Marginal product
In economics, the marginal product of a factor of production—such as labor, capital, or materials—refers to the additional output generated by employing one more unit of that factor while holding all other inputs constant.[1] This concept is central to understanding how firms transform inputs into outputs through a production function, typically expressed as Q = f(L, K), where Q is total output, L is labor, and K is capital.[2] The marginal product is calculated as the change in output divided by the change in the input, or MP = \Delta Q / \Delta X, where X represents the variable input; in continuous terms, it is the partial derivative of the production function with respect to that input.[2] For example, the marginal product of labor (MPL) measures the extra output from one additional worker, assuming fixed capital and technology.[3] A key feature is the law of diminishing marginal returns, which holds in the short run—when at least one input is fixed—that successive units of a variable input yield progressively smaller increases in output due to constraints like overcrowding or resource limitations.[4] Initially, marginal product may rise due to specialization, but it eventually declines, flattening the slope of the production function.[2] Marginal product plays a crucial role in firm decision-making, particularly in profit maximization, where businesses hire inputs up to the point where the value of the marginal product equals the input's cost—for labor, this means wage equals the marginal revenue product in competitive markets.[3] In imperfectly competitive settings, it influences employment levels by equating wage to the marginal revenue product, often resulting in lower hiring than in perfect competition.[3] In the long run, with all inputs variable, marginal products guide cost minimization along isoquants, where the optimal input mix occurs when the ratio of marginal products equals the ratio of input prices, affecting scale and efficiency.[4] Diminishing marginal returns also underpin rising marginal and average costs, shaping supply curves and market dynamics.[4]Definition and Formulation
Core Definition
The marginal product of an input, such as labor or capital, refers to the additional output produced by employing one more unit of that variable input while holding all other inputs constant.[1] This concept, often denoted in economic analysis as the marginal product of labor (MPL) or marginal product of capital (MPK), measures the incremental contribution of the extra input to total production in the short run, where at least one factor remains fixed.[5] The idea of marginal product traces its origins to classical economics in the 19th century, with significant development by American economist John Bates Clark in the late 1800s as part of the neoclassical revolution.[6] Clark's seminal work, The Distribution of Wealth (1899), formalized marginal productivity theory, arguing that factors of production receive remuneration equal to their marginal contributions to output under competitive conditions, building on earlier ideas from economists like John Stuart Mill.[7] This framework shifted economic thought from labor theories of value toward a productivity-based explanation of income distribution.[6] Marginal product typically focuses on the marginal physical product, which quantifies the physical increase in output units, distinct from the marginal revenue product that incorporates the monetary value of that output by multiplying the physical increment by the marginal revenue from selling additional units.[8] For instance, in agriculture, the marginal physical product of labor might manifest as the extra bushels of crops harvested when one additional worker is added to a fixed plot of land, assuming tools and weather remain unchanged.[9]Mathematical Formulation
The marginal product of an input, such as labor, measures the additional output produced by employing one more unit of that input, holding all other factors constant. In discrete terms, it is formulated as the ratio of the change in total product to the change in the input level:\text{MP} = \frac{\Delta \text{TP}}{\Delta L}
where TP denotes total product (output) and L represents the units of the variable input, such as labor. This formulation applies in scenarios where inputs are adjusted in whole units, common in empirical production analysis.[10] In continuous models, the marginal product is the derivative of the total product with respect to the input:
\text{MP} = \frac{d \text{TP}}{d L}
This captures the instantaneous rate of change in output as the input varies smoothly. For production processes modeled with multiple inputs, the marginal product of labor (MP_L) derives from the total production function Q = f(L, K), where K is fixed capital:
\text{MP}_L = \frac{\partial Q}{\partial L}
This partial derivative assumes ceteris paribus conditions, with other inputs held constant to isolate the effect of the variable input. The short-run context typically features one variable input, such as labor, while capital remains fixed, reflecting real-world constraints like plant capacity.[11][12] To illustrate the discrete formula, consider a firm where total output increases from 100 units to 115 units upon hiring one additional worker, raising labor from 5 to 6 units; the marginal product is then (115 - 100) / (6 - 5) = 15 units per worker. This calculation highlights how marginal product quantifies incremental productivity contributions under fixed other factors.[10]