Production function

From Hobson's Choice

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The production function in economics is an equation that purports to show the linkage between economic inputs (e.g., labor and capital) and output. Since the production function assumes the economic management remains the same, it does not include estimates of the effects of public policy on utilization of inputs.

1 The Production Function
2 Compared to the Utility Function
3 Marginal Rate of Technical Substitution
4 Notes
5 External Links
6 See Also

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The Production Function

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This is closely analogous to the utility function, and so we use a similar graphic with different labels.^[1] Mathematically, the functions used in production are almost identical to the ones used in utility. However, the production function enjoys a very important advantage: it can be measured empirically. Inputs of labor are carefully monitored by many countries using standardized methodologies^[2] Inputs of capital are likewise monitored by the International Monetary Fund using a single standard methodology.^[3] This means there is no reference to "observed preferences," or other question-begging that plagues efforts to measure utility.

Here, the tradeoff is between labor (L) and capital (K), or any other combination of inputs. While I've shown only two inputs in the diagram, it's possible to set up optimization equations involving as many inputs as you like... such as different capital structures (bonds versus bank loans versus equity), energy inputs, and so forth. One element that is new to the production curve here is the idea of technology: the possibility that output (X) can increase without an increase in L or K. In fact, economists simply treat technology as another input (A), and have long debated the role it plays.

KLEMS is an alternative to the conventional KL-format production function. Used by several organizations including the US Bureau of Economic Advisers, it measures and evaluates the contributions of capital, labor, energy, materials, and services. KLEMS, however, is calculated on the assumption that returns to factors are the same as what is paid for them; the actual volume of inputs themselves are not measured.

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Compared to the Utility Function

The differences between the utility function and the production function are subtle; much of the thinking, and indeed, the math, behind the utility function has been influenced by the production function. The assumptions of rationality, for example, are actual extrapolations of the assumptions of technical efficiency incorporated in the production function.

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Marginal Rate of Technical Substitution

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The equivalent to indifference curves is the marginal rate of technical substitution (MRTS). Assuming the vertical axis represents capital and the horizontal axis represents labor, the slope of the curve at any point L represents the amount of capital that is required to replace a unit of labor. This means

MRTS' = -(∂X/∂L) ÷ (∂X/∂K)

and the firm is supposed to take a position along the MRTS curve such that MRTS'(L) = -p_K/p_L (p refers to price, in units of either labor or equipment). The negative sign simply means that we're interested in a trade-off between productivity caused by more machinery versus productivity caused by more hours of labor; that means more of one thing corresponds to less of another, given the same total factor productivity and given the same output by the firm.

One significant difference is that there is no long-run budget line; the firm's budget line is determined by average and marginal cost functions, instead of a permanent budget. Either way, the firm/person maximizes utility by operating at the limits of their potential.

While a utility function is constrained by assumptions of rationality (such that the 2nd cross partial derivatives must be constantly positive, while the 2nd pure partial derivatives must be constantly negative), there is no compelling reason for a production function to have partial derivatives of consistent sign. Indeed, the history of steel and oil production include several phases where production function derivatives changed sign or were discontinuous. For example, there were points where plants simply had to be made larger and more centralized--more investment in the same technology would have reached a sinkhole of diminishing returns.

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Notes

↑ For the original graphic see Manfred Gärtner, eurMacro "Indifference curves"
↑ This data is freely available at the website of the International Labour Organisation (ILO; note British spelling), "Statistics and Databases."
↑ IMF, "International Financial Statistics Online"; available for a $670 annual subscription, but data for the USA is available for free from the Federal Reserve Board, "Economic research and data."

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