Q = f(k, L)
To increase output in the short run, the firm must increase the quantity of the variable factor labour. To know when the firm is employing an optimal number of units of labour it must calculate the effect on output of making a marginal change. The law of diminishing marginal returns (LDMR) states that: as a firm uses more of a variable input, with a given quantity of fixed inputs, the marginal product of the variable input eventually diminishes. (Parkin et Al, Economics 3rd e).
Diagram 1 below shows that given the production function above, as the firm employs more units of labour (L) total product (TP) – the total output produced with a given quantity of a fixed input - increases at a decreasing rate. Eventually it becomes negative with the addition of L1 units of labour. Units of capital (k) remain fixed. It has also been assumed that the price of labour (PL) and capital (PK) remain fixed.
In diagram 2, labour has been substituted for capital as the fixed factor of production. The slope of each TP curve is dependent on the LDMR for that factor. All points that lie on the TP curve are technically efficient.
The total product curves above can be joined to give the production surface in diagram 3 below. This shows all the possible combinations of labour and capital attainable to produce given levels of output.
However, when trying to evaluate what the optimal combination of inputs are for the firm, it is more useful to take a single, given level of output and analyse all the input combinations which yield that level of output. The curve that represents this information is known as an isoquant. The isoquant map shown in diagram 4 is a collection of all isoquants which correspond to the given production function above. Q1, Q2 and Q3 all represent a given level of output. and the axes correspond to the two inputs. The level of output increases as you move out from the origin; Q3 is a greater output level than Q1.
Isoquants can be used to determine the optimal factor combination in both the short and long run. Considering the short run first, assume the firm is producing output Q1 and capital is fixed at k1. The decision facing the firm is to make sure it uses the optimal amount of labour to produce this output. If it tried to use the input permutation k1 / L1 as shown by point a it would be unable to produce output Q1 hence the combination is not viable. Likewise, if the firm tried to use input permutation k1 / L3 as shown by point b, despite this being a feasible combination, it is using more labour than necessary to produce output Q1 and is failing to minimise costs. Hence the optimal combination of factors is achieved at point c, where the units of labour is L2 . This can be summarised by ensuring the firm satisfies the production function f(L, k1) = Q1.
It is also necessary to consider how a firm can achieve an optimal combination of the factors of production when it is possible to vary of its inputs. The consequence of this is that it may be possible to substitute one factor for another. This allows the firm to move along an isoquant without be affected by a fall in output. The slope of the isoquant tells us the rate at which the firm can substitute one factor for another while still producing the same output. This rate is known as the marginal rate of technical substitution (MRTS). (Defn: Katz and Rosen, Microeconomics 3rd e)
Diagram 5 below shows how a decrease in capital used in the production process from k1 to k2 is counterbalanced by an increase in the units of labour from L1 to L2. The combination of factors has changed from point A to point B but the total output has remained constant at Q1 because there has only been a movement along the isoquant. The rate at which it is possible for this exchange of factors to occur is given by the slope, as defined by MRTS.
In order to establish which combination of factors on the isoquant is economically efficient to the firm, we need to consider the cost of the individual factors. An input combination is economically efficient when it has the lowest opportunity cost of those input combinations that can be used to produce the desired output. (Defn: Katz and Rosen, Microeconomics 3rd e). In order to evaluate which input combination is the most cost efficient, isocost lines are used. These are lines representing input combinations of equal cost.
This is shown in diagram 6 below. PK represents the price of capital
PL represents the price of labour
B is the firms budget
Point A = the maximum number of workers the firm can employ.
Point B = the maximum units of capital the firm can employ.
By combining the isoquant and isocost lines of a firm from diagrams 5 and 6 in the diagram below, we are effectively determining the permutation of the factors of production subject to a budget constraint, thus achieving the optimal combination.
Point D lies to the right of the isocost and therefore cannot be achieved due to budget constraints. There will always be a more productive level than the one available to a firm given its budget constraint. Points A and B are not profit maximising levels of output, since X2 represents a higher level of output than X1. Therefore point C, where the isocost is tangential to the isoquant, is the optimal point of factor combination for a firm.
It is also possible for firms to derive this optimal point algebraically using Langrangian Multipliers. Given the production function Q = f(L, k) and combining it with the Langrangian multiplier λ subject to a constraint and equating using partial fractions, the outcome is the optimal point of production in algebraic form.
Given that: Q = f(L, k) c = F + PL + Pk
0 = λ(-c + F + PL + Pk)
Hence: Q = f(L, k) - λ(-c + F + PL + Pk)
= fk - λPL = 0 = fL - λPk = 0
= -c + F + PL + Pk = 0
Consequently: fk = rate of change in output with respect to capital = MPk
Rate of change in output with respect to labour MPL
Using this and the fact that MPk = Pk the equation can be rearranged to give: MPk = MPL
MPL PL Pk PL
The equation above shows the relationship between the cost of producing and production itself. From this the firm can find the value of one of the factors from another.
The optimal combination of factors can also be determined for a firm in the long run. Given that the firm only has two factors of production assumed to be k and L, then the initial approach that the firm will take is parallel to that taken in the short run. However, when analysing the long run, it is necessary to consider that there are other variable factors involved such as land. The extent to which this factor has a bearing on the output will depend on how significant it is in the production process. But consider the concept of increasing all the factors of production by the same amount (β). Due to the change in β, the output itself changes by a factor α. This is particularly important as the costs of production can be decreased over time in the long run if certain criteria are met.
For instance, if α • β then the firm will experience increasing returns to scale, or economies of scale, where average costs (being the sum of average variable costs and average fixed costs) in the long run will decrease as output rises. Hence the firm can maximise its profits over time by spending less than required. If α = β the firm will experience constant returns to scale where the long run average costs remain constant as output rises. Finally, if α > β, the firm experiences diseconomies of scale when long run average costs increase as output rises.
In reality it is rare to find a case where all the factors of production are increased by the same amount so most firms experience economies of size.
Bibliography
Begg, Fischer, Dornbusch, (1995), “Economics”, Fifth Edition, McGraw Hill
Katz M.L, Rosen H.S, (1998) “Microeconomics”, Third Edition, McGraw Hill
Parkin. M, Powell. M, Matthews. K, (1997) “Economics”, Third Edition, Addison Wesley
