Production of maximum output in the firm

Production of maximum output in the firm is possible in the given level of expenditure which can be studied with the help of Iso-cost curve and Iso-quant curve.

It is clear that nay desired level of output can be produced by a number of different combinations of inputs. But the manager of the firm most make major decisions such as which input combination to be used, what the input, combination will be optimal.

The firm can choose from among different combinations of capital (K) and Labor (L) to produce a given level of output or faced with specified input prices, it can choose from among many combinations of K and L that would lead to fixed level of cost i.e. expenditure. Thus the firm has to make either of two input choice decision.
  1. Choose the input combination that yields the maximum level of output possible with a fixed output (i.e. output maximization subject to cost constraint).
  2. Choose the input combination that leads to the lowest cost of producing a fixed level of output (i.e. cost minimization subject to output constraint).
The solution to any constrained maximization or minimization problem is obtained by choosing the level of each activity whereby the marginal benefits from each activity. Per rupee spend are equal. Here the profit maximizing firm has to choose the input combination for which the marginal product divided by input price is the same for all inputs used. The implication is that for two input case, a firm attains the highest level of output when

MPL/PL = MPK/PK or MPL/w = MPK/r

Where, w and r are respectively the prices of labor (PL) and capital (PK). Thus, the MRTS = MPL/MPK equals the factor price ratio (wr).

Input Prices and Iso-costs

The Iso-quant shows the desire of the producer. Usually a firm is supposed to have a fixed amount of money to buy resources. The Iso-cost line is the producer’s budget line. In determining the optimal input combination, a profit maximizing producing unit firm or producer has to pay attention to relative input prices of it is to minimize the cost of producing a given output or maximizing output for a given level of cost. Input prices are determined by the market forces.

The equation of total cost is C = rK + wL where all the terms have their usual meaning. Total cost is simply the sum of the cost of K units of capital at r rupees per unit of L units of labor at w rupees per units.

Suppose, capital costs Rs. 100 per month per unit (r = Rs. 100) and labor receives a wage of Rs. 200 per unit (w = Rs. 200). Then the firm’s total cost function is

C = 100 K + 200 L ……………………. (i)

Now, suppose that the fir decides to spend Rs. 2000 per month for capital and labor. Thus, equation becomes Rs. 2000 = Rs. 100 K + 200 L.
The process of solving this equation is

     2000 = 100 K + 200 L
or, K = 200 – 2 L

In a general situation, if a fixed amount Ḹ is to be spent, the firm can chose among the combinations given by

K = Ḹ/r – w/r . L

Production of a given Output at Minimum Cost

Whatever output a firm chooses to produce, the production manager is desirous of producing it at the lowest possible cost. To achieve this objective, the production process must not only to technically efficient but economically efficient, too. So, the production process has to organize in the most efficient manner.

For example, suppose that at given input prices r and w, a firm wishes to produce the output indicated by Iso-quant Q0 as shown in the figure.
In the figure, K1L1, K2L2 and K3L3 are the three Iso-cost lines from which the producer can choose at the given factor prices. The firm will choose the lowest level of expenditure that enables output level Q0 = 200 to be produced. As shown in the figure output level Q0 will be produced at the cost level Q0 will be produced at the cost represented by Iso-cost line K1L1.

Any cost outlay below that such as represented by KL is not feasible since it is impossible to produce output Q0 with these factor combinations. Any factor combinations above that represented by K1L1 are not considered because the firm seeks to produce the desired output at least cost. If combination A or B is chosen at the cost outlay represented by K2L2, the producer can reduce costs by moving along Q0 to point E. Point E shows the optimal resource combination, K0 units of capital and L0 units of labor. This is known as the level of cost combination of inputs.

This Iso-quant shows the desired rate of factor substitution and the Iso-cost is the actual rate of factor substitution. A firm reaches equilibrium and thus minimizes cost when the Iso-quant is tangent to the lowest possible Iso-cost line. Thus, equilibrium is reached when the Iso-quant representing the chosen output is just tangent to an Iso-cost line. At this tangent the slopes of the two curves are equal, production at least cost requires that the MRTS (Marginal Rate of Technical Substitution) of capital for labor be equal to the ratio of the price of labor to the price of capital.

      MPL/PMK = w/r

or, MPL/w = MPK/r

where, MPL = Marginal production of labor
MPK = Marginal productivity of capital
        r = Price of capital
       w = Price of labor

Production of Maximum Output with a given Level of Cost

It is an alternative technique but more preferable way of presenting the optimization problem, is to assume that the firm chooses a level of output and then select the factor combination that permits production of that output at least cost. This approach seems to be more practical that the previous one. It is assumed that the firm can spend only a fixed amount of money to spend and it seeks to attain the highest level of output consistent with that amount of outlay. It can be explained with the help of figure below:
The Iso-cost line KL shows all possible combination of the two inputs that can purchased with a fixed market prices. Three Iso-quants are shown in the figure. At the given level of cost, output level Q2 is unattainable. Neither output level Q1 nor level Q2 would be chosen, since higher levels of output can be produced with the fixed cost outlay. The highest possible output with the given level of cost is produced by using OL0 amount of labor and OK0 amount of capital. At point E, the highest attainable Iso-quant (i.e. Q) is just tangent to the given Iso-cost (KL). Thus, in the case of constrained output maximization, the MRTS of capital for labor equal the input-price ratio (the price of labor to the price of capital).

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