The demand for raw materials can also be treated as derived from the demand and supply conditions of the final commodities in the production of which the raw materials are used. Assume that the raw material is used in the production of one final good only, and that one unit of raw material and one unit of labor are used in the production of one unit of final good. Then if the only cost factors considered are raw materials and labor, it follows that:
where the p, p1, and w are the per unit cost in terms of dollars of the final good, raw material, and labor, respectively.
If in each area wages in terms of the currency of that area are Constant, the change in p due to devaluation is for the dollar area
and for the sterling area
The difference between (2.2) and (2.3) measures the advantage to the sterling area arising from the lower processing costs in terms of dollars after the devaluation. Formula (2.3) can be explained as follows. If w2 is the per unit labor cost in the sterling area, in terms of sterling, then this labor cost in terms of dollars is
from which it follows, since w2 is constant, that
Further, if ηS = elasticity of supply of final goods, and ηD = elasticity of demand for final goods (absolute value), it can be proved that
where the e’s, as previously defined, refer to the raw material and N is the number of dollars of raw material used per dollar of final commodity. Let the supply functions of the final good and the raw material be respectively
where qf and qr refer to quantities of finished goods and raw materials respectively. Since
because of the definition of the units, (2.4) can be proved by substituting (2.5) and (2.6) in (2.1), in which w is independent of q, and by taking the derivative with respect to q. The proof for the demand elasticities is similar.
If it is assumed that the two markets for final commodities are isolated from each other, then for the dollar area before devaluation
After devaluation the supply and demand curves are assumed to have the same elasticities but to shift by factors ζi and εi respectively, in the i-th area, so that for the dollar area
where the primes denote the value of the variable after devaluation. The dollar area’s demand curve does not shift at all, so that ζ1 = 0, but the supply curve is shifted by
because (2.2) implies that for any quantity supplied the corresponding price will be reduced by the reduction of the price for the raw material component of the finished good. It follows that
If (2.4) is used and derivatives of the second order and higher are neglected, it follows that
For the sterling area it follows, similarly, that
from (2.3) and (1.4)
From the definition of the units of quantities of raw materials and final goods, it follows that
The assumption is made that the markets for raw materials are not isolated, and that the equilibrium condition for these markets, following from (1.1) and (1.2), is
Then, as in the derivation of the formula (1.9) above, the left-hand member of this equation can be written
If (2.7), (2.8), and (2.9) are substituted for the right-hand member, then
Thus treating the demand for raw materials as a derived demand leads to an expression similar to (1.3) except that the factor
When (2.10) is solved for
which corresponds to (1.8).
When c = 0, (2.11) reduces to
which corresponds to the simplified formula (1.9).
It can be seen that (1.9) is a reasonable approximation of (2.11) and (2.12) for values of KS and KD satisfying –¼<KS–KD<¼.
Finally, the possibility may be taken into account that, contrary to the previous assumption, the elasticity of substitution for some raw materials as between the sterling area and the dollar area may be finite. In that case, the model can be changed as follows:
Pi.k = price of raw material produced in area i and consumed in area k in terms of the currency of area k;
Di = total demand for raw material produced in area i;
p= elasticity of substitution.
The definition of the elasticity of substitution implies that
The equilibrium conditions are
The supply equations are
The markets for each kind of raw material are not assumed to be isolated so that
If the demand for raw materials produced in the dollar and sterling areas, taken together, is fixed, then
From these equations it can be shown that
or if eS2 = eS1 = eS
This formula expresses the relative rise in sterling prices of raw materials produced in the sterling area if the elasticity of substitution is finite. This is most clearly the case with cotton, where substitution is limited by differences in length of staple. The elasticity of substitution between Indian and U.S. cotton has been estimated at –2.9 and that between Egyptian and Indian cotton at –1.1.1
If it is assumed in (2.14) that the ratio of the supply and substitution elasticities,
Mr. B. A. de Vries, economist in the Trade and Payments Division, Exchange Restrictions Department, was educated at the University of Utrecht and the University of Chicago, and was formerly a member of the staff of the Cowles Commission,
This study benefited considerably from the advice of Messrs. J. J. Polak and Sidney S. Alexander of the Research Department of the Fund.
The sterling and dollar prices before devaluation are presumed to be equivalent, at the rate of $4.03; after the rise in the sterling price and the fall in the dollar price, the prices are again presumed to be equivalent at $2.80. Under these conditions, the fall in dollar price associated with any given rise in sterling price is one that equates the prices of a commodity in the two currencies at the new rate of exchange. Thus, a 30.5 per cent devaluation of sterling implies that a 44 per cent rise of the sterling price would be associated with an unchanged dollar price; a 22 per cent rise in the sterling price with an 18 per cent fall in the dollar market; and a 30 per cent rise in the sterling price with a 10 per cent fall in the dollar price (1.44 × 1.00 = 1.22 × 1.18 = 1.30 × 1.10 = 1.44).
Defined to be positive if, at the reduced price, more is demanded than at the higher price.
An important exception might be rubber. In the computation of the effect of devaluation on the price of rubber (Table 1), synthetic rubber is included for the dollar area; synthetic rubber is produced under conditions which differ from those under which natural rubber is produced. On the demand side, commodities for which there are altogether different tastes in the two areas provide another exception.
J. Tinbergen, “Some Measurements of Elasticities of Substitution,” Review of Economics and Statistics, August, 1946.