Production Function

$$\begin{array}{rlr}x& =f(l)& \text{Short Run}\\ x& =f(l,k)& \text{Long Run}\end{array}$$

Terminology §

def. Marginal Product. The additional output in units of good $x$ giving additional input (labor or capital)

$$M{P}_l=\frac{df}{dl},M{P}_k=\frac{df}{dk}$$

def. Marginal Revenue Product. The additional output in units of dollars ($) given one unit of additional input (labor or capital)

def. Law of Diminishing Marginal Product. As the input increases over a certain point, the marginal product of the input decreases. This happens for both labour and capital in production functions.

• Mathematically, There exists some $l^{\ast }$ such that:

$$\frac{\partial M{P}_l}{\partial l}{|}_{l^{\ast }}<0$$

Returns to Scale §

Production functions are one of three types:

1. Decreasing Returns to scale: $tf(l,k)<f(tl,tk)$
2. Constant Returns to scale: $tf(l,k)=f(tl,tk)$
3. Increasing Returns to scale: $tf(l,k)>f(tl,tk)$

Micro Production Function §

Short Run §

def. SR Production function. $f(l,k)$ where $l$ denotes labor input, and $k$ denotes capital input. Range is the units of output good.

• Short Run (SR) is defined as the timeframe where $l$ can be varied, but $k$ is fixed.
• Also known as production frontiers or producer choice sets.
• While production frontiers can take on any shape, the most common shape is that of (b) (due to Law of diminishing marginal product).

def. SR Marginal Product of Labor (${\text{MP}}_l$) The marginal change in output with a unit more of labor.

• In the short-run production function, that is the gradient of the function, i.e. ${\text{MP}}_l=\frac{df}{dl}$.

Long Run §

Macro Production Function §

Homogenous Function