4.11 Summary

Things you should know

• A production function describes the relationship between inputs and output. For simplicity, we analyze simplistic functions using inputs labor ($L$) and capital ($K$) Examples of production functions include:
  • Linear: $f(L,K) = aL + bK$
  • Leontief (fixed proportions): $f(L,K) = \min{aL, bK}$
  • Cobb-Douglas: $f(L,K) = AL^aK^b$
• The marginal product of an input ($MP_L$ or $MP_K$) is the additional output generated per additional unit of input, holding other inputs constant. Mathematically, it is the partial derivative of the production function with respect to that input.
• An isoquant for some quantity $q$ shows all combinations of inputs that generate $q$ units of output.
• The slope of an isoquant is the marginal rate of technical substitution. It represents the rate at which inputs may be substituted while keeping output constant.
• The elasticity of substitution is a measure of how substitutable inputs are in a production process. The greater the elasticity of substitution, the closer substitutes they are. Mathematically, the elasticity of substitution is a measure of how quickly the MRTS changes along an isoquant.
• Not all inputs may be changed in the same time frame. The short run refers to a time horizon in which some inputs are fixed. The long run refers to a time horizon in which all inputs may be adjusted. As a shorthand convention, we often assume capital is fixed in the short run, so the firm’s short-run production function may be written as a function of $L$ only.

• The concept of “returns to scale” refers to how output increases as all inputs are scaled proportionally. Formally, for any $t > 1$, we say a production function $f(L,K)$ exhibits:

  • Decreasing returns to scale (DRS) if $f(tL, tK) < tf(L,K)$
  • Constant returns to scale (CRS) if $f(tL, tK) = tf(L,K)$
  • Increasing returns to scale (IRS) if $f(tL, tK) > tf(L,K)$

Things you should be able to do

Given any production function $f(L,K)$:

• Plot the isoquant for any quantity $q$
• Calculate the marginal products of labor and capital, in general and at a specific point
• Derive the MRTS, in general and at a specific point
• Plot the short-run production function for a fixed level of capital, $f(L | K = \overline K)$
• Determine whether the production function exhibits decreasing, constant, or increasing returns to scale