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Modern Economy, 2012, 3, 738-741
http://dx.doi.org/10.4236/me.2012.36094 Published Online October 2012 (http://www.SciRP.org/journal/me)
Study on the Production Technology, Elasticities and
Market Structure Impact on Profit Formation*
Jian Wang1#, Xuejun Zheng2
1College of Economics and Trade, Agricultural University of Hebei, Baoding City, China
2College of Foreign Language, Agricultural University of Hebei, Baoding City, China
Received July 5, 2012; revised August 10, 2012; accepted August 20, 2012
The size of the profit in a firm or a production system not only depends on the quantity of inputs and outputs, but also
depends on the market structure that means the market is perfect competition or imperfect competition. In general, the
relationship between output and inputs can be defined as the production structure, which is usually decided by produc-
tion technology. Therefore, under the market economy system, the production structure regulation has to follow the
market structure variation. Here we assume that production technology is a C-D function, and then to determine the
effects of different market structures, which we find, they are in close contacted with both production and market struc-
tures, especially some variations of elasticities. Through out a series of deduction and equilibrium analysis, the re-
stricted conditions of the maximum profit have been found. Therefore, the consequences show that the values of elastic-
ities have taken an important role in profit obtained for producer, the profits in a perfect competition market hardly de-
pends on market demand elasticity, in which production elasticity requires rather small. However, in the imperfect
competition market, monopoly make both price of demand and production elasticities impact on the profit. Those also
prove that market monopoly factors make production lose efficiency, or lead to the market failure. In the actual process
of production and management, production elasticities and related market information should be strengthened for
measurement, which will be useful for analysis price fluctuation risk and management decision.
Keywords: Elasticity; Profit Function; Supply & Demand Function; Homogeneous of Degree H
This study is about the production, market structure and
rules of revenue. Production technology often restricts
the profits obtained by producers. Here mainly refers to
the technical constraints of production elasticity. The
technical restriction is usually a production structural
problem, when considering the issue occurred in the
market and market structure will be in close contact with
it. In the production process, technology determines the
conversion of material and market structure makes this
transition become more complicated. Although these pro-
blems have been concerned by economists many years
ago, such as Harold Hotelling (1929), P. A. Samuelson
(1947), Jan Tinbergen (1959), Ragnar Frisch (1965), R.
Shephard (1970, 1981), Thomas J. Lutton and W. Erwin
Diewert (1982), Hal R. Varian (1984, 1997), etc. These
theories may include the production function, theory of
market and profit, and so on. However, many problems
generally exist in reality, which need to prove their seri-
ousness in theory and give more intuitive explanation. So
that we discuss those problems beginning with a homo-
geneous production function defined by the Hal R. Var-
ian (1984, p. 179), and attempt to demonstrate these is-
2. Definition of Production Function
The Cobb-Douglas technology describes can easily be
extended to the case of m inputs. A generalized C-D fun-
ction for a production process can be defined as follow-
12 m that is a vector with m-
dimensional input factors. y > 0 is a single output of the
production, and f is the function with twice different-
tiable and continuous. That A means technical coefficient
is used to keep in constant. bi is also a quantity of con-
*This paper is supported by China social science university humanity
roject (No. 12YJAZH138).
opyright © 2012 SciRes. ME
J. WANG, X. J. ZHENG 739
stant, which usually indicates the elasticity of production
with related input factor of xi, here
yyx xy xyx .
Therefore, y = f(x) is strictly quasi-concave and posi-
tively with homogenous of degree h which respect to
according to the significance of its economic definition.
Here, h can be defined as scale elasticity, so that h = ∑bi
for all x. If there is a t > 0, which makes all x time t, then
we find , it is easy to be
Let us define the scale elasticity as
to make the total differentiation
We get scale elasticity
3. Profit Function
3.1. Profit Function Definition
Suppose the study refers to the market system, the mar-
ket prices have to be decided by demand and supply. It is
well known that production and management decision
pursuits economic profit. The equilibrium analysis gen-
erally discusses how to decide the price, and to determine
the optimal input or output. Thus, let ω = (ω1, ω2, ···, ωm)
> 0 to present the price of input factors, and p as the price
of single input. In this case, a profit function can be given
as the following .
It is easy to prove the following two lemmas by (2)
yπp (Hotelling); xπ (Shephard).
3.2. Profit Maximization
The profit maximization requires determining the inputs
value of > 0, which make that
In general, during the analysis of profit maximization
the factors selection are used to be endogenous variables,
but prices (p, ω) and technologies are used to be exoge-
nous variables. Therefore, to obtain Max that means the
first order conditions has to be deviated by the defined
profit function and production function. When we con-
sider to pursuit the optimal profit in the product market,
we should use Formula (2), let i
πx0 to be the
first order condition.
for all i1,2,,m
Thus we can obtain 
Here, εp < 0 is defined as the demand price elasticity of
product y. Especially, in perfect competition market
when the producer is a price taker that p must
be required. However, in the market of imperfect or mo-
will be the requirement. Because,
is required according to properties of optimal profit. That
means y is increasing in
, which makes
p. Therefore, Formula (3) will be the condition of
profit maximization in monopoly market, which can be
written as following:
Considering Formula (1), using first-order partial de-
rivatives we obtain iii
, to combine it with
Formula (4), we get input set * as a solution for Max
in different markets at Formula (5)
In generally, in the perfect competition market we
have to determine the input i by (5-1), but in the im-
perfect market we have to determine the input i
(5-2). Because of
in monopoly market,
input x is usual less than it in perfect competition market.
If a maximum profit relies on the optimal input vector
, it is easy for us to use Formula (5) to find
,,x, for all p0,0,x0, with Formula (2).
y 1 h(6-1);
The important knowledge has been illustrated by For-
mula (6), basic information is about elasticity. In addition
Copyright © 2012 SciRes. ME
J. WANG, X. J. ZHENG
Copyright © 2012 SciRes.
to the revenue p·y, there is more important economic
signal that is the elasticity of production h. That shows
the stability of production or the sensitivity of input fac-
tors in production process will affect the quantity of
profit directively, particularly in the production function
the value of h needs to be measured and well controlled.
Certainly most of markets structures belong to im-
perfect competition, monopoly factors in real market
are existed generally. The influence of monopoly in
Formula (6) is indicated by price elasticity of demand
. Therefore, p1
is a basic requirement that
postulated to be reached in market of monopoly pro-
duction. To compare those two kinds of markets for
* ≥ 0, which requires h ≤ 1, but this only related to
the process of production, however, the sensitivity of
market demand variation with p will be measured by
. So that,
is the normal
case of imperfect competition market according to
Formula (4). Further discussion on the issues will be
launched as following.
4. Supply and Demand Functions
4.1. The Supply Function
If we take the results in Formula (5) into Function (1), a
supply function of producer for optimal profit can be
obtained. This result shows in Formula (7), which in dif-
ferent markets the issues will be different.
There are many properties in Formula (7), where (7-1)
is perfect competition market, and (7-2) is non perfect
competition market. Such as the output is monotoni-
cally increasing with both price p and
, the price
elasticity of supply function of the output that indicated
by h should be limited to small one, because if
h1h1 requires h12.  Moreover, in mono-
poly market also requires h12, and
4.2. The Demand Function
Further more, we can derivative demand function of the
input factors by (5) and (7), which are the optimal results.
We get the Formula (8):
As a well known property for demand function has
homogenous of degree 0 in p, ω, which is easy to dem-
onstrate in Formula (8). There is
, for a given t > 0.
Now we can use the above derived results, only when the
second order conditions 22
are satisfied, we
can obtain the optimal profit function.
ππp,, xπp,, yπ
for all p0, 0 and (i1,2,m). Then
The consequence of Formula (9) tell us that the opti-
can bring an optimal level of output y
which make maximum profit function (p, ω) as an en-
dogenous variable, then market prices (p, ω) can be
thought as exogenous variables. In the profit maximiza-
tion model, price and technology are exogenous variables,
and the factors of choice are endogenous variables, this
view is consistent with Hal R. Valrian (1992, p. 203).
Therefore, the properties of profit function
(p, ω) are
easy to be proved herewith. 
1) Above profit function is nondecreasing in p for
output, nonincreasing in ω for inputs. If
pp and , then πp, π
πp, py11 1hAp11b111h(9-2).
J. WANG, X. J. ZHENG 741
2) *(p, ω) is homogeneous of degree one in prices.
, for t0.
3) It is convex in (p, ω).
Let ptp1tp, for t
4) The profit function *(p, ω) is continuous in (p, ω),
at least for p > 0,
, and * is convergent condi-
tional with h
5) In addition, we are able to demonstrate
by Formula (4).
The classical economics suppose that a firm has com-
pleted information. A perfect market is able to utilize “a
spontaneous invisible hand” to produce free action be-
tween the demand and the supply to realize market clear.
The firm just like in a well planned system, the market is
with perfect competition and the producer is a price taker.
However, the ideal system is hardly existed, currently the
existences of monopolies factors are very objective, and
market information is general not enough according to
the information economics. As a matter of fact, the firm
usually makes optimal choice under different market
structures. Imperfect market with uncompleted or asym-
metric information is a general case. If an economy want
to use “a visible hand” to play an efficient role, then the
information and measurement from micro to macro be-
come necessary in all the cases. Such as this study, if a
firm considers optimal of its own profit, it has to analyze
the market structure, and to measure the price and the
production elasticities. That information collection should
give the institutional arrangement and solve the issues of
market to reduce uncertainty or avoid management risk,
and so on. REFERENCES
 H. R. Varian, “Microeconomic Analysis,” 2nd Edition, W.
W. Norton & Co., Inc., New York, 1984.
 H. R. Varian, “Microeconomic Analysis,” 3rd Edition, W.
W. Norton & Co., Inc., New York, 1992.
 J. Wang, “(M, E, I) Model as Multi-Input Production
Function with Homogeneousness of Degree H,” Anlyse de
Systemes, Vol. 23, No. 4, 1997, pp. 45-49.
 J. M. Price, “Price Elasticities Implied by Homogeneous
Production Functions,” The Journal of Agricultural Eco-
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 J. Tinbergen, “Production, Income and Welfare: The
Search for an Optimal Social Order,” Wheatsheaf Books,
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