2. The economists view: a multiproduct firm A multiproduct firm is - PowerPoint PPT Presentation

2. The economist’s view: a multiproduct firm • A multiproduct firm is a firm that produces more than one good or service – Products and services are considered distinct if they are produced in different time periods – Multiperiod firms are synonymous for multiproduct firms • Focus of this chapter – To what extent are our notions of cost • Total cost • Incremental cost • Marginal cost • Average cost applicable in a multiproduct setting? 1

Optimal production plan s.t. • Example 3.1 : selling prices: , factor prices: , , The third factor is used in producing both products and is limited to a total of . The first factor is specific to the first product and the second factor is specific to the second product. The technology is specified by and . • Profit maximization: s.t. • Optimal solution: 2

Constructing the cost function • Analogous to the single product firm, the cost function is obtained by calculating the cost from minimum factor input for each level of production quantities • Note that s.t. • Solving this problem results in , , and implying: 3

Cost function: example • Example 3.2 : By using the data from example 3.1 again but focusing on the cost function, we obtain s.t. • Table 3.1 displays selected values 4

5 Cost function for example 3.2

Cost function terminology • To what extend do our notions of cost from the single product firm extend to a multiproduct firm? • Definition 6 : In the multiproduct firm, the incremental cost of units of product at output quantity , is the difference between the cost of producing units and units, or where or . • Definition 7 : In the multiproduct firm, the marginal cost of output at output quantity , , denoted , is the rate at which cost changes with respect to change in quantity , or . • Definitions 4 and 5 referring to variable cost and fixed cost apply as in the single product world • What about average cost? – Concept is not applicable anymore – Denominator is not defined properly – Special case (exception): cost function is separable 6

Separability of a cost function • Definition 8 : The multiproduct firm’s cost function is separable if it can be expressed as the sum of single product cost functions. • If a cost function is separable – Product cost can be calculated – Average product cost can be calculated • For the cost function in example (3.2), separability does not apply • This is demonstrated in the table 7

Multiperiod Interpretation • Basic assumptions: – Two products are produced – q 1 units of the first product in period 1 – q 2 units of the second product in period 2 – Three input factors, z 1 and z 3 to be available at the beginning of the first period – The second factor has to be available at the beginning of the second period • Timeline: 8

Optimal production plan • Assuming a positive time value of money, the firm maximizes the present value (PV) of future cash flows subject to the familiar technology constraint s.t. 9

Multiperiod cost function • Cost now is the minimum present value of factor expenditures that renders a particular output schedule possible s.t. • Separability is critical in the same way it is in the multiproduct setting – Without separability, product cost cannot be measured in a multiproduct firm – Without separability, period cost cannot be measured in a multiperiod firm • Marginal cost is the only concept of product cost that can reasonably be applied 10