Returns to a Single Variable Input | Production | Microeconomics

If we keep all but one input fixed and vary only one, we can depict the effects on output in the form of a single input production function.

This is shown in Fig. 6.2, and may be explained as follows:

Suppose we are using water on an acre of land cultivated by some labour. No output is possible without water. As the first drops of water fall on the parched soil, output increases at an increasing rate, until w₀. The rate of increase in output is measured by the slope of the production function, which also is the marginal product of water (MP_w). It is evident from the figure that the marginal product of water rises initially, till w₀.

After w₀, the pressing thirst of land for water being slaked, output increases with water, but at a diminishing rate. That is to say, the marginal product of water declines after w₀. This is called diminishing marginal productivity. Finally, after w_s, when the land’s thirst for water is fully satiated, the marginal productivity of water turns negative. Further additions of water only reduce output.

From the above example, we see that the marginal product of an input may rise initially, and then eventually decline even to the point of turning negative. For the present, we note that the behaviour of the marginal product is intertwined with the behaviour of the total product and average product curves.

Relation between Marginal and Total Product:

These relations are illustrated in Fig. 6.2.

The following features stand out:

Firstly, where the marginal product of a variable input is positive, the total product is rising (w < w_s). Total product is constant or declining where the marginal product is zero or negative respectively. Secondly, where the marginal product is rising, total product increases at a rising rate (w < w₀). Thirdly, the total product curve changes curvature where the marginal product becomes constant (w₀) and switches direction. This point is called the point of inflexion.

Finally, as the marginal product begins to decline, the total product rises at a diminishing rate (w > w₀). Thus the single input production function gets an ‘S’ shape because marginal productivity first rises and then falls.

Relation between Marginal and Average Product:

The average product of a variable input is the total product divided by its quantity. We see in Fig. 6.2 that the average product of water is zero when no water is used. The movement of the average product depends upon its difference from the marginal product. This relation may be rendered as a general rule – So long as the marginal is greater than the average, it will pull up the average. The average will stay constant, if the marginal equals it. And the average will decline if the marginal is less than it. Thus, the marginal product pulls up or pulls down the average when it is above or below it.

Output Elasticity:

The output elasticity of a variable input (ε_i ↑) is the ratio of the percentage change in output to the percentage change in input. It can be measured at any point A on the single input production function by the ratio of the slope of the total product curve to the slope of the diagonal drawn through it. Thus, in Fig. 6.3,

It is evident that ε_i varies at different points of the production function. Algebraically, the output elasticity of any input at any point can be measured by the ratio of the marginal product of the input to its average product, at some level of input use.

Thus, the output elasticity of water e_w is given by:

This is a convenient result, because we know that when average product is rising/constant/falling, the marginal product is more than/equal to/less than it. Hence in these regions, e_w will be more than/equal to/less than one.

We can divide the single input production function into three parts based on the output elasticity, as in Fig. 6.4. In the first part of the production function, the output elasticity of the input is greater than one. That is to say, a one percent increases in the input results in a more than one percent increase in output. In the second part, the output elasticity is between zero and one. In the third part, the output elasticity is negative.

Single Input Variation – An Application:

Suppose we have an acre of cultivated land, where labour is the only variable input. Assuming that we are surplus maximizers, how much labour should we use? Let us answer this question in two phases. In the first phase, let labour be free. In the second phase, labour has to be paid for.

Free Labour:

If labour is free, its use does not increase the cost. But every extra man-day adds to the output, so long as its marginal product is positive. Hence we will expand employment when its marginal product is positive. We will continue to do so until the marginal product of labour becomes zero, and the need of land for labour is fully satiated. At this point, the surplus is maximum and the output elasticity of labour is zero (ε_l = 0 at l_s). We will not use more labour than l_s because more labour will reduce output. Hence, we will not produce in part III even if labour is free.

Wage Labour:

If labour is not free, its use increases cost. This increase in cost has to be compared with the marginal product before deciding whether to use or not to use, an extra man-day. Comparison however poses a problem. Labour has to be paid for in money, while the increase in output is physical, say – so many bags of corn. To compare the two, we must calculate what a man-day is worth in terms of bags of corn. This can be done by dividing the price of a man-day or the wage rate p_l by the price of corn p_c.

Suppose we find the ‘corn value’ of a man-day to be p_l/p_c, as shown in Fig. 6.5. In this case, the first man-days generate a deficit. They add less to the output than to the costs (f_l< p_l/p_c). The total deficit in this initial phase is shown by B. After l_l, man-days add more to the output than to cost. Since here, the marginal product of labour is greater than its cost (f_l > p_l/p_c), it pays to expand till l_π, when the two become equal. The total surplus generated by labour in this region (l_l → l_π) is given by A.

Employment Policy:

Only if A > B, is it worthwhile employing labour. That is to say, if labour is to be used at all, its use must generate a net surplus over its total cost. If this surplus is positive, we may fix the employment level at l_π. Here the marginal product of labour is equal to its cost measured in corn value.

Since the marginal product is declining at l_π, it falls below the cost of labour beyond this point. Hence after l_π extra man-days will be deficit making. They will reduce the surplus. The surplus from the use of labour is maximum at l_π, where its marginal product equals its corn-value and is declining.

Formally, the optimum level of labour use is characterized by – 

f_l = p_l/p_c

and f_ll < 0.

Thus we see that the decline in the marginal productivity plays a role of some importance in deciding the optimum level of labour use.