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

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• 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?

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Optimal production plan

• 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:
• Optimal solution:

s.t. s.t.

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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
• Solving this problem results in

, , and implying: s.t.

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Cost function: example

• Example 3.2: By using the data from example 3.1 again but focusing on the

cost function, we obtain

• Table 3.1 displays

selected values s.t.

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Cost function for example 3.2

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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

• r

.

• 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

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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

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Multiperiod Interpretation

• Basic assumptions:

– Two products are produced – q1 units of the first product in period 1 – q2 units of the second product in period 2 – Three input factors, z1 and z3 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:

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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.

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Multiperiod cost function

• Cost now is the minimum present value of factor expenditures that

renders a particular output schedule possible

• 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 s.t.