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Mathematical Representation of a Concave Production Function
A production function exhibits diminishing marginal productivity when it has a concave shape, meaning its slope decreases as more input is used. For a production process to be plausible, the function must be increasing, meaning more input yields more output (). The property of diminishing returns is mathematically captured by the function being strictly concave, which means its second derivative is negative (). For example, a technology for producing grain from hours of work () can be modeled by a production function , where the function must be both increasing and strictly concave.
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Mathematical Representation of a Concave Production Function
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A firm's production schedule shows how its total output changes as it adds more units of a single input, holding all other inputs constant. The table below shows the additional output (marginal product) generated by each successive unit of input.
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A production function that exhibits a continuously declining marginal product for each additional unit of input is an example of a(n) ________ function.
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Mathematical Representation of a Concave Production Function
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A production function
y = f(x)that models a process with diminishing marginal productivity must satisfy two conditions for all positive input levelsx: its first derivative must be positive (f'(x) > 0) and its second derivative must also be positive (f''(x) > 0).Interpreting the Derivatives of a Production Function
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