Production function

Specifying the production function

A production function can be expressed in a functional form as the right side of

${\displaystyle Q=f(X_{1},X_{2},X_{3},\dotsc ,X_{n})}$

where ${\displaystyle Q}$ is the quantity of output and ${\displaystyle X_{1},X_{2},X_{3},\dotsc ,X_{n}}$ are the quantities of factor inputs (such as capital, labour, land or raw materials). For ${\displaystyle X_{1}=X_{2}=...=X_{n}=0}$ it must be ${\displaystyle Q=0}$ since we cannot produce anything without inputs. The level sets of a production function with multiple inputs are known as isoquants.

If ${\displaystyle Q}$ is a scalar, then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if ${\displaystyle f}$ maps from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} ^{k}}$ then it is a joint production function expressing the determination of ${\displaystyle k}$ different types of output based on the joint usage of the specified quantities of the ${\displaystyle n}$ inputs.

One formulation is as a linear function:

${\displaystyle Q=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}+\dotsb +a_{n}X_{n}}$

where ${\displaystyle a_{1},\dots ,a_{n}}$ are parameters that are determined empirically. Linear functions imply that inputs are perfect substitutes in production. Another is as a Cobb–Douglas production function:

${\displaystyle Q=a_{0}X_{1}^{a_{1}}X_{2}^{a_{2}}\cdots X_{n}^{a_{n}}}$

where ${\displaystyle a_{0}}$ is the so-called total factor productivity. The Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, the output will not change. This production function is given by

${\displaystyle Q=\min(a_{1}X_{1},a_{2}X_{2},\dotsc ,a_{n}X_{n}).}$

Other forms include the constant elasticity of substitution production function (CES), which is a generalized form of the Cobb–Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters (${\displaystyle a_{0},\dots ,a_{n}}$) vary from company to company and industry to industry. In the short run, production function at least one of the ${\displaystyle X}$'s (inputs) is fixed. In the long run, all factor inputs are variable at the discretion of management.

Moysan and Senouci (2016) provide an analytical formula for all 2-input, neoclassical production functions.^[4]

Production function as a graph

Any of these equations can be plotted on a graph. 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 and Sickles and Zelenyuk (2019) for more extensive discussions of various production functions, their generalizations and estimations).

Stages of production

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. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 2. In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input.

Shifting a production function

By definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input. If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand. For example, you may only need to increase production by million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year.

If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit-maximizing output level will now be in stage 2.

Homogeneous and homothetic production functions

There are two special classes of production functions that are often analyzed. The production function ${\displaystyle Q=f(X_{1},X_{2},\dotsc ,X_{n})}$ is said to be homogeneous of degree ${\displaystyle m}$, if given any positive constant ${\displaystyle k}$, ${\displaystyle f(kX_{1},kX_{2},\dotsc ,kX_{n})=k^{m}f(X_{1},X_{2},\dotsc ,X_{n})}$. If ${\displaystyle m>1}$, the function exhibits increasing returns to scale, and it exhibits decreasing returns to scale if ${\displaystyle m<1}$. If it is homogeneous of degree ${\displaystyle 1}$, it exhibits constant returns to scale. The presence of increasing returns means that a one percent increase in the usage levels of all inputs would result in a greater than one percent increase in output; the presence of decreasing returns means that it would result in a less than one percent increase in output. Constant returns to scale is the in-between case. In the Cobb–Douglas production function referred to above, returns to scale are increasing if ${\displaystyle a_{1}+a_{2}+\dotsb +a_{n}>1}$, decreasing if ${\displaystyle a_{1}+a_{2}+\dotsb +a_{n}<1}$, and constant if ${\displaystyle a_{1}+a_{2}+\dotsb +a_{n}=1}$.

If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". A linearly homogeneous production function with inputs capital and labour has the properties that the marginal and average physical products of both capital and labour can be expressed as functions of the capital-labour ratio alone. Moreover, in this case, if each input is paid at a rate equal to its marginal product, the firm's revenues will be exactly exhausted and there will be no excess economic profit.^{[5]: pp.412–414}

Homothetic functions are functions whose marginal technical rate of substitution (the slope of the isoquant, a curve drawn through the set of points in say labour-capital space at which the same quantity of output is produced for varying combinations of the inputs) is homogeneous of degree zero. Due to this, along rays coming from the origin, the slopes of the isoquants will be the same. Homothetic functions are of the form ${\displaystyle F(h(X_{1},X_{2}))}$ where ${\displaystyle F(y)}$ is a monotonically increasing function (the derivative of ${\displaystyle F(y)}$ is positive (${\displaystyle \mathrm {d} F/\mathrm {d} y>0}$)), and the function ${\displaystyle h(X_{1},X_{2})}$ is a homogeneous function of any degree.

Aggregate production functions

In macroeconomics, aggregate production functions for whole nations are sometimes constructed. In theory, they are the summation of all the production functions of individual producers; however there are methodological problems associated with aggregate production functions, and economists have debated extensively whether the concept is valid.^[3]

Criticisms of the production function theory

There are two major criticisms^[which?] of the standard form of the production function.^[6]

The practice of production functions

The theory of the production function depicts the relation between physical outputs of a production process and physical inputs, i.e. factors of production. The practical application of production functions is obtained by valuing the physical outputs and inputs by their prices. The economic value of physical outputs minus the economic value of physical inputs is the income generated by the production process. By keeping the prices fixed between two periods under review we get the income change generated by a change of the production function. This is the principle how the production function is made a practical concept, i.e. measureable and understandable in practical situations.

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