Gold monetization and gold discipline

NBER WORKING PAPER SERIES GOLD MONETIZATION AND GOLD DISCIPLINE Robert P. Flood Peter M. Garber Working Paper No. 5I NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge MA 02138 September 1980 This research was partly supported by NSF grant SES—T926807. This paper was presented at the research meeting of the NBER Program on Economic Fluctuations, University of Rochester, July 214_25, 1980. The research reported here is part of the NBER's research program in Economic Fluctuations. Any opinions expressed are those of the author and not those of the National Bureau of Economic Research. NBER Working Paper #544 September, 1980 Gold Monetization and Gold Discipline ABSTRACT The paper is a study of the price level and relative price effects of a policy to monetize gold and fix its price at a given future time and at the then prevailing nominal price. Price movements are analyzed both during the transition to the gold standard and during the post—monetization period. The paper also explores the adjustments to fiat money which are necessary to ensure that this type of gold monetization is non—inflationary. Finally, some conditions which produce a run on the government's gold stock leading to the collapse of the gold standard and the timing of such a run are examined. Robert P. Flood, Jr. Department of Economics University of Virginia Charlottesville, Virginia 22901 Peter M. Carber Department of Economics University of Rochester Rochester, New York 14627 (804) 924—7894 (716) 275—4320 1 "Money" the is a theoretical concept whose imperfect realization has assumed form of a sequence of particular physical objects. Since the supply process for each particular physical object is unique, an object which is "money" in one epoch may he supplanted in another epoch by an object whose supply process is (perhaps temporarily) superior. While there exists a flow of objects through the category called "money", the sequence of "money" objects in modern times has been so limited that it can readily be characterized as a "gold-fiat-gold" cycle. Given the recurrent use of gold as a monetary standard, economic agents will believe that gold may be restored by a government as a money, even in an epoch in which gold is demonetized. Fluctuations in the intensity of such beliefs must then be reflected in gold price fluctuations. Recently, large movements in the price of gold have initiated two lines of thought concerning government gold market policy. First, Salant and Henderson 1978 (S-H hereafter), have developed a partial equilibrium gold market model to analyze the interaction between rapid gold price rises and government gold auctions. Second, Laffer (1979) has advocated a return to the monetary gold standard.' Simultaneously, Barro (1979) has constructed a model to study money and price dynamics under a gold standard. The present paper integrates these approaches, together with a method recently devised by Krugman (1979) to study exchange crises, to construct a framework for simul- taneously analyzing government gold and monetary policy. The model which we develop is suitable both for the study of historical gold standards and for the analysis of a future gold standard. Specifically, we wish to study three questions which arise in the context of transition to and from a monetary gold standard: (i) What is the inflationary effect of a policy to monetize gold at some future date? (ii) Given that a gold standard will be adopted, which monetary policy is consistent 2 with price stability and minimum disruption of the gold market during the transition to monetization? (iii) If a monetary authority implements a gold standard but does not adhere to the "discipline" of the standard, what is the anatomy of the gold standard's collapse? Qiiestions(i) and (ii) address recent plans for monetization while question(iifl is posed to provide both some perspective for realistic policy discussion and some tentative steps toward understanding the breakdown of the gold in the 1930's. standard technical the In addressing question (iii) we also propose a definition of the concept called allows us to definition "the discipline of the gold standard"; Uctermine if a particular monetary policy is consistent with "gold discipline". We present components agents path our analysis in four sections. Tn of our model and section I we introduce the develop equi I ibrjum price paths for a world where believe gold will never be money. In section ii we study gold's price following an announcement that gold will be monetized at a fixed future date, with gold's price pegged at the prevailing market level on that date. Section [II contains our analysis of the price level effects of moncti zation along with our model's section IV we study prescription for monetization without inflation. In the forced demonetization of gold, the collapse of a gold standard. Section IV is followed by some concluding remarks and our technical appendix. To maintain clarity of exposition most results will I)e introduced in the text without proof; the detailed algebra involved in the proofs will be left either to the appendix or to footnotes. 3 I) Th& World Without Monetary Gold We will first analyze a world in which gold has no monetary u.se. Such a world is very similar to that studied by S—H, except there will be no government auctions. Throughout the paper we employ a continuous time model in which agents have perfect foresight. To minimize technical complexity, we present our ideas in the context of a specific, linearized example. We divide our economy into two sectors, a gold sector and a monetary sector. Both the output of interest r are fixed at constant levels exogenously to the Sectors. a) goods other than gold and the real rate of gold and monetary 2 The Gold Sector The operation of the gold market can be described by equations (1)-(3): + G(t) (1) I (2) D(t) = v[D*(q(t)) (3) q(t) = [)(t) rq(t) — D(t)J for G(t) D*I , < 0 0. I is a fixed, total world stock of gold.3 D(t) is the quantity of gold which has been transformed for industrial or consumption purposes (or which remains in the ground). G(t) is the quantity of gold privately held as ingots or coins in sl)eculative hoards. E)*(q) is a target for the sum of consumption and industrial demand, which depends negatively on (j(t) the relative price of gold in terms of other goods.4'5 v is a positive, constant speed of adjustment which may, if desired, depend on r; and we assume Equation (1) states that the total supply of gold privately available is held either in speculative hoards as ingots or in tile form of jewelry or other consumption goods. Equation (2) is an adjustment equation which indicates the rate at which actual non-speculative holdings adjust to desired non-speculative holdings. Equation (3) is a requirement that the relative 4 price of gold must increase at the real rate of interest for any gold to he held in speculative Our gold We depart to IoanIs. sector model is similar to the gold model developed by S-Il. from S-Il in our equation (2) which allows consumers and producers disgorge gold from final uses. The S-Il model precludes the possibility of gold's return to speculative hoards after its conversion to final use. We have adopted this modification both because it captures the recently observed disgorging of gold from consumption stocks and because it is technically convenient to allow I) to hear no prior constraints.8 Equations (l)-(3) form a system of differential equations in l)(t) and q(t). We assume that (4) D*(q(t)) = is a constant. where In the remainder of this yields Equation (3) (lIt) = q(O)ert (5) section, we solve (l)-(4) for (1(t) and l)(t). when C(t) > 0. Substituting from (5) and (4) in (2) we (6) I)(t) which has the solution To complete and l)(O)e_Vt that t)(0) = equal zero, i.e. 0(v-r) dT + the solution in (7) we require an initial condition for l)(0) a condition on assume v -rt e = —vl)(t) + D(t) = (7) derive speculative gold holding to determine q(0). First, we 0. Second, if I the speculative is the date when l)(t) = 0, G(T) must gold stock must he exhausted when there is flO 5 further accumulation of non-speculative gold. This requirement is the terminal condition imposed by S-li, where speculative gold is held only in anticipation of future non-Speculative gold accumulation.'0 Since G(T) = 1 = D(T) = (8) 0, D*(q(T)) equations (4) and (8), we can determine the relative price of gold at the time that speculative hoards are exhausted, From (1(T) which, together with equation (5), yields 0-rT q(0) = (9) In addition, the condition that I = 1)(T), accumulation equation, (7), and our Substituting (ii) b) S j — log non-speculative gold assumed value D(0) = c (v-r)Tdi ye -vT (e 0 imply (v-r)T (v-r) -1)— we can solve for for q(0) in (10) = which with (12) rT 1 = ye -vi (10) the r - log v r-v equation (9) implies q(0) = iog r - r-v e log The lonetary Sector These results indicate that the relative price of gold and the quantity of gold held in various categories are determined by the operation of the gold sector alone. In order to determine nominal prices, we must appcn(1 a monetary sector to our model. The Operation of this sector will have no real effect in a model in which gold has no monetary function; we introduce the 6 sector at this point to serve as a preliminary to the analysis in Section II. monetary We assume that money market equilibrium P'1(t) (13) a[r — + is. given by'2 P(t) or (14) M(t) = (( - r)P(t) - aP(t) where P(t) is the price level for all goods, M(t) is the exogenous money 0. supply and (3—ar) > In this section we assume that M(t) =C, where C is the (constant) quantity of currency in circulation. The di ctatecl by market (15) fundamentals solution to (14) is13 P(t) = j3C -r The nominal price path for gold is obtained by multiplying q(t), from equation (5), by P(t), from (15) with q(O) given by equation (12). 7 IT) The World with an Announced Gold MonctizaIon The monetization policy studied in this section will consist of a government's unanticipated announcement that at a future time t=w it will fix the price of gold at the nominal level prevailing at w and that from.w onward speculative gold holdings can be used as money4'15The announcement is assumed to occur at time c, where This policy differs from the relative gold price pegging policy studied by S-H in that gold, which can be used for monetary purposes, will generate the same service return that any money yields and in that we assume that the nominal price is pegged. However, the method which analyze the problem will be qulte similar to that of S-Il: we determine a set of conditions which must prevail at the time of monetiza- we employ to first tion and then solve backward to find the initial by i.rnpljed relative price at time c profit maximizing behavior. a) The World after Gold is 1oney goal in this part is to derive an expression for Our the relative price of gold at the time of monetization w; such an expression will be essential in linking the gold standard world to the basic system studied in section 1. Because of the tedious algebra requi red to determine the path of the relativc price of gold, we will then text and only present a sketch of the required steps in the main skip to the relative price solution. We leave the details to the appendix. We for assume that oniy gold held in speculative hoards currency. a perfect substitute If Q(w) = Q is the nominal price of gold at t=w, equilibrium in the money and (16) is gold C + QG(t) P(t) = markets for t > (s-ar) - a P(t) I'(t) w requires 8 0(t) = vI---- - 0(t)] (17) (16) Equation money supply; the nominal the Substitution It is equation (13) with the additional gold comI)onent added to (18) i(t) = (19) G(t) = are for t>w. to transform (16)-(l7) into (ar) P(t) P(t) 0 which = Q(w) from equation (4) into equation (2) produces equation (17) convenient will be price of gold is pegged at Q - G(t) - vG(t) + - vi linear differential equations in G(t) and P(t) •17 Equation (19) two derived by substituting for E)(t) from equation (1'). is Since the determinant of the system (18) -(19) is negative, the characteristic polynomial has a positive root A1,and a negative root A2, therefore, the system exhibits saddlepoint + 1 J/(-ar+) = for stability. [I(-r) G* = and - /1/(13-r+) . The stable solution t>w obeys P(t) = (20) ((t) L G*] - + 2 2 where A2 —v/ (Q(v+A2) ) . lquation (20) , which we depict in Figure 1 as the line ss is the stable branch of (18) and We are 1:qmiat ion interested in finding (20) , and (;(w) . The wh i cli describes (lu) Steady-state values of the system are with t=w, provides system i s C(w) . 2 the relative the cvoltzt ion Some derivations.) price of gold at time w. one equation in completed by using, in the unknowns P (w) , addition to (20) , Q(w) = equation (17) of non—speculative gold holding and equation , money market equilibrium. P(w) = (19). (See appendix for these En solving the above we use the requi rement steps in the solution of this system are carried out in the appendix. Presently, we require only the expression for the relative 9 -1: c. o 0 C- Figure I Showing the Stable Gold Branch (Fquation is Monetized 20) After 10 price of gold at time w, which is Q(w) = P(w) (21) v6e (r-v)(w-c) -r6 (v-r)1)(c)e The relative price of gold depends on the quantity of gold used for non-speculative purposes at the time of the monetization announcement, on the speed of adjustment of non-speculative gold holdings, on the real rate of interest, and on the long run non-speculative gold holding parameter, (S. We wish to compare the relative price of gold in a world with monetary gold to the relative price derived in section 1. To do this we consider the relative prices at time c. jumps at In particular, we will show that Q(c)/P(i;) the time of the announcement as long as w<T. From equation (5) we need only discount Q(w)/P(w) by 0r( to find I) — — V0C (r—v)(w—ó) — (v-r)D(c)e -ra (r—v) (w-c) the relative price of gold in the instant after the impending gold standard is announced. Prom equation (9), the relative price at the instant before the announcement is q(c) = er( r-c)• and from equation (7) we have vi e ri (v-r)c D(E) = —[e -lie -vc . We nay use these expressions to write the ratio of the post- and pvc-announcement relative prices as Jhi s Q(c)/P(c) —- ye (r-v) (w-c) _____ —r q(c) }ei' (w-c) v[l-e logr- ______ logy = T. ratio equals unity when w = ______ Ihus, f gold is to he monetized after all gold enters non-speculative uses, the solution is identical to that in section 1. It can easily be shown that < 0; so if w<T, price q(c) > i, i.e. ti'e announcement of monetization causes the relative of gold to jump upward.'8 11 III) Inflation and the Gold Standard In this section we analyze the inflationary effects of the gold monetization policy proposed in section II. In addition we consider the monetary policy which must accompany gold monetization in order to prevent price level movements. To begin, we note in Figure I that the pre-announcement price, P(c); /(1-ar), the horizontal intercept of the P=0 schedule. The line ss is the system's saddlepoint path after gold monetization; therefore, it indicatcs the path for G(t), P(t.) for t>w. With these facts we can readily establish that P(w)>P(). Since P(w) is on ss and ss lies entirely to the right of P(c) it follows that P(w)>P(c). consider the behavior of P Next, we (c ,wJ . it and during the period While P may jump in an unforeseen manner at time c, agents cannot expect to move discontinuously in price at time c the future; otherwise, the expected rate of change would be infinite at the time of a jump, producing money market disequilibrium. Therefore, P moves smoothly (differentiably) to P(w) during Cc ,w We can easily show P<() that during the period (c,w] , P>0. If, on the contrary at some time in the interval, the money market can clear only if P<P(c) since C is unchanged before w. This precludes the smooth rise of price to P(w) . Since P(t) >0, tc (c ,wI , P(t) must lie between P(c) and P(w) ; hence the initial jump a ftcr the moneti zation announcement is also to a price between P(c) and P(w) To summari zc , we find that the jri cc level jumps s inni I taneously Wi th the monetization announcement and rises smoothly until the date of monetization. However, after time w, the path ss in the price level and Figure 1 . If G(w) <G , then monetary gold holdings will follow price will rise in a gold inflation, 1.2 ultimately converging to P*. if G(w)>G*, the price level will dec Inc in a gold deflation as gold is extracted from circulation and used for consumption. The price level will he stable after time w only if G(w)_G* Moncti zation without Inflation 1vidcntly, a government succeed announcement of eventual gold monetization cannot as a policy to avoid inflation in the absence of other measures. Therefore, we wish to employ our framework to explore how a government simul- taneously can announce the monetization of gold, avoid inflation and minimize the disruption of the gold market. The policy van able available C, which we tion-gold money, to the government have previously set at We require a time path for C which is the amount of the constant level C. will cause the nominal prices of gold and goods to follow (24a) P(t) = (1 (24b) 0(t) rQ(E)er(t (24c) Q(t) = t>w U given monetization at w. Condition (24a) requires P to remain constant indefinitely. (241)) and (24c) require Q both to move cont iniionsiy at the rate r between and w and to remain constant after w; in parti cul ar, (241)) prevents Q from jumping di scont i nuous ly at t ime c We can use the money ma rket to determine the requl red path for C: (25a) C(t) (25b) C(t) = P(s) [ -ar (25c) C(t) = Qflr(w-c)(;() = C(c) , - Q(c )e G(w) t=w t>w 13 In the above equations, G(w) equals I—[)(w); and I)(w) can he den ved from (7) as E)(w) = (26) D(c)e + :r)-vw (v-r)w (v-r)c-e In addition, for t>w, G(t) can be I determined from equation (19) by setting P(t)=P() From time c unti.1 w, the policy requires that C time w, remains constant. At must decline by the amount Q(w)(.(w)=Q(e)er(w-c) ('4w) to accommodate C monetized gold. Following w, the government stabilizes price by with balancing exactly currency injections or withdrawals the value of gold which leaks into or out of consumption and industrial use. The only component of the policy lacking in simplicity is (25h); this element's complexity arises from the goal of minimizing the disruption to the gold market. Indeed, any discrete reduction in C at time w would, when combined with (25a) and (25c), stabilize P. However, any reduction in C other than (25b) will disturb the gold market.'9 j_4 IV) Anatomy of a Crisis (Demonetization.) in the Cold Market Tn previous sections we did not account for government gold reserves since, except fo.r fixing the price of gold, the government was passive; merely its willingness to intervene was sufficient to produce the derived results. however, if the government adopts additional policies which potentially conflict with the maintenance of a gold standard, we may no longer treat gold market intervention as implicit; rather, we must directly consider the government's role in the gold market. In this section we introduce government gold reserves, R, and initially assume that the government ss;cs constant the overall money growth rate at the positive g. With finite R such a money growth policy must lead to a break- down of the gold stmdard.2° In addition, it is possible for the gold standard to break down even if money is decreasing, providing that prior to the inception of the gold standard there was "too much" money in cxi stence. A Breakdown with Constant Money Growth We can readily show that tile gold standard must collapse when money grows at a the If there were no breakdown and money grew constant, posItive rate. rate g, then P must rise at tile rate g. Since P rises, q=Q/P must at decline at the rate g; then i)*6/q rises at the rate g. Therefore, I)* and D increase toward infinity, which is incompatible with a finite R. It follows that the gold standard must collapse at some time. analysis of the our dynamics of a gold standard col lapse draws di rect ly from Krugman' s (1979) study of the breakdown of a fixed exchange rate system. In Krugman's paper, a small country's monetary authority fixes the price of foreign exchange and expands the domestic component of the money supply at a posit ive, constant rate. Eventual ly, a crisis must occur; there is a run on 15 the monetary authority's foreign exchange holdjjigs and an end to the fixed exchange rate. In Krugman's model the exchange rate and the price level are essentially the same variable; no relative price effects arise from the crisis. Our model of a gold market crisis displays the Krugman-type run on government stocks, but it extends his analysis by allowing for real effects of the CT! SI S. We now define I as the total stock of gold inclusive of government reserves, 1 = (27) R(t) + G(t) D(t) + While gold is money, the money supply is, as before, M(t) = If the government sets H on a "crisis path", rising at the. then C(t) + i)G(t). positive rate g, at some time z a crisis must occur in the gold market. At the time of the crisis QG(z) will be demonetized; also, R(z) in currency will be exchanged for government gold during the run on government stocks. In all, the money stock at the instant of the crisis must fall by Q(R(z) + (28) G(z)) = Q(I — D(z) where the equality follows from (27). (29a) H(t) = 1(0)gt (29b) H(t) = 1(0)gZ (29c) M(t) ()(t_.z) We Q1i—E)(z) j , t<z t=z , should emphas.i ze that while gold is money' the government may growth rate of N the , Thus, the time path of the money stock is hut may not control the division of M control the between C and Q(. Because size of C is indeterminate, the size of R is also indeterminate. However, for the problem of determining the time of the crisis , only the allocation 16 of I between 1) and (R÷G) is important. Next we will employ path and the that time of the (29a)-(29c) to determine both the pro-crisis price crisis. Crucial to our analysis is the recognition speculators will force the crisis to take place without any foreseen jumps in P The or Q. time path for P for P(t) = (30) e > 0. where A = M(T) -ctP(t) eTdT For t<z we use equations (29a)-(29c) in (.30) to obtain {j [(x)t} - t=z (31) becomes For P(z) = _±ic__ (31i) find lation these ö[iDjj} -{M(0)e Both (31) and (31a) depend on To P(t) [-rj J P(t) = (31) is obtained by solving the equation l'1(t) = the, values we use both our and the condition that Q as yet, unknown values of l)(z) and z. knowledge about consumption gold accumu- cannot jump at the moment of crisis. After the crisis, the relative price of gold can be determined from the model in section I as (32) where q(z) = S — e —r(T—z) T, as before, is the choke date. Since DF) = (33) 1 (33a) i I)(z) = -v( I -z) + 2.(J .v(T_z) ye + 1e 1, we require (v-r)T fF 1de (z)' dT v (r-v) (T-z) v-r [1 - l;quation (33a) follows directly from (33), and (33) is a reinterpreted version 17 of equation (7). Equation (33a) may, in principle,be solved for 'F as a funct ion of D(z) , z, and 1, i.e. (34) (D(z), T The (35) with 0. Q/P(z). q(z) is identical in (32) and (35), we substitute from (34) to derive z, I)-z] = (36) P(z) I:quat ions (31a) and (36) are two and z. The the 1 requirement that gold's price does not jump at time z is (1(Z) = Since z, 1). third equation, equati oiis in the three which can he used unknowns P(z) , D(z) to coniplete the solution, is accumulation equation fl(z) = D(0)e" (37) It + veZ is evident that closed form solutions to z are unlikely, even in simple models. In practice, numerical techniques would be required to derive z for particular parameter values. A Gold Standard Inconsistent Currency Love I That a gold standard must eventually collapse from an inflationary money creation process agrees with intuition. Less obvious is the possibility that a gold standard may collapse in a period of monetary and price decline. 18 As an example of such a case, we generalize the post-monet i zati dynamics equations (18) and (19) by allowing for government reserve holdings R(t). of government The only changes C fixed price Q; otherwise, it by exchanging curency does not intervene for its reserves in the money at the or gold markets. money supply at. any time that the gold standard exists is then The M(t) = (38) (C - QR&+ Q(R(t) + G(t)) where R0 is the quantity of gold reserves and C is the quantity of outstanding currency at tile beginning of the analysis (t=O). 21 The dynamic system analogous to (18)—(19) is -ar P(t) - (18a) P(t) = (19a) [R(t)+G(t)j = Q(R(t)÷ G(t)) - CQR() P(t) - v(R(t) + G(t)) + vi. Q The solution to the system is similar to that for (l8)-(19) . and (R÷G) P converge ultimately to the steady state values N and [R*+G*] ; however, nothing guarantees that (R*+I*) is positive. Indeed, [R*+G = [I (8-ar) can he negative if — 6 (C-QR0/Q] / (8 -ax'+6) the term 6(C-QR0)/ is large enough. We have depicted the case of a negative (R*+G*J in Figure II. The (R+G)=O and P=() loci are analogous to those in Figure 1, except that the steady state value for R*+C* is negative. Since the consumption demand for gold tinder tiìis gold standard eventually exceeds tile total available, the system must collapse at sonic I)Oint. At the time of the collapse, currency will he exchanged for all the governments reserves at the fixed rate Q; and he money. hence , the level post—collapse money stock will gold will cease to he C — QR() ; and the pr i cc will be (C-QR0)/(8—ar) , represented by point A in Figure II. In order j uinps, the to pri cc prevent the infinite profits associated with price level level must have attained (C — QR0)/(I3ar) at tile moment of 19 F N 0 A 41 1' (4G*) 0 Figure II p 20 the collapse. Similarly, to prevent a jump in the nominal gold price Q, there must still be some gold in the hands of the public at the time of the collapse. The gold price then gradually rises, thereby reducing the slope of the (R+G)=O locus and rotating it counter-clockwise until it intersects the P-axis at A. The lines through the points FIIA represent the path followed by the price level and (R÷G). The Fit segment, located to the left of the stable manifold ss, is a solution Path for the system (l8a)-(l9a). Thus, we have forced a collapse of the gold standard even when the money stock and prices continually fall; formal derivations of these results together in the with an equati on determining the timing of the collapse arc presented append:ix. Gold Discipline In the two preceding examples, we have demonstrated that a gold standard inflationary monetary policy is in effect and when money and prices are declining. In this section we explore the restrictions can collapse 1)0th when an on the money supply process which guarantee that a gold standard will survive indefinitely. We call such a restricted policy a strictly_gold-disciplined money 1)011 cy, and we seek a means of determi fling whether a particular policy is strictly golddiscijlined. One such method, presented in the appendix, consists of solving directly for z, the time of the collapse. If a finite solution for z exists, then the monetary policy can be classified as a z-year gold-disciplined policy; if z is infinite, then the monetary policy is strictly gold-disciplined. In the main text, we develop an equivalent method. A policy is a sequence of money growth rates g(t). We model government caused money growth, i.e. all changes in tinuous change associated the money supply except that discon- with the gold standard collapse, as a smooth The money supply at any time t is then 21 rt (39a) M(t) = M(O)e J 0 (39b) M(t) = g(h)dh M(0)eJ 0 ft -k(z) t=z g(h)dh t>z M(z)ez M(t) (39c) t<z where k(z) is the amount of money destroyed when the gold standard collapses at tfme z. Our model requires I> l)(t), Vt, which is a real resource constraint holding independently of the monetary regime. The real resource constraint applicable to the gold standard is I > D(t), Vt. A gold standard, once imposed, will collapse if and only if agents foresee that. in the absence of a collapse there would he some finite t when I = D(t). We define a strictly gold-disciplined money growth policy as a policy such that I > D(t) Vt where I)(t) is using the money growth policy and setting k(z)O. We emphasize that D(t) the value of D(t) constructed from equations ( 30) and ( 37) need not he less than or equal to I. The [)(t) values arc, for our model, a sequence of hypothetical values calculated by setting k(z)O. From previous P(r) = results we have eXTJ "fli? eX3dj and - D(t) = D(O)e -Vt + ye -Vt (S — JOQ or P(T)e VT dt 22 I)(t) = Vt D(O)e + t O (S vetf Aj eVTdT By setting k(z)O we find D(t) = D(O)et eJf J0j&0]3djTd1T + Thus, the policy in question is strictly gold-disciplined if and only I t Qe J -Vt > [)(O)e_Vt+ F all [M(O)e j 0, for if rj O g(h)dh I edi - }e t>O. We argued previously that a policy of money growth at the constant Positive rate g must lack guid discipline. To gain intuition about the applicability of our definition we will calculate fl(t) for the policy g(t)=g>() for all t. We find E)(t) = L)(O)e Vt gt + vó(e Q(X-g) Recall - e -Vt J (g+V) that we require A>g. Our expression for D(t) grows without bound as t rises. hence we can not have i>D(t) for all t. 23 V) Conclusion and Extensions A government may announce the fixing of the nominal price of gold in a number fiat of of different environments. Gold may currently circulate along with money as part of the money supply with gold. a freely floating nominal price Alternatively, gold may have no monetary role until the fixing of its nominal price. In this paper we have analyzed the movement of the price of gold in the latter case. Tn the context of our model we have shown that only a very Special money supply process will prevent inflation when gold is monetized. In addition we have devised a method for determining the timing of a gold standard's collapse and produced a formal definition of "the discipline of the gold standard." 'I'he model employed may be developed in a number of ways. The model specifies a fairly simple demand and supply behavior for money and gold rather than invoking explicit maximizing behavior. Also, agents are assumed to have perfect foresight. We think that it is worthwhile to place our model of a gold standard's collapse in a stochastic environment. Given such a framework it is possible to investigate the collapse of the standard to determine if it was vial)le when it was formed. 1920's go]d 24 References Barro, Robert J., 'Moneyand the Price Level Under the Gold Standard," Economic Journal, 89, (March, 1979), 13—33. Flood, Robert and Peter Garber, "Market Fundamentals vs. Price Level Bubbles: The First Tests," Journal of Political Economy, forthcoming, August, 1980. Friedman, Milton and Anna Schwartz, A Monetary History of the United States, 1867—1960, Princeton: Princeton University Press, 1963. Krugman, Paul, "A Model of Balance—of—Payments Crises", Journal of Money, Credit, and Banking, XI, no. 3, August, 1979, 311—325. Laffer, Arthur, "Making the Dollar 'as Good as Gold'" Los Angeles Times, October 30, 1979. Salant, Steven and Dale Henderson, "Market Anticipations of Government Policies and the Price of Cold," Journal of Political Economy, Vol. 86, no. 4, (August, 1978), 627—648. 25 Footnotes I Laf for proposes to have the government announce that gold will he mone- tized at the market price prevailing on a certain future date. To prevent inflation in the interim, the government woul.d follow an "austere" monetary policy and sell, a large portion of its gold holdings. 2 can easily relax the fixity of other goods, providing that other goods are given exogenously to the model developed here. Allowing for changes in the real rate of interest would greatly complicate our analysis; thus, the development of -our models is an exercise in partial equilibrium analysis. In (S-Il), r is given exogenously and output of other goods is ignored. l.n Barro, both r and goods production are exogenous. 3UnLil section IV we ignore government gold stocks. 4it is possible to interpret 1)*(q(t)) and E)(t) as desired and actual consumption holthngs of gold plus desired and actual gold remaining in the ground, respectively. T en the model allows for gold to be mined at increasing costs. (S-li). When and there i s no further change in specul at ive gold 5Our model coiitans an analogue to the choke price used by (q (t) 1 = [)(t) f)* , I)(t) = 0, holding. The (1(t) for which this condition holds is the analogue of the choke P i CC. n an opt imi zi iig model, the speed of adjustment, v, would depend on r. We avoid the complex problems that this causes by fixing r. If q/q < r, there would he no demand, for speculative gold holdings, and the pri cc of gold would discontinuously. If q/q > speculative demand would cause a discontinuous upward j ump i n gold pri cc. Since forsecable di scout i nuous jumps in q are not consi stent with speculative equi Ii hriiim, l must follow equation (3). fall there r, is no gold mining, equation (2) is identical to Barro's (1979) adjustment equation for non—monetary gol.d except that we don't include depreciation. To the extent that there is gold mining in our model., equation (2) differs from liarro's adjustment equation because our D is the difference between newly consumed and newly mined gold. We have also explored an alternative form for the D* function which includes anticipated capita.1 gains to gol.d holding as an argument; this form does not substantively change our results. See the appendix for this analysis. 9'Fhi s asstlmpt ion Precludes the O5S 1)1 ii ty of cost 1 y go] ci production. Gold as a pile of pure bars, all of which are contained initially in speciila— holdings. This is the same assumption that S—li employ, and I t greatly exists t i ye si algebra. An assumption that l)(0) > (1 a 11 ows for the i tiCS make no quail tat VC di fie relice effect of a monetization announcement on relative prices, it is helpful nip] i fi es the cx i stence in the to ignore ensul rig of gold nil ties; but S I rice such in them. G(T) > 0 equation (5) inipli es that q(t) continues to risc' for t>T. cont limes to rise after T, equation (2) indicates that l)(t)< 0, so G(t) (1(t) will ri so for t>T. But such speculative hoards will never be used for nonspeculative purposes since I)(t) remains negative. Therefore, G(T) = 0. If 101f 26 report in the text so1tions for rv. • Solutions are for the SCC jul case r=v but equations (10) and (11) become 12 (10') 1 (11') 1 = 1/v . An alternative money = easily attainable hr . . . M(t) demand function 1S P(t) = 8—a(r+ P(t) -----j-)1Y(t) where where y(t) is the amount of other goods produced in the economy. The solut ion for price (given a constant y(t) and M(t)) would then be P(t) = iM(t)/y(t)] /(—cLr) Alternatively, we can assume that M(t)/yt)6 moves exogenously; if so then most of the following analysis holds, except that LM(t)/y(t)1 place is used in of M(t) 13. like equation (14 ) and cannot reject the hypothesis that P'ice responds only to market fundamentals. 1100(1 and (arbcr (1 980) test a model very much 14That gold cannot be used as money prior to its monetization is a fairly extreme assumption, but it seems to characterize current gold use. 15The fixing of the pri cc of gold at a giving future time has a precedent of the U.S. In 1875, Congress passed a law requi ring Greenback period in the a return to the gold standard in .January, 1879 at the pre—Ci vii War pan ty. in this case gold circulated as money with a fluctuating greenback exchange For dctai ls, see Friedman and Schwartz (1963), Ch. 2. rate. 16Sjfl(e Laffer' s proposal for monetization includes intenini government studied here. it is hard reconcile Laffer's policy with interim price level stab1ity (see Section III). reserve sales, it is more complex than the policy to assume here that the quantity of gold in speculative hoards is eiiouh that merely the will ingness government to intervene i 11 cause the price of gold to be fixed; however, the government need not actually of the great intervene. hence , we can ignore movements i.n government reserves. In sect ion TV we generalize the model to the case in which the government must intervene directly with its reserves to preserve the gold standard. 81t is possible to extend the model to an uncertain world in which the timing such of the future fixing to gold is unknown. In the append i x , we explore a case and determine the time c relative price of gold that must prevai 1. 1 earl , we nni st as stime accommodate nominal that the i n i t i a 1 va I iie of C i s I a rge enough wi the des i red monet i zati on of p' i vate gold thout an increase in t) money balances. 20 A gold standard is said to break down when a private run on government reserves exhausts those reserves. As Krugnian has rioted a government may divide reserves i nto primary and secondar rese rves w i th on Iv primary reserves l)e i ng coimni t ted to the price fi xi iig p0 ii cy . Thus , a gold st.aiida rd maY break (town when primary reserves arc exhausted but secondary reserves remain intact 27 21 More genera Ely, we could begin the analysis at time c, the moment that the future monetization of gold .i s announced. Agents at time would real i ze that the gold standard to be implemented at w will not be permanently viable. They will prepare for the expected demonetization so that at time w the nominal and relative prices of goods and gold and the quantity of consumption gold holdings will be different from the results of section TI. Still more generally, the government may hold secondary and tertiary gold reserves in preparation for new gold standards to be established at some time after the collapse of the current standard; i.e. to defend the standard in the crisis, it will only expend a known part of its totalcurrent reserve. 'lit I s rhythm i c wi thdrawal from and return to the gold standard wi 11 reflect itself in yet different paths for the price level and gold consumption from those which we have examined thus far. 22 A possible (though very conjectural) application of this analysis is to study of the reestablishment and cot Iajse of the inter-war gold standard. The U.S. continuously used a gold standard until 1933, but other major countries, i.e,. Germany (1924), Great Britain (1925), France (1927), fixed their currencies to gold in the 1920's. It is often suggested that the gold parities which were established were "inappropriate". here we can interpret "I nappropri ate" to mean that there was too much currency outstanding in the the world to maintain the viability of the gold standard. 28 Append i x - [)erivation of Equation (21) Since G(w) = I - I)(w) with q (w) = (Al) Q(w) /P(w) P(w) = we use equation (7) and the condition q(c) = in equation (2U) to obtain - {Q(w) A2C + q(w)e I - E)(E)eV] - P(w)v [1(rv) (WE) v6Q(w)I + (v÷X2)Q(w)i(-r) + v [i3-cr+6] and we use P(w) = G(w) = G(w) — D(w) , equation and (7) , a(v+A 2 ) [Q(w)(TD(E)cV() t-c-a(v+A2 )IP(w) + of linear equations in P(w) and we found (A) which q(w)er along with - P(w)v (1C(r )(wc) = a(v+2) I Q(w) Since w-c is given, and D() and I arc given, pair q(c) = in equation (16) to obtain + (A2) I Q(w) . equations (Al) and (A2) are a After a lot of manipulations that = P(w) (r—v) (w—E) (v-r) is equation (21) in the text. An interesting aspect of (A3) , which do not have much intuition about, is that it is entirely independent of money market parameters. we 29 IT — Solution to the System (18)—(19) In this appendix we derive equation (20), the stable manifold for G(t) and P(t) 'after monetization. Equations (18) and (19) are a system of differential equations in G(t) and P(t). system are G* = — B—ar+6 — and The steady state values of the P* = C+QI The determinant of the homoegenous system is 8—ar —Q Det _!(8_ar+6) < 0. a —v Since the determinant is egative, the roots of the system are of opposite sign, indicating saddle point type stability. Assume that the roots are and A2<0. Complete solutions of (18) and (19) are then of the form C(t) = P(t) = C1A1eA1t C1eA1t + + C2A2eA2t C2eA2t + p* with A1 = —s---——- and A2 = Q(v+A1) Therefore, P(t) = + c* . For stability we require that C1 Q(v+A2) G(t) + (P* G*). 0. 30 III — Subjective Probabilities over Monetization In this section, we generalize, the model of section IT, to the case in which agents have a probability density function over the timing of monetization. Fortunately, most of the required drudgery has already been done in section II because the solution for the time c relative price of gold falls easily out of equation (22). This case is similar in form to the various policies studied by Salant and Henderson when agents are uncertain about the timing of policy implementation. The results which we derive here can be employed in an alternative interpretation of the relative gold price movements described in the first pages of Salant and Ilehderson. We will assume that our gold speculators arc risk neutral and that the subjective p.d.f. over the time w of monetization is the same exponential p. d. f. used by Salant and Henderson f(w) = ye -ó(w-) (a) in their appendix, i.e. for w > The relative l)rice solution in equation (22) is conditioned on a particular announcement the time c and monetization time w. government time announces that gold will l)e monetized at w with the p. d. f. over w given by (a). in solution (22) as Here we assume that at time E some uncertain future Let us denote the relative price Q/P(r/w), to indicate that it is conditional on the time of monet zat i on. i Risk neutral speculators will act to set actual relative price at time c equal to the weighted average of the Q/P(e/w) 's, where the weights arc in (a). given The expected return to speculative gold holding at c will then equal the real rate of interest. However, as long as the actual monetization does not occur, the relative real rate of interest. price of gold must rise at a higher rate than the 31 We proved in section II that if w>T, Q/P(/w) > q(c). The expected value of Q/P(/w) can be writtcn as (b) E 1T (c/w) ve-r -y(w-c) dw + q(c) ye (vr)E)()eW = yró J)(c)(v-r)(r-v+y) + q(c)e_1(T_ 10-(r_v+y) (T_c) + iT ye -y(w-e) dw 1)D() (l-e) - The determination of the time path for Q/P(c/w), conditional on monetization's not yet having occurred, has proven to be intractable (for us). 32 TV - Determination of z for the Case of too Much Initial Currency 1) The money SUPPlY is 2) (la) M(t) = (ib) M(t) (C-QR0) + (C(t) + R(t)) = - QR0 t<z t>z During the gold standard period, price must move along the path P(t) = (2a) where + C1et - C_QR0+QJ 13* = This . 13-ar + p* + C2eA2t t<z comes from the original P, G solution notes, and we will determine C1, C2 later. After z, the money supply is fixed so Price must he C P(t) (2b) 3) = - QR0 — t>z B-ar Just after the run, the relative price of gold is q(z) = to Then ( need = ____ z = - 1 log r 1QI(13-) F (OR) to solve for C1 and C2. From the solution for (G+R) we (G+R) (4a) (C-R0)/(B-ar), C-QR0) we can solve for 1— z: TWe - e1C I prevent price jumps. But since P(z) = _r(T_z) 4) - = I — C1A1e1t fl(O) + = C1A1 C7A2cA2t + C2A2 have: + (G*+R*) or at t=O + (G*+R*) where = 6 A1 Q(v+A1) 1(13-ar) - ,A= , and Q(v+A2) (C*+1*) = B - ar + 33 From (4a) we solve for C1 in terms of C2: -ö(v+A ) C = 1 v+A [I—D(o) vó Using (2a) we can solve for — (R*+G*)] — C 2v+A2 C2 explicitly: Q(v+A) p(z)_p* + = 2 [I — v6 Le v+A A,z — ____ _____ 2 Thus, we can solve for C1,C2 in terms of the parameters, the initial, and the terminal conditions. This determines 5) exactly the price path. Now we must find z. From equation (33a), 1= D(z) -vF I e v (r-v)F +—[1—e )] v-r so (5a) D(z) = From (5b) We ieVFti vV_r (l_e_V)) the accumutation equation for D(z), D(z) = D(O)e + veJZ J P(t)eTdr know what D(z) must be from (5a). Then we need only plug in our price path in (Sb), integrate and P(T)eTdT C1 j = Then C1 v+A (Sb) is e [e1_ij + solve for z. The integral is essentially l)TdT + C2 v+A 2 e2)TdT + p* [e2_l] + v VTd (5c) (le')] Ie'[l — + 5ve -vz 34 = D(O)e C 1 v+A1 C. [e1+h1_l] + 2 [e2"_l] v+A2 All we have to do is find the z which solves (5c) + v [e'-i)J 35 V — Analysis with an Alternative D* In the text we assumed that the target va1e of D is 6/q. D* = This simple form was chosen because of its tractability. In this appendix we demonstrate that none of the conclusions we reach in the text would be altered by changing our assumption about D* to conform with that assumed by Barro. In our notation, one functional form which has the 2+63 (q/q) + qualitative properties assumed by Barro (1979, pg. 15) is D* = q where y is real output, assumed constant in our paper, and the linearization is appropriate only over ranges where [62+tS3q/q] > 0. We differ from Barro in that he assumed unit income elasticity, but this does not seem substantive for present purposes. With the new specification of D* we may write the differential equation (19) from the text as system (18), —cxr (*) p = — 62 (**) G = v(—(-— Q — 63(B—cxr) _______ - — 63 (— + 1)G] + vii c — 6 — 0 6 53C 1 y — - —1 Qcz To derive (**) use (*) in place of P in the target and note that after monetization -P/P. qjq = If 62 (— — Q 63(—czr) ) is positive then the phase diagram of (* ) and (** ) is — Qa qualitatively identical to Figure I in the text and the analysis of the model is essentially unchanged. However, if 62 — — 63(1—ctr) — 9 < 0 then the Qci phase diagram appears as in Figure A. The important point about this figure is that, if positive, the slope of the G=0 schedule is less than that of the P=0 schedule. The slopes of these schedules are dG I3-cr0 - dP I =o C-) cz6 dG dP dp 2 36 P0 I=0 1+— 3 It is evident from Figure A that the stable branch, ss, is qualitatively unchanged from Figure I. Finally note that in the steady state q/q and hence -.P/P = for the steady state including the rate of return variable is irrelevant. 0 37 '3-0<- Figure A Viewpublicationstas