MATERIALS FOR "EDUCATION FINANCE AND OPTIMAL INVESTMENT WITH GESTATION LAGS"* A.R. 'Dobell University of Toronto. July 1973

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1 MATERIALS FOR EDUCATION FINANCE AND OPTIMAL INVESTMENT WITH GESTATION LAGS* by A.R. 'Dobell University of Toronto July 1973 These materials are preliminary, and circulated for discussion only. Please do not quote. ~ The simulation work reported in this document was initiated at the Institute for Policy Analysis, University of Toronto, with the active support and encouragement of Prof. D.G. Hartle, now Deputy Secretary, Treasury Board Secretariat, Ottawa. Initial model design, programming, and subsequent' development work has been carried out in collaboration with Morris A. Cohen.

2 1.' The Macroeco'nornic Approach [3,4,5, 7, 8,II,]. -- P 0 ~', Decision Block: Consumption ~ t' ( >determine e(t),s(t) T so as to maximize J ~ eq '. ~.-, so I~... 4 K Capital.,~ rt Evolution! Production 0 = F (K,W) J!l W Labour Force ia. Evolution... ~investment in h uman resoul;ces training - in vestment in ph ysical tesources equipment -..,j~ P. : Population Evolution max U(c(t»exp(- yt)dt s.t. c (t) = (l-s (t) -e (t» Q/P (t) = (l-s (t) -e(t» wf (k (t) /w (t) ) k = -uk + s!'lf(~/w) k (0) = ko,k (t f ) w < < 0 s(t) 1 0 <:,f; k f > = ewf/d - aw w (0) = wo,w(t f ) = w f e (t) < ~ c(t).- (l-s-e).f (k) < 0 't:w = < 1, 0,~ k

3 2 Mode'l II (more detail on population dynamics and skill structure)., Define the productivity-weighted measure of the effective labour force by: t W* (t) = S w(t, T ) A ( t, l' ) d T t-m.' where W(t,T) represents the fraction of the labour force entering employment at time l' surviving to time tl and A(t,T) represents the productivity factor attaching to them tt', criterion J ~.Su(c(t~exp[-yt] dt.: o,-;,....., -,', dynamics ;;:':~(t)' = ~aw)'~(t)' + b1u'(t)a(t) - b 1 ha'u(t-m)a(t-rh). ' k(t) ='- ok<t)..{s (thv~! (t)f(k (t) I w~:c (t) ).',,,. control variables A(t). set). u(i:).;...,':, \state variables k(t). ~~c(t)' '-: cpnstraints 'l~s~o, i~u~o~ c,;, [(l-s)w:,:cf(klw;'c>. - ~l ud(a)j e~ 0, '..', Sourcei Budelis [3]

4 3 Model III (expand labour force bloc~ to take account of gestation lags)... -, '...,.~ '.-:...: -;'.:'... ', '.,. \-.' ~,..,'... ~'..',.~..;:..'.... :,':'> '.. '.,.:. ~.., r-i---«r-..-:--,.-:,,,~,-.(,. '~.:,.;.,.;.,; \.Icrh~'IS 'A~j,,~\ '....,.... ~,...,' :,. ',.'...~..... :..: ~ ','.'.,... :... ;.,.,.:...~,.. '-':, '':.. '...~' - -~ ~... c.'.,': : ~.,.~. '.,, '..o. '.'.,:.1::,.. t '...:,./. L....,'~...,.-:._:.. i; j, :.:,-,~..'.,: _._--:--...:-.. -,.-,---,-.,-:.-:-----:;:::t --:------;---,-, :-:1 '..criterion,.' J1:: 'J'.,... :.. :,:,-, '. ;::0....,: - '::.:<>,,dy~amics ':W(t!, ~,~~ aw(t) + e~p[- ~~3] u(t-~3).',:.. ~-.,.' control variables.';'j,-..,:, net) :: ~~n(t) -e~p[-brn3]>(t-m~) + u(t) - - cr'k(t). -} ~(~)v:,(t)f(k/w)'.,.. ' '... ~... ' ~.... ', ::. t. ',... - /',... set), u(t), state variables.: k(t), n(t},w(t) -,.... :... 'constraints l~ s~ 0, 'w;: u, c :: (l-,s)wf-dn~ 0. I' ; w+n ','.,~.' ;. Source: Budelis [3] '...

5 Simulation' MOdel'(Schematic Diagram) lf~ r ~ II i A (stochastic i element) I I! I /Ki(t,a)~, I, J I II ' I i f ~ s. (t, 6) SSi (t, 6) i l> Vi (t, 6) ~6 Oi(t'6)~ / 1! I \ D. (t,a). 1 Micro Problem: Given a policy a, together with cost and income data, select 0 so as to maximize Vi(a) I ' Social Policy Decision t P (t) Macro Problem: Given population data (cohort sizes) and a statistical approximation to the individual decision rules', select the policy a yielding the optimal, configuration of cohort monetary flows, CSS, fund flows, F, and distributional results, H css ( a),h (a),f (a).. ~. 'r_'~'._~~~. ~. ~ _._...'~.. M......'..._~...,... _~._''''''' ~_..._._._._~~..._.._--..-_..._..,._.~_

6 BIBLIOGRAPHY ON CONTROL'-THEORETIC, ' APPROACHES TO INVESTMENT' IN EDUCATION' 1. Becker', G. S., Human Capital, New York : National Bureau of,economic Research, ColumbiCt: University Press, :;t., Ben-Porath, Y., The Production of Human Capital and the Life Cycle of Earnings, The Journal of Political, Economy, Vol. 74, No.4, August, 1967, pp Budelis, J.J., Optimal Economic Growth with Explicit Consideration of Human Capital Development: Optimal Paths for Some Differential-Difference Equations, unpublished Ph.D. Thesis, Harvard University, Dobell, A.R. and Y.C. Ho, An Optimal Unemployment Rate II, Quarte'rly Jourh'al of' Economics, LXXXI I November I 1967, pp (See also Comment and II Reply, 'Qu'ar't'erl'Y' 'Jou'rnalo'f' Economi'cs, LXXXIII, August, 1969.) 5. Dobell, A.R. an'd Y.C. Ho, Optimal Investment Policy: A Control Problem in Economic Theory,IEEE TRANS., AC 12, February 1967, pp , : 6. Eckaus, R.S., Economic Criteria for Education and Training, Review' '0' Economics'&' Stati'E)'tics 46, May I ' Lele I M.M., Jacobson, D. H., and McCabe, J. L., Qualitative Application of a Result in Control Theory to Problems of Economic Growth, Iht.' ECO'Il.' ReV. 12, No.2, June, Neher, P.A., and Hay, K.A.J., Education and Capital Misallocation in a Growing Economy,' The' Can'adian' Journal. 'o'f' Economics, I, No.4, November, Sheshipski, E., ~On the Individual's Lifetime AllocatioTh Between Education and Work, Metroeconomics, October, 1968.

7 Southwick, Lawrence, and Stanley Zionts, An Optimal Control Theory Approach to the Educational Investment Decision, Working Paper No. 104, School of Management, State University of New York at Buffalo, April, ; TU, Pier+e N.V., Optimal Educational Investment Program :tn an Economic Planning Model, Canadian Journal of Economics, II, No. l, February, 196$, pp Weiss, Yoram, uallocation of Time and Occupational Choice, Ph.D. Thesis,. Stanford University, August,