Question

A computer company produces hardware and software using the same plant and labor. The total cost of producing computer processing units $H$ and software programs $S$ is given by \[ \mathrm{TC}=a H+b S-\mathrm{cHS} \] where $a,b,$ and $c$ are positive. Is this total cost function consistent with the presence of economies or diseconomies of scale? With economies or diseconomies of scope?

Short answer

The given total cost function does not show economies or diseconomies of scale because the cost per unit stays constant as production increases. However, it does demonstrate economies of scope, as producing hardware and software together results in lower total costs compared to producing them separately.

Step-by-step solution

Understanding the Total Cost Equation

The Total Cost function provided is $TC=aH+bS-cHS$, where $H$ and $S$ are the quantities of hardware and software produced respectively, and $a$, $b$, and $c$ are positive constants representing the production costs. The terms $aH$ and $bS$ represent the individual costs of producing hardware and software, while $-cHS$ is the joint cost of producing both. It's negative because it reduces the total cost, indicating possible economies of scale or scope.

Analyzing Economies or Diseconomies of Scale

Economies of scale refer to a situation where the cost per unit decreases as the number of units produced increases. As both $aH$ and $bS$ are linear equations, if we double the production of either hardware or software, the total cost will also double. This implies there is neither economies nor diseconomies of scale in this total cost function because the cost per unit stays constant as production increases.

Analyzing Economies or Diseconomies of Scope

Economies of scope refer to a situation where it is smaller in cost to produce two or more goods together than to produce them separately. With a term like $-cHS$, if we produce both hardware and software together, the total cost will decrease more than if we produced them separately. This demonstrates that there are economies of scope within this total cost function. The presence of a negative term involving the product of $S$ and $H$ shows that the more of each good is jointly produced, the lower the total cost.

Key concepts

Total Cost Function in Economics

In microeconomics, understanding a company's expenses is critical for analyzing its efficiency and the potentials for cost saving. The total cost function signals how total cost changes with output levels. Specifically, it encapsulates the relationship between the cost incurred by a firm in the course of producing goods and the quantity of goods produced.

For our computer company, the total cost function is expressed mathematically as $\mathrm{TC}=aH+bS-cHS$, incorporating the variables for quantities of hardware $H$ and software $S$ produced. Here, $a$ and $b$ are the costs associated with producing one unit of hardware and software respectively. The term $-cHS$ points towards a decrease in total costs arising from producing hardware and software together, hinting at cost efficiencies that could either be related to economizing on scale or scope - a concept we will explore in further depth.

Microeconomic Production Costs

Within the framework of microeconomics, production costs are the expenses a business incurs to create a product or service. These can be broken down into fixed costs that do not change with the level of goods produced, and variable costs which flex with production volume. In our example, $aH$ and $bS$ could be representing the variable costs associated with each additional unit of hardware or software.

This is essential for businesses when making decisions about production levels, prices, and strategies for economies of scale and scope. For instance, understanding that each hardware unit produced involves a cost of $a$ and each software unit costs $b$ for production, helps managers price their products appropriately to cover costs and achieve desired profit margins. More importantly, identifying the interactions between these costs, as with the $-cHS$ term, can lead to strategic decisions about whether to diversify products or concentrate on scaling up.

Diseconomies of Scale

Conversely, diseconomies of scale emerge when a company grows to a point where the cost per unit starts to increase. This counterintuitive phenomenon can occur due to several factors such as increased complexity of management, inefficiency of large-scale production, or even rising material costs.

In analyzing the given total cost function, we do not see any inherent diseconomies of scale. That is, the costs related to producing hardware $aH$ and software $bS$ do not exhibit an increase per unit with greater quantities produced. Instead, the linear relationship suggests that the costs grow proportionately with output, maintaining a constant cost per unit. This implies a neutral effect on economies of scale; the company doesn't save or lose on cost per unit with an increased output. It is only through the consideration of the cost-reducing $-cHS$ term that we delve into potential efficiencies beyond simple scaling up, such as economies of scope.

Diseconomies of Scope

In contrast to economies of scope, diseconomies of scope occur when producing multiple products jointly is more costly than producing them separately. This could be due to the complexity of managing diverse product lines, reduced focus, or inefficiencies in shared resources. However, the exercise presented indicates the presence of economies of scope, not diseconomies. The joint production cost, represented by the term $-cHS$, illustrates cost savings when the firm produces both hardware and software jointly.

Why is this important? Understanding diseconomies of scope helps a company in making strategic choices about the variety of products to offer. In our example, there's no evidence of diseconomies of scope. In fact, the negative coefficient in the joint production term suggests that diversifying into both hardware and software could lead to cost efficiencies; the firm benefits from producing a broader product mix owing to shared use of the same plant and labor.