 1999-2000, sources of funding wereFederal 7.3 %State 49.5 %Local & other 43.2 %total funding was $373 billion (NCES 2002b)2) your tuition bills notwithstanding, higher educationis highly subsidized; in 1980-1 and 1995-6 thesources of revenue were1980-1 1995-6Tuition 21.0 % 27.9 %Federal 14.9 % 12.1 %State 30.7 % 23.1 %Local 2.7 % 2.8 %Other 30.8 % 34.0 %total funding was $66 billion in 1980-1 and $198billion in 1995-6 (NCES 2002a); public fundinghas decreased while private funding has increased3) according to the model, subsidies shouldVS r fSttS() ( ) ()=+−=+∞∑111VS VSfSrfS fSrrfSfS fSrSttSStStS()()()()()()()() ()()()()+− =−+++−+=+ − ++−+⎡⎣⎢⎤⎦⎥−=+∞−−−=+∞∑∑11111111212unambiguously increase schooling4) what is the rationale for subsidizing education? ata societal level, doesn’t subsidization lead to toomuch educational attainment? does this accountfor the fall in subsidies over time?4. Discrete time approacha. may be more realistic to think about discrete years ofschooling; how does this change the modelb. assuming there are no direct costs, the lifetime wealthassociated with S years of schooling can be expressedc. change in lifetime wealth associated with an additionalyear of schooling isd. optimal stopping rule1) continue in school while V(S+1)!V(S) is positive2) examining the term in brackets, the condition issimilar to that from the continuous time problem –continue in school while the current marginalforegone earnings are less than the presentdiscounted value of the marginal increment to()()()()**1111111+=++−=−=∑∑rES rEStttTtttTVfSetrtS=−∞∫() dlifetime earnings3) complication with this approach (relative tocontinuous time) is that it requires multipleevaluations of the value functionD. Rates of return1. Discrete time example: