We can use our knowledge of the relation between output elasticities of inputs, and the function coefficient to analyse the single input production function (Fig. 6.18). For convenience, we conduct our discussion with reference to the input production function for water.
To begin with, let us assume that the function coefficient is equal to one everywhere. That is to say, the production function is linear homogenous. Let the single input production function be divided into three parts on the basis of the output elasticity of water. Now let us try to determine in which part of the production function, a surplus maximizing agriculturist is likely to operate. We assume that water carries a non-negative price, and we measure its price in corn value as before.
The reader will recall, that even if water is free, its use will not be carried into the stage III. Here, the marginal product of water is negative. Using less water will therefore increase the output in stage III. So water use will be reduced in stage III, until we enter stage II. Stage III is not an economic stage of production.
The Limits of Stage I:
Then what about stage I? Stage I is also an uneconomic zone. When we begin applying water to land, the marginal product of water initially rises. It would be unwise to stop applying water where its marginal product is rising because we would be foregoing the surplus added by the later units A. Hence a surplus maximizing agriculturist will not operate in stage I either. Before Cassel, it was thought that stage I ends where the marginal product begins to decline (wm). Cassel shows that this was wrong in the case of a linear homogenous function.
At wm, the marginal product is higher than the average product, so that the output elasticity of water is greater than the function coefficient (which is equal to one). Hence, the marginal product of land at this level of water use must be negative.
Now, the use of land is ‘fixed’, in the sense that we cannot acquire more land. But land use can always be reduced. Since the marginal product of land is negative, reduction in its use increases the output at no extra cost. Hence, where the marginal product of land is negative, less land will be used. The reduced use of land will shift the average and marginal product curves of water. This process will go on until the marginal product of land ceases to be negative.
That is to say, the marginal and average product curves of water will shift until, in equilibrium, the marginal product of water equals or is less than its average product. Thus in equilibrium, the output elasticity of water cannot be more than one. Hence, the firm will not be found operating where the εw is greater than one. Cassel therefore concludes that the stage I of production extends up to we, where the εw equals one.
The Economic Stage of Production:
Cassel points out that the economic stage of production in a linear homogenous function is only stage II. Here the output elasticity of water varies between one and zero. In our figure, this extends from we to ws. These are the limits to the economic stage of production when the production function is homogenous of degree one, that is, when the production function shows constant returns to scale everywhere.