A model of the classical gold standard with depletion

Journal of Monetary Economics 16 (1985) 109-120. North-Holland

A MODEL

OF THE CLASSICAL GOLD STANDARD DEPLETION

WITH

Michael David BORDO and Richard Wayne ELLSON* University

of South

Carolina,

Columbia,

SC 29208,

USA

The operation and properties of the classical gold standard are well recognized. However, one aspect that has not been dealt with is that gold has the characteristics of a durable, but depletable resource. In this paper, we compare the simple classical model of the gold standard with a model of the gold standard that incorporates the durable, depletable nature of gold. Using numerical simulation techniques, we demonstrate an inescapable tendency to long-run deflation when account is taken of the resource constraint. These results are consistent, with and without technological progress and variable real rates of return.

1. Introduction

Recent dissatisfaction with high rates of inflation and real economic instability in the U.S. and elsewhere has led to criticism of the operation of the present fiat based monetary system. Some economists have advocated a return to the classical gold standard, based on a government maintained fixed price of gold in terms of the national currency, on the grounds that the gold standard would provide greater price stability than under current arrangements.’ Indeed, such interest led to the establishment of the U.S. Congressional Gold Commission in 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 by competitive market forces according to the classical commodity theory of money largely independent of government policy. The classical tradition of Thornton (1802), Mill (18654 Fisher (1922) and Friedman (1953) viewed the monetary gold stock and hence the money supply and the price level under the gold standard as determined by two offsetting sets of equilibrating forces producing a tendency to long-run price stability: the response of gold produc*The first author is also affiliated with the National Bureau of Economic Research, Cambridge, MA 02138. For helpful comments and suggestions, we would like to thank the following: John Chilton, Mike Connolly, Stephen Ferris, Milton Friedman Levis Kochin. John McDermott, Blame Roberts, Charlie Stuart, Anna Schwartz, and an anonymous referee. ’ For some historical evidence on the record of price level and real output stability of the gold standard for the U.S. and the U.K., see Bordo (1981). ‘For a discussion of the deliberations and conclusions of the Gold Commission, see Schwartz (1982). 0304-3923/85/$3.3001985,

Elsevier Science Publishers B.V. (North-Holland)

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gold standard

model

tion to changes in the real price of gold, and shifts between monetary and non-monetary uses of gold by households and firms in response to changes in the real price of gold.3 In a recent article, Barro (1979) has provided a lucid formal exposition of the operation of the classical gold standard. One important aspect of the operation of a commodity standard such as the gold standard not treated by Barro or elsewhere in the literature is that of gold as a durable, but depletable resource. This view takes above ground gold as a commodity that depreciates at a very slow rate and incorporates the potential exhaustion of gold mines. In the literature on exhaustible resources following Hotelling (1931) the long-run growth 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.4 Indeed, the long-run behavior of the monetary gold stock and the price level when depletion of below ground stocks of gold is accounted for will differ from that suggested by the classical model. In this paper we combine the treatment of gold as a durable depletable resource, following the recent approach taken by 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 gold standard and a model of the gold standard accounting for the durable depletable resource aspect of gold are: an inescapable tendency to long-run deflation when account is taken of the resource constraint, and a tendency for the equilibrating mechanism of the classical gold standard to be muted by the operation of the resource constraint. Our approach in section 2 is to construct a simple model of a closed economy gold standard accounting for the resource constraint. In addition we incorporate technological progress and a variable real rate of return into the model. In section 3 the model is then parameterized and simulated to generate hypothetical paths for its key endogenous variables: gold production, the growth of non-monetary gold demand, the monetary gold stock, the money supply, and the price level. Comparisons are then made between the performance of the model under both classical gold standard and resource model assumptions. Section 4 contains a brief conclusion. 2. The model 2.1. The classical model

We start with Barro’s treatment of the classical model for a closed economy adapting it slightly to account for a variable real rate of interest and to allow 3For a discussion of the traditional approach to the Gold Standard, see Bordo (1984). 41n the presence of rising costs and/or monopoly, the growth rate of net rent should equal the real interest rate. This holds for a non-durable resource with constant demand. In the presence of durability and growing demand the price path may differ from Hotelling’s rule. See Pindyck (1978) and Stewart (1980) for example.

M.D.

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111

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 money multiplier - the ratio of the sum of currency and deposits to the value of the monetary gold stock (it is the product of both the ratio of the money supply to the monetary base and the monetary base to the value of the monetary gold stock), PC the fixed nominal price per ounce of gold, and G, the monetary gold stock in ounces. Eq. (2) represents the income velocity of circulation. We assume it is a logarithmic function of the nominal interest rate: 5 V= VP.

Following

(2)

Fisher (1930) we define the nominal interest rate as

i=r+lr

2

(3)

where r represents the real rate of interest and rr the expected rate of change in the price level. Following Mundell(1970), we assume the real rate of interest to 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

(5)

in the money market requires that

P = W&,/Y,

(6)

given p, PG,the assumption of perfect foresight, and assuming a constant level of 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. These equations in combination with (1) to (6) determine a unique equilibrium 5 We depart from Barro who, in the text of his paper, assumes a constant real rate of return and makes the demand for money a function of the expected rate of change in the price level. But following Barro, eq. (2) assumes the real income and price elasticities of real money demand to be one. 6 We also tried an adaptive expectations scheme. As expected the adjustment path of the model differed under the two schemes of generating expectations, but the long-run equilibrium values of the endogenous variables were not affected.

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money supply and price level. We assume that gold production is characterized by 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 time trend to allow for exogenous technological progress.’ The demand for nonmonetary 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 the target or desired stock of non-monetary gold. G$ is defined as

The parameter E is a partial adjustment factor, 6 the depreciation rate or normal replacement flow, and G, represents the actual stock of non-monetary gold. Finally, on the assumption that the monetary authorities are committed to maintaining a fixed price of gold, the change in the monetary gold stock is simply 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 production at any point in time is equal to the depreciation rate multiplied by the desired non-monetary gold stock. 2.2. The resource model

The real sector of the classical gold model described above assumes that gold production, g, is a function of the real price of gold and exogenous technical change. However, if we treat gold as a durable finite resource, then gold production would not only be affected by the real price of gold but also by the cost characteristics of production. ‘Barr0 does not explicitly account for technological progress in his model but discusses the implications of accounting for it. We assume technological progress is exogenous to simplify the discussions. However, there is evidence that major technological changes in the gold industry were both induced and exogenous [Rockoff (1984)].

M.D.

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The simple Hotelling rule (1931) states that the (real) price of an exhaustible resource should rise at the rate r (the real interest rate), under conditions of certainty and zero marginal costs. As Levhari and Pindyck (1981) point out, the Hotelling rule properly defined implies that the resource rent (price minus marginal cost) increases by r, assuming perfect competition and certainty. This is an important distinction. In the classical model gold production is directly related 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 costs decline 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, where G, and G, represent non-monetary and monetary stocks, respectively. We have assumed a Cobb-Douglas specification for total costs accounting for both production and depletion of the resource: C = Ag”X- “,

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

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

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Bordo and R. W. Ellson,

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3. Parameters of the models and comparisons 3. I. Parameters

The initial values of the parameters in the model are given in panel A of table 1. The depreciation of gold, 6, is assumed to be 1.0 percent and the adjustment parameter, E, in the e, equation is 0.5. We further assume that the money multiplier, ~1,has a constant value of 10 over the simulation period, and that the rate of economic growth and the rate of. technological progress are exogenous and equal to 3.0 percent per time period. Finally, we assume the autonomous growth rate in desired demand for the resource to equal the real rate of interest. Thus, producers have no incentive to withhold or expand production based on a differential here. The remaining parameters are basically consistent with estimates found in the literature and were selected to correspond to the start values for the endogenous variables that are listed below in panel B. Table 1 Parameters and initial values of the model. (A) Parameters P= 1.1 a = 0.5 a = 0.1 g = 1.6 /3 = 0.6 GN = 0.03 8 = 1.2 lj = 1.0 8 = 0.1 F = 0.03 CT= 1.3 p = 1.75 Y = 0.02 (B) Initial

Values

PC = 20; fixed nominal price of gold, dollars per P = 20; real price of gold, dollars per ounce.

By = MS = G, = G,,,= g= X= r= Y= i=

ounce.

100; price index. 270; real output, $billions. 90; money supply, $billions. 550; non-monetary gold stock, millions of ounces. 450; monetary gold stock, millions of ounces. 30; gold production first period, millions of ounces. 1000; remaining stock, millions of ounces. 0.03; real interest rate. 3; velocity. 0.03; nominal interest rate.

M.D.

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Both the classical model and the integrated model were simulated over twenty-five time periods using the parameters discussed above and the start values. The initial values of the gold variables are hypothetical and were chosen for analytical convenience. However, they reasonably correspond to estimates for the world in the late 192O’s.s 3.2. Comparison of the models We now compare the performance of the classical model relative to the integrated model for the key endogenous variables. Table 2 represents the values 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 monetary gold stock, and the money supply and the price level. Four experiments are reported: (A) the two models assuming constant real interest rates, no expected change in the price level and the absence of technological change - the benchmark case (and the closest comparison with Barro’s model); (B) the two models assuming constant real interest rates, no expected change in the price level, with technological change at three percent per period; (C) the two models assuming variable real interest rates, expected change in the price level and no technological change; and (D) the two models with variable real interest rates, expected change in the price level and technological change at three percent. Column 1 shows the pattern of gold production, g. In the benchmark case A gold production rises over the whole period in the Classical Model. This reflects the influence of the assumed real growth of three percent per period creating an excess demand for money, which in turn produces deflation, raising the real price of gold and encouraging production. By contrast in the integrated model, production initially declines slightly because of increasing costs and declining resource rents. Introducing technological change in case B virtually doubles gold production in the Classical Model, whereas by contrast, in the resource model production increases by a much smaller amount reflecting the effects of depletion of the resource. Introducing a variable real rate of return and expected deflation produces virtually no change in the Classical Model,g but in the resource model production is increased relative to the *See the Statistical Compendium to the Report Gold in the Domestic

and International

Monetary

to the Congress of the Commission Systems, Vol. 1 (1982).

on rhe Role of

‘This result reflects the effect of our assumed interest elasticity of velocity (0.5). interest elasticity of nonmonetary gold demand (- 0.1) and a low rate of deflation.

32.0 33.1 34.4

31.8

10

15 20 25

Mean

Mean

15 20 25

10

5

1

30.9 35.0 40.9 47.7 55.8 65.1

(a)

46.0

(a)

19.8

16.5 18.4 19.2 20.1 21.2 22.4

(b)

28.2

16.5 22.9 27.4 30.5 32.8 34.6

(a)

606.6

463.5 511.9 571.1 630.2 689.6 749.6

(b)

510.2

463.5 499.2 519.3 522.3 517.3 511.6

(a)

121.3

92.7 102.4 114.2 126.0 137.9 149.9

(b)

102.0

g

(1)

35.0

30.1 30.5 32.0 34.9 39.1 45.0

(b)

27.8

16.8 20.8 24.6 29.0 34.2 40.4

(a)

(2)

30.2

16.5 23.0 27.9 32.2 36.5 41.3

(b)

660.1

464.0 519.0 595.9 684.5 186.4 903.5

(a)

(3)

530.8

463.6 500.9 527.5 542.9 555.6 571.6

(b)

180.7 132.1

92.7 103.8 119.2 136.9 157.3

(4

110.1 114.3 106.2

92.7 100.2 105.5 108.6

96.0 87.2 77.4 68.3 60.1

99.5 98.5 97.6 96.8 95.9

81.2

100

100

98.0

(b)

(a)

P

MS (b)

(5)

(4)

78.6

95.7 85.9 14.5 63.6 54.3

98.1 94.4 89.9 84.9 19.6 91.1

(b)

100

(a)

100

P

92.7 99.8 103.9 104.5 103.5 102.3

(3 -

(4)

MS

(B) Assuming a constant real rate of interest, no expected change in the price level and exogenous technological change

30.1

31.0

1

5

(b)

30.0 29.8 29.7 30.2 31.5 33.7

g

(4

6”

30.0 30.3

(3)

GM

(2)

(1)

(A) Assuming a constant real rate of interest, no expected change in the price level, no technical change

Table 2 Comparisons of (a) the Classical Model with (b) the integrated resource model.

(4

30.9

35.0 40.9 47.1 55.8 65.1

46.0

1

Mean

(a)

31.8

30.0 30.4 31.1 32.0 33.2 34.5

5 10 15 20 25

Mean

5 10 15 20 25

1

(b)

(4

35.8

30.7 32.5 35.7 46.4 46.9

30.1

27.8

20.8 24.6 29.0 34.2 40.4

16.9

-

G

(b)

27.3

29.6

16.5 22.4 26.9 31.3 36.0 41.6

(b)

(a)

661.0

GM

(3)

GM

525.5

540.1 544.7 547.2

463.5 501.1 527.8

0))

545.0

463.6 502.6 535.3 559.3 581.0 605.2

(b)

(a)

121.6

126.3 138.3 150.4

92.7 102.5 114.4

132.2

92.8 103.8 119.2 137.0 157.4 180.8

(a)

change

(a)

109.0

98.0

99.4 98.5 97.6 96.1 95.9

100.2

P

92.7 108.5 107.1 111.9 116.2 121.1

(5) MS (a)

90.9

(4)

(b)

technical

105.1

108.0 108.9 109.4

100 98.0 94.3 89.7 84.6 79.4

P

92.7 100.2 105.6

(5)

MS @I

change

(4)

level, no technical

level and exogenous

in the price

in the price

change (3)

change

463.9 519.1 596.2 684.8 786.7 903.9

expected

608.1

631.7 691.6 752.1

29.2 31.8 34.1

(a) 463.5 512.5 572.2

(b)

expected

2 (continued)

16.5 22.3 26.2

real rate of interest,

g

a variable

19.8

(2)

Assuming

(4

16.5 18.4 19.2 20.1 21.1 22.3

(1)

(D)

31.2

30.0 30.0 30.2 30.9 32.4 34.6

Gv

real rate of interest,

8

a variable

(2)

Assuming

(1)

(C)

Table

82.4

95.8 87.7 79.0 JO.8 63.7

100

(b)

79.9

100 95.5 86.4 76.2 66.2 57.4

(b)

> i$

; s 5. 5 m 8 3 E 9

$ 3 B 2 k 9 gw

5 6

118

hi. D. Bordo

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Clussicul

gold srutuhrd

model

benchmark case because of the additional, expected deflation which raises the real interest rate, and hence via Hotelling’s rule raises production. Column 2 shows GN, the net change in non-monetary demand for gold. In both models it is a function of real output and the real price of gold. We would expect G, to rise reflecting the direct effect of real growth but to decline reflecting the indirect effect of deflation producing substitution of monetary for non-monetary gold stocks. In the benchmark case A, GN rises above its corresponding values in the Classical Model because of the greater rise in the real price which in turn reflects the decline in production in that model as real rents decline. Technological change, case B, raises G;, in both models via increased gold production raising the monetary gold stock, the money supply, and the price level, and lowering the real price of gold, in turn encouraging substitution of monetary for non-monetary gold. This effect is considerably weaker in the integrated model because of the greater deflationary effect associated with the resource constraint. Finally, the assumptions of a variable real rate of interest and expected deflation have virtually no effect in the Classical Model but in the resource model they slightly reduce GN. This reflects the 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 benchmark case A rises at a much slower rate in the integrated model than in the Classical Model reflecting the forces described above. In addition technological change produces a smaller rise in G, in the integrated model than in the Classical Model. Introducing a variable real rate and expected deflation has virtually no effect on G, in the Classical Model, while it raises G, in the integrated model reflecting the effect of a rising real interest rate on gold production. Column 4 shows M”, the money supply whose movement is governed by and reflects the behavior of G,. Finally column 5 portrays movements in the price level. In the benchmark case, deflation prevails over the entire period in both models. However, deflation is much greater in the integrated model because of the decrease in gold production over the period. Thus, the offsetting influences to deflation of the Classical Model through the substitution between monetary and nonmonetary gold holding and the effects of a rising real price of gold on production are completely swamped when we account for the resource elfects. Furthermore, introducing technological progress at the same rate as the underlying growth rate of the economy (case B) almost fully offsets the deflation in the Classical Model - restoring price stability. However, this is definitely not the case in the resource model where deflation prevails. Moreover, 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 key parameters of the models (a, p, Y, a, a, e and @) are varied. Particularly

M.D.

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Classical

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119

important are a, a and @, for which ranges are suggested by the literature.‘O In addition, raising the rate of technological change from three percent to five percent per period reduces the rate of deflation in the resource model by a relatively 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 towards long-run deflation, a tendency which is not overcome by technological change or by a variable real rate of interest and expected deflation. Second the equilibrating mechanism of the Classical Model towards the long run equilibrium price path is muted by the operation of the resource constraint. These conclusions have important implications. The greater tendency towards long-run deflation suggests that the likelihood of gold discoveries and technological advances in gold production being sufficient to offset the tendency towards deflation are even more remote than would be suggested by the Classical Model in the presence of real growth. On the other hand, the rate of deflation which satisfies Hotelling’s rule would, if perfectly anticipated, also satisfy Friedman’s (1969) optimum quantity of money rule - giving the community the satisfaction level of real cash ba1ances.t’ References Barro. R., 1979, Money and the price level under the gold standard, Economic Journal 89,13-33. Bordo, M.D., 1981, The classical gold standard: Some lessons for today, Federal Reserve Bank of St. Louis Review 63. 2-17. Bordo, M.D., 1984, The gold standard: The traditional approach, in: M.D. Bordo and A.J. Schwartz, eds., A retrospective on the classical gold standard 1821-1931 (University of Chicago Press, Chicago,‘IL). Fisher, I., 1965. The purchasing power of money, 2nd cd. (Augustus M. Kelly, New York). Friedman, M., 1953, Commodity reserve currency, in: Essays in positive economics (University of Chicago Press, Chicago, IL). Friedman, M., 1969, The optimum quantity of money, in: Optimum quantity of money and other essays (Aldine, Chicago, IL). Hotelling, H., 1931. The economics of exhaustible resources, Journal of Political Economy 39, 137-175. Levhari, D. and R. Pindyck, 1981, The pricing of durable exhaustible resources, Quarterly Journal of Economics 96, 365-377. Mill, J.S., 1962. Principles of political economy (1865) (Augustus M. Kelly, New York). Pindyck, R.S.. 1978, The optimal exploration and production of nonrenewable resources, Journal pf Political Economy 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. The results of the sensitivity anafysis are available upon request from the authors. “See e.g., RockotT (1984, pp. 619-620).

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Report to the Congress of the Commission on the Role of Gold in the Domestic and International Monetary Systems, Vol. 1 (1982). RockotT, H., 1984, Some evidence on the real price of gold, its costs of production, and commodity prices, in: M.D. Bordo and A.J. Schwartz, eds., A retrospective on the classical gold standard 1821-1931 (University of Chicago Press, Chicago, IL). Schwartz, A.J., 1982, Reflections on the Gold Commission report, Journal of Money Credit and Banking 4. 538-551. Stewart, M.B., 1980, Monopoly and the international production of a durable extractable resource, Quarterly Journal of Economics 95, 99-111. Thornton, H., 1978, An inquiry into the nature and eRects of the paper credit of Great Britain (1802) (Augustus M. Kelly, New York).