1.3.2

Index

1. One Variable Factor Production Function

2. Two Variable Factor Production Function

The production function for a firm is the relationship between the quantities of inputs per time period and the maximum output that can be produced. It can be calculated for one or more than one variable factors of production. The one variable factor of production function corresponds to the short-run during which at least one factor of production is fixed (M&Y 10th Fig. 7.1; M&Y 11th Fig. 6.1; B&B Fig. 6.2; B&Z Fig. 7.1). The more than one variable factor function corresponds to either the short- or the long-run (M&Y 10th Fig. 7.7; M&Y 11th Fig. 6.7; B&B Fig. 6.8; B&Z Fig. 7.3).

Assuming that all but one factor of production are fixed, the production curve for total output shows an initially rising section that peaks and then declines if additional variable inputs are added (M&Y 10th Fig. 7.1; M&Y 11th Fig. 6.1; B&B Fig. 6.2; B&Z Fig. 7.1). No rational producer will go beyond the peak. Why does the curve peak and then turn down? Congestion. With fixed capital plant and equipment additional labour initially increases output but eventually an additional worker simply gets in the way of other workers and output actually declines.

As additional workers are added each contributes to output. If we take total output at each level of employment we can calculate both the average output per worker and the marginal or additional output contributed by one more worker (M&Y 10th Fig. 7.2; M&Y 11th Fig. 6.2; B&B Fig. 6.4; B&Z Fig. 7.1). As can be seen, the addition of a worker initially increases the average output per worker but then the average declines as the marginal output per additional worker gradually declines following the Law of Diminishing Marginal Returns: the marginal product of any input will (eventually) fall as the employment of that input increases - assuming other factors of production are held constant.

Given the production function for one variable factor, the average product can be calculated as the slope of the line measuring the distance from the point of origin to the total product curve (M&Y 10th Fig. 7.4; M&Y 11th Fig. 6.4; B&B Fig. 6.4; B&Z Fig. 7.2). Similarly, marginal product can be measured as the slope of the total product curve (M&Y 10th Fig. 7.5; M&Y 11th Fig. 6.5; B&B Fig. 6.4; B&Z Fig. 7.2).

As with the consumer utility function for two goods and services, a production function can be plotted for two variable factors of production, e.g. Q = f (k, l). Factors can be combined in different ways to generate the same level of output and graphed as an isoquant, that is a curve showing equal levels of production for different combinations of inputs (M&Y 10th Fig. 7.7; M&Y 11th Fig. 6.7; B&B Fig. 6.8; B&Z Fig. 7.3). The slope of an isoquant measures the Marginal Rate of Technical Substitution (MRTS) or the rate at which one input can be exchanged for another while maintaining the same level of output (M&Y 10th Fig. 7.8; M&Y 11th Fig. 6.9; B&B Fig. 6.10; B&Z Fig. 7.4). The rate is measured as an absolute value, that is the rate if actually negative but it is reported as a positive number. It is analogous to the Marginal Rate of Substitution for the consumer. A straight line drawn from the origin intercepting higher and higher isoquants marks a constant or fixed input ratio (K/L).

In the long-run a firm can increase the scale of its plant. As it does so it may experience constant, increasing or decreasing returns to scale (M&Y 10th Fig. 7.9; M&Y 11th Fig. 6.11; B&B Fig. 6.18; B&Z Fig. 7.5). Increasing returns to scale reflects that, for example, doubling employment and capital results in more than double the output. Similarly, decreasing returns to scale reflects that say a doubling of employment and capital results in less than double the output. Constant returns to scale reflects that a doubling of factors results in an exact doubling of output.

Economies or increasing returns of scale exist when the average cost falls as output rises. Economies of scale are due to specialization and division of labour. A related concept is economies of team production involving specialization in mutually supportive tasks or team production. Putting a designer together with an engineer and other specialists within the firm may be cheaper and much more effective than trying to buy such services on the market and then try to coordinate their various outputs.

On the other hand, diseconomies of scale occur when the average cost increases as output rises. Diseconomies of scale can occur as a firm grows in size and complexity. Thus some things are simply more cheaply done at a smaller scale of production, e.g. due to congestion. In fact, some industries are based on 'small scale' production, e.g. creative products like art, advertising and R&D. These activities are often more efficiently conducted in small rather than large firms. In entertainment and advertising the same result can sometimes be achieved by creating special small scale production units while the main administration of the enterprise handles marketing and other activities that benefits from economies of scale.

The artist, for example, tends by nature to be a risk-taking entrepreneur who does not readily submit to organizational goals.

In consequence, ... the artist functions as an independent entrepreneur ... or ... as a member of a very small firm which he can dominate or in which he can preserve the identity of his work. A few industries - the motion picture firms, television networks, the large advertising agencies - must, by their nature, associate artists with rather complex organization. All have a well-reported record of dissonance and conflict between the artists and the rest of the organization... Frequently the problem is solved by removing actors, actresses, scriptwriters, directors, composers, copywriters and creators of advertising commercials from the technostructure ... and reconstituting them in small independent companies. The large firm then confines itself to providing the appropriate facilities for producing and - more importantly - marketing, exhibiting or airing the product. Similarly painters, sculptors, concert pianists and novelists function, in effect, as one-man firms or, as in the case of rock, dance and folk music groups, as small partnerships and turn to larger organizations to market themselves or their products (J.K. Galbraith, Economics and the Public Purpose, Signet, 1973, 60).

Like consumer indifference curves that theoretically can be measured using revealed preference, isoquants can be measured by collecting relevant evidence from the 'real' world.

All cost considerations in a production function can be overturned due to technological change, e.g. information technology in the 1980s reduced the need for middle management and resulted in significant 'downsizing' of large firms and government. Why do such change takes place; why are something or process invented and others are not; why are some things and process that are innovated while others are not has been called the 'measure of our economic ignorance'. Virtually over night the entire production function may need to be re-written.