Suppose that a firm produces two different outputs, the quantities of which are represented by \(q_{1}\) and \(q_{2}\). In general, the firm's total costs can be represented by \(C\left(q_{1}, q_{2}\right) .\) This function exhibits economies of scope if \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\) for all output levels of either good. a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. b. If the two outputs are actually the same good, we can define total output as \(q=q_{1}+q_{2}\). Suppose that in this case average cost \((=C / q)\) decreases as \(q\) increases. Show that this firm also enjoys economies of scope under the definition provided here.

The mathematical function given for economies of scope is \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\). This condition implies that the combined cost of producing \(q_1\) and \(q_2\) in one firm (the multiproduct firm) is less than the cost of producing them separately in two single-product firms. In other words, the firm can more efficiently use its resources when producing both goods, so the joint production of \(q_1\) and \(q_2\) leads to cost savings. This is because the firm can utilize shared inputs, technologies, or other production processes when the two goods are produced together, making the overall cost lower than if they were produced separately.

If both outputs represent the same good, then total output is given by \(q=q_{1}+q_{2}\). The average cost, which is the total cost divided by the total output, \((=C / q)\), decreases as the output (\(q\)) increases. Now, let's show that the firm enjoys economies of scope. Since average cost decreases as output increases, we can write: \(\frac{C\left(q_{1}+q_{2}\right)}{q_{1}+q_{2}} < \frac{C\left(q_{1}\right)}{q_{1}}\) Multiplying both sides by \(q_{1}(q_{1}+q_{2})\), we have: \(C\left(q_{1}+q_{2}\right)q_{1} < C\left(q_{1}\right)(q_{1}+q_{2})\) Similarly, since average cost also decreases as output increases, we can write: \(\frac{C\left(q_{1}+q_{2}\right)}{q_{1}+q_{2}} < \frac{C\left(q_{2}\right)}{q_{2}}\) Multiplying both sides by \(q_{2}(q_{1}+q_{2})\), we have: \(C\left(q_{1}+q_{2}\right)q_{2} < C\left(q_{2}\right)(q_{1}+q_{2})\) Now, add these two inequalities: \(C\left(q_{1}+q_{2}\right)q_{1} + C\left(q_{1}+q_{2}\right)q_{2} < C\left(q_{1}\right)(q_{1}+q_{2}) + C\left(q_{2}\right)(q_{1}+q_{2})\) Factor the left side of the inequality: \(C\left(q_{1}+q_{2}\right)(q_{1}+q_{2}) < C\left(q_{1}\right)(q_{1}+q_{2}) + C\left(q_{2}\right)(q_{1}+q_{2})\) Divide both sides by \((q_{1}+q_{2})\) to get: \(C\left(q_{1}+q_{2}\right) < C\left(q_{1}\right)+C\left(q_{2}\right)\) The inequality above shows that the firm enjoys economies of scope, as the total cost of producing \(q_1\) and \(q_2\) in one firm is less than the sum of their separate production costs in two single-product firms.