employs. • Learn about production functions with a single input, points the firm gets less output from its labor than it could). Points such as C .. The relationship between marginal product and average product is the same as the relationship. Conversely, estimates of the production function as such for the same purpose .. it gives a more comprehensive picture of the relationship between variables. To satisfy the mathematical definition of a function, In the production function itself, the relationship of output.

We might propose a production function for a good y of the following general form, first proposed by Philip Wicksteed Note that in writing production functions in this form, we are excluding joint production, i. Using Ragnar Frisch 's terms, we are concentrating on "single-ware" rather than "multi-ware" production.

### The Production Function

Output Y is measured on the vertical axis. The two inputs, which we call L and K which, for mnemonic purposes, can be called labor and capital,are depicted on the horizontal axes. We ought to now warn that henceforth, throughout all our sections on the theory of production, all capital is assumed to be endowed, i.

The hill-shaped structure depicted in Figure 2.

### Production function - Wikipedia

Notice that it includes all the area on the surface and in the interior of the hill. The production set is essentially the set of technically feasible combinations of output Y and inputs, K and L. A production decision -- a feasible choice of inputs and output - is a particular point on or in this "hill".

It will be "on" the hill if it is technically efficient and "in" the hill if it is technically inefficient. Obviously, the hill-shape of the production function indicates that the more we use of the factors, the greater output is going to be at least up to the some maximum, the "top" of the hill.

The round contours along the production hill can be thought of as topographic contours as seen on maps and will serve as isoquants in our later analysis. The slope of the hill viewed from the origin captures the notion of returns to scale. Throughout the next few sections, we shall be outlining the technical properties of the production function.

The representation of production functions in the diagrammatic form of "hills" and the corresponding analysis of production theory in terms of isoquant contours, etc. In the s, the Neo-Walrasians e. Koopmans; Debreuapproached the analysis of the technical properties of production in a somewhat different spirit. Specifically, instead of focusing on the "production function" and its derivatives as the Paretians had done, the Neo-Walrasians preferred to analyze it via vector space methods and convex analysis.

On a more formal note, we should outline the properties of the production function, as normally assumed by Neoclassical economists. We shall define an input space as the acceptable set of inputs for our economy. Commonly, a bundle of factor inputs x is deemed "acceptable" if every entry in that vector, i. Let y be output, which is assumed to be a single, finite number, i.

These assumptions will be clarified as we go on. For the moment, let us just make the following notes. This is somewhat self-evident, at least for economists. Obviously, in other walks of life, one can produce something without inputs e.

The monotonicity assumption A.

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Although common, we will have more to say on this later. We shall have much more to say on this later.

In other words, adding more units of any factor input will increase output or at least not reduce it. This is the heart of assumption A. However, it is also common in Neoclassical theory to also impose A. It is often the case in economics that the quasi-concavity assumption implies that: It is worthwhile to spend a few moments on the diminishing marginal productivity assumption.

This means more we add of a particular factor input, all others factors remaining constant, the less the employment of an additional unit of that factor input contributes to output as a whole.

This concept performs the same function in production functions as diminishing marginal utility did in utility functions. Conceptually, however, they are quite distinct.

It was applied more generally to other factors of production by proto-marginalists such as J. The apotheosis of the concept is found in the work of John Bates Clark, and, more precisely, in Philip H. It was originally called the "Law of Diminishing Returns", although in order to keep this distinct from the idea of decreasing returns to scale, we shall refer to it henceforth as the "Law of Diminshing Marginal Productivity" Let us first be clear about the definition of the marginal productivity of a factor.

Mathematically, however, it is more convenient to assume that D x is infinitesimal. If we do not wish to assume that factor units are infinitely divisible or if we do not assume that the production function is differentiable, we cannot express the marginal product mathematically as a derivative.

Hobsondefined "marginal product" differently: This caused a problem for the "adding up" issue in the marginal productivity theory of distribution, although, as was clarified later, when marginal product is not defined so discretely, it does not make a difference which measure we use. For a useful discussion of the dilemma involving the measurement of the "marginal unit", see the discussion in Fritz Machlup Finally, we must note that a far more novel and interesting definition of marginal productivity was introduced by Joseph M.

Ostroywhere the concept is redefined in terms of contributions to tradeable surpluses, and thus both widened and deepened in scope. Taking Clark's famous analogy: Two laborers on the same ground will get less per man; and, if you enlarge the force to ten, the last man will perhaps get wages only.

The implication, then, is that as we increase the amount of labor applied to a particular fixed amount of land, each additional unit will increase total output but by smaller and smaller increments.

When the field is empty, the first laborer has absolutely free range and produces as much as his body can reasonably do, say ten bushels of corn.

When you add a second laborer to the same field, total output may increase, say to eighteen bushels of corn.

**Understanding the relationships between Total, Marginal and Average Product**

Thus, the marginal product is eight. The basic idea is that by adding the second man, the field gets "crowded" and the men begin to get in each other's way. If that explanation does not seem credible, think of the units of labor in terms of labor-hours for a single man: The diminution can be explained in this case as an "exhaustion" effect.

Taking another example, suppose we apply a man to a set of shoe-making tools and a given swathe of leather; let us say he can produce ten pairs of shoes in a day. Moysan and Senouci provide an analytical formula for all 2-input, neoclassical production functions.

A typical quadratic production function is shown in the following diagram under the assumption of a single variable input or fixed ratios of inputs so they can be treated as a single variable.

All points above the production function are unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified level of usage of the input. From point A to point C, the firm is experiencing positive but decreasing marginal returns to the variable input. As additional units of the input are employed, output increases but at a decreasing rate.

Point B is the point beyond which there are diminishing average returns, as shown by the declining slope of the average physical product curve APP beyond point Y. Point B is just tangent to the steepest ray from the origin hence the average physical product is at a maximum. Beyond point B, mathematical necessity requires that the marginal curve must be below the average curve See production theory basics for further explanation.

Stages of production[ edit ] To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 from the origin to point B the variable input is being used with increasing output per unit, the latter reaching a maximum at point B since the average physical product is at its maximum at that point.

Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.

In Stage 2, output increases at a decreasing rate, and the average and marginal physical product both decline. However, the average product of fixed inputs not shown is still rising, because output is rising while fixed input usage is constant.