Firm making decision with regard to the Optimum employment of two inputs

A profit maximizing firm (business unit) tries to minimize its cost for a given output or to maximize its output for a given total cost. The logic of iso-quant tells that a given output can be produced with different input combinations. Given the input prices, only one of the input combinations confirms to the least-cost criteria. We may now combine the iso-quants and iso-cost to determine the optimal input employment or the least cost combination of inputs. There are two conditions for the least-cost combination of inputs (a) the firm order condition, (b) the second order condition. The first order condition of the least-cost input condition of the least-cost input combinations can be expressed in both the physical and value terms.

Given the two inputs K and L, the first order condition in physical terms required that marginal rate of exchange (MRE) between K and L must equal the ratio of their marginal physical product (MPP) i.e., 

MPL/MPK = ∆K/∆L ……………… (i)

Where ∆K/∆L is the marginal rate of exchange (MRE) between K and L, and MPL/MPK is the ratio of marginal physical productivities of L and K i.e., on input combination at which factor exchange ratios equals the ratios of their marginal productivities is the least-cost input combination.

The first order criterion of the least-cost input combination is also expressed as the MRP ratio of K and L must equal their price ratio, i.e.,

MPL/MPL = PL/PK or, MPL/PL = MPK/PK ………………….. (ii)

Where, MPL and MPK are marginal products of labor and capital respectively, and PL and PK are prices of labor and capital respectively.

In equation (ii), (∆K/∆L) slope of the iso-quant, and MPL/MPK = slope of the iso-quants. It refers that the least cost combination exists at a point where iso-quant is tangent to the iso-cost. The least cost combination of K and L is graphically presented in the following figure.
The iso-quant Iq2 is targeted to iso-quant K2L2 at point E. At this combination of K and L equals OG of K plus OH of L. This combination of K and L is optimal as it satisfies the least-cost criterion, i.e.,

∆K/∆L = MPL/MPK

In case of second order condition, the second order combination requires that the first order condition be fulfilled at the highest possible iso-quant. The first order condition is satisfied also on points A and F, the points of intersection between Iq1 and iso-cost K2L2 in above figure. Since at these intersection points ∆K/∆L = MPL/MPK. But, points A, F and E are on the same iso-cost, point E is on an upper iso-quant while E is associated with on output of 400 units, points A and F being on a lower iso-quant, are associated with an output of 200 units. It means that the total cost, a firm can produce 200 units. It means that given the total cost, a firm can produce 200 units as well as 400 units. Therefore, a profit maximizing or cost minimizing firm chooses input combinations at point E rather than A or F.

In case of value terms, the following condition has to be operated for optimal employment of two points.

PL/PK = MRPL/ MRPK or MRPL/PL = MRPK/PK ………………… (iii)

Where, MRP = marginal revenue productivity of the factor.

It may be inferred from equation (iii) that least-cost or optimal input combination requires that the MRP ratios of inputs should be equal to their price ratios. In other way, the MRP and factor price ratios of all the inputs must be equal.

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