Group of Visualisations

Two Factor Cobb-Douglas Short-Run Cost

Introduction

Objectives

In general, Cobb-Douglas cost functions are derived from minimising the cost of production given a technology represented by a Cobb-Douglas production function. However, with only two factors of production (capital and labour), in the short-run we assume the quantity of capital deployed to be fixed. Consequently, there is no minimisation problem to be solved since the quantity of labour necessary to produce any given level of output is completely determined by the production function.

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 Cobb-Douglas cost function as well as the interpretation of points on these functions and the relationships that exist between average total and average marginal cost. Second, they aim to clarify how the various parameters of a Cobb-Douglas cost function affect the form of fixed cost, variable cost, total cost as well as the respective average cost functions and the marginal cost function.

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 Cobb-Douglas cost function can be varied. All functions are plotted twice, once as 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 by changing the value of the price of capital, as well as variable cost by changing the wage rate. In addition, you can change the values of the output elasticity of capital, the output elasticity of labour, and the efficiency parameter of the Cobb-Douglas production function.

Equations

The following equations drive the visualisation in this group.

Cobb-Douglas Technology

With two factors the Cobb-Douglas production function is given by:

where $q$ is the quantity of the output good produced, $K$ is the quantity of capital deployed, $L$ is the quantity of labour deployed, $\alpha$ is the output elasticity of capital, $\beta$ is the output elasticity of labour, and $A$ is the efficiency parameter.

Assuming capital to be fixed in the short-run (i.e. $K=\overline{K}$ ), we can express the quantity of labour required to produce the quantity of output $q$ as:

Total Cost (TC)

Hence, to find short-run total cost in the two factor case, there is no minimisation problem to solve. Short-run total cost is simply given by:

where $q$ is the quantity of the output good produced, $w_k$ is the price of capital, $\overline{K}$ is the fixed quantity of capital deployed, $w_l$ is the price of labour (wage rate), $\alpha$ is the output elasticity of capital, $\beta$ is the output elasticity of labour, and $A$ is the efficiency parameter.

Note that in the case of two factors short-run total cost is not dependant on $L$ since the quantity of labour required to produce any quantity of output $q$ is determined by $L^*=(\frac{q}{A\overline{K}^{\alpha }} )^{\frac{1}{\beta } }$ .

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:

Marginal Cost (MC)

Finally, marginal cost is given by:

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 View

Choose between sixteen different selections of combined coordinate systems.

Select a Parameter or Variable

Select one of the parameters $w_k$ , $w_l$ , $\alpha$ , $\beta$ or $A$ , 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 ( $w_k$ , $w_l$ , $\alpha$ , $\beta$ or $A$ ) or variable ( $q$ ). By keeping the button pressed, the value will continue to increase incrementally.

Decrease Value

Decrease the value of the selected parameter ( $w_k$ , $w_l$ , $\alpha$ , $\beta$ or $A$ ) or variable ( $q$ ). By keeping the button pressed, the value will continue to decrease incrementally.

Change Value

Set the value of the selected parameter ( $w_k$ , $w_l$ , $\alpha$ , $\beta$ or $A$ ) or variable ( $q$ ) by using a slider. This allows for faster changes than by using the plus and minus buttons.

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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