THE DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR AND THE INDUSTRY SUPPLY CURVE By RICHARD F. MUTH students of economics are familiar with Marshall's analysis of the forces influencing the elasticity of the derived demand for a productive factor in the case of fixed proportions.1 Many have worked through the derivation of Hicks's now famous formula, which generalized Marshall's earlier analysis to the case of variable proportions.2 The purpose of this paper is to show how from an analysis similar to that of Hicks one can derive the elasticity of the industry supply schedule as well as the coefficients of other variables which appear in its supply schedule and in its factor demand schedules. In section I, I discuss the assumptions underlying my model and ita structure. In the next section the industry's factor demand schedule is derived and some of its features are explored, while section III considers the industry's supply schedule. Section IV discusses some applications of my analysis to problems in the field of urban land economics. Finally, in the final section I point out some other possible applications of the analysis.

MOST

I I assume that the industry whose supply and factor demand schedules are to be derived consists of a group of actual or potential producers of a single, homogeneous product. These firms have identical production functions3 and use two productive factors which are unspecialized to any firm but which may be specialized to the industry. There are no external technological effects, so that the production possibilities open to each firm are independent of the industry output. All firms are assumed to be competitive or 'price-takers' in both the product and factor markets. Under these conditions, the average cost curve is the same for each firm actually producing at any time. Furthermore, the postulated existence of potential producers identical with those actually producing implies that each firm in the industry operates at a minimum average cost. If the entrepreneurial capacity of the firm or some other fixed factor is not limitational, each firm operates in a region where its average cost curve is horizontal. (In this case, however, its ouput is indeterminate.) So long 1 Alfred Marshall, Principles of Economics, 8th ed. (New York: The Macmillan Co., 1950), pp. 382-7 and Mathematical Appendix, Note XV, p. 853. « John R. Hicks, The Theory of Wages (New York: Peter Smith, 1948), Appendix, pp. 241-6. • This assumption enables me to neglect the problem of aggregation.

222

DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR

as factor prices and technological conditions remain unchanged, increases in industry output may come about either through expansion in the size of existing firms or from the entry of new firms. If, however, each firm has a unique output for which average cost is a minimum, it must operate at that output, and increases in the industry's output are brought about solely by the entry into the industry of new firms. Each of the latter produce the same output and use the same quantities of the two productive factors as those firms previously producing.1 In any case, under the assumptions made here I can express the industry output, Q, as depending only upon the inputs of the two factors used by the industry, A and B, and certain parameters describing the existing state of technology. Furthermore, Q is homogeneous of degree one; doubling the amounts of both factors used by the industry doubles the industry output, regardless of whether the expansion in output results from increases in the size of existing firms or from the entry of new firms into the industry. 2 Since each firm in the industry operates in a region where its production function is homogeneous of degree one, the marginal products of the two factore—which, in any case, must be the same for all firms—depend only upon the relative amounts of the two factors used by a firm. Finally, since it is assumed that all firms are competitive in both product and factor markets, the price per unit paid each productive factor is equal to the value of its marginal product. The equilibrium of the industry may be described by the following set of equations: Q=f(p), (1) Q = Q(A,B), (2) = A = g(pA),

(5)

B = h(ps).

(6)

PB

For simplicity I have not shown the dependence of these equations upon any exogenous variables explicitly. The effect of changes in any such variables will be treated as a shift in one or more of these functions. The endogenous variables of the model are industry output, Q; the amounts of the two factors used by the industry, A and B; the price per unit of 1 Of course, changes in factor prices or in technology may lead to changes in the output of a single firm and the factor inputs it uses. 1 Equation (2), below, can be considered to be the industry production function, since it gives the maximum output the industry can produce given the quantity of the inputs A and B used by the industry. This follows because under competition the marginal physical product of either factor is the same for all firms. Note that under the assumptions I have made the industry production function exhibits constant returns to scale, regardless of whether those of the individual firms do.

R. F. MUTH

223

the final product, p; and factor prices, pA and pB. Equation (1) is the demand schedule for the industry's output, while equations (5) and (6) are the factor supply schedules facing the industry. Equation (2) expresses the fact that industry output depends only upon the amounts of the two factors used by the industry, as noted above. Equations (3) and (4) expressthe fact that both factors are paid the value of their marginal products. I now consider displacements from the initial equilibrium which result from shifts in one or more of equations (1) through (6). Along a given demand schedule, the relative change in Q, or dQ*, which is the logarithmic differential of Q, is equal to the elasticity of the industry demand schedule, i), multiplied by the relative change in price, dp*. I find it convenient, however, to express shifts in equations (1), (5), and (6) in the direction oi the price axis, so where a is the relative increase in price at any given quantity on the new demand schedule, equation (1) becomes in differential form:1 , --dQ*+dp* = a. (1') V Similarly, where /? and y stand for shifts in the supply of factors A and B to the industry, and where eA and eB are the elasticities of these supply schedules, equations (5) and (6) can be rewritten as: dA*+d^

j8,

(5')

y.

(6')

Since I am measuring the shifts in the factor supply schedules in the direction of the price axis, a negative shift parameter denotes an increase in the supply of the factor. As long as equation (2) does not shift: dQ = QAdA + QBdB, or,

dQ*

= ^dA*

+ B^*dB*.

(7)

The partial derivatives of Q with respect to either factor, QA and QB, are simply the common marginal products of these factors for all firms. For, to the extent that increases in industry output result from the expansion of output by existing firms along horizontal segments of their average cost 1

If the demand schedule shifts because of an increase in consumer income, for example,

V where y denotes consumer income and E(Q, y) is the income elasticity of demand.

224 DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR curves, the change in the output of a firm is simply the marginal product of a factor multiplied by the increase in the amount used summed over both factors. To the extent that increases in the industry's output result from entry of new firms into the industry at a minimum average cost firm output, their contribution to incremental output can be expressed in the same way. The coefficient of dA* in the second of equations (7) is:

where kA is the proportion of total receipts for the industry or any firm which are imputed to factor A, or simply A'B share, and similarly for the coefficient of dB*. Shifts in equations (2) through (4) result from technological changes. It is convenient to express any given technological change as the sum of two separate kinds, a neutral technological change and a '5-saving' change. A neutral technological change is one which simply renumbers the isoquants of a firm's production function; it increases the marginal physical products of both factors proportionally, regardless of the relative proportions employed before the change. I define a iJ-saving technological change as one which increases the marginal physical product of A relative to that of B, but which leaves total output unchanged for the inputs of the two factors which were used prior to the change. Now, since (2) is homogeneous of degree one it can be written as: Q = AQA+BQB,

(9)

and, holding the inputs A and B constant: dQ = or,

AdQA+BdQB,

dQ* = kAdQ*+kBdQB.

(10)

If the marginal physical products of both factors increase by the same relative amount 8, then: dQ* = (kA+kB)8 = 8. (11) Also, for a jB-saving technological change which increases A'B marginal physical product by an amount e in relative terms:

or,

dQ%=-^-e.

(12)

KB

I can therefore rewrite (2) in differential form as: dQ+-kAdA+-kBdB* = 8. (2') Consider, now, equations (3) and (4). Given the level of technology, the relative change in, say, pA is equal to the relative change in product price

R. F. MUTH

225

plus the relative change in the marginal physical product of A. The latter dQ* = ^ddA* + B^dB*. VA

(13)

VA

Since each firm operates where its production function is homogeneous of degree one in A and B: QAA=-%QAB

and

a= § ^ ,

(14)

where a is the elasticity of substitution of a firm's production function. Making use of (14), equation (13) then becomes: dQ* =

-BQB^g-dA*+BQB^-dB* VA WJS

VA WB

+ a

(15) a

Adding in the shift in (3) which results from a change in technology: -dp* + hdA*-^dB*+dpS = 8+e, a

(3')

a

and similarly for equation (4):

=8 a

a

~re-

(4/)

KB

The system of equations (1') through (6') could be solved for all of the endogenous variables, obtaining reduced-form equations for each of them in terms of the shifts in the functions and the parameters 77, a, eA, eB, kA, andi B . (These reduced-forms are exhibited in the Appendix.) If, however, one treats pB, say, as exogenous and solves equations (1') through (5') for dB* one obtains the industry demand schedule for factor B. Alternatively, if one treats p as exogenous and solves equations (2') through (6') for dQ* one obtains the supply schedule of the industry. By so doing, one obtains not only the elasticity of the factor demand or output supply schedule but also the coefficients of all the shift parameters. I shall first consider the industry factor demand schedule in section II, below, and then the industry output supply schedule in section III. II There are various ways in which one could solve the system (1') through (5') for dB* when treating pB as exogenous. I have found that it is probably simplest first to eliminate dQ* using (1') and (2') and dpA using 4620.2

Q

226

DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR

(3') and (5'); doing so one obtains three equations in dp*, dA*, and dB*. Eliminating successively dp* and dA* one then obtains:

, jkfr+qtefc | l-(*+eA)(l+V))s | M-y+eA)){-kA \ (16) where

D =

hBa—kAT)-{-eA.

The common denominator of these coefficients, D, is positive under any empirically reasonable assumptions, namely that a > 0, TJ < 0, and eA ^ 0. On these same assumptions, the coefficient of dpB—which is, of course, the elasticity of the derived demand for factor B—is negative.1 Likewise, it is easily seen that an increase in product demand increases the derived demand for factor B, while a ^-saving technological change (e > 0) tends to reduce it. An increase in the supply of factor A {fi < 0) may either increase or decrease the demand for factor B, however, depending upon whether the elasticity of demand for the product is numerically larger or smaller than the elasticity of substitution in production.2 This is because an increase in the supply of factor A not only reduces its price and, hence, leads to a substitution of factor A for factor B, but also leads to a downward shift in the marginal and average cost curves of all firms and an increase in the supply of the product. The latter, of course, leads to a fall in price and to an expansion of industry output. The substitution effect of an increase in the supply of factor A outweighs the output-expansion effect on the demand for factor B if a exceeds — TJ. Finally, a neutral technological improvement may either increase or reduce the demand for a factor according to whether the demand for the industry's output is elastio or inelastic. While such a technological change reduces the input of a factor necessary to obtain any given output, it also reduces the marginal and average cost curves of all firms and increases the supply of the final product. If the industry demand is elastic the resulting 1 This elasticity can easily be rewritten in the form obtained by Hicks, ibid., p. 244, if one recalls that he defines, ij as the negative of the elasticity of product demand. 1 Cf. John R. Hicks's comment on M. Bronfenbrenner, 'Notes on the Elasticity of Derived Demand', Oxford Economic Papers, 13 (Oct. 1961). Hicks uses this fact in explaining why the elasticity of a factor demand curve increases with an increase in its relative importance only if the elasticity of demand is numerically larger than the elasticity of substitution. Note that when one considers the case in which eA tends to zero it is helpful to rewrite

the third term in (16) as —A\ a +1) ^

w here

/}' = — eAfi measures the shift in the supply

carve in the direction of the quantity axis. One then sees that as eA —>• 0 the statement in the text remains valid.

R. F. MUTH 227 expansion of output more than counteracts increased productivity insofar as its effects on factor demand curves are concerned. While the coefficients of (16) are rather complicated functions of the given parameters, they can be simplified considerably for the purposes of some problems. In aggregate models, for example, it is sometimes convenient to treat all firms in the economy as producing a single, homogeneous product. Also, some industries in some regions or countries produce only a small part of the total output sold in a national or world market. In such cases 77 ->- — 00 and (16) becomes:

(17) In (17) an increase in the supply of the other factor or a neutral technological improvement necessarily increases the demand for factor B. In other cases, all factors but one may be highly mobile as among industries, at least if sufficient time is allowed for adjustment. It seems quite likely, for example, in dealing with the real estate market in a given metropolitan area that labour and non-land capital are highly mobile as among areas in the long run. Land, however, while fixed in location has a positive elasticity of supply for urban uses since shifts between urban and rural uses are possible. Letting eA, where A refers to non-land inputs, in (16) go to +°o we have: dB* = (-kA+BV)p%q+{A(+])}p^(

+ ]) +

^ (18)

III I shall now consider the supply curve of the industry's output. As was noted earlier, by treating price as an exogenous variable and solving equations (2') through (6') the latter is obtained. These equations can be solved by eliminating dp* and dp*B from (3') and (5') and from (4') and (6'), respectively, and. then solving for dA* and dB*. Substituting for dA* and dB* in (2'), dQ* is then obtained:

• I—kB(er+eA)eB\

1 Ml+^fr+fc J ? e B )+fc i ? 3 i -r-^e i H-^e J ? l 8

|

(19) where

D' =

a-\-kBeA-\-kAeB.-

228

DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR

As before, it seems most reasonable to assume that a is positive and that the elasticities of factor supply are non-negative; the common denominator of the coefficients of (19) is therefore positive. On these same assumptions the elasticity of the industry supply curve is non-negative, and an increase in the supply of either factor or a neutral technological improvement will increase the supply of the product. A .B-saving technological change, however, will increase or decrease the industry supply depending upon whether the elasticity of supply of factor A to the industry is greater than or less than that of factor B. Examining the variation of the elasticity of the industry supply schedule with the elasticities of substitution and factor supply and with the relative importance of the two factors reveals no paradoxes comparable to that uncovered by Hicks.1 Differentiating one finds, where E(QS, P) is the elasticity of the industry supply schedule: 8E(QS,P) ks(a+eA)2 , . ., , , — 8g = D'i ' similarly for eA, 8E(QS,P) (a+eA)(o+eB)(es-eA) 8kB D'2 (

0)

Not surprisingly, the elasticity of supply of the industry's product increases with the elasticity of supply of either factor. The second of equations (20) indicates that E(QS, P) is greater the greater the relative importance of the factor whose supply is the more elastic. Finally, E( Q8, P) is greater the greater the substitutability of the two factors in production if the factor supply elasticities to the industry are different. If the two factor supply elasticities are the same there would be no incentive for any firm to substitute one factor for the other as industry output increases. Equation (19) simplifies considerably in the case where the supply of one of the factors, say A, is infinitely elastic to the industry. Taking the limit as eA goes to infinity I have:

£.. „„ As in the preceding section, the assumption that the supply of non-land factors is infinitely elastic might be appropriate for analysing long-run changes in the supply of real estate in a single metropolitan area. Or, 1 The Theory of Wagu, pp. 246-6. I refer to the fact that the derived factor demand schedule becomes less elastic as the relative importance of the factor declines only if the elasticity of final demand is numerically greater than the elasticity of substitution in production.

R. F. MUTH

229

regardless of the length of time allowed for adjustment, an infinitely elastio supply of non-land factors would seem appropriate for analysing differences in the elasticity of supply of real estate in different parts of a metropolitan area. IV I now wish to consider some applications of the analysis developed above to problems from my own field of special interest, housing and urban land economics. Using equations (18) and (21) above, I shall first present estimates of the elasticity of demand for land for the industry providing housing and this industry's supply elasticity. I shall then discuss several problems for which knowledge of these elasticities is helpful in providing solutions. These problems illustrate quite well, I think, how useful the analysis developed here can be in practical applications. Of course, it might find applications in many other specialized field of economics. As I have argued earner, for many problems in urban land economics it seems reasonable to assume that the supply elasticity of non-land factor inputs is infinite. To apply the formulas developed here, however, one needs information on the demand elasticity for housing, the elasticity of substitution of land for other inputs in producing housing, and on the relative importance of land in the production of housing. Previous work by Margaret Reid and by me would indicate that the elasticity of housing demand is about equal to —I. 1 Data published by the U.S. Housing and Home Finance Agency suggests that the elasticity of substitution is of the order of 0-75. From 1946 to 1960 the proportion of site to total property value for new FHA insured dwellings rose from 11-5 to 16-6 per cent.2 The Housing and Home Finance Agency also reports that 'prices of equivalent sites are more than 2-8 times their 1946 level [in I960]'.* Construction costs, however, went up only about 77 per cent over the same period.4 My inference about the elasticity of substitution is based 1 See Margaret G. Keid, Housing and Income (Chicago: University of Chicago Press, 1962), p. 381 and my "The Demand for Non-Farm Housing', Arnold C. Harberger, ed., The Demand for Durable Goods (Chicago: University of Chicago Press, 1960), p. 72. • Fourteenth Annual Report (Washington: U.S. Government Printing Office, 1961), Table HI-35, p. 110. • Ibid., p. 109. Site value is denned to include street improvements and utilities, rough grading, terracing, and retaining walls (p. 119). Since the costs of such capital improvements may easily be as much as the raw land value, unimproved land's factor share is of the order of 6 per cent. Furthermore, it is even smaller in considering housing services as contrasted with housing stocks since interest costs are only one-half the cost of providing housing services. 4 As measured by the U.S. Department of Commerce implicit residential non-farm construction deflator. The index for 1946 was computed from data in 1/.JS. Income and Output (Washington: U.S. Government Printing Office, 1958), Table I, pp. 118-rl9. The index for 1960 was obtained from 'National Income and Product in 1962', Survey of Current Business, 43 (July 1963), Table 6, 14.

230

DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR

upon the fact, demonstrated in the Appendix, that in the absence of jB-saving technological change dk*B = kJl-oWpBlpA)*.

(22)

The parameter values given in the above paragraph when substituted into equation (18) imply that the elasticity of demand for land on the part of the housing industry is slightly greater than —0-8. In addition to the parameter values in the paragraph above, one would need to know the elasticity of supply of land to the housing industry to determine the elasticity of supply of housing using equation (21). The latter, however, implies that the elasticity of supply of housing stocks in 1946 (1960) was about 5-8 (3-8) plus 8-7 (6-0) times the elasticity of land supply. Since the supply of land in all uses is perfectly inelastic, the elasticity of supply of land to the housing industry is the negative of the elasticity of demand for land in all other uses. If the elasticity of demand for land for uses other than housing were the same as for housing, for example, the elasticity of housing supply would have been 12-8 (8-6) in 1946 (1960). Not only is the elasticity of housing supply relatively high but it is also likely to be highly variable within a metropolitan area. Neglecting variability in the second term of the coefficient of dp* in (21), it can be readily shown that

dE(Q8,P) = (kAlkB)o(a-l)d(pBlpA)*.

(23)

Substituting the same values used above, the change in the elasticity of housing supply would be about —1-4 (—0-9) times the relative change in the ratio of land rents to construction costs in 1946 (1960). Assuming that construction costs are everywhere the same in a metropolitan area but that land rents vary on the order of ten times, the (absolute) variation in the elasticity of housing stock supply would be ten to fifteen, which is large relative to this elasticity's average value. Knowledge of the elasticities of the derived demand for urban land and the supply of housing have important implications for many problems in urban land economics. Urban land economists have long debated the effects of an improvement in urban transport on aggregate urban land values. An improvement in urban transport, in effect, increases the supply of urban land, and, as Herbert Mohring has recently pointed out, will increase or reduce aggregate urban land values depending upon whether the demand for urban land is elastio or inelastic.1 The discussion above indicates that the demand for land by the housing industry, the largest user of urban land, is inelastic. Furthermore, since land's share probably is small for other urban uses, the demand for land in these other uses 1

"Land Values and the Measurement of Highway Benefits', The Journal of Political Economy, lxix (June 1961), 243.

R. F. MUTH

231

depends primarily on the elasticity of substitution (equation (18)). In hia The Economics of Real Property, Ralph Turvey notes it is the opinion of experts in the real estate field that where the level of site values is particularly high—in the center of large towns— the capital value of a property, and hence alao ite annual value, though high in absolute terms, is particularly low in relation to the value of a site.1

Thus, it would appear that the elasticity of substitution is less than one for urban real property generally and that the demand for urban land is inelastic. It would seem, therefore, that an improvement in urban transport would reduce the aggregate of urban land values. The analysis developed in this paper is also helpful in explaining the fact that, at least in the past half-century or so in the U.S., as cities have grown in population their growth has tended to be relatively more rapid in their outer parts than in their inner zones. Part of the explanation for this fact, of course, lies in improvements in urban transport, especially those associated with the automobile and the express highway. However, in another paper I found that, even after taking account of differences in transport costs and other factors such as income and the spatial distribution of manufacturing and retail activities, there remains a strong negative partial correlation between the size of different metropolitan areas in the U.S. in 1950 and the rate at which population densities decline with distance from the city centre.2 An explanation may be found in the fact noted above that the elasticity of housing supply varies directly with distance from the city centre because land rents and land's share vary inversely with distance. As population and thus the demand for housing in a city grow, the output of housing tends to grow relatively more rapidly in the city's outer areas. The increase in population is therefore more rapid in the outer than in the inner part and the rate of decline in population densities with distance falls as population increases. The inverse variation of the elasticity of housing supply and distance from the city centre is helpful in explaining another phenomenon. Charles O. Hardy in his study The Housing Program of the City of Vienna noted that the housing standards of early twentieth-century Vienna were very low in comparison with those of many European cities and that Vienna was a very compact city with a residential section consisting mainly of apartment buildings.3 He attributed the city's poor housing quality to the 40 per cent tax on gross housing rentals, which was added to the taxes to which other industries as well were liable, but did not give a satisfactory 1 (London: George Allen & Unwin, Ltd., 1957), p. 86. • "The Spatial Structure of the Housing Market', Papers and Proceedings of the Regional Science Association, 7 (1961), 216. » (Washington: The Brookings Institution, 1934), pp. 1-2.

232

DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR

explanation for the compactness of the city.1 Given the elasticity of demand for housing, the smaller the elasticity of supply, the less will be the reduction in the output of housing resulting from the imposition of an excise tax such as the tax on housing rentals. But the elasticity of housing supply tends to be smallest near the city centre where land rents and land's share tend to be greatest. One would therefore expect a tax on the rental value of housing to result in the least reduction of housing output near the city centre and that the long run effect of the tax would be to make the city more compact. V In closing I would like to make a few comments about the analysis presented here: 1. From the reduced-form equations given in the Appendix it is relatively simple to derive certain other magnitudes in terms of the shifts in the functions (1) through (6). Thus, the relative change in the value of an industry's output, d(pQ)*, is simply dp*-{-dQ*, the relative change in total payments to factor B is dp%-\-dB*, and the relative change in factor B'B share is dp%-\-dB*—dp*—dQ*. (See the Appendix.) 2. Random variation can easily be introduced into the model. In fact, if we interpret the shift variables described earlier as random variables, then the reduced-forms in the Appendix can be used to derive the probability distributions of the endogenous variables in terms of those of the random shifts. 3. The model could easily be generalized to include specific consideration of two or more industries. For example, in the case of so-called 'raw material-oriented' industries,firmsat each material source could be treated as a separate industry and their supply schedules added to obtain the aggregate supply function for the commodity in question. Or, the demand schedules for land on the part of residential and the various non-residential users of land when added yield the aggregate demand schedule for urban land in a metropolitan area. 4. Finally, as illustrated in section IV above, the elasticity of supply or the elasticity of factor demand, as well as the coefficients of the shift parameters, can be derived from a knowledge of the k'a, -q, a, and the e's. The JC'B can often be determined readily on the basis of knowledge of the value of output and total factor payments, while demand elasticities have been estimated for some commodities. Estimates of substitution elasticities 1

Ibid., pp. 117-18. Also see Morton Bodfish, Vienna's Housing, a Preface to Urban Renewal, Indians Business Paper No. 3 (Bloomington, Ind., Foundation for Economic and Business Studies, Indiana University, 1961), pp. 10 and 25.

R. F. MUTH

233

1

have been published in two recent papers. Estimates of the elasticities of factor supply are the most difficult to obtain, but for some purposes it might be appropriate to assume that they are either zero or infinite.

APPENDIX The reduced-form equations for the six endogenous variables of the model are given here. In deriving them I find it convenient first to eliminate dp\ from (3') and (5') and dp$ from (4') and (6') as in deriving equation (19). Upon eliminating successively dQ* and dp* one can solve for dA* and dB*. Substituting these into (2')yields dQ*, whence dp*, dpf, and dp% can be determined from (1'), (5'), and (6')- The reduced forms are: kAr)eA(t —

D»

f.

(28)

^)et

(29)

XT 8

where 0

for

t) < 0, a > 0,

and e^ and eB > 0. 1 K. J. Arrow, H. B. Chenery, B. 8. Minhas, and R. M. Solow, ' Capital-Labor Substitution and Economic Efficiency', The Review of Economics and Statistics, xliii (Aug. 1961), 240, and J. B. Minasian, 'Elasticities of Substitution and Constant-Output Demand Curves for Labor', The Journal of Political Economy, lxir (June 1961), 261-70.

234 DERIVED DEMAND CURVE FOR A PRODUCTIVE FACTOR An equation for changes in factor B'a share is easily obtained by adding the coefficients of equations (20) and (29) and subtracting those of (24) and (27). The result is

^

^

«+

,

p-

,*k(l+g)(l-g)(^-« B ) g , Y+ 8 w + {kJkB)o{—r)(l+kAeB+kBeA)+kAeA+kBeB+eAeB)} ^j

T

e-

Similarly, the equation for the change in the factor price ratio is derived by subtracting equation (28) from (29); it is e)o

eB(

p

(l+y)(*A—eB)8

(°lkB)(-ii + kA6A+kBcB)

For € = 0, equation (30) is kA(\—o) times equation (31).

University of Chicago

c