OPTIMAL COMBINATION OF INPUTS

In the previous post we  learned that any desired level of output can be produced using a number of different combinations of inputs. One of the decision problems that concerns a production process manager is, which input combination to use. That is, what is the optimal input combination? While all the input combinations are technically efficient, the final decision to employ a particular input combination is purely an economic decision and rests on cost (expenditure). Thus, the production manager can make either of the following two input choice decisions:

  1. Choose the input combination that yields the maximum level of output with a given level of expenditure.
  2. Choose the input combination that leads to the lowest cost of producing a given level of output.

Thus, the decision is to minimize cost subject to an output constraint or maximize the output subject to a cost constraint. We will now discuss these two fundamental principles. Before doing this we will introduce the concept isocost, which shows all combinations of inputs that can be used for a given cost. 

Isocost Lines

Recall that a universally accepted objective of any firm is to maximise profit. If the firm maximises profit, it will necessarily minimise cost for producing a given level of output or maximise output for a given level of cost. Suppose there are 2 inputs: capital (K) and labour (L) that are variable in the relevant time period. What combination of (K,L) should the firm choose in order to maximise output for a given level of cost?

If there are 2 inputs, K,L, then given the price of capital (Pk) and the price of labour (PL), it is possible to determine the alternative combinations of (K,L) that can be purchased for a given level of expenditure. Suppose C is total expenditure, then

C= PL* L + PK* K

This linear function can be plotted on a graph.

fig-7-isocost-line

If only capital is purchased, then the maximum amount that can be bought is C/Pk shown by point A in figure-7. If only labour is purchased, then the maximum amount of labour that can be purchased is C/PL shown by point B in the figure. The 2 points A and B can be joined by a straight line. This straight line is called the isocost line or equal cost line. It shows the alternative combinations of (K,L) that can be purchased for the given expenditure level C. Any point to the right and above the isocost is not attainable as it involves a level of expenditure greater than C and any point to the left and below the isocost such as P is attainable, although it implies the firm is spending less than C. You should verify that the slope of the isocost is

equation-1-optimisation

consider only the case when the firm spends the entire budget of 200. The alternative combinations are shown in the figure -8.

fig-8-shifting-of-isocost

The slope of this isocost is –½. What will happen if labour becomes more expensive say PL increases to 20? Obviously with the same budget the firm can now purchase lesser units of labour. The isocost still meets the Y–axis at point A (because the price of capital is unchanged), but shifts inwards in the direction of the arrow to meet the X-axis at point C. The slope therefore changes to –1. Similarly we can work out the effect on the isocost curve on the following:

(i) decrease in the price of labour

(ii) increase in the price of capital

(iii) decrease in the price of capital

(iv) increase in the firms budget with no change in the price of labour and capital

[Hint: The slope of the isocost will not change in this case] 

Optimal Combination of Inputs: The Long Run

When both capital and labour are variable, determining the optimal input rates of capital and labour requires the technical information from the production function i.e. the isoquants be combined with market data on input prices i.e. the isocost function. If we superimpose the relevant isocost curve on the firm’s isoquant map, we can readily determine graphically as to which combination of inputs maximise the output for a given level of expenditure.

Consider the problem of minimising the cost of a given rate of output. Specifically if the firm wants to produce 50 units of output at minimum cost. Two production isocosts have been drawn in Figure-9. Three possible combinations (amongst a number of more combinations) are indicated by points A, Z and B in Figure-9. Obviously, the firm should pick the point on the lower isocost i.e point Z. In fact, Z is the minimum cost combination of capital and labour. At Z the isocost is tangent to the 50 unit isoquant.

Alternatively, consider the problem of maximising output subject to a given cost amount. You should satisfy yourself that among all possible output levels, the maximum amount will be represented by the isoquant that is tangent to the relevant isocost line. Suppose the budget of the firm increases to the amount shown by the higher of the two isocost lines in Figure-9, point Q or 100 units of output is the maximum attainable given the new cost constraint in Figure -9.

fig-9-optimal-combination-of-inputs

Regardless of the production objective, efficient production requires that the isoquant be tangent to the isocost function. If the problem is to maximize output, subject to a cost constraint or to minimise cost for a given level of output, the same efficiency condition holds true in both situations. Intuitively, if it is possible to substitute one input for another to keep output constant while reducing total cost, the firm is not using the least cost combination of inputs. In such a situation, the firm should substitute one input for another.

For example, if an extra rupee spent on capital generates more output than an extra rupee spent on labour, then more capital and less labour should be employed. At point Q in Figure-9, the marginal product of capital per rupee spent on capital is equal to the marginal product of labour per rupee spent on labour. Mathematically this can be shown as

equation-2-aequation-2-b

This cannot be an efficient input combination, because the firm is getting more output per rupee spent on labour than on capital. If one unit of capital is sold to obtain 2 units of labour (Pk = 20, PL = 10), net increase in output will be 60 *. Thus the substitution of labour for capital would result in a net increase in output at no additional cost. The inefficient combination corresponds to a point such as A in Figure-9. At that point two much capital is employed. The firm, in order to maximise profits will move down the isocost line by substituting labour for capital until it reaches point Q. Conversely, at a point such as B in figure-9 the reverse is true – there is too much labour and the inequality

equation-3-optimisation

[*Since the MPL = 50, 2 units of labour produce 100 units, while reducing capital by 1 unit decreases output by 40 units (MPk = 40). Therefore, net increase is 60 units. This, of course, assumes that MPL and MPk remain constant in the relevant range. We know that as more labour is employed in place of capital, MPL will decline and MPK will increase (this follows from the law of diminishing returns) and thus equation (1) will be satisfied.]

This means that the firm generates more output per rupee spent on capital than from rupees spent on labour. Thus a profit maximising firm should substitute capital for labour.

Suppose the firm was operating at point B in Figure-9. If the problem is to minimise cost for a given level of output (B is on the isoquant that corresponds to 50 units of output), the firm should move from B to Z along the 50-unit isoquant thereby reducing cost, while maintaining output at 50. Alternatively, if the firm wants to maximise output for given cost, it should more from B to Q, where the isocost is tangent to the 100-unit isoquant. In this case output will increase from 50 to 100 at no additional cost. Thus both the following decisions:

(a) the input combination that yields the maximum level of output with a given level of expenditure, and

(b) the input combination that leads to the lowest cost of producing a given level of output are satisfied at point Q in Figure -9.

You should be satisfied that this is indeed the case.

The isocost-isoquant framework described above lends itself to various applications. It demonstrates, simply and elegantly, when relative prices of inputs change, managers will respond by substituting the input that has become relatively less expensive for the input that has become relatively more expensive. On average, we know that compared to developed countries like the US, UK, Japan and Germany, labour in India is less expensive. It is not surprising therefore to find production techniques that on average, use more labour per unit of capital in India than in the developed world. For example, in construction activity you see around you in your city, inexpensive workers do the job that in developed countries is performed by machines.

RETURNS TO SCALE

Another important attribute of production function is how output responds in the long run to changes in the scale of the firm i.e. when all inputs are increased in the same proportion (by say 10%), how does output change. Clearly, there are 3 possibilities. If output increases by more than an increase in inputs (i.e. by more than 10%), then the situation is one of increasing returns to scale (IRS). If output increases by less than the increase in inputs, then it is a case of decreasing returns to scale (DRS). Lastly, output may increase by exactly the same proportion as inputs. For example a doubling of inputs may lead to a doubling of output. This is a case of constant returns to scale (CRS).

figure-10-isoquents-showing-return-to-scale

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