Materials for "Education finance and optimal investment with gestation lags"

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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 Please do not quote.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.

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1.' The Macroeco'nornic Approach [3,4,5, 7 , 8,II,]

--

P 0

.

~'", _{Decision Block:} _Consumption

~ >determine e(t) ,s(t) t' ( " so as to maximize J ~ eQ

'

. ~ . - , sO

"

I~

... ₄ K Capital .,~ rt Evolution ! T

~in

vestment in uman resoul;ces training h

-in ph

-vestment in ysical tesources equipment Production 0

₌

F (K ,W) J!l W Labour ForceiA ." Evolution

...

.,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) ) ,f; _k f k = -uk

+

s!'lf(~/w) k (0)

=

ko,k (tf ) w •

=

ewf/d - aw w (0)

₌

_{wo,w(tf )} = > w_f < < 0 s(t) 1 0 <: e (t) < 1 0 ~ c(t)

.-

(l-s-e ).f (k)

..

< 0 't:W

=

< 1, 0 ,~ k

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•

2

"

Mode'l II (more detail on population dynamics and skill structure)

_.

,

Define the productivity-weighted measure of the effect

ive 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. -. .

.

"

t t ' · ,

criterion J

~" .Su(c(t~exp[-yt]

dt . : o ,-; , , -,' . , ' : " " , . .. . .

, dynamics ;;:':~(t)' = ~aw)'~(t)'

+

b1u'(t)A(t) - b₁ha'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)' _'-":

'l~S~O, i~u~O~ c,;, [(l-s)w:,:cf(klw;'c>. - ~l ud(A)J e~ 0,

cpnstraints

" ' . "

.'

,

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3

Model III (expand labour force bloc~ to take account of

gestation lags) . "" ..- '. ... "."~ '" , .;. ..', .~. _.;:..' . , -;'.:'" .. ._,' , .-:

...

_:. .. "". ' . , ,. \-.' ~ " " ,.. ,' ...~' ....~, ....",' '"..' . ,.:". ~

,;

·\.Icrh~'I"S·'A~j ,,~\

'

.. ,

,.

:",':'>

"r-I

---«r-..-:--,.

-:,,,~,-.(

_'~ _._:,_.;. . " .... :",. " .' ...~.. .." ". :. .:~ ' , ' .' . .' .,: ", .. . :.... ".1:: , · .. t • • ' . . . . : " .. ;., . ,.:

..

.~, . " , '':. .'...~' --~ ~ .... c • . '-': " ,./ . L.· ". . .,'~ . " .' .,': : " ~ ., .~ . '. , , '. .o. ' • :.:,-,~ .. ,.-: ._: .." i; j , " '," .,: _._--:--....:-.. -,.-,- - - , - .,-:".-:---:;:::t£--:---;---,-,.-"---:-:1

.

.' ".' ,":..~-." . criterion, .' J1::

'J' .

, ...• : .. :,:,-, ' . ;::0 .,: - ""'::" .:<> ... '.

", ,dy~amics

_':W(t!,

~,~~

_aw(t)

₊

e~p[-~~3] u(t-~3)

.';'j

,-..

,:, net) ::

~~n(t) -e~p[-brn3]>(t-m~)

+

u(t)

- - cr'k(t). -}

~(~)v:,(t)f(k/w)'

" ; ,.'

.

, " ..' '...~ . " ..' .... ~ . ', .. .. " ..•.. . ::. t. " ,~.'

control variables 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] "' •...

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Simulation' MOdel'(Schematic Diagram) lf~

r---

---~---

---1

III

i

A (stochastic

i

I element) I I I

!

I

/Ki(t,a)~,

!

I I , J I II ' I

i

f

~

s.

(t, 6) SSi (t, 6)

i

l>

Vi (t, 6)

~6

Oi(t'6)~

/ 1 \ 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 css ( a) ,H (a) ,F (a)

Decision

t

P (t)

Macro Problem: Given population data (cohort sizes) and a statis tical 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•

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BIBLIOGRAPHY ON CONTROL'-THEORETIC , ' APPROACHES TO INVESTMENT' IN EDUCATION'

1. Becker', G. S., Human Capi tal, New York : National Bureau of ,Economic Research, ColumbiCt: University Press, :;t.,964. 2. 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. 352-365. 3. Budelis, J.J., "Optimal Economic Growth with Explicit Con

sideration of Human Capital Development: Optimal Paths for Some Differential-Difference Equations", unpublished Ph.D. Thesis, Harvard University, 1970.

4. Dobell, A.R. and Y.C. Ho, "An Optimal Unemployment RateII ,

Quarte'rly Jourh'al of' Economics, LXXXI I November I 1967,

pp. 675-683. (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. 4-14.

6. Eckaus, R.S., , :"

"Economic Criteria for Education and Training", Review' '0'£ Economics'&' Stati'E)'tics 46, May I 1964.

7.' 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, 1971.

8 •. Neher, P.A., and Hay, K.A.J., "Education and Capital Mis allocation in a Growing Economy",' The' Can'adian' Journal . 'o'f' Economics, I, No.4, November, 1968.

9. Sheshipski, E., ~On the Individual's Lifetime AllocatioTh Between Education and Work", Metroeconomics, October, 1968.

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•

10. 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, 1971.

11; TU, Pier+"e N.V., "Optimal Educational Investment Program :tn an Economic Planning Model", Canadian Journal of

Economics, II, No •. l, February, 196$, pp. 52-64

12. Weiss, Yoram, uAllocation of Time and Occupational Choice", Ph.D. Thesis,. Stanford University, August, 1968 .