The distinction between "supply by a firm" and "supply to a firm" is most important for produced factors of production, especially the vast array of capital goods. The importance comes from the possible confusion over which side of the factor market a firm is on. This confusion results as much as anything from the notion that firms are typically on the supply-side of the product markets and the demand-side of factor markets. However, firms also appear on the supply-side of factor markets.
To illustrate this notion of "supply by a firm," consider the production of Quadra 9000 computers by the innovative folks at Quadra DG Computer Works. Standard operating procedure for Quadra DG Computer Works is to produce Quadra 9000 computers according to the standard principles of production and cost (law of diminishing marginal returns, profit-maximization, etc.). It then supplies these computers to consumers, like Duncan Thurly, residing in households all over the economy, through product markets (stores, mail order, etc.). Quadra DG Computer Works does the supplying of Quadra 9000 computers and the relevant phrase is "supply by a firm."
The phrase "supply by a firm" is also relevant if Quadra DG Computer Works supplies Quadra 9000 computers as capital goods rather than consumer goods. When Quadra 9000 computers are used as capital goods and they are considered a factor of production. For example, Quadra 9000 computers are an important productive factor for cellular telephone communication services offered by Digital Distance Wireless Telecommunications, for the fabrication of Flex-Star Interactive Trophy Plaques, and for the automated assembly of OmniMotors XL GT 9000 Sports Coupe automobiles.
When Quadra DG Computer Works supplies Quadra 9000 computers to Digital Distance Wireless Telecommunications, Flex-Star Interactive Trophy Plaques, and OmniMotors XL GT 9000 Sports Coupe, then the relevant phrase is also "supply by a firm."
The phrase "supply to a firm," however, is relevant from the other side of this computer market. From the viewpoint of Digital Distance Wireless Telecommunications, computers are being supply TO, not FROM. They are the buyers in this factor market. This supply to a firm involves the "supply" of Quadra 9000 computers, with the "to a firm" being Digital Distance Wireless Telecommunications.
The distinction between "supply by a firm" and "supply to a firm" is particularly important in terms of market control. For a perfectly competitive seller, the supply by a firm, that is the firm's supply curve, is the firm's marginal cost curve above average variable cost. For a monopoly, in which the seller has complete control over the selling-side of the market, then the supply by a firm is whatever the monopoly supplies, which may or may not follow a standard positively-sloped supply curve. For imperfectly competitive firms, including oligopoly and monopolistic competition, supply by a firm is corresponding portions of the overall market supply controlled by each firm.
By way of comparison, for a perfectly competitive buyer, "supply to a firm" is a perfectly elastic, horizontal supply curve. For a monopsony, in which the buyer has complete control over the buying-side of the market, then the supply to a firm is the overall market supply, which is typically positively sloped. For imperfectly competitive firms, including oligopsony and monopsonistic competition, supply to a firm is corresponding portions of the overall market supply.
SUPPLY BY A FIRM, AmosWEB Encyclonomic WEB*pedia, http://www.AmosWEB.com, AmosWEB LLC, 2000-2018. [Accessed: September 19, 2018].