Question

A computer company's cost function, which relates its average cost of production AC to its cumulative output in thousands of computers \(Q\) and its plant size in terms of thousands of computers produced per year \(q\) (within the production range of 10,000 to 50,000 computers \(),\) is given by \\[ \mathrm{AC}=10-0.1 \mathrm{Q}+0.3 \mathrm{q} \\] a. Is there a learning-curve effect? b. Are there economies or diseconomies of scale? c. During its existence, the firm has produced a total of 40,000 computers and is producing 10,000 computers this year. Next year it plans to increase production to 12,000 computers. Will its average cost of production increase or decrease? Explain.

Short answer

a) Yes, there is a learning-curve effect in this problem as the coefficient of Q is negative in the cost function. b) The company experiences diseconomies of scale as the coefficient of q in the cost function is positive. c) The company's average cost of production decreases from 6 to 5.6 as it increases its annual production from 10,000 to 12,000 computers.

Step-by-step solution

Identify Learning Curve Effect

The learning curve effect is represented by the output in the cost function (Q). If the coefficient of Q is negative then it indicates a learning curve effect because costs decrease with increased production. In this case the coefficient of Q is -0.1, which is negative, so there is a learning curve effect.

Economies or Diseconomies of Scale

Looking at the cost function, we observe that the coefficient of the plant size (q) is positive or negative, that determines the economies or diseconomies of scale. If it is positive, then there are diseconomies of scale (costs rise as size increases) and if it is negative then there are economies of scale (costs decrease as size increases). Here the coefficient is 0.3 which is positive, so there are diseconomies of scale.

Average Cost of Production Increase or Decrease

Average Cost of production when the firm has produced a total of 40,000 computers and is producing 10,000 computers this year is calculated as: \[AC = 10-0.1(40)+0.3(10) = 6\]. Next year it plans to increase production to 12,000 computers so the average cost of production will be: \[ AC = 10-0.1(52)+0.3(12) = 5.6\]. The average cost of production decreases as the firm increase its production from 10,000 to 12,000.

Key concepts

Learning Curve

The learning curve illustrates how costs decline as a firm gains more experience in production. In the context of the computer company's cost function, when we see the cumulative output \( Q \), it directly influences the average cost (AC). A negative coefficient in front of \( Q \) indicates a learning curve effect.

This is because as cumulative output increases, the cost per unit decreases. In simpler terms, the more computers the company makes over time, the more efficient and skilled it becomes. This learning can come from various improvements, like better techniques or streamlined processes.

In the given cost function, \( ext{AC} = 10 - 0.1Q + 0.3q \), the coefficient of \( Q \) is \( -0.1 \). This negative number confirms that the company benefits from a learning curve effect. As output increases, the average cost decreases, indicating that the company is getting better at producing computers.

Economies of Scale

Economies of scale occur when increasing the size of production leads to a decrease in the cost per unit. However, this concept is closely tied to the coefficient in the cost function associated with the size of the plant, \( q \).

In our computer company example, we want a negative coefficient on \( q \) to signify this concept, because it would imply that as the plant produces more computers, costs go down. Unfortunately, in the analyzed cost function: \( ext{AC} = 10 - 0.1Q + 0.3q \), the coefficient for \( q \) is \( +0.3 \).

Thus, this cost function does not show economies of scale when increasing plant size, but rather the opposite, as explained further in the following section.

Diseconomies of Scale

Diseconomies of scale happen when a firm's production increase leads to higher average costs. This occurs when the plant's size grows beyond an optimal point, often due to factors like increased complexity or inefficiencies in communication.

In the given cost model \( ext{AC} = 10 - 0.1Q + 0.3q \), the positive coefficient for \( q \) of \( +0.3 \) indicates diseconomies of scale. This means that as the company's plant size increases, costs actually rise.

The company, therefore, faces a situation where, by expanding its production facility, it could inadvertently increase its costs due to operational challenges, possibly resulting in less effective management or slower decision-making.