Question

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.

Short answer

Answer: A multiproduct firm enjoys economies of scope when its total cost of producing two different outputs jointly is lower than the sum of the total costs of producing the outputs separately. This is shown in the mathematical formulation \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\), which represents the condition that the total cost of producing \(q_1\) units of output 1 separately and \(q_2\) units of output 2 separately is higher than the total cost of producing both outputs together.

Step-by-step solution

a. Explaining economies of scope in multiproduct firms

By definition, a firm exhibits economies of scope when its total cost of producing two different outputs together, i.e., \(C\left(q_{1}, q_{2}\right)\), is lower than the sum of total costs of producing the outputs separately, i.e., \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)\). In other words, the firm's total costs are lower when it produces multiple goods instead of each good independently. The mathematical formulation \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\) represents this condition, as it indicates that the total cost of producing \(q_{1}\) units of output 1 separately and \(q_{2}\) units of output 2 separately (\(C(q_1, 0) + C(0, q_2)\)) is higher than the total cost of producing both outputs together (\(C(q_1, q_2)\)). This means that the given multiproduct firm can save on total production costs by producing the two outputs jointly, rather than having each good produced separately by different firms. Hence, the multiproduct firm enjoys economies of scope.

b. Showing economies of scope in the case of average cost decreasing with q

Now, let's assume the two outputs are the same good. Therefore, the total output can be defined as \(q=q_{1}+q_{2}\). Suppose that the average cost \((=C / q)\) decreases as \(q\) increases. Recall that the definition of economies of scope is given by the inequality \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\) for all output levels of either good. So let's analyze the relationship between average cost and total cost for the firm when it produces both outputs together: 1. The average cost is given by \(AC=\frac{C}{q}\), where \(q=q_{1}+q_{2}\). 2. The total cost of producing output \(q\) is given by \(C(q)=AC\times q\). Now, let's denote the total cost of producing \(q_{1}\) units of output 1 and \(q_{2}\) units of output 2 separately by \(C_1\) and \(C_2\), respectively: 1. \(C_1 = C(q_1, 0) = AC_1\times q_1\) 2. \(C_2 = C(0, q_2) = AC_2\times q_2\) Since the average cost decreases as \(q\) (the sum of output 1 and output 2) increases, \(AC\) at output level \(q\) is lower than average costs for any output level strictly below \(q\). That means that \(AC<AC_1\) and \(AC<AC_2\). Therefore, \(C(q) = AC\times q < AC_1\times q_1 + AC_2\times q_2 = C_1+C_2\), which can be rewritten as \(C(q_1, q_2)< C(q_1, 0) + C(0, q_2)\), since \(C(q) = C(q_1, q_2)\). This is the definition of economies of scope. Thus, it can be concluded that when average cost decreases as total output increases, the firm also enjoys economies of scope.