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 AN OPTIMAL GROWTH PATH FOR THE MONEY 
 SUPPLY SUBJECT TO TARGET CONSTRAINTS 
 
 Dale J. Poirier 
 
 #146 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 

 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 November 2, 1973 
 
 AN OPTIMAL GROWTH PATH FOR THE MONEY 
 SUPPLY SUBJECT TO TARGET CONSTRAINTS 
 
 Dale J. Poirier 
 
 #146 
 
11 October, .1973 
 
 Ar. Optimal Growth Path for the Money 
 Supply Subject to Target Constraints 
 
 Dale J. Poirier* 
 
 Seemingly ingrained in the literature of monetary economics is the end- 
 less debate between the followers o£ Milton Friedman, who favor a steady and 
 moderate rate of growth in the money supply, and those who favor using the 
 money supply as an instrument together with fiscal policy to achieve desired 
 objectives in areas such as national income and employment, Somewhere be- 
 tween these two groups are many economises who are sympathetic to both 
 sides, acknowledging both the potential usefulness of using the money supply 
 as an instrument for policy matters, and the pot exit.! a 1 dangers (in terms of 
 induced cyclicality) of a widely fluctuating rate of growth in the money 
 supply. This paper suggests a growth path for the money supply which is op- 
 timal under a set of preferences likely to be representative of many econo- 
 mists in the middle-of-the-road group. 
 
 Suppose a policy maker desires the money supply (how-aver defined) at 
 
 yi 
 
 times t < t < ... <t, to equal e (i » 0, 1, ...,k). respectively, where . 
 k > 2. These target levels can be part of the instrument solution to a rig- 
 orous mathematical optimization problem, or merely reflect, tha policy maker '3 
 own subjective preferences. Their exact origin is not of concern here. In 
 
 *Th© author is Assistant Professor of Economics at the University of 
 Illinois at Urbana -Champaign. Acknowledgements are due Case Sprenkle for his 
 helpful comments on an earlier draft of this investigation. Any errors are 
 the sole responsibility of the author. 
 
 It is being implicitly assumed that the policy maker (e.g. the Fed) can 
 actually achieve these targets. In the words of Friedman [2]. "No serious stu- 
 dent of money— whatever bis political views—denies that the Fed can, if it wishes, 
 control the quantity of money. It cannot, of course, achieve a precise rate of- 
 growth from day to day or week to week. But it can come very close from month 
 to month and quarter to quarter." 
 

 
 
 
 
 
-2- 
 
 any case .suppose the policy maker decides to follow a "pseudo-Friedman" doc- 
 trine and requires , subject to achieving these target levels, that change? in 
 the rate of grov/tb of the money supply be "minimized." This lexiographic pref- 
 erence ordering in which a steady rate of growth is a secondary goal to achiev- 
 ing the target levels seems to capture the sentiments of many middle-of-the- 
 road economists.* 
 
 f(t) 
 
 The policy maker's dilemma of choosing an optimal path e which achieves 
 
 the targets, yet minimizes changes in the growth rate, can be formulated mathe- 
 matically as follows. 
 
 M (t) dt (X) 
 
 min f j , 
 
 t 
 
 9 
 
 subject to e ' - a e (a * 0, 1, . ..*k) 
 
 Less function (1) weighs changes in the rate of growth (i,e., f $ '(t}) 
 over the Zx:w period [t ,t, ] in the conventional quadratic fashion, and hence 
 seems to be a reasonable criterion function for the pseudo-Friedman doctrine 
 described previous!/, 
 
 One obvious candidate for the solution to this problem is to begin at 
 e* ° and let the money supply grew at a constant rate so as to hit the target 
 
 y. y 2 
 
 e' , then 1st it grow at the constant rate needed to achieve o , etc.. Such 
 a path would imply that f(t) take on the form of a continuous piecewise linear 
 function. While integral (1) is then zero, the growth rate is not defined at 
 t , t ,...,t, . since in general the growth rate has jump discontinuities at 
 these prints. These discontinuities raise two problems. First., it :<s ques- 
 tionable whether in practice it is possible to bring about such instantaneous 
 
 ges in the growth rate. Second, it is disturbing to envision the possible 
 ramifications of such sudden, abrupt changes in the growth rate. Hence, it 
 
•3- 
 
 seems desirable to restrict attention to the class of functions f ft) which 
 are "well-behaved," i.e., continuous functions with continuous first and second 
 derivatives over [t , t, } , hereafter denoted C ? "[t ,t,]. This insures a "smooth" 
 rate of growth. 
 
 Interestingly, this piecewise linear solution, as well as the solution when 
 attention is restricted to those functions in C 2 [t ,i, } are both members of the 
 family of spline functions. Specifically, the solution in the class C v [t : t, ] 
 of smooth functions is a natural^ cubic spl ine , i.e., (i) a piecewise function 
 s(t) in C [t ,t v ], (ii) whose pieces are cubic polynomials defined over the in- 
 
 tervals [t. ,,t.](j - 1, 2, . ,.,k) such that (iii) e ^^ « e Xi (i * 0,1,.. , ,k), 
 
 2 
 and (iv) S M (t ) ■ S ,5 (t«) •■■ 0. The piecewise linear solution (for obvious 
 
 reasons) is known as a linear spline . The linear spline and the cubic spline 
 
 axe simile: in that their first and third derivatives, respectively, are step 
 
 functions. 
 
 The proof that the natural cubic spline S(t) is the solution to (1) in the 
 
 7. 1 
 
 class C [t a ,t.j is well-known in approximation theory literature, The ingre* 
 cients of the proof are of no particular interest to the discussion here, and 
 henc$ will be omitted. Rather, for the purpose of illustration, a numerical 
 example is presented. 
 
 As stated earlier the exact origin of the targets s -*■ (i * l,2,.*,,kj is 
 not c£ importance here. Their selection is likely to reflect in a very compli - 
 
 '"Applications of spline functions in economics, both from a theoretical and 
 an empirical standpoint, have received the intensive attention, of the author. 
 The interested reader is encouraged to consult Poirier [3] and [4] for more de- 
 tails. 
 
 *ThQ adjective "natural" is used in spline theory to a cubic spline which 
 satisfies not only (i)-(iii), but which also satisfies (iv). 
 
 "'See for example Ahlberg, Nilson, and Walsh [1, pp. 3, 7$-7j. 
 

 
 
-4- 
 
 cated way the policy maker's philosophy concerning seasonal variations in the 
 demand for money and the desirability of integrated fiscal-monetary stabiliza- 
 tion measures. Here the following situation is considered. At time t (July, 
 
 o 
 
 1972) the policy maker's horizon is one year. He has target valuer for the 
 money supply (defined to be seasonally unadjusted M 2 ) at the end of the next 
 four quarters, i.e., at t ~' 3 (October, 1972), t « 6 (January, 1973). t « 
 
 ..2 2 3 
 
 9 (April, 1973), and t =12 (July, 1973), where time (t) is measured in months 
 
 <+ 
 
 Using hindsight the target values were selected as follows. The first and last 
 
 y y 
 
 targets , e ° * 503,6 and e * - $47,3. , measured in billions of dollars, are the 
 
 actual unadjusted M^ levels achieved in the respective periods. The interme- 
 
 Yi " Yz Ys 
 
 diate targets, e - 513.4, e * 532.0, and e ' * 540.1, reflect seasonal de- 
 mand variations from a constant rate of growth path over the period. These 
 targets were determined by (i) equating the adjusted and unadjusted series in 
 July, 1972 and July, 1973; (J.i) computing the resulting ratios of unadjusted to 
 adjusted, values in the intermediary quarters; and (iiij and then multiplying 
 ths.se ratios by the constant rate of growth amount (implied! by e and e ) in 
 the respective quarters. The sensitivity of the resulting optimal growth path 
 to the selection of these targets is discussed subsequently. 
 
 Given these targets t.h® optimal path S(t) which solves (1} is shown in 
 Figure 1, Also shown is the constant rate of growth path F(t), the targets 
 (denoted by m o")j and the actual observed unadjusted M values (denoted by "•"), 
 
 2 
 
 Although the natural cubic spline is a piecewise cubic function of time, 
 
 it is much easier to parameterize in terms of the targets. Poirier [3j pp 
 
 27-36] shows that at any point t in time, it is possible to write S(t) as a 
 Un sa r function of the log targets, i.e., 
 
 S{t) » WV (2) 
 
 where y = [y ,y , ..,,y, 3' is the vector of targets measured in natural ioga- 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
lnM a 
 6.32 
 
 FIGURE 1 
 
 Comparison of Growth Paths 
 
 6,30 
 
 6.28 
 
 6.26 
 
 6.24 
 
 6.22 
 
 A 
 
 «s 
 
 tf 
 
 
 r 
 
 or 
 
 / 
 
 / 
 
 s 
 
 '.1'' 
 
 y? 
 
 / /fit) 
 
 fy 
 
 12 
 
 Jwly 
 
 72 
 
 Jan 
 73 
 
 July 
 73 
 
rithms, and w = fw ,w .....w. 1 is a row vector whose elements depend oa t --aid 
 the "knots'' t ,t , ,..,t, . The linearity of 5(t) in terms of y makes it quite 
 easy to analyze the sensitivity of the optimal path to changes in the targets. 
 Specifically, letting A be the finite change operator, 
 
 S(t) - w Ay + w Ay + ,., + w.Ay, (3) 
 
 0. 12 K K 
 
 Since the weights w ,w , .,.,w, depend only on t and the knots, they do not have 
 
 1 »» 
 
 to be recomputed for each new set of targets. 
 
 Table 1 contains the monthly weights for the numerical problem considered 
 earlier. These weights are valid for any case involving a one year horizon and 
 quarterly targets. As to be expected, at quarter j, w. * 1, &x:.d all other 
 weights are zero. Note also that at other times the closest target receives 
 the largest weight. 
 
 In summary, this study has introduced a growth path for the money supply 
 which is optimal under a set of preferences likely to be representative of many 
 economist torn between a target oriented monetary policy and a constant rate of 
 growth path, from a spline theory viewpoint it has been shown once again 
 (c£. Poirier [3]) that economic theory can justify the use of spline functions. 
 In fact under the preferences implied by [1), a spline -function is the optimal 
 path . 
 

 • 
 
 
 
 
 
 
,E 1 
 
 Monthly Weights 
 
 '" - 
 
 t 
 
 w 
 
 * • 
 
 w 
 
 v./ 
 z 
 
 w 
 
 3 
 
 ><• 
 
 
 
 1 . 000 
 
 . 0000 
 
 , 0000 
 
 -- « HIM! II _ -u- J or— 
 
 . 0000 
 
 .0000 
 
 1 
 
 .5? 73 
 
 ,5132 
 
 -.1270 
 
 ,03175 
 
 -,005291 
 
 2 
 
 ,2341 
 
 .8915 
 
 -.1587 
 
 .5968 
 
 -.006614 
 
 3 
 
 . 0000 
 
 1 . OGO 
 
 ,0000 
 
 oOOOO 
 
 . 0000 
 
 4 
 
 -.07804 
 
 .7646 
 
 ,3862 
 
 -.08730 
 
 .014SS 
 
 5 
 
 -.0S291 
 
 «U4t> 
 
 .8042 
 
 --,1270 
 
 .02116 
 
 6 
 
 , 0000 
 
 1 
 
 , 0000 
 
 1.0000 
 
 . 0000 , 
 
 .0000 
 
 7 
 
 ,02U6 
 
 -,1270 
 
 .8042 
 
 ,354S 
 
 -.05291 
 
 8 
 
 .0I4S5 
 
 -.08730 
 
 ,3862 
 
 , 7646 
 
 -.07804 
 
 Q 
 
 , 0000 
 
 , 0000 
 
 ,0000 
 
 1.0000 
 
 ..0000 
 
 10 
 
 -.006624 
 
 .03568 
 
 -.1587 
 
 .,8515 
 
 .2341 
 
 11 
 
 -.005291 
 
 . 03175 
 
 -.1270 
 
 .5132 
 
 , 5873 
 
 liL 
 
 .0000 
 
 ' „0000 
 
 . 0000 
 
 .0000 
 
 1,000 
 

 
 
 
-8- 
 
 References 
 
 [1J Ahlberg, J.H«, E.N. Nil son, and J..L. Walsh, The Theory of Splines an d 
 
 Their Appl ica ticns , Mew York: Acadmsic Press, .1967, 
 
 [2] Friedman, Milton, "The Case for a Monetary Rule," Newsweek (February 7 
 1972), 67. 
 
 [3] Poirier, Dale J., "Applications of Spline Functions in Economics/' 
 Madison: unpublished University of Wisconsin Ph.D. thesis, 1973. 
 
 , "Piecewise Regression Using Cubic Splines," J ournal off 
 
 the Ame rican S ta tistical Assuci ation , LXVI 1 1 (September , 1 73) , 
 

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