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.,

MP_L / MP_L = P_L / P_K or, MP_L / P_L = MP_K / P_K ………………….. (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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Determination of Optimal Employment of an Input

To see how the economic productivity of an input, as defined by its marginal revenue product, is related to the factor for productive purposes, one needs only consider the simple question. If the price of input X in the production system is $100, how many units of X would a firm use? If the marginal revenue products exceed the related cost, that level of such input number could be employed.

The relationship between resource productivity as measured by the marginal revenue product and optimal employment or factors use can be maximized. If marginal revenue exceeds marginal cost, profits most increase. In the context of production decisions, this means that if the marginal revenue product of an input, i.e., the marginal revenue generated by its employment in a production system exceeds its marginal cost, then profits are increased as input employment increases.

Similarly, when the marginal revenue product is less than the cost of the factor, marginal profit is negative, so the firm would reduce employment of that factor.

This concept of optimal resource employment can be clarified by examining a very simple production system in which a single variable input labor (L), is used to produce a single product, Q. Profit maximization requires that production be at a level such that marginal revenue equals marginal cost. Since the only available factor in the system is input L, the marginal cost of production can be expressed as:

MC = ∆C / ∆Q = PL / MPL …………….. (i)

That is dividing PL, the price of a marginal unit of L, by MPL, the number of units output gained by the employment of an added unit of L, provides a measure of the marginal cost of producing each additional unit of the product.

Since MR must equal MC at the profit maximizing output level, MR can be substituted for MC in equation (i), resulting in the expression,

MR = PL / MPL ……………………. (ii)

Equation (ii) must hold for profit maximization since it was demonstrated immediately above that the right-handed side of equation (ii) is just another expression for MC. Solving equation (ii) for PL results in PL = MR x MPL which is defined as the marginal revenue product of L.

PL = MRPL …………………….. (iii)

Equation (iii) states the general result that a profit maximizing firm will always employs an input up to the point where its MRP is equal to its cost. If the MRP exceeds the cost of the input, profits are increased by employing additional units of the factor.

Similarly, when the resources’ price is greater than its MRP, profit is increased by using less of the factors.

Only at the level of usage where MRP = P are profits maximized than the costs incurred (PL). Only at L where PL = MRPL, will total profits be maximized. If PL were higher, the quantity of L demanded would be reduced. Similarly, if PL were lower the quantity of L purchased would be greater.

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Relationship of Price Elasticity of Demand with Total Revenue and Marginal Revenue

Price elasticity of demand is important in decision-making. It gives the measure of effect of price change in revenues. A given change in price leads to an increase or decrease or no change in total revenue. If the price elasticity accurately estimates, we can estimate accurately the new total revenue after price change.

When demand for a company’s product is elastic, a price cut leads to an increase in total revenue. Its total revenue increases marginal revenue is positive. Because the proportionate rise in the quantity demand more than offsets, the proportionate fall in price. Similarly, when demand is price inelastic, a price cut leads a fall in total revenue. This means that marginal revenue is negative. The reason now is that the percentage increase in the quantity, demanded is not enough to neutralize the percentage fall in price. Finally, if demand for a company’s product is unitary elastic, total revenue remains constant whether price rises or falls. In this situation MR is zero. The reason is easy to find out the percentage change in the quantity demanded and the price are equal but opposite, so that they cancel each other out. These relations can be expanded with the help of diagram.

 

The relations among TR, MR and price elasticity of demand are reviewed in the figure. The top half of the diagram shows a liner demand or AR curve and a corresponding straight line MR curve for a pure monopolist. The bottom of half of the diagram shows the monopolist’s TR curve. The figure shows that the demand is elastic over the range of output O to B. So, TR increases and MR is positive. At the output level of B, the demand is unitary elastic, TR is maximum (output) and MR is zero. Finally, over the range of output from B to T, the demand is inelastic. TR falls as price falls and sales volume increases. Hence MR is negative.

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