Chapter 4
/ Monday, September 30 | Production Functions

# 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

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