In production processes where some inputs, such as land or machinery, are fixed, adding more of a variable input like labor causes total output to increase at a slower and slower rate. This relationship results in the slope of the production function gradually decreasing as more labor is utilized, a key economic principle known as diminishing marginal product.

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For a mathematical expression to serve as a plausible production function, it must satisfy several key properties. Firstly, zero input should result in zero output, and any positive input should yield a positive amount of output. Secondly, the function must be increasing, meaning that as the quantity of input rises, the total output also increases. This implies that the function has a positive slope, and its derivative is always positive.

Properties of a Plausible Production Function

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CORE Econ - The Economy 2.0: Microeconomics

A key characteristic of a production function is its increasing nature. This property dictates that as the input, denoted by `x`, is increased, the total output, `y`, will also rise. Consequently, the slope of the production function is always positive. This is formally expressed by stating that the function's derivative is greater than zero, written as `$\frac{dy}{dx} > 0$` or, equivalently, `f'(x) > 0`.

Increasing Nature of the Production Function

Fixed Inputs and Diminishing Marginal Product

A farmer's production technology is modeled by the function $y=g(h)$, where $h$ represents non-negative daily work hours and $y$ is the resulting grain output. This function is characterized by being increasing and strictly concave, properties which can be confirmed through algebraic verification. It is also assumed that production is zero with zero work hours, so $g(0)=0$. For analytical purposes in economic models, this function is frequently expressed in a simple algebraic form.

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