## Producer Equilibrium in Long Run:

### Producer Equilibrium:

Producer
equilibrium is a situation where the producer produces maximum output from
given resources or inputs. It is a condition of profit maximization of the producer from the given level of inputs (i.e. labour and capital). Two conditions
must be fulfilled form producer equilibrium and they are:

#### 1. Necessary Condition:

The slope of ISO-QUANT must be equal to the ISO-COST
line i.e. MRTS

_{LK}= W/R.#### 2. Sufficient Condition:

ISO-QUANT must be convex to the origin at
the point of tangency to the ISO-COST line.

*Producer equilibrium can be explained in terms of cost minimization and output maximization with the help of a diagram.*

#### Assumptions:

1. There are two factors, labour
and capital.

2. All units of labour and capital
are homogeneous.

3. The prices of units of labour(w) and that of capital(r) is given and constant.

4. The cost outlay is given.

5. The firm produces a single
product.

6. The price of the product is
given and constant.

7. The firm aims at profit
maximization.

8. There is perfect competition in
the factor market.

### Cost Minimization:

Given
these assumptions, the point of the least-cost combination of factors for a given
level of output is where the isoquant curve is tangent to an iso-cost line. In the figure given below, the iso-cost line GH is tangent to the isoquant 200 at point
M.

The firm employs the combination of ОС of capital and OL of labour to produce 200
units of output at point M with the given cost-outlay GH. At this point, the
firm is minimizing its cost for producing 200 units.

Any
other combination on the isoquant 200, such as R or T, is on the higher
iso-cost line KP which shows a higher cost of production. The iso-cost line EF
shows lower cost but output 200 cannot be attained with it. Therefore, the firm
will choose the minimum cost point M which is the least-cost factor combination
for producing 200 units of output.

### Output Maximization:

The firm also maximizes its profits by maximizing its output, given its cost outlay
and the prices of the two factors. This analysis is based on the same
assumptions, as given above. The firm is in equilibrium at point P where the
isoquant curve 200 is tangent to the iso-cost line CL in Figure below.

At this point, the firm is maximizing its output level of 200 units by employing the
optimal combination of OM of capital and ON of labour, given its cost outlay CL.
But it cannot be at points E or F on the iso-cost line CL, since both points
give a smaller quantity of output, being on the iso-quant 100, than on the iso-quant
200.

The firm can reach the optimal factor combination level of maximum output by moving
along the iso-cost line CL from either point E or F to point P. This movement
involves no extra cost because the firm remains on the same iso-cost line.

The firm cannot attain a higher level of output such as isoquant 300 because of the
cost constraint. Thus the equilibrium point has to be P with optimal factor
combination OM + ON. At point P, the slope of the isoquant curve 200 is equal
to the slope of the iso-cost line CL. It implies that w/r = MP

_{L}/MPC = MRTS_{LC.}## Cobb-Douglas Production Function:

The Cobb-Douglas production function is based on the empirical study of the
American manufacturing industry made by Paul H. Douglas and Charles W. Cobb. It
is a linear homogeneous production function of degree one, which takes into
account two inputs, labour and capital, for the entire output of the manufacturing
industry.

The Cobb-Douglas production function is expressed
as:

*Q = A* L*^{α}** C**, where Q is output, L and С are inputs of labour and capital respectively. A, α and β are positive parameters where, α > 0, β > 0. The equation tells that output depends directly on L and C, and that part of output which cannot be explained by L and С is explained by A which is the ‘residual’, often called technical change.*^{β}
The
production function solved by Cobb-Douglas had 1/4 contribution of capital to
the increase in the manufacturing industry and 3/4 of labour so that the C-D production function is: Q = AL

^{3/4 }C^{1/4}which shows constant returns to scale because the total of the values of L and С is equal to one: (3/4 + 1/4), i.e. (α + β = 1). The coefficient of the labourer in the C-D function measures the percentage increase in Q that would result from a 1 per cent increase in L, while holding С as constant.
Similarly,
В is the percentage increase in Q that would result from a 1 per cent increase
in C while holding L as constant. The C-D production function showing constant
returns to scale is depicted in Figure below. Labor input is taken on the
horizontal axis and capital on the vertical axis.

To
produce 100 units of output, ОС, units of capital and OL units of labour are
used. If the output were to be doubled to 200, the inputs of labour and capital
would have to be doubled. ОС is exactly double of ОС

_{1}and of OL_{2 }is double of OL_{2.}_{}

Similarly,
if the output is to be raised three-fold to 300, the units of labour and capital
will have to be increased three-fold. OC

_{3}and OL_{3}are three times larger than ОС_{1}, and OL_{1}, respectively. Another method is to take the scale line or expansion path connecting the equilibrium points Q, P and R. OS is the scale line or expansion path joining these points.
It
shows that the isoquants 100, 200 and 300 are equidistant. Thus, on the OS
scale line OQ = QP = PR which shows that when capital and labour are increased
in equal proportions, the output also increases in the same proportion.

### Properties:

#### 1. Factor Intensity:

The
factor intensity can be measured by taking the ratio between a and b.

a. If a/b>1, there is an operation of labour-intensive production technique.

b. If a/b<1, there is an operation of capital
intensive production technique.

#### 2. The Efficiency of Production:

The efficiency of production can be measured by the coefficient A.

a. If the value of A is higher,
there is a higher degree of efficiency of production.

b. If the value of A is lower,
there is a lower degree of efficiency of production.

#### 3. Returns to Scale:

The
various degrees of returns to scale can be measured by taking the sum of a and b.

Let, a+b
= V

a. If V > 1, there is an operation
of increasing returns to scale.

b. If V = 1, there is an operation of
constant returns to scale.

c. If V < 1, there is an operation
of decreasing returns to scale.

#### 4. Average Productive of Inputs:

#### 5. Marginal Productivities of Inputs:

#### 6. The Marginal Rate of Technical Substitution:

#### 7. The Elasticity of Technical Substitution:

### Importance:

1. It has been used widely in
empirical studies of manufacturing industries and in inter-industry
comparisons.

2. It is used to determine the
relative shares of labour and capital in total output.

3. It is used to prove Euler’s
Theorem.

4. Its parameters a and b
represent elasticity coefficients that are used for inter-sectorial
comparisons.

5. This production function is
linear homogeneous of degree one which shows constant returns to scale If α +
β = 1, there are increasing returns to scale and if α + β < 1, there are
diminishing returns to scale.

6. Economists have extended this
production function to more than two variables.

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