One of the most important decisions a firm has to make involves the determination of the optimal combination of factors of production. This is because this is essential for the firm to reach the profit maximising point where marginal cost equals to marginal revenue. In order to reach this point, especially if the firm operates in a perfectly competitive market it has to produce where its factors of production cost as little as possible and produce as more as possible. In other words it has to minimise costs while maximising output.
The above illustrates some of the firm’s possible factor combinations using the total product curves of two factors of production and assuming the others are constant. However, the question remains, how is the firm to decide which one is the optimal factor combination?
In order to answer the above we have to derive the firm’s isoquant curves using the production surface formulated by the above total product functions and cutting slices from this surface as seen below.
In the above production surface points a and b lie on the Q1 isoquant which means they are points of equal output, thus both input combinations will yield the same output. The above curves therefore are essential for any firm to determine its optimal factor combination as they give all input combinations that yield a certain output. Three of the firm’s possible isoquants are given in the following diagram and are labelled Q1, Q2 and Q3. As we move from Q1 to Q3 the firm produces more output, as always we assume that we are operating within the ridge lines because if we were operating out of them that would mean that the firm isn’t being technically efficient
.
Considering the fact that the further out you go the greater the firms output it would make sense to think that a firm would produce at the isoquant that is as far out as possible. The further towards the left we move on an isoquant, the more capital we use and the further towards the right the more labour, this is called the marginal rate of technical substitution but, no matter what the combination though all points on an isoquant yield the exact same output. It is important for the firm to know the above because in order to be able to choose the long run input mix all possibilities of factor substitution must be known first before determining the optimal one. The marginal rate of technical substitution is illustrated by the slope of the isoquants above multiplied by -1 so that we get a positive number as opposed to a negative one because since isoquants are downward sloping they yield a negative derivative. However, firms do have constraints that force them to decide on which isoquant to operate. Like all resources a firms budget isn’t infinite thus it has to produce as much as possible without however exceeding the budget. This constraint is illustrated by the isocost curve below.
The reason why the isocost curve is a straight line is because the factor we use to employ labour and capital is money which is perfectly substitutable. The maximum amount of capital the firm can employ is denoted by B/PK which is the budget divided by the price of capital (interest) and the maximum amount of labour the firm can employ is denoted by B/PL which is the budget divided by the price of labour (wage). We derived this because if all the firm’s budget is spent on capital then:
B=PkK
Similarly if all the firms budget was used to employ labour then:
B=PlL
So we get the value of maximum quantity of each factor by solving the above two formulas for PK and PL respectively.
In order to determine the firm’s optimum combination of factors of production we are going to superimpose the isocost function on our isoquant map to determine the optimum level of production while taking into account our isocost curve as illustrated below.
In the above diagram we see that all points on Q3 are unattainable because the cost of production exceeds the firm’s budget. Points A and C are both within the firms budget but are technically inefficient because the firm could produce on a higher isoquant (Q2) which means it will produce a higher output with the same amount of money. B reflects the optimal combination of factors of production because at that point the slope of the isoquant curve and the isocost are equal i.e. the two lines are tangent to each other. This is because at this point the rule of optimal combination of factors holds. This rule states that the firm is using its factors of production optimally if the ratio of the prices of its factors equals to the ratio of their products. So (PK/PL)=(MPK/MPL) where PK/PL is the slope of the isocost curve and MPK/MPL is the slope of the isoquant curve. This explains why every firm in perfect competition operates where its price equals to its marginal costs.
The way a firm determines the optimal combination of factors of production can also be analysed using algebra and in particularly Lagrange Multipliers. First we assume that Q or our production function, is a function of capital and labour and the firms costs are its fixed costs and the prices of the factors of production multiplied by the units of each factor employed. So:
Q=f(K,L) and
C=F+PKK+PLL
In order to get a function that will give us the maximisation point given the constraint, we set our cost function to equal 0 so that adding it to the production function won’t affect its outcome and multiply it by a scalar λ. Then we subtract costs from our production function to determine when the production function is maximised given the cost constraint. We do this because this way we can find the point of technical efficiency, which can be found if for a given output we try to minimise inputs that generate this level of outputs or for a given level of inputs, maximise output. Thus the function we obtain looks something like this:
P=f(K,L)-λ(-C+F+PKK+PLL)
In order for the above to hold we have to have the following first order conditions, in other words the following functions have to be 0 so that the first derivative is 0 which gives us the maximum of this function i.e. the point where output is at a maximum given the constraint of the isocost curve. Thus we calculate the partial derivatives of the above function with respect to capital, labour and the scalar λ:
dl/dK= fk-Pk=0 so that the amount of capital used is maximised given the cost constraint
dl/dL= fl-PL=0 so that the amount of labour used is maximised given the cost constraint
dl/dλ= -C+F+PKK+PLL =0 so that your cost function is minimised.
From combining the first two equations we get that:
fk/fl=Pk/Pl
Since fl is the marginal product of labour and fk is the marginal product of capital than by rearranging, it works out that:
MPk/Pk=MPl/PL
Even though this is generally how firms according to economic theory determine the optimal combination of factors of production in reality we have to admit that very few, if any follow such a path in determining when to produce. This is for various reasons mainly revolving around the fact that this is very impractical. On the one hand isocost and isoquant lines change very frequently for a firm to have the ability to invest in research to determine which are the curves on which it operates. This is because prices of factors of production change very frequently and because each factor doesn’t have the exact same price. For example each unit of labour is usually paid a different wage and as a result isocost curves will change with every unit of labour employed. Moreover, technology changes too often for firms to assume that the productivity of their capital remains the same so the isoquant on which they operate changes with time.
Overall we see that a firm determines the optimal combination of factors of production by determining the point of tangency between its isoquant and isocost curves and we have illustrated this both by using diagrams and by illustrating it mathematically using Lagrange multipliers. Also I have shown how frequently firms in the real world don’t use the above analysis when determining the optimal combination of factors of production because it is too complicated and they often prefer cheaper less time consuming ways to determine the optimal combination of factors of production as it’s a routine they have to go through very frequently.