A production function relates the input of factors of production to the output of goods. In the basic production function inputs are typically capital and labor, though more expansive and complex production functions may include other variables such as land or natural resources. Output may be any consumer good produced by a firm. Cars, clothing, sandwiches, and toys are all examples of output.
Capital refers to the material objects necessary for production. Machinery, factory space, and tools are all types of capital. In the short run, economists assume that the level of capital is fixed - firms can't sell machinery the moment it's no longer needed, nor can they build a new factory and start producing goods there immediately. When looking at the production function in the short run, therefore, capital will be a constant rather than a variable. Although in reality a firm may own the capital that it uses, economists typically refer to the ongoing cost of employing capital as the rental rate because the opportunity cost of employing capital is the income that a firm could receive by renting it out. Thus, the price of capital is the rental rate.
Capital Goods
Capital equipment, like these motor graders, can vary in the long run but are fixed in the short run.
Labor refers to the human work that goes into production. Typically economists assume that labor is a variable factor of production; it can be increased or decreased in the short run in order to produce more or less output. The price of labor is the prevailing wage rate, since wages are the cost of hiring an additional unit of capital.
The marginal product of an input is the amount of output that is gained by using one additional unit of that input. It can be found by taking the derivative of the production function in terms of the relevant input. For example, if the production function is Q=3K+2L (where K represents units of capital and L represents units of labor), then the marginal product of capital is simply three; every additional unit of capital will produce an additional three units of output. Inputs are typically subject to the law of diminishing returns: as the amount of one factor of production increases, after a certain point the marginal product of that factor declines.