bordo a model of the classical gold standard

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Journal of Mon etary Eco nom ics 16 (1985) 109-120. North-Holland

A MODEL OF THE CLASSICAL GOLD STANDARD WITHDEPLETIONMichael David BORDO and Richard Wayne ELLSON*

Universi ty of South Carol ina, Columbia, SC 29208, US A

The operation and properties of the classical gold stand ard are well recognized. How eve r, oneaspe ct that ha s not been dealt with is that gold has the chara cteristics of a durable, but depletableresource. In this paper, w e compare the simple classical model of the gold standard wi th a modelof the gold standa rd that incorporates the durable, depletable nature of gold. Using numericalsimulation techniqu es, we dem onstrate an inescapable tende ncy to long-run deflation whenaccount is taken o f the resource constraint. These resul ts are consistent, wi th and wi thouttechnological progress and variable real rates of return.

1. IntroductionRecent dissatisfaction with high rates of inflation and real economic instabil-ity in the U.S. and elsewhere has led to criticism of the operation of the presentfiat based monetary system. Some economists have advocated a return to theclassical gold standard, based on a government maintained fixed price of goldin terms of the national currency, on the grounds that the gold standard wouldprovide greater price stability than under current arrangements. Indeed, suchinterest led to the establishment of the U.S. Congressional Gold Commissionin 1981.A second desirable attribute of the gold standard stressed by its advocates is

that the monetary gold stock and hence the money supply is determined bycompetitive market forces according to the classical commodity theory ofmoney largely independent of government policy. The classical tradition ofThornton (1802), Mill (18654 Fisher (1922) and Friedman (1953) viewed themonetary gold stock and hence the money supply and the price level under thegold standard as determined by two offsetting sets of equilibrating forcesproducing a tendency to long-run price stability: the response of gold produc-*The first author is also affi liated with the National Bureau o f Econ omic Res earch , Cam bridge,MA 02138. For helpful comm ents and suggestions, we would l ike to thank the fol lowing: JohnChi l ton, Mike Con nol ly, Stephen Ferr is, Mi l ton Fr iedman Levis Kochin. John McD ermo tt, BlameRoberts, Charl ie Stuart, Anna Schw artz, and an anonymous referee. For som e historical evidence on the record of price level and real output stabil i ty of the goldstandard for the U.S. and the U.K ., see Bordo (1981).For a discussion of the del iberations and conclusions of the Gold Com mission, see Schw artz(1982).

0304-3923/85/$3.3001 985, Elsevier Science Publishers B.V . (North-Holland)


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11 0 A4, D. Bordo and R. W. El lson, Classical gold standard mode ltion to changes in the real price of gold, and shifts between monetary andnon-monetary uses of gold by households and firms in response to changes inthe real price of gold.3In a recent article, Barro (1979) has provided a lucid formal exposition ofthe operation of the classical gold standard. One important aspect of theoperation of a commodity standard such as the gold standard not treated byBarro or elsewhere in the literature is that of gold as a durable, but depletableresource. This view takes above ground gold as a commodity that depreciatesat a very slow rate and incorporates the potential exhaustion of gold mines. Inthe literature on exhaustible resources following Hotelling (1931) the long-rungrowth rate of the real price of a resource, determined in a competitive market,and assuming zero marginal costs, should equal the real rate of interest.4Indeed, the long-run behavior of the monetary gold stock and the price levelwhen depletion of below ground stocks of gold is accounted for will differ fromthat suggested by the classical model. In this paper we combine the treatmentof gold as a durable depletable resource, following the recent approach takenby Levhari and Pindyck (1981), with that of the gold standard by Barro (1979).The key differences we find between the simple classical model of the goldstandard and a model of the gold standard accounting for the durabledepletable resource aspect of gold are: an inescapable tendency to long-rundeflation when account is taken of the resource constraint, and a tendency forthe equilibrating mechanism of the classical gold standard to be muted by theoperation of the resource constraint.Our approach in section 2 is to construct a simple model of a closedeconomy gold standard accounting for the resource constraint. In addition weincorporate technological progress and a variable real rate of return into themodel. In section 3 the model is then parameterized and simulated to generatehypothetical paths for its key endogenous variables: gold production, thegrowth of non-monetary gold demand, the monetary gold stock, the moneysupply, and the price level. Comparisons are then made between the perfor-mance of the model under both classical gold standard and resource modelassumptions. Section 4 contains a brief conclusion.2. The model2.1. The classical model

We start with Barros treatment of the classical model for a closed economyadapting it slightly to account for a variable real rate of interest and to allow3For a discussion of the traditional approach to the Gold Standard, see Bordo (1984).41n the presence of r ising cos ts and/or mon opoly, the growth rate of net rent should equal thereal interest rate. This holds for a non-durable resource with cons tant dem and. In the presence ofdurabil ity and growing demand the price path may differ from Hotell ings rule. See Pindy ck (1978)and Stewart (1980) for example.


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M.D. Bordo and R. W, El lson, Classical gold standard model 11 1

for technological progress. Eqs. (1) to (6) represent the money market. Eq. (1)represents the money supply:MS = pPGGM, 0)

where MS equals the money supply expressed in terms of dollars, p the moneymultiplier - the ratio of the sum of currency and deposits to the value of themonetary gold stock (it is the product of both the ratio of the money supply tothe monetary base and the monetary base to the value of the monetary goldstock), PC the fixed nominal price per ounce of gold, and G, the monetarygold stock in ounces. Eq. (2) represents the income velocity of circulation. Weassume it is a logarithmic function of the nominal interest rate: 5V= VP. (2)

Following Fisher (1930) we define the nominal interest rate asi=r+lr 2 (3)

where r represents the real rate of interest and rr the expected rate of changein the price level. Following Mundell(1970), we assume the real rate of interestto be a negative function of the expected rate of price change:

r=f-aa. (4)Finally, we assume perfect foresight, so that

IT = (P, - P,-,)/P,-1.

Equilibrium in the money market requires thatP = W&,/Y,

(5)

(6)given p, PG, he assumption of perfect foresight, and assuming a constant levelof real output, y, the price level is determined by the monetary gold stock.

Eqs. (7) to (9) represent the real conditions of the gold market. Theseequations in combination with (1) to (6) determine a unique equilibrium5 We depart from Barro who, in the text of his paper, assum es a constant real rate of return and

make s the demand for money a funct ion of the expected rate of change in the pr ice level . Butfol lowing Barro, eq. (2) assu me s the real income and price e lasticities of real mon ey dema nd to beone.6 We also tried an adaptive expectat ions schem e. As expected the adjustment path of the modeldiffered under the two schem es of generating e xpec tations, but the long-run equil ibrium values ofthe endogenous var iables were not affected.


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11 2 M.D. Bordo and R. W. Ellson . Classical gold standard mo delmoney supply and price level. We assume that gold production is characterizedby increasing costs and that the supply function for new gold is simply

g = gP/e 3 (7)where g equals production, Pg the real price of gold, P,/P, and t is a timetrend to allow for exogenous technological progress. The demand for non-monetary gold is assumed to be a flow function of the form

&=(e+6)(G;-G~)r (8)where G, equals the net change in the monetary gold stock, and G$ is thetarget or desired stock of non-monetary gold. G$ is defined as

The parameter E is a partial adjustment factor, 6 the depreciation rate ornormal replacement flow, and G, represents the actual stock of non-monetarygold.Finally, on the assumption that the monetary authorities are committed tomaintaining a fixed price of gold, the change in the monetary gold stock issimply the residual,c,=g- G,, (9)

where GM equals the net change in the monetary gold stock.Assuming y, PG, and ~1 are fixed, taking logs, and solving eqs. (i) to (9)simultaneously, the steady state solutions for the model are P = G,,,, = G, = 0(in terms of growth rates). This also implies that g = 6Gz - that gold produc-tion at any point in time is equal to the depreciation rate multiplied by thedesired non-monetary gold stock.

2.2. The resource modelThe real sector of the classical gold model described above assumes that goldproduction, g, is a function of the real price of gold and exogenous technicalchange. However, if we treat gold as a durable finite resource, then goldproduction would not only be affected by the real price of gold but also by thecost characteristics of production.Barr0 does not explicit ly accou nt for technological progress in his mode l but discu sses theimplications of accounting for i t. W e assu me technological progress is exogen ous to simp lify thediscus sions. How eve r, there is evidence that major technological chang es in the gold industry wereboth induced and exogenous [Roc koff (1984)].


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M.D. Bordo and R. W. Ellson. Classical gold standard model 113

The simple Hotelling rule (1931) states that the (real) price of an exhaustibleresource should rise at the rate r (the real interest rate), under conditions ofcertainty and zero marginal costs. As Levhari and Pindyck (1981) point out,the Hotelling rule properly defined implies that the resource rent (price minusmarginal cost) increases by r, assuming perfect competition and certainty. Thisis an important distinction. In the classical model gold production is directlyrelated to the real price of gold, whereas in the exhaustible resource literature,gold production can increase as the real price falls as long as marginal costsdecline more rapidly, thus resulting in an increase in resource rents.Our price and production equations are taken from Levhari and Pindyck.With respect to the former,kg= (r + a)p,,,-, -feAf, (10)

where j(Q)e is the marginal value of services from a stock of resource, Q,and ehr is the growth of real output. In our model, f(Q) = (G,,,/GM)-O, whereG, and G, represent non-monetary and monetary stocks, respectively.We have assumed a Cobb-Douglas specification for total costs accountingfor both production and depletion of the resource:C = AgX- ,

where C is total cost, X equals the remaining stock, and neutral technologicalprogress enters through A. Because of depletion, our production equationdiffers slightly from Levhari and Pindyck and also includes the effect ofdepletion on marginal costs:

with Cg marginal cost, CBg the derivative with respect to output, and Cs, thecross-effect. Although one would expect the latter to be positive, there issubstantial evidence that the discontinuities of gold deposits could have anegative effect. Thus, we have taken the term to be zero in our simulations.Eqs. (10) and (11) can then be solved simultaneously to determine theequilibrium real price and output paths for gold. The resource model can thenbe integrated with the monetary sector. This integrated model consists of fivesimultaneous equations from the resource sector including eqs. (8), (10) and(11) plus equations for C, and C,,. This is combined with the monetary sectordescribed by eqs. (1) through (6), and accordingly, G,,, and G,,, are alsodetermined.


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11 4 M.D. Bordo and R. W. Ellson, Classical gold srandurd model3. Parameters of the models and comparisons3. I. Parameters

The initial values of the parameters in the model are given in panel A oftable 1. The depreciation of gold, 6, is assumed to be 1.0 percent and theadjustment parameter, E, n the e, equation is 0.5. We further assume that themoney multiplier, ~1,has a constant value of 10 over the simulation period, andthat the rate of economic growth and the rate of. technological progress areexogenous and equal to 3.0 percent per time period. Finally, we assume theautonomous growth rate in desired demand for the resource to equal the realrate of interest. Thus, producers have no incentive to withhold or expandproduction based on a differential here. The remaining parameters are basi-cal ly consistent with estimates found in the literature and were selected tocorrespond to the start values for the endogenous variables that are listedbelow in panel B.

Table 1Param eters and initial values of the mod el.

(A ) ParametersP= 1.1a = 0.5a = 0.1g = 1.6/3 = 0.6GN = 0.038 = 1.2lj = 1.08 = 0.1F = 0.03CT= 1.3p = 1.75Y = 0.02

(B ) Initial ValuesPC = 20; f ixed nominal price of gold, dollars per ounce .P = 20; real price of gold, dollars per ounce.B- 100; price index.y = 270; real output, $bil l ions.

MS = 90; money supply, $bil l ions.G, = 55 0; non-monetary gold stock , mi l lions of ounces.G, , ,= 45 0; monetary gold stock , mi l lions of ounces.g = 30; gold production first period, mil l ions of ounc es.X = 1000; remaining sto ck, mil l ions of ounc es.r = 0.03; real interest rate.Y = 3; veloci ty.

i = 0.03; nominal interest rate.


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M.D. Bordo and R. W. Elison, Classical gold standard model 11 5Both the classical model and the integrated model were simulated overtwenty-five time periods using the parameters discussed above and the startvalues. The initial values of the gold variables are hypothetical and werechosen for analytical convenience. However, they reasonably correspond toestimates for the world in the late 192Os.s

3.2. Comparison of the modelsWe now compare the performance of the classical model relative to theintegrated model for the key endogenous variables. Table 2 represents thevalues at five period intervals as well as the overall period mean of: gold

production, the net change in the non-monetary demand for gold, the mone-tary gold stock, and the money supply and the price level. Four experimentsare reported:(A) the two models assuming constant real interest rates, no expected changein the price level and the absence of technological change - the bench-mark case (and the closest comparison with Barros model);(B) the two models assuming constant real interest rates, no expected changein the price level, with technological change at three percent per period;(C) the two models assuming variable real interest rates, expected change inthe price level and no technological change; and(D) the two models with variable real interest rates, expected change in theprice level and technological change at three percent.Column 1 shows the pattern of gold production, g. In the benchmark case Agold production rises over the whole period in the Classical Model. Thisreflects the influence of the assumed real growth of three percent per periodcreating an excess demand for money, which in turn produces deflation, raisingthe real price of gold and encouraging production. By contrast in the in-tegrated model, production initially declines slightly because of increasingcosts and declining resource rents. Introducing technological change in case Bvirtually doubles gold production in the Classical Model, whereas by contrast,in the resource model production increases by a much smaller amount reflect-ing the effects of depletion of the resource. Introducing a variable real rate ofreturn and expected deflation produces virtually no change in the ClassicalModel,g but in the resource model production is increased relative to the

*See the Stat istical Compendium to the Report to the Congress of the Commission on rhe Role ofGold in the Domestic and International Monetary Systems, Vol. 1 (1982).This result reflects the effe ct of our a ssum ed interest elasticity of ve locity (0.5). interestelasticity of nonm onetary gold demand (- 0.1) and a low rate of deflation.


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11 8 hi. D. Bordo und R. W. Elison . Clussicul gold srutuhrd mo del

benchmark case because of the additional, expected deflation which raises thereal interest rate, and hence via Hotellings rule raises production.Column 2 shows GN, the net change in non-monetary demand for gold. Inboth models it is a function of real output and the real price of gold. We wouldexpect G, to rise reflecting the direct effect of real growth but to declinereflecting the indirect effect of deflation producing substitution of monetary fornon-monetary gold stocks. In the benchmark case A, GN rises above itscorresponding values in the Classical Model because of the greater rise in thereal price which in turn reflects the decline in production in that model as realrents decline. Technological change, case B, raises G;, in both models viaincreased gold production raising the monetary gold stock, the money supply,and the price level, and lowering the real price of gold, in turn encouragingsubstitution of monetary for non-monetary gold. This effect is considerablyweaker in the integrated model because of the greater deflationary effectassociated with the resource constraint. Finally, the assumptions of a variablereal rate of interest and expected deflation have virtually no effect in theClassical Model but in the resource model they slightly reduce GN. This reflectsthe effects of a higher real interest rate in raising the real price of gold.Column 3 displays G,, the monetary gold stock, which in the benchmarkcase A rises at a much slower rate in the integrated model than in the ClassicalModel reflecting the forces described above. In addition technological changeproduces a smaller rise in G, in the integrated model than in the ClassicalModel. Introducing a variable real rate and expected deflation has virtually noeffect on G, in the Classical Model, while it raises G, in the integrated modelreflecting the effect of a rising real interest rate on gold production.Column 4 shows M, the money supply whose movement is governed by andreflects the behavior of G,.Finally column 5 portrays movements in the price level. In the benchmarkcase, deflation prevails over the entire period in both models. However,deflation is much greater in the integrated model because of the decrease ingold production over the period. Thus, the offsetting influences to deflation ofthe Classical Model through the substitution between monetary and non-monetary gold holding and the effects of a rising real price of gold onproduction are completely swamped when we account for the resource elfects.Furthermore, introducing technological progress at the same rate as theunderlying growth rate of the economy (case B) almost fully offsets thedeflation in the Classical Model - restoring price stability. However, this isdefinitely not the case in the resource model where deflation prevails. More-over, while accounting for a variable real rate of interest and expected deflation(case C) does offset some of the deflationary pressure in the integrated model,it does not negate it.The qualitative results above are not materially changed when the keyparameters of the models (a, p, Y, a, a, e and @) are varied. Particularly


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M.D. Bordo and R. W. El lson. Classical gold standard mode l 119important are a, a and @, for which ranges are suggested by the literature.OIn addition, raising the rate of technological change from three percent to fivepercent per period reduces the rate of deflation in the resource model by arelatively small amount.4. Conclusion

When account is taken of the durable, depletable resource property of gold,the operation of the classical gold standard is modified in two significant ways.First, in the presence of real growth there is an inescapable tendency towardslong-run deflation, a tendency which is not overcome by technological changeor by a variable real rate of interest and expected deflation. Second theequilibrating mechanism of the Classical Model towards the long run equi-librium price path is muted by the operation of the resource constraint.These conclusions have important implications. The greater tendency to-wards long-run deflation suggests that the likelihood of gold discoveries andtechnological advances in gold production being sufficient to offset the ten-dency towards deflation are even more remote than would be suggested by theClassical Model in the presence of real growth. On the other hand, the rate ofdeflation which satisfies Hotellings rule would, if perfectly anticipated, alsosatisfy Friedmans (1969) optimum quantity of money rule - giving the com-munity the satisfaction level of real cash ba1ances.tReferencesBarro. R. , 1979, Money and the price level under the gold standard, E conom ic Journal 89,13-33.Bordo, M.D ., 1981, The classical gold standard: Some lessons for toda y, Federal Reserve Bank ofSt. Louis Review 63. 2-17.Bordo, M.D ., 1984, The gold standard: The tradi tional approach, in: M.D . Bordo and A.J.Sch wa rtz, eds ., A retrospec tive on the classical gold standard 1821-1931 (Univers ity ofChicago Press, Chicago, IL).Fisher, I . , 1965. The purchasing power of money , 2nd cd. (Augustus M. Kel ly, New York) .Fr iedman, M. , 1953, Com mod ity reserve currency, in: E ssays in posi t ive econom ics (Universi ty ofChicago Press, Chicago, IL) .Fr iedman, M. , 1969, The opt imum quanti ty of money, in: Optimum quanti ty of money and otheressa ys (Aldine, Chicago, IL) .Hotel ling, H., 1931. The economics of exhaust ible resources, Journal of Pol it ical Econom y 39,137-175.Levhari, D. and R . Pind yck, 1981, The pricing of durable exhau stible re source s, Quarterly Journalof Economics 96, 365-377.Mil l , J.S ., 1962. Principles of poli t ical econ omy (1865) (Aug ustus M . Kelly, N ew York).Pind yck, R.S .. 1978, The optimal exploration and production of nonrenewable resource s, Journalpf Polit ical Econ om y 86. 841-861.

Model C was simulated over the following ranges of parameters: [ I , 1.3 to 1.4; p, 1.75 to 1.8;Y, 0.02 to 0.025; a, 0.1 to 0.5 to 0.6; a, 0.1 to 0 .5 to 1.0; @, 0.1 to 0.5 to 0.85; e, 0.25 to 0.75. Theresults of the sen sitivity ana fysis are available upon requ est from the authors.See e.g., Roc kotT (1984, pp. 619-620).


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12 0 hf. D. Bordo and R. W. Ellson. Classical gold srandurd mod elReport to the Congress of the Commission on the Role of Gold in the Domestic and Internat ionalMonetary Sys tems , Vol . 1 (1982).Roc kotT, H ., 1984, Some evidence on the real pr ice o f gold, i ts costs of production, and com mod itypr ices, in: M .D. Bordo and A.J. Schw artz, eds., A retrospective on the classical gold standard

1821-1931 (Universi ty of Chicago Press, Chicago, IL) .Schw artz, A.J., 1982, Reflect ions on the Gold Com mission report, Journal of Money Credi t andBanking 4. 538-551.Stew art, M.B ., 1980, Mono poly and the international production of a durable e xtractable resource,Quarterly Journal of Eco nom ics 95, 99-111.Thornton , H. , 1978, An inquiry into the nature and eR ects of the paper credit of Great Britain(1802) (Augustus M. Kel ly, N ew York).

bordo a model of the classical gold standard

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