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Oligopoly and derived demand
Economics Letters 5 (1980) 115-118 0 North-Holland Publishing Company
OLIGOPOLY
AND DERIVED DEMAND
Michael WATERSON * University of Newcastle Received
11 August
upon Tvne, Newcastle
upon Tym
NE1
7RlJ, England
1980
In this paper, a formula relating elasticity of demand for a factor to elasticity of final product demand, elasticity of substitution and share of the factor in cost of production is derived for the case of an oligopolistic industry.
The Marshall rules of derived demand are conventionally developed [see e.g. Allen (1938, pp. 372-373,504-508) Hicks (1963, pp. 241-246)] for the case of a competitive industry. One obvious question is whether these rules and the formulae from which they derive must be extensively modified to cater for more realistic market structures, such as oligopoly. We demonstrate that under reasonable simplifying assumptions no real modification is required (though some slight changes in interpretation arise) in applying the rules regarding elasticity of demand for the product, elasticity of substitution and share of the factor in cost of production to homogeneous oligopoly situations. Besides derivations of the rules under perfect competition, there have been analyses under various assumptions for monopoly. These range between Maurice and Ferguson (1973) who imposed neither linear homogeneity on the production function nor constant elasticity on the demand function, to Yeung (1972) who imposed both but did not make their assumption of fixed factor prices. Yeung found that the relevant formula under competition could be used for the case of monopoly. More recently Latham and Peel (1977) have looked at the case of oligopoly, imposing constant elasticity of demand, linear homogeneity and parametric factor prices. However, the formulae they derived were not straightforward extensions of those for competition (or monopoly). This paper shares their assumptions but reaches a far simpler and, we argue, more relevant resulting elasticity related to industry-wide changes in factor price. For simplicity of presentation, we choose not to use their method of analysis though.
* I should
like to thank
Martyn
Hill and Roger Latham 115
for helpful
comments
and suggestions.
Considel
a symmetric
stant elas!icity vi7.. x = zp
demand
Q.
oligopoly function
producin g a homogcneo~us relating
industry
output
product
with a con-
(X) to industry
price @).
r;(>O) being a parameter.
Each fir-m produces
its (identical)
of conslant
to scale, i.e..
returns
output
(1) .Y using factors
a and h under conditions
.Y = ,lja, b 1,
(2)
and the prices of these factors are exogenous to the industry. the firm’s cost function. using obvious notation. as PdX
c=&J,.
Thus, we
may
write
>
so average and marginal
costs arc
(‘ = g@,, 136) .
(3)
Now it is well known [see. e.g., Cowling and Waterson (1976)j metric oligopolistic indusiry of N firms producing a homogeneous price-cost
that in a symproduct the
margin may be written
C,,i/LJ= (I + A)/N7j )
0,
(4)
where c is the firm’s marginal cost and h is the output-weighted variation’ terms, d(C.ui)/d~, i # j. Thus, from (3) and (4) we have P = ‘7’
sum of ‘conjectural
(5)
&Pat Pb).
where
is the mark-up of price on marginal cosi, a function parameters so a constant in our model. By Shephard’s
lemma,
with a similar expression
the firm’s derived
demand
of things we here treat as for a is given by
for b.
Since all firms are operating at the same output level, they will all have equal ag/:lap, at equilibrium positions. Therefore, summing (6) to the level of the industry, we have (7) where ,4 is the industry
derived
demand
for the factor.
M. Waterson / Oligopoly
Differentiating
and derived demand
117
(7) with respect to pa (suppressing the arguments of g and X’),
(8) Now since the firm’s demand function prices, by Euler’s theorem,
a28
Pb .~
2=
apa -- Pa
Substituting
(6) is homogeneous
a2g
apa
%b
’
this in (8) then rewriting in elasticity form yields
ax P .ALPbb.xZg.aZg__._.
r:,E-4 A
apa
C
ab
aPaapb
ap
x
mp,a px
Defining k, =p,a/C, the share of A in factor payments, of the elasticity of substitution between two factors, g’ a2&.% ’ = (waPa)
of degree zero in factor
’ and recalling the definition
apb
’ (waPb)
’
en$bles us to rewrite the industry E, = kav + (1 - k,) u .
elasticity of derived demand for A as (10)
A similar expression, relating derived demand for a factor to the industry elasticity and the Allen elasticity of substitution, can be obtained in the more general case where there are more than two factors. Notice our formula (10) is identical to that for a competitive industry under the equivalent assumptions (e.g., pb fixed); see, for example, Layard and Walters (1978, p. 267) and Allen (1938, p. 373). It is also identical to that derived by Yeung (1972, p. 514) under those assumptions. The formulae Latham and Peel (1977, e.g., eq. 27) derive for oligopoly under the assumptions used here are somewhat more complex than our (10). In fact though, their formulae refer to the derived demand of an oligopoly firm when pa changes exogenously for that firm only, rather than the whole industry, so affecting the structure of the industry. As such, they are probably of more limited interest than our formula. Perhaps surprisingly then, the margin m does not appear directly in our formula for the derived demand of an oligopolistic industry, at least under our simplifying assumptions, and the rules consequently remain unchanged. Nevertheless it is important to note that the rule relating elasticity of derived demand to elasticity of final demand is valid only when holding the share of the factor in total cost constant, not the value share. Finally, note that the formula derived by Maurice and Ferguson (1973, eq. 16) for derived demand of a monopolist using more general demand and cost functions than ours will hold for any profit-maximising firm. Thus in a symmetric oligopoly
118
M. Waterson / Oligopoly and derived demand
it can be used to indicate factors determining the industry’s derived demand with appropriate reinterpretation. Unfortunately, interpretation is made particularly complex because the derived elasticity now depends inversely on the difference between the firm’s elasticity of its perceived marginal revenue curve and the elasticity of its marginal cost curve, something which would in general be very difficult to discern from industry data. However, when simplified using our assumptions, their formula yields our result (10).
References Allen, R.G.D., 1938, Mathematical analysis for economists (Macmillan, London). Cowling, K. and M. Waterson, 1976, Price-cost margins and market structure, Economica 43, 267-274. Hicks, J.R., 1963, The theory of wages, 2nd ed. (Macmillan, London). Latham, R.W. and D.A. Peel, 1977, Derived demand and oligopoly, Bulletin of Economic Research 29, 3-8. Layard, P.R.G. and A.A. Walters, 1978, Microeconomic theory (McGraw-Hill, Maidenhead). Maurice, S.C. and C.E. Ferguson, 1973, Factor demand elasticity under monopoly and monopsony, Economica 40,180-186. Yeung, P., 1972, A note on the rules of derived demand, Quarterly Journal of Economics 86, 511-517.
OLIGOPOLY
AND DERIVED DEMAND
Michael WATERSON * University of Newcastle Received
11 August
upon Tvne, Newcastle
upon Tym
NE1
7RlJ, England
1980
In this paper, a formula relating elasticity of demand for a factor to elasticity of final product demand, elasticity of substitution and share of the factor in cost of production is derived for the case of an oligopolistic industry.
The Marshall rules of derived demand are conventionally developed [see e.g. Allen (1938, pp. 372-373,504-508) Hicks (1963, pp. 241-246)] for the case of a competitive industry. One obvious question is whether these rules and the formulae from which they derive must be extensively modified to cater for more realistic market structures, such as oligopoly. We demonstrate that under reasonable simplifying assumptions no real modification is required (though some slight changes in interpretation arise) in applying the rules regarding elasticity of demand for the product, elasticity of substitution and share of the factor in cost of production to homogeneous oligopoly situations. Besides derivations of the rules under perfect competition, there have been analyses under various assumptions for monopoly. These range between Maurice and Ferguson (1973) who imposed neither linear homogeneity on the production function nor constant elasticity on the demand function, to Yeung (1972) who imposed both but did not make their assumption of fixed factor prices. Yeung found that the relevant formula under competition could be used for the case of monopoly. More recently Latham and Peel (1977) have looked at the case of oligopoly, imposing constant elasticity of demand, linear homogeneity and parametric factor prices. However, the formulae they derived were not straightforward extensions of those for competition (or monopoly). This paper shares their assumptions but reaches a far simpler and, we argue, more relevant resulting elasticity related to industry-wide changes in factor price. For simplicity of presentation, we choose not to use their method of analysis though.
* I should
like to thank
Martyn
Hill and Roger Latham 115
for helpful
comments
and suggestions.
Considel
a symmetric
stant elas!icity vi7.. x = zp
demand
Q.
oligopoly function
producin g a homogcneo~us relating
industry
output
product
with a con-
(X) to industry
price @).
r;(>O) being a parameter.
Each fir-m produces
its (identical)
of conslant
to scale, i.e..
returns
output
(1) .Y using factors
a and h under conditions
.Y = ,lja, b 1,
(2)
and the prices of these factors are exogenous to the industry. the firm’s cost function. using obvious notation. as PdX
c=&J,.
Thus, we
may
write
>
so average and marginal
costs arc
(‘ = g@,, 136) .
(3)
Now it is well known [see. e.g., Cowling and Waterson (1976)j metric oligopolistic indusiry of N firms producing a homogeneous price-cost
that in a symproduct the
margin may be written
C,,i/LJ= (I + A)/N7j )
0,
(4)
where c is the firm’s marginal cost and h is the output-weighted variation’ terms, d(C.ui)/d~, i # j. Thus, from (3) and (4) we have P = ‘7’
sum of ‘conjectural
(5)
&Pat Pb).
where
is the mark-up of price on marginal cosi, a function parameters so a constant in our model. By Shephard’s
lemma,
with a similar expression
the firm’s derived
demand
of things we here treat as for a is given by
for b.
Since all firms are operating at the same output level, they will all have equal ag/:lap, at equilibrium positions. Therefore, summing (6) to the level of the industry, we have (7) where ,4 is the industry
derived
demand
for the factor.
M. Waterson / Oligopoly
Differentiating
and derived demand
117
(7) with respect to pa (suppressing the arguments of g and X’),
(8) Now since the firm’s demand function prices, by Euler’s theorem,
a28
Pb .~
2=
apa -- Pa
Substituting
(6) is homogeneous
a2g
apa
%b
’
this in (8) then rewriting in elasticity form yields
ax P .ALPbb.xZg.aZg__._.
r:,E-4 A
apa
C
ab
aPaapb
ap
x
mp,a px
Defining k, =p,a/C, the share of A in factor payments, of the elasticity of substitution between two factors, g’ a2&.% ’ = (waPa)
of degree zero in factor
’ and recalling the definition
apb
’ (waPb)
’
en$bles us to rewrite the industry E, = kav + (1 - k,) u .
elasticity of derived demand for A as (10)
A similar expression, relating derived demand for a factor to the industry elasticity and the Allen elasticity of substitution, can be obtained in the more general case where there are more than two factors. Notice our formula (10) is identical to that for a competitive industry under the equivalent assumptions (e.g., pb fixed); see, for example, Layard and Walters (1978, p. 267) and Allen (1938, p. 373). It is also identical to that derived by Yeung (1972, p. 514) under those assumptions. The formulae Latham and Peel (1977, e.g., eq. 27) derive for oligopoly under the assumptions used here are somewhat more complex than our (10). In fact though, their formulae refer to the derived demand of an oligopoly firm when pa changes exogenously for that firm only, rather than the whole industry, so affecting the structure of the industry. As such, they are probably of more limited interest than our formula. Perhaps surprisingly then, the margin m does not appear directly in our formula for the derived demand of an oligopolistic industry, at least under our simplifying assumptions, and the rules consequently remain unchanged. Nevertheless it is important to note that the rule relating elasticity of derived demand to elasticity of final demand is valid only when holding the share of the factor in total cost constant, not the value share. Finally, note that the formula derived by Maurice and Ferguson (1973, eq. 16) for derived demand of a monopolist using more general demand and cost functions than ours will hold for any profit-maximising firm. Thus in a symmetric oligopoly
118
M. Waterson / Oligopoly and derived demand
it can be used to indicate factors determining the industry’s derived demand with appropriate reinterpretation. Unfortunately, interpretation is made particularly complex because the derived elasticity now depends inversely on the difference between the firm’s elasticity of its perceived marginal revenue curve and the elasticity of its marginal cost curve, something which would in general be very difficult to discern from industry data. However, when simplified using our assumptions, their formula yields our result (10).
References Allen, R.G.D., 1938, Mathematical analysis for economists (Macmillan, London). Cowling, K. and M. Waterson, 1976, Price-cost margins and market structure, Economica 43, 267-274. Hicks, J.R., 1963, The theory of wages, 2nd ed. (Macmillan, London). Latham, R.W. and D.A. Peel, 1977, Derived demand and oligopoly, Bulletin of Economic Research 29, 3-8. Layard, P.R.G. and A.A. Walters, 1978, Microeconomic theory (McGraw-Hill, Maidenhead). Maurice, S.C. and C.E. Ferguson, 1973, Factor demand elasticity under monopoly and monopsony, Economica 40,180-186. Yeung, P., 1972, A note on the rules of derived demand, Quarterly Journal of Economics 86, 511-517.