Modeling the Firm - Production Function
Overview
The production function is a mathematical abstraction of the amount of product a firm can produce
given a set of inputs. In reality, it is probably inaccurate to assume that a firms
production can be so easily modeled, however, it makes the concepts precise in a way that makes
it relatively easy to analyze.
The Production Function
Firms
can only produce goods after using a set of inputs, such as materials and worker time. The amount of product produced
is modeled as a function of those inputs.
{% q = f(input_1, input_2, ..., input_n) %}
(q is the standard variable used to represent quantity of product)
The inputs represent different quantities that affect the amount of product produced, such as product materials,
as well as capital goods such as factories.
As an example, the following might represent the production function for a simple clothing manufacturer.
{% q = f(cotton, thread, sewing \; machines, worker \; hours) %}
One simplification that is often used is to quote the amount of each input in dollars.
The Simplified Production Function
Often, the production function is presented in a simplified form as a function of capital, k, and labour, l.
{% q = f(k, l) %}
That is, for the purposes of economic modeling, all the inputs are aggregated together
into two categories, capital and labor. In addition, the inputs are measured in units of
total dollar amounts.
Furthermore, the function is chosen to measure the optimal
amount of output given the inputs.
The Production Function - Statistical Perspective
From a practical perspective, the output will not always be exactly the same,
even when provided with the exact same set of
inputs. That is, there will
always be some variance or noise in the process. From that perspective,
the function that is often referred to as the production function, as above,
is the average of the production function. that is
{% q = \mathbb{E}[f(input_1, input_2, ..., input_n)] %}
For most microeconomic models, the distinction is ignored. This is justified
especially for firms where the production process is fairly exact with little
variance. For firms where the production process can produce a wide
variance of output, the variance itself can be an issue that needs to be addressed,
and may be considered an element of
enterprise risk
Production Function Assumptions
As a general rule, knowing the value of the production function over
a range of inputs is a difficult exercise. However, in order for a firm
to make decisions about the optimal amount of product to produce, it
must have some feeling for what the cost curve looks like. In fact,
a firm could not maximize its profits without knowing exactly what
this curve looks like over the range of possible inputs.
As a matter of course, assuming that a firm could know its cost curve
is not practical. However, if we make certain assumptions about the cost
curve, the firm can optimize its profits without knowing the entire curve.
(see marginal product below)
- Smooth Production Function
The first assumption that is commonly made is that the cost curve is
smooth. (from the perspective of calculus, this means differetiable)
That is, if you draw the production function versus various
inputs on a chart, the resulting curves are smooth with no kinks.
This means that if you add a small amount of an input, the output
goes up by a small amount. As a general rule, inputs can only be increased
by certain discrete units, (You cant add half a worker, or half a factory)
but this usually ignored in the analysis, in order to make it easier to apply
standard mathematical tools.
- Efficient
is the assumption that a company uses its resources to optimal efficiency. The company
will product the maximum amount of goods and services for any given level of inputs.
Phrased alternatively,
efficiency is "the use of the least inputs (resources) to produce the most outputs (services)"
(see bogetoft chapt. 2)
Classical Theory - Descreasing Productivity
In the classical theory of costs, costs typically increase faster than output. To understand this, we must be careful to
be precise about our meaning. There are tow different ways to interpret this statement.
The first is an interpretation of the decreasing marginal productivity. That is if we hold all else fixed and we increase
one unit of one of the inputs, we get less output that the previous unit. As an example, suppose our production function
is a function of labor and capital
{% q = f(k,l) %}
If we add a single unit of labor, we get an increase in production
{% q_2 = f(k, l+1) %}
{% q_3 = f(k, l+2) %}
The additional units produced is less as we increase the amount of labor.
{% q_3-q_2 < q_2-q %}
This can be understood that as you add labor, there is a smaller amount of capital (land, computers, equipment)
supporting each unit of labor, so they are less productive.
The second way to interpret hte statement is as a statement about returns to scale. That is, what happens
if we scale up all factors of production by the same rate. For example, if we double both labor and capital, do
we double output?
{% 2q = f(2k,2l) %}
The classical assumption would be that we get less than double the output. This is because there is an assumption that
the best capital and labor is used first. That is, a farmer farms the most fertile soil first. If you double the amount of
land you farm, you will likely get less as you add less fertile soil.
This assumption underlies the supply curves upward slope as the number of units increases.
The assumption of diminishing returns to scale is somewhat controversial. It ignores economies of scale. Adam Smith himself
theorized that returns to scale is driven by multiple competing effects.
In math speak
marginal physical product of capital
{% \frac{\partial^2 f}{\partial k^2} < 0 %}
marginal physical product of capital
{% \frac{\partial^2 f}{\partial l^2} < 0 %}
The Experience Curve
The experience curve was an invention of the Boston Consulting Group in the early days of the rise of strategy consultants.
It challenges the classical theory outlined above. It asserts that the average cost per unit of output decreases as a company
increases its output. The reason is due to experience and economies of scale, that is, as the company produces more, it learns more about how to
produce its product efficiently and builds the necessary infrastructure to do so at scale.
(
Kiechel pg. 31)
Examples
