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 TC = aH + bS - 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?

The economies or diseconomies of scale occur with the change in output of only one product with a lesser or greater change in the proportion of input costs, respectively.

The total cost function for computer hardware units and software programs produced is given as:

TC = aH + bS – cHS, where aH is the cost of producing only hardware units and bS is the cost of producing only software programs, while cHS is the collective cost of hardware units and software programs.

The cost function of only hardware units says that the total production cost of hardware units increases in the same proportion as the output increases.

TC_H= aH

Similarly, the cost function for software units shows that a change in output will result in an equal change in the total production cost for software programs.

TC_S= bS

Thus, there are neither economies nor diseconomies of scale.

The economies or diseconomies of scope refer to the production cost of two products produced simultaneously by a single firm. It occurs when the joint output for the products produced by a single firm is more or less than that produced by different firms for separate products.

The following formula helps to check the degree of scope:

$SC=\frac{c\left(q_1\right)+c\left(q_2\right)+c\left(q_1,q_2\right)}{c\left(q_1,q_2\right)}$

Here, C(q1) is the cost for product one separately, C(q2) is the cost for product 2 separately, and C(q1q2) is the cost of producing both the products jointly.

If SC > 0: economies of scope.

If SC < 0: diseconomies of scope.

For the given equation,

$$\begin{gathered} SC=\frac{aH+bS–\left(aH+bS–cHS\right)}{aH+bS–cHS} \\ =\frac{cHS}{aH+bS–cHS} \\ =\frac{cHS}{TC} \end{gathered}$$

Since cHS and TC are positive, SC > 0.

Hence, there exist economies of scope.