What is a Cost Function?

The Input Price Versus the Output Quantity

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A cost function is a function of input prices and output quantity whose value is the cost of making that output given those input prices, often applied through the use of the cost curve by companies to minimize cost and maximize production efficiency. There are a variety of different applications to this cost curve which include the evaluation of marginal costs and sunk costs

In economics, the cost function is primarily used by businesses to determine which investments to make with capital used in the short and long term.


Short-run Average Total and Variable Costs

To account for the business expenses related to meeting the supply and demand model of the current market, analysists break short-run average costs into two categories: total and variable. The average variable cost model determines the variable cost (typically labor) per unit of output wherein the wage of the laborer is divided by the quantity of output produced. 

In the average total cost model, the relationship between the cost per unit of output and the level of output is depicted via a curve graph. It uses the unit price of physical capital per unit time multiplied by the price of labor per unit time and added to the product of the quantity of physical capital used multiplied by the quantity of labor used. The fixed costs (capital used) are stable in the short-run model, allowing for fixed costs to decrease as production increases depending on labor used.

In this way, companies can determine the opportunity cost of hiring more short-term laborers. 

Short- and Long-run Marginal Curves

Relying on the observation of flexible cost functions is pivotal to successful business planning in regards to market expenses. The short-run marginal curve depicts the relation between incremental (or marginal) cost incurred in the short-run of production as it compares to the output of product produced.

It holds technology and other resources constant, focusing on the marginal cost and level of output instead. Typically the cost starts high with low-level output and dips to its lowest as output increases before rising again toward the end of the curve. This intersects the average total and variable costs at its lowest point. When this curve is above the average cost, the average curve is seen as rising, if the opposite is true it is seen as falling.

On the other hand, the long-run marginal cost curve depicts how each output unit relates to the added total cost incurred over a long run — or the theoretical period when all production factors are considered variable to minimize long-term total cost. Therefore, this curve calculates the minimum a total cost will increase per additional output unit. Due to cost minimization over a long period, this curve typically appears more flat and less variable, accounting for the factors that help mediate a negative fluctuation in cost.