Group of Visualisations

Cubic Cost Function

Introduction

Objectives

Cubic cost functions are commonly used to introduce cost analysis. The visualisations in this group aim to achieve two basic objectives. First, they aim to clarify the forms of fixed cost, variable cost, total cost, average fixed cost, average variable cost, average total cost and marginal cost in the case of a cubic cost function as well as the interpretation of points on these functions. Second, they aim to clarify how the various parameters of a cubic cost function affect the graphs of fixed cost, variable cost, and total cost as well as the graphs of their respective averages and the graph of marginal cost.

Procedure

The visualisations achieve these objectives by plotting fixed cost, variable cost, total cost, average fixed cost, average variable cost, average total cost and marginal cost in individual coordinate systems and combining these coordinate systems in different views that are intended to clarify the relations that exist among these functions. Furthermore, to highlight a key result of cost analysis and to lay the ground for understanding a firm’s profit maximising choice of quantity, average variable cost, average total cost and marginal cost are combined in a single coordinate system. The quantity produced can be varied to highlight individual points on these functions.

Moreover, the values of all parameters driving a cubic cost function can be varied. All functions are plotted twice, once representing the original parameter values, and once representing the new parameter values following the changes you make. Comparing the two should enable you to clearly recognise the effects of your changes.

How you can proceed

Roadmap

The following remarks and questions can help you make the best use of the visualisation.

To achieve the first objective, change the quantity produced and make sure you understand the form of fixed cost, variable cost, total cost, average fixed cost, average variable cost, average total cost and marginal cost, as well as how these functions relate to one another. You can switch between views to focus on the relationships between fixed cost, variable cost, or total cost and their corresponding averages, between total cost and marginal cost or between average variable cost, average total cost and marginal cost.To achieve the second objective, change the values of the parameters whose effects you wish to better understand. You can change fixed cost, as well as the linear, quadratic and cubic components of variable cost.

Equations

The following equations drive the visualisation in this group.

Total Cost (TC)

Total cost is given by:

where $a$ , $b$ , $c$ and $d$ are Parameters. To ensure that the properties of a cost function are preserved, we require the following to hold for all possible parameter values: (1) $a\geq 0$ , (2) $b>0$ , (3) $c<0$ , (4) $d>0$ , and (5) $3db>c^2$ .

Fixed Cost (FC)

Given the expression for $TC$ , since $TC=FC+VC$ , it is easy to see that fixed cost is given by:

since $FC$ is the portion of $TC$ that does not vary with the quantity produced.

Variable Cost (VC)

Similarly, it should be clear that variable cost is given by:

since $VC$ is the portion of $TC$ that does vary with the quantity produced.

Average Fixed Cost (AFC)

Average fixed cost is given by:

Average Variable Cost (AVC)

Average variable cost is given by:

Average Total Cost (ATC)

Average total cost is given by:

Note that we have $ATC=AFC+AVC$ .

Marginal Cost (MC)

Marginal cost is the first derivative of total cost with respect to the quantity produced. It is given by:

Note that marginal cost is also equal to the first derivative of variable cost with respect to the quantity produced, i.e. $MC=\frac{dTC}{dq}=\frac{dVC}{dq}$ .

How to Use This Visualisation

The visualisation in this group offers the following modes of interaction.

The following values can be changed

Meaning of the icons

Refresh

Refresh the visualisation. It will reload with all its initial values as described below.

Select a Parameter or Variable

Select one of the parameters $a$ , $b$ , $c$ or $d$ or the variable $q$ in order to change its value and display the effects of the respective changes.

Increase Value

Increase the value of the selected parameter ( $a$ , $b$ , $c$ or $d$ ) or variable ( $q$ ). By keeping the button pressed, the value will continue to increase incrementally.

Decrease Value

Decrease the value of the selected parameter ( $a$ , $b$ , $c$ or $d$ ) or variable ( $q$ ). By keeping the button pressed, the value will continue to decrease incrementally.

Change Value

Set the value of the selected parameter ( $a$ , $b$ , $c$ or $d$ ) or variable ( $q$ ) by using a slider. This allows for faster changes than by using the plus and minus buttons.

Select a View

Choose between sixteen different selections of combined coordinate systems.

Display Details

Display a brief description as well as the equations driving the visualisation.

Visibilities

Toggle the visibilities of the graphical elements described below.

Initial view, values and visibilities.

Initialisation

1Visualisation

Description

Parameter Values

Highlighted Variable Value

Initial Visibilities

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Depicts fixed cost, variable cost, total cost, average fixed cost, average variable cost, average total cost and marginal cost of a two factor Cobb-Douglas short-run cost function in individual coordinate systems as well as combinations thereof in different views designed to clarify the relations among these functions. Clarifies the forms of these functions, the effects of alternative parameter values on those forms, as well as the interpretation of points on these functions.